Finite Group Databasebeta

← #14926812

Subgroups — #14926812

Order 2816.

2650 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
1C22×#66977C211yesmaximal, characteristic
2#13076964C214yesmaximal, complemented
3#13074732C214yesmaximal, complemented
4#13073997C214yesmaximal, complemented
5#13073996C214yesmaximal, complemented
6#13073095C212yesmaximal, complemented
7#13073090C212yesmaximal, complemented
8#13072261C212yesmaximal, complemented
9#13072256C212yesmaximal, complemented
10#13071228C214yesmaximal
11C2×C4×C22⋊Dic11C211yesmaximal, characteristic
12C22×C11×C22⋊Q8C211yesmaximal, characteristic
13C23×Q8×C11C2211yescharacteristic
14C24×C4×C11C2211yescharacteristic
15C23×Dic22C2211yescharacteristic
16C2×C11×C22⋊Q8—28no—
17C2×C11×C22⋊Q8C2212yes—
18C2×C11×C22⋊Q8C2212yes—
19C2×C11×C22⋊Q8C2212yes—
20C2×C11×C22⋊Q8C2212yes—
21C22×C11×C4⋊C4C2212yes—
22C22×C11×C4⋊C4C2211yescharacteristic
23C22×C11×C22⋊C4C2212yes—
24C22×C22⋊Dic11C2212yes—
25C22×C22⋊Dic11C2212yes—
26C2×#66997C2214yescomplemented
27C22×C4⋊Dic11C2211yescharacteristic
28C2×#66977—28no—
29C2×#66977C2212yes—
30C2×#66977C2212yes—
31C2×#66977C2212yes—
32C2×#66977C2212yes—
33C22×Dic11⋊C4C2212yes—
34C22×C4×Dic11C2212yes—
35#10622843C2214yescomplemented
36#10622842C2218yescomplemented
37#10622823C2218yes—
38#10622823C2218yes—
39C4×C22⋊Dic11C2212yes—
40C4×C22⋊Dic11C2212yes—
41C4×C22⋊Dic11C2212yes—
42C4×C22⋊Dic11C2212yes—
43C2×C11⋊#95C2212yes—
44#10622596C2218yescomplemented
45#10622593C2218yescomplemented
46#10622592C2218yescomplemented
47#10622589C2214yescomplemented
48#10622587C2214yescomplemented
49#10622585C22116yescomplemented
50#10622559C2218yes—
51#10622559C2218yes—
52#10622555C22116yes—
53C25×C11C2311yescharacteristic
54C22×Q8×C11—24no—
55C22×Q8×C11—44no—
56C22×Q8×C11C2312yes—
57C22×Q8×C11C2312yes—
58C23×C4×C11—44no—
59C23×C4×C11D418yes—
60C23×C4×C11C2311yescharacteristic
61C23×C4×C11C2311yescharacteristic
62C23×C4×C11C2311yescharacteristic
63C23×C4×C11C2312yes—
64C23×C4×C11C2312yes—
65C23×C4×C11C2311yescharacteristic
66C23×C4×C11C2311yescharacteristic
67C23×C4×C11C2311yescharacteristic
68C23×Dic11C2312yes—
69C23×Dic11C2312yes—
70C23×Dic11C2314yes—
71C22×Dic22—24no—
72C22×Dic22—44no—
73C22×Dic22C2312yes—
74C22×Dic22C2312yes—
75C11×C22⋊Q8—28no—
76C11×C22⋊Q8—28no—
77C11×C22⋊Q8—28no—
78C11×C22⋊Q8—28no—
79C2×C11×C4⋊C4—24no—
80C2×C11×C4⋊C4—28no—
81C2×C11×C4⋊C4C2314yes—
82C2×C11×C4⋊C4C2312yes—
83C2×C11×C4⋊C4C2314yes—
84C2×C11×C4⋊C4C2312yes—
85C2×C11×C22⋊C4—28no—
