Finite Group Databasebeta

← #377903

Subgroups — #377903

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2C2×#1945C211yesmaximal, characteristic, complemented
3#21147C211yesmaximal, characteristic, complemented
4C2×#1866C211yesmaximal, characteristic
5#20838C211yesmaximal, characteristic
6C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
7C2×#1764C211yesmaximal, characteristic
8#11291C211yesmaximal, characteristic, complemented
9#11233C211yesmaximal, characteristic, complemented
10#11068C211yesmaximal, characteristic
11#10929C211yesmaximal, characteristic
12#10923C211yesmaximal, characteristic
13#10909C211yesmaximal, characteristic, complemented
14#10882C211yesmaximal, characteristic, complemented
15#10878C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#104C2211yescharacteristic
25C22×#104C2211yescharacteristic
26C2×#416C2211yescharacteristic
27C2×#416C2211yescharacteristic
28C2×#408C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2297C2211yescharacteristic
33#2296C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#2068C412yescomplemented
36#2067C412yescomplemented
37#2066C412yescomplemented
38#2065C412yescomplemented
39#1945—24no—
40#1940—22no—
41#1939—22no—
42#1866—24no—
43#1851—22no—
44#1850—22no—
45C4×C4⋊C8—24no—
46#1764—24no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1728C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—22no—
67C2×#131—22no—
68C2×#131—41no—
69C2×#131—41no—
70C22×M16—21no—
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C23×C8—21no—
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—41no—
91#427—22no—
92#427—22no—
93C2×#107C2×C412yes—
94C2×#106C2×C412yes—
95#424—22no—
96#424—22no—
97#424—22no—
98#424—22no—
99#424—41no—
100#424—41no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2×C412yes—
110C2×C4⋊C8C2×C412yes—
111C2×C4⋊C8D412yes—
112C2×C4⋊C8D412yes—
113C2×#104—22no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—41no—
118C2×#104Q814yes—
119#416—22no—
120#416—22no—
121#416—22no—
122#416—22no—
123C2×#103C2×C411yescharacteristic
124C2×#103C2×C412yes—
125C2×#103C2×C411yescharacteristic
126C2×#102C2×C411yescharacteristic
127C2×#102C2×C412yes—
128C2×#102C2×C411yescharacteristic
129#408—22no—
130#408—41no—
131C2×C22⋊C8—22no—
132#405—22no—
133#405—41no—
134C2×#97—21no—
135C2×C4×C8—22no—
136C2×C4×C8—22no—
137C2×C4×C8—21no—
138C2×C4×C8—22no—
139C2×C4×C8—22no—
140C2×C4×C8—41no—
141C2×C4×C8—41no—
142#391C2×C412yes—
143#390C2×C412yes—
144#389C2×C412yes—
145#389C2×C412yes—
146#386—21no—
147#386—21no—
148#385—21no—
149#385—21no—
150C4×C4⋊C4—22no—
151C4×C22⋊C4—22no—
152C43—22no—
153C43—22no—
154#341—22no—
155#341—22no—
156#341—41no—
157#340C2×C412yes—
158#340C2×C412yes—
159#339—22no—
160#339—22no—
161#339—22no—
162#339—22no—
163#339—41no—
164#339D414yes—
165#336—21no—
166#336—21no—
167#336—22no—
168#336—22no—
169#336—22no—
170#336—22no—
171#336D414yes—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41C2×Q812yes—
183C2×#41#4112yes—
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4C22×C411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4#4111yescharacteristic
191C23×C4#4111yescharacteristic
192C23×C4C2411yescharacteristic
193#131—28no—
194#131—44no—
195#131—44no—
196#131—44no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
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—42no—
220C2×M16—42no—
221C2×M16—42no—
222C2×M16C2×D412yes—
223C2×M16C2×Q812yes—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
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—42no—
249C22×C8—42no—
250C22×C8—42no—
251C22×C8C22×C412yes—
252C22×C8C22×C411yescharacteristic
253C22×C8C22×C411yescharacteristic
254C22×C8C2×D412yes—
255C22×C8C2×D412yes—
256C22×C8C2×D412yes—
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8#4112yes—
260C22×C8#4112yes—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117C2×D412yes—
272#117#4112yes—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C22×C412yes—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—24no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—44no—
303C2×C42C22×C412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×Q812yes—
308C2×C42#4112yes—
309#108—44no—
310#107—24no—
311#106—24no—
312C4⋊C8—24no—
313C4⋊C8—24no—
314C4⋊C8—24no—
315C4⋊C8—24no—
316C4⋊C8—44no—
317C4⋊C8—44no—
318C4⋊C8—42no—
319C4⋊C8—42no—
320#104—28no—
321#104—44no—
322#103—22no—
323#103—22no—
