Finite Group Databasebeta

← #14682901

Subgroups — #14682901

Order 2816.

1310 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622843C212yesmaximal
2#13066873C214yesmaximal
3#13066869C212yesmaximal
4#13066850C214yesmaximal
5#13066843C212yesmaximal
6C2×C11×#391C211yesmaximal, characteristic
7C23×Q8×C11C2211yescharacteristic
8C22×C11×C4⋊C4C2211yescharacteristic
9C22×C42×C11C2211yescharacteristic
10C22×C4⋊Dic11C2212yes—
11C22×C4⋊Dic11C2212yes—
12C22×C4×Dic11C2212yes—
13C11×#391C418yescomplemented
14C2×C11×#95C2214yes—
15#10622843—28no—
16C2×C11⋊#95C2214yes—
17C2×C11⋊#95C2214yes—
18C2×C11⋊#95C2214yes—
19C2×C11⋊#95C2214yes—
20C2×C11⋊#95C2212yes—
21C2×C11⋊#95C2212yes—
22C2×C11⋊#95C2212yes—
23C22×Q8×C11—41no—
24C22×Q8×C11—41no—
25C22×Q8×C11—41no—
26C22×Q8×C11—41no—
27C22×Q8×C11C2×C414yes—
28C22×Q8×C11D414yes—
29C22×Q8×C11Q814yes—
30C23×C4×C11C2312yes—
31C23×C4×C11C2312yes—
32C23×C4×C11C2312yes—
33C23×C4×C11C2311yescharacteristic
34C23×Dic11C2314yes—
35C23×Dic11C2314yes—
36C2×C11×C4⋊C4—41no—
37C2×C11×C4⋊C4—41no—
38C2×C11×C4⋊C4C2×C414yes—
39C2×C42×C11—41no—
40C2×C42×C11—41no—
41C2×C42×C11C2×C414yes—
42C2×C4⋊Dic11—24no—
43C2×C4⋊Dic11—28no—
44C2×C4⋊Dic11—42no—
45C2×C4⋊Dic11—42no—
46C2×C4⋊Dic11—42no—
47C2×C4×Dic11—24no—
48C2×C4×Dic11—42no—
49C2×C4×Dic11—42no—
50C11×#95C2×C4116yes—
51C11⋊#95—28no—
52C11⋊#95—28no—
53C11⋊#95—28no—
54C11⋊#95—28no—
55C11⋊#95—42no—
56C11⋊#95—42no—
57C11⋊#95—42no—
58#14501—111nomaximal, Sylow
59C24×C11C2411yescharacteristic
60C2×Q8×C11—44no—
61C2×Q8×C11—44no—
62C2×Q8×C11—44no—
63C2×Q8×C11—44no—
64C2×Q8×C11—44no—
65C2×Q8×C11—44no—
66C2×Q8×C11C4⋊C4116yescomplemented
67C22×C4×C11—22no—
68C22×C4×C11—22no—
69C22×C4×C11—22no—
70C22×C4×C11—22no—
71C22×C4×C11—21no—
72C22×C4×C11—21no—
73C22×C4×C11—21no—
74C22×C4×C11—21no—
75C22×C4×C11—24no—
76C22×C4×C11—24no—
77C22×C4×C11—24no—
78C22×C4×C11—22no—
79C22×C4×C11—22no—
80C22×C4×C11—22no—
81C22×C4×C11—22no—
82C22×C4×C11—24no—
83C22×C4×C11C22×C414yes—
84C22×C4×C11C22×C414yes—
85C22×C4×C11C22×C414yes—
86C22×C4×C11C22×C412yes—
87C22×C4×C11C2×D412yes—
88C22×C4×C11C2×D414yes—
89C22×C4×C11C2×Q814yes—
90C22×C4×C11C2×Q812yes—
91C22×Dic11—24no—
92C22×Dic11—24no—
93C22×Dic11—24no—
94C22×Dic11—24no—
95C22×Dic11—24no—
96C22×Dic11—24no—
97C22×Dic11—24no—
98C22×Dic11—24no—
99C22×Dic11—24no—
100C22×Dic11—24no—
101C22×Dic11—24no—
102C22×Dic11—24no—
103C22×Dic11—24no—
104C22×Dic11#4118yes—
105C11×C4⋊C4—44no—
106C42×C11—44no—
107C4⋊Dic11—48no—
108C4⋊Dic11—48no—
109C4×Dic11—48no—
110C2×#391—112no—
111C2×#391D1111yescharacteristic
112#1442—114no—
113#1441—112no—
114#1436—114no—
115#1434—112no—
116C23×C11C23×C411yescharacteristic
117C23×C11C22×D411yescharacteristic
118C23×C11C22×Q811yescharacteristic
