Finite Group Databasebeta

← #14770522

Subgroups — #14770522

Order 2816.

1412 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊Dic11C212yesmaximal, complemented
2#13069453C216yesmaximal, complemented
3#13068604C213yesmaximal, complemented
4C4×#66898C213yesmaximal, complemented
5C2×C11×#548C211yesmaximal, characteristic
6C22×C11×#41C411yescharacteristic, complemented
7C23×Q8×C11C2211yescharacteristic
8C11×#548—22no—
9C11×#548—26no—
10C2×C11×#124C413yescomplemented
11C2×C11×C22⋊Q8C413yescomplemented
12C2×C11×#121C2213yes—
13C2×C4×Q8×C11C411yescharacteristic, complemented
14C2×C4×D4×C11C2213yes—
15C2×Q8×Dic11—22no—
16#10622843—26no—
17C2×#66898C2216yes—
18C2×#66898C2216yes—
19#10622825C2216yes—
20C4×C22⋊Dic11C2216yes—
21#10622164C2213yes—
22#10622164C2211yescharacteristic
23C2×C11×#41—23no—
24C2×C11×#41—21no—
25C2×C11×#41—44no—
26C22×Q8×C11—44no—
27C22×Q8×C11C2×C411yescharacteristic
28C22×Q8×C11C2×C413yes—
29C22×Q8×C11D412yes—
30C22×Q8×C11D416yes—
31C22×D4×C11C2×C413yes—
32C22×D4×C11C2313yes—
33C23×C4×C11C2×C413yes—
34C23×C4×C11C2313yes—
35C11×#124—26no—
36C11×#124—26no—
37C11×C22⋊Q8—26no—
38C11×C22⋊Q8—26no—
39C11×#121—26no—
40C11×#121—26no—
41C4×Q8×C11—23no—
42C4×Q8×C11—21no—
43C4×D4×C11—26no—
44C4×D4×C11—26no—
45C2×C11×C4⋊C4C2×C413yes—
46C2×C11×C4⋊C4C2×C413yes—
47C2×C11×C22⋊C4C2×C413yes—
48C2×C11×C22⋊C4C2×C416yes—
49C2×C11×C22⋊C4C2311yescharacteristic
50C2×C42×C11C2×C413yes—
51C2×C22⋊Dic11C2316yes—
52C2×C22⋊Dic11C2312yes—
53Q8×Dic11—48no—
54C2×C4⋊Dic11—26no—
55C2×C4×Dic11—26no—
56#66898—212no—
57#66898—46no—
58C11⋊#95—26no—
59C11⋊#95—26no—
60#22534—111nomaximal, Sylow
61C24×C11C22×C413yes—
62C24×C11C2411yescharacteristic
63C11×#41—412no—
64C11×#41—44no—
65C2×Q8×C11—28no—
66C2×Q8×C11—24no—
67C2×Q8×C11—46no—
68C2×Q8×C11—412no—
69C2×Q8×C11C22⋊C413yes—
70C2×Q8×C11C22⋊C416yes—
71C2×Q8×C11C22⋊C416yes—
72C2×Q8×C11C22⋊C411yescharacteristic
73C2×D4×C11—26no—
74C2×D4×C11—26no—
75C2×D4×C11—23no—
76C2×D4×C11—23no—
77C2×D4×C11—23no—
78C2×D4×C11—23no—
79C2×D4×C11—46no—
80C22×C4×C11—23no—
81C22×C4×C11—23no—
82C22×C4×C11—46no—
83C22×C4×C11—46no—
84C22×C4×C11C22×C413yes—
85C22×C4×C11C22×C413yes—
86C22×C4×C11C22×C411yescharacteristic
87C22×C4×C11C22×C411yescharacteristic
88C22×C4×C11C22×C413yes—
89C22×C4×C11C2×D416yes—
90C22×C4×C11C2×D416yes—
91C22×Dic11—22no—
92C22×Dic11—26no—
93C11×C4⋊C4—26no—
94C11×C4⋊C4—23no—
95C11×C4⋊C4—23no—
96C11×C22⋊C4—22no—
97C11×C22⋊C4—26no—
98C11×C22⋊C4—212no—
99C42×C11—23no—
100C42×C11—23no—
101C22⋊Dic11—26no—
102C22⋊Dic11—26no—
103C22⋊Dic11—22no—
104C22⋊Dic11#4114yescomplemented
105C4⋊Dic11—412no—
106C4×Dic11—412no—
107C2×#548D1111yescharacteristic
108Q8×C22⋊C4—112no—
109#1964—116no—
110#1799—113no—
111C4×#99—113no—
112C23×C11—23no—
113C23×C11—23no—
114C23×C11—23no—
115C23×C11—41no—
116C23×C11—41no—
117C23×C11—43no—
118C23×C11—43no—
