Finite Group Databasebeta

← #14764322

Subgroups — #14764322

Order 2816.

806 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622298C211yesmaximal, characteristic
2#13071022C211yesmaximal, characteristic
3#13071009C211yesmaximal, characteristic
4#13069378C212yesmaximal, complemented
5#13069373C212yesmaximal, complemented
6#13068818C212yesmaximal
7#13068816C212yesmaximal
8#13068780C211yesmaximal, characteristic, complemented
9#13068778C211yesmaximal, characteristic, complemented
10#13068746C211yesmaximal, characteristic, complemented
11C11×#1835C211yesmaximal, characteristic, complemented
12C2×#66945C2212yes—
13C2×C42×D11C2211yescharacteristic
14C2×C4×C8×C11C2211yescharacteristic
15C11×#382C2211yescharacteristic
16C11×#382C2211yescharacteristic
17C2×(C2×D11)⋊C8C2211yescharacteristic
18C2×(C2×D11)⋊C8C2211yescharacteristic
19C2×#66882C2212yes—
20C2×#66881C2212yes—
21C2×Dic11⋊C8C2211yescharacteristic
22C2×Dic11⋊C8C2211yescharacteristic
23#10622600C2212yes—
24#10622589C2212yes—
25C4⋊C4×Dic11C2212yes—
26C2×#66872C2212yes—
27C2×#66871C2212yes—
28#10622549C2211yescharacteristic
29#10622549C2211yescharacteristic
30C2×C4×C11⋊C8C2211yescharacteristic
31#10622298C418yescomplemented
32#10622280—21no—
33#10622280—21no—
34#10622279—21no—
35#10622277—21no—
36#10622264—21no—
37#10622262—21no—
38#10622262—21no—
39#10622261—21no—
40C11×#340C2212yes—
41C11×#340C2212yes—
42#10622167C2212yes—
43#10622167C2212yes—
44C22×C4×D11D411yescharacteristic
45C22×C4×D11D411yescharacteristic
46C22×C4×D11C2311yescharacteristic
47C22×C8×C11C2311yescharacteristic
48C22×C8×C11C2311yescharacteristic
49C2×C11×C4⋊C4C2312yes—
50C2×C11×C4⋊C4C2312yes—
51C2×C42×C11C2311yescharacteristic
52C2×(C2×D11)⋊C4—22no—
53C2×(C2×D11)⋊C4—22no—
54C2×C4⋊Dic11C2312yes—
55C2×C4⋊Dic11C2312yes—
56C2×Dic11⋊C4—22no—
57C2×Dic11⋊C4—22no—
58C2×C4×Dic11—22no—
59C2×C4×Dic11—22no—
60C2×C4×Dic11D411yescharacteristic
61C2×C4×Dic11D411yescharacteristic
62C2×C4×Dic11C2311yescharacteristic
63C22×C11⋊C8C2311yescharacteristic
64C22×C11⋊C8C2311yescharacteristic
65#66945—42no—
66#66945—42no—
67C42×D11C2×C414yes—
68C42×D11Q814yes—
69C4×C8×C11C2×C414yes—
70C11×#95—21no—
71C11×#95—21no—
72C11×#95—21no—
73C11×#95—21no—
74C11⋊#95—21no—
75C11⋊#95—21no—
76C11⋊#95—21no—
77C11⋊#95—21no—
78C11⋊#95—22no—
79C11⋊#95—22no—
80(C2×D11)⋊C8C2×C414yes—
81(C2×D11)⋊C8C2×C414yes—
82#66882—42no—
83#66881—42no—
84Dic11⋊C8C2×C414yes—
85Dic11⋊C8C2×C414yes—
86#66872—42no—
87#66871—42no—
88C4×C11⋊C8C2×C414yes—
89#21985—111nomaximal, Sylow
90C23×D11C2×D411yescharacteristic
91C22×C4×C11—22no—
92C22×C4×C11—22no—
93C22×C4×C11C2×D411yescharacteristic
94C22×C4×C11C2×D411yescharacteristic
95C22×C4×C11C2411yescharacteristic
96C22×Dic11—22no—
97C22×Dic11—22no—
98C22×Dic11C2×D411yescharacteristic
99C22×Dic11C2×D411yescharacteristic
100C22×Dic11C2×D411yescharacteristic
101C2×C4×D11—24no—
102C2×C4×D11—21no—
103C2×C4×D11—21no—
104C2×C4×D11—21no—
105C2×C4×D11—21no—
106C2×C4×D11—42no—
107C2×C4×D11C4⋊C414yes—
108C2×C4×D11C4⋊C414yes—
109C2×C4×D11C22×C412yes—
110C2×C4×D11C2×Q812yes—
111C2×C8×C11—21no—
112C2×C8×C11—21no—
113C2×C8×C11—21no—
114C2×C8×C11—21no—
115C2×C8×C11C22×C412yes—
116C2×C8×C11C22×C412yes—
117C11×C4⋊C4—22no—
118C11×C4⋊C4—22no—
119C11×C4⋊C4—22no—
120C11×C4⋊C4—22no—
121C42×C11C22×C412yes—
122C42×C11C2×Q812yes—
123(C2×D11)⋊C4—44no—
124(C2×D11)⋊C4—44no—
125C4⋊Dic11—22no—
126C4⋊Dic11—22no—
