Finite Group Databasebeta

← C2×#36432 #7630112

Subgroups — C2×#36432 #7630112

Order 512.

9170 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#548C211yesmaximal, characteristic
2C22×C4×#41C211yesmaximal, characteristic, complemented
3C2×#2855C213yesmaximal, complemented
4#36432C2132yesmaximal, direct product factor
5C2×#2499C213yesmaximal, complemented
6C2×#2485C213yesmaximal, complemented
7C2×#2447C213yesmaximal, complemented
8C2×#2447C213yesmaximal, complemented
9C2×#2420C213yesmaximal, complemented
10C2×#2422C213yesmaximal, complemented
11C2×#2351C211yesmaximal, characteristic, complemented
12C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
13C2×#2341C211yesmaximal, characteristic, complemented
14C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
15C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
16C2×C4×#123C213yesmaximal, complemented
17C23×#41C2211yescharacteristic
18C24×Q8C2211yescharacteristic
19C2×#548—216no—
20C2×#548C22116yes—
21C22×#124C2213yes—
22C22×#123C2213yes—
23C22×C22⋊Q8C2213yes—
24C22×#121C2213yes—
25C2×C4×#41—216no—
26C2×C4×#41C22116yescomplemented
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×C4×D4C2213yes—
33C22×C4×D4C2213yes—
34C22×C4×D4C2213yes—
35C22×#117C2213yes—
36C23×C42C2213yes—
37#2855C22148yescomplemented
38#2499C22148yescomplemented
39#2485C22148yescomplemented
40#2447C22148yescomplemented
41#2447C22148yescomplemented
42#2422C22148yescomplemented
43#2420C22148yescomplemented
44C2×#397C2216yes—
45C2×#397C2216yes—
46C2×#397C2216yes—
47C2×#394C2216yes—
48#2351C22116yescomplemented
49Q8×C4⋊C4C22116yescomplemented
50#2341C22116yescomplemented
51Q8×C22⋊C4C22116yescomplemented
52Q8×C22⋊C4C22116yescomplemented
53C4×#123C22148yescomplemented
54C2×#391C2213yes—
55C2×#391C2213yes—
56C2×#391C2213yes—
57C2×#391C2213yes—
58C2×#390C2213yes—
59C2×#390C2213yes—
60C2×#389C2213yes—
61C2×#389C2213yes—
62C2×#389C2213yes—
63C2×#389C2213yes—
64C2×#388C2213yes—
65C2×#388C2211yescharacteristic
66C2×#387C2213yes—
67C2×#387C2216yes—
68C2×#387C2213yes—
69C2×#386C2213yes—
70C2×#385C2216yes—
71C2×#385C2213yes—
72C2×#384C2213yes—
73C2×#382C2213yes—
74C2×#381C2216yes—
75C2×#381C2213yes—
76C2×#380C2213yes—
77C2×#380C2213yes—
78C2×#380C2213yes—
79C2×C4×C4⋊C4C2213yes—
80C2×C4×C4⋊C4C2213yes—
81C2×C4×C22⋊C4C2213yes—
82C2×C4×C22⋊C4C2213yes—
83C2×C4×C22⋊C4C2213yes—
84C22×#41—28no—
85C22×#41—28no—
86C22×#41D4116yes—
87C22×#41C2318yes—
88C23×Q8—28no—
89C23×Q8—28no—
90C23×Q8—28no—
91C23×Q8C2318yes—
92C23×Q8C2313yes—
93C23×Q8C2311yescharacteristic
94C23×D4C2313yes—
95C23×D4C2313yes—
96C24×C4C2313yes—
97C24×C4C2313yes—
98C24×C4C2313yes—
99C24×C4C2313yes—
100#548—2128no—
