Finite Group Databasebeta

← #13072001

Subgroups — #13072001

Order 1408.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 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×C11×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#67006C211yesmaximal, characteristic
3C2×#66982C211yesmaximal, characteristic, complemented
4C2×#66979C211yesmaximal, characteristic, complemented
5#10622860C211yesmaximal, characteristic, complemented
6#10622826C211yesmaximal, characteristic, complemented
7#10622823C211yesmaximal, characteristic, complemented
8#10622574C212yesmaximal, complemented
9#10622570C212yesmaximal, complemented
10#10622561C212yesmaximal, complemented
11#10622558C212yesmaximal, complemented
12C25×C11C2211yescharacteristic
13C23×C4×C11C2211yescharacteristic
14C23×C4×C11C2211yescharacteristic
15C22×C22⋊D11C2211yescharacteristic
16C22×C22⋊D11C2212yes—
17C2×C11×C22⋊C4—24no—
18C2×C11×C22⋊C4C2212yes—
19C2×C11×C22⋊C4C2212yes—
20#67006—24no—
21C2×C22⋊Dic11C2211yescharacteristic
22C2×C22⋊Dic11C2211yescharacteristic
23C2×C22⋊Dic11C2212yes—
24C2×C22⋊Dic11C2212yes—
25C2×C22⋊Dic11C2212yes—
26C2×C22⋊Dic11C2211yescharacteristic
27#66982—24no—
28C2×(C2×D11)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D11)⋊C4C2211yescharacteristic, complemented
30C2×(C2×D11)⋊C4C2212yescomplemented
31#66979—24no—
32C2×C4⋊Dic11C2212yescomplemented
33C2×Dic11⋊C4C2211yescharacteristic, complemented
34C2×Dic11⋊C4C2211yescharacteristic, complemented
35C2×Dic11⋊C4C2212yescomplemented
36C2×C4×Dic11C2212yescomplemented
37C11⋊#95C2212yescomplemented
38C11⋊#95C2212yescomplemented
39C24×C11—22no—
40C24×C11—24no—
41C24×C11—22no—
42C24×C11—22no—
43C24×C11—42no—
44C24×C11C2312yes—
45C24×C11C2311yescharacteristic
46C23×D11C2311yescharacteristic
47C22×C4×C11—22no—
48C22×C4×C11—22no—
49C22×C4×C11—22no—
50C22×C4×C11—42no—
51C22×C4×C11D414yescomplemented
52C22×C4×C11C2312yes—
53C22×C4×C11C2311yescharacteristic
54C22×C4×C11C2311yescharacteristic
55C2×C22⋊D11—24no—
56C2×C22⋊D11—22no—
57C2×C22⋊D11—22no—
58C2×C22⋊D11—24no—
59C2×C22⋊D11—22no—
60C2×C22⋊D11—42no—
61C22×Dic11C2311yescharacteristic
62C22×Dic11C2311yescharacteristic
63C22×Dic11C2312yes—
64C22×Dic11C2311yescharacteristic
65C22×Dic11C2312yes—
66C11×C22⋊C4—24no—
67C11×C22⋊C4—24no—
68C22⋊Dic11—24no—
69C22⋊Dic11—22no—
70C22⋊Dic11—22no—
71C22⋊Dic11—24no—
72C22⋊Dic11—42no—
73C22⋊Dic11D414yes—
74(C2×D11)⋊C4—22no—
75(C2×D11)⋊C4—22no—
76(C2×D11)⋊C4—42no—
77C4⋊Dic11—24no—
78Dic11⋊C4—22no—
79Dic11⋊C4—22no—
80Dic11⋊C4—42no—
81C4×Dic11—24no—
82#2411—111nomaximal, Sylow
83C23×C11—22no—
84C23×C11—22no—
85C23×C11—28no—
86C23×C11—22no—
87C23×C11—22no—
88C23×C11—22no—
89C23×C11—22no—
90C23×C11—22no—
