Finite Group Databasebeta

← #13072178

Subgroups — #13072178

Order 1408.

632 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C11×#121C211yesmaximal, characteristic
2C2×#66989C211yesmaximal, characteristic, complemented
3C2×#66977C211yesmaximal, characteristic, complemented
4C2×#66940C211yesmaximal, characteristic, complemented
5#10622835C212yesmaximal, complemented
6#10622834C212yesmaximal, complemented
7#10622834C211yesmaximal, characteristic, complemented
8C4×C22⋊Dic11C211yesmaximal, characteristic, complemented
9#10622598C211yesmaximal, characteristic, complemented
10#10622561C212yesmaximal, complemented
11#10622557C212yesmaximal, complemented
12C22×D4×C11C2211yescharacteristic
13C22×D4×C11C2212yes—
14C23×C4×C11C2211yescharacteristic
15C22×Dic22C2211yescharacteristic
16C11×#121—24no—
17C2×C11×C4⋊C4C2211yescharacteristic
18C2×C11×C22⋊C4C2212yes—
19C2×C22⋊Dic11C2212yes—
20C2×C22⋊Dic11C2212yes—
21C2×C22⋊Dic11C2212yes—
22C2×C22⋊Dic11C2212yes—
23C2×C22⋊Dic11C2211yescharacteristic
24C2×C22⋊Dic11C2211yescharacteristic
25C2×C22⋊Dic11C2212yes—
26#66989—24no—
27C2×C4⋊Dic11C2211yescharacteristic
28C2×C4⋊Dic11C2212yes—
29#66977—24no—
30C2×Dic11⋊C4C2212yes—
31C2×Dic11⋊C4C2212yes—
32C2×C4×Dic11C2212yes—
33C2×C4×Dic11C2211yescharacteristic
34C2×C4×Dic11C2211yescharacteristic
35#66940—24no—
36C11⋊#95C2211yescharacteristic
37C11⋊#95C2212yes—
38C11⋊#95C2211yescharacteristic
39C24×C11C2312yes—
40C24×C11C2311yescharacteristic
41C2×D4×C11—24no—
42C2×D4×C11—22no—
43C2×D4×C11—24no—
44C2×D4×C11—22no—
45C2×D4×C11—22no—
46C2×D4×C11—42no—
47C22×C4×C11—22no—
48C22×C4×C11—22no—
49C22×C4×C11—22no—
50C22×C4×C11C2311yescharacteristic
51C22×C4×C11C2312yes—
52C22×C4×C11C2311yescharacteristic
53C22×Dic11C2311yescharacteristic
54C22×Dic11C2311yescharacteristic
55C22×Dic11C2311yescharacteristic
56C22×Dic11C2312yes—
57C22×Dic11C2312yes—
58C22×Dic11C2311yescharacteristic
59C2×Dic22—22no—
60C2×Dic22—22no—
61C2×Dic22D414yescomplemented
62C11×C4⋊C4D414yes—
63C11×C22⋊C4—24no—
64C22⋊Dic11—24no—
65C22⋊Dic11—24no—
66C22⋊Dic11—24no—
67C22⋊Dic11—22no—
68C22⋊Dic11—24no—
69C22⋊Dic11—22no—
70C22⋊Dic11—42no—
71C4⋊Dic11—22no—
72C4⋊Dic11—42no—
73Dic11⋊C4—24no—
74Dic11⋊C4—24no—
75C4×Dic11—24no—
76C4×Dic11—22no—
77C4×Dic11—22no—
78#2426—111nomaximal, Sylow
79C23×C11—21no—
80C23×C11—21no—
81C23×C11—21no—
82C23×C11—21no—
83C23×C11—21no—
84C23×C11—21no—
85C23×C11—22no—
86C23×C11—22no—
87C23×C11—22no—
88C23×C11—22no—
89C23×C11—22no—
90C23×C11—22no—
91C23×C11—22no—
92C23×C11#4112yes—
93C23×C11C2411yescharacteristic
94D4×C11—48no—
95D4×C11—44no—
96C2×C4×C11—22no—
97C2×C4×C11—22no—
98C2×C4×C11—21no—
99C2×C4×C11—21no—
100C2×C4×C11—21no—
101C2×C4×C11—44no—
102C2×C4×C11C2×D412yes—
103C2×C4×C11C2×D414yes—
104C2×C4×C11C2×D412yes—
105C2×C4×C11#4112yes—
106C2×Dic11—21no—
107C2×Dic11—22no—
108C2×Dic11—22no—
109C2×Dic11—21no—
110C2×Dic11—21no—
111C2×Dic11—21no—
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—21no—
122C2×Dic11C2×D414yes—
123C2×Dic11#4112yes—
124Dic22—28no—
125C2×#124—111no—
126C2×C22⋊Q8—111no—
127C2×#121D1111yescharacteristic
128C2×C4×Q8—111no—
129#396—112no—
130#394—112no—
131#390—111no—
132#390—112no—
133#389—111no—
134#388—112no—
135C4×C22⋊C4—111no—
136C22×C11—22no—
137C22×C11—22no—
138C22×C11—22no—
139C22×C11—41no—
140C22×C11—41no—
141C22×C11—41no—
142C22×C11—41no—
143C22×C11—42no—
144C22×C11—42no—
145C22×C11—42no—
146C22×C11—42no—
147C22×C11—42no—
148C22×C11—42no—
149C22×C11—42no—
150C22×C11C22×D411yescharacteristic
151C22×C11C22×D411yescharacteristic
152C22×C11C2×#4111yescharacteristic
153C22×C11C2×#4111yescharacteristic
154C22×C11C2×#4111yescharacteristic
155C22×C11H(5,𝔽2)11yescharacteristic