101C2×#124—224no—
102C2×#124C23124yes—
103C2×#123—224no—
104C2×#123C23124yes—
105C2×C22⋊Q8—224no—
106C2×C22⋊Q8C23124yes—
107C2×#121—224no—
108C2×#121C23124yes—
109C4×#41—2128no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—28no—
113C2×C4×Q8—28no—
114C2×C4×Q8—28no—
115C2×C4×Q8C2318yes—
116C2×C4×Q8C2318yes—
117C2×C4×Q8C2318yes—
118C2×C4×Q8C2318yes—
119C2×C4×Q8C2318yes—
120C2×C4×D4—224no—
121C2×C4×D4—224no—
122C2×C4×D4—224no—
123C2×C4×D4C23124yes—
124C2×C4×D4C23124yes—
125C2×C4×D4C23124yes—
126C2×#117—224no—
127C2×#117C23124yes—
128C22×C4⋊C4C2313yes—
129C22×C4⋊C4C2313yes—
130C22×C4⋊C4C2313yes—
131C22×C4⋊C4C2313yes—
132C22×C4⋊C4C2313yes—
133C22×C4⋊C4C2313yes—
134C22×C4⋊C4C2313yes—
135C22×C4⋊C4C2313yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2316yes—
138C22×C4⋊C4C2313yes—
139C22×C22⋊C4C2313yes—
140C22×C22⋊C4C2316yes—
141C22×C22⋊C4C2313yes—
142C22×C22⋊C4C2316yes—
143C22×C22⋊C4C2313yes—
144C22×C22⋊C4C2311yescharacteristic
145C22×C22⋊C4C2311yescharacteristic
146C22×C22⋊C4C2313yes—
147C22×C22⋊C4C2316yes—
148C22×C22⋊C4C2313yes—
149C22×C22⋊C4C2311yescharacteristic
150C22×C42—224no—
151C22×C42C23124yes—
152C22×C42C2311yescharacteristic
153C22×C42C2313yes—
154C22×C42C2313yes—
155C22×C42C2313yes—
156C22×C42C2313yes—
157C22×C42C2313yes—
158C22×C42C2313yes—
159#397C23148yes—
160#397C23148yes—
161#397C23148yes—
162#394C23148yes—
163#391C23124yes—
164#391C23124yes—
165#391C23124yes—
166#391C23124yes—
167#390C23124yes—
168#390C23124yes—
169#389C23124yes—
170#389C23124yes—
171#389C23124yes—
172#389C23124yes—
173#388C23124yes—
174#388C2318yes—
175#387C23124yes—
176#387C23148yes—
177#387C23124yes—
178#386C23124yes—
179#385C23148yes—
180#385C23124yes—
181#384C23124yes—
182#382C23124yes—
183#381C23148yes—
184#381C23124yes—
185#380C23124yes—
186#380C23124yes—
187#380C23124yes—
188C4×C4⋊C4C23124yes—
189C4×C4⋊C4C23124yes—
190C4×C22⋊C4C23124yes—
191C4×C22⋊C4C23124yes—
192C4×C22⋊C4C23124yes—
193C2×#95C2316yes—
194C2×#95C2313yes—
195C2×#95C2313yes—
196C2×#95C2313yes—
197C2×#95C2313yes—
198C2×#95C2316yes—
199C2×#95C2316yes—
200C2×#95C2311yescharacteristic
201C2×#95C2313yes—
202C2×#95C2313yes—
203C2×#95C2313yes—
204C2×#95C2311yescharacteristic
205C2×#95C2313yes—
206C25C2413yes—
207C25C2411yescharacteristic
208C2×#41—2128no—
209C2×#41—232no—
210C2×#41—232no—
211C2×#41C2×D4164yes—
212C22×Q8—232no—
213C22×Q8—232no—
214C22×Q8—232no—
215C22×Q8—212no—
216C22×Q8—212no—
217C22×Q8—212no—
218C22×Q8—464no—
219C22×Q8C2×D418yes—
220C22×Q8#4118yes—
221C22×Q8#4118yes—
222C22×Q8C24112yes—
223C22×Q8C2414yes—
224C22×D4—212no—
225C22×D4—212no—
226C22×D4—212no—
227C22×D4—212no—
228C22×D4—46no—
229C22×D4C2×D4124yes—
230C22×D4C24112yes—