86C2×C11×C22⋊C4C2314yes—
87C2×C11×C22⋊C4C2314yes—
88C2×C22⋊Dic11—28no—
89C2×C22⋊Dic11Q8116yescomplemented
90C2×C22⋊Dic11C2314yes—
91C2×C22⋊Dic11C2314yes—
92C2×C22⋊Dic11C2314yes—
93C2×C22⋊Dic11C2314yes—
94#66997—48no—
95C2×C4⋊Dic11—24no—
96C2×C4⋊Dic11C2312yes—
97C2×C4⋊Dic11C2312yes—
98#66977—28no—
99#66977—28no—
100#66977—28no—
101#66977—28no—
102C2×Dic11⋊C4—28no—
103C2×Dic11⋊C4C2314yes—
104C2×Dic11⋊C4C2314yes—
105C2×Dic11⋊C4C23116yes—
106C2×Dic11⋊C4C23116yes—
107C2×C4×Dic11—44no—
108C2×C4×Dic11C2314yes—
109C2×C4×Dic11C2314yes—
110C11⋊#95C23116yes—
111C11⋊#95C23116yes—
112C11⋊#95C2314yes—
113C11⋊#95C2314yes—
114#35011—111nomaximal, Sylow
115C24×C11—42no—
116C24×C11—42no—
117C24×C11C2×D414yes—
118C24×C11C2×Q818yes—
119C24×C11C2411yescharacteristic
120C24×C11C2411yescharacteristic
121C24×C11C2411yescharacteristic
122C2×Q8×C11—24no—
123C2×Q8×C11—24no—
124C2×Q8×C11—28no—
125C2×Q8×C11—44no—
126C2×Q8×C11—416no—
127C22×C4×C11—24no—
128C22×C4×C11—24no—
129C22×C4×C11—24no—
130C22×C4×C11—24no—
131C22×C4×C11—24no—
132C22×C4×C11—416no—
133C22×C4×C11—42no—
134C22×C4×C11—42no—
135C22×C4×C11—42no—
136C22×C4×C11—44no—
137C22×C4×C11—44no—
138C22×C4×C11C2×D414yes—
139C22×C4×C11C2×D414yes—
140C22×C4×C11C2×D414yes—
141C22×C4×C11C2×D414yes—
142C22×C4×C11C2×D414yes—
143C22×C4×C11C2×D414yes—
144C22×C4×C11C2414yes—
145C22×C4×C11C2414yes—
146C22×C4×C11C2411yescharacteristic
147C22×C4×C11C2411yescharacteristic
148C22×C4×C11C2411yescharacteristic
149C22×C4×C11C2411yescharacteristic
150C22×Dic11—24no—
151C22×Dic11—24no—
152C22×Dic11—44no—
153C22×Dic11—44no—
154C22×Dic11—44no—
155C22×Dic11—44no—
156C22×Dic11C2×Q8116yescomplemented
157C22×Dic11C2414yes—
158C22×Dic11C2414yes—
159C22×Dic11C2418yes—
160C2×Dic22—24no—
161C2×Dic22—24no—
162C2×Dic22—28no—
163C2×Dic22—44no—
164C2×Dic22—416no—
165C11×C4⋊C4—24no—
166C11×C4⋊C4—24no—
167C11×C4⋊C4—28no—
168C11×C4⋊C4—28no—
169C11×C22⋊C4—28no—
170C11×C22⋊C4—28no—
171C22⋊Dic11—28no—
172C22⋊Dic11—28no—
173C22⋊Dic11C2×Q8116yescomplemented
174C22⋊Dic11C2×Q8116yescomplemented
175C4⋊Dic11—24no—
176C4⋊Dic11—24no—
177Dic11⋊C4—28no—
178Dic11⋊C4—28no—
179Dic11⋊C4—416no—
180Dic11⋊C4—416no—
181C4×Dic11—28no—
182C4×Dic11—44no—
183C22×C22⋊Q8—111no—
184C22×C22⋊Q8D1111yescharacteristic
185#2857—114no—
186#2659—114no—
187#2583—114no—
188#2583—114no—
189#2502—112no—
190#2502—112no—
191#2434—112no—
192#2434—112no—
193#2315—114no—
194C2×C4×C22⋊C4—111no—
195C23×C11—22no—
196C23×C11—22no—
197C23×C11—22no—
