Finite Group Databasebeta

← #30604469

Subgroups — #30604469

Order 15616.

378 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 112 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#21462179C211yesmaximal, characteristic, complemented
2C4×D61⋊C16C211yesmaximal, characteristic, complemented
3#21460786C212yesmaximal, complemented
4D4⋊#11758187C212yesmaximal, complemented
5C8×#11758224C211yesmaximal, characteristic
6C2×D61⋊C16—21no—
7C4⋊#11758187C2212yes—
8#16840657—21no—
9#16840648—21no—
10#16840648—21no—
11C4×#11758187C2212yes—
12C4×#11758224C411yescharacteristic
13C8×Dic122C411yescharacteristic
14#16839886C2211yescharacteristic
15D4×#92418C2211yescharacteristic
16C8×C22⋊D61—21no—
17#16839825—21no—
18#16839814C411yescharacteristic
19#16839812—21no—
20C2×C8×Dic61—21no—
21C4×C8×D61C2211yescharacteristic
22C8×D4×C61C411yescharacteristic
23D61⋊C16—24no—
24C2×#11758187—22no—
25C2×#11758187—22no—
26C2×#11758224C811yescharacteristic
27C22×#92418—21no—
28C2×C8×D61D411yescharacteristic
29C2×C8×D61D411yescharacteristic
30C4×Dic122C2×C411yescharacteristic
31#14519742C811yescharacteristic
32D4×Dic61C811yescharacteristic
33C4×C22⋊D61—21no—
34#14519737—21no—
35#14519735C811yescharacteristic
36#14519733—21no—
37C2×C4×Dic61—21no—
38C42×D61C2×C411yescharacteristic
39C4⋊#92418C2×C411yescharacteristic
40#14519706—21no—
41#14519706—21no—
42C22⋊#92418—21no—
43#14519689—21no—
44C4×#92418C2311yescharacteristic
45C8×Dic61—21no—
46C8×Dic61—21no—
47C8×Dic61—21no—
48C8×Dic61D411yescharacteristic
49C8×Dic61D411yescharacteristic
50C22×C8×C61—21no—
51C4×D4×C61C2×C411yescharacteristic
52C4⋊C8×C61C2×C411yescharacteristic
53C22⋊C8×C61—21no—
54C4×C8×C61C2×C411yescharacteristic
55C2×D4×C61C2×C811yescharacteristic
56C22×C4×C61—21no—
57C2×C22⋊D61—21no—
58C22×Dic61—21no—
59C22×Dic61—21no—
60#11758224—22no—
61#11758224C1614yescomplemented
62C2×C4×D61C22⋊C411yescharacteristic
63C2×C4×D61C2×C811yescharacteristic
64C2×C4×D61M1611yescharacteristic
65C2×Dic122C2×C811yescharacteristic
66C2×C8×C61—21no—
67C2×C8×C61—42no—
68C2×C8×C61C22⋊C411yescharacteristic
69C2×C8×C61C22⋊C411yescharacteristic
70C4⋊C4×C61C2×C811yescharacteristic
71C22⋊C4×C61—21no—
72C42×C61C22×C411yescharacteristic
73C22⋊Dic61—21no—
74#11758198—21no—
75C4⋊Dic61C2×C811yescharacteristic
76#11758196—21no—
77#11758196—21no—
78C4×Dic61—21no—
79C4×Dic61—21no—
80C4×Dic61—21no—
81C4×Dic61C22⋊C411yescharacteristic
82C4×Dic61C2×C811yescharacteristic
83C4×Dic61M1611yescharacteristic
84C2×#92418—21no—
85C2×#92418—42no—
86C2×#92418C2×D411yescharacteristic
87C2×#92418C2×D411yescharacteristic
88C8×D61—22no—
89C8×D61—22no—
90#11758187—24no—
91#11758187—42no—
92C23×C61—21no—
93C22×D61C22⋊C811yescharacteristic
94D4×C61—21no—
95D4×C61C2×C1612yescomplemented
96C2×C4×C61—21no—
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61C2×C22⋊C411yescharacteristic
100C2×C4×C61C22×C811yescharacteristic
101C2×C4×C61C2×M1611yescharacteristic
102C22⋊D61—24no—
103C2×Dic61—21no—
104C2×Dic61—24no—
105C2×Dic61—21no—
106C2×Dic61—42no—
107C2×Dic61C22⋊C811yescharacteristic
108C2×Dic61C22⋊C811yescharacteristic
109C2×Dic61C22⋊C811yescharacteristic
110C4×D61—21no—
111C4×D61—21no—
112C4×D61—22no—
113C4×D61C2×C1612yes—
114C4×D61M3212yes—
115Dic122—21no—
116Dic122C2×C1612yescomplemented
117C8×C61—21no—
118C8×C61—41no—
119C8×C61#10412yes—
120#92418—21no—
121#92418—41no—
122#92418#13512yes—
123#11613—611nomaximal, Sylow
124C22×C61—22no—
125C22×C61—41no—
126C22×C61—41no—
127C22×C61C2×C22⋊C811yescharacteristic
128C2×D61—21no—
129C2×D61—21no—
130C2×D61C22⋊C1612yes—
131C4×C61—21no—
132C4×C61—41no—
133C4×C61#41611yescharacteristic
134C4×C61C2×#10411yescharacteristic
135C4×C61C22×C1611yescharacteristic
136C4×C61C2×M3211yescharacteristic
