Finite Group Databasebeta

← C2×#35801 #7629630

Subgroups — C2×#35801 #7629630

Order 512.

8612 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#548C211yesmaximal, characteristic
2C22×#519C211yesmaximal, characteristic, complemented
3#35801C2132yesmaximal, direct product factor
4C2×#2500C213yesmaximal, complemented
5C2×#2485C213yesmaximal, complemented
6C2×#2448C213yesmaximal, complemented
7C2×#2448C213yesmaximal, complemented
8C2×#2421C213yesmaximal, complemented
9C2×#2423C213yesmaximal, complemented
10C2×#2624C213yesmaximal, complemented
11C2×#2395C213yesmaximal, complemented
12C2×#2351C211yesmaximal, characteristic, complemented
13C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
14C2×#2341C211yesmaximal, characteristic, complemented
15C2×#2341C211yesmaximal, characteristic, complemented
16C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
17C23×#41C2211yescharacteristic
18C24×Q8C2211yescharacteristic
19C2×#548—216no—
20C2×#548C22116yes—
21C22×#124C2213yes—
22C22×C22⋊Q8C2213yes—
23C22×#121C2213yes—
24C2×#519—216no—
25C2×#519C22116yescomplemented
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2213yes—
32C22×C4×Q8C2213yes—
33C22×C4×D4C2213yes—
34C22×#117C2213yes—
35C22×#117C2213yes—
36#2624C22148yescomplemented
37#2500C22148yescomplemented
38#2485C22148yescomplemented
39#2448C22148yescomplemented
40#2448C22148yescomplemented
41#2423C22148yescomplemented
42#2421C22148yescomplemented
43#2395C22148yescomplemented
44C2×#400C2213yes—
45C2×#399C2216yes—
46C2×#399C2213yes—
47C2×#398C2213yes—
48C2×#397C2216yes—
49C2×#397C2213yes—
50C2×#397C2213yes—
51C2×#397C2211yescharacteristic
52C2×#394C2213yes—
53C2×#394C2216yes—
54C2×#394C2213yes—
55#2351C22116yescomplemented
56Q8×C4⋊C4C22116yescomplemented
57#2341C22116yescomplemented
58#2341C22116yescomplemented
59Q8×C22⋊C4C22116yescomplemented
60C2×#391C2213yes—
61C2×#391C2213yes—
62C2×#390C2213yes—
63C2×#390C2213yes—
64C2×#390C2213yes—
65C2×#390C2213yes—
66C2×#389C2213yes—
67C2×#389C2213yes—
68C2×#389C2213yes—
69C2×#389C2213yes—
70C2×#388C2216yes—
71C2×#388C2216yes—
72C2×#388C2216yes—
73C2×#387C2216yes—
74C2×#384C2213yes—
75C2×#382C2213yes—
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×#41—28no—
87C22×#41C2318yes—
88C23×Q8—28no—
89C23×Q8—28no—
90C23×Q8D4116yes—
91C23×Q8C2318yes—
92C23×Q8C2313yes—
93C23×Q8C2311yescharacteristic
94C23×D4C2313yes—
95C23×D4C2313yes—
96C24×C4C2313yes—
97C24×C4C2313yes—
98#548—2128no—
99C2×#124—224no—
100C2×#124C23124yes—
101C2×C22⋊Q8—224no—
102C2×C22⋊Q8C23124yes—
103C2×#121—224no—
104C2×#121C23124yes—
105#519—2128no—
106C2×C4×Q8—224no—
107C2×C4×Q8—224no—
108C2×C4×Q8—28no—
109C2×C4×Q8—28no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—28no—
113C2×C4×Q8C2318yes—
114C2×C4×Q8C2318yes—
115C2×C4×Q8C2318yes—
116C2×C4×Q8C2318yes—
117C2×C4×Q8C2318yes—
118C2×C4×Q8C23124yes—
119C2×C4×Q8C23124yes—
120C2×C4×D4—224no—
121C2×C4×D4C23124yes—
122C2×#117—224no—
123C2×#117—224no—
124C2×#117C23124yes—
125C2×#117C23124yes—
126C22×C4⋊C4C2313yes—
127C22×C4⋊C4C2313yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2313yes—
130C22×C4⋊C4C2313yes—
131C22×C4⋊C4C2313yes—
132C22×C4⋊C4C2313yes—
133C22×C4⋊C4C2313yes—
134C22×C4⋊C4C2313yes—
135C22×C4⋊C4C2313yes—
