Finite Group Databasebeta

← #377627

Subgroups — #377627

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 772 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic, complemented
3C2×#1929C211yesmaximal, characteristic
4C2×#1929C211yesmaximal, characteristic
5#21051C211yesmaximal, characteristic
6C2×#1772C211yesmaximal, characteristic, complemented
7#20439C211yesmaximal, characteristic, complemented
8#11221C211yesmaximal, characteristic, complemented
9#11209C211yesmaximal, characteristic, complemented
10#11068C211yesmaximal, characteristic
11#10933C211yesmaximal, characteristic
12#10929C211yesmaximal, characteristic
13#10910C211yesmaximal, characteristic, complemented
14#10876C211yesmaximal, characteristic, complemented
15#10809C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2297C2211yescharacteristic
34#2296C2211yescharacteristic
35#2068C411yescharacteristic, complemented
36#2068C411yescharacteristic, complemented
37#2067C411yescharacteristic, complemented
38#2067C411yescharacteristic, complemented
39#2066C411yescharacteristic, complemented
40#2066C411yescharacteristic, complemented
41#2065C411yescharacteristic, complemented
42#2065C411yescharacteristic, complemented
43#1950—24no—
44#1929—24no—
45#1929—24no—
46#1924—21no—
47#1924—21no—
48#1923—21no—
49#1923—21no—
50#1772—24no—
51#1771—21no—
52#1771—21no—
53#1770—21no—
54C8×C4⋊C4—21no—
55C2×#341C2211yescharacteristic
56C2×#341C2211yescharacteristic
57#1735C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62C2×#339C2211yescharacteristic
63C2×#339C2211yescharacteristic
64#1729C2211yescharacteristic
65#1729C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C2×#336C2211yescharacteristic
70C4×#95C2211yescharacteristic
71C22×#41C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—22no—
91C2×#108D414yes—
92#427—22no—
93#427—22no—
94C2×#107C2×C411yescharacteristic
95C2×#107C2×C411yescharacteristic
96C2×#106C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98#424—22no—
99#424—22no—
100#423—22no—
101#423—41no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8C2×C411yescharacteristic
109C2×C4⋊C8C2×C411yescharacteristic
110C2×C4⋊C8C2×C411yescharacteristic
111C2×C4⋊C8C2×C411yescharacteristic
112C2×#104—22no—
113C2×#104—22no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—41no—
118#416—22no—
119#416—22no—
120#416—22no—
121#416—22no—
122C2×#103C2×C411yescharacteristic
123C2×#103C2×C411yescharacteristic
124C2×#103C2×C411yescharacteristic
125C2×#103C2×C411yescharacteristic
126C2×#102C2×C411yescharacteristic
127C2×#102C2×C411yescharacteristic
128C2×#102C2×C411yescharacteristic
129C2×#102C2×C411yescharacteristic
130#408—22no—
131#408—22no—
132#408—22no—
133#408—41no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138#405—22no—
139#405—22no—
140#405—22no—
141#405—22no—
142#405—41no—
143#405D414yes—
144C4×M16—22no—
145C4×M16—41no—
146C2×#97—21no—
147C2×#97—21no—
148C2×#97—21no—
149C2×#97—21no—
150C2×C4×C8—22no—
151C2×C4×C8—21no—
152C2×C4×C8—22no—
153C2×C4×C8—22no—
154C2×C4×C8—21no—
155C2×C4×C8—21no—
156C2×C4×C8—21no—
157#391C2×C411yescharacteristic
158#391C2×C411yescharacteristic
159#390C2×C411yescharacteristic
160#390C2×C411yescharacteristic
161#389C2×C411yescharacteristic
162#389C2×C411yescharacteristic
163#389C2×C411yescharacteristic
164#389C2×C411yescharacteristic
165#386—21no—
166#386—21no—
167#385—21no—
168#385—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173C2×#95—21no—
174C2×#95—21no—
175#341—22no—
176#341—22no—
177#340C2×C411yescharacteristic
178#340C2×C411yescharacteristic
179#340C2×C411yescharacteristic
