Finite Group Databasebeta

← #13072272

Subgroups — #13072272

Order 1408.

852 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×Dic22C211yesmaximal, characteristic, complemented
2C2×#66992C211yesmaximal, characteristic
3C2×#66938C212yesmaximal, complemented
4C11×#386C211yesmaximal, characteristic, complemented
5#10622833C211yesmaximal, characteristic, complemented
6#10622826C211yesmaximal, characteristic, complemented
7#10622571C211yesmaximal, characteristic, complemented
8#10622560C212yesmaximal, complemented
9C22⋊C4×Dic11C211yesmaximal, characteristic, complemented
10#10622286C212yesmaximal, complemented
11#10622259C212yesmaximal, complemented
12C22×D4×C11C2211yescharacteristic
13C23×C4×C11C2211yescharacteristic
14C22×C22⋊D11C2211yescharacteristic
15C22×C22⋊D11C2211yescharacteristic
16C23×Dic11C2211yescharacteristic
17C23×Dic11C2211yescharacteristic
18C22×Dic22—24no—
19C22×Dic22—28no—
20C22×Dic22C2212yes—
21C22×Dic22C2212yes—
22C2×C11×C22⋊C4C2212yes—
23C2×C11×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic11C2212yes—
25C2×C22⋊Dic11C2211yescharacteristic
26C2×C22⋊Dic11C2211yescharacteristic
27#66992—24no—
28C2×(C2×D11)⋊C4C2212yes—
29C2×(C2×D11)⋊C4C2212yes—
30C2×(C2×D11)⋊C4C2211yescharacteristic
31C2×Dic11⋊C4C2211yescharacteristic
32C2×C4×Dic11C2212yes—
33#66938—28no—
34C11×#95C2212yes—
35C11⋊#95C2212yes—
36C11⋊#95C2212yes—
37C11⋊#95C2212yes—
38C11⋊#95C2212yes—
39C24×C11C2311yescharacteristic
40C24×C11C2311yescharacteristic
41C23×D11C2311yescharacteristic
42C2×D4×C11—22no—
43C2×D4×C11—22no—
44C2×D4×C11—22no—
45C22×C4×C11—22no—
46C22×C4×C11—24no—
47C22×C4×C11C2311yescharacteristic
48C22×C4×C11C2312yes—
49C22×C4×C11C2312yes—
50C2×C22⋊D11—24no—
51C2×C22⋊D11—22no—
52C2×C22⋊D11—22no—
53C2×C22⋊D11—22no—
54C2×C22⋊D11—22no—
55C22×Dic11—24no—
56C22×Dic11—24no—
57C22×Dic11—22no—
58C22×Dic11D414yes—
59C22×Dic11C2311yescharacteristic
60C22×Dic11C2312yes—
61C22×Dic11C2311yescharacteristic
62C22×Dic11C2312yes—
63C22×Dic11C2311yescharacteristic
64C2×Dic22—232no—
65C2×Dic22—24no—
66C2×Dic22—24no—
67C2×Dic22—24no—
68C2×Dic22—24no—
69C2×Dic22—24no—
70C2×Dic22D418yescomplemented
71C11×C22⋊C4—22no—
72C11×C22⋊C4—24no—
73C22⋊Dic11—24no—
74C22⋊Dic11—22no—
75C22⋊Dic11—22no—
76(C2×D11)⋊C4—24no—
77(C2×D11)⋊C4—22no—
78(C2×D11)⋊C4—24no—
79Dic11⋊C4—22no—
80C4×Dic11—24no—
81#2436—111nomaximal, Sylow
82C23×C11—22no—
83C23×C11—22no—
84C23×C11—22no—
85C23×C11—21no—
86C23×C11—21no—
87C23×C11—21no—
88C23×C11—22no—
89C23×C11C2×D412yes—
90C23×C11#4112yes—
91C23×C11#4112yes—
92C23×C11C2411yescharacteristic
93C22×D11—22no—
94C22×D11—21no—
95C22×D11—21no—
96C22×D11—21no—
97C22×D11—22no—
98D4×C11—44no—
99C2×C4×C11—28no—
100C2×C4×C11—22no—
101C2×C4×C11—22no—
102C2×C4×C11—22no—
103C2×C4×C11—21no—
104C2×C4×C11—22no—
105C2×C4×C11—21no—
106C2×C4×C11—21no—
107C2×C4×C11—22no—
108C2×C4×C11C2×D414yes—
109C22⋊D11—44no—
110C22⋊D11—44no—
111C2×Dic11—28no—
112C2×Dic11—22no—
113C2×Dic11—22no—
114C2×Dic11—22no—
115C2×Dic11—28no—
116C2×Dic11—22no—
117C2×Dic11—21no—
118C2×Dic11—22no—
119C2×Dic11—21no—
120C2×Dic11—21no—
121C2×Dic11—22no—
122C2×Dic11C2×D412yes—
123C2×Dic11C2×D414yes—
124C2×Dic11C2×D414yes—
125C2×Dic11C2×D412yes—
126Dic22—216no—
127Dic22—216no—
128C23×Q8—111no—
129C2×#124—112no—
130C2×#121—111no—
131#393—112no—
132#393—112no—
133#391—112no—
134#386—111no—
135#386—111no—
136#386—111no—
137#386D1111yescharacteristic, complemented
138C4×C22⋊C4—111no—
139C22×C11—22no—
140C22×C11—22no—
141C22×C11—24no—
142C22×C11—24no—
143C22×C11—41no—
144C22×C11—41no—
145C22×C11—41no—
146C22×C11—42no—
147C22×C11—42no—
148C22×C11#12414yescomplemented
149C22×C11C22×D411yescharacteristic
150C22×C11C22×D412yes—
151C22×C11C2×#4111yescharacteristic
152C22×C11C2×#4111yescharacteristic
