Finite Group Databasebeta

← #30604822

Subgroups — #30604822

Order 15616.

448 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461518C211yesmaximal, characteristic
2#21461052C211yesmaximal, characteristic
3#21460814C214yesmaximal
4#21451641C211yesmaximal, characteristic
5C2×#14519856—21no—
6#16840778—21no—
7#16840640—21no—
8#16840640C2212yes—
9#16840608—21no—
10#16840606C2212yes—
11C2×#14519734C2211yescharacteristic
12#16839700—21no—
13#16839697—21no—
14#16839681C412yescomplemented
15#16839673C2212yes—
16#16839658—21no—
17#16839628C411yescharacteristic, complemented
18#16839619—21no—
19C2×C4⋊Q8×C61C411yescharacteristic, complemented
20#14519856—24no—
21#14519856—42no—
22C2×C4⋊#6923—41no—
23C2×C22⋊#6923—22no—
24C2×C4×#6923—41no—
25C4⋊#92420—41no—
26C22⋊#92420—22no—
27C4×#92420—41no—
28C22×C4×D61D411yescharacteristic
29#14519734—41no—
30#14519734—41no—
31C2×C4⋊Dic61—21no—
32C2×#11758196—22no—
33C2×#11758196C2×C411yescharacteristic
34C2×#11758196D411yescharacteristic
35C2×#11758198—22no—
36C2×#11758198D411yescharacteristic
37C2×#11758198C2311yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39#14519684—21no—
40#14519684—22no—
41#14519684—22no—
42#14519684C2×C412yes—
43C22×Q8×C61C2×C412yes—
44C4⋊Q8×C61—41no—
45C4⋊Q8×C61—41no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C42×C61C2×C411yescharacteristic
49C23×D61C2×D411yescharacteristic
50C22×#6923—42no—
51C2×Q8×C61—42no—
52C2×Q8×C61—82no—
53C22×C4×C61—22no—
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22×C411yescharacteristic
57C22×Dic61—22no—
58C22×Dic61C22⋊C411yescharacteristic
59C22×Dic61C22⋊C411yescharacteristic
60C22×Dic61C2×D411yescharacteristic
61C2×C4×D61—21no—
62C2×C4×D61—21no—
63C2×C4×D61—22no—
64C2×C4×D61SD1612yes—
65C2×C4×D61Q1612yes—
66C22⋊#6923—24no—
67C22⋊#6923—42no—
68C2×#92420—42no—
69C4⋊#6923—44no—
70C4×#6923—44no—
71C4⋊C4×C61—42no—
72C4⋊C4×C61—41no—
73C4⋊C4×C61—41no—
74C42×C61—21no—
75C42×C61—21no—
76#11758198—41no—
77#11758198—41no—
78#11758198—82no—
79C4⋊Dic61—42no—
80#11758196—41no—
81#11758196—41no—
82#11758196—82no—
83C4×Dic61—21no—
84C4×Dic61—21no—
85C23×C61C2×C22⋊C411yescharacteristic
86C22×D61—21no—
87C22×D61—21no—
88C22×D61—41no—
89C22×D61—41no—
90C22×D61C2×SD1611yescharacteristic
91C22×D61C2×Q1611yescharacteristic
92C2×#6923—44no—
93C2×#6923—82no—
94C2×#6923—82no—
95Q8×C61—84no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61#9911yescharacteristic
104C2×C4×C61#9911yescharacteristic
105C2×C4×C61#10312yes—
106C2×C4×C61#10412yes—
107C2×Dic61—21no—
108C2×Dic61—21no—
109C2×Dic61—21no—
110C2×Dic61—21no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61#10312yes—
117C4×D61—22no—
118C4×D61—22no—
119C4×D61—22no—
120C4×D61—22no—
121#92420—82no—
122#11717—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61C2×#9911yescharacteristic
126C22×C61C2×#10311yescharacteristic
127C22×C61C2×#10411yescharacteristic
128C2×D61—21no—
129C2×D61—21no—
130C2×D61—41no—
131C2×D61—41no—
132C2×D61—41no—
133C2×D61—41no—
134C2×D61—81no—
135#6923—84no—
136C4×C61—41no—
137C4×C61—41no—
138C4×C61—82no—
139C4×C61#32812yes—
140C4×C61#32812yes—
141Dic61—22no—
142Dic61—41no—
143Dic61—41no—
144Dic61—82no—
145#2628—611no—
146#1740—611no—
147#1458—611no—
