Finite Group Databasebeta

← #373428

Subgroups — #373428

Order 512.

1009 conjugacy classes of subgroups.

The subgroup lattice has 776 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1929C211yesmaximal, characteristic
2#21058C211yesmaximal, characteristic
3#20838C211yesmaximal, characteristic
4#20431C211yesmaximal, characteristic, complemented
5C2×#1765C211yesmaximal, characteristic
6#11326C211yesmaximal, characteristic, complemented
7#11221C211yesmaximal, characteristic, complemented
8#11125C211yesmaximal, characteristic
9#11053C211yesmaximal, characteristic
10#10910C211yesmaximal, characteristic, complemented
11#10887C211yesmaximal, characteristic, complemented
12#10877C211yesmaximal, characteristic, complemented
13#10860C211yesmaximal, characteristic, complemented
14#10808C211yesmaximal, characteristic, complemented
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C22×#104C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#416C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2297C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1929—24no—
34#1928C411yescharacteristic, complemented
35#1928C411yescharacteristic, complemented
36#1927C411yescharacteristic, complemented
37#1927C411yescharacteristic, complemented
38#1926C411yescharacteristic, complemented
39#1926C411yescharacteristic, complemented
40#1925C411yescharacteristic, complemented
41#1925C411yescharacteristic, complemented
42#1851—21no—
43#1851—21no—
44#1850—21no—
45#1850—21no—
46#1768—21no—
47#1768—21no—
48#1767—21no—
49C4×C4⋊C8—21no—
50#1765—24no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53#1735C2211yescharacteristic
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—22no—
95C2×#108—41no—
96C2×#108—41no—
97C2×#108—41no—
98#427—41no—
99#427—41no—
100C2×#107C2×C411yescharacteristic
101C2×#107C2×C411yescharacteristic
102C2×#106C2×C411yescharacteristic
103C2×#106C2×C411yescharacteristic
104#424—22no—
105#424—41no—
106#423—22no—
107#423—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8D411yescharacteristic
115C2×C4⋊C8D411yescharacteristic
116C2×C4⋊C8D411yescharacteristic
117C2×C4⋊C8D411yescharacteristic
118C2×#104—22no—
119C2×#104—22no—
120C2×#104—22no—
121C2×#104—22no—
122C2×#104—22no—
123C2×#104—41no—
124#416—41no—
125#416—41no—
126#416D414yes—
127#416Q814yes—
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136#408—22no—
137#408—41no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140#405—22no—
141#405—22no—
142#405—41no—
143#405—41no—
144C2×#97—22no—
145C2×#97—21no—
146C2×#97—21no—
147C2×#97—22no—
148C2×#97—41no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151#391C2×C411yescharacteristic
152#391C2×C411yescharacteristic
153#390C2×C411yescharacteristic
154#390C2×C411yescharacteristic
155#386—21no—
156#386—21no—
157C4×C4⋊C4C2×C411yescharacteristic
158C4×C4⋊C4C2×C411yescharacteristic
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C4⋊C4C2×C411yescharacteristic
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165#376—21no—
166C2×#95—21no—
167C2×#95—21no—
168C43—21no—
169#341—22no—
170#341—41no—
171#340C2×C411yescharacteristic
172#340C2×C411yescharacteristic
173#340C2×C411yescharacteristic
174#340C2×C411yescharacteristic
175#339—22no—
176#339—22no—
177#339—22no—
178#339—22no—
179#339—22no—
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×D412yes—
207C2×#41C2×Q812yes—
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—21no—
241C2×M16—21no—
242C2×M16—42no—
243C2×M16—42no—
244C2×M16—42no—
245C2×M16#4112yes—
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×C8C2×Q812yes—
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#117C2×D412yes—
298#117C2×Q812yes—
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—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×C42#4112yes—
339#108—44no—
340#108—44no—
341#107—41no—
342#107—41no—
343#106—41no—
344#106—41no—
345C4⋊C8—24no—
346C4⋊C8—24no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8—42no—
352C4⋊C8—42no—
353#104—28no—
354#104—44no—
355#103—41no—
356#103—41no—
357#103C4⋊C414yes—
358#103C4⋊C414yes—
359#102—41no—
360#102—41no—
361#102C4⋊C414yes—
362#102C4⋊C414yes—
363C22⋊C8—42no—
364C22⋊C8—42no—
365#97—42no—
366#97—42no—
367#97—44no—
368C4×C8—42no—
369C4×C8—42no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—42no—
379#95—42no—
380C24—21no—
381C24C2×#4111yescharacteristic
382#41—22no—
383#41—22no—
384#41—22no—
385#41—22no—
386#41—22no—
387#41—22no—
388#41—22no—
389#41—22no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8C2×C4⋊C412yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C2×C4⋊C412yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
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×C4C4×D411yescharacteristic
452C22×C4C4×D411yescharacteristic
453C22×C4C4×D411yescharacteristic
454C22×C4C4×D411yescharacteristic
