Finite Group Databasebeta

← #29842875

Subgroups — #29842875

Order 15616.

2410 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449000C212yesmaximal, complemented
2#21448996C212yesmaximal, complemented
3#21448931C214yesmaximal, complemented
4#21448748C218yesmaximal, complemented
5#21448586C218yesmaximal, complemented
6#21448411C214yesmaximal, complemented
7#21448217C212yesmaximal, complemented
8C2×C61×#532C211yesmaximal, characteristic
9C2×#14519736C2214yes—
10C22×C4⋊Dic61C2214yes—
11#16839669C2218yes—
12#16839665C2218yes—
13#16839655C22116yes—
14#16839648C22116yes—
15#16839642C22116yes—
16#16839642C2218yes—
17#16839642C2218yes—
18#16839630C2214yes—
19#16839628C2214yes—
20#16839625C2214yes—
21#16839622C2214yes—
22#16839622C2218yes—
23C4×C4⋊Dic61C2214yes—
24C22⋊C4×Dic61C22116yes—
25C4×C22⋊Dic61C2214yes—
26#16839608C2214yes—
27C22×#41×C61C2212yes—
28C61×#532C4116yescomplemented
29C2×C4⋊Q8×C61C2211yescharacteristic
30C2×#127×C61C2211yescharacteristic
31C2×#124×C61C2212yes—
32C2×#121×C61C2214yes—
33C2×C4×D4×C61C2214yes—
34C22×C42×C61C2211yescharacteristic
35C23×Dic61C2318yes—
36#14519736—48no—
37C2×C4⋊Dic61—48no—
38C2×C4⋊Dic61C2318yes—
39C2×C4⋊Dic61C2318yes—
40C2×C22⋊Dic61C2318yes—
41C2×C22⋊Dic61C2318yes—
42C2×C22⋊Dic61C23116yes—
43C2×C4×Dic61C2318yes—
44C2×C4×Dic61C2318yes—
45#14519684C2318yes—
46#14519684C2318yes—
47#14519684C2318yes—
48#14519684C23116yes—
49#14519684C2318yes—
50C2×#41×C61—44no—
51C2×#41×C61—44no—
52C2×#41×C61C2×C4116yescomplemented
53C22×Q8×C61C2312yes—
54C22×D4×C61C2312yes—
55C22×D4×C61C2314yes—
56C22×D4×C61C2314yes—
57C23×C4×C61C2314yes—
58C23×C4×C61C2311yescharacteristic
59C23×C4×C61C2312yes—
60C4⋊Q8×C61C2×C418yescomplemented
61#127×C61C2×C418yes—
62#124×C61C2×C4116yescomplemented
63#121×C61C2×C4132yescomplemented
64C4×D4×C61C2×C4132yescomplemented
65C2×C4⋊C4×C61C2314yes—
66C2×C22⋊C4×C61C2318yes—
67C2×C42×C61C2×C418yescomplemented
68C2×C42×C61C2311yescharacteristic
69C2×C42×C61C2311yescharacteristic
70C2×C42×C61C2312yes—
71C24×C61C2414yes—
72C24×C61C2411yescharacteristic
73#41×C61—432no—
74C2×Q8×C61—42no—
75C2×Q8×C61—42no—
76C2×Q8×C61C22×C418yes—
77C2×D4×C61—44no—
78C2×D4×C61—44no—
79C2×D4×C61—44no—
80C2×D4×C61—44no—
81C2×D4×C61—42no—
82C2×D4×C61—42no—
83C2×D4×C61C22×C4116yes—
84C2×D4×C61C22×C4116yes—
85C2×D4×C61C22×C418yes—
86C22×C4×C61—24no—
87C22×C4×C61—28no—
88C22×C4×C61—28no—
89C22×C4×C61—44no—
90C22×C4×C61C22×C414yes—
91C22×C4×C61C22×C4116yes—
92C22×C4×C61C22×C418yes—
93C22×C4×C61C2412yes—
94C22×C4×C61C2412yes—
95C22×C4×C61C2411yescharacteristic
96C22×C4×C61C2414yes—
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—416no—
99C22×Dic61—48no—
100C22×Dic61C24116yes—
101C4⋊C4×C61C22×C4116yes—
102C22⋊C4×C61C22×C4132yes—
103C42×C61C22×C418yes—
