Finite Group Databasebeta

← #30603849

Subgroups — #30603849

Order 15616.

526 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 104 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#21461050C212yesmaximal
2#21460750C214yesmaximal, complemented
3#21448403C211yesmaximal, characteristic
4#16840778—22no—
5#16840607C2214yes—
6#16840605—22no—
7C23×Dic122C411yescharacteristic
8C2×#14519736C2211yescharacteristic
9#16839668C412yes—
10#16839668C2212yes—
11#16839667—22no—
12#16839623—22no—
13C2×#117×C61C411yescharacteristic
14C4⋊#92420—42no—
15C22⋊#92420—24no—
16C4×#92420—42no—
17C22×Dic122—21no—
18C22×Dic122—22no—
19C22×Dic122—42no—
20C22×Dic122—42no—
21C22×Dic122C814yescomplemented
22C22×Dic122C2×C412yes—
23C23×Dic61—21no—
24#14519736—42no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2311yescharacteristic
27C2×C22⋊Dic61D412yes—
28C2×C4×Dic61D412yes—
29#14519684—24no—
30#14519684—24no—
31C23×C4×C61C2×C411yescharacteristic
32#117×C61—42no—
33C2×C4⋊C4×C61—21no—
34C2×C22⋊C4×C61—21no—
35C2×C42×C61C2×C412yes—
36C24×C61C22⋊C411yescharacteristic
37C22×C4×C61—22no—
38C22×C4×C61—22no—
39C22×C4×C61—22no—
40C22×C4×C61C22⋊C411yescharacteristic
41C22×C4×C61C2×C812yes—
42C22×C4×C61M1612yes—
43C22×C4×C61C22×C411yescharacteristic
44C22×Dic61—24no—
45C22×Dic61—22no—
46C22×Dic61—42no—
47C22×Dic61—42no—
48C22×Dic61C22⋊C412yes—
49C22×Dic61C2×D412yes—
50C2×Dic122—24no—
51C2×Dic122—22no—
52C2×Dic122—48no—
53C2×Dic122—42no—
54C2×Dic122—42no—
55C2×Dic122—42no—
56C2×Dic122—42no—
57C2×Dic122—48no—
58C2×Dic122C2×C814yes—
59C2×#92420—44no—
60C4⋊C4×C61—42no—
61C22⋊C4×C61—42no—
62C42×C61—42no—
63C22⋊Dic61—42no—
64C4⋊Dic61—41no—
65C4⋊Dic61—41no—
66C4×Dic61—22no—
67C4×Dic61—22no—
68C23×C61—21no—
69C23×C61—21no—
70C23×C61—41no—
71C23×C61—41no—
72C23×C61C22⋊C812yes—
73C23×C61C2×C22⋊C411yescharacteristic
74C2×C4×C61—21no—
75C2×C4×C61—21no—
76C2×C4×C61—24no—
77C2×C4×C61—24no—
78C2×C4×C61—42no—
79C2×C4×C61—42no—
80C2×C4×C61—42no—
81C2×C4×C61C22⋊C812yes—
82C2×C4×C61#9912yes—
83C2×C4×C61#10012yes—
84C2×C4×C61C22×C811yescharacteristic
85C2×C4×C61C2×M1611yescharacteristic
86C2×Dic61—24no—
87C2×Dic61—22no—
88C2×Dic61—22no—
89C2×Dic61—48no—
90C2×Dic61—42no—
91C2×Dic61—42no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94Dic122—44no—
95Dic122—48no—
96Dic122—44no—
97#92420—84no—
98#11434—611nomaximal, Sylow
99C22×C61—21no—
100C22×C61—21no—
101C22×C61—22no—
102C22×C61—42no—
103C22×C61—42no—
104C22×C61—41no—
105C22×C61—41no—
106C22×C61C2×C22⋊C811yescharacteristic
107C22×C61C2×#9911yescharacteristic
108C22×C61C2×#10011yescharacteristic
109C4×C61—21no—
110C4×C61—21no—
111C4×C61—82no—
112C4×C61#32314yes—
113Dic61—42no—
114Dic61—42no—
115Dic61—44no—
116#2310—611no—
117#1458—612no—
118#1319—614no—
119C2×C61—21no—
120C2×C61—21no—
121C2×C61—42no—
122C2×C61C2×#32311yescharacteristic
123C2×C61#212911yescharacteristic
124C2×C61#213511yescharacteristic
125C23×Q8—611no—
126C2×#117—611no—
127C2×#117#692311yescharacteristic
128#431—1222no—
129#391—612no—
130#391—612no—
131#391—1222no—
132#381—1222no—
133#324—614no—
134#323—1222no—
135C61#1143411yescharacteristic, complemented, Sylow
136C22×Q8—614no—
137C22×Q8—612no—
138C22×Q8—1222no—
139C22×Q8—1221no—
140C22×Q8—2442no—
141C22×Q8—2442no—
142C23×C4—1221no—
143C23×C4C2×#692311yescharacteristic
144#117—42no—
145#117—2442no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—611no—
148C2×C4⋊C4—611no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—612no—
151C2×C42—612no—
152C2×C42C2×#692312yes—
153C4⋊C8—2442no—
154C22⋊C8—1224no—
155C4×C8—2442no—
156#95—1224no—
157#95—1224no—
158C24C22⋊#692311yescharacteristic
159C2×Q8—614no—
160C2×Q8—1224no—
161C2×Q8—1222no—
162C2×Q8—2448no—
163C2×Q8—2448no—
164C2×Q8—2442no—
165C2×Q8—2442no—
166C2×Q8—2442no—
167C2×Q8—2442no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—612no—
172C22×C4—612no—
173C22×C4—1224no—
174C22×C4—1222no—
175C22×C4—2442no—
176C22×C4—2442no—
177C22×C4D61⋊C812yes—
178C22×C4#1175821312yes—
179C22×C4C22⋊#692311yescharacteristic
180C22×C4C22×#692311yescharacteristic
181C2×C8—2444no—
182C4⋊C4—42no—
183C4⋊C4—2441no—
184C4⋊C4—2441no—
185C22⋊C4—42no—
186C22⋊C4—2442no—
187C42—42no—
188C42—1222no—
189C42—1222no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23#1451983412yes—
195C23C2×C22⋊#692311yescharacteristic
196Q8—2444no—
197Q8—2444no—
198Q8—2448no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—1224no—
207C2×C4—1222no—
208C2×C4—1222no—
209C2×C4—2448no—
210C2×C4—2442no—
211C2×C4—2442no—
212C2×C4—2442no—
213C2×C4—2442no—
214C2×C4#1451983412yes—
215C2×C4#1451983612yes—
216C2×C4#1451983712yes—
217C2×C4C2×D61⋊C811yescharacteristic
218C2×C4C2×#1175821311yescharacteristic
219C8—4884no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×#1451983411yescharacteristic
228C22C2×#1451983611yescharacteristic
229C22C2×#1451983711yescharacteristic
230C4—21no—
231C4—21no—
232C4—82no—
233C4—2444no—
234C4—2442no—
235C4—2442no—
236C4#1684060614yes—
237C2—21no—
238C2—21no—
239C2—42no—
240C2C2×#1684060611yescharacteristic
241C2#2146210311yescharacteristic
242C2#2146212311yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×C4⋊Dic61C23
hypercenterC2×#117#6923
hypocenterC61#11434
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #21460750C2semidirect
21. C22×Dic122C8semidirect
135. C61#11434semidirect

Maximal subgroups

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

subgroupindex
#214610502
#214607502
#214484032
#1143461