Finite Group Databasebeta

← C22×#3524 #10585970

Subgroups — C22×#3524 #10585970

Order 512.

35014 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3524C2196yesmaximal, direct product factor
2C22×#559C213yesmaximal, complemented
3C22×#557C211yesmaximal, characteristic, complemented
4C22×#555C213yesmaximal, complemented
5C22×#553C213yesmaximal, complemented
6C22×#552C213yesmaximal, complemented
7C22×#550C211yesmaximal, characteristic, complemented
8C22×#548C211yesmaximal, characteristic, complemented
9C22×#548C211yesmaximal, characteristic, complemented
10C22×#542C213yesmaximal, complemented
11C22×#540C213yesmaximal, complemented
12C22×#529C213yesmaximal, complemented
13C22×#529C213yesmaximal, complemented
14C22×#520C211yesmaximal, characteristic, complemented
15C22×#519C211yesmaximal, characteristic, complemented
16C22×C4×#41C211yesmaximal, characteristic, complemented
17C23×#41C2211yescharacteristic
18C23×#41C2211yescharacteristic
19C24×Q8C2211yescharacteristic
20#3524C2211024yesdirect product factor
21C2×#559C221144yescomplemented
22C2×#557C22148yescomplemented
23C2×#555C221144yescomplemented
24C2×#553C221144yescomplemented
25C2×#552C221144yescomplemented
26C2×#550C22148yescomplemented
27C2×#548C22148yescomplemented
28C2×#548C22148yescomplemented
29C2×#542C221144yescomplemented
30C2×#540C221144yescomplemented
31C22×C4⋊Q8C2213yes—
32C22×#127C2213yes—
33C2×#529C221144yescomplemented
34C2×#529C221144yescomplemented
35C22×#126C2211yescharacteristic
36C22×#126C2213yes—
37C22×#126C2216yes—
38C22×#125C2213yes—
39C22×#125C2213yes—
40C22×#125C2213yes—
41C22×#124C2213yes—
42C22×#124C2213yes—
43C22×#124C2213yes—
44C22×#124C2216yes—
45C22×#123C2216yes—
46C22×#123C2213yes—
47C22×#123C2213yes—
48C22×#123C2216yes—
49C22×C22⋊Q8C2213yes—
50C22×C22⋊Q8C2213yes—
51C22×C22⋊Q8C2213yes—
52C22×#121C2213yes—
53C22×#121C2213yes—
54C22×#121C2213yes—
55C22×#121C2213yes—
56C22×#121C2216yes—
57C22×#121C2213yes—
58C2×#520C22148yescomplemented
59C2×#519C22148yescomplemented
60C2×C4×#41C22148yescomplemented
61C22×C4×Q8C2211yescharacteristic
62C22×C4×Q8C2211yescharacteristic
63C22×C4×Q8C2213yes—
64C22×C4×Q8C2211yescharacteristic
65C22×C4×Q8C2213yes—
66C22×C4×Q8C2211yescharacteristic
67C22×C4×Q8C2213yes—
68C22×C4×D4C2213yes—
69C22×C4×D4C2213yes—
70C22×C4×D4C2213yes—
71C22×C4×D4C2213yes—
72C22×C4×D4C2213yes—
73C22×C4×D4C2213yes—
74C22×C4×D4C2213yes—
75C22×C4×D4C2216yes—
76C22×#117C2213yes—
77C22×#117C2213yes—
78C22×#117C2213yes—
79C22×#117C2213yes—
80C22×#117C2216yes—
81C23×C4⋊C4C2213yes—
82C23×C42C2213yes—
83C22×#41—216no—
84C22×#41—216no—
85C22×#41C23124yes—