455C22×C4C23×C411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×Q811yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4111yescharacteristic
465C22×C4H(5,𝔽2)11yescharacteristic
466C22×C4H(5,𝔽2)11yescharacteristic
467C22×C4H(5,𝔽2)11yescharacteristic
468C22×C42−1+411yescharacteristic
469M16—22no—
470M16—22no—
471M16—22no—
472M16—22no—
473M16—22no—
474M16—22no—
475M16—48no—
476M16—41no—
477M16—41no—
478M16—41no—
479M16—41no—
480M16—41no—
481M16—41no—
482M16—41no—
483M16—41no—
484M16—41no—
485M16—41no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—22no—
517C2×C8—48no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
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×C8C2×C4⋊C412yes—
536C2×C8C2×C4⋊C412yes—
537C2×C8#12312yes—
538C2×C8#12312yes—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4C2×C4⋊C412yes—
550C4⋊C4C2×C4⋊C412yes—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—42no—
554C22⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4—42no—
559C22⋊C4—81no—
560C22⋊C4—81no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—42no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—42no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—81no—
586C42—81no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—22no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23C2×C4×D411yescharacteristic
599C23#52911yescharacteristic
600C23#52911yescharacteristic
601C23#54711yescharacteristic
602C23#55311yescharacteristic
603C23#55411yescharacteristic
604C23#55511yescharacteristic
605Q8—22no—
606Q8—22no—
607Q8—41no—
608Q8—41no—
609D4—22no—
610D4—22no—
611D4—42no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—41no—
616D4—41no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
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—42no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—81no—
668C2×C4—81no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4#44412yes—
672C2×C4C22×C4⋊C411yescharacteristic
673C2×C4C4×#4111yescharacteristic
674C2×C4C4×#4111yescharacteristic
675C2×C4C4×#4111yescharacteristic
676C2×C4#51811yescharacteristic
677C2×C4#52011yescharacteristic
678C2×C4C2×#12311yescharacteristic
679C2×C4#52511yescharacteristic
680C2×C4#52511yescharacteristic
681C2×C4#52511yescharacteristic
682C2×C4#53311yescharacteristic
683C2×C4#53311yescharacteristic
684C2×C4#53811yescharacteristic
685C2×C4#53811yescharacteristic
686C2×C4#54211yescharacteristic
687C2×C4#54211yescharacteristic
688C2×C4D4211yescharacteristic
689C2×C4#54611yescharacteristic
690C2×C4#54611yescharacteristic
691C2×C4#54611yescharacteristic
692C2×C4D4×Q811yescharacteristic
693C2×C4#55011yescharacteristic
694C2×C4#55111yescharacteristic
695C2×C4#55211yescharacteristic
696C2×C4#55311yescharacteristic
697C2×C4#55411yescharacteristic
698C2×C4#55411yescharacteristic
699C2×C4#55611yescharacteristic
700C8—22no—
701C8—22no—
702C8—22no—
703C8—22no—
704C8—41no—
705C8—41no—
706C8—41no—
707C8—41no—
708C8—42no—
709C8—42no—
710C8—42no—
711C8—81no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—21no—
717C22—21no—
718C22—21no—
719C22—21no—
720C22—21no—
721C22—21no—
722C22—21no—
723C22—21no—
724C22—42no—
725C22—42no—
726C22—42no—
727C22—81no—
728C22—81no—
729C22—81no—
730C22—81no—
731C22C4×#12311yescharacteristic
732C22D4×C4⋊C411yescharacteristic
733C22#236311yescharacteristic
734C22#246811yescharacteristic
735C22#247411yescharacteristic
736C22#257411yescharacteristic
737C22#258711yescharacteristic
738C22C2×#44411yescharacteristic
739C22#295511yescharacteristic
740C22#295611yescharacteristic
741C22#295711yescharacteristic
742C4—22no—
743C4—42no—
744C4—41no—
745C4—41no—
746C4—41no—
747C4—42no—
748C4—42no—
749C4—82no—
750C4—82no—
751C4—81no—
752C4—81no—
753C4—81no—
754C4—81no—
755C4#229611yescharacteristic
756C4#233711yescharacteristic
757C4#234911yescharacteristic
758C4#237311yescharacteristic
759C4#241811yescharacteristic
760C4#242311yescharacteristic
761C4#245611yescharacteristic
762C4#246511yescharacteristic
763C2—21no—
764C2—21no—
765C2—21no—
766C2—21no—
767C2—82no—
768C2#1411811yescharacteristic
769C2#2159311yescharacteristic
770C2#2159411yescharacteristic
771C2#2268611yescharacteristic
772C2#2268711yescharacteristic
773C2#2311411yescharacteristic
774C2#2311511yescharacteristic
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
4. #20431C2semidirect
6. #11326C2semidirect
7. #11221C2semidirect
10. #10910C2semidirect
11. #10887C2semidirect
12. #10877C2semidirect
13. #10860C2semidirect
14. #10808C2semidirect
34. #1928C4semidirect
35. #1928C4semidirect
36. #1927C4semidirect
37. #1927C4semidirect
38. #1926C4semidirect
39. #1926C4semidirect
40. #1925C4semidirect
41. #1925C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19292
#210582
#208382
#204312
C2×#17652
#113262
#112212
#111252
#110532
#109102
#108872
#108772
#108602
#108082
#107912