Finite Group Databasebeta

← #30605152

Subgroups — #30605152

Order 15616.

402 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 66 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
1#21461716C211yesmaximal, characteristic
2#21461072C211yesmaximal, characteristic
3#21460816C212yesmaximal
4#21460814C212yesmaximal
5#21449089C211yesmaximal, characteristic
6#16840782—21no—
7#16840705—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840608—21no—
11#16840606C2212yes—
12C2×#14519734C2211yescharacteristic
13#16839700C411yescharacteristic, complemented
14#16839697C2211yescharacteristic
15#16839695—21no—
16#16839684—21no—
17#16839681C411yescharacteristic, complemented
18#16839673C2211yescharacteristic
19C4×#11758196—21no—
20C4×C4⋊Dic61C411yescharacteristic, complemented
21C4×#11758198—21no—
22C2×C4×Q8×C61C411yescharacteristic, complemented
23C2×C22⋊#6923—22no—
24#14519844—41no—
25C22⋊#92420—22no—
26#14519832—41no—
27(C2×#6923)⋊C4—41no—
28(C2×#6923)⋊C4—41no—
29C22×C4×D61D411yescharacteristic
30#14519734—41no—
31#14519734—41no—
32C2×C4⋊Dic61—21no—
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×#11758196C2×C411yescharacteristic
36C2×#11758196D411yescharacteristic
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198D411yescharacteristic
40C2×#11758198C2311yescharacteristic
41C2×C4×Dic61—21no—
42C2×C4×Dic61D411yescharacteristic
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684C2×C411yescharacteristic
48#14519684C2×C411yescharacteristic
49C22×Q8×C61C2×C411yescharacteristic
50C4×Q8×C61—41no—
51C4×Q8×C61—41no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61C2×C411yescharacteristic
54C2×C42×C61—21no—
55C2×C42×C61C2×C411yescharacteristic
56C23×D61C2×D411yescharacteristic
57C22×#6923—42no—
58C2×Q8×C61—42no—
59C2×Q8×C61—41no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×Dic61—21no—
66C22×Dic61—21no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C2×C4×D61—22no—
71C2×C4×D61—21no—
72C2×C4×D61—21no—
73C2×C4×D61—41no—
74C22⋊#6923—42no—
75C22⋊#6923—42no—
76C2×#92420—42no—
77C4⋊C4×C61—42no—
78C4⋊C4×C61—41no—
79C42×C61—21no—
80C42×C61—21no—
81C42×C61—41no—
82C42×C61—41no—
83#11758198—41no—
84#11758198—41no—
85#11758198—42no—
86#11758198—42no—
87C4⋊Dic61—42no—
88#11758196—41no—
89#11758196—41no—
90#11758196—42no—
91#11758196—42no—
92C4×Dic61—21no—
93C4×Dic61—21no—
94C4×Dic61—42no—
95C23×C61C2×C22⋊C411yescharacteristic
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61#13712yes—
101C2×#6923—82no—
102C2×#6923—82no—
103C2×#6923—82no—
104Q8×C61—42no—
105Q8×C61—42no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61#9911yescharacteristic
119C2×C4×C61#9911yescharacteristic
120C2×Dic61—22no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C4×D61—42no—
134C4×D61—42no—
135#92420—82no—
136#11810—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—21no—
139C22×C61C2×#9911yescharacteristic
140C22×C61#41911yescharacteristic
141C22×C61#42111yescharacteristic
142C2×D61—22no—
143C2×D61—41no—
144C2×D61—41no—
145C2×D61—42no—
146C2×D61—81no—
147#6923—162no—
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—42no—
151C4×C61—41no—
152C4×C61—41no—
153C4×C61—81no—
154Dic61—42no—
155Dic61—41no—
156Dic61—41no—
157Dic61—41no—
158Dic61—81no—
159#2396—611no—
160#1820—611no—
161#1464—611no—
162#1352—612no—
