Finite Group Databasebeta

← #30595478

Subgroups — #30595478

Order 15616.

492 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 118 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#21461734C212yesmaximal, complemented
2#21460639C212yesmaximal, complemented
3#21460635C212yesmaximal
4#124×Dic61C211yesmaximal, characteristic
5C2×C4⋊#92420—22no—
6#16840628—22no—
7#16840628C2214yes—
8C2×Q8×Dic61C411yescharacteristic
9C2×D4×Dic61C411yescharacteristic
10#16839669C2211yescharacteristic
11#16839665C2211yescharacteristic
12#16839655—22no—
13#16839628C411yescharacteristic
14C22⋊C4×Dic61—22no—
15C42×Dic61C2211yescharacteristic
16C2×#124×C61C411yescharacteristic
17C22×#92420—24no—
18C4⋊#92420—44no—
19C23×Dic61—21no—
20Q8×Dic61—42no—
21D4×Dic61—42no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61C2×C411yescharacteristic
24C2×C4⋊Dic61C2×C411yescharacteristic
25C2×C22⋊Dic61—21no—
26C2×C22⋊Dic61—22no—
27C2×C4×Dic61—21no—
28C2×C4×Dic61C2×C411yescharacteristic
29C2×C4×Dic61D412yes—
30C2×C4×Dic61D412yes—
31C2×C4×Dic61C2311yescharacteristic
32#14519684—22no—
33#14519684—22no—
34C22×Q8×C61C2×C411yescharacteristic
35C22×D4×C61C2×C411yescharacteristic
36#124×C61—22no—
37#124×C61C814yescomplemented
38C2×C22⋊C4×C61—22no—
39C2×C42×C61C2×C411yescharacteristic
40C24×C61—21no—
41C2×Q8×C61—21no—
42C2×Q8×C61—41no—
43C2×Q8×C61—41no—
44C2×Q8×C61C2×C812yes—
45C2×D4×C61—21no—
46C2×D4×C61—41no—
47C2×D4×C61—41no—
48C2×D4×C61C2×C812yes—
49C22×C4×C61—21no—
50C22×C4×C61C22⋊C412yes—
51C22×C4×C61C22×C411yescharacteristic
52C22×Dic61—21no—
53C22×Dic61—21no—
54C22×Dic61—44no—
55C22×Dic61—42no—
56C22×Dic61C22⋊C411yescharacteristic
57C22×Dic61C22⋊C411yescharacteristic
58C22×Dic61C2×D412yes—
59C2×#92420—44no—
60C2×#92420—44no—
61C2×#92420—44no—
62C22⋊C4×C61—28no—
63C42×C61C2×C812yes—
64C42×C61M1612yes—
65C22⋊Dic61—42no—
66C22⋊Dic61—42no—
67C22⋊Dic61—42no—
68C4⋊Dic61—42no—
69C4⋊Dic61—41no—
70C4⋊Dic61—41no—
71C4×Dic61—22no—
72C4×Dic61—22no—
73C4×Dic61—22no—
74C4×Dic61—42no—
75C4×Dic61—41no—
76C4×Dic61—42no—
77C23×C61—22no—
78C23×C61—41no—
79C23×C61—41no—
80C23×C61—42no—
81C23×C61—42no—
82C23×C61C2×C22⋊C411yescharacteristic
83Q8×C61—42no—
84Q8×C61—42no—
85D4×C61—42no—
86D4×C61—42no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—22no—
90C2×C4×C61—22no—
91C2×C4×C61—22no—
92C2×C4×C61—41no—
93C2×C4×C61—41no—
94C2×C4×C61C22⋊C814yes—
95C2×C4×C61C22×C811yescharacteristic
96C2×C4×C61C2×M1611yescharacteristic
97C2×Dic61—22no—
98C2×Dic61—22no—
99C2×Dic61—22no—
100C2×Dic61—21no—
101C2×Dic61—22no—
102C2×Dic61—41no—
103C2×Dic61—41no—
104C2×Dic61—42no—
105C2×Dic61—42no—
106C2×Dic61—84no—
107C2×Dic61#10012yes—
108C2×Dic61AGL(1,ℤ/8)12yes—
109C2×Dic61#13712yes—
110#92420—84no—
111#7601—611nomaximal, Sylow
112C22×C61—42no—
113C22×C61—44no—
114C22×C61—81no—
115C22×C61—82no—
116C22×C61—81no—
117C22×C61C2×C22⋊C811yescharacteristic
118C22×C61#41011yescharacteristic
119C22×C61C2×#10011yescharacteristic
120C22×C61#41712yes—
121C22×C61#42112yes—
122C4×C61—21no—
123C4×C61—21no—
124C4×C61—24no—
125C4×C61—42no—
126Dic61—22no—
127Dic61—22no—
128Dic61—22no—
129Dic61—41no—
130Dic61—81no—