324#103—24no—
325#102—22no—
326#102—22no—
327#102—24no—
328C22⋊C8—44no—
329#97—42no—
330C4×C8—44no—
331C4×C8—44no—
332C4×C8—42no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337C24—21no—
338C24C2×#4111yescharacteristic
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—22no—
346#41—22no—
347#41—41no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353#41—41no—
354#41—41no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8C4×Q812yes—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4C4×Q812yes—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4C4×D412yes—
401C22×C4C4×D412yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×Q811yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C42−1+411yescharacteristic
415C22×C42−1+411yescharacteristic
416M16—22no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16—48no—
421M16—41no—
422M16—41no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—21no—
441C2×C8—22no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—48no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8C4×D412yes—
477C2×C8C4×D412yes—
478C2×C8#12112yes—
479C2×C8#12112yes—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—44no—
488C4⋊C4—82no—
489C4⋊C4C4×D414yes—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C42—216no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—48no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42C22⋊Q814yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23C2×C4×D411yescharacteristic
526C23#52511yescharacteristic
527C23#52911yescharacteristic
528C23#54711yescharacteristic
529C23#55211yescharacteristic
530C23#55311yescharacteristic
531C23#55411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—22no—
537D4—22no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—41no—
543D4—41no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4#44314yes—
581C2×C4C2×C4×D411yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×C4×Q811yescharacteristic
584C2×C4C4×#4111yescharacteristic
585C2×C4#52011yescharacteristic
586C2×C4#52011yescharacteristic
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×C22⋊Q811yescharacteristic
589C2×C4#52511yescharacteristic
590C2×C4#52911yescharacteristic
591C2×C4#53211yescharacteristic
592C2×C4#53311yescharacteristic
593C2×C4#53811yescharacteristic
594C2×C4#53911yescharacteristic
595C2×C4#54111yescharacteristic
596C2×C4#54211yescharacteristic
597C2×C4D4211yescharacteristic
598C2×C4D4211yescharacteristic
599C2×C4#54611yescharacteristic
600C2×C4#54611yescharacteristic
601C2×C4#54711yescharacteristic
602C2×C4#54811yescharacteristic
603C2×C4#54811yescharacteristic
604C2×C4D4×Q811yescharacteristic
605C2×C4D4×Q811yescharacteristic
606C2×C4#55211yescharacteristic
607C2×C4#55311yescharacteristic
608C2×C4#55411yescharacteristic
609C8—22no—
610C8—22no—
611C8—22no—
612C8—22no—
613C8—42no—
614C8—42no—
615C8—42no—
616C8—42no—
617C8—81no—
618C8—81no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—42no—
628C22—44no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22C4×#12111yescharacteristic
633C22#236011yescharacteristic
634C22#236111yescharacteristic
635C22#244911yescharacteristic
636C22#247011yescharacteristic
637C22#247511yescharacteristic
638C22#258711yescharacteristic
639C22C2×#44312yes—
640C22#295612yes—
641C4—28no—
642C4—24no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—81no—
648C4—81no—
649C4—82no—
650C4C4×C22⋊Q811yescharacteristic
651C4#233411yescharacteristic
652C4#236011yescharacteristic
653C4#236111yescharacteristic
654C4#242411yescharacteristic
655C4#242911yescharacteristic
656C4#245211yescharacteristic
657C4#255811yescharacteristic
658C2—21no—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—82no—
663C2#1420411yescharacteristic
664C2#2154411yescharacteristic
665C2#2155111yescharacteristic
666C2#2298212yes—
667C2#2301512yes—
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×#1945C2semidirect
3. #21147C2semidirect
6. C2×C4×C4⋊C8C2semidirect
8. #11291C2semidirect
9. #11233C2semidirect
13. #10909C2semidirect
14. #10882C2semidirect
15. #10878C2semidirect
35. #2068C4semidirect
36. #2067C4semidirect
37. #2066C4semidirect
38. #2065C4semidirect

Maximal subgroups

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

subgroupindex
#236092
C2×#19452
#211472
C2×#18662
#208382
C2×C4×C4⋊C82
C2×#17642
#112912
#112332
#110682
#109292
#109232
#109092
#108822
#108782