198C23×C11—22no—
199C23×C11—22no—
200C23×C11—22no—
201C23×C11—44no—
202C23×C11—44no—
203C23×C11—416no—
204C23×C11C4⋊Q8116yescomplemented
205C23×C11C22×D412yes—
206C23×C11C22×D412yes—
207C23×C11C22×D412yes—
208C23×C11C22×Q814yes—
209C23×C11C22×Q814yes—
210C23×C11C22×Q814yes—
211C23×C11C2511yescharacteristic
212Q8×C11—48no—
213Q8×C11—48no—
214C2×C4×C11—22no—
215C2×C4×C11—22no—
216C2×C4×C11—22no—
217C2×C4×C11—28no—
218C2×C4×C11—28no—
219C2×C4×C11—28no—
220C2×C4×C11—24no—
221C2×C4×C11—24no—
222C2×C4×C11—28no—
223C2×C4×C11—28no—
224C2×C4×C11—44no—
225C2×C4×C11—44no—
226C2×C4×C11—44no—
227C2×C4×C11—48no—
228C2×C4×C11—48no—
229C2×C4×C11C22×D412yes—
230C2×C4×C11C22×D412yes—
231C2×C4×C11C22×D412yes—
232C2×C4×C11C22×D412yes—
233C2×C4×C11H(5,𝔽2)14yes—
234C2×Dic11—28no—
235C2×Dic11—28no—
236C2×Dic11—24no—
237C2×Dic11—24no—
238C2×Dic11—28no—
239C2×Dic11—28no—
240C2×Dic11—416no—
241C2×Dic11—48no—
242C2×Dic11—48no—
243C2×Dic11C22×Q8116yes—
244Dic22—48no—
245Dic22—48no—
246C23×Q8—111no—
247C23×Q8C2×D1111yescharacteristic
248C24×C4C2×D1111yescharacteristic
249C2×C4⋊Q8—114no—
250C2×C22⋊Q8—28no—
251C2×C22⋊Q8—112no—
252C2×C22⋊Q8—112no—
253C2×C22⋊Q8—112no—
254C2×C22⋊Q8—112no—
255C2×C22⋊Q8—228no—
256C2×C22⋊Q8C2×D1112yes—
257C2×C22⋊Q8C2×D1112yes—
258C2×C22⋊Q8C2×D1112yes—
259C2×C22⋊Q8C2×D1112yes—
260C22×C4⋊C4—112no—
261C22×C4⋊C4—111no—
262C22×C4⋊C4C2×D1112yes—
263C22×C4⋊C4C2×D1111yescharacteristic
264C22×C22⋊C4—112no—
265C22×C22⋊C4—112no—
266C22×C22⋊C4C2×D1112yes—
267C22×C42—112no—
268#398—118no—
269#398—118no—
270#395—118no—
271#395—118no—
272#391—114no—
273#391—114no—
274#389—1116no—
275#385—118no—
276#385—118no—
277#385—118no—
278#385—118no—
279#384—114no—
280#380—1116no—
281C4×C22⋊C4—112no—
282C4×C22⋊C4—112no—
283C4×C22⋊C4—112no—
284C4×C22⋊C4—112no—
285C2×#95—112no—
286C22×C11—22no—
287C22×C11—22no—
288C22×C11—22no—
289C22×C11—22no—
290C22×C11—22no—
291C22×C11—22no—
292C22×C11—28no—
293C22×C11—28no—
294C22×C11—28no—
295C22×C11—416no—
296C22×C11C2×C4⋊Q814yes—
297C22×C11C2×C4⋊Q814yes—
298C22×C11C2×C4⋊Q814yes—
299C22×C11C23×D411yescharacteristic
300C22×C11C23×Q812yes—
301C22×C11C2×H(5,𝔽2)12yes—
302C22×C11#58512yes—
303C4×C11—22no—
304C4×C11—22no—
305C4×C11—22no—
306C4×C11—44no—
307C4×C11—44no—
308C4×C11#53414yes—
309Dic11—216no—
310Dic11—44no—
311Dic11—44no—
312C25C22×D1111yescharacteristic
313C22×Q8—24no—
314C22×Q8—44no—
315C22×Q8—112no—
316C22×Q8—112no—
317C22×Q8—224no—
318C22×Q8—444no—
319C22×Q8C22×D1112yes—
320C22×Q8C22×D1112yes—
