Finite Group Databasebeta

← #30598251

Subgroups — #30598251

Order 15616.

1030 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 160 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#21461675C212yesmaximal
2C2×#16840622C214yesmaximal, complemented
3C22×#14519775C211yesmaximal, characteristic
4C2×#14519844C2214yes—
5C2×C22⋊#92420—22no—
6#16840628—22no—
7#16840622—28no—
8C23×Dic122C411yescharacteristic
9C2×#14519775—28no—
10C2×#14519775—22no—
11C2×#14519775C412yes—
12C2×#14519775C2212yes—
13C22×#11758196—22no—
14C22×C4×Dic61C2211yescharacteristic
15C23×Q8×C61C411yescharacteristic
16C22×#92420—24no—
17#14519844—28no—
18C22⋊#92420—44no—
19C22×Dic122—21no—
20C22×Dic122—24no—
21C22×Dic122—42no—
22C22×Dic122—42no—
23C22×Dic122C2×C412yes—
24C23×Dic61—21no—
25C23×Dic61D412yes—
26#14519775—216no—
27#14519775—24no—
28#14519775—24no—
29#14519775—24no—
30#14519775D418yes—
31C2×#11758196—216no—
32C2×#11758196—24no—
33C2×#11758196—24no—
34C2×C4×Dic61—24no—
35C2×C4×Dic61C2×C411yescharacteristic
36C2×C4×Dic61D412yes—
37C2×C4×Dic61C2311yescharacteristic
38C22×Q8×C61—24no—
39C22×Q8×C61—21no—
40C22×Q8×C61—42no—
41C22×Q8×C61—42no—
42C22×Q8×C61C2×C412yes—
43C23×C4×C61—21no—
44C23×C4×C61C2×C411yescharacteristic
45C24×C61C22⋊C411yescharacteristic
46C2×Q8×C61—24no—
47C2×Q8×C61—22no—
48C2×Q8×C61—416no—
49C2×Q8×C61—42no—
50C2×Q8×C61—42no—
51C2×Q8×C61—42no—
52C2×Q8×C61—42no—
53C2×Q8×C61C22⋊C414yes—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—24no—
58C22×C4×C61—22no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22×C411yescharacteristic
63C22×Dic61—24no—
64C22×Dic61—24no—
65C22×Dic61—24no—
66C22×Dic61—22no—
67C22×Dic61—42no—
68C22×Dic61—42no—
69C22×Dic61C22⋊C412yes—
70C22×Dic61SD1614yes—
71C22×Dic61Q1614yes—
72C22×Dic61C2×D412yes—
73C2×Dic122—24no—
74C2×Dic122—22no—
75C2×Dic122—42no—
76C2×Dic122—416no—
77C2×Dic122—42no—
78C2×Dic122—42no—
79C2×Dic122—42no—
80C2×Dic122C22⋊C414yes—
81C2×#92420—44no—
82C2×#92420—44no—
83C2×#92420—44no—
84#11758196—216no—
85#11758196—216no—
86C4×Dic61—22no—
87C4×Dic61—24no—
88C4×Dic61C2×D414yes—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—22no—
92C23×C61#10012yes—
93C23×C61#10314yes—
94C23×C61C2×C22⋊C411yescharacteristic
95Q8×C61—48no—
96Q8×C61—44no—
97Q8×C61—44no—
98C2×C4×C61—28no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—24no—
104C2×C4×C61—48no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61#10012yes—
108C2×C4×C61C2×C22⋊C412yes—
109C2×Dic61—22no—
110C2×Dic61—22no—
111C2×Dic61—28no—
112C2×Dic61—28no—
113C2×Dic61—24no—
114C2×Dic61—48no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117C2×Dic61#10314yes—
118C2×Dic61#12014yes—
119C2×Dic61C2×SD1612yes—
120C2×Dic61C2×Q1612yes—
121C2×Dic61#13714yes—
122Dic122—48no—
123Dic122—44no—
124Dic122—44no—
125#92420—84no—
126#8342—611nomaximal, Sylow
127C22×C61—24no—
128C22×C61—22no—
129C22×C61—22no—
130C22×C61—24no—
131C22×C61#33314yescomplemented
132C22×C61#37911yescharacteristic
133C22×C61C2×#10011yescharacteristic
134C22×C61C2×#10011yescharacteristic
135C22×C61C2×#10312yes—
136C22×C61#41912yes—
137C4×C61—22no—
138C4×C61—22no—
139C4×C61—44no—
140Dic61—24no—
141Dic61—44no—
142Dic61#44814yes—
143Dic61#45114yes—
144C22×C4⋊Q8—611no—
145#1797—612no—