180#340C2×C411yescharacteristic
181#339—22no—
182#339—22no—
183#339—22no—
184#339—22no—
185#339—22no—
186#339—41no—
187#336—21no—
188#336—21no—
189#336—22no—
190#336—22no—
191#336—21no—
192#336—21no—
193#336—22no—
194#336D411yescharacteristic
195#336D411yescharacteristic
196#336D411yescharacteristic
197#336D411yescharacteristic
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×#41—21no—
207C2×#41—21no—
208C2×#41—21no—
209C2×#41—21no—
210C22×Q8C22×C411yescharacteristic
211C22×D4—21no—
212C22×D4C22×C411yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4#4111yescharacteristic
217C23×C4#4111yescharacteristic
218C23×C4C2411yescharacteristic
219C2×M16—21no—
220C2×M16—21no—
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×M16C2×D412yes—
239C2×M16C2×D412yes—
240C2×M16C2×D412yes—
241C2×M16#4112yes—
242C2×M16#4112yes—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
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×C8C22×C411yescharacteristic
268C22×C8C22×C411yescharacteristic
269C22×C8C22×C411yescharacteristic
270C22×C8C22×C411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8C2×D411yescharacteristic
273C22×C8C2×D412yes—
274C22×C8C2×D412yes—
275C22×C8C2×D411yescharacteristic
276C22×C8C2×D411yescharacteristic
277C22×C8#4112yes—
278C22×C8#4112yes—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117C2×D412yes—
290#117#4112yes—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—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×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42#4112yes—
331#108—28no—
332#108—44no—
333#107—22no—
334#107—22no—
335#106—22no—
336#106—22no—
337C4⋊C8—24no—
338C4⋊C8—24no—
339C4⋊C8—22no—
340C4⋊C8—22no—
341C4⋊C8—22no—
342C4⋊C8—22no—
343C4⋊C8—42no—
344C4⋊C8—42no—
345#104—44no—
346#104—44no—
347#103—22no—
348#103—22no—
349#103—22no—
350#103—22no—
351#102—22no—
352#102—22no—
353#102—22no—
354#102—22no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357C22⋊C8—42no—
358C22⋊C8—42no—
359#97—24no—
360#97—42no—
361#97—42no—
362#97—42no—
363C4×C8—28no—
364C4×C8—24no—
365C4×C8—42no—
366C4×C8—42no—
367C4×C8—42no—
368#95—24no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—42no—
378C24—21no—
379C24C2×#4111yescharacteristic
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—41no—
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×C4C4×D411yescharacteristic
450C22×C4C4×D411yescharacteristic
451C22×C4C4×D411yescharacteristic
452C22×C4C4×D411yescharacteristic
453C22×C4C23×C411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4111yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4H(5,𝔽2)11yescharacteristic
464C22×C42−1+411yescharacteristic
465C22×C42−1+411yescharacteristic
466C22×C42−1+411yescharacteristic
467M16—22no—
468M16—22no—
469M16—22no—
470M16—22no—
471M16—22no—
472M16—22no—
473M16—22no—
474M16—41no—
475M16—41no—
476M16—41no—
477M16—41no—
478M16—41no—
479M16—41no—
480M16—41no—
481M16—41no—
482M16#12114yes—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—21no—
496C2×C8—21no—
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—21no—
512C2×C8—21no—
513C2×C8—22no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—41no—
517C2×C8—41no—
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×C8C4×D412yes—
532C2×C8C4×D412yes—
533C2×C8#12112yes—
534C2×C8#12112yes—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—81no—
548C4⋊C4—81no—
549C4⋊C4C4×D412yes—
550C4⋊C4C4×D412yes—
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—22no—
566C42—22no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—22no—
