Finite Group Databasebeta

← #30603797

Subgroups — #30603797

Order 15616.

666 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 116 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#21462046C212yesmaximal, complemented
2#21460740C214yesmaximal, complemented
3C22×#14519689C211yesmaximal, characteristic
4C2×#14519851—22no—
5C22⋊#11758187—22no—
6C22⋊#11758187C2214yescomplemented
7C23×C4×D61C411yescharacteristic
8C23×#92418C2211yescharacteristic
9C2×#14519689—22no—
10C2×#14519689—22no—
11C2×#14519689C412yes—
12C2×#14519689C2212yes—
13C23×C8×C61C411yescharacteristic
14#14519851—44no—
15C2×#11758187—24no—
16C24×D61C811yescharacteristic
17C23×Dic61C811yescharacteristic
18C22×C4×D61—21no—
19C22×C4×D61—21no—
20C22×C4×D61—21no—
21C22×C4×D61—21no—
22C22×C4×D61—42no—
23C22×C4×D61—42no—
24C22×C4×D61C812yes—
25C22×C4×D61C2×C412yes—
26C22×#92418—42no—
27C22×#92418C2×C411yescharacteristic
28C22×#92418D412yes—
29C22×#92418D412yes—
30C22×#92418C2311yescharacteristic
31#14519689—24no—
32#14519689—24no—
33C23×C4×C61C2×C411yescharacteristic
34C22×C8×C61—21no—
35C22×C8×C61—21no—
36C22×C8×C61—42no—
37C22×C8×C61C2×C412yes—
38C24×C61C2×C811yescharacteristic
39C23×D61—21no—
40C23×D61—21no—
41C23×D61—24no—
42C23×D61—42no—
43C23×D61—42no—
44C23×D61C2×C812yes—
45C22×C4×C61—22no—
46C22×C4×C61—22no—
47C22×C4×C61C22⋊C411yescharacteristic
48C22×C4×C61C22⋊C411yescharacteristic
49C22×C4×C61C2×C811yescharacteristic
50C22×C4×C61M1611yescharacteristic
51C22×C4×C61M1611yescharacteristic
52C22×C4×C61C22×C411yescharacteristic
53C22×Dic61—21no—
54C22×Dic61—21no—
55C22×Dic61—42no—
56C22×Dic61C2×C812yes—
57C2×C4×D61—22no—
58C2×C4×D61—22no—
59C2×C4×D61—22no—
60C2×C4×D61—22no—
61C2×C4×D61—44no—
62C2×C4×D61—44no—
63C2×C4×D61—42no—
64C2×C4×D61—42no—
65C2×C4×D61—42no—
66C2×C4×D61—42no—
67C2×C4×D61—44no—
68C2×C4×D61—42no—
69C2×C4×D61—42no—
70C2×C8×C61—22no—
71C2×C8×C61—42no—
72C2×C8×C61—42no—
73C2×C8×C61—42no—
74C2×#92418—24no—
75C2×#92418—41no—
76C2×#92418—42no—
77C2×#92418—41no—
78C2×#92418C22⋊C412yes—
79C2×#92418C2×D412yes—
80#11758187—44no—
81C23×C61—22no—
82C23×C61—22no—
83C23×C61—22no—
84C23×C61C22⋊C811yescharacteristic
85C23×C61C22⋊C811yescharacteristic
86C23×C61C22×C811yescharacteristic
87C22×D61—24no—
88C22×D61—24no—
89C22×D61—22no—
90C22×D61—22no—
91C22×D61—42no—
92C22×D61—416no—
93C22×D61—42no—
94C22×D61—42no—
95C22×D61—42no—
96C22×D61—42no—
97C22×D61—42no—
98C22×D61M3214yescomplemented
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61C22⋊C811yescharacteristic
107C2×C4×C61C22⋊C811yescharacteristic
108C2×C4×C61#9912yes—
109C2×C4×C61#10012yes—
110C2×C4×C61C2×C22⋊C411yescharacteristic
111C2×C4×C61C2×M1611yescharacteristic
112C2×Dic61—22no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C4×D61—44no—
117C4×D61—44no—
118C4×D61—44no—
119C4×D61—42no—
120C4×D61—42no—
121C8×C61—42no—
122#92418—22no—
123#92418—22no—
124#11418—611nomaximal, Sylow
125C22×C61—28no—
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—21no—
132C22×C61#32312yes—
133C22×C61#34912yes—
134C22×C61C2×C22⋊C811yescharacteristic
135C22×C61C2×M3212yes—
136C2×D61—24no—
137C2×D61—48no—
138C2×D61—48no—
139C2×D61—44no—
140C2×D61—42no—
141C2×D61—42no—
142C2×D61—44no—
143C4×C61—21no—
144C4×C61—21no—
145C4×C61#32312yes—
146C4×C61C2×#9911yescharacteristic
147C4×C61C2×#10011yescharacteristic
148Dic61—42no—
149C22×C22⋊C8—611no—
150#2112—612no—
151#1314—614no—
152C2×C61—22no—
153C2×C61—22no—
154C2×C61—22no—
155C2×C61C2×#32311yescharacteristic
156C2×C61#211211yescharacteristic
