Finite Group Databasebeta

← #14763690

Subgroups — #14763690

Order 2816.

1118 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623259C211yesmaximal, characteristic
2C2×D11×C4⋊C8C211yesmaximal, characteristic
3#13071012C211yesmaximal, characteristic
4#13069568C211yesmaximal, characteristic, complemented
5#13069567C211yesmaximal, characteristic, complemented
6#13069540C211yesmaximal, characteristic, complemented
7#13069036C214yesmaximal
8#13068629C214yesmaximal, complemented
9C11×#1981C211yesmaximal, characteristic, complemented
10C2×C11×C4⋊Q8C2211yescharacteristic
11C2×#66999C2212yes—
12C2×Q8×Dic11C2212yes—
13C22×C8×D11C2212yes—
14#10623259—24no—
15#10623259C418yescomplemented
16C2×#66950—21no—
17C2×#66945—21no—
18C2×#66943—21no—
19C2×#66940—21no—
20C2×C42×D11C2211yescharacteristic
21C2×#66922C2211yescharacteristic
22C2×C11×C4⋊C8C2211yescharacteristic
23C2×C11×#103C2214yes—
24C4×C11×C4⋊C4C2211yescharacteristic
25C2×#66899C2214yes—
26C2×Dic11⋊C8C2212yes—
27#10622600C2212yes—
28#10622590C2212yes—
29C4×(C2×D11)⋊C4—21no—
30C4×C4⋊Dic11C2211yescharacteristic
31#10622545—21no—
32C2×C4⋊(C11⋊C8)C2211yescharacteristic
33D11×C4⋊C8—24no—
34#10622277—21no—
35#10622264—21no—
36#10622189C2214yes—
37#10622167C2214yes—
38C22×Q8×C11C2312yes—
39C22×C4×D11D411yescharacteristic
40C22×C4×D11D411yescharacteristic
41C22×C4×D11C2311yescharacteristic
42C22×Dic22—21no—
43C22×C8×C11C2312yes—
44C11×C4⋊Q8—22no—
45C11×C4⋊Q8C2×C414yes—
46C2×C11×C4⋊C4—21no—
47C2×C11×C4⋊C4—21no—
48C2×C11×C4⋊C4C2312yes—
49C2×C42×C11—21no—
50C2×C42×C11C2311yescharacteristic
51#66999—24no—
52#66999C2×C418yes—
53Q8×Dic11—24no—
54Q8×Dic11C2×C418yes—
55C2×(C2×D11)⋊C4—22no—
56C2×(C2×D11)⋊C4—22no—
57C2×C4⋊Dic11—21no—
58C2×C4⋊Dic11—22no—
59C2×C4⋊Dic11C2312yes—
60C2×C4⋊Dic11C2312yes—
61C2×Dic11⋊C4—22no—
62C2×C4×Dic11—22no—
63C2×C4×Dic11—21no—
64C2×C4×Dic11D411yescharacteristic
65C2×C4×Dic11D411yescharacteristic
66C2×C4×Dic11C2311yescharacteristic
67C22×C11⋊C8C2312yes—
68C2×C8×D11—24no—
69C2×C8×D11—44no—
70#66950—28no—
71#66945—28no—
72#66943—28no—
73#66940—28no—
74C42×D11C2×C414yes—
75C42×D11D414yes—
76#66922—22no—
77#66922C2×C414yes—
78C11×C4⋊C8—22no—
79C11×#103—44no—
80C11×#95—21no—
81#66899—44no—
82C11⋊#95—21no—
83C11⋊#95—22no—
84C11⋊#95—22no—
85C11⋊#95—22no—
86Dic11⋊C8—24no—
87C4⋊(C11⋊C8)—22no—
88#21889—111nomaximal, Sylow
89C23×D11C2×D411yescharacteristic
90C2×Q8×C11—22no—
91C2×Q8×C11—24no—
92C2×Q8×C11—24no—
93C2×Q8×C11C22×C414yes—
94C22×C4×C11—22no—
95C22×C4×C11—22no—
96C22×C4×C11C2×D411yescharacteristic
97C22×C4×C11C2×D411yescharacteristic
98C22×C4×C11C2411yescharacteristic
99C22×Dic11—22no—
100C22×Dic11—22no—
101C22×Dic11C2×D411yescharacteristic
102C22×Dic11C2×D411yescharacteristic
103C22×Dic11C2×D411yescharacteristic
104C2×C4×D11—24no—
105C2×C4×D11—22no—
106C2×C4×D11—22no—
107C2×C4×D11—22no—
108C2×C4×D11—22no—
109C2×C4×D11C22⋊C414yes—
110C2×C4×D11C22⋊C414yes—
111C2×C4×D11C22×C412yes—
112C2×C4×D11C2×D412yes—
113C2×Dic22—24no—
114C2×Dic22—42no—
115C2×Dic22—42no—
116C2×C8×C11—22no—
117C2×C8×C11—24no—
118C11×C4⋊C4—24no—
119C11×C4⋊C4—24no—
120C11×C4⋊C4—22no—
121C11×C4⋊C4—22no—
122C42×C11—42no—
123C42×C11C22×C412yes—
124C42×C11C2×D412yes—
125(C2×D11)⋊C4—28no—
126(C2×D11)⋊C4—44no—