119C23×C11C23×C411yescharacteristic
120C23×C11C22×D412yes—
121C23×C11C2×#4112yes—
122C23×C11H(5,𝔽2)11yescharacteristic
123C23×C112−1+411yescharacteristic
124Q8×C11—212no—
125Q8×C11—26no—
126Q8×C11—46no—
127Q8×C11#9914yescomplemented
128D4×C11—46no—
129D4×C11—46no—
130D4×C11—46no—
131D4×C11—46no—
132C2×C4×C11—212no—
133C2×C4×C11—26no—
134C2×C4×C11—21no—
135C2×C4×C11—23no—
136C2×C4×C11—21no—
137C2×C4×C11—23no—
138C2×C4×C11—23no—
139C2×C4×C11—43no—
140C2×C4×C11—46no—
141C2×C4×C11—46no—
142C2×C4×C11—46no—
143C2×C4×C11—41no—
144C2×C4×C11—41no—
145C2×C4×C11C2×C22⋊C413yes—
146C2×C4×C11C2×C22⋊C416yes—
147C2×C4×C11C2×C22⋊C413yes—
148C2×Dic11—24no—
149C2×Dic11—42no—
150C2×Dic11—42no—
151C2×Dic11—46no—
152C2×Dic11—46no—
153C2×Dic11—46no—
154C22×#41Dic1111yescharacteristic, complemented
155C23×Q8C2×D1111yescharacteristic
156#548—22no—
157#548—26no—
158C2×#124Dic1113yescomplemented
159C2×C22⋊Q8Dic1113yescomplemented
160C2×#121C2×D1113yes—
161C2×C4×Q8—222no—
162C2×C4×Q8Dic1111yescharacteristic, complemented
163C2×C4×D4C2×D1113yes—
164C2×#99—116no—
165C2×#99—116no—
166#391—226no—
167#380—116no—
168C4×C22⋊C4—116no—
169#342—113no—
170#342—111no—
171C22×C11—21no—
172C22×C11—22no—
173C22×C11—21no—
174C22×C11—22no—
175C22×C11—42no—
176C22×C11—41no—
177C22×C11—41no—
178C22×C11—43no—
179C22×C11—43no—
180C22×C11—43no—
181C22×C11—83no—
182C22×C11—83no—
183C22×C11C22×C22⋊C411yescharacteristic
184C22×C11C4×#4111yescharacteristic
185C22×C11#52011yescharacteristic
186C22×C11#54812yes—
187C22×C11#55012yes—
188C4×C11—23no—
189C4×C11—26no—
190C4×C11—43no—
191C4×C11—41no—
192C4×C11—41no—
193C4×C11C2×#9916yes—
194Dic11—44no—
195Dic11—86no—
196C2×#41—23no—
197C2×#41—21no—
198C2×#41—44no—
199C22×Q8—44no—
200C22×Q8C2×Dic1111yescharacteristic
201C22×Q8C2×Dic1113yes—
202C22×Q8C22⋊D1112yes—
203C22×Q8C22⋊D1116yes—
204C22×D4C2×Dic1113yes—
205C22×D4C22×D1113yes—
206C23×C4C2×Dic1113yes—
207C23×C4C22×D1113yes—
208#124—26no—
209#124—26no—
210C22⋊Q8—26no—
211C22⋊Q8—26no—
212#121—26no—
213#121—26no—
214C4×Q8—23no—
215C4×Q8—21no—
216C4×Q8—448no—
217C4×D4—26no—
218C4×D4—26no—
219C2×C4⋊C4—226no—
220C2×C4⋊C4C2×Dic1113yes—
221C2×C4⋊C4C2×Dic1113yes—
222C2×C22⋊C4—112no—
223C2×C22⋊C4—116no—
224C2×C22⋊C4C2×Dic1113yes—
225C2×C22⋊C4C2×Dic1116yes—
226C2×C22⋊C4C22×D1111yescharacteristic
227C2×C42—226no—
228C2×C42C2×Dic1113yes—
229#99—2212no—
230#99—446no—
231#95—226no—
232#95—226no—
233C2×C11—22no—
234C2×C11—22no—
235C2×C11—21no—
236C2×C11—21no—
237C2×C11—83no—
238C2×C11#234111yescharacteristic
239C2×C11C22×#9911yescharacteristic
240C2×C11#288411yescharacteristic
241C24C22×Dic1113yes—
242C24C23×D1111yescharacteristic
243#41—412no—
244#41—44no—
245C2×Q8—28no—
246C2×Q8—24no—
247C2×Q8—46no—
248C2×Q8—412no—
249C2×Q8C22⋊Dic1113yes—
250C2×Q8C22⋊Dic1116yes—
251C2×Q8C22⋊Dic1116yes—
252C2×Q8C22⋊Dic1111yescharacteristic
253C2×D4—26no—
254C2×D4—23no—
