Finite Group Databasebeta

← #30601223

Subgroups — #30601223

Order 15616.

458 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 76 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#21462499C211yesmaximal, characteristic
2#21461163C212yesmaximal, complemented
3#21461071C211yesmaximal, characteristic
4#21460678C212yesmaximal, complemented
5#21451397C211yesmaximal, characteristic
6#16840786—24no—
7#16840783C2212yes—
8#16840765—21no—
9#16840732—21no—
10C4×#11758217C2212yes—
11#16840612—24no—
12#16840608—21no—
13#16840605—21no—
14C2×#14519742C2211yescharacteristic
15C2×#14519733C2211yescharacteristic
16#16839693C411yescharacteristic
17#16839683C411yescharacteristic
18#16839674C411yescharacteristic
19#16839655—21no—
20#16839645—21no—
21#16839644—21no—
22#16839634—21no—
23#16839629C2211yescharacteristic
24C61×#397C411yescharacteristic
25C2×#11758217—22no—
26#14519844—22no—
27C4⋊#92420—22no—
28C4⋊#92420—22no—
29C22⋊#92420—24no—
30C22⋊#92420—22no—
31#14519832—22no—
32#14519832—22no—
33C4×#92420D414yes—
34C22×Dic122C2×C411yescharacteristic
35C23×Dic61D411yescharacteristic
36#14519742—22no—
37#14519742—22no—
38#14519733—22no—
39#14519733—22no—
40C2×C4⋊Dic61—21no—
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758196C2×C411yescharacteristic
44C2×#11758196D411yescharacteristic
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61D411yescharacteristic
47C2×C4×Dic61—21no—
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61C2311yescharacteristic
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684C2×C411yescharacteristic
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61C2×C411yescharacteristic
59#95×C61—21no—
60#95×C61C2×C411yescharacteristic
61C24×C61C22⋊C411yescharacteristic
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—41no—
71C22×Dic61—41no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×Dic122—22no—
77C2×Dic122—42no—
78#11758217—42no—
79#11758217—42no—
80C2×#92420—24no—
81C2×#92420—42no—
82C4⋊C4×C61—22no—
83C22⋊C4×C61—22no—
84C22⋊C4×C61—42no—
85C22⋊Dic61—22no—
86C22⋊Dic61—42no—
87C4⋊Dic61—42no—
88#11758196—22no—
89#11758196—22no—
90#11758196—42no—
91#11758196—41no—
92#11758196—41no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C4×Dic61—42no—
96C4×Dic61C2×D412yes—
97C4×Dic61#4112yes—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61C2×C22⋊C411yescharacteristic
104C2×C4×C61—21no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61C4×D412yes—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—22no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—41no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61#12011yescharacteristic
131C2×Dic61#12112yes—
132C2×Dic61#12311yescharacteristic
133Dic122—24no—
134Dic122—42no—
135#92420—44no—
136#92420—82no—
137#9493—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—42no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—81no—
145C22×C61#38611yescharacteristic
146C22×C61#42111yescharacteristic
147C22×C61#42111yescharacteristic
148C4×C61—22no—
149C4×C61—41no—
150C4×C61—81no—
151C4×C61—81no—
152Dic61—21no—
153Dic61—21no—
154Dic61—22no—
155Dic61—42no—
156Dic61—41no—
157Dic61—41no—
158Dic61—81no—
159#2921—611no—
160#2599—611no—
161#1491—612no—
162#1464—611no—
163#1288—612no—
164C2×C61—21no—
165C2×C61—21no—
166C2×C61—81no—
167C2×C61#189111yescharacteristic
168C2×C61#189711yescharacteristic
169C2×C61#190911yescharacteristic
170C2×C4×Q8—611no—
171C2×#117—611no—
172#433—1224no—
173#432—612no—
174#423—1221no—
175#407—1221no—
176C4×M16—612no—
177#398—611no—
178#397#692311yescharacteristic
179#395—611no—
180#393—611no—
181#388—1221no—
182#387—1221no—
183#387—1221no—
184#385—1221no—
185#383—611no—
186Q8⋊C8—1224no—
187#324—1221no—
188#323—1221no—
189C61#949311yescharacteristic, complemented, Sylow
190C22×Q8—611no—
191C23×C4—611no—
192C2×M16—1222no—
193C4×Q8—1222no—
194C4×Q8—1222no—
195#117—1222no—
196#117—1222no—
197C2×C4⋊C4—611no—
198C2×C4⋊C4—611no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4—1221no—
202C2×C4⋊C4C2×#692311yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—611no—
205C2×C22⋊C4—1221no—
206C2×C22⋊C4C2×#692311yescharacteristic
207C2×C42—611no—
208C2×C42—611no—
209C2×C42—1221no—
210C4⋊C8—1222no—
211C4⋊C8—1222no—
212C4⋊C8—1222no—
213C22⋊C8—1224no—
214C22⋊C8—1222no—
215#97—1222no—
216#97—1222no—
217C4×C8—614no—
218#95—21no—
219#95—611no—
220#95—1221no—
221#95—1221no—
222#95—1221no—
223#95—1221no—
224#95—1221no—
225#95C2×#692311yescharacteristic
226C24C22⋊#692311yescharacteristic
227C2×Q8—1222no—
228C2×Q8—2442no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—611no—
232C22×C4—611no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4—2441no—
240C22×C4—2441no—
241C22×C4C22⋊#692311yescharacteristic
242C22×C4C22×#692311yescharacteristic
243M16—2442no—
244M16—2442no—
245C2×C8—1224no—
246C2×C8—2442no—
247C4⋊C4—22no—
248C4⋊C4—1222no—
249C4⋊C4—1222no—
250C4⋊C4—2441no—
251C4⋊C4—2441no—
252C4⋊C4—2442no—
253C4⋊C4—2442no—
254C22⋊C4—22no—
255C22⋊C4—42no—
256C22⋊C4—1222no—
257C22⋊C4—2442no—
258C42—612no—
259C42—612no—
260C42—1221no—
261C42—1221no—
262C42—2442no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23C2×C22⋊#692311yescharacteristic
269Q8—1224no—
270Q8—2442no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—611no—
281C2×C4—612no—
282C2×C4—611no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1222no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—2441no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4D4×#692312yes—
300C8—4882no—
301C8—2444no—
302C22—21no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—81no—
309C22#1684071011yescharacteristic
310C22#1684075911yescharacteristic
311C22#1684076011yescharacteristic
312C4—22no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—1221no—
317C4—1221no—
318C4—1222no—
319C4—4881no—
320C4—2441no—
321C4—2442no—
322C4—2441no—
323C2—21no—
324C2—21no—
325C2—81no—
326C2#2146183111yescharacteristic
327C2#2146183911yescharacteristic
328C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#397C4
socleC22×C61#386
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#397#6923
hypocenterC61#9493
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #21461163C2semidirect
4. #21460678C2semidirect
189. C61#9493semidirect

Maximal subgroups

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

subgroupindex
#214624992
#214611632
#214610712
#214606782
#214513972
#949361