Finite Group Databasebeta

← C2×#36467 #7630150

Subgroups — C2×#36467 #7630150

Order 512.

8624 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C22×#548C211yesmaximal, characteristic
3C2×#2861C213yesmaximal, complemented
4#36467C2132yesmaximal, direct product factor
5C2×#2592C213yesmaximal, complemented
6C2×#2562C213yesmaximal, complemented
7C2×#2490C213yesmaximal, complemented
8C2×#2489C213yesmaximal, complemented
9C2×#2492C213yesmaximal, complemented
10C2×#2491C213yesmaximal, complemented
11C2×#2341C211yesmaximal, characteristic, complemented
12C2×#2341C211yesmaximal, characteristic, complemented
13C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
14C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
15C2×C4×#124C213yesmaximal, complemented
16C2×C42×Q8C211yesmaximal, characteristic, complemented
17C23×#41C2211yescharacteristic
18C24×Q8C2211yescharacteristic
19C2×#557—216no—
20C2×#557C22116yescomplemented
21C2×#548—216no—
22C2×#548C22116yes—
23C22×C4⋊Q8C2213yes—
24C22×#125C2213yes—
25C22×#125C2213yes—
26C22×#124C2213yes—
27C22×#124C2213yes—
28C22×C22⋊Q8C2213yes—
29C22×#121C2213yes—
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33C22×C4×Q8C2211yescharacteristic
34C22×C4×Q8C2213yes—
35C22×C4×Q8C2211yescharacteristic
36C22×C4×D4C2213yes—
37#2861C22148yescomplemented
38#2592C22148yescomplemented
39#2562C22148yescomplemented
40#2492C22148yescomplemented
41#2491C22148yescomplemented
42#2490C22148yescomplemented
43#2489C22148yescomplemented
44C2×#400C2216yes—
45C2×#397C2216yes—
46C2×#394C2216yes—
47C2×#393C2216yes—
48#2341C22116yescomplemented
49#2341C22116yescomplemented
50Q8×C22⋊C4C22116yescomplemented
51Q8×C22⋊C4C22116yescomplemented
52C4×#124C22148yescomplemented
53C2×#391C2213yes—
54C2×#391C2213yes—
55C2×#391C2213yes—
56C2×#391C2213yes—
57C2×#391C2213yes—
58C2×#390C2213yes—
59C2×#390C2213yes—
60C2×#390C2213yes—
61C2×#390C2213yes—
62C2×#390C2213yes—
63C2×#388C2213yes—
64C2×#388C2216yes—
65C2×#388C2216yes—
66C2×#388C2216yes—
67C2×#388C2213yes—
68C2×#388C2213yes—
69C2×#388C2211yescharacteristic
70C2×#382C2213yes—
71C2×#380C2213yes—
72C2×#380C2213yes—
73C2×#380C2213yes—
74C2×#380C2213yes—
75C42×Q8C22116yescomplemented
76C2×C4×C4⋊C4C2213yes—
77C2×C4×C4⋊C4C2213yes—
78C2×C4×C4⋊C4C2213yes—
79C2×C4×C22⋊C4C2213yes—
80C2×C4×C22⋊C4C2213yes—
81C2×C4×C22⋊C4C2213yes—
82C2×C4×C22⋊C4C2213yes—
83C2×C43C2213yes—
84C22×#41—28no—
85C22×#41—28no—
86C22×#41—28no—
87C22×#41C2318yes—
88C23×Q8—28no—
89C23×Q8—28no—
90C23×Q8—28no—
91C23×Q8C2318yes—
92C23×Q8C2313yes—
93C23×Q8C2313yes—
94C23×Q8C2311yescharacteristic
95C23×D4C2313yes—
96C23×D4C2313yes—
97C24×C4C2313yes—
98C24×C4C2313yes—
99#557—2128no—
100#548—2128no—
