Finite Group Databasebeta

← #30599217

Subgroups — #30599217

Order 15616.

642 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 144 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#21461709C212yesmaximal
2#21460657C214yesmaximal
3C2×C4⋊C4×Dic61C211yesmaximal, characteristic
4C2×C22⋊#92420C2214yes—
5#16840628—22no—
6#16840628—22no—
7C22×C4⋊Dic61C411yescharacteristic
8C22×C4×Dic61C412yes—
9C22×C4×Dic61C2211yescharacteristic
10C4⋊C4×Dic61—22no—
11C4⋊C4×Dic61—22no—
12C2×#14519684C2212yes—
13C22×C4⋊C4×C61C411yescharacteristic
14C22×#92420—24no—
15C22⋊#92420—28no—
16C23×Dic61C2×C411yescharacteristic
17C23×Dic61C2×C412yes—
18C23×Dic61C2311yescharacteristic
19C2×C4⋊Dic61—22no—
20C2×C4⋊Dic61—24no—
21C2×C4×Dic61—24no—
22C2×C4×Dic61—22no—
23C2×C4×Dic61—22no—
24C2×C4×Dic61—44no—
25C2×C4×Dic61D412yes—
26C2×C4×Dic61D412yes—
27#14519684—24no—
28C23×C4×C61C2×C412yes—
29C23×C4×C61C2×C411yescharacteristic
30C2×C4⋊C4×C61—22no—
31C2×C4⋊C4×C61—22no—
32C2×C4⋊C4×C61C814yescomplemented
33C24×C61C22×C411yescharacteristic
34C22×C4×C61—22no—
35C22×C4×C61—22no—
36C22×C4×C61—22no—
37C22×C4×C61—22no—
38C22×C4×C61—42no—
39C22×C4×C61—42no—
40C22×C4×C61C22⋊C412yes—
41C22×C4×C61C2×C812yes—
42C22×C4×C61C2×C814yes—
43C22×C4×C61M1612yes—
44C22×Dic61—22no—
45C22×Dic61—22no—
46C22×Dic61—22no—
47C22×Dic61—22no—
48C22×Dic61—22no—
49C22×Dic61—41no—
50C22×Dic61—41no—
51C22×Dic61—42no—
52C22×Dic61—42no—
53C22×Dic61C22⋊C411yescharacteristic
54C22×Dic61C22⋊C411yescharacteristic
55C22×Dic61M1614yes—
56C22×Dic61C2×D412yes—
57C22×Dic61#4114yes—
58C2×#92420—44no—
59C2×#92420—44no—
60C2×#92420—44no—
61C4⋊C4×C61—28no—
62C4⋊Dic61—28no—
63C4×Dic61—24no—
64C4×Dic61—24no—
65C4×Dic61—44no—
66C4×Dic61—44no—
67C23×C61—21no—
68C23×C61—21no—
69C23×C61—21no—
70C23×C61—21no—
71C23×C61C2×C22⋊C411yescharacteristic
72C23×C61#11712yes—
73C23×C61C22×C811yescharacteristic
74C23×C61C2×M1611yescharacteristic
75C23×C61C2×M1612yes—
76C2×C4×C61—22no—
77C2×C4×C61—22no—
78C2×C4×C61—24no—
79C2×C4×C61—24no—
80C2×C4×C61—24no—
81C2×C4×C61—44no—
82C2×C4×C61—42no—
83C2×C4×C61—42no—
84C2×C4×C61—44no—
85C2×C4×C61C22⋊C814yes—
86C2×Dic61—28no—
87C2×Dic61—21no—
88C2×Dic61—21no—
89C2×Dic61—22no—
90C2×Dic61—22no—
91C2×Dic61—21no—
92C2×Dic61—21no—
93C2×Dic61—24no—
94C2×Dic61—44no—
95C2×Dic61—44no—
96C2×Dic61—42no—
97C2×Dic61—42no—
98C2×Dic61—44no—
99C2×Dic61#10012yes—
100C2×Dic61#10112yes—
101C2×Dic61#12314yes—
102#92420—84no—
103#8701—611nomaximal, Sylow
104C22×C61—22no—
105C22×C61—22no—
106C22×C61—21no—
107C22×C61—21no—
108C22×C61—21no—
109C22×C61—21no—
110C22×C61—22no—
111C22×C61—22no—
112C22×C61#38111yescharacteristic
113C22×C61C2×C22⋊C811yescharacteristic
114C22×C61#40711yescharacteristic
115C22×C61#41011yescharacteristic
116C22×C61#41011yescharacteristic
117C22×C61C2×#10011yescharacteristic
118C22×C61C2×#10111yescharacteristic
119C22×C61#43112yes—
120C22×C61#43212yes—
121C4×C61—24no—
122C4×C61—44no—
123Dic61—24no—
124Dic61—22no—
125Dic61—22no—
126Dic61—44no—
127C2×C4×C4⋊C4—611no—
128#1817—612no—
129#1280—614no—
130C2×C61—22no—
131C2×C61—22no—
