Finite Group Databasebeta

← #13072012

Subgroups — #13072012

Order 1408.

814 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C11×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#67006C211yesmaximal, characteristic
3C22×C22⋊Dic11C211yesmaximal, characteristic, complemented
4C2×#66982C211yesmaximal, characteristic, complemented
5#10622826C211yesmaximal, characteristic, complemented
6#10622824C211yesmaximal, characteristic, complemented
7#10622823C211yesmaximal, characteristic, complemented
8#10622572C212yesmaximal, complemented
9#10622571C212yesmaximal, complemented
10#10622565C212yesmaximal, complemented
11#10622555C212yesmaximal, complemented
12C25×C11C2211yescharacteristic
13C23×C4×C11C2211yescharacteristic
14C23×C4×C11C2211yescharacteristic
15C22×C22⋊D11C2211yescharacteristic
16C22×C22⋊D11C2211yescharacteristic
17C22×C22⋊D11C2211yescharacteristic
18C23×Dic11C2212yes—
19C2×C11×C22⋊C4—24no—
20C2×C11×C22⋊C4C2212yes—
21C2×C11×C22⋊C4C2212yes—
22#67006—24no—
23C2×C22⋊Dic11—24no—
24C2×C22⋊Dic11C2211yescharacteristic
25C2×C22⋊Dic11C2211yescharacteristic
26C2×C22⋊Dic11C2212yes—
27C2×C22⋊Dic11C2211yescharacteristic
28C2×C22⋊Dic11C2211yescharacteristic
29C2×C22⋊Dic11C2211yescharacteristic
30#66982—24no—
31C2×(C2×D11)⋊C4C2212yescomplemented
32C2×(C2×D11)⋊C4C2212yescomplemented
33C2×Dic11⋊C4C2212yescomplemented
34C2×Dic11⋊C4C2212yescomplemented
35C11⋊#95C2212yescomplemented
36C11⋊#95C2212yescomplemented
37C11⋊#95C2212yescomplemented
38C11⋊#95C2212yescomplemented
39C24×C11—22no—
40C24×C11—22no—
41C24×C11—24no—
42C24×C11—42no—
43C24×C11D414yes—
44C24×C11C2311yescharacteristic
45C24×C11C2311yescharacteristic
46C24×C11C2311yescharacteristic
47C23×D11C2311yescharacteristic
48C22×C4×C11—22no—
49C22×C4×C11—22no—
50C22×C4×C11—22no—
51C22×C4×C11—22no—
52C22×C4×C11—42no—
53C22×C4×C11C2312yes—
54C22×C4×C11C2312yes—
55C2×C22⋊D11—22no—
56C2×C22⋊D11—22no—
57C2×C22⋊D11—22no—
58C2×C22⋊D11—22no—
59C2×C22⋊D11—24no—
60C2×C22⋊D11—22no—
61C2×C22⋊D11—42no—
62C22×Dic11—24no—
63C22×Dic11—24no—
64C22×Dic11—42no—
65C22×Dic11C2311yescharacteristic
66C22×Dic11C2312yes—
67C22×Dic11C2311yescharacteristic
68C22×Dic11C2311yescharacteristic
69C22×Dic11C2312yes—
70C11×C22⋊C4—24no—
71C11×C22⋊C4—24no—
72C22⋊Dic11—24no—
73C22⋊Dic11—22no—
74C22⋊Dic11—22no—
75C22⋊Dic11—22no—
76C22⋊Dic11—22no—
77C22⋊Dic11D414yescomplemented
78(C2×D11)⋊C4—24no—
79(C2×D11)⋊C4—42no—
80Dic11⋊C4—24no—
81Dic11⋊C4—42no—
82#2412—111nomaximal, Sylow
83C23×C11—21no—
84C23×C11—22no—
85C23×C11—21no—
86C23×C11—22no—
87C23×C11—22no—
88C23×C11—28no—
89C23×C11—28no—
90C23×C11—21no—
91C23×C11—22no—
92C23×C11—21no—
93C23×C11—22no—
94C23×C11—44no—
95C23×C11—48no—
96C23×C11—48no—
97C23×C11C2×D412yes—
98C23×C11C2×D412yes—