86C22×#41C23124yes—
87C23×Q8—216no—
88C23×Q8C2313yes—
89C23×Q8C23124yes—
90C23×Q8C2313yes—
91C23×Q8C2311yescharacteristic
92C23×D4C2313yes—
93C23×D4C2313yes—
94C23×D4C2313yes—
95C23×D4C2313yes—
96C23×D4C2316yes—
97C23×D4C2313yes—
98C24×C4C2313yes—
99C24×C4C2313yes—
100C24×C4C2313yes—
101C24×C4C2313yes—
102C24×C4C2313yes—
103C24×C4C2316yes—
104#559C231768yescomplemented
105#557C231256yescomplemented
106#555C231768yescomplemented
107#553C231768yescomplemented
108#552C231768yescomplemented
109#550C231256yescomplemented
110#548C231256yescomplemented
111#548C231256yescomplemented
112#542C231768yescomplemented
113#540C231768yescomplemented
114C2×C4⋊Q8C23172yes—
115C2×#127C23172yes—
116#529C231768yescomplemented
117#529C231768yescomplemented
118C2×#126C23124yes—
119C2×#126C23172yes—
120C2×#126C231144yes—
121C2×#125C23172yes—
122C2×#125C23172yes—
123C2×#125C23172yes—
124C2×#124C23172yes—
125C2×#124C23172yes—
126C2×#124C23172yes—
127C2×#124C231144yes—
128C2×#123C231144yes—
129C2×#123C23172yes—
130C2×#123C23172yes—
131C2×#123C231144yes—
132C2×C22⋊Q8C23172yes—
133C2×C22⋊Q8C23172yes—
134C2×C22⋊Q8C23172yes—
135C2×#121C23172yes—
136C2×#121C23172yes—
137C2×#121C23172yes—
138C2×#121C23172yes—
139C2×#121C231144yes—
140C2×#121C23172yes—
141#520C231256yescomplemented
142#519C231256yescomplemented
143C4×#41C231256yescomplemented
144C2×C4×Q8C23124yes—
145C2×C4×Q8C23124yes—
146C2×C4×Q8C23172yes—
147C2×C4×Q8C23124yes—
148C2×C4×Q8C23172yes—
149C2×C4×Q8C23124yes—
150C2×C4×Q8C23172yes—
151C2×C4×D4C23172yes—
152C2×C4×D4C23172yes—
153C2×C4×D4C23172yes—
154C2×C4×D4C23172yes—
155C2×C4×D4C23172yes—
156C2×C4×D4C23172yes—
157C2×C4×D4C23172yes—
158C2×C4×D4C231144yes—
159C2×#117C23172yes—
160C2×#117C23172yes—
161C2×#117C23172yes—
162C2×#117C23172yes—
163C2×#117C231144yes—
164C22×C4⋊C4C2313yes—
165C22×C4⋊C4C2313yes—
166C22×C4⋊C4C2313yes—
167C22×C4⋊C4C23172yes—
168C22×C4⋊C4C2311yescharacteristic
169C22×C4⋊C4C2313yes—
170C22×C4⋊C4C2313yes—
171C22×C4⋊C4C2313yes—
172C22×C4⋊C4C2313yes—
173C22×C4⋊C4C2313yes—
174C22×C4⋊C4C2316yes—
175C22×C4⋊C4C2311yescharacteristic
176C22×C4⋊C4C2313yes—
177C22×C4⋊C4C2313yes—
178C22×C4⋊C4C2311yescharacteristic
179C22×C4⋊C4C2313yes—
180C22×C22⋊C4C2311yescharacteristic
181C22×C22⋊C4C2313yes—
182C22×C22⋊C4C2313yes—
183C22×C22⋊C4C2313yes—
184C22×C22⋊C4C2316yes—
185C22×C22⋊C4C2313yes—
186C22×C22⋊C4C2316yes—
187C22×C22⋊C4C2311yescharacteristic
188C22×C22⋊C4C2313yes—
189C22×C22⋊C4C2311yescharacteristic
190C22×C22⋊C4C2313yes—
191C22×C22⋊C4C2316yes—
192C22×C22⋊C4C2313yes—
193C22×C42C2311yescharacteristic
194C22×C42C2313yes—
195C22×C42C2313yes—