119C23×C11C2×#4114yes—
120C23×C11H(5,𝔽2)11yescharacteristic
121C23×C11H(5,𝔽2)11yescharacteristic
122C23×C11H(5,𝔽2)11yescharacteristic
123C23×C11H(5,𝔽2)11yescharacteristic
124C23×C112−1+414yes—
125Q8×C11—416no—
126C2×C4×C11—22no—
127C2×C4×C11—22no—
128C2×C4×C11—24no—
129C2×C4×C11—24no—
130C2×C4×C11—24no—
131C2×C4×C11—24no—
132C2×C4×C11—22no—
133C2×C4×C11—22no—
134C2×C4×C11—24no—
135C2×C4×C11—22no—
136C2×C4×C11—24no—
137C2×C4×C11—22no—
138C2×C4×C11—24no—
139C2×C4×C11—28no—
140C2×C4×C11—24no—
141C2×C4×C11—24no—
142C2×C4×C11—24no—
143C2×C4×C11—44no—
144C2×C4×C11—44no—
145C2×C4×C11—42no—
146C2×C4×C11—42no—
147C2×C4×C11—44no—
148C2×C4×C11C2×C4⋊C418yes—
149C2×C4×C11C2×C4⋊C414yes—
150C2×Dic11—28no—
151C2×Dic11—28no—
152C2×Dic11—28no—
153C2×Dic11—44no—
154C2×Dic11—44no—
155C2×Dic11—44no—
156C2×Dic11—44no—
157C2×Dic11—44no—
158C2×Dic11—44no—
159C2×Dic11—44no—
160C2×Dic11—44no—
161C2×Dic11—44no—
162C2×Dic11—44no—
163C2×Dic11—44no—
164C23×Q8C2×D1111yescharacteristic
165C22×C4⋊C4—112no—
166C22×C4⋊C4—112no—
167C22×C4⋊C4C2×D1111yescharacteristic
168C22×C42—112no—
169C22×C42C2×D1111yescharacteristic
170#391—228no—
171#391Dic1118yescomplemented
172C2×#95—114no—
173C2×#95—114no—
174C2×#95—114no—
175C2×#95—114no—
176C2×#95—112no—
177C2×#95—112no—
178C2×#95—112no—
179C2×#95C2×D1114yes—
180C22×C11C22×C4⋊C411yescharacteristic
181C22×C11C4×#4112yes—
182C22×C11#51811yescharacteristic
183C22×C11#51811yescharacteristic
184C22×C11#51912yes—
185C22×C11#53411yescharacteristic
186C22×C11#53411yescharacteristic
187C22×C11#53914yes—
188C22×C11#54214yes—
189C22×C11#54311yescharacteristic
190C22×C11#54311yescharacteristic
191C22×C11#54814yes—
192C22×C11#55112yes—
193C22×C11#55512yes—
194C22×C11#55714yes—
195C22×C11#56312yes—
196C22×C11#56312yes—
197C4×C11—24no—
198C4×C11—28no—
199C4×C11—44no—
200C4×C11—44no—
201Dic11—48no—
202Dic11—84no—
203C22×Q8—41no—
204C22×Q8—41no—
205C22×Q8—41no—
206C22×Q8—41no—
207C22×Q8Dic2214yes—
208C22×Q8D4414yes—
209C22×Q8C2×Dic1114yes—
210C23×C4—114no—
211C23×C4—114no—
212C23×C4C22×D1112yes—
213C23×C4C22×D1112yes—
214C23×C4C22×D1112yes—
215C23×C4C22×D1111yescharacteristic
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—228no—
219C2×C4⋊C4—224no—
220C2×C4⋊C4—442no—
221C2×C4⋊C4—442no—
222C2×C4⋊C4—442no—
223C2×C4⋊C4C2×Dic1114yes—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—224no—
227C2×C42—442no—
228C2×C42—442no—
229C2×C42C2×Dic1114yes—
230#95—228no—
231#95—228no—
232#95—228no—
233#95—228no—
234#95—442no—
235#95—442no—
236#95—442no—
237#95C2×Dic11116yes—
238C2×C11#231511yescharacteristic
239C2×C11#233212yes—
240C2×C11#235112yes—
241C2×C11#237211yescharacteristic
242C2×C11#238111yescharacteristic
