Finite Group Databasebeta

← #30604012

Subgroups — #30604012

Order 15616.

460 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 86 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#21461056C212yesmaximal
2#21460762C212yesmaximal, complemented
3#21460749C212yesmaximal, complemented
4#21448526C211yesmaximal, characteristic
5D61⋊(C22⋊C8)—22no—
6#16840607C2214yes—
7#16840606—22no—
8C2×#14519766C411yescharacteristic
9C2×C4×D244C2211yescharacteristic
10#16839669C2211yescharacteristic
11#16839663C411yescharacteristic
12#16839653—22no—
13#16839633—22no—
14#16839631C2211yescharacteristic
15#16839608C411yescharacteristic
16C2×C4×Q8×C61C411yescharacteristic
17C2×D61⋊C8—42no—
18#14519834—42no—
19C22⋊#92420—24no—
20C22×D244C2×C411yescharacteristic
21C22×C4×D61D412yes—
22#14519766—22no—
23#14519766C814yescomplemented
24C4×D244—42no—
25C2×C4⋊Dic61C2311yescharacteristic
26C2×#11758196—22no—
27C2×#11758198—22no—
28C2×#11758198—22no—
29C2×#11758198D412yes—
30C2×C4×Dic61C2×C411yescharacteristic
31C2×C4×Dic61C2×C411yescharacteristic
32#14519684—21no—
33#14519684—21no—
34#14519684—22no—
35C22×Q8×C61C2×C411yescharacteristic
36C4×Q8×C61—42no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61C2×C411yescharacteristic
39C2×C42×C61—21no—
40C2×C42×C61C2×C411yescharacteristic
41C23×D61C22⋊C412yes—
42C2×Q8×C61—21no—
43C2×Q8×C61—41no—
44C2×Q8×C61—41no—
45C2×Q8×C61C2×C812yescomplemented
46C22×C4×C61—21no—
47C22×C4×C61—21no—
48C22×C4×C61C22⋊C411yescharacteristic
49C22×C4×C61C22⋊C411yescharacteristic
50C22×C4×C61C22×C411yescharacteristic
51C22×Dic61—21no—
52C22×Dic61—21no—
53C22×Dic61C2×D412yes—
54C2×D244—41no—
55C2×D244—41no—
56C2×D244C2×C812yes—
57C2×D244M1612yes—
58C2×C4×D61—22no—
59C2×C4×D61—22no—
60C2×C4×D61—82no—
61C2×#92420—44no—
62D61⋊C8—84no—
63C4⋊C4×C61—41no—
64C4⋊C4×C61—42no—
65C42×C61—22no—
66C42×C61—42no—
67#11758198—28no—
68#11758198—42no—
69#11758198—82no—
70C4⋊Dic61—41no—
71#11758196—82no—
72C4×Dic61—21no—
73C4×Dic61—41no—
74C4×Dic61C2×C812yescomplemented
75C23×C61C2×C22⋊C411yescharacteristic
76C22×D61—22no—
77C22×D61—22no—
78C22×D61—42no—
79C22×D61—42no—
80C22×D61C22⋊C814yes—
81Q8×C61—42no—
82Q8×C61—42no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—22no—
86C2×C4×C61—21no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61—41no—
90C2×C4×C61—41no—
91C2×C4×C61—42no—
92C2×C4×C61—41no—
93C2×C4×C61#9912yes—
94C2×C4×C61C22×C811yescharacteristic
95C2×C4×C61C2×M1611yescharacteristic
96C2×Dic61—22no—
97C2×Dic61—22no—
98C2×Dic61—41no—
99C2×Dic61—41no—
100C2×Dic61—42no—
101C2×Dic61—41no—
102C2×Dic61—41no—
103C2×Dic61—41no—
104D244—42no—
105D244—42no—
106C4×D61—44no—
107C4×D61—82no—
108#92420—84no—
109#11481—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—21no—
112C22×C61C2×C22⋊C811yescharacteristic
113C22×C61C2×#9911yescharacteristic
114C22×C61#41311yescharacteristic
115C2×D61—24no—
116C2×D61—42no—
117C2×D61—42no—
118C2×D61—44no—
119C2×D61—82no—
120C4×C61—22no—
121C4×C61—21no—
122C4×C61—21no—
123C4×C61—41no—
124C4×C61—42no—
125C4×C61—81no—
126Dic61—42no—
127Dic61—42no—