572C42—22no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23C2×C4×D411yescharacteristic
595C23#52911yescharacteristic
596C23#53211yescharacteristic
597C23#54811yescharacteristic
598C23#55211yescharacteristic
599C23#55311yescharacteristic
600C23#55611yescharacteristic
601Q8—41no—
602Q8—41no—
603Q8—41no—
604Q8—41no—
605D4—41no—
606D4—41no—
607D4—41no—
608D4—41no—
609D4—81no—
610D4—81no—
611D4—81no—
612D4—81no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—24no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
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—81no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4#44312yes—
668C2×C4C2×C4×D411yescharacteristic
669C2×C4C2×C4×D411yescharacteristic
670C2×C4C4×#4111yescharacteristic
671C2×C4C4×#4111yescharacteristic
672C2×C4#51911yescharacteristic
673C2×C4#52011yescharacteristic
674C2×C4C2×#12111yescharacteristic
675C2×C4C2×#12111yescharacteristic
676C2×C4#52511yescharacteristic
677C2×C4#52511yescharacteristic
678C2×C4#52911yescharacteristic
679C2×C4#53211yescharacteristic
680C2×C4#53611yescharacteristic
681C2×C4#53711yescharacteristic
682C2×C4#54111yescharacteristic
683C2×C4#54211yescharacteristic
684C2×C4D4211yescharacteristic
685C2×C4#54711yescharacteristic
686C2×C4#54711yescharacteristic
687C2×C4#54711yescharacteristic
688C2×C4#54811yescharacteristic
689C2×C4#54811yescharacteristic
690C2×C4#55111yescharacteristic
691C2×C4#55211yescharacteristic
692C2×C4#55211yescharacteristic
693C2×C4#55311yescharacteristic
694C2×C4#55611yescharacteristic
695C2×C4#55611yescharacteristic
696C8—22no—
697C8—22no—
698C8—24no—
699C8—22no—
700C8—22no—
701C8—42no—
702C8—41no—
703C8—41no—
704C8—42no—
705C8—41no—
706C8—41no—
707C8—81no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—21no—
717C22—21no—
718C22—21no—
719C22—21no—
720C22—81no—
721C22—81no—
722C22—81no—
723C22—81no—
724C22—81no—
725C22—81no—
726C22—81no—
727C22C4×#12111yescharacteristic
728C22#236111yescharacteristic
729C22#236211yescharacteristic
730C22#251911yescharacteristic
731C22#252511yescharacteristic
732C22#254511yescharacteristic
733C22#260111yescharacteristic
734C22C2×#44311yescharacteristic
735C22#295511yescharacteristic
736C22#295611yescharacteristic
737C22#295711yescharacteristic
738C4—22no—
739C4—44no—
740C4—42no—
741C4—42no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—82no—
746C4—81no—
747C4—81no—
748C4—81no—
749C4—81no—
750C4—81no—
751C4C4×#12111yescharacteristic
752C4#232311yescharacteristic
753C4#236211yescharacteristic
754C4#237311yescharacteristic
755C4#241311yescharacteristic
756C4#241611yescharacteristic
757C4#246111yescharacteristic
758C4#246911yescharacteristic
759C2—21no—
760C2—21no—
761C2—21no—
762C2—21no—
763C2—161no—
764C2#1420011yescharacteristic
765C2#2151711yescharacteristic
766C2#2151911yescharacteristic
767C2#2301811yescharacteristic
768C2#2302111yescharacteristic
769C2#2308011yescharacteristic
770C2#2308211yescharacteristic
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 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1950C2semidirect
6. C2×#1772C2semidirect
7. #20439C2semidirect
8. #11221C2semidirect
9. #11209C2semidirect
13. #10910C2semidirect
14. #10876C2semidirect
15. #10809C2semidirect
35. #2068C4semidirect
36. #2068C4semidirect
37. #2067C4semidirect
38. #2067C4semidirect
39. #2066C4semidirect
40. #2066C4semidirect
41. #2065C4semidirect
42. #2065C4semidirect

Maximal subgroups

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

subgroupindex
#236092
C2×#19502
C2×#19292
#210512
C2×#17722
#204392
#112212
#112092
#110682
#109332
#109292
#109102
#108762
#108092