137Dic61—22no—
138Dic61—21no—
139Dic61—41no—
140Dic61C22⋊C1612yes—
141Dic61#35012yes—
142C8×#41—611no—
143#2162—611no—
144C2×C4×C16—611no—
145Q8⋊C16—612no—
146D4⋊C16—612no—
147C2×C61—42no—
148C2×C61#147611yescharacteristic
149C2×C61C2×C22⋊C1611yescharacteristic
150C2×C61C2×#35011yescharacteristic
151D61—22no—
152C4×#41—611no—
153C22×C16—1221no—
154C8×Q8—611no—
155C8×Q8—611no—
156C8×D4—611no—
157C8×D4—1221no—
158C8×D4—1221no—
159C8×D4#692311yescharacteristic
160#431—611no—
161#431—1221no—
162C2×C4×C8—611no—
163C2×C4×C8—1221no—
164C4⋊C16—612no—
165C22⋊C16—1221no—
166C4×C16—612no—
167C4×C16—1221no—
168C4×C16—1221no—
169C61#1161311yescharacteristic, complemented, Sylow
170C2×#41—611no—
171C22×C8—21no—
172C22×C8—611no—
173C22×C8—611no—
174C22×C8—1221no—
175C4×Q8—611no—
176C4×Q8—611no—
177C4×D4—611no—
178C4×D4—1221no—
179C4×D4—1221no—
180C4×D4C2×#692311yescharacteristic
181#117—611no—
182#117—1221no—
183C2×C42—611no—
184C2×C42—1221no—
185C2×C16—1222no—
186C2×C16—1224no—
187C2×C16—1222no—
188C4⋊C8—611no—
189C4⋊C8—1221no—
190C4⋊C8—1221no—
191C4⋊C8C2×#692311yescharacteristic
192C22⋊C8—21no—
193C22⋊C8—1221no—
194C22⋊C8—1221no—
195C4×C8—611no—
196C4×C8—611no—
197C4×C8—611no—
198C4×C8—1221no—
199C4×C8—1221no—
200C4×C8—1221no—
201C4×C8C2×#692311yescharacteristic
202#41—614no—
203#41—1222no—
204C2×Q8—611no—
205C2×D4—1221no—
206C2×D4D61⋊C811yescharacteristic
207C22×C4—21no—
208C22×C4—611no—
209C22×C4—611no—
210C22×C4—611no—
211C22×C4—1221no—
212C22×C4—1221no—
213C2×C8—21no—
214C2×C8—42no—
215C2×C8—611no—
216C2×C8—611no—
217C2×C8—1222no—
218C2×C8—1222no—
219C2×C8—1221no—
220C2×C8—2442no—
221C2×C8C22⋊#692311yescharacteristic
222C2×C8C22⋊#692311yescharacteristic
223C4⋊C4—611no—
224C4⋊C4—1221no—
225C4⋊C4—1221no—
226C4⋊C4D61⋊C811yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—1221no—
229C22⋊C4—1221no—
230C42—611no—
231C42—611no—
232C42—611no—
233C42—1221no—
234C42—1221no—
235C42—1221no—
236C42C22×#692311yescharacteristic
237C16—1224no—
238C16—2442no—
239C23—21no—
240C23—611no—
241Q8—612no—
242Q8—1221no—
243D4—21no—
244D4—1224no—
245D4D61⋊C1612yescomplemented
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—611no—
250C2×C4—611no—
251C2×C4—612no—
252C2×C4—612no—
253C2×C4—611no—
254C2×C4—1222no—
255C2×C4—1224no—
256C2×C4—1221no—
257C2×C4—1221no—
258C2×C4—1221no—
259C2×C4—1221no—
260C2×C4—2442no—
261C2×C4C2×C22⋊#692311yescharacteristic
262C2×C4C2×D61⋊C811yescharacteristic
263C2×C4D61⋊M1611yescharacteristic
264C8—21no—
265C8—41no—
266C8—612no—
267C8—1221no—
268C8—2441no—
269C8#1451984112yes—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—612no—
274C22—1221no—
275C22—1221no—
276C22D61⋊(C22⋊C8)11yescharacteristic
277C4—21no—
278C4—41no—
279C4—612no—
280C4—612no—
281C4—1222no—
282C4—1221no—
283C4—2441no—
284C4#1684075211yescharacteristic
285C4C2×#1451984111yescharacteristic
286C4C2×D61⋊C1611yescharacteristic
287C4D61⋊M3211yescharacteristic
288C2—42no—
289C2—1222no—
290C2#2146111111yescharacteristic
291C2D61⋊(C22⋊C16)11yescharacteristic
292C2D61⋊#35011yescharacteristic
subgroupG/H
centerC2×C8C22⋊#6923
commutatorC4×C61C22×C16
FrattiniC42C22×#6923
FittingC8×D4×C61C4
socleC22×C61C2×C22⋊C8
prime radicalC2×#11758224C8
cosocleC4×#92418C23
hypercenterC8×D4#6923
hypocenterC61#11613
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #21462179C2semidirect
2. C4×D61⋊C16C2semidirect
3. #21460786C2semidirect
4. D4⋊#11758187C2semidirect
61. #11758224C16semidirect
95. D4×C61C2×C16semidirect
116. Dic122C2×C16semidirect
169. C61#11613semidirect
245. D4D61⋊C16semidirect

Maximal subgroups

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

subgroupindex
#214621792
C4×D61⋊C162
#214607862
D4⋊#117581872
C8×#117582242
#1161361