128Dic61—81no—
129#2326—611no—
130#1459—612no—
131#1325—612no—
132#1319—612no—
133C2×C61—21no—
134C2×C61—21no—
135C2×C61#145911yescharacteristic
136C2×C61#212711yescharacteristic
137C2×C61#213511yescharacteristic
138D61—44no—
139C2×#124—611no—
140C2×C4×Q8#692311yescharacteristic
141C2×C4×D4—611no—
142C2×C22⋊C8—1222no—
143#391—611no—
144#390—611no—
145#388—1222no—
146#385—1222no—
147#384—611no—
148#376—611no—
149#324—614no—
150#323—1222no—
151C61#1148111yescharacteristic, complemented, Sylow
152C22×Q8C2×#692311yescharacteristic
153C22×D4—611no—
154C23×C4—612no—
155C22×C8—2442no—
156#124—614no—
157#124—1222no—
158C4×Q8—42no—
159C4×D4—2442no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—611no—
162C2×C4⋊C4—1222no—
163C2×C4⋊C4C2×#692311yescharacteristic
164C2×C22⋊C4—612no—
165C2×C22⋊C4—1222no—
166C2×C22⋊C4—1222no—
167C2×C42—21no—
168C2×C42—611no—
169C2×C42—611no—
170C2×C42C2×#692311yescharacteristic
171C22⋊C8—1224no—
172C22⋊C8—2442no—
173#95—1222no—
174#95—1221no—
175#95—1221no—
176C24—612no—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8D61⋊C812yescomplemented
181C2×D4—612no—
182C2×D4—612no—
183C2×D4—2441no—
184C2×D4—2441no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—612no—
188C22×C4—1221no—
189C22×C4—1222no—
190C22×C4—1222no—
191C22×C4—1221no—
192C22×C4—4882no—
193C22×C4C22⋊#692311yescharacteristic
194C22×C4C22⋊#692311yescharacteristic
195C22×C4C22×#692311yescharacteristic
196C2×C8—4884no—
197C2×C8—2444no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—4882no—
201C4⋊C4—2441no—
202C22⋊C4—1228no—
203C22⋊C4—4882no—
204C22⋊C4—2442no—
205C42—22no—
206C42—42no—
207C42—612no—
208C42—1221no—
209C42—2441no—
210C23—614no—
211C23—1222no—
212C23—1222no—
213C23—2442no—
214C23—2442no—
215C23C2×C22⋊#692311yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—2442no—
219D4—2442no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—1222no—
231C2×C4—1222no—
232C2×C4—4882no—
233C2×C4—2444no—
234C2×C4—2441no—
235C2×C4—2441no—
236C2×C4—2441no—
237C2×C4—2441no—
238C2×C4—2441no—
239C2×C4—2442no—
240C2×C4#1451983512yes—
241C2×C4C2×D61⋊C811yescharacteristic
242C2×C4D61⋊M1611yescharacteristic
243C8—4884no—
244C22—21no—
245C22—21no—
246C22—1224no—
247C22—4882no—
248C22—2444no—
249C22—2442no—
250C22—2442no—
251C22D61⋊(C22⋊C8)11yescharacteristic
252C22C2×#1451983511yescharacteristic
253C22#1684074811yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—41no—
259C4—81no—
260C4—4881no—
261C4—2442no—
262C4—2442no—
263C2—21no—
264C2—21no—
265C2—2444no—
266C2#2146105411yescharacteristic
267C2#2146209411yescharacteristic
268C2#2146212111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC2×C4⋊Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#11481
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #21460762C2semidirect
3. #21460749C2semidirect
23. #14519766C8semidirect
45. C2×Q8×C61C2×C8semidirect
74. C4×Dic61C2×C8semidirect
151. C61#11481semidirect
180. C2×Q8D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214610562
#214607622
#214607492
#214485262
#1148161