Finite Group Databasebeta

← #29572459

Subgroups — #29572459

Order 15616.

736 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 147 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#21446789C212yesmaximal, complemented
2#21446636C211yesmaximal, characteristic
3C2×#16839504C212yesmaximal, complemented
4C2×#16839502C211yesmaximal, characteristic
5C2×C61×#563C211yesmaximal, characteristic
6C2×#14519701—21no—
7C2×#14519698—21no—
8C2×#14519688C2212yes—
9C2×#14519688C2211yescharacteristic
10#16839632C2211yescharacteristic
11#16839524—22no—
12#16839524—22no—
13#16839504—24no—
14#16839502—22no—
15C61×#563—24no—
16C61×#563—22no—
17C61×#563C414yescomplemented
18C2×C4⋊Q8×C61—21no—
19C2×#126×C61—22no—
20C2×#125×C61C2212yes—
21C2×#124×C61C2211yescharacteristic
22C2×#123×C61—21no—
23C2×C22⋊Q8×C61—22no—
24C22×#92418—21no—
25C22×#92418—22no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—22no—
28#14519701—42no—
29#14519698—42no—
30#14519688—22no—
31#14519688—22no—
32#14519688—22no—
33C22×Q8×C61—21no—
34C22×Q8×C61D411yescharacteristic
35C22×D4×C61D411yescharacteristic
36C23×C4×C61—21no—
37C4⋊Q8×C61—24no—
38C4⋊Q8×C61—24no—
39#126×C61—28no—
40#126×C61—28no—
41#125×C61—22no—
42#125×C61—22no—
43#125×C61C2×C414yes—
44#125×C61D414yes—
45#124×C61—22no—
46#124×C61C2×C412yes—
47#124×C61D412yes—
48#123×C61—24no—
49#123×C61—24no—
50C22⋊Q8×C61—28no—
51C22⋊Q8×C61—28no—
52C2×C4⋊C4×C61—22no—
53C2×C4⋊C4×C61—22no—
54C2×C4⋊C4×C61—22no—
55C2×C4⋊C4×C61D412yes—
56C2×C4⋊C4×C61D412yes—
57C2×C22⋊C4×C61—22no—
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61—22no—
60C2×C42×C61—21no—
61C2×C42×C61C2311yescharacteristic
62C24×C61—21no—
63C2×Q8×C61—24no—
64C2×Q8×C61—21no—
65C2×Q8×C61—41no—
66C2×Q8×C61—41no—
67C2×Q8×C61—81no—
68C2×Q8×C61—81no—
69C2×Q8×C61C22⋊C412yes—
70C2×D4×C61—21no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61C22⋊C412yes—
74C22×C4×C61—22no—
75C22×C4×C61—21no—
76C22×C4×C61—22no—
77C22×C4×C61—24no—
78C22×C4×C61—81no—
79C22×C4×C61—81no—
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D411yescharacteristic
82C22×Dic61—41no—
83C4⋊C4×C61—28no—
84C4⋊C4×C61—24no—
85C4⋊C4×C61—24no—
86C4⋊C4×C61—24no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61C22⋊C414yes—
91C4⋊C4×C61C22⋊C414yes—
92C22⋊C4×C61—28no—
93C22⋊C4×C61—24no—
94C22⋊C4×C61—24no—
95C22⋊C4×C61—24no—
96C42×C61—24no—
97C42×C61C22×C411yescharacteristic
98C42×C61C2×D412yes—
99C42×C61C2×D411yescharacteristic
100C4⋊Dic61—41no—
101C4⋊Dic61—41no—
102C4⋊Dic61—44no—
103C2×#92418—22no—
104C2×#92418—24no—
105C2×#92418—42no—
106C2×#92418—41no—
107C2×#92418—42no—
108C2×#92418—41no—
109C23×C61—22no—
110C23×C61—41no—
111C23×C61—42no—
112C23×C61—41no—
113C23×C61—42no—
114C23×C61#12011yescharacteristic
115Q8×C61—42no—
116Q8×C61—81no—
117Q8×C61—84no—
118D4×C61—42no—
119D4×C61—81no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—21no—
123C2×C4×C61—22no—
124C2×C4×C61—24no—
125C2×C4×C61—22no—
126C2×C4×C61—24no—
127C2×C4×C61—42no—
128C2×C4×C61—41no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—41no—
132C2×C4×C61—42no—
133C2×C4×C61—84no—
134C2×C4×C61C2×C22⋊C412yes—
135C2×C4×C61C2×C22⋊C411yescharacteristic
136C2×C4×C61#12012yes—
137C2×C4×C61#12011yescharacteristic
138C2×Dic61—81no—
139C2×Dic61—82no—
140#92418—42no—
141#92418—44no—
142#8507—611nomaximal, Sylow
143C22×C61—42no—
144C22×C61—44no—
145C22×C61—81no—
146C22×C61—81no—
147C22×C61—82no—
148C22×C61#37911yescharacteristic
149C22×C61#45311yescharacteristic
150C22×C61#45412yes—
151C22×C61#45511yescharacteristic
152C22×C61#45612yes—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—22no—
