Finite Group Databasebeta

← #14726566

Subgroups — #14726566

Order 2816.

970 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13078742C212yesmaximal, complemented
2#13071900C211yesmaximal, characteristic
3#13067607C214yesmaximal, complemented
4#13067573C214yesmaximal, complemented
5#13067128C212yesmaximal, complemented
6#13066957C211yesmaximal, characteristic
7C11×#3501C211yesmaximal, characteristic
8C11×#554—22no—
9C11×#554—24no—
10D4×Q8×C11—22no—
11C11×#544—21no—
12C11×#543C2214yes—
13C11×#533C2212yes—
14C2×C11×C22⋊Q8—22no—
15C11×#518—21no—
16C2×C4×D4×C11C2211yescharacteristic
17C2×C4×Dic22C2211yescharacteristic
18#10622835—21no—
19C4×C22⋊Dic11—22no—
20#10622707—22no—
21#10622703—22no—
22#10622663C2214yes—
23#10622656C2214yes—
24#10622643—24no—
25#10622595—21no—
26#10622566C2212yes—
27#10622560C2212yes—
28#10622545C2211yescharacteristic
29#10622539C2212yes—
30#10622176C2218yes—
31#10622163C2214yes—
32C22×Q8×C11—22no—
33C22×D4×C11D411yescharacteristic
34C23×C4×C11D412yes—
35C22×Dic22C2311yescharacteristic
36C11×C4⋊Q8—24no—
37C11×C4⋊Q8D412yes—
38C11×C4⋊Q8D412yes—
39C11×#125—24no—
40C11×#125—22no—
41C11×C22⋊Q8—24no—
42C11×C22⋊Q8—24no—
43C11×C22⋊Q8—28no—
44C11×C22⋊Q8—22no—
45C11×C22⋊Q8—24no—
46C11×C22⋊Q8—28no—
47C11×C22⋊Q8C2314yes—
48C4×Q8×C11—22no—
49C4×Q8×C11D414yes—
50C4×D4×C11—24no—
51C4×D4×C11—24no—
52C4×D4×C11—22no—
53C4×D4×C11—22no—
54C11×#117—22no—
55C2×C11×C4⋊C4—22no—
56C2×C11×C4⋊C4—24no—
57C2×C11×C4⋊C4D411yescharacteristic
58C2×C11×C22⋊C4—24no—
59C2×C11×C22⋊C4C2312yes—
60C2×C42×C11C2311yescharacteristic
61C2×C22⋊Dic11—22no—
62C2×C22⋊Dic11—22no—
63C2×#4326—24no—
64C2×C4⋊Dic11C2311yescharacteristic
65C2×Dic11⋊C4—21no—
66C2×C4×Dic11—21no—
67C2×C4×Dic11—21no—
68C4×Dic22—22no—
69C4×Dic22—22no—
70#66899—24no—
71C11⋊#95—21no—
72C11⋊#95—22no—
73C11⋊#95—22no—
74C11⋊#95C2312yes—
75C22⋊(C11⋊C8)C2314yes—
76#66874—24no—
77#66872—24no—
78C4⋊(C11⋊C8)—22no—
79#66867—22no—
80#18181—111nomaximal, Sylow
81C24×C11C2×D412yes—
82C2×Q8×C11—24no—
83C2×Q8×C11—22no—
84C2×Q8×C11—84no—
85C2×Q8×C11C2×D414yes—
86C2×D4×C11—24no—
87C2×D4×C11—22no—
88C22×C4×C11—24no—
89C22×C4×C11—24no—
90C22×C4×C11—22no—
91C22×C4×C11—22no—
92C22×C4×C11—44no—
93C22×C4×C11C2×D411yescharacteristic
94C22×C4×C11C2×D411yescharacteristic
95C22×C4×C11C2×D412yes—
96C22×C4×C11C2411yescharacteristic
97C22×Dic11—21no—
98C22×Dic11—22no—
99C22×Dic11—21no—
100C2×Dic22—22no—
101C2×Dic22—42no—
102C11×C4⋊C4—24no—
103C11×C4⋊C4—28no—
104C11×C4⋊C4—22no—
105C11×C4⋊C4—24no—
106C11×C4⋊C4—22no—
107C11×C4⋊C4—22no—
108C11×C4⋊C4—22no—
109C11×C4⋊C4—24no—
110C11×C4⋊C4C2×D414yes—
