Finite Group Databasebeta

← #30600543

Subgroups — #30600543

Order 15616.

478 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 84 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#21461328C212yesmaximal, complemented
3#21461051C211yesmaximal, characteristic
4#21460672C212yesmaximal, complemented
5#21451584C211yesmaximal, characteristic
6#16840786—24no—
7#16840779C2212yes—
8#16840765C2212yes—
9#16840732—21no—
10C4×#11758217—21no—
11#16840621—24no—
12#16840608—21no—
13#16840605—21no—
14C2×#14519776C2211yescharacteristic
15C2×#14519754—21no—
16C2×#14519733C2211yescharacteristic
17#16839696—21no—
18#16839694C411yescharacteristic
19#16839693C411yescharacteristic
20#16839690—21no—
21#16839687C411yescharacteristic
22#16839666—21no—
23#16839626C2211yescharacteristic
24C61×#397C411yescharacteristic
25C2×#11758217—22no—
26#14519844—22no—
27#14519844D414yes—
28C4⋊#92420—22no—
29C22⋊#92420—24no—
30C22⋊#92420—22no—
31#14519832—22no—
32C4×#92420—22no—
33C4×#92420—22no—
34C22×Dic122—21no—
35C23×Dic61D411yescharacteristic
36#14519776—22no—
37#14519776—22no—
38#14519754—44no—
39#14519733—22no—
40#14519733—22no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196C2×C411yescharacteristic
47C2×#11758196D411yescharacteristic
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61D411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61C2311yescharacteristic
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684C2×C411yescharacteristic
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61C2×C411yescharacteristic
60#95×C61—21no—
61#95×C61C2×C411yescharacteristic
62C24×C61C22⋊C411yescharacteristic
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C22×Dic61C2×D411yescharacteristic
77C2×Dic122—41no—
78C2×Dic122—41no—
79C2×Dic122—42no—
80C2×Dic122—42no—
81#11758217—42no—
82#11758217—42no—
83C2×#92420—24no—
84C2×#92420—42no—
85C4⋊C4×C61—22no—
86C22⋊C4×C61—22no—
87C22⋊C4×C61—42no—
88C22⋊Dic61—22no—
89C22⋊Dic61—42no—
90C4⋊Dic61—22no—
91C4⋊Dic61—42no—
92#11758196—22no—
93#11758196—22no—
94#11758196—42no—
95#11758196—42no—
96#11758196—42no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C4×Dic61C2×D412yes—
100C4×Dic61#4112yes—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61C2×C22⋊C411yescharacteristic
107C2×C4×C61—21no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61C4×D412yes—
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—41no—
131C2×Dic61#12011yescharacteristic
132C2×Dic61#12112yes—
133C2×Dic61#12311yescharacteristic
134C2×Dic61#13512yes—
135C2×Dic61AGL(1,ℤ/8)12yes—
136Dic122—82no—
137Dic122—82no—
138#92420—44no—
139#92420—82no—
140#9196—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—42no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—41no—
147C22×C61—81no—
148C22×C61#38611yescharacteristic
149C22×C61#41611yescharacteristic
150C22×C61#41811yescharacteristic
151C4×C61—22no—
152C4×C61—41no—
153C4×C61—81no—
154C4×C61—81no—
155Dic61—22no—
156Dic61—41no—
157Dic61—41no—
158Dic61—81no—
159Dic61—81no—
160Dic61#44912yes—
161Dic61#48212yes—
162#2921—611no—
163#2621—611no—
164#1546—612no—
165#1458—611no—
166#1286—612no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61—81no—
170C2×C61#187511yescharacteristic
171C2×C61#190911yescharacteristic
172C2×C61#191511yescharacteristic
173C2×C4⋊Q8—611no—
174C2×C22⋊Q8—1221no—
175C2×#117—611no—
176#433—1224no—
177#431—612no—
178#423—612no—
179#407—1221no—
180C4×M16—1221no—
181#398—611no—
182#398—611no—
183#398—1221no—
184#397—611no—
185#397—1221no—
186#397#692311yescharacteristic
187#391—1221no—
188#382—611no—
189#332—1224no—
190#324—1221no—
191#323—1221no—
192C61#919611yescharacteristic, complemented, Sylow
193C22×Q8—1221no—
194C23×C4—611no—
195C2×M16—1222no—
196C4⋊Q8—1222no—
197C4⋊Q8—1222no—
198C22⋊Q8—2444no—
199#117—1222no—
200#117—1222no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4—611no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4C2×#692311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—611no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4C2×#692311yescharacteristic
213C2×C42—611no—
214C2×C42—611no—
215C4⋊C8—614no—
216C4⋊C8—1222no—
217C4⋊C8—1222no—
218C22⋊C8—1224no—
219C22⋊C8—1222no—
220#97—1222no—
221C4×C8—1222no—
222C4×C8—1222no—
223#95—21no—
224#95—611no—
225#95—1221no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95C2×#692311yescharacteristic
230C24C22⋊#692311yescharacteristic
231C2×Q8—2442no—
232C2×Q8—2441no—
233C2×Q8—2441no—
234C2×Q8—2442no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—1222no—
244C22×C4—1222no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4C22⋊#692311yescharacteristic
248C22×C4C22×#692311yescharacteristic
249M16—2442no—
250M16—2442no—
251C2×C8—1224no—
252C2×C8—2442no—
253C4⋊C4—22no—
254C4⋊C4—1222no—
255C4⋊C4—1222no—
256C4⋊C4—1222no—
257C4⋊C4—2442no—
258C4⋊C4—2442no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C22⋊C4—22no—
262C22⋊C4—42no—
263C22⋊C4—1222no—
264C22⋊C4—2442no—
265C42—612no—
266C42—612no—
267C42—1221no—
268C42—1221no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23C2×C22⋊#692311yescharacteristic
275Q8—4882no—
276Q8—4882no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—611no—
287C2×C4—611no—
288C2×C4—612no—
289C2×C4—612no—
290C2×C4—612no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4D4×#692312yes—
306C8—4882no—
307C8—2444no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—81no—
315C22#1684071011yescharacteristic
316C22#1684075211yescharacteristic
317C22#1684075411yescharacteristic
318C4—22no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—612no—
323C4—612no—
324C4—1222no—
325C4—4881no—
326C4—4881no—
327C4—2441no—
328C4—2441no—
329C2—21no—
330C2—21no—
331C2—81no—
332C2#2146180711yescharacteristic
333C2#2146186011yescharacteristic
334C2#2146187111yescharacteristic
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#9196
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. #21461328C2semidirect
4. #21460672C2semidirect
192. C61#9196semidirect

Maximal subgroups

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

subgroupindex
#214624992
#214613282
#214610512
#214606722
#214515842
#919661