321C23×C4—44no—
322C23×C4—114no—
323C23×C4—112no—
324C23×C4—112no—
325C23×C4C22⋊D1118yes—
326C23×C4C22×D1111yescharacteristic
327C23×C4C22×D1111yescharacteristic
328C23×C4C22×D1111yescharacteristic
329C23×C4C22×D1112yes—
330C23×C4C22×D1112yes—
331C23×C4C22×D1111yescharacteristic
332C23×C4C22×D1111yescharacteristic
333C23×C4C22×D1111yescharacteristic
334C4⋊Q8—448no—
335C22⋊Q8—28no—
336C22⋊Q8—28no—
337C22⋊Q8—28no—
338C22⋊Q8—28no—
339C22⋊Q8—228no—
340C22⋊Q8—228no—
341C22⋊Q8—228no—
342C22⋊Q8—228no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—28no—
345C2×C4⋊C4—112no—
346C2×C4⋊C4—114no—
347C2×C4⋊C4—114no—
348C2×C4⋊C4—112no—
349C2×C4⋊C4—1116no—
350C2×C4⋊C4—1116no—
351C2×C4⋊C4—228no—
352C2×C4⋊C4—224no—
353C2×C4⋊C4C22×D1112yes—
354C2×C4⋊C4C22×D1114yes—
355C2×C4⋊C4C22×D1114yes—
356C2×C4⋊C4C22×D1112yes—
357C2×C22⋊C4—28no—
358C2×C22⋊C4—1116no—
359C2×C22⋊C4—114no—
360C2×C22⋊C4—114no—
361C2×C22⋊C4—114no—
362C2×C22⋊C4—114no—
363C2×C22⋊C4—228no—
364C2×C22⋊C4C22×D1114yes—
365C2×C22⋊C4C22×D1114yes—
366C2×C42—114no—
367C2×C42—114no—
368C2×C42—444no—
369#95—114no—
370#95—114no—
371#95—1116no—
372#95—1116no—
373C2×C11—24no—
374C2×C11—24no—
375C2×C11—24no—
376C2×C11C22×C4⋊Q811yescharacteristic
377C2×C11C2×#53411yescharacteristic
378C2×C11#344911yescharacteristic
379C2×C11#350114yes—
380C24—42no—
381C24—42no—
382C24Q8×D1118yes—
383C24C2×C22⋊D1114yes—
384C24C23×D1111yescharacteristic
385C24C23×D1111yescharacteristic
386C24C23×D1111yescharacteristic
387C2×Q8—28no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—44no—
391C2×Q8—416no—
392C2×Q8—228no—
393C2×Q8—224no—
394C2×Q8—224no—
395C2×Q8—444no—
396C2×Q8—4416no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—416no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—118no—
409C22×C4—1116no—
410C22×C4—114no—
411C22×C4—114no—
412C22×C4—224no—
413C22×C4—224no—
414C22×C4—444no—
415C22×C4—444no—
416C22×C4—444no—
417C22×C4—444no—
418C22×C4C2×C22⋊D1114yes—
419C22×C4C2×C22⋊D1114yes—
420C22×C4C2×C22⋊D1114yes—
421C22×C4C2×C22⋊D1114yes—
422C22×C4C2×C22⋊D1114yes—
423C22×C4C2×C22⋊D1114yes—
424C22×C4C23×D1114yes—
425C22×C4C23×D1114yes—
426C22×C4C23×D1111yescharacteristic
427C22×C4C23×D1111yescharacteristic
428C22×C4C23×D1111yescharacteristic
429C22×C4C23×D1111yescharacteristic
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—228no—
435C4⋊C4—228no—
436C4⋊C4—224no—
437C4⋊C4—224no—
438C4⋊C4—4416no—
439C4⋊C4—4416no—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—1116no—