196C22×C42C2313yes—
197C22×C42C23172yes—
198C22×C42C2316yes—
199C22×C42C2313yes—
200C22×C42C2313yes—
201C25C2411yescharacteristic
202C25C2413yes—
203C25C2413yes—
204C2×#41—2192no—
205C2×#41—2192no—
206C2×#41C24164yes—
207C2×#41C24164yes—
208C22×Q8—2192no—
209C22×Q8—224no—
210C22×Q8—412no—
211C22×Q8#41116yes—
212C22×Q8C24136yes—
213C22×Q8C24164yes—
214C22×Q8C24136yes—
215C22×Q8C24112yes—
216C22×D4—224no—
217C22×D4—224no—
218C22×D4—412no—
219C22×D4—412no—
220C22×D4—412no—
221C22×D4—424no—
222C22×D4C24136yes—
223C22×D4C24136yes—
224C22×D4C24136yes—
225C22×D4C24136yes—
226C22×D4C24172yes—
227C22×D4C24136yes—
228C23×C4—224no—
229C23×C4—224no—
230C23×C4—224no—
231C23×C4—412no—
232C23×C4—424no—
233C23×C4—412no—
234C23×C4C24136yes—
235C23×C4C24136yes—
236C23×C4C24136yes—
237C23×C4C2411yescharacteristic
238C23×C4C2413yes—
239C23×C4C2411yescharacteristic
240C23×C4C2413yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2413yes—
244C23×C4C2411yescharacteristic
245C23×C4C2413yes—
246C23×C4C2413yes—
247C23×C4C2411yescharacteristic
248C23×C4C24172yes—
249C23×C4C24136yes—
250C23×C4C24136yes—
251C23×C4C2413yes—
252C4⋊Q8C241192yes—
253#127C241192yes—
254#126C24164yes—
255#126C241192yes—
256#126C241384yes—
257#125C241192yes—
258#125C241192yes—
259#125C241192yes—
260#124C241192yes—
261#124C241192yes—
262#124C241192yes—
263#124C241384yes—
264#123C241384yes—
265#123C241192yes—
266#123C241192yes—
267#123C241384yes—
268C22⋊Q8C241192yes—
269C22⋊Q8C241192yes—
270C22⋊Q8C241192yes—
271#121C241192yes—
272#121C241192yes—
273#121C241192yes—
274#121C241192yes—
275#121C241384yes—
276#121C241192yes—
277C4×Q8C24164yes—
278C4×Q8C24164yes—
279C4×Q8C241192yes—
280C4×Q8C24164yes—
281C4×Q8C241192yes—
282C4×Q8C24164yes—
283C4×Q8C241192yes—
284C4×D4C241192yes—
285C4×D4C241192yes—
286C4×D4C241192yes—
287C4×D4C241192yes—
288C4×D4C241192yes—
289C4×D4C241192yes—
290C4×D4C241192yes—
291C4×D4C241384yes—
292#117C241192yes—
293#117C241192yes—
294#117C241192yes—
295#117C241192yes—
296#117C241384yes—
297C2×C4⋊C4C24136yes—
298C2×C4⋊C4C24136yes—
299C2×C4⋊C4C24136yes—
300C2×C4⋊C4C241192yes—
301C2×C4⋊C4C24112yes—
302C2×C4⋊C4C24136yes—
303C2×C4⋊C4C24136yes—
304C2×C4⋊C4C24136yes—
305C2×C4⋊C4C24136yes—
306C2×C4⋊C4C24136yes—
307C2×C4⋊C4C24172yes—
308C2×C4⋊C4C24112yes—
309C2×C4⋊C4C24136yes—
310C2×C4⋊C4C24136yes—
311C2×C4⋊C4C24112yes—
312C2×C4⋊C4C24136yes—
313C2×C22⋊C4C24112yes—
314C2×C22⋊C4C24136yes—
315C2×C22⋊C4C24136yes—
316C2×C22⋊C4C24136yes—
317C2×C22⋊C4C24172yes—
318C2×C22⋊C4C24136yes—
319C2×C22⋊C4C24172yes—
320C2×C22⋊C4C24112yes—
321C2×C22⋊C4C24136yes—
322C2×C22⋊C4C24112yes—
323C2×C22⋊C4C24136yes—
