Finite Group Databasebeta

← #13072068

Subgroups — #13072068

Order 1408.

648 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C11×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#66999C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D11C211yesmaximal, characteristic
4C22×C4⋊Dic11C211yesmaximal, characteristic, complemented
5#10622843C211yesmaximal, characteristic, complemented
6#10622607C212yesmaximal, complemented
7#10622600C211yesmaximal, characteristic, complemented
8#10622597C212yesmaximal, complemented
9#10622586C211yesmaximal, characteristic, complemented
10#10622571C212yesmaximal
11#10622555C212yesmaximal, complemented
12C22×Q8×C11C2211yescharacteristic
13C23×C4×C11C2211yescharacteristic
14C22×C22⋊D11C2211yescharacteristic
15C23×Dic11C2212yes—
16C22×C4×D11C2211yescharacteristic
17C11×C22⋊Q8—24no—
18C2×C11×C4⋊C4C2211yescharacteristic
19C2×C11×C4⋊C4C2212yes—
20C2×C11×C22⋊C4C2212yes—
21C2×C22⋊Dic11C2211yescharacteristic
22#66999—24no—
23C4×C22⋊D11—24no—
24C2×(C2×D11)⋊C4C2211yescharacteristic
25C2×(C2×D11)⋊C4C2212yes—
26C2×(C2×D11)⋊C4C2212yes—
27C2×C4⋊Dic11—24no—
28C2×C4⋊Dic11C2211yescharacteristic
29C2×C4⋊Dic11C2212yes—
30C2×C4⋊Dic11C2211yescharacteristic
31C2×Dic11⋊C4C2211yescharacteristic
32C2×Dic11⋊C4C2212yes—
33C2×Dic11⋊C4C2212yes—
34C2×C4×Dic11C2211yescharacteristic
35C11⋊#95C2212yes—
36C11⋊#95C2212yes—
37C11⋊#95C2212yes—
38C11⋊#95C2212yes—
39C24×C11C2311yescharacteristic
40C23×D11C2311yescharacteristic
41C2×Q8×C11—22no—
42C2×Q8×C11—22no—
43C2×Q8×C11—22no—
44C22×C4×C11—22no—
45C22×C4×C11—22no—
46C22×C4×C11D414yes—
47C22×C4×C11C2312yes—
48C22×C4×C11C2312yes—
49C22×C4×C11C2311yescharacteristic
50C22×C4×C11C2311yescharacteristic
51C2×C22⋊D11—22no—
52C2×C22⋊D11—22no—
53C2×C22⋊D11Q814yescomplemented
54C22×Dic11—24no—
55C22×Dic11—24no—
56C22×Dic11—42no—
57C22×Dic11C2312yes—
58C22×Dic11C2311yescharacteristic
59C22×Dic11C2311yescharacteristic
60C22×Dic11C2312yes—
61C22×Dic11C2311yescharacteristic
62C2×C4×D11—22no—
63C2×C4×D11—22no—
64C2×C4×D11—22no—
65C11×C4⋊C4—22no—
66C11×C4⋊C4—24no—
67C11×C22⋊C4—24no—
68C22⋊Dic11—22no—
69(C2×D11)⋊C4—24no—
70(C2×D11)⋊C4—22no—
71(C2×D11)⋊C4—42no—
72C4⋊Dic11—24no—
73C4⋊Dic11—22no—
74C4⋊Dic11D414yescomplemented
75Dic11⋊C4—24no—
76Dic11⋊C4—22no—
77Dic11⋊C4—42no—
78C4×Dic11—22no—
79#2417—111nomaximal, Sylow
80C23×C11—21no—
81C23×C11—21no—
82C23×C11—21no—
83C23×C11—21no—
84C23×C11C2×D412yes—
85C23×C11C2×Q812yes—
86C23×C11#4112yes—
87C23×C11C2411yescharacteristic
88C22×D11—21no—
89C22×D11—21no—
90C22×D11—21no—
91C22×D11—21no—
92C22×D11—21no—
93C22×D11—21no—
94C22×D11C2×Q812yes—
95Q8×C11—44no—
96C2×C4×C11—21no—
97C2×C4×C11—21no—
98C2×C4×C11—28no—
99C2×C4×C11—22no—