91C23×C11—22no—
92C23×C11—48no—
93C23×C11—48no—
94C23×C11—44no—
95C23×C11—44no—
96C23×C11C2×D412yes—
97C23×C11C2×D412yes—
98C23×C11#4114yes—
99C23×C11#4112yes—
100C23×C11C2411yescharacteristic
101C22×D11—21no—
102C22×D11—21no—
103C22×D11—22no—
104C22×D11—21no—
105C22×D11—22no—
106C2×C4×C11—28no—
107C2×C4×C11—21no—
108C2×C4×C11—21no—
109C2×C4×C11—21no—
110C2×C4×C11—21no—
111C2×C4×C11—22no—
112C2×C4×C11—22no—
113C2×C4×C11—22no—
114C2×C4×C11—44no—
115C2×C4×C11C2×D412yes—
116C2×C4×C11C2×D412yes—
117C22⋊D11—48no—
118C22⋊D11—44no—
119C2×Dic11—22no—
120C2×Dic11—22no—
121C2×Dic11—22no—
122C2×Dic11—21no—
123C2×Dic11—21no—
124C2×Dic11—21no—
125C2×Dic11—21no—
126C2×Dic11—22no—
127C2×Dic11—22no—
128C2×Dic11—21no—
129C2×Dic11—22no—
130C2×Dic11—22no—
131C2×Dic11C2×D414yes—
132C2×#123—111no—
133C2×#120—111no—
134C2×#117—111no—
135C22×C22⋊C4D1111yescharacteristic, complemented
136#396—112no—
137#394—112no—
138#390—112no—
139#388—112no—
140#386—111no—
141#385—111no—
142#379—111no—
143C22×C11—22no—
144C22×C11—22no—
145C22×C11—24no—
146C22×C11—24no—
147C22×C11—24no—
148C22×C11—24no—
149C22×C11—24no—
150C22×C11—416no—
151C22×C11—42no—
152C22×C11—42no—
153C22×C11—42no—
154C22×C11—42no—
155C22×C11#12114yes—
156C22×C11C22×D411yescharacteristic
157C22×C11C22×D411yescharacteristic
158C22×C11C2×#4112yes—
159C22×C11C2×#4111yescharacteristic
160C22×C11H(5,𝔽2)12yes—
161C2×D11—41no—
162C2×D11—41no—
163C2×D11—41no—
164C2×D11—42no—
165C2×D11—42no—
166C4×C11—22no—
167C4×C11—22no—
168C4×C11—42no—
169Dic11—24no—
170Dic11—41no—
171Dic11—41no—
172Dic11—41no—
173Dic11—42no—
174C25C2×D1111yescharacteristic
175C22×D4—111no—
176C22×D4—112no—
177C23×C4C2×D1111yescharacteristic
178C23×C4C2×D1111yescharacteristic
179#123—224no—
180#120—224no—
181#117—224no—
182C2×C4⋊C4—112no—
183C2×C4⋊C4—111no—
184C2×C4⋊C4—111no—
185C2×C4⋊C4—112no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—111no—
188C2×C22⋊C4—111no—
189C2×C22⋊C4—112no—
190C2×C22⋊C4—111no—
191C2×C22⋊C4—112no—
192C2×C22⋊C4—111no—
193C2×C22⋊C4—111no—
194C2×C22⋊C4—112no—
195C2×C22⋊C4—112no—
196C2×C22⋊C4C2×D1112yes—
197C2×C22⋊C4C2×D1112yes—
198C2×C42—112no—
199#95—112no—
200#95—112no—
201C2×C11—24no—
202C2×C11—44no—
203C2×C11C2×#12111yescharacteristic
204C2×C11#52511yescharacteristic
205C2×C11#53511yescharacteristic
206C2×C11#54612yes—
207C2×C11#55112yes—
208D11—81no—
209C24—24no—
210C24—22no—
211C24—22no—
212C24—22no—
213C24—42no—
214C24—111no—
215C24C22×D1112yes—
216C24C22×D1111yescharacteristic
217C2×D4—224no—
218C2×D4—224no—
219C2×D4—222no—
220C2×D4—222no—
221C2×D4—222no—
222C2×D4—442no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—112no—
228C22×C4—111no—
229C22×C4—111no—
