Finite Group Databasebeta

← #30601213

Subgroups — #30601213

Order 15616.

544 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462383C211yesmaximal, characteristic
2#21461161C212yesmaximal, complemented
3#21461071C211yesmaximal, characteristic
4#21460678C212yesmaximal, complemented
5#21451389C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840783C2212yes—
8#16840726—24no—
9C4×#11758217—21no—
10C4×#11758217C2212yes—
11Q8⋊#92420—24no—
12#16840608—21no—
13#16840605—21no—
14C2×Q8×Dic61C2211yescharacteristic
15C2×#14519733C2211yescharacteristic
16#16839694C411yescharacteristic
17#16839687C411yescharacteristic
18#16839681C411yescharacteristic
19#16839656—21no—
20#16839649—21no—
21#16839645—21no—
22#16839637—21no—
23#16839629C2211yescharacteristic
24C61×#393C411yescharacteristic
25C2×#11758217—22no—
26C2×#11758217—24no—
27C2×#11758217—24no—
28C2×#11758217—22no—
29C22×#92420—22no—
30C4⋊#92420—22no—
31C22⋊#92420—22no—
32#14519832—22no—
33#14519832—22no—
34#14519832—22no—
35C4×#92420—22no—
36C4×#92420D414yes—
37C23×Dic61D411yescharacteristic
38Q8×Dic61—22no—
39Q8×Dic61—22no—
40#14519733—22no—
41#14519733—22no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196C2×C411yescharacteristic
47C2×#11758196D411yescharacteristic
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61D411yescharacteristic
50C2×C4×Dic61—21no—
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61C2311yescharacteristic
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684C2×C411yescharacteristic
59C22×Q8×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61C2×C411yescharacteristic
62#95×C61—21no—
63#95×C61C2×C411yescharacteristic
64C24×C61C22⋊C411yescharacteristic
65C2×Q8×C61—22no—
66C2×Q8×C61—42no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×Dic61—22no—
72C22×Dic61—22no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C2×D411yescharacteristic
80C22×Dic61C2×D411yescharacteristic
81#11758217—28no—
82#11758217—24no—
83#11758217—24no—
84C2×#92420—28no—
85C2×#92420—24no—
86C2×#92420—42no—
87C22⋊C4×C61—22no—
88C22⋊C4×C61—42no—
89C22⋊Dic61—22no—
90C22⋊Dic61—42no—
91C4⋊Dic61—22no—
92C4⋊Dic61—42no—
93C4⋊Dic61—41no—
94C4⋊Dic61—41no—
95#11758196—22no—
96#11758196—22no—
97#11758196—42no—
98C4×Dic61—21no—
99C4×Dic61—21no—
100C4×Dic61—42no—
101C4×Dic61C2×D412yes—
102C4×Dic61#4112yes—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61#10412yes—
108C23×C61C2×C22⋊C411yescharacteristic
109Q8×C61—24no—
110Q8×C61—42no—
111C2×C4×C61—22no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61C4×D412yes—
121C2×Dic61—22no—
122C2×Dic61—22no—
123C2×Dic61—24no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61#10412yes—
138C2×Dic61#12011yescharacteristic
139C2×Dic61#12112yes—
140C2×Dic61#12311yescharacteristic
141#92420—28no—
142#92420—82no—
143#9491—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—42no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—81no—
151C22×C61#38611yescharacteristic
152C22×C61C2×#10411yescharacteristic
153C22×C61C2×#10411yescharacteristic
154C4×C61—22no—
155C4×C61—42no—
156C4×C61—41no—
157C4×C61—81no—
158Dic61—21no—
159Dic61—22no—
160Dic61—21no—
161Dic61—22no—
162Dic61—41no—
163Dic61—81no—
164Dic61—81no—
165#2873—611no—
166#2599—611no—
167#1491—612no—
168#1464—611no—
169#1288—612no—
170C2×C61—21no—
171C2×C61—21no—
172C2×C61—42no—
173C2×C61#189111yescharacteristic
174C2×C61#189611yescharacteristic
175C2×C61#190911yescharacteristic
176C22×M16—1221no—
177C2×C4×Q8—611no—
178C2×#117—611no—
179#432—612no—
180#405—1224no—
181C4×M16—612no—
182C4×M16—1221no—
183#398—611no—
184#397—611no—
185#395—611no—
186#393#692311yescharacteristic
187#388—1221no—
188#387—1221no—
189#387—1221no—
190#385—1221no—
191#383—611no—
192Q8⋊C8—1224no—
193#324—1221no—
194#323—1221no—
195C61#949111yescharacteristic, complemented, Sylow
196C22×Q8C2×#692311yescharacteristic
197C23×C4—611no—
198C2×M16—1222no—
199C2×M16—1224no—
200C2×M16—1224no—
201C2×M16—1222no—
202C22×C8—1222no—
203C4×Q8—1222no—
204C4×Q8—1222no—
205#117—1222no—
206#117—1222no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—1221no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—611no—
215C2×C22⋊C4—1221no—
216C2×C22⋊C4C2×#692311yescharacteristic
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—1221no—
220C4⋊C8—1222no—
221C22⋊C8—1222no—
222#97—1222no—
223#97—1222no—
224#97—1222no—
225C4×C8—614no—
226C4×C8—1222no—
227#95—21no—
228#95—611no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95C2×#692311yescharacteristic
235C24C22⋊#692311yescharacteristic
236C2×Q8—22no—
237C2×Q8—42no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1222no—
247C22×C4—1222no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4C22⋊#692311yescharacteristic
251C22×C4C22×#692311yescharacteristic
252M16—1228no—
253M16—1224no—
254M16—1224no—
255C2×C8—1224no—
256C2×C8—1228no—
257C2×C8—2442no—
258C4⋊C4—1222no—
259C4⋊C4—1222no—
260C4⋊C4—1222no—
261C4⋊C4—2442no—
262C4⋊C4—2442no—
263C4⋊C4—2441no—
264C4⋊C4—2441no—
265C22⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—1222no—
268C22⋊C4—2442no—
269C42—612no—
270C42—612no—
271C42—1221no—
272C42—1221no—
273C42—2442no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23#1451984212yes—
279C23C2×C22⋊#692311yescharacteristic
280Q8—24no—
281Q8—42no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—611no—
292C2×C4—612no—
293C2×C4—611no—
294C2×C4—612no—
295C2×C4—1222no—
296C2×C4—1222no—
297C2×C4—1224no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4D4×#692312yes—
312C8—1228no—
313C8—4882no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1684071011yescharacteristic
322C22D61⋊#10411yescharacteristic
323C22C2×#1451984211yescharacteristic
324C4—22no—
325C4—42no—
326C4—41no—
327C4—81no—
328C4—1221no—
329C4—1222no—
330C4—1221no—
331C4—1222no—
332C4—4881no—
333C4—4881no—
334C4—2441no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2146183011yescharacteristic
339C2#2146183611yescharacteristic
340C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#393C4
socleC22×C61#386
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#393#6923
hypocenterC61#9491
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #21461161C2semidirect
4. #21460678C2semidirect
195. C61#9491semidirect

Maximal subgroups

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

subgroupindex
#214623832
#214611612
#214610712
#214606782
#214513892
#949161