99C23×C11C2×D412yes—
100C23×C11C2×D412yes—
101C23×C11#4114yes—
102C23×C11#4112yes—
103C23×C11C2411yescharacteristic
104C22×D11—21no—
105C22×D11—21no—
106C22×D11—22no—
107C22×D11—21no—
108C22×D11—22no—
109C2×C4×C11—22no—
110C2×C4×C11—22no—
111C2×C4×C11—22no—
112C2×C4×C11—22no—
113C2×C4×C11—22no—
114C2×C4×C11—22no—
115C2×C4×C11—44no—
116C2×C4×C11—44no—
117C22⋊D11—44no—
118C22⋊D11—44no—
119C22⋊D11—44no—
120C2×Dic11—22no—
121C2×Dic11—22no—
122C2×Dic11—21no—
123C2×Dic11—22no—
124C2×Dic11—21no—
125C2×Dic11—21no—
126C2×Dic11—21no—
127C2×Dic11—22no—
128C2×Dic11—22no—
129C2×Dic11—21no—
130C2×Dic11—22no—
131C2×Dic11—22no—
132C2×Dic11—48no—
133C2×Dic11C2×D414yes—
134C2×#123—111no—
135C2×#120—111no—
136C22×C22⋊C4—111no—
137C22×C22⋊C4D1111yescharacteristic, complemented
138#397—112no—
139#394—112no—
140#386—111no—
141#386—112no—
142#385—111no—
143#381—111no—
144#380—112no—
145C22×C11—22no—
146C22×C11—22no—
147C22×C11—22no—
148C22×C11—22no—
149C22×C11—24no—
150C22×C11—24no—
151C22×C11—24no—
152C22×C11—24no—
153C22×C11—22no—
154C22×C11—22no—
155C22×C11—22no—
156C22×C11—416no—
157C22×C11—42no—
158C22×C11—42no—
159C22×C11#12114yes—
160C22×C11#12314yescomplemented
161C22×C11C22×D411yescharacteristic
162C22×C11C22×D411yescharacteristic
163C22×C11C2×#4112yes—
164C22×C11C2×#4111yescharacteristic
165C22×C11H(5,𝔽2)12yes—
166C2×D11—41no—
167C2×D11—41no—
168C2×D11—41no—
169C2×D11—42no—
170C2×D11—42no—
171C4×C11—42no—
172C4×C11—42no—
173Dic11—24no—
174Dic11—41no—
175Dic11—41no—
176Dic11—41no—
177Dic11—42no—
178C25C2×D1111yescharacteristic
179C22×D4—111no—
180C22×D4—111no—
181C22×D4—111no—
182C23×C4—112no—
183C23×C4C2×D1111yescharacteristic
184C23×C4C2×D1111yescharacteristic
185#123—224no—
186#120—224no—
187C2×C4⋊C4—112no—
188C2×C4⋊C4—112no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—111no—
191C2×C22⋊C4—111no—
192C2×C22⋊C4—112no—
193C2×C22⋊C4—111no—
194C2×C22⋊C4—111no—
195C2×C22⋊C4—111no—
196C2×C22⋊C4—112no—
197C2×C22⋊C4—112no—
198C2×C22⋊C4—224no—
199C2×C22⋊C4C2×D1112yes—
200C2×C22⋊C4C2×D1112yes—
201#95—112no—
202#95—112no—
203#95—112no—
204#95—112no—
205C2×C11—24no—
206C2×C11—24no—
207C2×C11—42no—
208C2×C11C2×#12111yescharacteristic
209C2×C11C2×#12311yescharacteristic
210C2×C11#53411yescharacteristic
211C2×C11#54612yes—
212C2×C11#55112yes—
213D11—81no—
214C24—24no—
215C24—22no—
216C24—22no—
217C24—42no—
218C24—111no—
219C24C22⋊D1114yes—
220C24C22×D1111yescharacteristic
221C24C22×D1111yescharacteristic
222C24C22×D1111yescharacteristic
223C2×D4—224no—
224C2×D4—222no—
225C2×D4—222no—
226C2×D4—222no—
227C2×D4—222no—
228C2×D4—222no—
229C2×D4—442no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—112no—
236C22×C4—111no—
237C22×C4—112no—
238C22×C4—111no—
