Finite Group Databasebeta

← #13072237

Subgroups — #13072237

Order 1408.

952 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4343C211yesmaximal, characteristic, complemented
2C2×#67000C211yesmaximal, characteristic, complemented
3C2×#66948C211yesmaximal, characteristic, complemented
4C2×#66927C211yesmaximal, characteristic, complemented
5C11×#391C211yesmaximal, characteristic, complemented
6#10622601C211yesmaximal, characteristic, complemented
7C4×(C2×D11)⋊C4C211yesmaximal, characteristic, complemented
8#10622284C214yesmaximal, complemented
9#10622278C214yesmaximal, complemented
10C22×Q8×C11C2211yescharacteristic
11C2×#4343—24no—
12C2×#4343—24no—
13C2×#4343—24no—
14C22×D44C2211yescharacteristic
15C22×D44C2212yes—
16C22×D44C2214yes—
17C22×D44C2211yescharacteristic
18C22×D44C2212yes—
19C22×C4×D11C2212yes—
20C22×C4×D11C2211yescharacteristic
21C2×C11×C4⋊C4C2211yescharacteristic
22C2×C42×C11C2211yescharacteristic
23#67000—24no—
24C2×(C2×D11)⋊C4C2212yes—
25C2×(C2×D11)⋊C4C2214yes—
26C2×(C2×D11)⋊C4C2212yes—
27C2×(C2×D11)⋊C4C2214yes—
28C2×C4×Dic11C2211yescharacteristic
29#66948—24no—
30#66927—24no—
31C11×#95C2214yes—
32C11⋊#95C2212yes—
33C23×D11C2312yes—
34C23×D11C2311yescharacteristic
35C23×D11C2312yes—
36C2×Q8×C11—22no—
37C2×Q8×C11—22no—
38C2×Q8×C11D414yescomplemented
39C22×C4×C11C2312yes—
40C22×C4×C11C2312yes—
41C22×C4×C11C2312yes—
42C22×C4×C11C2311yescharacteristic
43C22×Dic11C2312yes—
44C22×Dic11C2311yescharacteristic
45#4343—232no—
46C2×D44—24no—
47C2×D44—24no—
48C2×D44—28no—
49C2×D44—24no—
50C2×D44—24no—
51C2×D44—22no—
52C2×D44—22no—
53C2×D44—22no—
54C2×D44—22no—
55C2×D44—22no—
56C2×D44—28no—
57C2×D44—44no—
58C2×D44—44no—
59C2×D44D414yescomplemented
60C2×C4×D11—24no—
61C2×C4×D11—24no—
62C2×C4×D11—24no—
63C2×C4×D11—22no—
64C2×C4×D11—22no—
65C2×C4×D11D414yescomplemented
66C11×C4⋊C4—22no—
67C42×C11—22no—
68(C2×D11)⋊C4—24no—
69(C2×D11)⋊C4—28no—
70(C2×D11)⋊C4—24no—
71(C2×D11)⋊C4—44no—
72C4×Dic11—22no—
73#2431—111nomaximal, Sylow
74C23×C11C2411yescharacteristic
75C22×D11—22no—
76C22×D11—21no—
77C22×D11—22no—
78C22×D11—21no—
79C22×D11—22no—
80C22×D11—24no—
81C22×D11—24no—
82C22×D11—22no—
83C22×D11—22no—
84C22×D11—22no—
85C22×D11—22no—
86C22×D11—24no—
87C22×D11—24no—
88C22×D11C2×D414yes—
89C22×D11C2×D412yes—
90Q8×C11—28no—
91C2×C4×C11—22no—
92C2×C4×C11—22no—
93C2×C4×C11—22no—
94C2×C4×C11—22no—
95C2×C4×C11—22no—
96C2×C4×C11—24no—
97C2×C4×C11C2×D414yes—
98C2×C4×C11C2×D412yes—
99C2×C4×C11C2×D412yes—
100C2×C4×C11C2×D412yes—
101C2×C4×C11#4114yescomplemented
102C2×Dic11—21no—
103C2×Dic11—21no—
104C2×Dic11—22no—
105C2×Dic11—22no—
106C2×Dic11—22no—
107C2×Dic11C2×D412yes—
108D44—216no—
109D44—28no—
110D44—416no—
111D44—48no—
112D44—44no—
113C4×D11—216no—
114C4×D11—28no—
115C22×#41—111no—
116C2×#127—111no—
117C2×#124—111no—
118C2×#121—111no—
119#392—114no—
120#391D1111yescharacteristic, complemented
121#390—111no—
122#386—114no—
123C4×C22⋊C4—111no—
124C22×C11C22×D411yescharacteristic
125C22×C11C22×D411yescharacteristic
126C22×C11C22×D411yescharacteristic
127C22×C11C2×#4112yes—
128C22×C11H(5,𝔽2)12yes—
129C2×D11—24no—
130C2×D11—22no—
131C2×D11—22no—
132C2×D11—22no—
133C2×D11—28no—
134C2×D11—42no—
135C2×D11—42no—
136C2×D11—42no—
137C2×D11—42no—
138C2×D11—44no—
139C2×D11—42no—
140C2×D11—42no—
141C2×D11—42no—
142C2×D11—44no—
143C2×D11—44no—
144C4×C11—24no—
145C4×C11—24no—
146C4×C11—42no—
147C4×C11#12014yes—
148Dic11—22no—
149Dic11—42no—
