Finite Group Databasebeta

← #29707421

Subgroups — #29707421

Order 15616.

1506 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1506 conjugacy classes of subgroups — too many to draw. Only its 275 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#21448394C213yesmaximal
2#21446985C214yesmaximal, complemented
3#21446982C214yesmaximal, complemented
4C2×#16839529C213yesmaximal, complemented
5C2×C61×#543C211yesmaximal, characteristic, complemented
6C23×C22⋊D61C2211yescharacteristic
7C22×C22⋊Dic61C2213yes—
8#16839641—26no—
9#16839639—26no—
10#16839623C2216yes—
11#16839617C2216yes—
12#16839529C22112yes—
13C61×#543—26no—
14C61×#543C2214yes—
15C2×C22⋊Q8×C61—26no—
16C22×C22⋊C4×C61C2213yes—
17C24×D61C2311yescharacteristic
18C22×C22⋊D61—23no—
19C22×C22⋊D61—23no—
20C22×C22⋊D61—26no—
21C22×C22⋊D61Q814yescomplemented
22C23×Dic61—23no—
23C23×Dic61C2311yescharacteristic
24C2×C22⋊Dic61—212no—
25C2×C22⋊Dic61—23no—
26C2×C22⋊Dic61C2316yes—
27C2×#11758198—212no—
28#14519684—212no—
29C25×C61C2311yescharacteristic
30C22×Q8×C61—22no—
31C23×C4×C61D416yes—
32C22⋊Q8×C61—224no—
33C22⋊Q8×C61—224no—
34C2×C4⋊C4×C61—26no—
35C2×C22⋊C4×C61—26no—
36C2×C22⋊C4×C61—26no—
37C2×C22⋊C4×C61D416yes—
38C2×C22⋊C4×C61C2316yes—
39C24×C61—82no—
40C24×C61C2×D413yes—
41C24×C61C2×D413yes—
42C24×C61C2×Q812yes—
43C24×C61#4116yes—
44C24×C61C2411yescharacteristic
45C23×D61—23no—
46C23×D61—23no—
47C23×D61—82no—
48C23×D61C2×Q812yescomplemented
49C2×Q8×C61—28no—
50C2×Q8×C61—84no—
51C22×C4×C61—26no—
52C22×C4×C61—26no—
53C22×C4×C61—86no—
54C22×C4×C61C2×D4112yes—
55C2×C22⋊D61—212no—
56C2×C22⋊D61—212no—
57C2×C22⋊D61—43no—
58C2×C22⋊D61—41no—
59C2×C22⋊D61—46no—
60C2×C22⋊D61—46no—
61C22×Dic61—23no—
62C22×Dic61—23no—
63C22×Dic61—26no—
64C22×Dic61—46no—
65C22×Dic61—86no—
66C22×Dic61C2×Q812yescomplemented
67C4⋊C4×C61—224no—
68C22⋊C4×C61—212no—
69C22⋊C4×C61—212no—
70C22⋊Dic61—212no—
71C22⋊Dic61—46no—
72#11758198—812no—
73C23×C61—46no—
74C23×C61—46no—
75C23×C61—42no—
76C23×C61—82no—
77C23×C61—86no—
78C23×C61#12013yes—
79C23×C61#12011yescharacteristic
80C23×C61C22⋊Q816yes—
81C23×C61C22⋊Q816yes—
82C23×C61#12416yes—
83C23×C61#12416yes—
84C23×C61C22×D413yes—
85C23×C61C22×Q811yescharacteristic
86C23×C61C2×#4113yes—
87C22×D61—26no—
88C22×D61—412no—
89C22×D61—41no—
90C22×D61—43no—
91C22×D61—82no—
92C22×D61—86no—
93Q8×C61—88no—
94C2×C4×C61—212no—
95C2×C4×C61—412no—
96C2×C4×C61—412no—
97C2×C4×C61—86no—
98C22⋊D61—412no—
99C22⋊D61—44no—
100C2×Dic61—26no—
101C2×Dic61—41no—
102C2×Dic61—43no—
103C2×Dic61—412no—
104C2×Dic61—46no—
105C2×Dic61—812no—
106#23468—611nomaximal, Sylow
107C22×C61—26no—
108C22×C61—22no—
109C22×C61—26no—
110C22×C61—46no—
111C22×C61—41no—
112C22×C61—43no—
113C22×C61—46no—
114C22×C61—46no—
115C22×C61—46no—
116C22×C61#39316yes—
117C22×C61#39312yes—
118C22×C61C2×#12011yescharacteristic
119C22×C61C2×C22⋊Q813yes—
120C22×C61C2×#12413yes—
121C2×D61—412no—
122C2×D61—42no—
123C2×D61—86no—
124C2×D61—82no—
125C4×C61—412no—
126Dic61—42no—
127Dic61—86no—
128C2×#543D6111yescharacteristic, complemented
129#2308—613no—
130#2030—614no—
131#2029—614no—
132C2×#342—613no—
133C2×C61—23no—