156C22×C112−1+411yescharacteristic
157C4×C11—24no—
158C4×C11—41no—
159C4×C11#12114yes—
160Dic11—24no—
161Dic11—22no—
162Dic11—41no—
163Dic11—41no—
164Dic11—41no—
165Dic11—42no—
166C22×Q8—111no—
167C22×D4C2×D1111yescharacteristic
168C22×D4C2×D1112yes—
169C23×C4C2×D1111yescharacteristic
170#124—224no—
171C22⋊Q8—224no—
172#121—24no—
173C4×Q8—224no—
174C2×C4⋊C4—112no—
175C2×C4⋊C4—112no—
176C2×C4⋊C4—111no—
177C2×C4⋊C4—112no—
178C2×C4⋊C4C2×D1111yescharacteristic
179C2×C22⋊C4—112no—
180C2×C22⋊C4—112no—
181C2×C22⋊C4—112no—
182C2×C22⋊C4—111no—
183C2×C22⋊C4—111no—
184C2×C22⋊C4—112no—
185C2×C22⋊C4—112no—
186C2×C22⋊C4C2×D1112yes—
187C2×C42—111no—
188C2×C42—112no—
189C2×C42—111no—
190#95—111no—
191#95—111no—
192#95—112no—
193C2×C11—42no—
194C2×C11—82no—
195C2×C11C2×#12111yescharacteristic
196C2×C11#52911yescharacteristic
197C2×C11#54211yescharacteristic
198C2×C11#54611yescharacteristic
199C2×C11#54811yescharacteristic
200C2×C11#54811yescharacteristic
201C2×C11#55011yescharacteristic
202C24C22×D1112yes—
203C24C22×D1111yescharacteristic
204C2×Q8—114no—
205C2×Q8—222no—
206C2×Q8—222no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—42no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—111no—
217C22×C4—112no—
218C22×C4—111no—
219C22×C4—111no—
220C22×C4—111no—
221C22×C4—112no—
222C22×C4C22×D1111yescharacteristic
223C22×C4C22×D1112yes—
224C22×C4C22×D1111yescharacteristic
225C4⋊C4—224no—
226C4⋊C4—222no—
227C4⋊C4—224no—
228C4⋊C4—442no—
229C4⋊C4C22⋊D1114yes—
230C22⋊C4—24no—
231C22⋊C4—224no—
232C22⋊C4—224no—
233C22⋊C4—224no—
234C22⋊C4—222no—
235C22⋊C4—224no—
236C22⋊C4—222no—
237C22⋊C4—442no—
238C42—222no—
239C42—224no—
240C42—222no—
241C11#242611yescharacteristic, complemented, Sylow
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23#434112yes—
256C23C23×D1111yescharacteristic
257Q8—228no—
258D4—48no—
259D4—44no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—44no—
266C2×C4—114no—
267C2×C4—112no—
268C2×C4—222no—
269C2×C4—221no—
270C2×C4—221no—
271C2×C4—222no—
272C2×C4—221no—
273C2×C4—221no—
274C2×C4—222no—
275C2×C4—221no—
276C2×C4—221no—
277C2×C4—221no—
278C2×C4—222no—
279C2×C4—221no—
280C2×C4—221no—
281C2×C4—222no—
282C2×C4—221no—
283C2×C4—221no—
284C2×C4D4×D1112yes—
285C2×C4#434112yes—
286C2×C4C2×C22⋊D1114yes—
287C2×C4C2×C22⋊D1112yes—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22C2×D4×D1111yescharacteristic
303C22C2×#434111yescharacteristic
304C22C2×#434111yescharacteristic
305C22#6703711yescharacteristic
306C22#41×D1111yescharacteristic
307C22#6704311yescharacteristic
308C22C22×C22⋊D1111yescharacteristic
309C4—24no—
310C4—41no—
311C4—224no—
312C4—222no—
313C4—441no—
314C4—441no—
315C4—441no—
316C4—442no—
317C4#6699114yes—
318C2—42no—
319C2—82no—
320C2#1062312911yescharacteristic
321C2#1062313011yescharacteristic
322C2#1062313111yescharacteristic
323C2#1062313411yescharacteristic
324C2C2×#6699111yescharacteristic
325C2#1062333711yescharacteristic
326C2#1062336211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
Fitting, prime radicalC2×C11×#121C2
hypercenterC2×#121D11
hypocenterC11#2426
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#66989C2semidirect
3. C2×#66977C2semidirect
4. C2×#66940C2semidirect
5. #10622835C2semidirect
6. #10622834C2semidirect
7. #10622834C2semidirect
8. C4×C22⋊Dic11C2semidirect
9. #10622598C2semidirect
10. #10622561C2semidirect
11. #10622557C2semidirect
61. C2×Dic22D4semidirect
241. C11#2426semidirect

Maximal subgroups

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

subgroupindex
C2×C11×#1212
C2×#669892
C2×#669772
C2×#669402
#106228352
#106228342
C4×C22⋊Dic112
#106225982
#106225612
#106225572
#242611