136C22×C4⋊C4C2313yes—
137C22×C4⋊C4C2316yes—
138C22×C4⋊C4C2313yes—
139C22×C22⋊C4C2311yescharacteristic
140C22×C22⋊C4C2313yes—
141C22×C22⋊C4C2311yescharacteristic
142C22×C22⋊C4C2313yes—
143C22×C22⋊C4C2311yescharacteristic
144C22×C22⋊C4C2313yes—
145C22×C22⋊C4C2316yes—
146C22×C22⋊C4C2313yes—
147C22×C22⋊C4C2313yes—
148C22×C22⋊C4C2313yes—
149C22×C22⋊C4C2313yes—
150C22×C22⋊C4C2316yes—
151C22×C22⋊C4C2316yes—
152C22×C42C2313yes—
153C22×C42C2313yes—
154C22×C42C2316yes—
155C22×C42C2313yes—
156C22×C42C2313yes—
157C22×C42C2313yes—
158#400C23124yes—
159#399C23148yes—
160#399C23124yes—
161#398C23124yes—
162#397C23148yes—
163#397C2318yes—
164#397C23124yes—
165#397C23124yes—
166#394C23124yes—
167#394C23148yes—
168#394C23124yes—
169#391C23124yes—
170#391C23124yes—
171#390C23124yes—
172#390C23124yes—
173#390C23124yes—
174#390C23124yes—
175#389C23124yes—
176#389C23124yes—
177#389C23124yes—
178#389C23124yes—
179#388C23148yes—
180#388C23148yes—
181#388C23148yes—
182#387C23148yes—
183#384C23124yes—
184#382C23124yes—
185#380C23124yes—
186#380C23124yes—
187#380C23124yes—
188C4×C4⋊C4C23124yes—
189C4×C4⋊C4C23124yes—
190C4×C22⋊C4C23124yes—
191C4×C22⋊C4C23124yes—
192C4×C22⋊C4C23124yes—
193C2×#95C2313yes—
194C2×#95C2316yes—
195C2×#95C2313yes—
196C2×#95C2311yescharacteristic
197C2×#95C2313yes—
198C2×#95C2311yescharacteristic
199C2×#95C2313yes—
200C2×#95C2316yes—
201C2×#95C2311yescharacteristic
202C2×#95C2313yes—
203C2×#95C2313yes—
204C2×#95C2313yes—
205C25C2413yes—
206C25C2411yescharacteristic
207C2×#41—232no—
208C2×#41—232no—
209C2×#41—232no—
210C2×#41—464no—
211C22×Q8—2128no—
212C22×Q8—232no—
213C22×Q8—232no—
214C22×Q8—212no—
215C22×Q8—212no—
216C22×Q8C2×D4124yes—
217C22×Q8C2×D4164yes—
218C22×Q8C2×D418yes—
219C22×Q8#4118yes—
220C22×Q8#4118yes—
221C22×Q8C24112yes—
222C22×Q8C2414yes—
223C22×D4—212no—
224C22×D4—212no—
225C22×D4—212no—
226C22×D4—212no—
227C22×D4—46no—
228C22×D4—46no—
229C22×D4C24112yes—
230C22×D4C24112yes—
231C23×C4—212no—
232C23×C4—212no—
233C23×C4—212no—
234C23×C4—212no—
235C23×C4—212no—
236C23×C4C2×D4124yes—
237C23×C4C24112yes—
238C23×C4C2411yescharacteristic
239C23×C4C2413yes—
240C23×C4C2413yes—
241C23×C4C2413yes—
242C23×C4C2411yescharacteristic
243C23×C4C2413yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2413yes—
247C23×C4C2411yescharacteristic
248C23×C4C24112yes—
249C23×C4C2413yes—
250C23×C4C2411yescharacteristic
251C23×C4C2413yes—
252#124—296no—
253C22⋊Q8—296no—
254#121—296no—
255C4×Q8—296no—
256C4×Q8—296no—
257C4×Q8—232no—
258C4×Q8—232no—
259C4×Q8—232no—
260C4×Q8—232no—
261C4×Q8—232no—
262C4×D4—296no—
263#117—296no—
264#117—296no—
265C2×C4⋊C4—212no—
266C2×C4⋊C4—212no—
267C2×C4⋊C4—212no—
268C2×C4⋊C4—212no—
269C2×C4⋊C4—212no—
270C2×C4⋊C4—212no—
271C2×C4⋊C4—212no—
272C2×C4⋊C4—212no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—46no—
275C2×C4⋊C4—46no—
276C2×C4⋊C4—46no—
277C2×C4⋊C4—412no—
278C2×C4⋊C4C24112yes—
279C2×C4⋊C4C24112yes—
280C2×C4⋊C4C24112yes—
281C2×C4⋊C4C24112yes—
282C2×C4⋊C4C24112yes—
283C2×C4⋊C4C24112yes—
284C2×C4⋊C4C24112yes—
285C2×C4⋊C4C24112yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C24112yes—
288C2×C4⋊C4C24112yes—
289C2×C4⋊C4C24112yes—
290C2×C4⋊C4C24124yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—212no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—212no—
