Finite Group Databasebeta

← C22×#3242 #10575696

Subgroups — C22×#3242 #10575696

Order 512.

6170 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C22×#544C211yesmaximal, characteristic
3C2×#3242C2148yesmaximal, direct product factor
4C22×#501C211yesmaximal, characteristic, complemented
5C22×#499C211yesmaximal, characteristic, complemented
6C22×#491C211yesmaximal, characteristic, complemented
7C22×#487C211yesmaximal, characteristic, complemented
8C22×#484C211yesmaximal, characteristic
9C22×#485C211yesmaximal, characteristic
10C22×#485C211yesmaximal, characteristic
11C22×#483C211yesmaximal, characteristic
12C22×#479C211yesmaximal, characteristic, complemented
13C22×#479C211yesmaximal, characteristic, complemented
14C22×#477C211yesmaximal, characteristic, complemented
15C22×#475C211yesmaximal, characteristic, complemented
16C22×#433C211yesmaximal, characteristic
17C2×#544—212no—
18C2×#544C22124yes—
19C2×#541—212no—
20C2×#541C22124yes—
21C22×C4⋊Q8—21no—
22C22×C4⋊Q8C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C22×C4⋊Q8C2211yescharacteristic
25C22×#126—21no—
26C22×#126—21no—
27C22×#125—21no—
28C22×#125—21no—
29C22×#125—21no—
30C22×#125C2211yescharacteristic
31C22×#125C2211yescharacteristic
32C22×#125C2211yescharacteristic
33C22×C22⋊Q8—21no—
34C22×C22⋊Q8—21no—
35C22×C22⋊Q8C2211yescharacteristic
36C22×C22⋊Q8C2211yescharacteristic
37C22×C4×Q8C2211yescharacteristic
38C22×C4×Q8C2211yescharacteristic
39C22×#117C2211yescharacteristic
40C22×#117C2211yescharacteristic
41C23×C4⋊C4C2211yescharacteristic
42#3242C221256yesdirect product factor
43C2×#501C22124yescomplemented
44C2×#499C22124yescomplemented
45C2×#491C22124yescomplemented
46C2×#487C22124yescomplemented
47C2×#484C22124yes—
48C2×#485C22124yes—
49C2×#485C22124yes—
50C2×#483C22124yes—
51C2×#479C22124yescomplemented
52C2×#479C22124yescomplemented
53C2×#477C22124yescomplemented
54C2×#475C22124yescomplemented
55C2×#433C22124yes—
56C22×#107C2211yescharacteristic
57C22×#107C2211yescharacteristic
58C22×#107C2211yescharacteristic
59C22×#107C2211yescharacteristic
60C22×#106C2211yescharacteristic
61C22×#106C2211yescharacteristic
62C22×#106C2211yescharacteristic
63C22×#106C2211yescharacteristic
64C22×C4⋊C8C2211yescharacteristic
65C22×C4⋊C8C2211yescharacteristic
66C22×#103C2211yescharacteristic
67C22×#103C2211yescharacteristic
68C22×#103C2211yescharacteristic
69C22×#103C2211yescharacteristic
70C22×#103C2211yescharacteristic
71C22×#103C2211yescharacteristic
72C22×#103C2211yescharacteristic
73C22×#103C2211yescharacteristic
74C22×C22⋊C8C2211yescharacteristic
75C22×C22⋊C8C2211yescharacteristic
76C22×#97C2211yescharacteristic
77C22×C4×C8C2211yescharacteristic
78C23×Q8—21no—
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88#544—296no—
