Finite Group Databasebeta

← #14809946

Subgroups — #14809946

Order 2816.

1338 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622694C212yesmaximal, complemented
2#13076968C211yesmaximal, characteristic
3#13069859C214yesmaximal, complemented
4#13069419C212yesmaximal, complemented
5#13068551C214yesmaximal
6C4×C22⋊(C11⋊C8)C211yesmaximal, characteristic
7C22×C11×C4⋊Q8C211yesmaximal, characteristic
8C23×Q8×C11C2212yes—
9C2×C11×C4⋊Q8—28no—
10C2×C11×C4⋊Q8—21no—
11C2×C11×C4⋊Q8—21no—
12C2×C11×C4⋊Q8C2212yes—
13C2×C11×C4⋊Q8C2212yes—
14C22×C11×C4⋊C4—21no—
15C22×C11×C4⋊C4—21no—
16C22×C42×C11C2211yescharacteristic
17C2×#66977—21no—
18C2×#66922C2211yescharacteristic
19C2×#66899C2218yes—
20C2×#66899C2218yes—
21#10622825—21no—
22#10622825C2212yes—
23C2×C22⋊(C11⋊C8)C2212yes—
24#10622694—28no—
25#10622548C2211yescharacteristic
26#10622542—21no—
27#10622541—21no—
28#10622541C2212yes—
29C2×C4×C11⋊C8C2212yes—
30#10622162C2212yes—
31C22×Q8×C11—28no—
32C22×Q8×C11—22no—
33C22×Q8×C11—28no—
34C22×Q8×C11—44no—
35C22×Q8×C11C2314yes—
36C23×C4×C11—22no—
37C23×C4×C11D411yescharacteristic
38C23×C4×C11D411yescharacteristic
39C23×C4×C11C2311yescharacteristic
40C22×Dic22—21no—
41C11×C4⋊Q8—28no—
42C11×C4⋊Q8—28no—
43C11×C4⋊Q8—28no—
44C11×C4⋊Q8—24no—
45C11×C4⋊Q8D418yes—
46C2×C11×C4⋊C4—28no—
47C2×C11×C4⋊C4—28no—
48C2×C11×C4⋊C4—22no—
49C2×C11×C4⋊C4—22no—
50C2×C11×C4⋊C4—22no—
51C2×C11×C4⋊C4—22no—
52C2×C42×C11—24no—
53C2×C42×C11D411yescharacteristic
54C2×C42×C11D411yescharacteristic
55C2×C42×C11C2311yescharacteristic
56C2×C42×C11C2311yescharacteristic
57C2×C22⋊Dic11—22no—
58C2×C4⋊Dic11—21no—
59C2×C4⋊Dic11—21no—
60C2×C4⋊Dic11—21no—
61C2×C4⋊Dic11C2312yes—
62C2×C4⋊Dic11C2312yes—
63#66977—44no—
64C2×Dic11⋊C4—22no—
65C22×C11⋊C8C2314yes—
66#66922—24no—
67#66899—216no—
68#66899—48no—
69C11⋊#95—22no—
70C11⋊#95—22no—
71C22⋊(C11⋊C8)—24no—
72C4×C11⋊C8—24no—
73#25856—111nomaximal, Sylow
74C24×C11C2×D411yescharacteristic
75C2×Q8×C11—28no—
76C2×Q8×C11—24no—
77C2×Q8×C11—28no—
78C2×Q8×C11—28no—
79C2×Q8×C11—432no—
80C2×Q8×C11—44no—
81C2×Q8×C11—44no—
82C2×Q8×C11C2×D418yes—
83C22×C4×C11—22no—
84C22×C4×C11—22no—
85C22×C4×C11—22no—
86C22×C4×C11—22no—
87C22×C4×C11—22no—
88C22×C4×C11—22no—
89C22×C4×C11—28no—
90C22×C4×C11—24no—
91C22×C4×C11—44no—
92C22×C4×C11—44no—
93C22×C4×C11SD1614yes—
94C22×C4×C11Q1614yes—
95C22×C4×C11C2×D411yescharacteristic
96C22×C4×C11C2×D411yescharacteristic
97C22×C4×C11C2×D411yescharacteristic
98C22×C4×C11C2×D411yescharacteristic
99C22×C4×C11C2×D411yescharacteristic
100C22×C4×C11#4114yes—
101C22×C4×C11C2411yescharacteristic
102C22×Dic11—22no—
103C22×Dic11—22no—
104C2×Dic22—42no—
105C2×Dic22—42no—
106C2×Dic22—42no—
107C11×C4⋊C4—28no—
108C11×C4⋊C4—28no—
109C11×C4⋊C4—28no—
110C11×C4⋊C4—28no—
111C42×C11—22no—
112C42×C11—22no—
113C42×C11—22no—
114C42×C11C2×D414yes—
115C22⋊Dic11—44no—
116C4⋊Dic11—24no—
117C4⋊Dic11—24no—
118C4⋊Dic11—42no—
119C4⋊Dic11—42no—
120C4⋊Dic11—42no—
121Dic11⋊C4—44no—
122C2×C11⋊C8—24no—
123C2×C11⋊C8—24no—
124C2×C11⋊C8—24no—