295C2×C22⋊C4—224no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—212no—
298C2×C22⋊C4—46no—
299C2×C22⋊C4—46no—
300C2×C22⋊C4—412no—
301C2×C22⋊C4—46no—
302C2×C22⋊C4—46no—
303C2×C22⋊C4—412no—
304C2×C22⋊C4C2414yes—
305C2×C22⋊C4C24112yes—
306C2×C22⋊C4C2414yes—
307C2×C22⋊C4C24112yes—
308C2×C22⋊C4C24124yes—
309C2×C22⋊C4C2414yes—
310C2×C22⋊C4C24112yes—
311C2×C22⋊C4C24112yes—
312C2×C22⋊C4C24112yes—
313C2×C22⋊C4C24112yes—
314C2×C22⋊C4C24112yes—
315C2×C22⋊C4C24124yes—
316C2×C22⋊C4C24124yes—
317C2×C42—224no—
318C2×C42—212no—
319C2×C42—212no—
320C2×C42—212no—
321C2×C42—212no—
322C2×C42—212no—
323C2×C42C24112yes—
324C2×C42C24112yes—
325C2×C42C24124yes—
326C2×C42C24112yes—
327C2×C42C24112yes—
328C2×C42C24112yes—
329#95C2414yes—
330#95C2414yes—
331#95C24112yes—
332#95C24112yes—
333#95C24112yes—
334#95C24112yes—
335#95C2414yes—
336#95C24112yes—
337#95C24112yes—
338#95C24112yes—
339#95C24124yes—
340#95C24124yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—26no—
348C24—26no—
349C24—26no—
350C24—26no—
351C24—26no—
352C24—26no—
353C24—26no—
354C24C22×D414yes—
355C24C2516yes—
356C24C2512yes—
357C24C2511yescharacteristic
358#41—4128no—
359C2×Q8—2256no—
360C2×Q8—296no—
361C2×Q8—224no—
362C2×Q8—224no—
363C2×Q8#123164yes—
364C2×Q8C22×D4148yes—
365C2×Q8C22×D4116yes—
366C2×Q8C2×#41116yes—
367C2×Q8C2×#41116yes—
368C2×D4—224no—
369C2×D4—224no—
370C2×D4—224no—
371C2×D4—224no—
372C2×D4—412no—
373C2×D4—412no—
374C2×D4—448no—
375C2×D4—448no—
376C22×C4—22no—
377C22×C4—26no—
378C22×C4—26no—
379C22×C4—22no—
380C22×C4—296no—
381C22×C4—224no—
382C22×C4—224no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—26no—
386C22×C4—26no—
387C22×C4—22no—
388C22×C4—26no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—26no—
392C22×C4—26no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—26no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—26no—
403C22×C4—26no—
404C22×C4—26no—
405C22×C4—26no—
406C22×C4—26no—
407C22×C4—26no—
408C22×C4—26no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—26no—
412C22×C4—26no—
413C22×C4—224no—
414C22×C4—224no—
415C22×C4—224no—
416C22×C4—448no—
417C22×C4C22×D4112yes—
418C22×C4C22×D4148yes—
419C22×C4C22×D4112yes—
420C22×C4C2×#41112yes—
421C22×C4C2×#41112yes—
422C22×C42−1+414yes—
423C22×C4C2512yes—
424C22×C4C2512yes—
425C22×C4C2516yes—
426C22×C4C2516yes—
427C22×C4C2512yes—
428C22×C4C2516yes—
429C22×C4C2512yes—
430C22×C4C2516yes—
431C22×C4C2512yes—
432C22×C4C2516yes—
433C22×C4C2516yes—
434C22×C4C2512yes—
435C22×C4C2516yes—
436C4⋊C4—224no—
437C4⋊C4—224no—
438C4⋊C4—224no—
439C4⋊C4—224no—
440C4⋊C4—224no—
441C4⋊C4—224no—
442C4⋊C4—224no—
443C4⋊C4—224no—
444C4⋊C4—28no—
445C4⋊C4—412no—
446C4⋊C4—412no—
447C4⋊C4—412no—
448C4⋊C4—424no—
449C22⋊C4—28no—
450C22⋊C4—224no—
451C22⋊C4—28no—
452C22⋊C4—224no—
453C22⋊C4—248no—
454C22⋊C4—28no—
455C22⋊C4—224no—
456C22⋊C4—412no—
457C22⋊C4—412no—
458C22⋊C4—424no—
459C22⋊C4—412no—
460C22⋊C4—412no—
461C22⋊C4—424no—
462C42—248no—
463C42—224no—
464C42—224no—
465C42—224no—
466C42—224no—
467C42—224no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—26no—
478C23—26no—
479C23—26no—
480C23—26no—
481C23—26no—
482C23—26no—