443C22⋊C4—1116no—
444C22⋊C4—228no—
445C22⋊C4—228no—
446C42—228no—
447C42—444no—
448C11#3501111yescharacteristic, complemented, Sylow
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—44no—
456C23—416no—
457C23—44no—
458C23#66997116yescomplemented
459C23C2×Q8×D1114yes—
460C23C2×Q8×D1114yes—
461C23C2×Q8×D1114yes—
462C23C22×C22⋊D1112yes—
463C23C22×C22⋊D1112yes—
464C23C22×C22⋊D1112yes—
465C23C24×D1111yescharacteristic
466Q8—48no—
467Q8—48no—
468Q8—448no—
469Q8—448no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—1116no—
486C2×C4—228no—
487C2×C4—228no—
488C2×C4—228no—
489C2×C4—228no—
490C2×C4—224no—
491C2×C4—224no—
492C2×C4—4416no—
493C2×C4—448no—
494C2×C4—448no—
495C2×C4#6703714yes—
496C2×C4C22×C22⋊D1112yes—
497C2×C4C22×C22⋊D1112yes—
498C2×C4C22×C22⋊D1112yes—
499C2×C4C22×C22⋊D1112yes—
500C22—28no—
501C22—28no—
502C22—28no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—416no—
510C22C2×#6699714yes—
511C22C2×#6699714yes—
512C22C2×#6699714yes—
513C22C2×#6703712yes—
514C22C22×Q8×D1112yes—
515C22#1062345312yes—
516C22C23×C22⋊D1111yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—44no—
521C4—44no—
522C4—2216no—
523C4—444no—
524C4—444no—
525C4#1062333514yes—
526C2—24no—
527C2—24no—
528C2—24no—
529C2C22×#6699711yescharacteristic
530C2C2×#1062333511yescharacteristic
531C2#1308194511yescharacteristic
532C2#1308241214yes—
subgroupG/H
center, FrattiniC23C24×D11
commutator, socle, cosocleC23×C11C25
FittingC22×C11×C22⋊Q8C2
prime radicalC25×C11C23
hypercenterC22×C22⋊Q8D11
hypocenterC11#35011
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #13076964C2semidirect
3. #13074732C2semidirect
4. #13073997C2semidirect
5. #13073996C2semidirect
6. #13073095C2semidirect
7. #13073090C2semidirect
8. #13072261C2semidirect
9. #13072256C2semidirect
26. C2×#66997C22semidirect
35. #10622843C22semidirect
36. #10622842C22semidirect
44. #10622596C22semidirect
45. #10622593C22semidirect
46. #10622592C22semidirect
47. #10622589C22semidirect
48. #10622587C22semidirect
49. #10622585C22semidirect
89. C2×C22⋊Dic11Q8semidirect
156. C22×Dic11C2×Q8semidirect
173. C22⋊Dic11C2×Q8semidirect
174. C22⋊Dic11C2×Q8semidirect
204. C23×C11C4⋊Q8semidirect
448. C11#35011semidirect
458. C23#66997semidirect

Maximal subgroups

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

subgroupindex
C22×#669772
#130769642
#130747322
#130739972
#130739962
#130730952
#130730902
#130722612
#130722562
#130712282
C2×C4×C22⋊Dic112
C22×C11×C22⋊Q82
#3501111