Finite Group Databasebeta

← #29611069

Subgroups — #29611069

Order 15616.

1530 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 411 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
1C2×#16839632C211yesmaximal, characteristic, complemented
2C2×#16839630C211yesmaximal, characteristic, complemented
3#21444072C216yesmaximal, complemented
4#21444064C216yesmaximal
5C22×#126×C61C211yesmaximal, characteristic
6C22×C4⋊Dic61C2216yes—
7C22×C22⋊Dic61C2214yes—
8#16839632—24no—
9#16839630—24no—
10C2×#14519684C2216yes—
11C2×#14519684C22112yes—
12C2×#126×C61—24no—
13C2×#126×C61—24no—
14C2×#126×C61C418yescomplemented
15C22×C4⋊C4×C61C2213yes—
16C22×C22⋊C4×C61C2213yes—
17C22×C42×C61C2211yescharacteristic
18C23×Dic61C2314yes—
19C23×Dic61C2314yes—
20C2×C4⋊Dic61—212no—
21C2×C4⋊Dic61—412no—
22C2×C22⋊Dic61—412no—
23#14519684—212no—
24#14519684—412no—
25C25×C61C2311yescharacteristic
26C23×C4×C61C2313yes—
27C23×C4×C61C2313yes—
28#126×C61—232no—
29C2×C4⋊C4×C61—26no—
30C2×C4⋊C4×C61—26no—
31C2×C4⋊C4×C61C2×C4112yes—
32C2×C22⋊C4×C61—26no—
33C2×C22⋊C4×C61—26no—
34C2×C22⋊C4×C61C2×C4112yes—
35C2×C42×C61C2×C414yes—
36C2×C42×C61D414yes—
37C2×C42×C61D414yes—
38C24×C61—21no—
39C24×C61—21no—
40C24×C61—26no—
41C24×C61—26no—
42C24×C61C22×C412yes—
43C24×C61C2411yescharacteristic
44C22×C4×C61—26no—
45C22×C4×C61—26no—
46C22×C4×C61—23no—
47C22×C4×C61—23no—
48C22×C4×C61—26no—
49C22×C4×C61—26no—
50C22×C4×C61C22×C416yes—
51C22×C4×C61C22×C416yes—
52C22×C4×C61C2×D416yes—
53C22×C4×C61C2×D416yes—
54C22×Dic61—24no—
55C22×Dic61—24no—
56C22×Dic61—212no—
57C22×Dic61—212no—
58C22×Dic61—212no—
59C22×Dic61—212no—
60C4⋊C4×C61—224no—
61C22⋊C4×C61—224no—
62C42×C61C22⋊C4116yescomplemented
63C22⋊Dic61—416no—
64C4⋊Dic61—424no—
65C23×C61—22no—
66C23×C61—212no—
67C23×C61—43no—
68C23×C61—43no—
69C23×C61—46no—
70C23×C61—44no—
71C23×C61—412no—
72C23×C61C23×C411yescharacteristic
73C23×C61C22×D411yescharacteristic
74C23×C61C22×D411yescharacteristic
75C23×C61H(5,𝔽2)16yes—
76C23×C612−1+416yes—
77C2×C4×C61—212no—
78C2×C4×C61—26no—
79C2×C4×C61—26no—
80C2×C4×C61—212no—
81C2×C4×C61—26no—
82C2×C4×C61—212no—
83C2×C4×C61—43no—
84C2×C4×C61—43no—
85C2×C4×C61—46no—
86C2×C4×C61C2×C22⋊C4112yes—
87C2×Dic61—412no—
88C2×Dic61—412no—
89C2×Dic61—44no—
90C2×Dic61—44no—
91C2×Dic61—412no—
92C2×Dic61—412no—
93#14665—611nomaximal, Sylow
94C22×C61—46no—
95C22×C61—48no—
96C22×C61—81no—
97C22×C61—83no—
98C22×C61—84no—
99C22×C61C22×C22⋊C411yescharacteristic
100C22×C61#52016yes—
101C22×C61#53513yes—
102C22×C61#53613yes—
103C22×C61#53716yes—
104C22×C61#56014yes—
105C22×C61#563112yes—
106C4×C61—212no—
107C4×C61—46no—
108Dic61—84no—
109Dic61—84no—
110C22×#126D6111yescharacteristic
111C2×#384—611no—
112C2×#383—611no—
113#1441—616no—
114#1439—616no—
115C2×C61—82no—
116C2×C61#231113yes—
117C2×C61#238314yes—