101C2×C4⋊Q8—224no—
102C2×C4⋊Q8C23124yes—
103C2×#125—224no—
104C2×#125—224no—
105C2×#125C23124yes—
106C2×#125C23124yes—
107C2×#124—224no—
108C2×#124—224no—
109C2×#124C23124yes—
110C2×#124C23124yes—
111C2×C22⋊Q8—224no—
112C2×C22⋊Q8C23124yes—
113C2×#121—224no—
114C2×#121C23124yes—
115C2×C4×Q8—28no—
116C2×C4×Q8—28no—
117C2×C4×Q8—28no—
118C2×C4×Q8—28no—
119C2×C4×Q8—224no—
120C2×C4×Q8D4116yes—
121C2×C4×Q8C23124yes—
122C2×C4×Q8C2318yes—
123C2×C4×Q8C2318yes—
124C2×C4×Q8C2318yes—
125C2×C4×Q8C2318yes—
126C2×C4×Q8C2318yes—
127C2×C4×D4—224no—
128C2×C4×D4C23124yes—
129C22×C4⋊C4C2316yes—
130C22×C4⋊C4C2313yes—
131C22×C4⋊C4C2313yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2313yes—
134C22×C4⋊C4C2313yes—
135C22×C4⋊C4C2313yes—
136C22×C4⋊C4C2313yes—
137C22×C4⋊C4C2313yes—
138C22×C4⋊C4C2313yes—
139C22×C22⋊C4C2313yes—
140C22×C22⋊C4C2316yes—
141C22×C22⋊C4C2313yes—
142C22×C22⋊C4C2311yescharacteristic
143C22×C22⋊C4C2313yes—
144C22×C22⋊C4C2313yes—
145C22×C22⋊C4C2311yescharacteristic
146C22×C22⋊C4C2313yes—
147C22×C22⋊C4C2316yes—
148C22×C22⋊C4C2313yes—
149C22×C22⋊C4C2316yes—
150C22×C22⋊C4C2313yes—
151C22×C22⋊C4C2311yescharacteristic
152C22×C42C2313yes—
153C22×C42C2313yes—
154C22×C42C2313yes—
155C22×C42C2313yes—
156C22×C42C2313yes—
157C22×C42C2311yescharacteristic
158C22×C42C2313yes—
159C22×C42C2313yes—
160C22×C42C2313yes—
161#400C23148yes—
162#397C23148yes—
163#394C23148yes—
164#393C23148yes—
165#391C23124yes—
166#391C23124yes—
167#391C23124yes—
168#391C23124yes—
169#391C23124yes—
170#390C23124yes—
171#390C23124yes—
172#390C23124yes—
173#390C23124yes—
174#390C23124yes—
175#388C23124yes—
176#388C23148yes—
177#388C23124yes—
178#388C23124yes—
179#388C23148yes—
180#388C23148yes—
181#388C2318yes—
182#382C23124yes—
183#380C23124yes—
184#380C23124yes—
185#380C23124yes—
186#380C23124yes—
187C4×C4⋊C4C23124yes—
188C4×C4⋊C4C23124yes—
189C4×C4⋊C4C23124yes—
190C4×C22⋊C4C23124yes—
191C4×C22⋊C4C23124yes—
192C4×C22⋊C4C23124yes—
193C4×C22⋊C4C23124yes—
194C2×#95C2313yes—
195C2×#95C2316yes—
196C2×#95C2316yes—
197C2×#95C2313yes—
198C2×#95C2316yes—
199C2×#95C2313yes—
200C2×#95C2313yes—
201C2×#95C2311yescharacteristic
202C2×#95C2311yescharacteristic
203C2×#95C2313yes—
204C2×#95C2313yes—
205C43C23124yes—
206C25C2413yes—
207C25C2411yescharacteristic
208C2×#41—232no—
209C2×#41—232no—
210C2×#41—232no—
211C2×#41—464no—
212C22×Q8—232no—
213C22×Q8—232no—
214C22×Q8—232no—
215C22×Q8—212no—
216C22×Q8—212no—
217C22×Q8—212no—
218C22×Q8—212no—
219C22×Q8—212no—
220C22×Q8—46no—
221C22×Q8—464no—
222C22×Q8C2×D418yes—
223C22×Q8#4118yes—
224C22×Q8#4118yes—
225C22×Q8C24112yes—
226C22×Q8C24112yes—
227C22×Q8C2414yes—
228C22×D4—212no—
229C22×D4—212no—
230C22×D4—212no—