89#544C23164yes—
90#541—296no—
91#541C23164yes—
92C2×C4⋊Q8—224no—
93C2×C4⋊Q8—26no—
94C2×C4⋊Q8—26no—
95C2×C4⋊Q8—26no—
96C2×C4⋊Q8C23112yes—
97C2×C4⋊Q8C23112yes—
98C2×C4⋊Q8C23112yes—
99C2×#126—224no—
100C2×#126—224no—
101C2×#125—224no—
102C2×#125—224no—
103C2×#125—224no—
104C2×#125—26no—
105C2×#125—26no—
106C2×#125—26no—
107C2×#125C23112yes—
108C2×#125C23112yes—
109C2×#125C23112yes—
110C2×C22⋊Q8—26no—
111C2×C22⋊Q8—26no—
112C2×C22⋊Q8—224no—
113C2×C22⋊Q8—224no—
114C2×C22⋊Q8C23112yes—
115C2×C22⋊Q8C23112yes—
116C2×C4×Q8—26no—
117C2×C4×Q8—26no—
118C2×C4×Q8C23112yes—
119C2×C4×Q8C23112yes—
120C2×#117—26no—
121C2×#117D4112yes—
122C2×#117C23112yes—
123C2×#117C23112yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—26no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4D411yescharacteristic
141C22×C4⋊C4D411yescharacteristic
142C22×C4⋊C4C23112yes—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C22⋊C4—21no—
150C22×C22⋊C4—21no—
151C22×C22⋊C4—21no—
152C22×C22⋊C4D411yescharacteristic
153C22×C22⋊C4D411yescharacteristic
154C22×C42—21no—
155C22×C42—21no—
156C22×C42—21no—
157C22×C42C2311yescharacteristic
158C22×C42C2311yescharacteristic
159#501C23164yescomplemented
160#499C23164yescomplemented
161#491C23164yescomplemented
162#487C23164yescomplemented
163#485C23164yes—
164#485C23164yes—
165#484C23164yes—
166#483C23164yes—
167#479C23164yescomplemented
168#479C23164yescomplemented
169#477C23164yescomplemented
170#475C23164yescomplemented
171#433C23164yes—
172C2×#107C23112yes—
173C2×#107C23112yes—
174C2×#107C23112yes—
175C2×#107C23112yes—
176C2×#106C23112yes—
177C2×#106C23112yes—
178C2×#106C23112yes—
179C2×#106C23112yes—
180C2×C4⋊C8C23112yes—
181C2×C4⋊C8C23112yes—
182C2×#103C23112yes—
183C2×#103C23112yes—
184C2×#103C23112yes—
185C2×#103C23112yes—
186C2×#103C23112yes—
187C2×#103C23112yes—
188C2×#103C23112yes—
189C2×#103C23112yes—
190C2×C22⋊C8C23112yes—
191C2×C22⋊C8C23112yes—
192C2×#97C23112yes—
193C2×C4×C8C23112yes—
194C25C2×D411yescharacteristic
195C22×Q8—23no—
196C22×Q8—23no—
197C22×Q8—212no—
198C22×Q8—212no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—84no—
204C22×Q8—84no—
205C22×Q8C2416yes—
206C22×Q8C2416yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—212no—
218C23×C4—84no—
219C23×C4C2×D416yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2416yes—
226C23×C4C2411yescharacteristic
227C22×C8—24no—
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8C2416yes—
232C22×C8C2416yes—
233C22×C8C2416yes—
234C22×C8C2416yes—
235C4⋊Q8—264no—
236C4⋊Q8—224no—
237C4⋊Q8—224no—
238C4⋊Q8—224no—
239C4⋊Q8C24116yes—
240C4⋊Q8C24116yes—
241C4⋊Q8C24116yes—
242#126—264no—
243#126—264no—
244#125—264no—
245#125—264no—
246#125—264no—
247#125—224no—
248#125—224no—
249#125—224no—
250#125C24116yes—
251#125C24116yes—
