Finite Group Databasebeta

← #376374

Subgroups — #376374

Order 512.

1015 conjugacy classes of subgroups.

The subgroup lattice has 779 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23730C211yesmaximal, characteristic
2C2×#1922C211yesmaximal, characteristic, complemented
3#20838C211yesmaximal, characteristic
4#20441C211yesmaximal, characteristic, complemented
5C2×#1764C211yesmaximal, characteristic
6#11359C211yesmaximal, characteristic, complemented
7#11221C211yesmaximal, characteristic, complemented
8#11125C211yesmaximal, characteristic
9#10934C211yesmaximal, characteristic
10#10933C211yesmaximal, characteristic
11#10911C211yesmaximal, characteristic, complemented
12#10909C211yesmaximal, characteristic, complemented
13#10876C211yesmaximal, characteristic, complemented
14#10859C211yesmaximal, characteristic, complemented
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2297C2211yescharacteristic
32#2285C2211yescharacteristic
33#2097C411yescharacteristic, complemented
34#2097C411yescharacteristic, complemented
35#2096C411yescharacteristic, complemented
36#2095C411yescharacteristic, complemented
37#2094C411yescharacteristic, complemented
38#2093C411yescharacteristic, complemented
39#2092C411yescharacteristic, complemented
40#2092C411yescharacteristic, complemented
41#1922—24no—
42#1851—21no—
43#1851—21no—
44#1850—21no—
45#1850—21no—
46#1771—21no—
47#1771—21no—
48#1770—21no—
49C8×C4⋊C4—21no—
50#1764—24no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62#1729C2211yescharacteristic
63#1728C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C4×#95C2211yescharacteristic
69C22×#41C2311yescharacteristic
70C2×#131—22no—
71C2×#131—22no—
72C2×#131—22no—
73C2×#131—22no—
74C2×#131—41no—
75C2×#131—41no—
76C22×M16—21no—
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C23×C8—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C2×#108—22no—
93C2×#108—22no—
94C2×#108—41no—
95#427—22no—
96#427—22no—
97#427—41no—
98#427—41no—
99C2×#107C2×C411yescharacteristic
100C2×#107C2×C411yescharacteristic
101C2×#107C2×C411yescharacteristic
102C2×#107C2×C411yescharacteristic
103C2×#106C2×C411yescharacteristic
104C2×#106C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×#106C2×C411yescharacteristic
107#424—22no—
108#424—41no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8D411yescharacteristic
114C2×C4⋊C8D411yescharacteristic
115C2×C4⋊C8D411yescharacteristic
116C2×C4⋊C8D411yescharacteristic
117C2×#104—22no—
118C2×#104—22no—
119C2×#104—22no—
120C2×#104—22no—
121C2×#104—41no—
122C2×#104Q814yes—
123#416—22no—
124#416—22no—
125#416—41no—
126#416—41no—
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135#408—22no—
136#408—41no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139#405—22no—
140#405—22no—
141#405—41no—
142#405—41no—
143C4×M16—22no—
144C4×M16—22no—
145C2×#97—21no—
146C2×#97—21no—
147C2×#97—21no—
148C2×C4×C8—21no—
149C2×C4×C8—22no—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8—22no—
153C2×C4×C8—41no—
154#391C2×C411yescharacteristic
155#391C2×C411yescharacteristic
156#390C2×C411yescharacteristic
157#390C2×C411yescharacteristic
158#386—21no—
159#386—21no—
160#384C2×C411yescharacteristic
161#384C2×C411yescharacteristic
162#382C2×C411yescharacteristic
163#382C2×C411yescharacteristic
164#380—21no—
165#380—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4—21no—
170C2×#95—21no—
171C2×#95—21no—
172#341—22no—
173#341—41no—
174#341D414yes—
175#339—22no—
176#339—22no—
177#339—22no—
178#339—41no—
179#339—41no—
180#339—41no—
181#336—21no—
182#336—21no—
183#336—22no—
184#336—21no—
185#336—21no—
186#336—21no—
187#336—21no—
188#336—22no—
189#336—21no—
190#336—21no—
191#336—41no—
192#336C2×C411yescharacteristic