483C23—26no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—412no—
489C23—412no—
490C23—412no—
491C23—412no—
492C23—412no—
493C23—412no—
494C23—412no—
495C23C23×D414yes—
496C23C23×D412yes—
497C23C22×#4112yes—
498C23C22×#4112yes—
499C23C2×2−1+412yes—
500C23#58512yes—
501C23#58512yes—
502C23#58512yes—
503C23C2611yescharacteristic
504Q8—296no—
505Q8C2×#123164yes—
506D4—448no—
507D4—448no—
508C2×C4—22no—
509C2×C4—26no—
510C2×C4—26no—
511C2×C4—22no—
512C2×C4—224no—
513C2×C4—28no—
514C2×C4—296no—
515C2×C4—22no—
516C2×C4—26no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—26no—
520C2×C4—26no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—26no—
524C2×C4—26no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—26no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—26no—
535C2×C4—26no—
536C2×C4—26no—
537C2×C4—26no—
538C2×C4—26no—
539C2×C4—26no—
540C2×C4—26no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—26no—
544C2×C4—26no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—412no—
548C2×C4—412no—
549C2×C4—44no—
550C2×C4—412no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—412no—
554C2×C4—412no—
555C2×C4—448no—
556C2×C4C2×#123148yes—
557C2×C4C23×D4112yes—
558C2×C4C23×D4112yes—
559C2×C4C22×#41112yes—
560C2×C4C22×#41112yes—
561C2×C4C2×2−1+414yes—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—416no—
570C22—46no—
571C22—46no—
572C22—46no—
573C22—46no—
574C22—46no—
575C22—46no—
576C22—46no—
577C22—824no—
578C22C22×#12314yes—
579C22#348714yes—
580C22#349714yes—
581C22#351614yes—
582C22#351814yes—
583C22#352414yes—
584C22#352414yes—
585C22C24×D411yescharacteristic
586C22C23×#4111yescharacteristic
587C22C23×#4111yescharacteristic
588C22C22×2−1+411yescharacteristic
589C22C2×#58511yescharacteristic
590C22C2×#58511yescharacteristic
591C22C2×#58511yescharacteristic
592C4—24no—
593C4—212no—
594C4—42no—
595C4—46no—
596C4—42no—
597C4—42no—
598C4—46no—
599C4—46no—
600C4—42no—
601C4—46no—
602C4—46no—
603C4—42no—
604C4C22×#123124yes—
605C2—44no—
606C2—86no—
607C2#3580118yesdirect product factor
608C2C23×#12311yescharacteristic
609C2C2×#349711yescharacteristic
610C2C2×#351611yescharacteristic
611C2C2×#348711yescharacteristic
612C2C2×#351811yescharacteristic
613C2C2×#352411yescharacteristic
614C2C2×#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
2. C22×#519C2semidirect
3. #35801C2direct (N × H)
4. C2×#2500C2semidirect
5. C2×#2485C2semidirect
6. C2×#2448C2semidirect
7. C2×#2448C2semidirect
8. C2×#2421C2semidirect
9. C2×#2423C2semidirect
10. C2×#2624C2semidirect
11. C2×#2395C2semidirect
12. C2×#2351C2semidirect
13. C2×Q8×C4⋊C4C2semidirect
14. C2×#2341C2semidirect
15. C2×#2341C2semidirect
16. C2×Q8×C22⋊C4C2semidirect
25. C2×#519C22semidirect
36. #2624C22semidirect
37. #2500C22semidirect
38. #2485C22semidirect
39. #2448C22semidirect
40. #2448C22semidirect
41. #2423C22semidirect
42. #2421C22semidirect
43. #2395C22semidirect
55. #2351C22semidirect
56. Q8×C4⋊C4C22semidirect
57. #2341C22semidirect
58. #2341C22semidirect
59. Q8×C22⋊C4C22semidirect
607. C2#35801direct (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×#5192
#358012
C2×#25002
C2×#24852
C2×#24482
C2×#24212
C2×#24232
C2×#26242
C2×#23952
C2×#23512
C2×Q8×C4⋊C42
C2×#23412
C2×Q8×C22⋊C42