Finite Group Databasebeta

← #30595575

Subgroups — #30595575

Order 15616.

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

normal subgroups only

This group has 658 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#21461545C212yesmaximal
2#21460639C214yesmaximal, complemented
3C2×#16839669C211yesmaximal, characteristic
4C2×C22⋊#92420—22no—
5C2×#14519832—22no—
6#16840628C2214yes—
7C22×C4⋊Dic61C411yescharacteristic
8C22×C4×Dic61C2211yescharacteristic
9#16839669—22no—
10#16839669C412yes—
11#16839669C2212yes—
12C2×#14519684—22no—
13C23×Q8×C61C411yescharacteristic
14C22×#92420—24no—
15C22⋊#92420—28no—
16#14519832—44no—
17C23×Dic61—21no—
18C23×Dic61D412yes—
19C2×C4⋊Dic61—21no—
20C2×C4⋊Dic61—42no—
21C2×C4⋊Dic61C2×C412yes—
22C2×C4×Dic61—42no—
23C2×C4×Dic61C2×C411yescharacteristic
24C2×C4×Dic61D412yes—
25C2×C4×Dic61C2311yescharacteristic
26#14519684—24no—
27#14519684—24no—
28C22×Q8×C61—24no—
29C22×Q8×C61—21no—
30C22×Q8×C61—42no—
31C22×Q8×C61—42no—
32C22×Q8×C61C2×C412yes—
33C23×C4×C61—21no—
34C23×C4×C61C2×C411yescharacteristic
35C24×C61C22⋊C411yescharacteristic
36C2×Q8×C61—24no—
37C2×Q8×C61—22no—
38C2×Q8×C61—416no—
39C2×Q8×C61—42no—
40C2×Q8×C61—42no—
41C2×Q8×C61—42no—
42C2×Q8×C61—42no—
43C2×Q8×C61M1614yes—
44C22×C4×C61—22no—
45C22×C4×C61—22no—
46C22×C4×C61—22no—
47C22×C4×C61—24no—
48C22×C4×C61—22no—
49C22×C4×C61—42no—
50C22×C4×C61—42no—
51C22×C4×C61C22⋊C411yescharacteristic
52C22×C4×C61C22×C411yescharacteristic
53C22×Dic61—24no—
54C22×Dic61—24no—
55C22×Dic61—22no—
56C22×Dic61—42no—
57C22×Dic61—42no—
58C22×Dic61—44no—
59C22×Dic61C22⋊C412yes—
60C22×Dic61C2×D412yes—
61C2×#92420—28no—
62C2×#92420—44no—
63C2×#92420—44no—
64C4⋊Dic61—42no—
65C4⋊Dic61—42no—
66C4×Dic61—24no—
67C4×Dic61—41no—
68C4×Dic61—41no—
69C23×C61—21no—
70C23×C61—21no—
71C23×C61—22no—
72C23×C61#10012yes—
73C23×C61#10414yes—
74C23×C61C2×C22⋊C411yescharacteristic
75Q8×C61—48no—
76Q8×C61—44no—
77Q8×C61—44no—
78C2×C4×C61—28no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—24no—
84C2×C4×C61—48no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61#9912yes—
88C2×C4×C61C2×M1612yes—
89C2×Dic61—22no—
90C2×Dic61—22no—
91C2×Dic61—22no—
92C2×Dic61—22no—
93C2×Dic61—22no—
94C2×Dic61—48no—
95C2×Dic61—42no—
96C2×Dic61—44no—
97C2×Dic61—84no—
98C2×Dic61#13714yes—
99#92420—48no—
100#7629—611nomaximal, Sylow
101C22×C61—22no—
102C22×C61—22no—
103C22×C61—24no—
104C22×C61—24no—
105C22×C61#32914yes—
106C22×C61#40711yescharacteristic
107C22×C61C2×#9911yescharacteristic
108C22×C61C2×#10011yescharacteristic
109C22×C61#41912yes—
110C22×C61C2×#10412yes—
111C4×C61—22no—
112C4×C61—22no—
113C4×C61—44no—
114Dic61—24no—
115Dic61—42no—
116Dic61—82no—
117C2×#391—611no—
118#1749—612no—
119#1272—614no—
120C2×C61—24no—
121C2×C61#146211yescharacteristic
122C2×C61#149112yes—
123C2×C61#150812yes—
124C2×C61C2×#32911yescharacteristic
125C2×C61#155811yescharacteristic
126C23×Q8#692311yescharacteristic
127C22×C4⋊C4—611no—
128C22×C42—611no—
129C2×C22⋊C8—1222no—
130C2×#97—1222no—
131#391—612no—
132#391—612no—
133#391—1222no—
134C2×#95—1222no—
135#336—614no—
136C61#762911yescharacteristic, complemented, Sylow
137C22×Q8—24no—
138C22×Q8—21no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8C2×#692312yes—
142C23×C4—21no—
143C23×C4—612no—
144C23×C4—1221no—
145C23×C4C2×#692311yescharacteristic
146C22×C8—1224no—
147C2×C4⋊C4—612no—
148C2×C4⋊C4—1221no—
149C2×C4⋊C4—2442no—
150C2×C42—612no—
151C2×C42—611no—
152C2×C42—611no—
153C2×C42—2442no—
154C22⋊C8—1228no—
155#97—2444no—
156#95—1224no—
157#95—1224no—
158C24C22⋊#692311yescharacteristic
159C2×Q8—24no—
160C2×Q8—22no—
161C2×Q8—416no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8#1175821714yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—612no—
175C22×C4—612no—
176C22×C4—1224no—
177C22×C4—1224no—
178C22×C4—1222no—
179C22×C4—2442no—
180C22×C4—2442no—
181C22×C4—2444no—
182C22×C4C22⋊#692311yescharacteristic
183C22×C4C22×#692311yescharacteristic
184C2×C8—1228no—
185C2×C8—2444no—
186C2×C8—2444no—
187C4⋊C4—2442no—
188C4⋊C4—2442no—
189C42—1224no—
190C42—2441no—
191C42—2441no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23#1451983712yes—
196C23#1451984214yes—
197C23C2×C22⋊#692311yescharacteristic
198Q8—44no—
199Q8—48no—
200Q8—44no—
201C2×C4—28no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—48no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—614no—
211C2×C4—1222no—
212C2×C4—1222no—
213C2×C4—1222no—
214C2×C4—1222no—
215C2×C4—1222no—
216C2×C4—4884no—
217C2×C4—2448no—
218C2×C4—2442no—
219C2×C4—2444no—
220C2×C4#1451983512yes—
221C2×C4C2×#1175821712yes—
222C8—2448no—
223C22—22no—
224C22—22no—
225C22—24no—
226C22—24no—
227C22#1684061814yes—
228C22#1684073211yescharacteristic
229C22C2×#1451983511yescharacteristic
230C22C2×#1451983711yescharacteristic
231C22#1684075512yes—
232C22C2×#1451984212yes—
233C4—22no—
234C4—22no—
235C4—44no—
236C4—1224no—
237C4—4882no—
238C4—2442no—
239C2—24no—
240C2#2146106411yescharacteristic
241C2#2146116112yes—
242C2#2146121912yes—
243C2C2×#1684061811yescharacteristic
244C2#2146135211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC23×Q8#6923
hypocenterC61#7629
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #21460639C2semidirect
136. C61#7629semidirect

Maximal subgroups

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

subgroupindex
#214615452
#214606392
C2×#168396692
#762961