243C2×C11#272812yes—
244C2×C11#273112yes—
245C2×C11#279912yes—
246C2×C11#280912yes—
247C24C23×D1111yescharacteristic
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8C4⋊Dic11116yescomplemented
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—118no—
272C22×C4—224no—
273C22×C4—224no—
274C22×C4—224no—
275C22×C4—224no—
276C22×C4—224no—
277C22×C4—224no—
278C22×C4—224no—
279C22×C4—224no—
280C22×C4—224no—
281C22×C4—224no—
282C22×C4—224no—
283C22×C4—224no—
284C22×C4—224no—
285C22×C4C2×Dic2214yes—
286C22×C4C2×Dic2212yes—
287C22×C4C2×D4412yes—
288C22×C4C2×D4414yes—
289C22×C4C22×Dic1114yes—
290C22×C4C22×Dic1114yes—
291C22×C4C22×Dic1114yes—
292C22×C4C22×Dic1112yes—
293C4⋊C4—44no—
294C4⋊C4—448no—
295C4⋊C4—448no—
296C42—44no—
297C42—448no—
298C11#1450111yescharacteristic, complemented, Sylow
299C23C22×Dic2211yescharacteristic
300C23C22×D4411yescharacteristic
301C23#6703711yescharacteristic
302C23#6703711yescharacteristic
303C23#6703711yescharacteristic
304C23#6703711yescharacteristic
305C23#6704014yes—
306C23#41×D1114yes—
307C23C23×Dic1111yescharacteristic
308Q8—416no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—228no—
332C2×C4—228no—
333C2×C4—228no—
334C2×C4—444no—
335C2×C4—444no—
336C2×C4—444no—
337C2×C4—444no—
338C2×C4—444no—
339C2×C4—444no—
340C2×C4—444no—
341C2×C4—444no—
342C2×C4—444no—
343C2×C4—444no—
344C2×C4—444no—
345C2×C4C2×C4⋊Dic1114yes—
346C2×C4C2×C4⋊Dic1118yes—
347C22#1062301411yescharacteristic
348C22#1062301411yescharacteristic
349C22#1062302411yescharacteristic
350C22#1062302411yescharacteristic
351C22#1062310014yes—
352C22#1062310714yes—
353C22#1062316914yes—
354C22#1062317114yes—
355C22#1062320112yes—
356C22#1062321012yes—
357C22#1062326612yes—
358C22#1062326612yes—
359C22C22×C4⋊Dic1111yescharacteristic
360C22#1062332911yescharacteristic
361C22#1062332911yescharacteristic
362C22#1062334312yes—
363C22#41×Dic1112yes—
364C4—28no—
365C4—24no—
366C4—44no—
367C4—44no—
368C4—448no—
369C4—884no—
370C2#1307123011yescharacteristic
371C2#1307137412yes—
372C2#1307153712yes—
373C2#1307172811yescharacteristic
374C2#1307179711yescharacteristic
375C2#1307558512yes—
376C2#1307562512yes—
377C2#1307650012yes—
378C2#1307660212yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC2×C11×#391C2
socle, prime radical, cosocleC24×C11C24
hypercenterC2×#391D11
hypocenterC11#14501
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
13. C11×#391C4semidirect
66. C2×Q8×C11C4⋊C4semidirect
171. #391Dic11semidirect
254. C2×Q8C4⋊Dic11semidirect
298. C11#14501semidirect

Maximal subgroups

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

subgroupindex
C2×#106228432
#130668732
#130668692
#130668502
#130668432
C2×C11×#3912
#1450111