Finite Group Databasebeta

← #30600263

Subgroups — #30600263

Order 15616.

566 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 116 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#21462666C211yesmaximal, characteristic
2#21461123C212yesmaximal, complemented
3#21461059C211yesmaximal, characteristic
4#21460643C212yesmaximal, complemented
5#21448529C211yesmaximal, characteristic
6D4×#92420—24no—
7#16840778C2212yes—
8#16840778C2212yes—
9C2×C22⋊#92420—22no—
10#16840612—24no—
11#16840605—22no—
12C2×#14519764C411yescharacteristic
13C2×#14519742C2211yescharacteristic
14C2×D4×Dic61C2211yescharacteristic
15#16839668C411yescharacteristic
16#16839656—22no—
17#16839634—22no—
18#16839631C411yescharacteristic
19#16839608C2211yescharacteristic
20C61×#390C411yescharacteristic
21C22×#92420—24no—
22C4⋊#92420—22no—
23C4⋊#92420—22no—
24C22⋊#92420—24no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C4×#92420—22no—
28C4×#92420D414yes—
29C22×Dic122C2×C411yescharacteristic
30C23×Dic61D412yes—
31#14519764—22no—
32#14519764C814yescomplemented
33#14519742—22no—
34#14519742—22no—
35D4×Dic61—22no—
36D4×Dic61—22no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—22no—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61D412yes—
42C2×C4×Dic61—21no—
43C2×C4×Dic61C2×C411yescharacteristic
44C2×C4×Dic61C2311yescharacteristic
45#14519684—22no—
46#14519684—21no—
47#14519684—21no—
48C22×D4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—22no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61C22⋊C412yes—
53C2×D4×C61—22no—
54C2×D4×C61—22no—
55C2×D4×C61C2×C812yes—
56C2×D4×C61M1612yes—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—21no—
61C22×Dic61—21no—
62C22×Dic61—22no—
63C22×Dic61—22no—
64C22×Dic61—82no—
65C22×Dic61C22⋊C412yes—
66C22×Dic61C2×D411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C2×Dic122—22no—
69C2×Dic122—22no—
70C2×Dic122—21no—
71C2×Dic122C2×C812yescomplemented
72C2×#92420—24no—
73C2×#92420—42no—
74C2×#92420—84no—
75C4⋊C4×C61—22no—
76C22⋊C4×C61—82no—
77C42×C61—22no—
78C22⋊Dic61—28no—
79C22⋊Dic61—24no—
80C4⋊Dic61—22no—
81#11758196—41no—
82#11758196—41no—
83#11758196—82no—
84C4×Dic61—21no—
85C4×Dic61—42no—
86C4×Dic61C2×C812yescomplemented
87C4×Dic61C2×D412yes—
88C4×Dic61#4112yes—
89C23×C61—22no—
90C23×C61—22no—
91C23×C61—42no—
92C23×C61—42no—
93C23×C61C22⋊C814yes—
94C23×C61C2×C22⋊C411yescharacteristic
95D4×C61—24no—
96D4×C61—24no—
97C2×C4×C61—42no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61C4×D412yes—
103C2×C4×C61C22×C811yescharacteristic
104C2×C4×C61C2×M1611yescharacteristic
105C2×C4×C61#13112yes—
106C2×Dic61—22no—
107C2×Dic61—22no—
108C2×Dic61—22no—
109C2×Dic61—41no—
110C2×Dic61—41no—
111C2×Dic61—44no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61—82no—
118C2×Dic61#12011yescharacteristic
119C2×Dic61#12112yes—
120C2×Dic61#12311yescharacteristic
121Dic122—24no—
122Dic122—42no—
123#92420—44no—
124#92420—82no—
125#9099—611nomaximal, Sylow
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—24no—
129C22×C61—44no—
130C22×C61—42no—
131C22×C61—42no—
132C22×C61—82no—
133C22×C61#38611yescharacteristic
134C22×C61C2×C22⋊C811yescharacteristic
135C22×C61#40811yescharacteristic
136C4×C61—81no—
137C4×C61—81no—
138C4×C61C8×D412yes—
139C4×C61#43512yes—
140Dic61—22no—
141Dic61—21no—
142Dic61—21no—
143Dic61—42no—
144Dic61—42no—
145Dic61—42no—
146#2972—611no—
