Finite Group Databasebeta

← #14914013

Subgroups — #14914013

Order 2816.

2350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13071573C216yesmaximal, complemented
2Q8×C4⋊Dic11C212yesmaximal, complemented
3#13071485C216yesmaximal
4Q8×C22⋊Dic11C212yesmaximal
5#13071267C216yesmaximal, complemented
6#13071023C213yesmaximal
7#13071014C213yesmaximal
8#13070981C212yesmaximal, complemented
9C22×C4×Q8×C11C211yesmaximal, characteristic
10C23×Q8×C11C2211yescharacteristic
11C2×C4×Q8×C11C4116yescomplemented
12C2×C4×Q8×C11C2211yescharacteristic
13C2×C4×Q8×C11C2213yes—
14C2×C4×Q8×C11C2211yescharacteristic
15C2×C4×Q8×C11C2213yes—
16C22×C11×C4⋊C4C2213yes—
17C22×C42×C11C2213yes—
18C2×Q8×Dic11C2214yes—
19C2×#66979C2216yes—
20#10622843C22112yes—
21#10622825C2216yes—
22#10622825C2211yescharacteristic
23#10622825C2213yes—
24#10622824C2216yes—
25C4×C22⋊Dic11C2216yes—
26#10622598C22112yes—
27#10622594C22124yes—
28C4⋊C4×Dic11C22112yes—
29#10622550C2216yes—
30#10622549C2216yes—
31#10622549C2216yes—
32#10622548C2216yes—
33C4×C4⋊Dic11C2216yes—
34C4×C4⋊Dic11C2216yes—
35C4×C4⋊Dic11C2216yes—
36#10622545C2216yes—
37C22×Q8×C11—28no—
38C22×Q8×C11C2×C418yes—
39C22×Q8×C11C2313yes—
40C22×Q8×C11C2311yescharacteristic
41C23×C4×C11C2313yes—
42C23×C4×C11C2313yes—
43C23×C4×C11C2311yescharacteristic
44C4×Q8×C11C2×C4124yes—
45C4×Q8×C11C2×C4124yes—
46C4×Q8×C11C2×C418yes—
47C4×Q8×C11C2×C418yes—
48C2×C11×C4⋊C4C2×C4124yes—
49C2×C11×C4⋊C4C2313yes—
50C2×C11×C4⋊C4C2316yes—
51C2×C11×C4⋊C4C2313yes—
52C2×C42×C11C2×C4124yes—
53C2×C42×C11C2313yes—
54C2×C42×C11C2313yes—
55C2×C42×C11C2313yes—
56C2×C42×C11C2313yes—
57C2×C22⋊Dic11C2316yes—
58C2×C22⋊Dic11C2312yes—
59Q8×Dic11—216no—
60#66979—412no—
61C2×C4⋊Dic11C2316yes—
62C2×C4⋊Dic11C2312yes—
63C2×C4⋊Dic11C23112yes—
64C2×C4⋊Dic11C23112yes—
65C2×C4×Dic11C23112yes—
66C2×C4×Dic11C23112yes—
67C11⋊#95C2312yes—
68C11⋊#95C2316yes—
69C11⋊#95C23112yes—
70C11⋊#95C23112yes—
71C11⋊#95C23112yes—
72C11⋊#95C23112yes—
73#34196—111nomaximal, Sylow
74C24×C11C2411yescharacteristic
75C2×Q8×C11—232no—
76C2×Q8×C11—212no—
77C2×Q8×C11C22×C4112yes—
78C2×Q8×C11C22×C414yes—
79C2×Q8×C11#4118yes—
80C22×C4×C11—26no—
81C22×C4×C11—26no—
82C22×C4×C11—212no—
83C22×C4×C11—24no—
84C22×C4×C11C22×C414yes—
85C22×C4×C11C22×C4112yes—
86C22×C4×C11C22×C4112yes—
87C22×C4×C11C2411yescharacteristic
88C22×C4×C11C2413yes—
89C22×C4×C11C2413yes—
90C22×C4×C11C2413yes—
91C22×C4×C11C2413yes—
92C22×C4×C11C2411yescharacteristic
93C22×Dic11C2414yes—
94C22×Dic11C24112yes—
95C11×C4⋊C4C22×C4124yes—
96C11×C4⋊C4C22×C4112yes—
97C11×C4⋊C4C22×C4112yes—
98C42×C11C22×C4112yes—
99C42×C11C22×C4112yes—
100C42×C11C22×C4112yes—
101C42×C11C22×C4112yes—
102C22⋊Dic11—212no—
103C22⋊Dic11—24no—
104C4⋊Dic11—224no—
105C4⋊Dic11—212no—