134C2×C61—23no—
135C2×C61—22no—
136C2×C61—26no—
137C2×C61C2×#39311yescharacteristic
138C2×C61#302612yes—
139D61—82no—
140C23×D4—611no—
141#543—26no—
142#543C2×D6114yes—
143C2×C22⋊Q8—26no—
144C22×C22⋊C4—613no—
145C22×C22⋊C4C2×D6113yes—
146#386—1226no—
147#386—1226no—
148#381—616no—
149#379—616no—
150#342—6112no—
151C61#2346811yescharacteristic, complemented, Sylow
152C25—611no—
153C25C22×D6111yescharacteristic
154C22×Q8—22no—
155C22×D4—614no—
156C22×D4—1226no—
157C22×D4—1223no—
158C22×D4—1223no—
159C23×C4—611no—
160C23×C4—1223no—
161C23×C4C22⋊D6116yes—
162C22⋊Q8—224no—
163C22⋊Q8—224no—
164C2×C4⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—616no—
168C2×C22⋊C4—12212no—
169C2×C22⋊C4—12212no—
170C2×C22⋊C4—1223no—
171C2×C22⋊C4C22⋊D6116yes—
172C2×C22⋊C4C22×D6116yes—
173#95—12212no—
174C24—82no—
175C24—612no—
176C24—1223no—
177C24—1223no—
178C24—4882no—
179C24Q8×D6112yes—
180C24#1175822616yes—
181C24C2×C22⋊D6113yes—
182C24C2×C22⋊D6113yes—
183C24C23×D6111yescharacteristic
184C2×Q8—28no—
185C2×Q8—84no—
186C2×D4—12212no—
187C2×D4—12212no—
188C2×D4—2443no—
189C2×D4—2441no—
190C2×D4—2446no—
191C2×D4—2446no—
192C22×C4—26no—
193C22×C4—26no—
194C22×C4—86no—
195C22×C4—612no—
196C22×C4—1223no—
197C22×C4—1223no—
198C22×C4—1226no—
199C22×C4—4886no—
200C22×C4—2446no—
201C22×C4C2×C22⋊D61112yes—
202C4⋊C4—224no—
203C22⋊C4—212no—
204C22⋊C4—212no—
205C22⋊C4—12212no—
206C22⋊C4—48812no—
207C22⋊C4—2446no—
208C23—46no—
209C23—46no—
210C23—42no—
211C23—82no—
212C23—86no—
213C23—1226no—
214C23—4886no—
215C23—4882no—
216C23—24412no—
217C23—2441no—
218C23—2443no—
219C23#1451974613yes—
220C23#1451974611yescharacteristic
221C23#1451975316yes—
222C23#1451975316yes—
223C23#1451976616yes—
224C23#1451976616yes—
225C23C22×C22⋊D6113yes—
226C23C2×Q8×D6111yescharacteristic
227C23C2×#1175822613yes—
228Q8—88no—
229D4—24412no—
230D4—2444no—
231C2×C4—212no—
232C2×C4—412no—
233C2×C4—412no—
234C2×C4—86no—
235C2×C4—1226no—
236C2×C4—48812no—
237C2×C4—24412no—
238C2×C4—2446no—
239C2×C4—2441no—
240C2×C4—2443no—
241C22—22no—
242C22—26no—
243C22—26no—
244C22—46no—
245C22—41no—
246C22—43no—
247C22—46no—
248C22—46no—
249C22—46no—
250C22—4886no—
251C22—4882no—
252C22—24412no—
253C22—2442no—
254C22#1683967316yes—
255C22#1683967312yes—
256C22C2×#1451974611yescharacteristic
257C22C2×#1451975313yes—
258C22C2×#1451976613yes—
259C4—412no—
260C4—4886no—
261C4—2442no—
262C2—23no—
263C2—23no—
264C2—22no—
265C2—26no—
266C2—4882no—
267C2C2×#1683967311yescharacteristic
268C2#2145521612yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC2×C61×#543C2
socleC22×C61C2×#120
prime radicalC23×C22⋊D61C22
hypercenterC2×#543D61
hypocenterC61#23468
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #21446985C2semidirect
3. #21446982C2semidirect
4. C2×#16839529C2semidirect
5. C2×C61×#543C2semidirect
21. C22×C22⋊D61Q8semidirect
48. C23×D61C2×Q8semidirect
66. C22×Dic61C2×Q8semidirect
128. C2×#543D61semidirect
151. C61#23468semidirect

Maximal subgroups

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

subgroupindex
#214483942
#214469852
#214469822
C2×#168395292
C2×C61×#5432
#2346861