Finite Group Databasebeta

← #377308

Subgroups — #377308

Order 512.

866 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23627C211yesmaximal, characteristic
2#21052C211yesmaximal, characteristic
3#20441C211yesmaximal, characteristic, complemented
4#11370C211yesmaximal, characteristic, complemented
5#11317C211yesmaximal, characteristic, complemented
6#11125C211yesmaximal, characteristic
7#10934C211yesmaximal, characteristic
8#10934C211yesmaximal, characteristic
9#10933C211yesmaximal, characteristic
10#10930C211yesmaximal, characteristic
11#10911C211yesmaximal, characteristic, complemented
12#10910C211yesmaximal, characteristic, complemented
13#10877C211yesmaximal, characteristic, complemented
14#10809C211yesmaximal, characteristic, complemented
15#10707C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×#104C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C2×#416C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#408C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×#97C2211yescharacteristic
29#2297C2211yescharacteristic
30#2285C2211yescharacteristic
31#2074C411yescharacteristic, complemented
32#2074C411yescharacteristic, complemented
33#2073C411yescharacteristic, complemented
34#2073C411yescharacteristic, complemented
35#2072C411yescharacteristic, complemented
36#2072C411yescharacteristic, complemented
37#2071C411yescharacteristic, complemented
38#2071C411yescharacteristic, complemented
39#1924—21no—
40#1924—21no—
41#1923—21no—
42#1923—21no—
43#1771—21no—
44#1771—21no—
45#1770—21no—
46C8×C4⋊C4—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51#1735C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#108—22no—
83C2×#108—22no—
84C2×#108—41no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8D411yescharacteristic
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×#104—22no—
92C2×#104—22no—
93C2×#104—22no—
94C2×#104—22no—
95C2×#104—22no—
96C2×#104—41no—
97#416—41no—
98#416—41no—
99#416—41no—
100#416—41no—
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109#408—22no—
110#408—22no—
111#408—22no—
112#408—41no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117#405—22no—
118#405—22no—
119#405—22no—
120#405—41no—
121#405—41no—
122#405—41no—
123C4×M16—22no—
124C4×M16—22no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—21no—
128C2×#97—21no—
129C2×#97—22no—
130C2×#97—41no—
131C2×#97—41no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136#391C2×C411yescharacteristic
137#391C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#386—21no—
141#386—21no—
142#384C2×C411yescharacteristic
143#384C2×C411yescharacteristic
144#382C2×C411yescharacteristic
145#382C2×C411yescharacteristic
146#380—21no—
147#380—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C2×#95—21no—
153C2×#95—21no—
154#341—22no—
155#341—22no—
156#341—41no—
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#340C2×C411yescharacteristic
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#339—22no—
166#339—22no—
167#339—22no—
168#339—41no—
169#339—41no—
170#339—41no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—22no—
178#336—21no—
179#336—21no—
180#336—41no—
181#336—41no—
182#336C2×C411yescharacteristic
183#336C2×C411yescharacteristic
184#336C2×C411yescharacteristic
185#336C2×C411yescharacteristic
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C22×Q8C22×C411yescharacteristic
199C22×D4—21no—
200C22×D4C22×C411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4#4111yescharacteristic
205C23×C4#4111yescharacteristic
206C23×C4C2411yescharacteristic
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16#4112yes—
227C2×M16#4112yes—
228C2×M16#4112yes—
229C2×M16#4112yes—
230C2×M16#4112yes—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
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×C8C22×C411yescharacteristic
258C22×C8C22×C411yescharacteristic
259C22×C8C22×C411yescharacteristic
260C22×C8C22×C411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8#4112yes—
266C22×C8#4112yes—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42#4112yes—
319#108—44no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324#104—44no—
325#104—44no—
326#103—41no—
327#103—41no—
328#103—41no—
329#103—41no—
330#102—41no—
331#102—41no—
332#102—41no—
333#102—41no—