156C4×C61—22no—
157C4×C61—44no—
158C4×C61—42no—
159C4×C61—44no—
160Dic61—161no—
161C2×#563D6111yescharacteristic
162#1992—612no—
163#1959—611no—
164C2×#330—612no—
165C2×#329—611no—
166C2×C61—82no—
167C2×C61#165711yescharacteristic
168C2×C61#166011yescharacteristic
169C2×C61#166412yes—
170C2×C61#179211yescharacteristic
171C2×C61#179312yes—
172#563—24no—
173#563—22no—
174#563Dic6114yescomplemented
175C2×C4⋊Q8—21no—
176C2×#126—22no—
177C2×#125C2×D6112yes—
178C2×#124C2×D6111yescharacteristic
179C2×#123—21no—
180C2×C22⋊Q8—22no—
181C2×#103—1221no—
182C2×#102—1221no—
183C2×#97—611no—
184C2×#97—612no—
185#384—611no—
186#340—1222no—
187#340—1222no—
188#330—1224no—
189#329—1222no—
190C61#850711yescharacteristic, complemented, Sylow
191C22×Q8—21no—
192C22×Q8C22⋊D6111yescharacteristic
193C22×D4C22⋊D6111yescharacteristic
194C23×C4—21no—
195C22×C8—1222no—
196C22×C8—1221no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199#126—28no—
200#126—28no—
201#125—22no—
202#125—22no—
203#125C2×Dic6114yes—
204#125C22⋊D6114yes—
205#124—22no—
206#124C2×Dic6112yes—
207#124C22⋊D6112yes—
208#123—24no—
209#123—24no—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—1222no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4C22⋊D6112yes—
218C2×C4⋊C4C22⋊D6112yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C42—21no—
223C2×C42C22×D6111yescharacteristic
224#103—2442no—
225#102—2442no—
226#97—1222no—
227#97—1222no—
228#97—1222no—
229C24—21no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—81no—
235C2×Q8—81no—
236C2×Q8C22⋊Dic6112yes—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C22⋊Dic6112yes—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—2441no—
248C22×C4C2×C22⋊D6112yes—
249C22×C4C2×C22⋊D6111yescharacteristic
250C2×C8—1222no—
251C2×C8—1224no—
252C2×C8—2441no—
253C2×C8—2442no—
254C2×C8—2442no—
255C2×C8—2441no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—2441no—
264C4⋊C4—2441no—
265C4⋊C4—2444no—
266C4⋊C4C22⋊Dic6114yes—
267C4⋊C4C22⋊Dic6114yes—
268C22⋊C4—28no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C42—24no—
273C42C22×Dic6111yescharacteristic
274C42C2×C22⋊D6112yes—
275C42C2×C22⋊D6111yescharacteristic
276C23—22no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23#1451974611yescharacteristic
282Q8—42no—
283Q8—81no—
284Q8—84no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—84no—
301C2×C4—4881no—
302C2×C4—4882no—
303C2×C4C2×C22⋊Dic6112yes—
304C2×C4C2×C22⋊Dic6111yescharacteristic
305C2×C4#1451974612yes—
306C2×C4#1451974611yescharacteristic
307C8—2442no—
308C8—2444no—
309C22—42no—
310C22—44no—
311C22—81no—
312C22—82no—
313C22—81no—
314C22#1683961811yescharacteristic
315C22#1683993011yescharacteristic
316C22#1683993512yes—
317C22#1683994011yescharacteristic
318C22#1683994512yes—
319C4—21no—
320C4—21no—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—42no—
325C4—44no—
326C4—9761no—
327C2—82no—
328C2#2144511711yescharacteristic
329C2#2144513011yescharacteristic
330C2#2144515312yes—
331C2#2144577311yescharacteristic
332C2#2144577812yes—
subgroupG/H
centerC23#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#563C2
socleC23×C61#120
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenterC2×#563D61
hypocenterC61#8507
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. #21446789C2semidirect
3. C2×#16839504C2semidirect
17. C61×#563C4semidirect
174. #563Dic61semidirect
190. C61#8507semidirect

Maximal subgroups

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

subgroupindex
#214467892
#214466362
C2×#168395042
C2×#168395022
C2×C61×#5632
#850761