Finite Group Databasebeta

← #30595358

Subgroups — #30595358

Order 15616.

528 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 126 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#21461533C212yesmaximal, 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—28no—
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—42no—
55C22×Dic61—44no—
56C22×Dic61C22⋊C411yescharacteristic
57C22×Dic61C22⋊C411yescharacteristic
58C22×Dic61C2×D412yes—
59C2×#92420—28no—
60C2×#92420—44no—
61C2×#92420—44no—
62C22⋊C4×C61—28no—
63C42×C61C2×C812yes—
64C42×C61M1612yes—
65C22⋊Dic61—42no—
66C22⋊Dic61—44no—
67C4⋊Dic61—42no—
68C4⋊Dic61—41no—
69C4⋊Dic61—41no—
70C4×Dic61—22no—
71C4×Dic61—24no—
72C4×Dic61—24no—
73C4×Dic61—41no—
74C4×Dic61—42no—
75C23×C61—22no—
76C23×C61—41no—
77C23×C61—41no—
78C23×C61—42no—
79C23×C61—42no—
80C23×C61C2×C22⋊C411yescharacteristic
81Q8×C61—42no—
82Q8×C61—42no—
83D4×C61—42no—
84D4×C61—42no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61—22no—
90C2×C4×C61—41no—
91C2×C4×C61—41no—
92C2×C4×C61C22⋊C814yes—
93C2×C4×C61C22×C811yescharacteristic
94C2×C4×C61C2×M1611yescharacteristic
95C2×Dic61—22no—
96C2×Dic61—22no—
97C2×Dic61—21no—
98C2×Dic61—22no—
99C2×Dic61—41no—
100C2×Dic61—41no—
101C2×Dic61—42no—
102C2×Dic61—42no—
103C2×Dic61—84no—
104C2×Dic61#10012yes—
105C2×Dic61#10414yes—
106C2×Dic61#13514yes—
107#92420—48no—
108#7567—611nomaximal, Sylow
109C22×C61—42no—
110C22×C61—44no—
111C22×C61—81no—
112C22×C61—82no—
113C22×C61—81no—
114C22×C61C2×C22⋊C811yescharacteristic
115C22×C61#41011yescharacteristic
116C22×C61C2×#10011yescharacteristic
117C22×C61#41612yes—
118C22×C61C2×#10412yes—
119C4×C61—21no—
120C4×C61—21no—
121C4×C61—24no—
122C4×C61—42no—
123Dic61—24no—
124Dic61—41no—
125Dic61—81no—
126Dic61—81no—
127Dic61#32914yes—
128C4×#124—611no—
129#1744—612no—
130#1272—612no—
131#1270—612no—
132C2×C61—82no—
133C2×C61#146011yescharacteristic
134C2×C61#147612yes—
135C2×C61#150012yes—
136C2×C61C2×#32911yescharacteristic
137C2×C61#154311yescharacteristic
138C2×#124#692311yescharacteristic
139C2×C4×Q8—611no—
140C2×C4×D4—611no—
141C2×C4×C8—1222no—
142#391—611no—
143#390—611no—
144#388—1222no—
145#382—611no—
146C4×C22⋊C4—1222no—
147C43—611no—
148#336—614no—
149#336—1222no—
150C61#756711yescharacteristic, complemented, Sylow
151C22×Q8C2×#692311yescharacteristic
152C22×D4C2×#692311yescharacteristic
153C23×C4—1221no—
154C22×C8—1224no—
155#124—22no—
156#124#9242014yescomplemented
157C4×Q8—2442no—
158C4×D4—2442no—
159C2×C4⋊C4—611no—
160C2×C4⋊C4—611no—
161C2×C4⋊C4—1221no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—1221no—
164C2×C22⋊C4—1222no—
165C2×C42—612no—
166C2×C42—612no—
167C2×C42—611no—
168C2×C42—611no—
169C2×C42—1221no—
170C2×C42C2×#692311yescharacteristic
171C4×C8—1228no—
172#95—1222no—
173#95—1222no—
174C24—21no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8C2×#9242012yes—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4C2×#9242012yes—
183C22×C4—21no—
184C22×C4—611no—
185C22×C4—612no—
186C22×C4—611no—
187C22×C4—1221no—
188C22×C4—1221no—
189C22×C4—2442no—
190C22×C4—2444no—
191C22×C4C22⋊#692312yes—
192C22×C4C22×#692311yescharacteristic
193C2×C8—1228no—
194C2×C8—2444no—
195C2×C8—2444no—
196C4⋊C4—2441no—
197C4⋊C4—2441no—
198C4⋊C4—2442no—
199C22⋊C4—28no—
200C22⋊C4—2444no—
201C22⋊C4—2442no—
202C42—1224no—
203C42—1224no—
204C42—1222no—
205C42—2441no—
206C42—2442no—
207C42C2×#9242012yes—
208C42#1175821712yes—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23C2×C22⋊#692311yescharacteristic
215Q8—42no—
216Q8—42no—
217D4—42no—
218D4—42no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—612no—
227C2×C4—614no—
228C2×C4—614no—
229C2×C4—1222no—
230C2×C4—1222no—
231C2×C4—1221no—
232C2×C4—1222no—
233C2×C4—4884no—
234C2×C4—2441no—
235C2×C4—2441no—
236C2×C4—2442no—
237C2×C4—2442no—
238C2×C4C22⋊#9242014yes—
239C2×C4C22×#9242011yescharacteristic
240C2×C4C2×#1175821711yescharacteristic
241C8—2448no—
242C22—42no—
243C22—44no—
244C22—81no—
245C22—81no—
246C22—82no—
247C22C2×C22⋊#9242011yescharacteristic
248C22#1684074211yescharacteristic
249C22D61⋊#10011yescharacteristic
250C22#1684075212yes—
251C22D61⋊#10412yes—
252C4—21no—
253C4—21no—
254C4—24no—
255C4—42no—
256C4—614no—
257C4—1224no—
258C4—4881no—
259C4—4881no—
260C4—2441no—
261C2—82no—
262C2#2146105911yescharacteristic
263C2#2146111012yes—
264C2#2146119112yes—
265C2D61⋊#32911yescharacteristic
266C2#2146132311yescharacteristic
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#7567
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. #21461533C2semidirect
2. #21460639C2semidirect
37. #124×C61C8semidirect
150. C61#7567semidirect
156. #124#92420semidirect

Maximal subgroups

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

subgroupindex
#214615332
#214606392
#214606352
#124×Dic612
#756761