Finite Group Databasebeta

← #29590691

Subgroups — #29590691

Order 15616.

402 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 87 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#21445896C211yesmaximal, characteristic
2#21444153C211yesmaximal, characteristic
3#21443819C214yesmaximal
4C61×#2621C211yesmaximal, characteristic
5#16839819—21no—
6#16839625—21no—
7#16839529—21no—
8#16839529C2212yes—
9#16839488—21no—
10#16839486C2212yes—
11C2×C4⋊Q8×C61C411yescharacteristic, complemented
12C2×C22⋊Q8×C61—21no—
13C2×#117×C61C2211yescharacteristic
14C61×#398—21no—
15C61×#398C412yescomplemented
16C61×#397—21no—
17C61×#397C2212yes—
18C61×#391—21no—
19C61×#382C411yescharacteristic, complemented
20C2×C22⋊Dic61—22no—
21C4⋊#92418—41no—
22C22⋊#92418—22no—
23#14519688—41no—
24#14519684—41no—
25#14519684—41no—
26C22×Q8×C61—21no—
27C23×C4×C61D411yescharacteristic
28C4⋊Q8×C61—41no—
29C4⋊Q8×C61—41no—
30C22⋊Q8×C61—44no—
31#117×C61—41no—
32#117×C61—41no—
33C2×C4⋊C4×C61—21no—
34C2×C4⋊C4×C61—21no—
35C2×C4⋊C4×C61—22no—
36C2×C4⋊C4×C61C2×C412yes—
37C2×C4⋊C4×C61C2×C411yescharacteristic
38C2×C4⋊C4×C61D411yescharacteristic
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61D411yescharacteristic
41C2×C22⋊C4×C61C2311yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43C2×C42×C61D411yescharacteristic
44#95×C61—21no—
45#95×C61—22no—
46#95×C61—22no—
47#95×C61C2×C412yes—
48C24×C61C2×D411yescharacteristic
49C2×Q8×C61—42no—
50C2×Q8×C61—42no—
51C2×Q8×C61—41no—
52C2×Q8×C61—41no—
53C22×C4×C61—22no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61—41no—
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C22×Dic61—42no—
66C4⋊C4×C61—41no—
67C4⋊C4×C61—41no—
68C4⋊C4×C61—42no—
69C4⋊C4×C61—44no—
70C4⋊C4×C61—42no—
71C4⋊C4×C61—42no—
72C22⋊C4×C61—41no—
73C22⋊C4×C61—41no—
74C22⋊C4×C61—44no—
75C42×C61—21no—
76C42×C61—21no—
77C42×C61—21no—
78C42×C61—21no—
79C22⋊Dic61—42no—
80C22⋊Dic61—42no—
81C2×#92418—42no—
82C23×C61—21no—
83C23×C61—21no—
84C23×C61—41no—
85C23×C61—41no—
86C23×C61C2×C22⋊C411yescharacteristic
87C23×C61AGL(1,ℤ/8)12yes—
88Q8×C61—82no—
89Q8×C61—82no—
90C2×C4×C61—21no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—42no—
100C2×C4×C61—42no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61#9911yescharacteristic
112C2×C4×C61#9911yescharacteristic
113C2×Dic61—82no—
114C2×Dic61—82no—
115C2×Dic61—82no—
116#92418—82no—
117#11801—611nomaximal, Sylow
118C22×C61—21no—
119C22×C61—21no—
120C22×C61—22no—
121C22×C61—42no—
122C22×C61—41no—
123C22×C61—41no—
124C22×C61—81no—
125C22×C61C2×#9911yescharacteristic
126C22×C61#41811yescharacteristic
127C22×C61#42111yescharacteristic
128C4×C61—21no—
129C4×C61—21no—
130C4×C61—41no—
131C4×C61—41no—
132C4×C61—41no—
133C4×C61—41no—
134C4×C61—41no—
135C4×C61—82no—
136C4×C61—82no—
137Dic61—162no—
138#2621D6111yescharacteristic
139#1820—611no—
140#1464—611no—
141#1344—614no—
142C2×C61—21no—
143C2×C61—21no—
144C2×C61—42no—
145C2×C61#151011yescharacteristic
146C2×C61#212311yescharacteristic
147C2×C61#212911yescharacteristic
148C2×C4⋊Q8Dic6111yescharacteristic, complemented
149C2×C22⋊Q8—21no—
150C2×#117C2×D6111yescharacteristic
151#432—1221no—
152#398—21no—
153#398Dic6112yescomplemented
154#397—21no—
155#397C2×D6112yes—
156#391—21no—
157#382Dic6111yescharacteristic, complemented
158#381—1221no—
159#342—612no—
160#342—1221no—
161#324—1221no—
162#323—612no—
163C61#1180111yescharacteristic, complemented, Sylow
164C22×Q8—21no—
165C23×C4C22⋊D6111yescharacteristic
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C22⋊Q8—44no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C2×Dic6112yes—
175C2×C4⋊C4C2×Dic6111yescharacteristic
176C2×C4⋊C4C22⋊D6111yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1222no—
179C2×C22⋊C4C22⋊D6111yescharacteristic
180C2×C22⋊C4C22×D6111yescharacteristic
181C2×C42C2×Dic6111yescharacteristic
182C2×C42C22⋊D6111yescharacteristic
183C4⋊C8—2441no—
184C22⋊C8—1222no—
185#97—2441no—
186#95—21no—
187#95—22no—
188#95—22no—
189#95—2441no—
190#95—2441no—
191#95C2×Dic6112yes—
192C24C2×C22⋊D6111yescharacteristic
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—41no—
196C2×Q8—41no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—2442no—
204C22×C4C22⋊Dic6111yescharacteristic
205C22×C4C22⋊Dic6111yescharacteristic
206C22×C4C22⋊Dic6111yescharacteristic
207C22×C4C22⋊Dic6111yescharacteristic
208C22×C4C22×Dic6111yescharacteristic
209C22×C4C2×C22⋊D6111yescharacteristic
210C2×C8—2442no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—44no—
220C22⋊C4—2442no—
221C22⋊C4—2442no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23C2×C22⋊Dic6111yescharacteristic
231C23#1451980312yes—
232Q8—82no—
233Q8—82no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—4882no—
256C2×C4—4882no—
257C2×C4—4882no—
258C2×C4#1451969211yescharacteristic
259C2×C4#1451969211yescharacteristic
260C8—4882no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#1451969211yescharacteristic
269C22#1683976711yescharacteristic
270C22#1683977911yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—82no—
280C4—9762no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2144434811yescharacteristic
285C2#2144747311yescharacteristic
286C2#2144749411yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2621C2
socleC22×C61C2×#99
prime radicalC24×C61C2×D4
cosocleC2×C22⋊C4×C61C23
hypercenter#2621D61
hypocenterC61#11801
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C61C4semidirect
15. C61×#398C4semidirect
19. C61×#382C4semidirect
148. C2×C4⋊Q8Dic61semidirect
153. #398Dic61semidirect
157. #382Dic61semidirect
163. C61#11801semidirect

Maximal subgroups

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

subgroupindex
#214458962
#214441532
#214438192
C61×#26212
#1180161