193#336C2×C411yescharacteristic
194#336C2×C411yescharacteristic
195#336C2×C411yescharacteristic
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—21no—
204C2×#41—21no—
205C2×#41—21no—
206C2×#41C2×Q812yes—
207C2×#41#4112yes—
208C22×Q8C22×C411yescharacteristic
209C22×D4—21no—
210C22×D4C22×C411yescharacteristic
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4#4111yescharacteristic
215C23×C4#4111yescharacteristic
216C23×C4C2411yescharacteristic
217#131—28no—
218#131—44no—
219#131—44no—
220#131—44no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—42no—
241C2×M16—42no—
242C2×M16—42no—
243C2×M16C2×D412yes—
244C2×M16C2×D412yes—
245C2×M16C2×Q812yes—
246C2×M16#4112yes—
247C2×M16#4112yes—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—42no—
273C22×C8—42no—
274C22×C8—42no—
275C22×C8C22×C411yescharacteristic
276C22×C8C22×C411yescharacteristic
277C22×C8C22×C411yescharacteristic
278C22×C8C22×C411yescharacteristic
279C22×C8C2×D411yescharacteristic
280C22×C8C2×D411yescharacteristic
281C22×C8C2×D412yes—
282C22×C8C2×D411yescharacteristic
283C22×C8C2×D411yescharacteristic
284C22×C8#4112yes—
285C22×C8#4112yes—
286C22×C8#4112yes—
287#117—21no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C4⋊C4C22×C411yescharacteristic
306C2×C4⋊C4C22×C411yescharacteristic
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C4⋊C4C22×C411yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42C22×C411yescharacteristic
335C2×C42C22×C411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×Q812yes—
339#108—44no—
340#107—22no—
341#107—22no—
342#107—41no—
343#107—41no—
344#106—22no—
345#106—22no—
346#106—41no—
347#106—41no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8—22no—
352C4⋊C8—42no—
353C4⋊C8—42no—
354#104—28no—
355#104—44no—
356#103—22no—
357#103—22no—
358#103—41no—
359#103—41no—
360#102—22no—
361#102—22no—
362#102—41no—
363#102—41no—
364C22⋊C8—42no—
365C22⋊C8—42no—
366#97—24no—
367#97—42no—
368#97—42no—
369C4×C8—24no—
370C4×C8—44no—
371C4×C8—42no—
372C4×C8—42no—
373#95—24no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—42no—
383C24—21no—
384C24C2×#4111yescharacteristic
385#41—22no—
386#41—22no—
387#41—22no—
388#41—22no—
389#41—22no—
390#41—22no—
391#41—22no—
392#41—22no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8C4×Q812yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4C4×Q812yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4C4×D411yescharacteristic
455C22×C4C4×D411yescharacteristic
456C22×C4C4×D411yescharacteristic
457C22×C4C4×D411yescharacteristic
458C22×C4C23×C411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×Q811yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4111yescharacteristic
465C22×C4C2×#4111yescharacteristic
466C22×C4C2×#4111yescharacteristic
467C22×C4C2×#4111yescharacteristic
468C22×C4H(5,𝔽2)11yescharacteristic
469C22×C4H(5,𝔽2)11yescharacteristic
470C22×C4H(5,𝔽2)11yescharacteristic
471C22×C42−1+411yescharacteristic
472M16—22no—
473M16—22no—
474M16—22no—
475M16—22no—
476M16—22no—
477M16—22no—
478M16—22no—
479M16—48no—
480M16—41no—
481M16—41no—
482M16—41no—
483M16—41no—
484M16—41no—
485M16—41no—
486M16—41no—
487M16—41no—
488M16C22⋊Q814yes—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—22no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—22no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—48no—
523C2×C8—41no—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8—41no—
527C2×C8—41no—
528C2×C8—41no—
529C2×C8—41no—
530C2×C8—41no—
531C2×C8—41no—
532C2×C8—41no—
533C2×C8—41no—
534C2×C8—41no—
535C2×C8—41no—
536C2×C8—41no—
537C2×C8—41no—
538C2×C8C4×D412yes—
539C2×C8C4×D412yes—
540C2×C8#12312yes—
541C2×C8#12312yes—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—81no—
557C4⋊C4—81no—