324C2×C22⋊C4C24172yes—
325C2×C22⋊C4C24136yes—
326C2×C42C24136yes—
327C2×C42C24136yes—
328C2×C42C24172yes—
329C2×C42C24136yes—
330C2×C42C24136yes—
331C2×C42C24136yes—
332C2×C42C241192yes—
333C2×C42C24112yes—
334C24—24no—
335C24—212no—
336C24—212no—
337C24—212no—
338C24—212no—
339C24—212no—
340C24—24no—
341C24—24no—
342C24—212no—
343C24C2516yes—
344C24C25118yes—
345C24C25118yes—
346C24C2511yescharacteristic
347#41—2256no—
348#41—2256no—
349C2×Q8—2256no—
350C2×Q8—2144no—
351C2×Q8—472no—
352C2×Q8C2×#41196yes—
353C2×Q8C25148yes—
354C2×Q8C25148yes—
355C2×Q8C25116yes—
356C2×D4—2144no—
357C2×D4—2144no—
358C2×D4—472no—
359C2×D4—472no—
360C2×D4—472no—
361C2×D4—4144no—
362C2×D4C25148yes—
363C2×D4C25148yes—
364C2×D4C25148yes—
365C2×D4C25148yes—
366C2×D4C25196yes—
367C2×D4C25148yes—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—212no—
371C22×C4—24no—
372C22×C4—212no—
373C22×C4—24no—
374C22×C4—212no—
375C22×C4—212no—
376C22×C4—24no—
377C22×C4—212no—
378C22×C4—24no—
379C22×C4—2144no—
380C22×C4—2144no—
381C22×C4—2144no—
382C22×C4—472no—
383C22×C4—4144no—
384C22×C4—472no—
385C22×C4C2×#41124yes—
386C22×C4C2516yes—
387C22×C4C25118yes—
388C22×C4C2516yes—
389C22×C4C2516yes—
390C22×C4C2516yes—
391C22×C4C25118yes—
392C22×C4C2516yes—
393C22×C4C25118yes—
394C22×C4C25118yes—
395C22×C4C25148yes—
396C22×C4C25118yes—
397C22×C4C2516yes—
398C22×C4C25148yes—
399C22×C4C25148yes—
400C22×C4C25148yes—
401C22×C4C25148yes—
402C22×C4C25196yes—
403C22×C4C25118yes—
404C4⋊C4C25148yes—
405C4⋊C4C25148yes—
406C4⋊C4C25148yes—
407C4⋊C4C25116yes—
408C4⋊C4C25148yes—
409C4⋊C4C25148yes—
410C4⋊C4C25148yes—
411C4⋊C4C25148yes—
412C4⋊C4C25148yes—
413C4⋊C4C25196yes—
414C4⋊C4C25116yes—
415C4⋊C4C25148yes—
416C4⋊C4C25148yes—
417C4⋊C4C25116yes—
418C4⋊C4C25148yes—
419C22⋊C4C25116yes—
420C22⋊C4C25148yes—
421C22⋊C4C25148yes—
422C22⋊C4C25148yes—
423C22⋊C4C25196yes—
424C22⋊C4C25148yes—
425C22⋊C4C25196yes—
426C22⋊C4C25116yes—
427C22⋊C4C25148yes—
428C22⋊C4C25116yes—
429C22⋊C4C25148yes—
430C22⋊C4C25196yes—
431C22⋊C4C25148yes—
432C42C25116yes—
433C42C25148yes—
434C42C25148yes—
435C42C25148yes—
436C42C25196yes—
437C42C25148yes—
438C42C25148yes—
439C23—212no—
440C23—236no—
441C23—236no—
442C23—236no—
443C23—236no—
444C23—236no—
445C23—212no—
446C23—212no—
447C23—236no—
448C23—416no—
449C23—448no—
450C23—448no—
451C23C22×#4114yes—
452C23#58514yes—
453C23#58514yes—
454C23C2614yes—
455C23C26112yes—
456C23C26112yes—
457C23C2613yes—
458Q8—296no—
459Q8—448no—
460Q8C22×#41164yes—
461D4—296no—
462D4—296no—
463D4—448no—
464D4—448no—
465D4—448no—
466D4—496no—
467C2×C4—212no—
468C2×C4—212no—