146C2×#333—614no—
147C2×C61—24no—
148C2×C61C2×#33311yescharacteristic
149C2×C61#154812yes—
150C2×C61#155811yescharacteristic
151C2×C61#178311yescharacteristic
152C2×C61#178712yes—
153C23×Q8—611no—
154C23×Q8#692311yescharacteristic
155C2×C4⋊Q8—612no—
156C2×C4⋊Q8—612no—
157C2×C4⋊Q8—1222no—
158C2×C4⋊Q8—1228no—
159C22×C4⋊C4—1222no—
160C22×C42—611no—
161C2×C4⋊C8—614no—
162C2×C22⋊C8—1222no—
163#336—1222no—
164#333—1228no—
165C61#834211yescharacteristic, complemented, Sylow
166C22×Q8—24no—
167C22×Q8—21no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—612no—
171C22×Q8—1224no—
172C22×Q8—1221no—
173C22×Q8—2442no—
174C22×Q8—2442no—
175C22×Q8C2×#692312yes—
176C23×C4—21no—
177C23×C4—612no—
178C23×C4—1221no—
179C23×C4C2×#692311yescharacteristic
180C22×C8—1224no—
181C4⋊Q8—618no—
182C4⋊Q8—1224no—
183C4⋊Q8—1224no—
184C4⋊Q8—12216no—
185C4⋊Q8—1224no—
186C2×C4⋊C4—1224no—
187C2×C4⋊C4—1224no—
188C2×C4⋊C4—12216no—
189C2×C42—611no—
190C2×C42—612no—
191C2×C42—611no—
192C2×C42—1224no—
193C4⋊C8—1228no—
194C22⋊C8—2444no—
195C24C22⋊#692311yescharacteristic
196C2×Q8—24no—
197C2×Q8—22no—
198C2×Q8—416no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—614no—
204C2×Q8—1224no—
205C2×Q8—1222no—
206C2×Q8—24416no—
207C2×Q8—2442no—
208C2×Q8—2442no—
209C2×Q8—2442no—
210C2×Q8—2442no—
211C2×Q8C22⋊#692314yes—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—612no—
220C22×C4—614no—
221C22×C4—614no—
222C22×C4—612no—
223C22×C4—1224no—
224C22×C4—1224no—
225C22×C4—1224no—
226C22×C4—1222no—
227C22×C4—2442no—
228C22×C4—2442no—
229C22×C4C22⋊#692311yescharacteristic
230C22×C4C22×#692311yescharacteristic
231C2×C8—2444no—
232C2×C8—2444no—
233C2×C8—2444no—
234C4⋊C4—12216no—
235C4⋊C4—12216no—
236C42—614no—
237C42—1224no—
238C42—1222no—
239C23—21no—
240C23—22no—
241C23—21no—
242C23#1451983712yes—
243C23#1451984014yes—
244C23C2×C22⋊#692311yescharacteristic
245Q8—44no—
246Q8—48no—
247Q8—44no—
248Q8—2444no—
249Q8—2448no—
250Q8—2444no—
251C2×C4—28no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—48no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—612no—
261C2×C4—612no—
262C2×C4—614no—
263C2×C4—614no—
264C2×C4—614no—
265C2×C4—1228no—
266C2×C4—1228no—
267C2×C4—1222no—
268C2×C4—1222no—
269C2×C4—1224no—
270C2×C4—2448no—
271C2×C4—2442no—
272C2×C4—2442no—
273C2×C4#1451983712yes—
274C2×C4C2×C22⋊#692312yes—
275C8—4884no—
276C22—24no—
277C22—22no—
278C22—22no—
279C22—24no—
280C22#1684062214yescomplemented
281C22#1684070311yescharacteristic
282C22C2×#1451983711yescharacteristic
283C22C2×#1451983711yescharacteristic
284C22C2×#1451984012yes—
285C22#1684075512yes—
286C4—22no—
287C4—22no—
288C4—44no—
289C4—614no—
290C4—614no—
291C4—1224no—
292C4—2444no—
293C2—24no—
294C2C2×#1684062211yescharacteristic
295C2#2146133212yes—
296C2#2146135211yescharacteristic
297C2#2146165111yescharacteristic
298C2#2146165812yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC23×Q8#6923
hypocenterC61#8342
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16840622C2semidirect
131. C22×C61#333semidirect
165. C61#8342semidirect
280. C22#16840622semidirect

Maximal subgroups

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

subgroupindex
#214616752
C2×#168406222
C22×#145197752
#834261