231C22×D4—212no—
232C22×D4—212no—
233C22×D4—46no—
234C22×D4C24112yes—
235C22×D4C24112yes—
236C23×C4—212no—
237C23×C4—212no—
238C23×C4—212no—
239C23×C4—212no—
240C23×C4—212no—
241C23×C4—212no—
242C23×C4C24112yes—
243C23×C4C24112yes—
244C23×C4C2413yes—
245C23×C4C2413yes—
246C23×C4C2411yescharacteristic
247C23×C4C2413yes—
248C23×C4C2413yes—
249C23×C4C2413yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2413yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2413yes—
257C4⋊Q8—296no—
258#125—296no—
259#125—296no—
260#124—296no—
261#124—296no—
262C22⋊Q8—296no—
263#121—296no—
264C4×Q8—232no—
265C4×Q8—232no—
266C4×Q8—232no—
267C4×Q8—232no—
268C4×Q8—296no—
269C4×Q8C2×D4164yes—
270C4×D4—296no—
271C2×C4⋊C4—212no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—212no—
274C2×C4⋊C4—212no—
275C2×C4⋊C4—212no—
276C2×C4⋊C4—212no—
277C2×C4⋊C4—224no—
278C2×C4⋊C4—212no—
279C2×C4⋊C4—212no—
280C2×C4⋊C4C2×D4124yes—
281C2×C4⋊C4C24112yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C24112yes—
284C2×C4⋊C4C24112yes—
285C2×C4⋊C4C24112yes—
286C2×C4⋊C4C24112yes—
287C2×C4⋊C4C24112yes—
288C2×C4⋊C4C24112yes—
289C2×C4⋊C4C24124yes—
290C2×C4⋊C4C24112yes—
291C2×C22⋊C4—212no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—212no—
294C2×C22⋊C4—224no—
295C2×C22⋊C4—212no—
296C2×C22⋊C4—212no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—212no—
299C2×C22⋊C4—46no—
300C2×C22⋊C4—412no—
301C2×C22⋊C4—46no—
302C2×C22⋊C4—412no—
303C2×C22⋊C4#4118yescomplemented
304C2×C22⋊C4C2414yes—
305C2×C22⋊C4C24112yes—
306C2×C22⋊C4C24124yes—
307C2×C22⋊C4C24112yes—
308C2×C22⋊C4C24112yes—
309C2×C22⋊C4C24112yes—
310C2×C22⋊C4C2414yes—
311C2×C22⋊C4C24112yes—
312C2×C22⋊C4C24112yes—
313C2×C22⋊C4C24112yes—
314C2×C22⋊C4C24124yes—
315C2×C22⋊C4C24124yes—
316C2×C22⋊C4C2414yes—
317C2×C42—212no—
318C2×C42—212no—
319C2×C42—212no—
320C2×C42—212no—
321C2×C42—212no—
322C2×C42—24no—
323C2×C42—46no—
324C2×C42—46no—
325C2×C42C2×D4124yes—
326C2×C42C24112yes—
327C2×C42C24112yes—
328C2×C42C24112yes—
329C2×C42C24112yes—
330C2×C42C24112yes—
331C2×C42C2414yes—
332C2×C42C24112yes—
333C2×C42C24112yes—
334C2×C42C24112yes—
335#95C2414yes—
336#95C2414yes—
337#95C24112yes—
338#95C24112yes—
339#95C24112yes—
340#95C24112yes—
341#95C24112yes—
342#95C24112yes—
343#95C24124yes—
344#95C24124yes—
345#95C24124yes—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—26no—
353C24—26no—
354C24—26no—
355C24—26no—
356C24—26no—
357C24—26no—
358C24—26no—
359C24C2×#4114yes—
360C24C2516yes—
361C24C2512yes—
362C24C2511yescharacteristic
363#41—4128no—
364C2×Q8—224no—
365C2×Q8—224no—
366C2×Q8—224no—
367C2×Q8—224no—
368C2×Q8—224no—
369C2×Q8—412no—
370C2×Q8—448no—
371C2×Q8—4128no—
372C2×Q8—448no—