148#1351—614no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61C2×#32811yescharacteristic
152C2×C61#212211yescharacteristic
153C2×C61#212911yescharacteristic
154D61—41no—
155D61—41no—
156C2×C4⋊Q8#692311yescharacteristic, complemented
157C2×#117—611no—
158C2×#117—1221no—
159#431—1221no—
160#400—1221no—
161#399—1221no—
162#395—612no—
163#393—612no—
164#389—1221no—
165#382—611no—
166#380—1221no—
167#342—612no—
168#342—1221no—
169#324—1221no—
170#323—612no—
171C61#1171711yescharacteristic, complemented, Sylow
172C22×Q8C2×#692312yes—
173C23×C4—611no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—1224no—
177#117—2441no—
178#117—2441no—
179#117—2442no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—611no—
183C2×C4⋊C4—611no—
184C2×C4⋊C4—1222no—
185C2×C4⋊C4—1221no—
186C2×C4⋊C4—2441no—
187C2×C22⋊C4—611no—
188C2×C22⋊C4—611no—
189C2×C22⋊C4—1222no—
190C2×C22⋊C4—1222no—
191C2×C42—611no—
192C2×C42—2441no—
193C2×C42C2×#692311yescharacteristic
194C4⋊C8—2441no—
195C22⋊C8—1222no—
196C4×C8—2441no—
197#95—612no—
198#95—1222no—
199#95—1222no—
200#95—1221no—
201C24—611no—
202C2×Q8—42no—
203C2×Q8—82no—
204C22×C4—22no—
205C22×C4—611no—
206C22×C4—612no—
207C22×C4—612no—
208C22×C4—611no—
209C22×C4—611no—
210C22×C4—1221no—
211C22×C4—1221no—
212C22×C4—1222no—
213C22×C4—1222no—
214C22×C4—2442no—
215C22×C4C22⋊#692311yescharacteristic
216C22×C4C22⋊#692311yescharacteristic
217C22×C4C22×#692311yescharacteristic
218C2×C8—2442no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—4882no—
223C4⋊C4—2441no—
224C4⋊C4—2441no—
225C4⋊C4—2444no—
226C4⋊C4—2442no—
227C22⋊C4—1224no—
228C22⋊C4—4882no—
229C22⋊C4—2441no—
230C22⋊C4—2441no—
231C22⋊C4—2442no—
232C42—21no—
233C42—21no—
234C42—1221no—
235C42—1221no—
236C42—2444no—
237C23—611no—
238C23—611no—
239C23—1221no—
240C23—1221no—
241C23—2441no—
242C23—2441no—
243C23C2×C22⋊#692311yescharacteristic
244Q8—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—612no—
253C2×C4—1221no—
254C2×C4—1221no—
255C2×C4—1222no—
256C2×C4—1222no—
257C2×C4—1222no—
258C2×C4—1222no—
259C2×C4—1221no—
260C2×C4—1221no—
261C2×C4—4882no—
262C2×C4—4882no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2444no—
266C2×C4—2442no—
267C2×C4—2441no—
268C2×C4—2441no—
269C2×C4#1451983511yescharacteristic
270C2×C4#1451983511yescharacteristic
271C2×C4#1451984012yes—
272C2×C4#1451984212yes—
273C8—4882no—
274C22—21no—
275C22—21no—
276C22—1221no—
277C22—1221no—
278C22—4881no—
279C22—2441no—
280C22—2441no—
281C22—2441no—
282C22—2441no—
283C22C2×#1451983511yescharacteristic
284C22C2×#1451984011yescharacteristic
285C22C2×#1451984211yescharacteristic
286C4—41no—
287C4—41no—
288C4—82no—
289C4—1222no—
290C4—4882no—
291C4—4884no—
292C4—2441no—
293C4—2441no—
294C4#1684061612yes—
295C4#1684061612yes—
296C2—21no—
297C2—21no—
298C2—2441no—
299C2—2441no—
300C2C2×#1684061611yescharacteristic
301C2#2146207811yescharacteristic
302C2#2146210111yescharacteristic
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#11717
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
14. #16839681C4semidirect
17. #16839628C4semidirect
19. C2×C4⋊Q8×C61C4semidirect
156. C2×C4⋊Q8#6923semidirect
171. C61#11717semidirect

Maximal subgroups

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

subgroupindex
#214615182
#214610522
#214608142
#214516412
#1171761