106C4⋊Dic11—24no—
107C4⋊Dic11—412no—
108C4×Dic11—224no—
109C4×Dic11—412no—
110C22×C4×Q8D1111yescharacteristic
111#2355—116no—
112Q8×C4⋊C4—112no—
113#2344—116no—
114Q8×C22⋊C4—112no—
115#2320—116no—
116#2285—113no—
117#2283—113no—
118#2276—112no—
119C23×C11—22no—
120C23×C11—22no—
121C23×C11C23×C412yes—
122C23×C11H(5,𝔽2)12yes—
123C23×C112−1+412yes—
124C23×C11C2511yescharacteristic
125Q8×C11—224no—
126Q8×C11#117116yescomplemented
127C2×C4×C11—224no—
128C2×C4×C11—26no—
129C2×C4×C11—23no—
130C2×C4×C11—23no—
131C2×C4×C11—22no—
132C2×C4×C11—23no—
133C2×C4×C11—23no—
134C2×C4×C11—224no—
135C2×C4×C11—28no—
136C2×C4×C11C23×C412yes—
137C2×C4×C11C23×C416yes—
138C2×C4×C11C23×C416yes—
139C2×C4×C11C23×C416yes—
140C2×C4×C11C23×C416yes—
141C2×C4×C11C23×C412yes—
142C2×C4×C11C2×#41112yes—
143C2×C4×C112−1+414yes—
144C2×Dic11—24no—
145C2×Dic11—212no—
146C2×Dic11—212no—
147C2×Dic11—212no—
148C2×Dic11—24no—
149C2×Dic11—24no—
150C23×Q8C2×D1111yescharacteristic
151C2×C4×Q8—114no—
152C2×C4×Q8Dic11116yescomplemented
153C2×C4×Q8C2×D1113yes—
154C2×C4×Q8C2×D1111yescharacteristic
155C2×C4×Q8C2×D1113yes—
156C2×C4×Q8C2×D1111yescharacteristic
157C2×#117—116no—
158C22×C4⋊C4C2×D1113yes—
159C22×C42C2×D1113yes—
160#391—1112no—
161#389—1112no—
162#387—1124no—
163#384—116no—
164#383—116no—
165#382—116no—
166#382—116no—
167#381—116no—
168#380—116no—
169#380—111no—
170#380—113no—
171C4×C4⋊C4—1112no—
172C4×C4⋊C4—116no—
173C4×C4⋊C4—116no—
174C4×C4⋊C4—116no—
175C4×C22⋊C4—116no—
176#376—116no—
177C22×C11—22no—
178C22×C11—22no—
179C22×C11—24no—
180C22×C11—24no—
181C22×C11#52014yes—
182C22×C11C24×C411yescharacteristic
183C22×C11C22×#4112yes—
184C22×C11C2×H(5,𝔽2)11yescharacteristic
185C22×C11C2×2−1+412yes—
186C22×C11C2×2−1+411yescharacteristic
187C4×C11—26no—
188C4×C11—22no—
189C4×C11—26no—
190C4×C11—26no—
191C4×C11C2×#117112yes—
192C4×C11#51914yes—
193Dic11—412no—
194Dic11—44no—
195C22×Q8—28no—
196C22×Q8C2×Dic1118yes—
197C22×Q8C22×D1113yes—
198C22×Q8C22×D1111yescharacteristic
199C23×C4C22×D1113yes—
200C23×C4C22×D1113yes—
201C23×C4C22×D1111yescharacteristic
202C4×Q8—2216no—
203C4×Q8C2×Dic11124yes—
204C4×Q8C2×Dic11124yes—
205C4×Q8C2×Dic1118yes—
206C4×Q8C2×Dic1118yes—
207#117—4412no—
208C2×C4⋊C4—1112no—
209C2×C4⋊C4—112no—
210C2×C4⋊C4—116no—
211C2×C4⋊C4—1112no—
212C2×C4⋊C4C2×Dic11124yes—
213C2×C4⋊C4C22×D1113yes—
214C2×C4⋊C4C22×D1116yes—
215C2×C4⋊C4C22×D1113yes—
216C2×C22⋊C4—116no—
217C2×C22⋊C4—112no—
218C2×C42—1112no—
219C2×C42—1112no—
220C2×C42C2×Dic11124yes—
221C2×C42C22×D1113yes—
222C2×C42C22×D1113yes—
223C2×C42C22×D1113yes—
224C2×C42C22×D1113yes—
225#95—112no—
226#95—116no—
227#95—1112no—
228#95—1112no—
229#95—1112no—
230#95—1112no—
231C2×C11—24no—
232C2×C11C22×#11711yescharacteristic