252#125C24116yes—
253C22⋊Q8—224no—
254C22⋊Q8—224no—
255C22⋊Q8—264no—
256C22⋊Q8—264no—
257C22⋊Q8C24116yes—
258C22⋊Q8C24116yes—
259C4×Q8—224no—
260C4×Q8—224no—
261C4×Q8C24116yes—
262C4×Q8C24116yes—
263#117—224no—
264#117C2×D4148yes—
265#117C24116yes—
266#117C24116yes—
267C2×C4⋊C4—212no—
268C2×C4⋊C4—23no—
269C2×C4⋊C4—23no—
270C2×C4⋊C4—224no—
271C2×C4⋊C4—212no—
272C2×C4⋊C4—212no—
273C2×C4⋊C4—212no—
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—212no—
283C2×C4⋊C4—212no—
284C2×C4⋊C4—212no—
285C2×C4⋊C4—23no—
286C2×C4⋊C4—23no—
287C2×C4⋊C4—23no—
288C2×C4⋊C4—23no—
289C2×C4⋊C4C2×D4112yes—
290C2×C4⋊C4C2×D4112yes—
291C2×C4⋊C4C2416yes—
292C2×C4⋊C4C2416yes—
293C2×C4⋊C4C24116yes—
294C2×C4⋊C4C2416yes—
295C2×C4⋊C4C2416yes—
296C2×C4⋊C4C2416yes—
297C2×C4⋊C4C2416yes—
298C2×C22⋊C4—212no—
299C2×C22⋊C4—212no—
300C2×C22⋊C4—212no—
301C2×C22⋊C4C2×D4112yes—
302C2×C22⋊C4C2×D4112yes—
303C2×C42—212no—
304C2×C42—212no—
305C2×C42—212no—
306C2×C42C2×D416yes—
307C2×C42C2×D416yes—
308C2×C42C2416yes—
309C2×C42C2416yes—
310#107C24116yes—
311#107C24116yes—
312#107C24116yes—
313#107C24116yes—
314#106C24116yes—
315#106C24116yes—
316#106C24116yes—
317#106C24116yes—
318C4⋊C8C24116yes—
319C4⋊C8C24116yes—
320#103C24116yes—
321#103C24116yes—
322#103C24116yes—
323#103C24116yes—
324#103C24116yes—
325#103C24116yes—
326#103C24116yes—
327#103C24116yes—
328C22⋊C8C24116yes—
329C22⋊C8C24116yes—
330#97C24116yes—
331C4×C8C24116yes—
332C24—24no—
333C24—24no—
334C24—24no—
335C24C22×D416yes—
336C24C22×D411yescharacteristic
337C2×Q8—26no—
338C2×Q8—26no—
339C2×Q8—216no—
340C2×Q8—216no—
341C2×Q8—212no—
342C2×Q8—212no—
343C2×Q8—212no—
344C2×Q8—212no—
345C2×Q8—412no—
346C2×Q8—412no—
347C2×Q8—824no—
348C2×Q8—824no—
349C2×Q8C2514yes—
350C2×Q8C2514yes—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—224no—
357C22×C4—212no—
358C22×C4—212no—
359C22×C4—26no—
360C22×C4—26no—
361C22×C4—26no—
362C22×C4—26no—
363C22×C4—26no—
364C22×C4—26no—
365C22×C4—26no—
366C22×C4—26no—
367C22×C4—216no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—824no—
377C22×C4C22×D416yes—
378C22×C4C22×D416yes—
379C22×C4C22×D416yes—
380C22×C4C22×D416yes—
381C22×C4C22×D416yes—
382C22×C4C22×D4112yes—
383C22×C4C22×D413yes—
384C22×C42−1+414yes—
385C22×C42−1+414yes—
386C22×C4C2514yes—
387C22×C4C2513yes—
388C2×C8—212no—
389C2×C8—212no—
390C2×C8—212no—
391C2×C8—212no—
392C2×C8C2514yes—
393C2×C8C2514yes—
394C2×C8C2514yes—
395C2×C8C2514yes—
396C4⋊C4—216no—
397C4⋊C4—26no—
398C4⋊C4—26no—
399C4⋊C4—216no—
400C4⋊C4—216no—
401C4⋊C4—216no—
402C4⋊C4—216no—
403C4⋊C4—216no—
404C4⋊C4—216no—
405C4⋊C4—216no—
406C4⋊C4—216no—
407C4⋊C4—216no—
408C4⋊C4—216no—
409C4⋊C4—216no—
410C4⋊C4—216no—
411C4⋊C4—216no—
412C4⋊C4—216no—
413C4⋊C4—26no—
414C4⋊C4—26no—
415C4⋊C4—26no—
416C4⋊C4—26no—