131Dic61—81no—
132C4×#124—611no—
133#1831—612no—
134#1272—612no—
135#1270—612no—
136C2×C61—82no—
137C2×C61#146011yescharacteristic
138C2×C61#147712yes—
139C2×C61#151212yes—
140C2×C61#153311yescharacteristic
141C2×C61#156011yescharacteristic
142C2×#124#692311yescharacteristic
143C2×C4×Q8—611no—
144C2×C4×D4—611no—
145C2×C4⋊C8—1222no—
146#391—611no—
147#390—611no—
148#388—1222no—
149#382—611no—
150C4×C22⋊C4—1222no—
151C43—611no—
152#336—614no—
153#336—1222no—
154C61#760111yescharacteristic, complemented, Sylow
155C22×Q8C2×#692311yescharacteristic
156C22×D4C2×#692311yescharacteristic
157C23×C4—1221no—
158C22×C8—1224no—
159#124—22no—
160#124#9242014yescomplemented
161C4×Q8—2442no—
162C4×D4—2442no—
163C2×C4⋊C4—611no—
164C2×C4⋊C4—611no—
165C2×C4⋊C4—1221no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—1221no—
168C2×C22⋊C4—1222no—
169C2×C42—612no—
170C2×C42—611no—
171C2×C42—612no—
172C2×C42—611no—
173C2×C42—1221no—
174C2×C42C2×#692311yescharacteristic
175C4⋊C8—2444no—
176#95—1222no—
177#95—1222no—
178C24—21no—
179C2×Q8—21no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C2×#9242012yes—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4C2×#9242012yes—
187C22×C4—21no—
188C22×C4—612no—
189C22×C4—611no—
190C22×C4—611no—
191C22×C4—1221no—
192C22×C4—1221no—
193C22×C4—2442no—
194C22×C4—2444no—
195C22×C4C22⋊#692312yes—
196C22×C4C22×#692311yescharacteristic
197C2×C8—2444no—
198C2×C8—2444no—
199C2×C8—2444no—
200C4⋊C4—2441no—
201C4⋊C4—2441no—
202C4⋊C4—2442no—
203C22⋊C4—28no—
204C22⋊C4—2442no—
205C22⋊C4—2442no—
206C22⋊C4—2442no—
207C42—1222no—
208C42—1222no—
209C42—1222no—
210C42—2441no—
211C42—2442no—
212C42—2442no—
213C42C2×#9242012yes—
214C42#1175821712yes—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23C2×C22⋊#692311yescharacteristic
221Q8—42no—
222Q8—42no—
223D4—42no—
224D4—42no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—612no—
233C2×C4—612no—
234C2×C4—612no—
235C2×C4—1222no—
236C2×C4—1222no—
237C2×C4—1222no—
238C2×C4—1221no—
239C2×C4—1222no—
240C2×C4—4884no—
241C2×C4—2441no—
242C2×C4—2441no—
243C2×C4—2442no—
244C2×C4—2442no—
245C2×C4C22⋊#9242014yes—
246C2×C4C22×#9242011yescharacteristic
247C2×C4C2×#1175821711yescharacteristic
248C8—4884no—
249C22—42no—
250C22—44no—
251C22—81no—
252C22—81no—
253C22—82no—
254C22C2×C22⋊#9242011yescharacteristic
255C22#1684074211yescharacteristic
256C22D61⋊#10011yescharacteristic
257C22#1684075312yes—
258C22#1684075912yes—
259C4—21no—
260C4—21no—
261C4—24no—
262C4—42no—
263C4—1222no—
264C4—1222no—
265C4—1222no—
266C4—4881no—
267C4—4881no—
268C4—2441no—
269C2—82no—
270C2#2146105911yescharacteristic
271C2#2146111412yes—
272C2#2146123112yes—
273C2#2146129911yescharacteristic
274C2#2146135711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenterC2×#124#6923
hypocenterC61#7601
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #21461734C2semidirect
2. #21460639C2semidirect
37. #124×C61C8semidirect
154. C61#7601semidirect
160. #124#92420semidirect

Maximal subgroups

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

subgroupindex
#214617342
#214606392
#214606352
#124×Dic612
#760161