334C22⋊C8—42no—
335C22⋊C8—42no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338#97—24no—
339#97—42no—
340#97—42no—
341#97—42no—
342#97—44no—
343C4×C8—24no—
344C4×C8—42no—
345C4×C8—42no—
346C4×C8—42no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—42no—
356#95—42no—
357C24—21no—
358C24C2×#4111yescharacteristic
359#41—41no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C23×C411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C42−1+411yescharacteristic
446M16—22no—
447M16—22no—
448M16—22no—
449M16—22no—
450M16—22no—
451M16—22no—
452M16—22no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16#12614yes—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—22no—
488C2×C8—21no—
489C2×C8—22no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8#12312yes—
513C2×C8#12312yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—81no—
529C4⋊C4—81no—
530C22⋊C4—42no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—42no—
538C22⋊C4—81no—
539C22⋊C4—81no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—22no—
549C42—22no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23C2×C4×D411yescharacteristic
574C23#52911yescharacteristic
575C23#52911yescharacteristic
576C23#55311yescharacteristic
577C23#55511yescharacteristic
578C23#55611yescharacteristic
579C23#55911yescharacteristic
580Q8—41no—
581Q8—41no—
582Q8—41no—
583Q8—41no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—81no—
589D4—81no—
590D4—81no—
591D4—81no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4#44412yes—
647C2×C4C4×#4111yescharacteristic
648C2×C4C4×#4111yescharacteristic
649C2×C4C4×#4111yescharacteristic
650C2×C4C4×#4111yescharacteristic
651C2×C4#51811yescharacteristic
652C2×C4#52011yescharacteristic
653C2×C4C2×#12311yescharacteristic
654C2×C4#52511yescharacteristic
655C2×C4C2×#12611yescharacteristic
656C2×C4#52911yescharacteristic
657C2×C4#52911yescharacteristic
658C2×C4#52911yescharacteristic
659C2×C4#53811yescharacteristic
660C2×C4#53911yescharacteristic
661C2×C4#54011yescharacteristic
662C2×C4#54211yescharacteristic
663C2×C4#54611yescharacteristic
664C2×C4#54611yescharacteristic
665C2×C4#54811yescharacteristic
666C2×C4#55011yescharacteristic
667C2×C4#55111yescharacteristic
668C2×C4#55111yescharacteristic
669C2×C4#55211yescharacteristic
670C2×C4#55311yescharacteristic
671C2×C4#55311yescharacteristic
672C2×C4#55311yescharacteristic
673C2×C4#55611yescharacteristic
674C2×C4#55911yescharacteristic
675C8—22no—
676C8—22no—
677C8—24no—
678C8—41no—
679C8—41no—
680C8—41no—
681C8—41no—
682C8—41no—
683C8—41no—
684C8—42no—
685C8—42no—
686C8—81no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22C4×#12311yescharacteristic
707C22#236311yescharacteristic
708C22#236711yescharacteristic
709C22#251311yescharacteristic
710C22#253211yescharacteristic
711C22#259411yescharacteristic
712C22#259711yescharacteristic
713C22C2×#44411yescharacteristic
714C22#295511yescharacteristic
715C22#295611yescharacteristic
716C22#295711yescharacteristic
717C4—22no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—82no—
722C4—82no—
723C4—81no—
724C4—81no—
725C4—81no—
726C4—81no—
727C4—81no—
728C4—81no—
729C4—81no—
730C4C4×#12611yescharacteristic
731C4#233511yescharacteristic
732C4#237311yescharacteristic
733C4#237411yescharacteristic
734C4#244411yescharacteristic
735C4#244811yescharacteristic
736C4#247111yescharacteristic
737C4#248111yescharacteristic
738C2—21no—
739C2—21no—
740C2—21no—
741C2—21no—
742C2—161no—
743C2#1419311yescharacteristic
744C2#2159311yescharacteristic
745C2#2159411yescharacteristic
746C2#2311411yescharacteristic
747C2#2311511yescharacteristic
748C2#2323511yescharacteristic
749C2#2323611yescharacteristic
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
3. #20441C2semidirect
4. #11370C2semidirect
5. #11317C2semidirect
11. #10911C2semidirect
12. #10910C2semidirect
13. #10877C2semidirect
14. #10809C2semidirect
15. #10707C2semidirect
31. #2074C4semidirect
32. #2074C4semidirect
33. #2073C4semidirect
34. #2073C4semidirect
35. #2072C4semidirect
36. #2072C4semidirect
37. #2071C4semidirect
38. #2071C4semidirect

Maximal subgroups

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

subgroupindex
#236272
#210522
#204412
#113702
#113172
#111252
#109342
#109332
#109302
#109112
#109102
#108772
#108092
#107072