230C22×C4—112no—
231C22×C4—111no—
232C22×C4C22⋊D1114yescomplemented
233C22×C4C22×D1112yes—
234C22×C4C22×D1111yescharacteristic
235C22×C4C22×D1111yescharacteristic
236C4⋊C4—224no—
237C4⋊C4—222no—
238C4⋊C4—222no—
239C4⋊C4—442no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—114no—
243C22⋊C4—224no—
244C22⋊C4—224no—
245C22⋊C4—222no—
246C22⋊C4—222no—
247C22⋊C4—222no—
248C22⋊C4—222no—
249C22⋊C4—442no—
250C22⋊C4—442no—
251C42—224no—
252C11#241111yescharacteristic, complemented, Sylow
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—222no—
267C23—221no—
268C23—221no—
269C23—222no—
270C23—221no—
271C23#433914yes—
272C23D4×D1112yes—
273C23#434112yes—
274C23C2×C22⋊D1112yes—
275C23C23×D1111yescharacteristic
276D4—448no—
277D4—444no—
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—114no—
288C2×C4—222no—
289C2×C4—222no—
290C2×C4—222no—
291C2×C4—222no—
292C2×C4—222no—
293C2×C4—221no—
294C2×C4—221no—
295C2×C4—222no—
296C2×C4—222no—
297C2×C4—221no—
298C2×C4—221no—
299C2×C4—221no—
300C2×C4C2×C22⋊D1112yes—
301C2×C4C2×C22⋊D1112yes—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—442no—
315C22—442no—
316C22—441no—
317C22—441no—
318C22—441no—
319C22#6699214yes—
320C22C2×#433912yes—
321C22C2×D4×D1111yescharacteristic
322C22C2×#434111yescharacteristic
323C22#6703712yes—
324C22C22×C22⋊D1111yescharacteristic
325C4—22no—
326C4—22no—
327C4—42no—
328C4—224no—
329C4—442no—
330C4—441no—
331C4—441no—
332C4—441no—
333C2—24no—
334C2—44no—
335C2—881no—
336C2#1062302112yes—
337C2#1062302612yes—
338C2#1062332211yescharacteristic
339C2C2×#6699211yescharacteristic
340C2#1062333611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC22×C11×C22⋊C4C2
prime radicalC2×#67006C2
hypercenterC22×C22⋊C4D11
hypocenterC11#2411
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C11×C22⋊C4C2semidirect
3. C2×#66982C2semidirect
4. C2×#66979C2semidirect
5. #10622860C2semidirect
6. #10622826C2semidirect
7. #10622823C2semidirect
8. #10622574C2semidirect
9. #10622570C2semidirect
10. #10622561C2semidirect
11. #10622558C2semidirect
28. C2×(C2×D11)⋊C4C22semidirect
29. C2×(C2×D11)⋊C4C22semidirect
30. C2×(C2×D11)⋊C4C22semidirect
32. C2×C4⋊Dic11C22semidirect
33. C2×Dic11⋊C4C22semidirect
34. C2×Dic11⋊C4C22semidirect
35. C2×Dic11⋊C4C22semidirect
36. C2×C4×Dic11C22semidirect
37. C11⋊#95C22semidirect
38. C11⋊#95C22semidirect
51. C22×C4×C11D4semidirect
135. C22×C22⋊C4D11semidirect
232. C22×C4C22⋊D11semidirect
252. C11#2411semidirect

Maximal subgroups

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

subgroupindex
C22×C11×C22⋊C42
C2×#670062
C2×#669822
C2×#669792
#106228602
#106228262
#106228232
#106225742
#106225702
#106225612
#106225582
#241111