100C2×C4×C11—22no—
101C2×C4×C11—22no—
102C2×C4×C11—22no—
103C2×C4×C11—22no—
104C2×C4×C11—22no—
105C2×C4×C11C2×D412yes—
106C2×C4×C11C2×D412yes—
107C2×C4×C11C2×D412yes—
108C2×C4×C11#4112yes—
109C22⋊D11—28no—
110C2×Dic11—21no—
111C2×Dic11—22no—
112C2×Dic11—21no—
113C2×Dic11—22no—
114C2×Dic11—21no—
115C2×Dic11—21no—
116C2×Dic11—21no—
117C2×Dic11—22no—
118C2×Dic11—21no—
119C2×Dic11—22no—
120C2×Dic11—21no—
121C2×Dic11—22no—
122C2×Dic11—21no—
123C2×Dic11—48no—
124C2×Dic11C2×D414yes—
125C2×Dic11C2×Q812yes—
126C4×D11—44no—
127C2×C22⋊Q8—111no—
128C2×C22⋊Q8D1111yescharacteristic, complemented
129C2×C4×D4—111no—
130C22×C4⋊C4—111no—
131#398—112no—
132#396—112no—
133#391—111no—
134#389—111no—
135#386—112no—
136#380—111no—
137#380—112no—
138C22×C11—22no—
139C22×C11—22no—
140C22×C11—22no—
141C22×C11—22no—
142C22×C11—22no—
143C22×C11—22no—
144C22×C11C22⋊Q814yescomplemented
145C22×C11C22×D411yescharacteristic
146C22×C11C22×D411yescharacteristic
147C22×C11C22×Q811yescharacteristic
148C22×C11C2×#4111yescharacteristic
149C22×C11C2×#4111yescharacteristic
150C22×C11H(5,𝔽2)11yescharacteristic
151C22×C112−1+411yescharacteristic
152C2×D11—22no—
153C2×D11—22no—
154C2×D11—22no—
155C2×D11—41no—
156C2×D11—41no—
157C2×D11—41no—
158C2×D11—41no—
159C4×C11—22no—
160C4×C11—42no—
161C4×C11—42no—
162C4×C11#12114yes—
163Dic11—22no—
164Dic11—24no—
165Dic11—41no—
166Dic11—41no—
167Dic11—42no—
168C22×Q8C2×D1111yescharacteristic
169C22×D4—111no—
170C23×C4—112no—
171C23×C4—111no—
172C23×C4C2×D1111yescharacteristic
173C22⋊Q8—24no—
174C22⋊Q8—224no—
175C4×D4—224no—
176C2×C4⋊C4—111no—
177C2×C4⋊C4—111no—
178C2×C4⋊C4—112no—
179C2×C4⋊C4—111no—
180C2×C4⋊C4—112no—
181C2×C4⋊C4—112no—
182C2×C4⋊C4—224no—
183C2×C4⋊C4C2×D1111yescharacteristic
184C2×C4⋊C4C2×D1112yes—
185C2×C22⋊C4—111no—
186C2×C22⋊C4—111no—
187C2×C22⋊C4—112no—
188C2×C22⋊C4—112no—
189C2×C22⋊C4C2×D1112yes—
190C2×C42—111no—
191#95—112no—
192#95—112no—
193#95—112no—
194#95—112no—
195C2×C11—24no—
196C2×C11C2×#12111yescharacteristic
197C2×C11C2×C22⋊Q811yescharacteristic
198C2×C11#53711yescharacteristic
199C2×C11#54611yescharacteristic
200C2×C11D4×Q811yescharacteristic
201C2×C11#55211yescharacteristic
202C2×C11#55411yescharacteristic
203D11—42no—
204C24—111no—
205C24C22×D1111yescharacteristic
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×D4—114no—
210C2×D4—222no—
211C2×D4—222no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—111no—
215C22×C4—112no—
216C22×C4—112no—
217C22×C4—111no—
218C22×C4—111no—
219C22×C4—224no—
220C22×C4—224no—
221C22×C4—222no—
222C22×C4—222no—
223C22×C4—222no—
224C22×C4—442no—
225C22×C4C22⋊D1114yes—
226C22×C4C22×D1112yes—
227C22×C4C22×D1112yes—