469C2×C4—236no—
470C2×C4—212no—
471C2×C4—236no—
472C2×C4—212no—
473C2×C4—236no—
474C2×C4—236no—
475C2×C4—212no—
476C2×C4—236no—
477C2×C4—212no—
478C2×C4—296no—
479C2×C4—296no—
480C2×C4—296no—
481C2×C4—448no—
482C2×C4—496no—
483C2×C4—448no—
484C2×C4C22×#41172yes—
485C2×C4C2614yes—
486C2×C4C2614yes—
487C2×C4C2614yes—
488C2×C4C26112yes—
489C2×C4C2614yes—
490C2×C4C26112yes—
491C2×C4C2614yes—
492C2×C4C26112yes—
493C2×C4C26112yes—
494C2×C4C26112yes—
495C2×C4C2614yes—
496C2×C4C26112yes—
497C22—24no—
498C22—212no—
499C22—212no—
500C22—212no—
501C22—212no—
502C22—212no—
503C22—24no—
504C22—24no—
505C22—212no—
506C22—424no—
507C22—472no—
508C22—472no—
509C22#3524116yesdirect product factor
510C22C23×#4116yes—
511C22C2×#58516yes—
512C22C2×#58516yes—
513C22C2711yescharacteristic
514C4—24no—
515C4—24no—
516C4—212no—
517C4—24no—
518C4—212no—
519C4—24no—
520C4—212no—
521C4—212no—
522C4—24no—
523C4—212no—
524C4—24no—
525C4C23×#41124yes—
526C2—44no—
527C2—412no—
528C2—412no—
529C2C2×#3524112yesdirect product factor
530C2C24×#4111yescharacteristic
531C2C22×#58511yescharacteristic
532C2C22×#58511yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC22C27
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3524C2direct (N × H)
2. C22×#559C2semidirect
3. C22×#557C2semidirect
4. C22×#555C2semidirect
5. C22×#553C2semidirect
6. C22×#552C2semidirect
7. C22×#550C2semidirect
8. C22×#548C2semidirect
9. C22×#548C2semidirect
10. C22×#542C2semidirect
11. C22×#540C2semidirect
12. C22×#529C2semidirect
13. C22×#529C2semidirect
14. C22×#520C2semidirect
15. C22×#519C2semidirect
16. C22×C4×#41C2semidirect
20. #3524C22direct (N × H)
21. C2×#559C22semidirect
22. C2×#557C22semidirect
23. C2×#555C22semidirect
24. C2×#553C22semidirect
25. C2×#552C22semidirect
26. C2×#550C22semidirect
27. C2×#548C22semidirect
28. C2×#548C22semidirect
29. C2×#542C22semidirect
30. C2×#540C22semidirect
33. C2×#529C22semidirect
34. C2×#529C22semidirect
58. C2×#520C22semidirect
59. C2×#519C22semidirect
60. C2×C4×#41C22semidirect
104. #559C23semidirect
105. #557C23semidirect
106. #555C23semidirect
107. #553C23semidirect
108. #552C23semidirect
109. #550C23semidirect
110. #548C23semidirect
111. #548C23semidirect
112. #542C23semidirect
113. #540C23semidirect
116. #529C23semidirect
117. #529C23semidirect
141. #520C23semidirect
142. #519C23semidirect
143. C4×#41C23semidirect
509. C22#3524direct (N × H)
529. C2C2×#3524direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35242
C22×#5592
C22×#5572
C22×#5552
C22×#5532
C22×#5522
C22×#5502
C22×#5482
C22×#5422
C22×#5402
C22×#5292
C22×#5202
C22×#5192
C22×C4×#412