111C11×C22⋊C4—28no—
112C11×C22⋊C4—24no—
113C11×C22⋊C4—24no—
114C42×C11—24no—
115C42×C11C2×D412yes—
116C42×C11C2×D412yes—
117C22⋊Dic11—44no—
118C22⋊Dic11—44no—
119#4326—48no—
120C4⋊Dic11—22no—
121Dic11⋊C4—42no—
122C4×Dic11—42no—
123C4×Dic11—41no—
124C4×Dic11—41no—
125C2×C11⋊C8—24no—
126#3501D1111yescharacteristic
127#3113—112no—
128#2398—111no—
129#1608—114no—
130#1602—114no—
131#1506—112no—
132#1466—111no—
133C23×C11—22no—
134C23×C11—22no—
135C23×C11—42no—
136C23×C11—42no—
137C23×C11#13714yes—
138C23×C11C22×D411yescharacteristic
139Q8×C11—44no—
140Q8×C11—44no—
141Q8×C11—84no—
142D4×C11—28no—
143C2×C4×C11—22no—
144C2×C4×C11—22no—
145C2×C4×C11—24no—
146C2×C4×C11—24no—
147C2×C4×C11—22no—
148C2×C4×C11—41no—
149C2×C4×C11—44no—
150C2×C4×C11—42no—
151C2×C4×C11—82no—
152C2×C4×C11—84no—
153C2×C4×C11—84no—
154C2×C4×C11#12011yescharacteristic
155C2×C4×C11#12012yes—
156C2×C4×C11#12011yescharacteristic
157C2×C4×C11C22×D412yes—
158C2×Dic11—22no—
159C2×Dic11—41no—
160C2×Dic11—41no—
161C2×Dic11—42no—
162C2×Dic11—42no—
163C2×Dic11—41no—
164C2×Dic11—42no—
165C2×Dic11—42no—
166Dic22—24no—
167Dic22—42no—
168C11⋊C8—44no—
169#554—22no—
170#554—24no—
171D4×Q8—22no—
172#544—21no—
173#543C2×D1114yes—
174#533C2×D1112yes—
175C2×C22⋊Q8—22no—
176#518—21no—
177C2×C4×Q8—111no—
178C2×C4×D4C2×D1111yescharacteristic
179#483—114no—
180#475—222no—
181#462—222no—
182#451—114no—
183#441—224no—
184#432—112no—
185#400—221no—
186#397—112no—
187#394—221no—
188#393—112no—
189C4×C22⋊C4—222no—
190#376—111no—
191#328—118no—
192#323—114no—
193C22×C11—21no—
194C22×C11—21no—
195C22×C11—24no—
196C22×C11—44no—
197C22×C11—42no—
198C22×C11—42no—
199C22×C11—82no—
200C22×C11C2×#12011yescharacteristic
201C22×C11C2×#13712yes—
202C4×C11—22no—
203C4×C11—21no—
204C4×C11—22no—
205C4×C11—42no—
206C4×C11—44no—
207C4×C11—44no—
208C4×C11#45512yes—
209Dic11—42no—
210Dic11—82no—
211Dic11—81no—
212C22×Q8—22no—
213C22×Q8—111no—
214C22×D4C22⋊D1111yescharacteristic
215C23×C4C22⋊D1112yes—
216C2×Q16—224no—
217C4⋊Q8—24no—
218C4⋊Q8C22⋊D1112yes—
219C4⋊Q8C22⋊D1112yes—
220#125—24no—
221#125—22no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—28no—
225C22⋊Q8—22no—
226C22⋊Q8—24no—
227C22⋊Q8—28no—
228C22⋊Q8C22×D1114yes—
229C4×Q8—22no—
230C4×Q8—222no—
231C4×Q8—222no—
232C4×Q8C22⋊D1114yes—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—22no—
236C4×D4—22no—
237#117—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—111no—
241C2×C4⋊C4—221no—
242C2×C4⋊C4C22⋊D1111yescharacteristic
243C2×C22⋊C4—24no—
244C2×C22⋊C4—222no—
245C2×C22⋊C4—222no—
246C2×C22⋊C4C22×D1112yes—
247C2×C42—221no—
248C2×C42—221no—
