Finite Group Databasebeta

← #30594751

Subgroups — #30594751

Order 15616.

342 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 52 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#21461798C211yesmaximal, characteristic
2#21461681C211yesmaximal, characteristic
3#21444158C211yesmaximal, characteristic
4C22×#11758217C411yescharacteristic, complemented
5C2×#14519835C412yes—
6C2×#14519834—21no—
7C2×C22⋊#92420C411yescharacteristic, complemented
8#16840704—21no—
9#16840640—21no—
10#16840628—21no—
11C2×#14519737C2211yescharacteristic
12#16839816—21no—
13#16839486C411yescharacteristic, complemented
14#16839485—21no—
15C61×#323C411yescharacteristic, complemented
16C2×#11758217—42no—
17C2×#11758217—41no—
18C22×#92420—21no—
19C22×#92420—21no—
20C2×C4⋊#6923—41no—
21C2×C22⋊#6923—22no—
22#14519835—42no—
23#14519834—42no—
24C22⋊#92420—41no—
25(C2×#6923)⋊C4—41no—
26C22×C22⋊D61C2×C411yescharacteristic
27C23×Dic61C2×C411yescharacteristic
28C22×C4×D61D411yescharacteristic
29#14519737—41no—
30#14519737—41no—
31C2×#11758196Q811yescharacteristic
32C2×#11758198D411yescharacteristic
33C2×C4×Dic61D411yescharacteristic
34#14519706—41no—
35C22⋊#92418—21no—
36#14519687—41no—
37C2×C22⋊C4×C61C2×C411yescharacteristic
38C22⋊C8×C61—21no—
39C24×C61C4211yescharacteristic
40C23×D61C22⋊C411yescharacteristic
41C22×#6923—42no—
42C22×C4×C61C22⋊C411yescharacteristic
43C22×C4×C61C4⋊C411yescharacteristic
44C2×C22⋊D61—21no—
45C2×C22⋊D61—21no—
46C2×C22⋊D61—41no—
47C2×C22⋊D61—41no—
48C22×Dic61—21no—
49C22×Dic61—21no—
50C22×Dic61—41no—
51C22×Dic61—41no—
52C22×Dic61C22⋊C411yescharacteristic
53C22×Dic61C4⋊C411yescharacteristic
54C22×Dic61C4⋊C411yescharacteristic
55C2×C4×D61—21no—
56C2×C4×D61—21no—
57C2×C4×D61—41no—
58C2×C4×D61—41no—
59C22⋊#6923—42no—
60C22⋊#6923—42no—
61#11758217—42no—
62#11758217—42no—
63C2×#92420—22no—
64C2×#92420—41no—
65C2×#92420—41no—
66C2×#92420—41no—
67C2×#92420—41no—
68C2×#92420—41no—
69C4⋊#6923—81no—
70C4⋊#6923—81no—
71C2×C8×C61—41no—
72C22⋊C4×C61—41no—
73#11758198—41no—
74#11758196—41no—
75C4×Dic61—21no—
76C4×Dic61—21no—
77C2×#92418—41no—
78C23×C61—21no—
79C23×C61—21no—
80C23×C61—21no—
81C23×C61—41no—
82C23×C61—41no—
83C23×C61#9511yescharacteristic
84C22×D61—21no—
85C22×D61—21no—
86C22×D61—41no—
87C22×D61—41no—
88C22×D61#10011yescharacteristic
89C22×D61#10111yescharacteristic
90C2×#6923—82no—
91C2×#6923—82no—
92C2×#6923—82no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C22⋊D61—42no—
98C22⋊D61—42no—
99C2×Dic61—21no—
100C2×Dic61—21no—
101C2×Dic61—21no—
102C2×Dic61—21no—
103C2×Dic61—21no—
104C2×Dic61—21no—
105C2×Dic61—42no—
106C2×Dic61—41no—
107C2×Dic61—41no—
108C2×Dic61—41no—
109C2×Dic61#9911yescharacteristic
110C2×Dic61#9911yescharacteristic
111C4×D61—41no—
112C4×D61—41no—
113C4×D61—81no—
114#92420—42no—
115#92420—81no—
116C8×C61—81no—
117#92418—81no—
118#7320—611nomaximal, Sylow
119C22×C61—21no—
120C22×C61—21no—
121C22×C61—41no—
122C22×C61—41no—
123C22×C61—41no—
124C22×C61—81no—
125C22×C61—81no—
126C22×C61#34211yescharacteristic
127C22×C61#34311yescharacteristic
128C22×C61#34411yescharacteristic
129C2×D61—21no—
130C2×D61—21no—
131C2×D61—41no—
132C2×D61—41no—
133C2×D61—41no—
134C2×D61—41no—
135C2×D61—81no—
136#6923—162no—
137C4×C61—41no—
138C4×C61—41no—
139Dic61—21no—
140Dic61—41no—
141Dic61—41no—
142Dic61AΣL(1,𝔽16)11yescharacteristic
143Dic61#35211yescharacteristic
144#1870—611no—
145#1799—611no—