127C4⋊Dic11—22no—
128C4⋊Dic11—22no—
129Dic11⋊C4—44no—
130Dic11⋊C4—44no—
131C4×Dic11—44no—
132C4×Dic11—44no—
133C4×Dic11C4⋊C414yes—
134C4×Dic11C4⋊C414yes—
135C4×Dic11C22×C412yes—
136C4×Dic11C2×Q812yes—
137C2×C11⋊C8—21no—
138C2×C11⋊C8—21no—
139C2×C11⋊C8—21no—
140C2×C11⋊C8—21no—
141C2×C11⋊C8C22×C412yes—
142C2×C11⋊C8C22×C412yes—
143C2×#431—111no—
144#2285—111no—
145#2283—111no—
146#1948—112no—
147#1947—112no—
148#1854—112no—
149#1854—112no—
150#1846—111no—
151#1846—111no—
152#1835—111no—
153#1835D1111yescharacteristic, complemented
154C23×C11C22×D411yescharacteristic
155C22×D11—21no—
156C22×D11—21no—
157C22×D11—22no—
158C22×D11—22no—
159C22×D11C2×C4⋊C412yes—
160C2×C4×C11—22no—
161C2×C4×C11—22no—
162C2×C4×C11—42no—
163C2×C4×C11—42no—
164C2×C4×C11—42no—
165C2×C4×C11—42no—
166C2×C4×C11—42no—
167C2×C4×C11—42no—
168C2×C4×C11C2×C4⋊C412yes—
169C2×C4×C11C2×C4⋊C412yes—
170C2×C4×C11C23×C411yescharacteristic
171C2×C4×C11C22×Q811yescharacteristic
172C2×C4×C11H(5,𝔽2)11yescharacteristic
173C2×C4×C11H(5,𝔽2)11yescharacteristic
174C2×C4×C112−1+411yescharacteristic
175C2×C4×C112−1+411yescharacteristic
176C2×Dic11—21no—
177C2×Dic11—21no—
178C2×Dic11—22no—
179C2×Dic11—42no—
180C2×Dic11—42no—
181C2×Dic11—42no—
182C2×Dic11—42no—
183C2×Dic11—42no—
184C2×Dic11—42no—
185C2×Dic11C2×C4⋊C412yes—
186C2×Dic11C2×C4⋊C412yes—
187C2×Dic11C2×C4⋊C412yes—
188C2×Dic11#13514yes—
189C4×D11—22no—
190C4×D11—22no—
191C4×D11—22no—
192C4×D11—22no—
193C4×D11—28no—
194C4×D11—44no—
195C8×C11—22no—
196C8×C11—22no—
197C11⋊C8—22no—
198C11⋊C8—22no—
199C2×#117—112no—
200C22×C42—111no—
201#431—118no—
202C2×#107—112no—
203C2×#107—112no—
204C2×#106—112no—
205C2×#106—112no—
206C2×C4⋊C8—111no—
207C2×C4⋊C8—111no—
208C2×C22⋊C8—111no—
209C2×C22⋊C8—111no—
210C2×C4×C8—111no—
211C2×C4×C8C2×D1111yescharacteristic
212#384—112no—
213#383—221no—
214#382—111no—
215#382—111no—
216#382C2×D1111yescharacteristic
217#382C2×D1111yescharacteristic
218#381—221no—
219#380—112no—
220#380—221no—
221#380—221no—
222C4×C4⋊C4—112no—
223C4×C4⋊C4—221no—
224C4×C4⋊C4—221no—
225C4×C22⋊C4—221no—
226#376—221no—
227#340—112no—
228#340—112no—
229#340C2×D1112yes—
230#340C2×D1112yes—
231C22×C11C22×C4⋊C411yescharacteristic
232C22×C11#53511yescharacteristic
233C22×C11#53611yescharacteristic
234C22×C11C2×#13512yes—
235C22×C11#57512yes—
236C2×D11—22no—
237C2×D11—22no—
238C2×D11—22no—
239C2×D11—42no—
240C2×D11—42no—
241C4×C11—22no—
242C4×C11—22no—
243C4×C11—82no—
244C4×C11—82no—
245C4×C11#52011yescharacteristic
246C4×C11#52011yescharacteristic
247C4×C11#54411yescharacteristic
248C4×C11#54411yescharacteristic
249Dic11—22no—
250Dic11—22no—
251Dic11—82no—
252Dic11—82no—
253Dic11#42714yes—
254C23×C4—111no—
255C23×C4—111no—
256C23×C4—111no—
257C22×C8—111no—
258C22×C8—111no—
259C22×C8C22×D1111yescharacteristic
260C22×C8C22×D1111yescharacteristic
261#117—442no—
262#117—442no—
263C2×C4⋊C4—112no—
264C2×C4⋊C4—112no—
265C2×C4⋊C4—222no—
266C2×C4⋊C4—222no—
267C2×C4⋊C4C22×D1112yes—
268C2×C4⋊C4C22×D1112yes—
269C2×C22⋊C4—222no—
270C2×C22⋊C4—222no—
271C2×C42—111no—
272C2×C42—111no—
273C2×C42—111no—
274C2×C42—114no—
275C2×C42—114no—
276C2×C42—222no—
277C2×C42—222no—
278C2×C42C22×D1111yescharacteristic
279#107—442no—
280#107—442no—
281#106—442no—
282#106—442no—
283C4⋊C8—114no—
284C4⋊C8—114no—
285C22⋊C8—114no—
286C22⋊C8—114no—
287C4×C8—114no—
288C4×C8C4×D1114yes—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—222no—