228C22×C4C22×D1111yescharacteristic
229C22×C4C22×D1111yescharacteristic
230C4⋊C4—22no—
231C4⋊C4—24no—
232C4⋊C4—114no—
233C4⋊C4—224no—
234C4⋊C4—224no—
235C4⋊C4—222no—
236C4⋊C4—222no—
237C4⋊C4—442no—
238C22⋊C4—24no—
239C22⋊C4—224no—
240C22⋊C4—222no—
241C22⋊C4—222no—
242C22⋊C4—442no—
243C42—222no—
244C11#241711yescharacteristic, complemented, Sylow
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—112no—
250C23—221no—
251C23—221no—
252C23—221no—
253C23—221no—
254C23—221no—
255C23—221no—
256C23Q8×D1112yes—
257C23#434312yes—
258C23C2×C22⋊D1112yes—
259C23C23×D1111yescharacteristic
260Q8—44no—
261D4—228no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—28no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—114no—
272C2×C4—112no—
273C2×C4—222no—
274C2×C4—221no—
275C2×C4—222no—
276C2×C4—221no—
277C2×C4—221no—
278C2×C4—222no—
279C2×C4—221no—
280C2×C4—222no—
281C2×C4—222no—
282C2×C4—221no—
283C2×C4—221no—
284C2×C4—221no—
285C2×C4—221no—
286C2×C4—448no—
287C2×C4—444no—
288C2×C4D4×D1112yes—
289C2×C4#434112yes—
290C2×C4C2×C22⋊D1112yes—
291C2×C4C2×C22⋊D1112yes—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—222no—
299C22—222no—
300C22—222no—
301C22—441no—
302C22—441no—
303C22—441no—
304C22—441no—
305C22#6699914yescomplemented
306C22C2×D4×D1111yescharacteristic
307C22C2×#434111yescharacteristic
308C22C2×Q8×D1111yescharacteristic
309C22C2×#434311yescharacteristic
310C22#6704211yescharacteristic
311C22#6704311yescharacteristic
312C22C22×C22⋊D1111yescharacteristic
313C4—22no—
314C4—42no—
315C4—42no—
316C4—222no—
317C4—224no—
318C4—441no—
319C4—441no—
320C4—442no—
321C4#6699214yes—
322C2—24no—
323C2—442no—
324C2#1062316111yescharacteristic
325C2#1062316311yescharacteristic
326C2#1062316611yescharacteristic
327C2#1062317311yescharacteristic
328C2C2×#6699211yescharacteristic
329C2C2×#6699911yescharacteristic
330C2#1062336411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC2×C11×C22⋊Q8C2
prime radicalC22×C22⋊D11C22
hypercenterC2×C22⋊Q8D11
hypocenterC11#2417
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C11×C22⋊Q8C2semidirect
2. C2×#66999C2semidirect
4. C22×C4⋊Dic11C2semidirect
5. #10622843C2semidirect
6. #10622607C2semidirect
7. #10622600C2semidirect
8. #10622597C2semidirect
9. #10622586C2semidirect
11. #10622555C2semidirect
53. C2×C22⋊D11Q8semidirect
74. C4⋊Dic11D4semidirect
128. C2×C22⋊Q8D11semidirect
144. C22×C11C22⋊Q8semidirect
244. C11#2417semidirect
305. C22#66999semidirect

Maximal subgroups

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

subgroupindex
C2×C11×C22⋊Q82
C2×#669992
C2×C4×C22⋊D112
C22×C4⋊Dic112
#106228432
#106226072
#106226002
#106225972
#106225862
#106225712
#106225552
#241711