147#2326—611no—
148#1479—612no—
149#1460—611no—
150#1274—612no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61—44no—
154C2×C61#186911yescharacteristic
155C2×C61#190911yescharacteristic
156C2×C61#191111yescharacteristic
157C2×#124—611no—
158C2×C4×Q8—611no—
159C2×C4×D4—611no—
160C8×D4—1224no—
161#431—612no—
162#431—612no—
163C2×C22⋊C8—1222no—
164#391—611no—
165#390#692311yescharacteristic
166#388—1222no—
167#385—1222no—
168#384—611no—
169#376—611no—
170Q8⋊C8—1224no—
171#323—1222no—
172C61#909911yescharacteristic, complemented, Sylow
173C22×Q8—611no—
174C22×D4C2×#692311yescharacteristic
175C23×C4—612no—
176C22×C8—1224no—
177#124—614no—
178#124—1222no—
179C4×Q8—1222no—
180C4×Q8—1222no—
181C4×D4—1222no—
182C4×D4—1222no—
183C2×C4⋊C4—611no—
184C2×C4⋊C4—1222no—
185C2×C4⋊C4—1221no—
186C2×C4⋊C4C2×#692311yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—612no—
189C2×C22⋊C4—1222no—
190C2×C42—611no—
191C2×C42—611no—
192C2×C42—1221no—
193C2×C42C2×#692311yescharacteristic
194C4⋊C8—1222no—
195C4⋊C8—1222no—
196C22⋊C8—1222no—
197C22⋊C8—1224no—
198C22⋊C8—1222no—
199C4×C8—614no—
200C4×C8—1222no—
201#95—1221no—
202#95—1221no—
203#95—1222no—
204C24C22⋊#692312yes—
205C2×Q8—612no—
206C2×Q8—1221no—
207C2×Q8—1222no—
208C2×Q8—1222no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4D61⋊C812yes—
212C2×D4#1175821312yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—611no—
216C22×C4—611no—
217C22×C4—612no—
218C22×C4—1222no—
219C22×C4—1222no—
220C22×C4—1221no—
221C22×C4—1221no—
222C22×C4—4882no—
223C22×C4C22×#692311yescharacteristic
224C2×C8—1224no—
225C2×C8—4884no—
226C2×C8—2442no—
227C4⋊C4—22no—
228C4⋊C4—1222no—
229C4⋊C4—4882no—
230C4⋊C4—2441no—
231C4⋊C4—2441no—
232C22⋊C4—82no—
233C22⋊C4—1224no—
234C22⋊C4—1228no—
235C42—22no—
236C42—612no—
237C42—612no—
238C42—612no—
239C42—1221no—
240C42—2442no—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23#1451983414yes—
246C23C2×C22⋊#692311yescharacteristic
247Q8—1224no—
248Q8—2442no—
249D4—24no—
250D4—24no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—611no—
257C2×C4—612no—
258C2×C4—611no—
259C2×C4—1222no—
260C2×C4—1222no—
261C2×C4—1222no—
262C2×C4—4882no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4—2444no—
266C2×C4—2441no—
267C2×C4—2441no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4—2442no—
271C2×C4D4×#692312yes—
272C2×C4C2×D61⋊C811yescharacteristic
273C2×C4C2×#1175821311yescharacteristic
274C2×C4#1451986412yes—
275C8—4882no—
276C8—2444no—
277C22—21no—
278C22—21no—
279C22—24no—
280C22—44no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22#1684071011yescharacteristic
285C22C2×#1451983411yescharacteristic
286C22#1684073511yescharacteristic
287C4—81no—
288C4—81no—
289C4—1222no—
290C4—1221no—
291C4—1221no—
292C4—2442no—
293C4—2442no—
294C4—2442no—
295C4#1684078912yes—
296C4#1684079112yes—
297C2—21no—
298C2—21no—
299C2—44no—
300C2#2146179411yescharacteristic
301C2#2146186011yescharacteristic
302C2#2146186511yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#9099
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #21461123C2semidirect
4. #21460643C2semidirect
32. #14519764C8semidirect
71. C2×Dic122C2×C8semidirect
86. C4×Dic61C2×C8semidirect
172. C61#9099semidirect

Maximal subgroups

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

subgroupindex
#214626662
#214611232
#214610592
#214606432
#214485292
#909961