125C22×C4⋊Q8D1111yescharacteristic
126C2×#487—112no—
127#2857—111no—
128#2043—114no—
129#1956—112no—
130#1787—114no—
131C4×C22⋊C8—111no—
132C23×C11—21no—
133C23×C11—21no—
134C23×C11—21no—
135C23×C11—21no—
136C23×C11#12412yes—
137C23×C11C2×SD1612yes—
138C23×C11C2×Q1612yes—
139C23×C11C22×D411yescharacteristic
140Q8×C11—216no—
141Q8×C11—416no—
142Q8×C11—48no—
143C2×C4×C11—21no—
144C2×C4×C11—21no—
145C2×C4×C11—21no—
146C2×C4×C11—21no—
147C2×C4×C11—21no—
148C2×C4×C11—21no—
149C2×C4×C11—28no—
150C2×C4×C11—28no—
151C2×C4×C11—28no—
152C2×C4×C11—28no—
153C2×C4×C11—416no—
154C2×C4×C11—44no—
155C2×C4×C11—44no—
156C2×C4×C11#12012yes—
157C2×C4×C11#12012yes—
158C2×C4×C11#12412yes—
159C2×C4×C11C2×SD1612yes—
160C2×C4×C11C2×SD1612yes—
161C2×C4×C11C2×Q1612yes—
162C2×C4×C11C2×Q1612yes—
163C2×C4×C11#13712yes—
164C2×C4×C11#13712yes—
165C2×C4×C11C22×D412yes—
166C2×C4×C11C2×#4112yes—
167C2×C4×C11H(5,𝔽2)12yes—
168C2×Dic11—42no—
169C2×Dic11—42no—
170C2×Dic11—42no—
171C2×Dic11—42no—
172C2×Dic11—42no—
173C2×Dic11—42no—
174Dic22—84no—
175C11⋊C8—44no—
176C23×Q8C2×D1112yes—
177C2×C4⋊Q8—28no—
178C2×C4⋊Q8—21no—
179C2×C4⋊Q8—21no—
180C2×C4⋊Q8—111no—
181C2×C4⋊Q8C2×D1112yes—
182C2×C4⋊Q8C2×D1112yes—
183C2×C22⋊Q8—221no—
184C22×C4⋊C4—21no—
185C22×C4⋊C4—21no—
186C22×C42C2×D1111yescharacteristic
187#487—228no—
188C2×#103—118no—
189C2×#103—118no—
190C2×C22⋊C8—112no—
191C2×C4×C8—112no—
192#391—221no—
193#389—112no—
194#389—221no—
195#384—111no—
196#380—112no—
197#380—221no—
198#336—112no—
199C22×C11—22no—
200C22×C11—22no—
201C22×C11—22no—
202C22×C11—22no—
203C22×C11—22no—
204C22×C11—22no—
205C22×C11#48714yescomplemented
206C22×C11C2×#12011yescharacteristic
207C22×C11C2×#12411yescharacteristic
208C22×C11#53511yescharacteristic
209C22×C11C22×SD1611yescharacteristic
210C22×C11C22×Q1611yescharacteristic
211C22×C11C2×#13711yescharacteristic
212C22×C11C2×#13711yescharacteristic
213C4×C11—22no—
214C4×C11—22no—
215C4×C11—22no—
216C4×C11—48no—
217C4×C11#44814yes—
218C4×C11#45114yes—
219C4×C11#54614yes—
220Dic11—82no—
221Dic11—82no—
222C22×Q8—28no—
223C22×Q8—22no—
224C22×Q8—28no—
225C22×Q8—44no—
226C22×Q8—221no—
227C22×Q8C22×D1114yes—
228C23×C4—22no—
229C23×C4C22⋊D1111yescharacteristic
230C23×C4C22⋊D1111yescharacteristic
231C23×C4C22×D1111yescharacteristic
232C22×C8—114no—
233C4⋊Q8—28no—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236C4⋊Q8—24no—
237C4⋊Q8—224no—
238C4⋊Q8C22⋊D1118yes—
239C22⋊Q8—444no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—112no—
247C2×C4⋊C4—112no—
248C2×C4⋊C4—221no—
249C2×C4⋊C4—222no—
250C2×C4⋊C4—221no—
251C2×C4⋊C4—221no—
252C2×C22⋊C4—222no—
253C2×C42—24no—
254C2×C42C22⋊D1111yescharacteristic
255C2×C42C22⋊D1111yescharacteristic
256C2×C42C22×D1111yescharacteristic
257C2×C42C22×D1111yescharacteristic
258#103—2216no—
259#103—448no—
260C22⋊C8—224no—
261C4×C8—224no—
262#95—222no—
263#95—222no—
264C2×C11—24no—
265C2×C11#243511yescharacteristic
266C2×C11C2×#44811yescharacteristic
267C2×C11C2×#45111yescharacteristic
268C2×C11C2×#48711yescharacteristic