104C42×C61C22×C414yes—
105C42×C61C22×C414yes—
106C22⋊Dic61—416no—
107C22⋊Dic61—48no—
108C22⋊Dic61—48no—
109C4⋊Dic61—216no—
110C4⋊Dic61—48no—
111C4×Dic61—48no—
112C4×Dic61—48no—
113C23×C61—24no—
114C23×C61—24no—
115C23×C61—22no—
116C23×C61—24no—
117C23×C61—28no—
118C23×C61—28no—
119C23×C61C23×C418yes—
120C23×C61C23×C412yes—
121C23×C61C2511yescharacteristic
122Q8×C61—48no—
123D4×C61—416no—
124D4×C61—48no—
125D4×C61—416no—
126C2×C4×C61—24no—
127C2×C4×C61—24no—
128C2×C4×C61—22no—
129C2×C4×C61—24no—
130C2×C4×C61—24no—
131C2×C4×C61—232no—
132C2×C4×C61—28no—
133C2×C4×C61—48no—
134C2×C4×C61C23×C414yes—
135C2×C4×C61C23×C414yes—
136C2×C4×C61C23×C412yes—
137C2×C4×C61C23×C418yes—
138C2×C4×C61C23×C412yes—
139C2×C4×C61H(5,𝔽2)14yes—
140C2×Dic61—216no—
141C2×Dic61—216no—
142C2×Dic61—216no—
143C2×Dic61—816no—
144#34369—611nomaximal, Sylow
145C22×C61—22no—
146C22×C61—28no—
147C22×C61—24no—
148C22×C61—216no—
149C22×C61—44no—
150C22×C61—48no—
151C22×C61—44no—
152C22×C61—44no—
153C22×C61C24×C411yescharacteristic
154C22×C61C2×H(5,𝔽2)12yes—
155C22×C61#58514yes—
156C4×C61—22no—
157C4×C61—28no—
158C4×C61—24no—
159C4×C61—24no—
160C4×C61#51814yes—
161Dic61—416no—
162C2×#532D6111yescharacteristic
163#2381—612no—
164#2380—612no—
165#2372—614no—
166#2352—618no—
167#2333—618no—
168#2311—614no—
169#2283—612no—
170C2×C61—42no—
171C2×C61—48no—
172C2×C61C2×#51811yescharacteristic
173C2×C61#342812yes—
174C2×C61#355114yes—
175C22×#41C2×D6112yes—
176#532Dic61116yescomplemented
177C2×C4⋊Q8C2×D6111yescharacteristic
178C2×#127C2×D6111yescharacteristic
179C2×#124C2×D6112yes—
180C2×#121C2×D6114yes—
181C2×C4×D4C2×D6114yes—
182C2×#117—614no—
183C22×C4⋊C4—614no—
184C22×C42C2×D6111yescharacteristic
185#391—618no—
186#390—618no—
187#388—6116no—
188#387—6116no—
189#386—618no—
190#386—618no—
191#386—6116no—
192#383—614no—
193#382—614no—
194#381—614no—
195#380—614no—
196#380—618no—
197C4×C4⋊C4—614no—
198C4×C22⋊C4—614no—
199C4×C22⋊C4—6116no—
200#376—614no—
201C61#3436911yescharacteristic, complemented, Sylow
202C2×#41—44no—
203C2×#41—44no—
204C2×#41C2×Dic61116yescomplemented
205C22×Q8C22×D6112yes—
206C22×D4C22×D6114yes—
207C22×D4C22×D6114yes—
208C22×D4C22×D6112yes—
209C23×C4—618no—
210C23×C4C22×D6114yes—
211C23×C4C22×D6111yescharacteristic
212C23×C4C22×D6112yes—
213C4⋊Q8C2×Dic6118yescomplemented
214#127C2×Dic6118yes—
215#124C2×Dic61116yescomplemented
216#121C2×Dic61132yescomplemented
217C4×D4C2×Dic61132yescomplemented
218#117—2448no—
219C2×C4⋊C4—618no—
220C2×C4⋊C4—618no—
221C2×C4⋊C4—2448no—
222C2×C4⋊C4C22×D6114yes—
223C2×C22⋊C4—618no—
224C2×C22⋊C4—618no—
225C2×C22⋊C4—6116no—
226C2×C22⋊C4C22×D6118yes—
227C2×C42—618no—
228C2×C42—618no—
229C2×C42C2×Dic6118yescomplemented
230C2×C42C22×D6112yes—
231C2×C42C22×D6111yescharacteristic