157C2×C61C2×#34911yescharacteristic
158D61—44no—
159C24×C4—611no—
160C23×C8—611no—
161C23×C8#692311yescharacteristic
162C2×M32—1222no—
163C2×C22⋊C8—612no—
164C2×C22⋊C8—612no—
165C2×C22⋊C8—1222no—
166C2×C22⋊C8—1222no—
167C22⋊C16—614no—
168C22⋊C16—1222no—
169C61#1141811yescharacteristic, complemented, Sylow
170C25—611no—
171C23×C4—611no—
172C23×C4—612no—
173C23×C4—612no—
174C23×C4—1221no—
175C23×C4—1221no—
176C23×C4—1221no—
177C23×C4—1221no—
178C23×C4—2442no—
179C23×C4—2442no—
180C23×C4C2×#692311yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8—612no—
185C22×C8—612no—
186C22×C8—611no—
187C22×C8—611no—
188C22×C8—2442no—
189C22×C8C2×#692312yes—
190M32—2444no—
191C2×C16—1224no—
192C22⋊C8—1224no—
193C22⋊C8—1224no—
194C24—612no—
195C24—1224no—
196C24—1221no—
197C24—1221no—
198C24—2442no—
199C24—2442no—
200C24D61⋊C811yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—612no—
204C22×C4—1221no—
205C22×C4—1221no—
206C22×C4—1222no—
207C22×C4—1222no—
208C22×C4—1222no—
209C22×C4—1222no—
210C22×C4—2444no—
211C22×C4—2442no—
212C22×C4—2442no—
213C22×C4—2444no—
214C22×C4—2442no—
215C22×C4—2442no—
216C22×C4—2444no—
217C22×C4—2442no—
218C22×C4—2442no—
219C22×C4—2442no—
220C22×C4D61⋊C811yescharacteristic
221C22×C4#1175821311yescharacteristic
222C22×C4#1175821311yescharacteristic
223C22×C4C22⋊#692311yescharacteristic
224C22×C4C22⋊#692311yescharacteristic
225C22×C4C22×#692311yescharacteristic
226C2×C8—22no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—612no—
231C2×C8—612no—
232C2×C8—1224no—
233C2×C8—2441no—
234C2×C8—2442no—
235C2×C8—2441no—
236C16—2444no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—614no—
241C23—1224no—
242C23—1224no—
243C23—1222no—
244C23—1222no—
245C23—2442no—
246C23—2442no—
247C23—2442no—
248C23—24416no—
249C23—2442no—
250C23—2442no—
251C23—2442no—
252C23#1451983411yescharacteristic
253C23#1451983411yescharacteristic
254C23C2×D61⋊C811yescharacteristic
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—1222no—
263C2×C4—2444no—
264C2×C4—2444no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4—2442no—
268C2×C4—2442no—
269C2×C4—2444no—
270C2×C4—2442no—
271C2×C4#1451983411yescharacteristic
272C2×C4#1451983411yescharacteristic
273C2×C4#1451983612yes—
274C2×C4#1451983712yes—
275C2×C4C2×C22⋊#692311yescharacteristic
276C2×C4C2×#1175821311yescharacteristic
277C8—42no—
278C8—1222no—
279C8—1222no—
280C22—28no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—1224no—
288C22—2444no—
289C22—2444no—
290C22—2448no—
291C22—2448no—
292C22—2442no—
293C22—2442no—
294C22#1684060612yes—
295C22#1684065912yes—
296C22C2×#1451983411yescharacteristic
297C22D61⋊M3212yes—
298C4—21no—
299C4—21no—
300C4—2442no—
301C4#1684060612yes—
302C4C2×#1451983611yescharacteristic
303C4C2×#1451983711yescharacteristic
304C2—22no—
305C2—22no—
306C2—22no—
307C2—2444no—
308C2C2×#1684060611yescharacteristic
309C2#2146204811yescharacteristic
310C2C2×#1684065911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC23×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC23×C8×C61C4
socleC22×C61C2×C22⋊C8
prime radicalC24×D61C8
cosocleC22×#92418C23
hypercenterC23×C8#6923
hypocenterC61#11418
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. #21462046C2semidirect
2. #21460740C2semidirect
6. C22⋊#11758187C22semidirect
98. C22×D61M32semidirect
169. C61#11418semidirect

Maximal subgroups

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

subgroupindex
#214620462
#214607402
C22×#145196892
#1141861