269C2×C11#313911yescharacteristic
270C2×C11#320611yescharacteristic
271C2×C11#320911yescharacteristic
272C24C2×C22⋊D1111yescharacteristic
273C2×Q8—28no—
274C2×Q8—24no—
275C2×Q8—28no—
276C2×Q8—28no—
277C2×Q8—432no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—442no—
281C2×Q8—442no—
282C2×Q8—442no—
283C2×Q8C2×C22⋊D1118yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—222no—
295C22×C4—222no—
296C22×C4#432514yes—
297C22×C4#432614yes—
298C22×C4#434114yes—
299C22×C4C2×C22⋊D1111yescharacteristic
300C22×C4C2×C22⋊D1111yescharacteristic
301C22×C4C2×C22⋊D1111yescharacteristic
302C22×C4C2×C22⋊D1111yescharacteristic
303C22×C4C2×C22⋊D1111yescharacteristic
304C22×C4C23×D1111yescharacteristic
305C2×C8—224no—
306C2×C8—224no—
307C2×C8—224no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—224no—
313C4⋊C4—224no—
314C4⋊C4—444no—
315C4⋊C4—442no—
316C4⋊C4—442no—
317C4⋊C4—442no—
318C22⋊C4—444no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42C2×C22⋊D1114yes—
323C11#2585611yescharacteristic, complemented, Sylow
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23#6698912yes—
329C23C2×#432512yes—
330C23C2×#432612yes—
331C23C22×C22⋊D1111yescharacteristic
332Q8—216no—
333Q8—416no—
334Q8—48no—
335Q8—884no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—416no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—442no—
350C2×C4—442no—
351C2×C4—442no—
352C2×C4—442no—
353C2×C4—442no—
354C2×C4—442no—
355C2×C4#6698912yes—
356C2×C4C2×#432512yes—
357C2×C4C2×#432512yes—
358C2×C4#6699512yes—
359C2×C4#6699512yes—
360C2×C4C2×#432612yes—
361C2×C4C2×#432612yes—
362C2×C4#6700612yes—
363C2×C4#6700612yes—
364C2×C4C2×#434112yes—
365C2×C4#6703712yes—
366C2×C4C22×C22⋊D1112yes—
367C8—444no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22#1062269414yescomplemented
375C22C2×#6698911yescharacteristic
376C22#1062333611yescharacteristic
377C22C22×#432511yescharacteristic
378C22C2×#6699511yescharacteristic
379C22C2×#6699511yescharacteristic
380C22C22×#432611yescharacteristic
381C22C2×#6700611yescharacteristic
382C4—22no—
383C4—22no—
384C4—22no—
385C4—48no—
386C4—882no—
387C4—882no—
388C4#1062284014yes—
389C4#1062284114yes—
390C4#1062333714yes—
391C2—24no—
392C2#1307226311yescharacteristic
393C2C2×#1062284011yescharacteristic
394C2C2×#1062284111yescharacteristic
395C2C2×#1062269411yescharacteristic
396C2#1307899611yescharacteristic
397C2#1307954111yescharacteristic
398C2#1307956411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC22×C11×C4⋊Q8C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
hypercenterC22×C4⋊Q8D11
hypocenterC11#25856
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10622694C2semidirect
3. #13069859C2semidirect
4. #13069419C2semidirect
205. C22×C11#487semidirect
323. C11#25856semidirect
374. C22#10622694semidirect

Maximal subgroups

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

subgroupindex
C2×#106226942
#130769682
#130698592
#130694192
#130685512
C4×C22⋊(C11⋊C8)2
C22×C11×C4⋊Q82
#2585611