232C2×C42C22×D6111yescharacteristic
233#95—618no—
234#95—618no—
235#95—6116no—
236#95—618no—
237#95—618no—
238C24C23×D6111yescharacteristic
239C24C23×D6114yes—
240#41—432no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8C22×Dic6118yes—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4C22×Dic61116yes—
251C2×D4C22×Dic61116yes—
252C2×D4C22×Dic6118yes—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—6116no—
258C22×C4—2448no—
259C22×C4—24416no—
260C22×C4C22×Dic6114yes—
261C22×C4C22×Dic61116yes—
262C22×C4C22×Dic6118yes—
263C22×C4C23×D6112yes—
264C22×C4C23×D6111yescharacteristic
265C22×C4C23×D6112yes—
266C22×C4C23×D6114yes—
267C22×C4C23×D6111yescharacteristic
268C4⋊C4—12216no—
269C4⋊C4—2448no—
270C4⋊C4C22×Dic61116yes—
271C22⋊C4—2448no—
272C22⋊C4—24416no—
273C22⋊C4—2448no—
274C22⋊C4C22×Dic61132yes—
275C42—2448no—
276C42—2448no—
277C42C22×Dic6118yes—
278C42C22×Dic6114yes—
279C42C22×Dic6114yes—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—24no—
284C23—28no—
285C23—28no—
286C23C23×Dic6118yes—
287C23C23×Dic6112yes—
288C23C24×D6111yescharacteristic
289Q8—48no—
290D4—416no—
291D4—48no—
292D4—416no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—232no—
299C2×C4—28no—
300C2×C4—48no—
301C2×C4—12216no—
302C2×C4—12216no—
303C2×C4—12216no—
304C2×C4—48816no—
305C2×C4C23×Dic6114yes—
306C2×C4C23×Dic6114yes—
307C2×C4C23×Dic6112yes—
308C2×C4C23×Dic6118yes—
309C2×C4C23×Dic6112yes—
310C2×C4#1451982214yes—
311C22—22no—
312C22—28no—
313C22—24no—
314C22—216no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—44no—
319C22C24×Dic6111yescharacteristic
320C22C2×#1451982212yes—
321C22#1684059814yes—
322C4—22no—
323C4—28no—
324C4—24no—
325C4—24no—
326C4—24416no—
327C4#1684023714yes—
328C2—42no—
329C2—48no—
330C2C2×#1684023711yescharacteristic
331C2#2145902212yes—
332C2#2146024614yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
Fitting, prime radicalC2×C61×#532C2
socle, cosocleC23×C61C25
hypercenterC2×#532D61
hypocenterC61#34369
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. #21449000C2semidirect
2. #21448996C2semidirect
3. #21448931C2semidirect
4. #21448748C2semidirect
5. #21448586C2semidirect
6. #21448411C2semidirect
7. #21448217C2semidirect
28. C61×#532C4semidirect
52. C2×#41×C61C2×C4semidirect
60. C4⋊Q8×C61C2×C4semidirect
62. #124×C61C2×C4semidirect
63. #121×C61C2×C4semidirect
64. C4×D4×C61C2×C4semidirect
67. C2×C42×C61C2×C4semidirect
176. #532Dic61semidirect
201. C61#34369semidirect
204. C2×#41C2×Dic61semidirect
213. C4⋊Q8C2×Dic61semidirect
215. #124C2×Dic61semidirect
216. #121C2×Dic61semidirect
217. C4×D4C2×Dic61semidirect
229. C2×C42C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214490002
#214489962
#214489312
#214487482
#214485862
#214484112
#214482172
C2×C61×#5322
#3436961