373C2×Q8#124164yescomplemented
374C2×Q8C22×D4116yes—
375C2×Q8C2×#41116yes—
376C2×Q8C2×#41116yes—
377C2×D4—224no—
378C2×D4—224no—
379C2×D4—224no—
380C2×D4—224no—
381C2×D4—224no—
382C2×D4—412no—
383C2×D4—448no—
384C2×D4—448no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—26no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—26no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—26no—
397C22×C4—26no—
398C22×C4—26no—
399C22×C4—22no—
400C22×C4—26no—
401C22×C4—26no—
402C22×C4—26no—
403C22×C4—26no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—26no—
407C22×C4—26no—
408C22×C4—26no—
409C22×C4—22no—
410C22×C4—26no—
411C22×C4—26no—
412C22×C4—26no—
413C22×C4—26no—
414C22×C4—22no—
415C22×C4—26no—
416C22×C4—224no—
417C22×C4—224no—
418C22×C4—224no—
419C22×C4—224no—
420C22×C4—224no—
421C22×C4—224no—
422C22×C4—448no—
423C22×C4—448no—
424C22×C4C22×D414yes—
425C22×C4C22×D4112yes—
426C22×C4C22×D4112yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#41112yes—
430C22×C4C2×#41112yes—
431C22×C42−1+414yes—
432C22×C4C2516yes—
433C22×C4C2512yes—
434C22×C4C2516yes—
435C22×C4C2516yes—
436C22×C4C2512yes—
437C22×C4C2516yes—
438C22×C4C2516yes—
439C22×C4C2516yes—
440C22×C4C2512yes—
441C22×C4C2512yes—
442C22×C4C2512yes—
443C22×C4C2512yes—
444C22×C4C2516yes—
445C4⋊C4—224no—
446C4⋊C4—28no—
447C4⋊C4—224no—
448C4⋊C4—224no—
449C4⋊C4—224no—
450C4⋊C4—224no—
451C4⋊C4—248no—
452C4⋊C4—224no—
453C4⋊C4—224no—
454C4⋊C4C22×D4148yes—
455C22⋊C4—224no—
456C22⋊C4—28no—
457C22⋊C4—224no—
458C22⋊C4—248no—
459C22⋊C4—224no—
460C22⋊C4—224no—
461C22⋊C4—28no—
462C22⋊C4—224no—
463C22⋊C4—412no—
464C22⋊C4—412no—
465C22⋊C4—424no—
466C22⋊C4—424no—
467C22⋊C4C2×#41116yescomplemented
468C42—224no—
469C42—224no—
470C42—224no—
471C42—224no—
472C42—224no—
473C42—28no—
474C42—412no—
475C42—412no—
476C42C22×D4148yes—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—26no—
487C23—26no—
488C23—26no—
489C23—26no—
490C23—26no—
491C23—26no—
492C23—26no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—412no—
498C23—412no—
499C23—412no—
500C23—412no—
501C23—412no—
502C23—412no—
503C23—412no—
504C23C23×D412yes—
505C23C22×#4114yes—
506C23C22×#4112yes—
507C23C22×#4112yes—
508C23C22×#4112yes—
509C23C2×2−1+412yes—
510C23#58512yes—
511C23#58512yes—
512C23C2611yescharacteristic
513Q8—448no—
514Q8—448no—
515Q8C2×#124164yescomplemented
516D4—448no—
517D4—448no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—26no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—26no—
527C2×C4—22no—
528C2×C4—224no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—26no—
535C2×C4—26no—
536C2×C4—26no—
537C2×C4—22no—
538C2×C4—26no—
539C2×C4—26no—
540C2×C4—26no—