558C22⋊C4—42no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—42no—
566C22⋊C4—81no—
567C22⋊C4—81no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—21no—
576C42—22no—
577C42—22no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—22no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23C2×C4×D411yescharacteristic
602C23#52911yescharacteristic
603C23#52911yescharacteristic
604C23#55011yescharacteristic
605C23#55211yescharacteristic
606C23#55411yescharacteristic
607C23#55911yescharacteristic
608Q8—22no—
609Q8—22no—
610Q8—41no—
611Q8—41no—
612D4—22no—
613D4—22no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—41no—
619D4—41no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—81no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—81no—
672C2×C4—81no—
673C2×C4—81no—
674C2×C4#44312yes—
675C2×C4C2×C4×D411yescharacteristic
676C2×C4C2×C4×Q811yescharacteristic
677C2×C4C4×#4111yescharacteristic
678C2×C4C4×#4111yescharacteristic
679C2×C4#51811yescharacteristic
680C2×C4#52011yescharacteristic
681C2×C4C2×C22⋊Q811yescharacteristic
682C2×C4C2×#12311yescharacteristic
683C2×C4#52511yescharacteristic
684C2×C4#52511yescharacteristic
685C2×C4#52911yescharacteristic
686C2×C4#53311yescharacteristic
687C2×C4#53811yescharacteristic
688C2×C4#53911yescharacteristic
689C2×C4#54011yescharacteristic
690C2×C4#54211yescharacteristic
691C2×C4D4211yescharacteristic
692C2×C4#54611yescharacteristic
693C2×C4#54611yescharacteristic
694C2×C4#54711yescharacteristic
695C2×C4D4×Q811yescharacteristic
696C2×C4#55011yescharacteristic
697C2×C4#55111yescharacteristic
698C2×C4#55111yescharacteristic
699C2×C4#55111yescharacteristic
700C2×C4#55211yescharacteristic
701C2×C4#55411yescharacteristic
702C2×C4#55411yescharacteristic
703C8—22no—
704C8—22no—
705C8—22no—
706C8—22no—
707C8—24no—
708C8—41no—
709C8—41no—
710C8—41no—
711C8—41no—
712C8—42no—
713C8—42no—
714C8—81no—
715C22—21no—
716C22—21no—
717C22—21no—
718C22—21no—
719C22—21no—
720C22—21no—
721C22—21no—
722C22—21no—
723C22—21no—
724C22—21no—
725C22—21no—
726C22—21no—
727C22—42no—
728C22—42no—
729C22—42no—
730C22—81no—
731C22—81no—
732C22—81no—
733C22—81no—
734C22C4×#12311yescharacteristic
735C22#236011yescharacteristic
736C22#236611yescharacteristic
737C22#245411yescharacteristic
738C22#247911yescharacteristic
739C22#257511yescharacteristic
740C22#258911yescharacteristic
741C22C2×#44311yescharacteristic
742C22#295511yescharacteristic
743C22#295611yescharacteristic
744C22#295711yescharacteristic
745C4—22no—
746C4—44no—
747C4—41no—
748C4—41no—
749C4—41no—
750C4—42no—
751C4—82no—
752C4—81no—
753C4—81no—
754C4—81no—
755C4—81no—
756C4—81no—
757C4—81no—
758C4C4×C22⋊Q811yescharacteristic
759C4#233511yescharacteristic
760C4#235911yescharacteristic
761C4#236811yescharacteristic
762C4#241811yescharacteristic
763C4#242311yescharacteristic
764C4#244911yescharacteristic
765C4#247111yescharacteristic
766C2—21no—
767C2—21no—
768C2—21no—
769C2—21no—
770C2—82no—
771C2#1417811yescharacteristic
772C2#2159211yescharacteristic
773C2#2159411yescharacteristic
774C2#2298211yescharacteristic
775C2#2298411yescharacteristic
776C2#2317911yescharacteristic
777C2#2318111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1922C2semidirect
4. #20441C2semidirect
6. #11359C2semidirect
7. #11221C2semidirect
11. #10911C2semidirect
12. #10909C2semidirect
13. #10876C2semidirect
14. #10859C2semidirect
33. #2097C4semidirect
34. #2097C4semidirect
35. #2096C4semidirect
36. #2095C4semidirect
37. #2094C4semidirect
38. #2093C4semidirect
39. #2092C4semidirect
40. #2092C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237302
C2×#19222
#208382
#204412
C2×#17642
#113592
#112212
#111252
#109342
#109332
#109112
#109092
#108762
#108592
#107912