233C2×C11C2×#51911yescharacteristic
234C2×C11C2×#52011yescharacteristic
235C2×C11#352212yes—
236C2×C11#352312yes—
237C24C23×D1111yescharacteristic
238C2×Q8—232no—
239C2×Q8—212no—
240C2×Q8#433918yes—
241C2×Q8C22×Dic11112yes—
242C2×Q8C22×Dic1114yes—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—212no—
246C22×C4—24no—
247C22×C4—114no—
248C22×C4—1112no—
249C22×C4C22×Dic1114yes—
250C22×C4C22×Dic11112yes—
251C22×C4C22×Dic11112yes—
252C22×C4C23×D1111yescharacteristic
253C22×C4C23×D1113yes—
254C22×C4C23×D1113yes—
255C22×C4C23×D1113yes—
256C22×C4C23×D1113yes—
257C22×C4C23×D1111yescharacteristic
258C4⋊C4—2224no—
259C4⋊C4—224no—
260C4⋊C4—2212no—
261C4⋊C4—4412no—
262C4⋊C4C22×Dic11124yes—
263C4⋊C4C22×Dic11112yes—
264C4⋊C4C22×Dic11112yes—
265C22⋊C4—2212no—
266C22⋊C4—224no—
267C42—2224no—
268C42—4412no—
269C42C22×Dic11112yes—
270C42C22×Dic11112yes—
271C42C22×Dic11112yes—
272C42C22×Dic11112yes—
273C11#3419611yescharacteristic, complemented, Sylow
274C23—22no—
275C23—22no—
276C23#6704212yes—
277C23#6704312yes—
278C23C23×Dic1112yes—
279C23C24×D1111yescharacteristic
280Q8—224no—
281Q8#66979116yescomplemented
282C2×C4—224no—
283C2×C4—26no—
284C2×C4—22no—
285C2×C4—23no—
286C2×C4—23no—
287C2×C4—23no—
288C2×C4—23no—
289C2×C4—224no—
290C2×C4—28no—
291C2×C4—224no—
292C2×C4—2212no—
293C2×C4—2212no—
294C2×C4—224no—
295C2×C4—2212no—
296C2×C4—224no—
297C2×C4C2×#4339112yes—
298C2×C4#6704014yes—
299C2×C4C23×Dic1112yes—
300C2×C4C23×Dic1116yes—
301C2×C4C23×Dic1116yes—
302C2×C4C23×Dic1116yes—
303C2×C4C23×Dic1116yes—
304C2×C4C23×Dic1112yes—
305C22—24no—
306C22—24no—
307C22—22no—
308C22—22no—
309C22#1062335814yes—
310C22C22×#433912yes—
311C22C2×#6704012yes—
312C22C2×#6704211yescharacteristic
313C22C2×#6704311yescharacteristic
314C22C24×Dic1111yescharacteristic
315C4—26no—
316C4—22no—
317C4—26no—
318C4—26no—
319C4—4412no—
320C4—444no—
321C4C2×#66979112yes—
322C4#1062334314yes—
323C2—24no—
324C2C22×#6697911yescharacteristic
325C2C2×#1062334311yescharacteristic
326C2C2×#1062335811yescharacteristic
327C2#1308270112yes—
328C2#1308271212yes—
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC22×C4×Q8×C11C2
socle, cosocleC23×C11C25
prime radicalC24×C11C24
hypercenterC22×C4×Q8D11
hypocenterC11#34196
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #13071573C2semidirect
2. Q8×C4⋊Dic11C2semidirect
5. #13071267C2semidirect
8. #13070981C2semidirect
11. C2×C4×Q8×C11C4semidirect
126. Q8×C11#117semidirect
152. C2×C4×Q8Dic11semidirect
273. C11#34196semidirect
281. Q8#66979semidirect

Maximal subgroups

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

subgroupindex
#130715732
Q8×C4⋊Dic112
#130714852
Q8×C22⋊Dic112
#130712672
#130710232
#130710142
#130709812
C22×C4×Q8×C112
#3419611