231C22×D4C24112yes—
232C23×C4—212no—
233C23×C4—212no—
234C23×C4—212no—
235C23×C4—212no—
236C23×C4—212no—
237C23×C4—212no—
238C23×C4—212no—
239C23×C4—212no—
240C23×C4—212no—
241C23×C4—46no—
242C23×C4—46no—
243C23×C4C2×D4124yes—
244C23×C4C24112yes—
245C23×C4C2413yes—
246C23×C4C2413yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2413yes—
250C23×C4C2413yes—
251C23×C4C2413yes—
252C23×C4C2413yes—
253C23×C4C24112yes—
254C23×C4C24112yes—
255C23×C4C24112yes—
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2413yes—
261#124—296no—
262#123—296no—
263C22⋊Q8—296no—
264#121—296no—
265C4×Q8—232no—
266C4×Q8—232no—
267C4×Q8—232no—
268C4×Q8—232no—
269C4×Q8—232no—
270C4×D4—296no—
271C4×D4—296no—
272C4×D4—296no—
273#117—296no—
274C2×C4⋊C4—212no—
275C2×C4⋊C4—212no—
276C2×C4⋊C4—212no—
277C2×C4⋊C4—212no—
278C2×C4⋊C4—212no—
279C2×C4⋊C4—212no—
280C2×C4⋊C4—212no—
281C2×C4⋊C4—212no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—46no—
284C2×C4⋊C4—412no—
285C2×C4⋊C4C24112yes—
286C2×C4⋊C4C24112yes—
287C2×C4⋊C4C24112yes—
288C2×C4⋊C4C24112yes—
289C2×C4⋊C4C24112yes—
290C2×C4⋊C4C24112yes—
291C2×C4⋊C4C24112yes—
292C2×C4⋊C4C24112yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C24112yes—
295C2×C4⋊C4C24124yes—
296C2×C22⋊C4—224no—
297C2×C22⋊C4—212no—
298C2×C22⋊C4—224no—
299C2×C22⋊C4—212no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—212no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—412no—
304C2×C22⋊C4—46no—
305C2×C22⋊C4—46no—
306C2×C22⋊C4#4118yescomplemented
307C2×C22⋊C4C24124yes—
308C2×C22⋊C4C2414yes—
309C2×C22⋊C4C24112yes—
310C2×C22⋊C4C24124yes—
311C2×C22⋊C4C24112yes—
312C2×C22⋊C4C24112yes—
313C2×C22⋊C4C2414yes—
314C2×C22⋊C4C24112yes—
315C2×C22⋊C4C24112yes—
316C2×C22⋊C4C24124yes—
317C2×C22⋊C4C2414yes—
318C2×C42—212no—
319C2×C42—212no—
320C2×C42—212no—
321C2×C42—212no—
322C2×C42—296no—
323C2×C42—24no—
324C2×C42—212no—
325C2×C42—212no—
326C2×C42C24112yes—
327C2×C42C24112yes—
328C2×C42C24112yes—
329C2×C42C24112yes—
330C2×C42C24112yes—
331C2×C42C24112yes—
332C2×C42C2414yes—
333#95C2414yes—
334#95C24112yes—
335#95C24112yes—
336#95C24112yes—
337#95C24112yes—
338#95C24112yes—
339#95C2414yes—
340#95C24112yes—
341#95C24112yes—
342#95C24112yes—
343#95C24124yes—
344#95C24124yes—
345#95C24124yes—
346C24—22no—
347C24—26no—
348C24—22no—
349C24—22no—
350C24—26no—
351C24—26no—
352C24—22no—
353C24—26no—
354C24—22no—
355C24—26no—
356C24—26no—
357C24—22no—
358C24C22×D4112yes—
359C24C2×#4114yes—
360C24C2516yes—
361C24C2512yes—
362C24C2511yescharacteristic
363#41—2256no—
364C2×Q8—224no—
365C2×Q8—224no—
366C2×Q8—224no—
367C2×Q8—4128no—
368C2×Q8—448no—
369C2×Q8#123164yescomplemented
370C2×Q8C22×D4116yes—
371C2×Q8C2×#41116yes—