294#95—222no—
295#95—221no—
296#95—221no—
297#95—221no—
298#95—221no—
299C2×C11#231711yescharacteristic
300C2×C11C2×#42711yescharacteristic
301C2×C11#291111yescharacteristic
302C2×C11#321212yes—
303C2×C11#323312yes—
304D11—42no—
305C24—111no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—111no—
309C22×C4—111no—
310C22×C4—111no—
311C22×C4—114no—
312C22×C4—112no—
313C22×C4—112no—
314C22×C4—114no—
315C22×C4—224no—
316C22×C4—221no—
317C22×C4—221no—
318C22×C4—221no—
319C22×C4—221no—
320C22×C4—222no—
321C22×C4—222no—
322C22×C4—442no—
323C22×C4D4×D1111yescharacteristic
324C22×C4D4×D1111yescharacteristic
325C22×C4C23×D1111yescharacteristic
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—112no—
331C2×C8—112no—
332C2×C8—221no—
333C2×C8—221no—
334C2×C8—221no—
335C2×C8—221no—
336C2×C8C2×C4×D1112yes—
337C2×C8C2×C4×D1112yes—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—222no—
343C4⋊C4—222no—
344C4⋊C4—222no—
345C4⋊C4—222no—
346C4⋊C4—444no—
347C4⋊C4—444no—
348C22⋊C4—444no—
349C22⋊C4—444no—
350C42—114no—
351C42—114no—
352C42—112no—
353C42—112no—
354C42—444no—
355C42—444no—
356C42C2×C4×D1112yes—
357C42Q8×D1112yes—
358C11#2198511yescharacteristic, complemented, Sylow
359C23—112no—
360C23—222no—
361C23—222no—
362C23—221no—
363C23—221no—
364C23C2×D4×D1111yescharacteristic
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—112no—
374C2×C4—114no—
375C2×C4—112no—
376C2×C4—112no—
377C2×C4—228no—
378C2×C4—222no—
379C2×C4—221no—
380C2×C4—221no—
381C2×C4—222no—
382C2×C4—222no—
383C2×C4—222no—
384C2×C4—222no—
385C2×C4—444no—
386C2×C4—442no—
387C2×C4—442no—
388C2×C4—442no—
389C2×C4—442no—
390C2×C4—442no—
391C2×C4—442no—
392C2×C4C4⋊C4×D1112yes—
393C2×C4C4⋊C4×D1112yes—
394C2×C4C22×C4×D1111yescharacteristic
395C2×C4C2×Q8×D1111yescharacteristic
396C2×C4#6704211yescharacteristic
397C2×C4#6704211yescharacteristic
398C2×C4#6704311yescharacteristic
399C2×C4#6704311yescharacteristic
400C8—22no—
401C8—22no—
402C8—222no—
403C8—222no—
404C22—222no—
405C22—222no—
406C22—222no—
407C22—442no—
408C22—442no—
409C22C2×C4⋊C4×D1111yescharacteristic
410C22#1062320511yescharacteristic
411C22#1062320611yescharacteristic
412C22D11×#13512yes—
413C22#1062329912yes—
414C4—22no—
415C4—22no—
416C4—82no—
417C4—82no—
418C4—114no—
419C4—222no—
420C4—222no—
421C4—882no—
422C4—882no—
423C4#1062305411yescharacteristic
424C4#1062305411yescharacteristic
425C4#1062322011yescharacteristic
426C4#1062322011yescharacteristic
427C2—442no—
428C2#1307124111yescharacteristic
429C2D11×#42711yescharacteristic
430C2#1307720411yescharacteristic
431C2#1307959312yes—
432C2#1307979012yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1835C2
socleC23×C11C22×D4
prime radicalC23×D11C2×D4
cosocleC22×C4×C11C24
hypercenter#1835D11
hypocenterC11#21985
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1835

Complements

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

NH ≅ G/Ntype
4. #13069378C2semidirect
5. #13069373C2semidirect
8. #13068780C2semidirect
9. #13068778C2semidirect
10. #13068746C2semidirect
11. C11×#1835C2semidirect
31. #10622298C4semidirect
153. #1835D11semidirect
358. C11#21985semidirect

Maximal subgroups

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

subgroupindex
C2×#106222982
#130710222
#130710092
#130693782
#130693732
#130688182
#130688162
#130687802
#130687782
#130687462
C11×#18352
#2198511