163#1351—612no—
164C2×C61—21no—
165C2×C61—21no—
166C2×C61#151111yescharacteristic
167C2×C61#212011yescharacteristic
168C2×C61#212611yescharacteristic
169D61—42no—
170C2×C4×Q8#692311yescharacteristic, complemented
171C2×#117—611no—
172#432—1221no—
173#400—611no—
174#399—611no—
175#398—1221no—
176#396—1221no—
177#395—611no—
178#393—611no—
179#381—1221no—
180C4×C4⋊C4—611no—
181C4×C4⋊C4—1221no—
182C4×C22⋊C4—1221no—
183#342—612no—
184#342—1221no—
185#324—1221no—
186#323—612no—
187C61#1181011yescharacteristic, complemented, Sylow
188C22×Q8C2×#692311yescharacteristic
189C23×C4—611no—
190C4×Q8—41no—
191C4×Q8—41no—
192#117—2441no—
193#117—2441no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—611no—
196C2×C4⋊C4—611no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4C2×#692311yescharacteristic
201C2×C22⋊C4—611no—
202C2×C22⋊C4—611no—
203C2×C22⋊C4—1221no—
204C2×C22⋊C4—1221no—
205C2×C22⋊C4—1222no—
206C2×C42—21no—
207C2×C42—611no—
208C2×C42—1221no—
209C2×C42C2×#692311yescharacteristic
210C4⋊C8—2441no—
211C22⋊C8—1222no—
212#97—2441no—
213#95—611no—
214#95—611no—
215#95—1221no—
216#95—1221no—
217#95—1221no—
218#95—1221no—
219#95—2441no—
220#95—2441no—
221C24—611no—
222C2×Q8—42no—
223C2×Q8—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—1222no—
230C22×C4—1221no—
231C22×C4—1221no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—2441no—
235C22×C4—2442no—
236C22×C4C22⋊#692311yescharacteristic
237C22×C4C22⋊#692311yescharacteristic
238C22×C4C22×#692311yescharacteristic
239C2×C8—2442no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—2441no—
243C4⋊C4—2441no—
244C4⋊C4—2442no—
245C4⋊C4—2442no—
246C4⋊C4—2442no—
247C22⋊C4—2441no—
248C22⋊C4—2441no—
249C22⋊C4—2442no—
250C22⋊C4—2442no—
251C22⋊C4—2442no—
252C22⋊C4—2442no—
253C42—21no—
254C42—21no—
255C42—41no—
256C42—41no—
257C42—1221no—
258C42—1221no—
259C42—2442no—
260C23—612no—
261C23—1221no—
262C23—1221no—
263C23—2441no—
264C23—2441no—
265C23C2×C22⋊#692311yescharacteristic
266Q8—42no—
267Q8—42no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—1222no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—4882no—
287C2×C4—4882no—
288C2×C4—4882no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4#1451983511yescharacteristic
299C2×C4#1451983511yescharacteristic
300C8—4882no—
301C22—21no—
302C22—21no—
303C22—1222no—
304C22—4881no—
305C22—2442no—
306C22—2441no—
307C22—2441no—
308C22C2×#1451983511yescharacteristic
309C22#1684075511yescharacteristic
310C22#1684076111yescharacteristic
311C4—21no—
312C4—21no—
313C4—42no—
314C4—41no—
315C4—41no—
316C4—81no—
317C4—9762no—
318C4—4881no—
319C4—2442no—
320C4—2441no—
321C4—2441no—
322C4—2441no—
323C2—21no—
324C2—21no—
325C2—2442no—
326C2#2146122911yescharacteristic
327C2#2146207011yescharacteristic
328C2#2146209011yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C2×#99
prime radicalC23×D61C2×D4
cosocleC2×#11758198C23
hypercenterC2×C4×Q8#6923
hypocenterC61#11810
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
13. #16839700C4semidirect
17. #16839681C4semidirect
20. C4×C4⋊Dic61C4semidirect
22. C2×C4×Q8×C61C4semidirect
170. C2×C4×Q8#6923semidirect
187. C61#11810semidirect

Maximal subgroups

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

subgroupindex
#214617162
#214610722
#214608162
#214608142
#214490892
#1181061