118C2×C61#281811yescharacteristic
119C2×C61#282113yes—
120C2×C61#282314yes—
121C2×#126—24no—
122C2×#126—24no—
123C2×#126Dic6118yescomplemented
124C22×C4⋊C4—616no—
125C22×C4⋊C4C2×D6113yes—
126C22×C22⋊C4—614no—
127C22×C22⋊C4C2×D6113yes—
128C22×C42C2×D6111yescharacteristic
129#384—1224no—
130#383—1224no—
131C2×#95—616no—
132C2×#95—6112no—
133C61#1466511yescharacteristic, complemented, Sylow
134C25C22×D6111yescharacteristic
135C23×C4—614no—
136C23×C4—614no—
137C23×C4C22×D6113yes—
138C23×C4C22×D6113yes—
139#126—232no—
140C2×C4⋊C4—26no—
141C2×C4⋊C4—26no—
142C2×C4⋊C4—12212no—
143C2×C4⋊C4—24412no—
144C2×C4⋊C4C2×Dic61112yes—
145C2×C22⋊C4—26no—
146C2×C22⋊C4—26no—
147C2×C22⋊C4—24412no—
148C2×C22⋊C4C2×Dic61112yes—
149C2×C42C2×Dic6114yes—
150C2×C42C22⋊D6114yes—
151C2×C42C22⋊D6114yes—
152#95—12212no—
153#95—24412no—
154C24—21no—
155C24—21no—
156C24—26no—
157C24—26no—
158C24C22×Dic6112yes—
159C24C23×D6111yescharacteristic
160C22×C4—26no—
161C22×C4—26no—
162C22×C4—23no—
163C22×C4—23no—
164C22×C4—26no—
165C22×C4—26no—
166C22×C4—1224no—
167C22×C4—1224no—
168C22×C4—12212no—
169C22×C4—12212no—
170C22×C4—12212no—
171C22×C4—12212no—
172C22×C4C22×Dic6116yes—
173C22×C4C22×Dic6116yes—
174C22×C4C2×C22⋊D6116yes—
175C22×C4C2×C22⋊D6116yes—
176C4⋊C4—224no—
177C4⋊C4—24424no—
178C22⋊C4—224no—
179C22⋊C4—24416no—
180C42C22⋊Dic61116yescomplemented
181C23—22no—
182C23—212no—
183C23—44no—
184C23—412no—
185C23—43no—
186C23—43no—
187C23—46no—
188C23C23×Dic6111yescharacteristic
189C23C22×C22⋊D6111yescharacteristic
190C23C22×C22⋊D6111yescharacteristic
191C23#1451982316yes—
192C23#1451982416yes—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—26no—
196C2×C4—212no—
197C2×C4—26no—
198C2×C4—212no—
199C2×C4—43no—
200C2×C4—43no—
201C2×C4—46no—
202C2×C4—2444no—
203C2×C4—2444no—
204C2×C4—24412no—
205C2×C4—24412no—
206C2×C4—24412no—
207C2×C4—24412no—
208C2×C4C2×C22⋊Dic61112yes—
209C22—48no—
210C22—46no—
211C22—81no—
212C22—83no—
213C22—84no—
214C22C22×C22⋊Dic6111yescharacteristic
215C22#1684024516yes—
216C22#1684032613yes—
217C22#1684033313yes—
218C22#1684034016yes—
219C22#1684050314yes—
220C22#16840514112yes—
221C4—212no—
222C4—46no—
223C4—4884no—
224C4—4884no—
225C2—82no—
226C2#2144841113yes—
227C2#2144901114yes—
228C2#2145385911yescharacteristic
229C2#2145387713yes—
230C2#2145389114yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC22×#126×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC22×#126D61
hypocenterC61#14665
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C2×#16839632C2semidirect
2. C2×#16839630C2semidirect
3. #21444072C2semidirect
14. C2×#126×C61C4semidirect
62. C42×C61C22⋊C4semidirect
123. C2×#126Dic61semidirect
133. C61#14665semidirect
180. C42C22⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396322
C2×#168396302
#214440722
#214440642
C22×#126×C612
#1466561