150C2×#41—224no—
151C2×#41—224no—
152C2×#41—224no—
153C22×Q8C2×D1111yescharacteristic
154C22×D4—111no—
155C22×D4—111no—
156C22×D4—112no—
157C22×D4—114no—
158C22×D4—112no—
159C23×C4—111no—
160C23×C4—112no—
161#127—224no—
162#124—224no—
163#121—224no—
164C2×C4⋊C4C2×D1111yescharacteristic
165C2×C22⋊C4—112no—
166C2×C22⋊C4—114no—
167C2×C22⋊C4—112no—
168C2×C22⋊C4—114no—
169C2×C42—111no—
170C2×C42C2×D1111yescharacteristic
171#95—112no—
172#95C2×D1114yes—
173C2×C11C2×#12011yescharacteristic
174C2×C11#53211yescharacteristic
175C2×C11#53511yescharacteristic
176C2×C11#54612yes—
177C2×C11#55012yes—
178D11—42no—
179D11—44no—
180D11—82no—
181C24—111no—
182C24—112no—
183C24—112no—
184#41—2232no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8D4414yescomplemented
188C2×D4—114no—
189C2×D4—224no—
190C2×D4—224no—
191C2×D4—224no—
192C2×D4—228no—
193C2×D4—224no—
194C2×D4—222no—
195C2×D4—222no—
196C2×D4—222no—
197C2×D4—222no—
198C2×D4—222no—
199C2×D4—228no—
200C2×D4—444no—
201C2×D4—444no—
202C22×C4—111no—
203C22×C4—114no—
204C22×C4—112no—
205C22×C4—224no—
206C22×C4—224no—
207C22×C4—224no—
208C22×C4—222no—
209C22×C4—222no—
210C22×C4C22×D1112yes—
211C22×C4C22×D1112yes—
212C22×C4C22×D1112yes—
213C22×C4C22×D1111yescharacteristic
214C4⋊C4—22no—
215C22⋊C4—224no—
216C22⋊C4—228no—
217C22⋊C4—224no—
218C22⋊C4—444no—
219C42—22no—
220C42—222no—
221C11#243111yescharacteristic, complemented, Sylow
222C23—112no—
223C23—114no—
224C23—222no—
225C23—221no—
226C23—222no—
227C23—221no—
228C23—222no—
229C23—224no—
230C23—224no—
231C23—222no—
232C23—222no—
233C23—222no—
234C23—222no—
235C23—224no—
236C23—224no—
237C23C23×D1111yescharacteristic
238Q8—28no—
239D4—2216no—
240D4—228no—
241D4—4416no—
242D4—448no—
243D4—444no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—112no—
251C2×C4—2216no—
252C2×C4—221no—
253C2×C4—221no—
254C2×C4—228no—
255C2×C4—222no—
256C2×C4—222no—
257C2×C4—222no—
258C2×C4C2×D4414yes—
259C2×C4C2×D4412yes—
260C2×C4D4×D1112yes—
261C2×C4D4×D1112yes—
262C2×C4#434314yescomplemented
263C22—222no—
264C22—224no—
265C22—222no—
266C22—228no—
267C22—222no—
268C22—442no—
269C22—442no—
270C22—444no—
271C22—442no—
272C22—442no—
273C22—442no—
274C22—442no—
275C22—444no—
276C22—444no—
277C22—442no—
278C22C22×D4411yescharacteristic
279C22C2×D4×D1111yescharacteristic
280C22C2×D4×D1111yescharacteristic
281C22C2×#434312yes—
282C22#6704212yes—
283C4—24no—
284C4—24no—
285C4—42no—
286C4—222no—
287C4—442no—
288C4#6693514yes—
289C2—442no—
290C2—444no—
291C2—882no—
292C2C2×#6693511yescharacteristic
293C2#1062310812yes—
294C2#1062316112yes—
295C2#1062320511yescharacteristic
296C2#1062325611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#391C2
hypercenter#391D11
hypocenterC11#2431
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×#4343C2semidirect
2. C2×#67000C2semidirect
3. C2×#66948C2semidirect
4. C2×#66927C2semidirect
5. C11×#391C2semidirect
6. #10622601C2semidirect
7. C4×(C2×D11)⋊C4C2semidirect
8. #10622284C2semidirect
9. #10622278C2semidirect
38. C2×Q8×C11D4semidirect
59. C2×D44D4semidirect
65. C2×C4×D11D4semidirect
101. C2×C4×C11#41semidirect
120. #391D11semidirect
187. C2×Q8D44semidirect
221. C11#2431semidirect
262. C2×C4#4343semidirect

Maximal subgroups

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

subgroupindex
C22×#43432
C2×#670002
C2×#669482
C2×#669272
C11×#3912
#106226012
C4×(C2×D11)⋊C42
#106222842
#106222782
#243111