146#1466—611no—
147C2×C61—21no—
148C2×C61—21no—
149C2×C61—81no—
150C2×C61#130511yescharacteristic
151C2×C61#139011yescharacteristic
152C2×C61#139111yescharacteristic
153D61—41no—
154D61—41no—
155C22×M16—611no—
156C2×C4×D4—611no—
157#432—1221no—
158C2×#99—612no—
159C2×C22⋊C8—611no—
160C2×C22⋊C8—1221no—
161#380—1221no—
162#342—1221no—
163#336—1221no—
164#323—611no—
165#323—1221no—
166#323#692311yescharacteristic, complemented
167C61#732011yescharacteristic, complemented, Sylow
168C22×D4—611no—
169C23×C4—611no—
170C23×C4—611no—
171C2×M16—2442no—
172C2×M16—2441no—
173C22×C8—1221no—
174C22×C8—1221no—
175C4×D4—2441no—
176C4×D4—2441no—
177C2×C4⋊C4—611no—
178C2×C4⋊C4—2441no—
179C2×C22⋊C4—611no—
180C2×C22⋊C4—1222no—
181C2×C22⋊C4C2×#692311yescharacteristic
182C2×C42—611no—
183C4⋊C8—2441no—
184#99—2442no—
185C22⋊C8—21no—
186C22⋊C8—1221no—
187C22⋊C8—2442no—
188C22⋊C8—2441no—
189#97—2441no—
190#95—2441no—
191C24—611no—
192C24C4×#692311yescharacteristic
193C2×D4—1221no—
194C2×D4—1221no—
195C2×D4—2441no—
196C2×D4—2441no—
197C22×C4—611no—
198C22×C4—611no—
199C22×C4—611no—
200C22×C4—1221no—
201C22×C4—1221no—
202C22×C4—1221no—
203C22×C4—1221no—
204C22×C4—2441no—
205C22×C4—2441no—
206C22×C4—2441no—
207C22×C4—2441no—
208C22×C4—2442no—
209C22×C4C4⋊#692311yescharacteristic
210C22×C4C22⋊#692311yescharacteristic
211M16—2442no—
212M16—2442no—
213C2×C8—41no—
214C2×C8—1222no—
215C2×C8—2441no—
216C2×C8—2441no—
217C2×C8—2441no—
218C2×C8—2441no—
219C2×C8—2441no—
220C2×C8—2441no—
221C4⋊C4—4881no—
222C4⋊C4—4881no—
223C4⋊C4—2441no—
224C22⋊C4—41no—
225C22⋊C4—2441no—
226C22⋊C4—2442no—
227C22⋊C4—2442no—
228C42—1221no—
229C42—1221no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23—611no—
236C23—611no—
237C23—1221no—
238C23—1221no—
239C23—2441no—
240C23—2441no—
241C23(C2×#6923)⋊C411yescharacteristic
242D4—2442no—
243D4—2442no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—611no—
249C2×C4—611no—
250C2×C4—1221no—
251C2×C4—1221no—
252C2×C4—1221no—
253C2×C4—1221no—
254C2×C4—1221no—
255C2×C4—1221no—
256C2×C4—4881no—
257C2×C4—4882no—
258C2×C4—4882no—
259C2×C4—4882no—
260C2×C4—2441no—
261C2×C4—2441no—
262C2×C4—2441no—
263C2×C4—2441no—
264C2×C4—2442no—
265C2×C4—2441no—
266C8—81no—
267C8—4881no—
268C8—4881no—
269C8—2442no—
270C22—21no—
271C22—21no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—81no—
276C22—81no—
277C22—1221no—
278C22—1221no—
279C22—4881no—
280C22—2441no—
281C22—2441no—
282C22—2441no—
283C22—2441no—
284C22#1684063911yescharacteristic
285C22#1684064111yescharacteristic
286C22#1684064411yescharacteristic
287C4—41no—
288C4—41no—
289C4—611no—
290C4—611no—
291C4—1221no—
292C4—9762no—
293C4—2441no—
294C4—2441no—
295C2—21no—
296C2—21no—
297C2—81no—
298C2—2441no—
299C2—2441no—
300C2#2146072111yescharacteristic
301C2#2146090511yescharacteristic
302C2#2146090611yescharacteristic
subgroupG/H
centerC22#16840639
commutatorC24×C61C42
FrattiniC2×C22⋊C4C2×#6923
FittingC61×#323C4
socleC22×C61#342
prime radicalC22×C22⋊D61C2×C4
cosocleC2×#14519737C22
hypercenter#323#6923
hypocenterC61#7320
LCSG ▷ C24×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
4. C22×#11758217C4semidirect
7. C2×C22⋊#92420C4semidirect
13. #16839486C4semidirect
15. C61×#323C4semidirect
166. #323#6923semidirect
167. C61#7320semidirect

Maximal subgroups

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

subgroupindex
#214617982
#214616812
#214441582
#732061