541C2×C4—26no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—26no—
545C2×C4—26no—
546C2×C4—22no—
547C2×C4—26no—
548C2×C4—26no—
549C2×C4—26no—
550C2×C4—26no—
551C2×C4—22no—
552C2×C4—26no—
553C2×C4—26no—
554C2×C4—44no—
555C2×C4—412no—
556C2×C4—412no—
557C2×C4—412no—
558C2×C4—44no—
559C2×C4—412no—
560C2×C4—412no—
561C2×C4—448no—
562C2×C4—448no—
563C2×C4C2×#124148yes—
564C2×C4C23×D414yes—
565C2×C4C23×D4112yes—
566C2×C4C23×D4112yes—
567C2×C4C22×#4114yes—
568C2×C4C22×#4114yes—
569C2×C4C22×#41112yes—
570C2×C4C22×#41112yes—
571C2×C4C2×2−1+414yes—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—416no—
579C22—42no—
580C22—46no—
581C22—46no—
582C22—46no—
583C22—46no—
584C22—46no—
585C22—46no—
586C22—46no—
587C22—824no—
588C22C22×#12414yes—
589C22D4×#4114yes—
590C22#348014yes—
591C22#348214yes—
592C22#349714yes—
593C22#351614yes—
594C22#352414yes—
595C22C24×D411yescharacteristic
596C22C23×#4111yescharacteristic
597C22C23×#4111yescharacteristic
598C22C23×#4111yescharacteristic
599C22C22×2−1+411yescharacteristic
600C22C2×#58511yescharacteristic
601C22C2×#58511yescharacteristic
602C4—24no—
603C4—24no—
604C4—212no—
605C4—24no—
606C4—24no—
607C4—42no—
608C4—46no—
609C4—46no—
610C4—42no—
611C4—46no—
612C4—46no—
613C4—46no—
614C4C22×#124124yes—
615C2—44no—
616C2—86no—
617C2#3646718yesdirect product factor
618C2C23×#12411yescharacteristic
619C2C2×D4×#4111yescharacteristic
620C2C2×#348011yescharacteristic
621C2C2×#348211yescharacteristic
622C2C2×#349711yescharacteristic
623C2C2×#351611yescharacteristic
624C2C2×#352411yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC22×#548C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#557C2semidirect
3. C2×#2861C2semidirect
4. #36467C2direct (N × H)
5. C2×#2592C2semidirect
6. C2×#2562C2semidirect
7. C2×#2490C2semidirect
8. C2×#2489C2semidirect
9. C2×#2492C2semidirect
10. C2×#2491C2semidirect
11. C2×#2341C2semidirect
12. C2×#2341C2semidirect
13. C2×Q8×C22⋊C4C2semidirect
14. C2×Q8×C22⋊C4C2semidirect
15. C2×C4×#124C2semidirect
16. C2×C42×Q8C2semidirect
20. C2×#557C22semidirect
37. #2861C22semidirect
38. #2592C22semidirect
39. #2562C22semidirect
40. #2492C22semidirect
41. #2491C22semidirect
42. #2490C22semidirect
43. #2489C22semidirect
48. #2341C22semidirect
49. #2341C22semidirect
50. Q8×C22⋊C4C22semidirect
51. Q8×C22⋊C4C22semidirect
52. C4×#124C22semidirect
75. C42×Q8C22semidirect
303. C2×C22⋊C4#41semidirect
373. C2×Q8#124semidirect
467. C22⋊C4C2×#41semidirect
515. Q8C2×#124semidirect
617. C2#36467direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5572
C22×#5482
C2×#28612
#364672
C2×#25922
C2×#25622
C2×#24902
C2×#24892
C2×#24922
C2×#24912
C2×#23412
C2×Q8×C22⋊C42
C2×C4×#1242
C2×C42×Q82