255C2×D4—23no—
256C2×D4—26no—
257C2×D4—23no—
258C2×D4—23no—
259C2×D4—46no—
260C22×C4—23no—
261C22×C4—23no—
262C22×C4—46no—
263C22×C4—46no—
264C22×C4—222no—
265C22×C4—226no—
266C22×C4C22×Dic1113yes—
267C22×C4C22×Dic1113yes—
268C22×C4C22×Dic1111yescharacteristic
269C22×C4C22×Dic1111yescharacteristic
270C22×C4C22×Dic1113yes—
271C22×C4C2×C22⋊D1116yes—
272C22×C4C2×C22⋊D1116yes—
273C4⋊C4—26no—
274C4⋊C4—23no—
275C4⋊C4—23no—
276C4⋊C4—4412no—
277C22⋊C4—22no—
278C22⋊C4—26no—
279C22⋊C4—212no—
280C22⋊C4—114no—
281C22⋊C4—222no—
282C22⋊C4—226no—
283C22⋊C4—226no—
284C42—23no—
285C42—23no—
286C42—4412no—
287C11#2253411yescharacteristic, complemented, Sylow
288C23—23no—
289C23—23no—
290C23—23no—
291C23—41no—
292C23—41no—
293C23—43no—
294C23—43no—
295C23#41×D1112yes—
296C23#6704211yescharacteristic
297C23#6704311yescharacteristic
298C23C23×Dic1111yescharacteristic
299C23C22×C22⋊D1112yes—
300Q8—212no—
301Q8—26no—
302Q8—46no—
303Q8#6689814yescomplemented
304D4—46no—
305D4—46no—
306D4—46no—
307D4—46no—
308C2×C4—212no—
309C2×C4—26no—
310C2×C4—23no—
311C2×C4—23no—
312C2×C4—21no—
313C2×C4—23no—
314C2×C4—21no—
315C2×C4—43no—
316C2×C4—46no—
317C2×C4—46no—
318C2×C4—46no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—224no—
322C2×C4—442no—
323C2×C4—442no—
324C2×C4—446no—
325C2×C4—446no—
326C2×C4—446no—
327C2×C4C2×C22⋊Dic1113yes—
328C2×C4C2×C22⋊Dic1116yes—
329C2×C4C2×C22⋊Dic1113yes—
330C22—21no—
331C22—22no—
332C22—21no—
333C22—22no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—43no—
338C22—43no—
339C22—43no—
340C22—83no—
341C22—83no—
342C22#41×Dic1111yescharacteristic
343C22#1062335811yescharacteristic
344C22#1062336212yes—
345C22#1062336512yes—
346C22C22×C22⋊Dic1111yescharacteristic
347C4—23no—
348C4—26no—
349C4—41no—
350C4—41no—
351C4—43no—
352C4—444no—
353C4—886no—
354C4C2×#6689816yes—
355C2—22no—
356C2—22no—
357C2—21no—
358C2—21no—
359C2—83no—
360C2#1307145711yescharacteristic
361C2C22×#6689811yescharacteristic
362C2#1307708011yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic11
commutatorC23×C11C23×C4
FrattiniC24C23×D11
Fitting, prime radicalC2×C11×#548C2
socleC22×C11C22×C22⋊C4
cosocleC24×C11C24
hypercenterC2×#548D11
hypocenterC11#22534
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×Q8 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊Dic11C2semidirect
2. #13069453C2semidirect
3. #13068604C2semidirect
4. C4×#66898C2semidirect
6. C22×C11×#41C4semidirect
10. C2×C11×#124C4semidirect
11. C2×C11×C22⋊Q8C4semidirect
13. C2×C4×Q8×C11C4semidirect
104. C22⋊Dic11#41semidirect
127. Q8×C11#99semidirect
154. C22×#41Dic11semidirect
158. C2×#124Dic11semidirect
159. C2×C22⋊Q8Dic11semidirect
162. C2×C4×Q8Dic11semidirect
287. C11#22534semidirect
303. Q8#66898semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊Dic112
#130694532
#130686042
C4×#668982
C2×C11×#5482
#2253411