127C4⋊Dic11—28no—
128C4⋊Dic11—24no—
129C4⋊Dic11—22no—
130C4⋊Dic11—22no—
131C4⋊Dic11—22no—
132C4⋊Dic11C22×C414yes—
133Dic11⋊C4—28no—
134C4×Dic11—28no—
135C4×Dic11—42no—
136C4×Dic11C22⋊C414yes—
137C4×Dic11C22⋊C414yes—
138C4×Dic11C22×C412yes—
139C4×Dic11C2×D412yes—
140C2×C11⋊C8—22no—
141C2×C11⋊C8—24no—
142C8×D11—48no—
143C2×#533—111no—
144C22×C4⋊C8—111no—
145#2283—111no—
146#1988—111no—
147#1988—111no—
148#1981—111no—
149#1981D1111yescharacteristic, complemented
150#1892—114no—
151#1805—114no—
152C23×C11C22×D411yescharacteristic
153C22×D11—22no—
154C22×D11—21no—
155C22×D11—21no—
156C22×D11C2×C22⋊C412yes—
157C22×D11#13714yes—
158Q8×C11—28no—
159Q8×C11—42no—
160Q8×C11—42no—
161C2×C4×C11—22no—
162C2×C4×C11—24no—
163C2×C4×C11—44no—
164C2×C4×C11—42no—
165C2×C4×C11—42no—
166C2×C4×C11—42no—
167C2×C4×C11C2×C22⋊C412yes—
168C2×C4×C11C2×C22⋊C412yes—
169C2×C4×C11#13512yes—
170C2×C4×C11#13512yes—
171C2×C4×C11C23×C411yescharacteristic
172C2×C4×C11C22×D411yescharacteristic
173C2×C4×C11H(5,𝔽2)12yes—
174C2×C4×C112−1+412yes—
175C2×Dic11—22no—
176C2×Dic11—21no—
177C2×Dic11—21no—
178C2×Dic11—22no—
179C2×Dic11—24no—
180C2×Dic11—42no—
181C2×Dic11—42no—
182C2×Dic11—42no—
183C2×Dic11—42no—
184C2×Dic11—42no—
185C2×Dic11C2×C22⋊C412yes—
186C2×Dic11C2×C22⋊C412yes—
187C2×Dic11C2×C22⋊C412yes—
188C4×D11—28no—
189C4×D11—24no—
190C4×D11—22no—
191C4×D11—22no—
192C4×D11—28no—
193Dic22—48no—
194C8×C11—42no—
195C11⋊C8—42no—
196C23×C8—112no—
197#533—118no—
198#533—224no—
199C2×C4⋊Q8—111no—
200C2×C4⋊Q8C2×D1111yescharacteristic
201C2×#125—221no—
202C2×C22⋊Q8—112no—
203C2×C22⋊Q8—221no—
204C2×C4×Q8—112no—
205C2×C4×Q8—221no—
206C2×#117—221no—
207C22×C42—111no—
208C2×C4⋊C8—112no—
209C2×C4⋊C8—111no—
210C2×C4⋊C8—224no—
211C2×C4⋊C8C2×D1111yescharacteristic
212C2×#103—114no—
213C2×#103C2×D1114yes—
214#383—221no—
215#382—112no—
216#381—221no—
217#380—112no—
218C4×C4⋊C4—111no—
219C4×C4⋊C4C2×D1111yescharacteristic
220C4×C22⋊C4—221no—
221#376—221no—
222#340—114no—
223#340—114no—
224C22×C11C22×C22⋊C411yescharacteristic
225C22×C11#53712yes—
226C22×C11C2×#13511yescharacteristic
227C22×C11C2×#13511yescharacteristic
228C22×C11C2×#13712yes—
229C2×D11—22no—
230C2×D11—22no—
231C2×D11—24no—
232C2×D11—24no—
233C2×D11#41914yes—
234C4×C11—22no—
235C4×C11—44no—
236C4×C11—82no—
237C4×C11#41614yes—
238C4×C11#52012yes—
239C4×C11#53511yescharacteristic
240C4×C11#53611yescharacteristic
241Dic11—22no—
242Dic11—22no—
243Dic11—22no—
244Dic11—44no—
245Dic11—82no—
246C22×Q8—221no—
247C22×Q8C22×D1112yes—
248C23×C4—111no—
249C23×C4—111no—
250C23×C4—111no—
251C22×C8—112no—
252C22×C8—224no—
253C22×C8—444no—
254C22×C8C22×D1112yes—
255C4⋊Q8—22no—
256C4⋊Q8—114no—
257C4⋊Q8—222no—
258C4⋊Q8C4×D1114yes—
259#125—228no—
260C22⋊Q8—118no—
261C22⋊Q8—228no—
262C22⋊Q8—224no—
263C4×Q8—118no—
264C4×Q8—228no—
265C4×Q8—224no—
266#117—228no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—112no—
270C2×C4⋊C4—112no—
271C2×C4⋊C4—222no—
272C2×C4⋊C4—222no—
273C2×C4⋊C4—221no—
274C2×C4⋊C4C22×D1112yes—
275C2×C22⋊C4—222no—
276C2×C22⋊C4—222no—
277C2×C42—21no—
278C2×C42—111no—
279C2×C42—111no—
280C2×C42—111no—
281C2×C42—114no—
282C2×C42—114no—
283C2×C42—222no—
284C2×C42—221no—
285C2×C42C22×D1111yescharacteristic
286C4⋊C8—22no—
287C4⋊C8—224no—