239C22×C4—111no—
240C22×C4—224no—
241C22×C4—224no—
242C22×C4—442no—
243C22×C4C22×D1112yes—
244C22×C4C22×D1112yes—
245C4⋊C4—224no—
246C4⋊C4—442no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—114no—
250C22⋊C4—224no—
251C22⋊C4—224no—
252C22⋊C4—222no—
253C22⋊C4—222no—
254C22⋊C4—222no—
255C22⋊C4—222no—
256C22⋊C4—442no—
257C11#241211yescharacteristic, complemented, Sylow
258C23—22no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—28no—
264C23—28no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—44no—
270C23—48no—
271C23—48no—
272C23—222no—
273C23—221no—
274C23—221no—
275C23—222no—
276C23—221no—
277C23#433914yes—
278C23D4×D1112yes—
279C23#434112yes—
280C23C2×C22⋊D1112yes—
281C23C2×C22⋊D1112yes—
282C23C2×C22⋊D1112yes—
283C23C23×D1111yescharacteristic
284D4—444no—
285D4—444no—
286D4—444no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—114no—
296C2×C4—222no—
297C2×C4—222no—
298C2×C4—222no—
299C2×C4—222no—
300C2×C4—222no—
301C2×C4—221no—
302C2×C4—221no—
303C2×C4—222no—
304C2×C4—222no—
305C2×C4—221no—
306C2×C4—221no—
307C2×C4—221no—
308C2×C4—448no—
309C22—24no—
310C22—24no—
311C22—24no—
312C22—24no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—416no—
321C22—42no—
322C22—42no—
323C22—442no—
324C22—442no—
325C22—441no—
326C22—441no—
327C22—441no—
328C22#6698214yescomplemented
329C22#6699214yes—
330C22C2×#433912yes—
331C22C2×D4×D1111yescharacteristic
332C22C2×#434111yescharacteristic
333C22#6703712yes—
334C22C22×C22⋊D1111yescharacteristic
335C4—42no—
336C4—42no—
337C4—224no—
338C4—442no—
339C4—441no—
340C4—441no—
341C4—441no—
342C2—24no—
343C2—24no—
344C2—42no—
345C2—881no—
346C2#1062302112yes—
347C2#1062302612yes—
348C2C2×#6698211yescharacteristic
349C2C2×#6699211yescharacteristic
350C2#1062333511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC22×C11×C22⋊C4C2
prime radicalC2×#67006C2
hypercenterC22×C22⋊C4D11
hypocenterC11#2412
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. C22×C22⋊Dic11C2semidirect
4. C2×#66982C2semidirect
5. #10622826C2semidirect
6. #10622824C2semidirect
7. #10622823C2semidirect
8. #10622572C2semidirect
9. #10622571C2semidirect
10. #10622565C2semidirect
11. #10622555C2semidirect
31. C2×(C2×D11)⋊C4C22semidirect
32. C2×(C2×D11)⋊C4C22semidirect
33. C2×Dic11⋊C4C22semidirect
34. C2×Dic11⋊C4C22semidirect
35. C11⋊#95C22semidirect
36. C11⋊#95C22semidirect
37. C11⋊#95C22semidirect
38. C11⋊#95C22semidirect
77. C22⋊Dic11D4semidirect
137. C22×C22⋊C4D11semidirect
160. C22×C11#123semidirect
257. C11#2412semidirect
328. C22#66982semidirect

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
C22×C22⋊Dic112
C2×#669822
#106228262
#106228242
#106228232
#106225722
#106225712
#106225652
#106225552
#241211