249C2×C42C22×D1111yescharacteristic
250#106—224no—
251C4⋊C8—222no—
252#103—224no—
253#103—224no—
254C22⋊C8—114no—
255#97—222no—
256#95—112no—
257#95—221no—
258#95—222no—
259#95—222no—
260C2×C11—22no—
261C2×C11—44no—
262C2×C11#300411yescharacteristic
263C2×C11C2×#45511yescharacteristic
264C2×C11#302511yescharacteristic
265C24C2×C22⋊D1112yes—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—84no—
269C2×Q8—222no—
270C2×Q8—442no—
271C2×Q8C2×C22⋊D1114yes—
272C2×D4—24no—
273C2×D4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—44no—
279C22×C4—221no—
280C22×C4—222no—
281C22×C4—221no—
282C22×C4C2×C22⋊D1111yescharacteristic
283C22×C4C2×C22⋊D1111yescharacteristic
284C22×C4C2×C22⋊D1112yes—
285C22×C4C23×D1111yescharacteristic
286Q16—448no—
287C2×C8—224no—
288C4⋊C4—24no—
289C4⋊C4—28no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—222no—
297C4⋊C4—442no—
298C4⋊C4C2×C22⋊D1114yes—
299C22⋊C4—28no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—444no—
303C22⋊C4—444no—
304C42—24no—
305C42—442no—
306C42—441no—
307C42—441no—
308C42C2×C22⋊D1112yes—
309C42C2×C22⋊D1112yes—
310C11#1818111yescharacteristic, complemented, Sylow
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23#6699514yes—
316C23C22×C22⋊D1111yescharacteristic
317Q8—44no—
318Q8—44no—
319Q8—84no—
320Q8—224no—
321Q8—442no—
322D4—28no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—222no—
335C2×C4—441no—
336C2×C4—441no—
337C2×C4—442no—
338C2×C4—442no—
339C2×C4—442no—
340C2×C4—442no—
341C2×C4—441no—
342C2×C4#6700612yes—
343C2×C4#6700611yescharacteristic
344C2×C4#6700611yescharacteristic
345C2×C4C22×C22⋊D1112yes—
346C8—444no—
347C22—21no—
348C22—21no—
349C22—24no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22C2×#6699512yes—
355C22C2×#6700611yescharacteristic
356C4—22no—
357C4—21no—
358C4—22no—
359C4—42no—
360C4—44no—
361C4—44no—
362C4—442no—
363C4—882no—
364C4—881no—
365C4#1062285512yes—
366C2—22no—
367C2—44no—
368C2#1307785011yescharacteristic
369C2C2×#1062285511yescharacteristic
370C2#1307800811yescharacteristic
subgroupG/H
centerC22C2×#67006
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#3501C2
socleC22×C11C2×#120
prime radicalC22×D4×C11D4
hypercenter#3501D11
hypocenterC11#18181
LCSG ▷ C22×C4×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #13078742C2semidirect
3. #13067607C2semidirect
4. #13067573C2semidirect
5. #13067128C2semidirect
310. C11#18181semidirect

Maximal subgroups

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

subgroupindex
#130787422
#130719002
#130676072
#130675732
#130671282
#130669572
C11×#35012
#1818111