372C2×Q8C2×#41116yes—
373C2×D4—224no—
374C2×D4—224no—
375C2×D4—296no—
376C2×D4—224no—
377C2×D4—224no—
378C2×D4—412no—
379C2×D4—448no—
380C2×D4C22×D4148yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—296no—
390C22×C4—224no—
391C22×C4—224no—
392C22×C4—22no—
393C22×C4—26no—
394C22×C4—26no—
395C22×C4—224no—
396C22×C4—224no—
397C22×C4—26no—
398C22×C4—26no—
399C22×C4—22no—
400C22×C4—26no—
401C22×C4—26no—
402C22×C4—26no—
403C22×C4—26no—
404C22×C4—26no—
405C22×C4—26no—
406C22×C4—26no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—26no—
410C22×C4—22no—
411C22×C4—26no—
412C22×C4—26no—
413C22×C4—26no—
414C22×C4—26no—
415C22×C4—22no—
416C22×C4—26no—
417C22×C4—26no—
418C22×C4—224no—
419C22×C4—224no—
420C22×C4—224no—
421C22×C4—224no—
422C22×C4—224no—
423C22×C4—412no—
424C22×C4—412no—
425C22×C4—448no—
426C22×C4—448no—
427C22×C4—448no—
428C22×C4C22×D414yes—
429C22×C4C22×D4148yes—
430C22×C4C22×D4112yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#41112yes—
434C22×C4C2×#41112yes—
435C22×C42−1+414yes—
436C22×C4C2512yes—
437C22×C4C2516yes—
438C22×C4C2516yes—
439C22×C4C2516yes—
440C22×C4C2516yes—
441C22×C4C2512yes—
442C22×C4C2516yes—
443C22×C4C2516yes—
444C22×C4C2512yes—
445C22×C4C2512yes—
446C22×C4C2512yes—
447C22×C4C2512yes—
448C22×C4C2516yes—
449C4⋊C4—224no—
450C4⋊C4—224no—
451C4⋊C4—224no—
452C4⋊C4—224no—
453C4⋊C4—224no—
454C4⋊C4—224no—
455C4⋊C4—224no—
456C4⋊C4—224no—
457C4⋊C4—28no—
458C4⋊C4—412no—
459C4⋊C4—424no—
460C22⋊C4—248no—
461C22⋊C4—224no—
462C22⋊C4—248no—
463C22⋊C4—224no—
464C22⋊C4—28no—
465C22⋊C4—224no—
466C22⋊C4—28no—
467C22⋊C4—424no—
468C22⋊C4—412no—
469C22⋊C4—412no—
470C22⋊C4C2×#41116yescomplemented
471C42—28no—
472C42—224no—
473C42—224no—
474C42—224no—
475C42—224no—
476C42—224no—
477C42—224no—
478C23—22no—
479C23—26no—
480C23—22no—
481C23—22no—
482C23—26no—
483C23—26no—
484C23—22no—
485C23—26no—
486C23—22no—
487C23—26no—
488C23—26no—
489C23—22no—
490C23—224no—
491C23—224no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—224no—
496C23—44no—
497C23—412no—
498C23—412no—
499C23—44no—
500C23—44no—
501C23—412no—
502C23—44no—
503C23—412no—
504C23C23×D4112yes—
505C23C23×D412yes—
506C23C22×#4114yes—
507C23C22×#4112yes—
508C23C22×#4112yes—
509C23C22×#4112yes—
510C23C2×2−1+412yes—
511C23#58512yes—
512C23#58512yes—
513C23C2611yescharacteristic
514Q8—448no—
515Q8C2×#123164yescomplemented
516D4—296no—
517D4—448no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—296no—
531C2×C4—22no—
532C2×C4—26no—
533C2×C4—26no—
534C2×C4—26no—
535C2×C4—26no—
536C2×C4—22no—
537C2×C4—26no—
538C2×C4—26no—
539C2×C4—26no—
540C2×C4—26no—
541C2×C4—26no—
542C2×C4—26no—
543C2×C4—26no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—26no—