288C4⋊C8—222no—
289#103—44no—
290#103—444no—
291#95—21no—
292#95—222no—
293#95—222no—
294#95—222no—
295#95—221no—
296C2×C11#231111yescharacteristic
297C2×C11C2×#41611yescharacteristic
298C2×C11C2×#41911yescharacteristic
299C2×C11#324812yes—
300C2×C11#324812yes—
301D11—24no—
302C24—111no—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—224no—
307C2×Q8—442no—
308C2×Q8—442no—
309C2×Q8C2×C4×D1114yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—111no—
313C22×C4—111no—
314C22×C4—111no—
315C22×C4—114no—
316C22×C4—114no—
317C22×C4—112no—
318C22×C4—112no—
319C22×C4—224no—
320C22×C4—222no—
321C22×C4—222no—
322C22×C4—222no—
323C22×C4—222no—
324C22×C4—222no—
325C22×C4—222no—
326C22×C4D4×D1111yescharacteristic
327C22×C4D4×D1111yescharacteristic
328C22×C4C23×D1111yescharacteristic
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—222no—
332C2×C8—224no—
333C2×C8—448no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—114no—
339C4⋊C4—228no—
340C4⋊C4—228no—
341C4⋊C4—224no—
342C4⋊C4—222no—
343C4⋊C4—222no—
344C4⋊C4—222no—
345C22⋊C4—228no—
346C22⋊C4—444no—
347C42—42no—
348C42—114no—
349C42—114no—
350C42—112no—
351C42—112no—
352C42—228no—
353C42—442no—
354C42C2×C4×D1112yes—
355C42D4×D1112yes—
356C11#2188911yescharacteristic, complemented, Sylow
357C23—112no—
358C23—114no—
359C23—221no—
360C23—221no—
361C23—222no—
362C23C2×D4×D1111yescharacteristic
363Q8—28no—
364Q8—42no—
365Q8—42no—
366Q8—448no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—112no—
374C2×C4—112no—
375C2×C4—112no—
376C2×C4—228no—
377C2×C4—221no—
378C2×C4—221no—
379C2×C4—222no—
380C2×C4—222no—
381C2×C4—222no—
382C2×C4—224no—
383C2×C4—228no—
384C2×C4—222no—
385C2×C4—224no—
386C2×C4—442no—
387C2×C4—442no—
388C2×C4—442no—
389C2×C4—442no—
390C2×C4—442no—
391C2×C4C22⋊C4×D1112yes—
392C2×C4C22⋊C4×D1112yes—
393C2×C4#6696912yes—
394C2×C4#6697212yes—
395C2×C4C22×C4×D1111yescharacteristic
396C2×C4C2×D4×D1111yescharacteristic
397C2×C4#6703712yes—
398C2×C4#6704312yes—
399C8—42no—
400C8—442no—
401C22—114no—
402C22—222no—
403C22—222no—
404C22—224no—
405C22—224no—
406C22C2×C22⋊C4×D1111yescharacteristic
407C22#1062313912yes—
408C22C2×#6696911yescharacteristic
409C22C2×#6697211yescharacteristic
410C22D11×#13712yes—
411C4—22no—
412C4—44no—
413C4—82no—
414C4—222no—
415C4—222no—
416C4—222no—
417C4—444no—
418C4—882no—
419C4#1062241714yes—
420C4#1062307712yes—
421C4#1062320611yescharacteristic
422C4#1062324611yescharacteristic
423C2—224no—
424C2#1307120611yescharacteristic
425C2C2×#1062241711yescharacteristic
426C2D11×#41911yescharacteristic
427C2#1307994912yes—
428C2#1307995112yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1981C2
socleC23×C11C22×D4
prime radicalC23×D11C2×D4
cosocleC22×C4×C11C24
hypercenter#1981D11
hypocenterC11#21889
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
4. #13069568C2semidirect
5. #13069567C2semidirect
6. #13069540C2semidirect
8. #13068629C2semidirect
9. C11×#1981C2semidirect
15. #10623259C4semidirect
149. #1981D11semidirect
356. C11#21889semidirect

Maximal subgroups

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

subgroupindex
C2×#106232592
C2×D11×C4⋊C82
#130710122
#130695682
#130695672
#130695402
#130690362
#130686292
C11×#19812
#2188911