153C22×C112−1+412yes—
154C2×D11—41no—
155C2×D11—41no—
156C2×D11—41no—
157C2×D11—42no—
158C2×D11—42no—
159C4×C11—24no—
160C4×C11—41no—
161C4×C11—42no—
162Dic11—22no—
163Dic11—24no—
164Dic11—41no—
165Dic11—42no—
166Dic11#12014yes—
167C22×Q8—112no—
168C22×Q8—112no—
169C22×Q8—224no—
170C22×Q8—228no—
171C22×D4—111no—
172C22×D4—111no—
173C22×D4C2×D1111yescharacteristic
174C23×C4—111no—
175C23×C4—111no—
176C23×C4C2×D1111yescharacteristic
177#124—228no—
178#121—224no—
179C2×C4⋊C4—111no—
180C2×C22⋊C4—111no—
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×C22⋊C4C2×D1111yescharacteristic
188C2×C42—112no—
189#95—112no—
190#95—112no—
191#95—112no—
192#95—112no—
193#95C2×D1112yes—
194C2×C11—24no—
195C2×C11—81no—
196C2×C11C2×#12011yescharacteristic
197C2×C11C2×#12411yescharacteristic
198C2×C11#53611yescharacteristic
199C2×C11#54812yes—
200C2×C11#54812yes—
201D11—81no—
202C24—111no—
203C24C22×D1111yescharacteristic
204C24C22×D1111yescharacteristic
205C2×Q8—118no—
206C2×Q8—224no—
207C2×Q8—224no—
208C2×Q8—224no—
209C2×Q8—224no—
210C2×Q8—2232no—
211C2×Q8—224no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—222no—
216C2×D4—224no—
217C2×D4—222no—
218C2×D4—222no—
219C2×D4—222no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—111no—
223C22×C4—114no—
224C22×C4—111no—
225C22×C4—112no—
226C22×C4—111no—
227C22×C4—112no—
228C22×C4—224no—
229C22×C4—224no—
230C22×C4—222no—
231C22×C4C22×D1111yescharacteristic
232C22×C4C22×D1112yes—
233C22×C4C22×D1112yes—
234C4⋊C4—222no—
235C22⋊C4—22no—
236C22⋊C4—24no—
237C22⋊C4—222no—
238C22⋊C4—224no—
239C22⋊C4—222no—
240C22⋊C4—224no—
241C22⋊C4—222no—
242C22⋊C4—224no—
243C42—224no—
244C11#243611yescharacteristic, complemented, Sylow
245C23—22no—
246C23—22no—
247C23—21no—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—221no—
253C23—222no—
254C23—221no—
255C23—221no—
256C23—222no—
257C23#433912yes—
258C23D4×D1112yes—
259C23#434112yes—
260C23C23×D1111yescharacteristic
261Q8—2216no—
262Q8—2216no—
263D4—44no—
264D4—444no—
265D4—444no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—114no—
276C2×C4—112no—
277C2×C4—112no—
278C2×C4—114no—
279C2×C4—228no—
280C2×C4—228no—
281C2×C4—222no—
282C2×C4—222no—
283C2×C4—222no—
284C2×C4—222no—
285C2×C4—221no—
286C2×C4—222no—
287C2×C4—221no—
288C2×C4—221no—
289C2×C4—222no—
290C2×C4D4×D1114yes—
291C22—24no—
292C22—22no—
293C22—24no—
294C22—22no—
295C22—41no—
296C22—42no—
297C22—41no—
298C22—42no—
299C22—41no—
300C22—441no—
301C22—442no—
302C22—441no—
303C22—442no—
304C22—441no—
305C22#6693814yescomplemented
306C22C2×#433911yescharacteristic
307C22C2×D4×D1111yescharacteristic
308C22C2×D4×D1112yes—
309C22C2×#434111yescharacteristic
310C22#6704312yes—
311C4—24no—
312C4—41no—
313C4—42no—
314C4—114no—
315C4—222no—
316C4—224no—
317C4—441no—
318C4—442no—
319C2—24no—
320C2—81no—
321C2—881no—
322C2C2×#6693811yescharacteristic
323C2#1062308312yes—
324C2D11×#12011yescharacteristic
325C2#1062312912yes—
326C2#1062317911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#386C2
prime radicalC2×#66992C2
hypercenter#386D11
hypocenterC11#2436
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C23×Dic22C2semidirect
3. C2×#66938C2semidirect
4. C11×#386C2semidirect
5. #10622833C2semidirect
6. #10622826C2semidirect
7. #10622571C2semidirect
8. #10622560C2semidirect
9. C22⋊C4×Dic11C2semidirect
10. #10622286C2semidirect
11. #10622259C2semidirect
70. C2×Dic22D4semidirect
137. #386D11semidirect
148. C22×C11#124semidirect
244. C11#2436semidirect
305. C22#66938semidirect

Maximal subgroups

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

subgroupindex
C23×Dic222
C2×#669922
C2×#669382
C11×#3862
#106228332
#106228262
#106225712
#106225602
C22⋊C4×Dic112
#106222862
#106222592
#243611