417C4⋊C4C22×D4116yes—
418C4⋊C4C22×D4116yes—
419C4⋊C4C2514yes—
420C4⋊C4C2514yes—
421C4⋊C4C2514yes—
422C4⋊C4C2514yes—
423C4⋊C4C2514yes—
424C4⋊C4C2514yes—
425C22⋊C4—216no—
426C22⋊C4—216no—
427C22⋊C4—216no—
428C22⋊C4C22×D4116yes—
429C22⋊C4C22×D4116yes—
430C42—216no—
431C42—216no—
432C42—216no—
433C42C22×D4112yes—
434C42C22×D4112yes—
435C42C2514yes—
436C42C2514yes—
437C23—212no—
438C23—212no—
439C23—212no—
440C23—416no—
441C23#53614yes—
442C23#57714yes—
443C23#57814yes—
444C23C23×D414yes—
445C23C23×D413yes—
446Q8—24no—
447Q8—24no—
448Q8—24no—
449Q8—24no—
450Q8—412no—
451Q8—412no—
452Q8—816no—
453Q8—816no—
454C2×C4—212no—
455C2×C4—212no—
456C2×C4—212no—
457C2×C4—212no—
458C2×C4—212no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—224no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—412no—
471C2×C4—412no—
472C2×C4—412no—
473C2×C4—412no—
474C2×C4—412no—
475C2×C4—412no—
476C2×C4—412no—
477C2×C4—412no—
478C2×C4—816no—
479C2×C4#536112yes—
480C2×C4C23×D414yes—
481C2×C4C23×D414yes—
482C2×C4C23×D414yes—
483C2×C4C23×D414yes—
484C2×C4C23×D414yes—
485C2×C4C23×D413yes—
486C2×C4C2×2−1+416yes—
487C2×C4C2×2−1+416yes—
488C2×C4C2611yescharacteristic
489C8—24no—
490C8—24no—
491C8—24no—
492C8—24no—
493C22—24no—
494C22—24no—
495C22—24no—
496C22—424no—
497C22#3242116yesdirect product factor
498C22C2×#53616yes—
499C22C2×#57716yes—
500C22C2×#57816yes—
501C22C24×D411yescharacteristic
502C4—24no—
503C4—24no—
504C4—24no—
505C4—24no—
506C4—24no—
507C4—44no—
508C4—44no—
509C4—44no—
510C4—44no—
511C4—44no—
512C4—44no—
513C4—44no—
514C4—44no—
515C4C2×#53616yes—
516C4C22×2−1+411yescharacteristic
517C4C22×2−1+411yescharacteristic
518C2—44no—
519C2C2×#3242112yesdirect product factor
520C2C22×#53611yescharacteristic
521C2C22×#57711yescharacteristic
522C2C22×#57811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC2×C4C26
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#3242C2direct (N × H)
4. C22×#501C2semidirect
5. C22×#499C2semidirect
6. C22×#491C2semidirect
7. C22×#487C2semidirect
12. C22×#479C2semidirect
13. C22×#479C2semidirect
14. C22×#477C2semidirect
15. C22×#475C2semidirect
42. #3242C22direct (N × H)
43. C2×#501C22semidirect
44. C2×#499C22semidirect
45. C2×#491C22semidirect
46. C2×#487C22semidirect
51. C2×#479C22semidirect
52. C2×#479C22semidirect
53. C2×#477C22semidirect
54. C2×#475C22semidirect
159. #501C23semidirect
160. #499C23semidirect
161. #491C23semidirect
162. #487C23semidirect
167. #479C23semidirect
168. #479C23semidirect
169. #477C23semidirect
170. #475C23semidirect
497. C22#3242direct (N × H)
519. C2C2×#3242direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5412
C22×#5442
C2×#32422
C22×#5012
C22×#4992
C22×#4912
C22×#4872
C22×#4842
C22×#4852
C22×#4832
C22×#4792
C22×#4772
C22×#4752
C22×#4332