132C2×C61#146011yescharacteristic
133C2×C61#146111yescharacteristic
134C2×C61#146611yescharacteristic
135C2×C61#146711yescharacteristic
136C2×C61#181711yescharacteristic
137C2×C61#182011yescharacteristic
138C2×C61#182111yescharacteristic
139C22×C4⋊C4—611no—
140C22×C4⋊C4#692311yescharacteristic
141C22×C42—611no—
142C22×C42—612no—
143C2×C22⋊C8—614no—
144C4×C4⋊C4—1222no—
145C4×C4⋊C4—1222no—
146C2×#95—612no—
147#336—1222no—
148#336—1222no—
149C61#870111yescharacteristic, complemented, Sylow
150C23×C4—611no—
151C23×C4—611no—
152C23×C4—612no—
153C23×C4C2×#692312yes—
154C23×C4C2×#692311yescharacteristic
155C22×C8—1224no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—1224no—
159C2×C4⋊C4—1222no—
160C2×C4⋊C4#9242014yescomplemented
161C2×C42—612no—
162C2×C42—612no—
163C2×C42—1224no—
164C2×C42—1222no—
165C2×C42—1222no—
166C2×C42—2444no—
167C22⋊C8—1228no—
168#95—1224no—
169C24C22×#692311yescharacteristic
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—612no—
177C22×C4—611no—
178C22×C4—611no—
179C22×C4—614no—
180C22×C4—614no—
181C22×C4—1222no—
182C22×C4—1222no—
183C22×C4—1222no—
184C22×C4—1222no—
185C22×C4—1222no—
186C22×C4—2441no—
187C22×C4—2441no—
188C22×C4—2442no—
189C22×C4—2442no—
190C22×C4C2×#9242012yes—
191C22×C4C2×#9242014yes—
192C22×C4#1175821712yes—
193C22×C4C22⋊#692312yes—
194C2×C8—2444no—
195C2×C8—2444no—
196C2×C8—2444no—
197C4⋊C4—28no—
198C4⋊C4—1228no—
199C42—1224no—
200C42—1224no—
201C42—2444no—
202C42—2444no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23C2×C22⋊#692311yescharacteristic
208C23#1451985612yes—
209C23C22×#9242011yescharacteristic
210C23C2×#1175821711yescharacteristic
211C23D61⋊M1612yes—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—614no—
222C2×C4—612no—
223C2×C4—612no—
224C2×C4—1228no—
225C2×C4—1221no—
226C2×C4—1221no—
227C2×C4—1222no—
228C2×C4—1222no—
229C2×C4—1221no—
230C2×C4—1221no—
231C2×C4—1224no—
232C2×C4—2444no—
233C2×C4—2444no—
234C2×C4—2442no—
235C2×C4—2442no—
236C2×C4—2444no—
237C2×C4C22⋊#9242014yes—
238C8—4884no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22#1684070511yescharacteristic
248C22C2×C22⋊#9242011yescharacteristic
249C22#1684073411yescharacteristic
250C22#1684074211yescharacteristic
251C22#1684074311yescharacteristic
252C22D61⋊#10011yescharacteristic
253C22D61⋊#10111yescharacteristic
254C22#1684077812yes—
255C22#1684078212yes—
256C4—24no—
257C4—44no—
258C4—1224no—
259C4—1222no—
260C4—1222no—
261C4—2444no—
262C2—22no—
263C2—22no—
264C2#2146105911yescharacteristic
265C2#2146106211yescharacteristic
266C2#2146108011yescharacteristic
267C2#2146108311yescharacteristic
268C2#2146170911yescharacteristic
269C2#2146171711yescharacteristic
270C2#2146171911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22×C4
cosocleC23×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#8701
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
32. C2×C4⋊C4×C61C8semidirect
149. C61#8701semidirect
160. C2×C4⋊C4#92420semidirect

Maximal subgroups

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

subgroupindex
#214617092
#214606572
C2×C4⋊C4×Dic612
#870161