Finite Group Databasebeta

← #30604899

Subgroups — #30604899

Order 15616.

400 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 68 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#21461572C211yesmaximal, characteristic, complemented
2#21461063C211yesmaximal, characteristic, complemented
3#21460819C212yesmaximal, complemented
4#21460801C212yesmaximal
5#21451684C211yesmaximal, characteristic
6#16840778—21no—
7C2×#14519838—21no—
8C4×#11758217—21no—
9#16840642C2212yes—
10#16840608—21no—
11#16840608C2212yes—
12C2×#14519764C411yescharacteristic
13C2×#14519742C2211yescharacteristic
14#16839698—21no—
15#16839683C2211yescharacteristic
16#16839682C411yescharacteristic
17#16839676C2211yescharacteristic
18#16839674—21no—
19#16839656—21no—
20#16839655—21no—
21#16839629C411yescharacteristic
22C61×#394C411yescharacteristic
23C2×#11758217—22no—
24C4⋊#92420—41no—
25#14519838—24no—
26C22⋊#92420—22no—
27#14519832—41no—
28C4×#92420—41no—
29C4×#92420—41no—
30C22×Dic122C2×C411yescharacteristic
31C22×Dic122D411yescharacteristic
32#14519764—41no—
33#14519764—41no—
34#14519742—41no—
35#14519742—41no—
36C2×C4⋊Dic61—21no—
37C2×#11758196—21no—
38C2×#11758196D411yescharacteristic
39C2×#11758196C2311yescharacteristic
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684C2×C411yescharacteristic
52C22×D4×C61C2×C411yescharacteristic
53C2×C4⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61—21no—
56#95×C61C2×C411yescharacteristic
57C24×C61—21no—
58C2×D4×C61—41no—
59C2×D4×C61—81no—
60C22×C4×C61—21no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C22×Dic61—21no—
65C22×Dic61—21no—
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C2×D411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C2×Dic122—22no—
71C2×Dic122—22no—
72C2×Dic122—41no—
73C2×Dic122—41no—
74C2×Dic122—81no—
75#11758217—24no—
76#11758217—42no—
77C2×#92420—42no—
78C2×#92420—42no—
79C4⋊C4×C61—41no—
80C22⋊C4×C61—42no—
81C22⋊C4×C61—81no—
82C22⋊Dic61—42no—
83C22⋊Dic61—41no—
84C22⋊Dic61—41no—
85C22⋊Dic61—42no—
86C22⋊Dic61—81no—
87C4⋊Dic61—81no—
88#11758196—41no—
89#11758196—41no—
90#11758196—81no—
91C4×Dic61—21no—
92C4×Dic61—21no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C4×Dic61—21no—
96C4×Dic61—21no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61C2×C22⋊C411yescharacteristic
105D4×C61—82no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61#10412yes—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61#9911yescharacteristic
131C2×Dic61#9911yescharacteristic
132C2×Dic61#13512yes—
133Dic122—44no—
134Dic122—82no—
135#92420—44no—
136#92420—82no—
137#11738—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—81no—
141C22×C61—81no—
142C22×C61—81no—
143C22×C61—81no—
144C22×C61—81no—
145C22×C61—81no—
146C22×C61—81no—
147C22×C61C2×#9911yescharacteristic
148C22×C61#41611yescharacteristic
149C22×C61C2×#10411yescharacteristic
150C4×C61—22no—
151C4×C61—41no—
152C4×C61—41no—
153C4×C61—81no—
154Dic61—21no—
155Dic61—21no—
156Dic61—22no—
157Dic61—41no—
158Dic61—41no—
159Dic61—81no—
160Dic61—81no—
161#2632—611no—
162C4×#101—611no—
163#1461—611no—
164#1354—612no—
165#1345—612no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—161no—
169C2×C61#150111yescharacteristic
170C2×C61#212211yescharacteristic
171C2×C61#213311yescharacteristic
172C2×#124—611no—
173C2×C4×Q8—611no—
174#431—1221no—
175C2×#101—1221no—
176C4×M16—1221no—
177#399—1221no—
178#395—611no—
179#395—611no—
180#394—611no—
181#394#692311yescharacteristic
182#393—1221no—
183#388—1221no—
184#388—1221no—
185#383—611no—
186#343—612no—
187#324—612no—
188#324—1221no—
189C61#1173811yescharacteristic, complemented, Sylow
190C22×Q8—611no—
191C22×Q8—611no—
192C22×D4C2×#692311yescharacteristic
193C2×M16—1222no—
194#124—2441no—
195#124—2441no—
196C4×Q8—2441no—
197C4×Q8—2441no—
198C2×C4⋊C4—611no—
199C2×C4⋊C4—611no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4—1221no—
202C2×C4⋊C4C2×#692311yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—1221no—
206C2×C22⋊C4—1221no—
207C2×C22⋊C4—1221no—
208C2×C22⋊C4—1221no—
209C2×C42—611no—
210C2×C42—611no—
211C2×C42—611no—
212C4⋊C8—2441no—
213#101—1224no—
214C22⋊C8—1222no—
215#97—2441no—
216C4×C8—2441no—
217C4×C8—2441no—
218#95—611no—
219#95—1221no—
220#95—1221no—
221#95—1221no—
222#95—1221no—
223#95C2×#692311yescharacteristic
224C24—21no—
225C2×Q8—1222no—
226C2×Q8—1222no—
227C2×Q8—4881no—
228C2×Q8—2441no—
229C2×Q8—2441no—
230C2×D4—41no—
231C2×D4—81no—
232C22×C4—21no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4C22⋊#692311yescharacteristic
240C22×C4C22⋊#692311yescharacteristic
241C22×C4C22×#692311yescharacteristic
242M16—1224no—
243M16—2442no—
244C2×C8—2442no—
245C2×C8—2442no—
246C4⋊C4—41no—
247C4⋊C4—4881no—
248C4⋊C4—4881no—
249C4⋊C4—2441no—
250C4⋊C4—2441no—
251C22⋊C4—42no—
252C22⋊C4—81no—
253C22⋊C4—4881no—
254C22⋊C4—2442no—
255C22⋊C4—2441no—
256C22⋊C4—2441no—
257C22⋊C4—2442no—
258C42—1221no—
259C42—1221no—
260C42—1221no—
261C42—1221no—
262C42—1221no—
263C42—1221no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23C2×C22⋊#692311yescharacteristic
272Q8—4882no—
273Q8—2444no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—611no—
285C2×C4—611no—
286C2×C4—612no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4#1451984112yes—
302C8—4882no—
303C8—2444no—
304C22—21no—
305C22—21no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22D61⋊#9911yescharacteristic
314C22#1684075211yescharacteristic
315C22C2×#1451984111yescharacteristic
316C4—22no—
317C4—41no—
318C4—41no—
319C4—81no—
320C4—1221no—
321C4—1221no—
322C4—1222no—
323C4—4881no—
324C4—4881no—
325C4—2441no—
326C4—2441no—
327C2—21no—
328C2—21no—
329C2—161no—
330C2#2146119511yescharacteristic
331C2#2146207911yescharacteristic
332C2#2146211511yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#394C4
socleC22×C61C2×#99
prime radicalC22×D4×C61C2×C4
cosocleC2×#11758196C23
hypercenter#394#6923
hypocenterC61#11738
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #21461572C2semidirect
2. #21461063C2semidirect
3. #21460819C2semidirect
189. C61#11738semidirect

Maximal subgroups

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

subgroupindex
#214615722
#214610632
#214608192
#214608012
#214516842
#1173861