547C2×C4—22no—
548C2×C4—26no—
549C2×C4—26no—
550C2×C4—26no—
551C2×C4—26no—
552C2×C4—22no—
553C2×C4—26no—
554C2×C4—26no—
555C2×C4—448no—
556C2×C4—448no—
557C2×C4—412no—
558C2×C4—412no—
559C2×C4—412no—
560C2×C4—412no—
561C2×C4—44no—
562C2×C4—412no—
563C2×C4—44no—
564C2×C4—412no—
565C2×C4—448no—
566C2×C4C2×#123148yes—
567C2×C4C23×D414yes—
568C2×C4C23×D4112yes—
569C2×C4C22×#4114yes—
570C2×C4C22×#4114yes—
571C2×C4C22×#41112yes—
572C2×C4C22×#41112yes—
573C2×C4C2×2−1+414yes—
574C22—212no—
575C22—212no—
576C22—212no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—42no—
581C22—46no—
582C22—42no—
583C22—46no—
584C22—448no—
585C22—416no—
586C22—42no—
587C22—42no—
588C22—46no—
589C22—46no—
590C22C22×#12314yes—
591C22D4×#4114yes—
592C22#348014yes—
593C22#348214yes—
594C22#348914yes—
595C22#351814yes—
596C22#352414yes—
597C22C24×D411yescharacteristic
598C22C23×#4111yescharacteristic
599C22C23×#4111yescharacteristic
600C22C23×#4111yescharacteristic
601C22C22×2−1+411yescharacteristic
602C22C2×#58511yescharacteristic
603C22C2×#58511yescharacteristic
604C4—24no—
605C4—24no—
606C4—24no—
607C4—24no—
608C4—42no—
609C4—46no—
610C4—42no—
611C4—46no—
612C4—46no—
613C4—46no—
614C4—46no—
615C4—46no—
616C4C22×#123124yes—
617C2—412no—
618C2—44no—
619C2#3643218yesdirect product factor
620C2C23×#12311yescharacteristic
621C2C2×D4×#4111yescharacteristic
622C2C2×#348011yescharacteristic
623C2C2×#348211yescharacteristic
624C2C2×#351811yescharacteristic
625C2C2×#352411yescharacteristic
626C2C2×#348911yescharacteristic
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
2. C22×C4×#41C2semidirect
3. C2×#2855C2semidirect
4. #36432C2direct (N × H)
5. C2×#2499C2semidirect
6. C2×#2485C2semidirect
7. C2×#2447C2semidirect
8. C2×#2447C2semidirect
9. C2×#2420C2semidirect
10. C2×#2422C2semidirect
11. C2×#2351C2semidirect
12. C2×Q8×C4⋊C4C2semidirect
13. C2×#2341C2semidirect
14. C2×Q8×C22⋊C4C2semidirect
15. C2×Q8×C22⋊C4C2semidirect
16. C2×C4×#123C2semidirect
26. C2×C4×#41C22semidirect
37. #2855C22semidirect
38. #2499C22semidirect
39. #2485C22semidirect
40. #2447C22semidirect
41. #2447C22semidirect
42. #2422C22semidirect
43. #2420C22semidirect
48. #2351C22semidirect
49. Q8×C4⋊C4C22semidirect
50. #2341C22semidirect
51. Q8×C22⋊C4C22semidirect
52. Q8×C22⋊C4C22semidirect
53. C4×#123C22semidirect
306. C2×C22⋊C4#41semidirect
369. C2×Q8#123semidirect
470. C22⋊C4C2×#41semidirect
515. Q8C2×#123semidirect
619. C2#36432direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5482
C22×C4×#412
C2×#28552
#364322
C2×#24992
C2×#24852
C2×#24472
C2×#24202
C2×#24222
C2×#23512
C2×Q8×C4⋊C42
C2×#23412
C2×Q8×C22⋊C42
C2×C4×#1232