Finite Group Databasebeta

← #29661339

Subgroups — #29661339

Order 15616.

938 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 171 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#21456822C212yesmaximal, complemented
2#21456368C211yesmaximal, characteristic, complemented
3#21445136C218yesmaximal, complemented
4#21444446C211yesmaximal, characteristic, complemented
5#21444417C212yesmaximal, complemented
6C2×C61×#562C211yesmaximal, characteristic
7#16840319C2211yescharacteristic
8#16840172C2214yescomplemented
9#16840161—22no—
10#16840157C2214yescomplemented
11#16840121C2214yescomplemented
12#16839981—24no—
13#16839918—22no—
14#16839913—22no—
15#16839819C2212yes—
16#16839802—22no—
17C4×#11758194C2211yescharacteristic
18#16839504C2214yescomplemented
19#16839502C2218yescomplemented
20C61×#562—22no—
21C61×#562—24no—
22C61×#562C2214yes—
23C2×#125×C61C2211yescharacteristic
24C2×#124×C61C2212yes—
25C2×#123×C61—22no—
26C2×C22⋊Q8×C61—22no—
27C2×#121×C61—22no—
28C2×#11758202—24no—
29C2×#11758200—24no—
30C2×#11758194—21no—
31#14519777C2312yes—
32#14519769—21no—
33#14519756—22no—
34C4×Dic122—22no—
35#14519736—21no—
36#14519710—24no—
37#14519708—24no—
38C4⋊#92418—22no—
39#14519701—24no—
40#14519698—24no—
41C22⋊#92418—22no—
42#14519688C2312yes—
43#14519688C2314yes—
44C4×#92418—21no—
45C22×Q8×C61D412yes—
46C22×D4×C61—22no—
47C22×D4×C61D412yes—
48C23×C4×C61—22no—
49#125×C61—22no—
50#125×C61D412yes—
51#125×C61C2312yes—
52#124×C61—22no—
53#124×C61—22no—
54#124×C61D414yes—
55#124×C61C2314yes—
56#123×C61—28no—
57#123×C61—28no—
58C22⋊Q8×C61—28no—
59C22⋊Q8×C61—28no—
60#121×C61—28no—
61#121×C61—28no—
62C2×C4⋊C4×C61—22no—
63C2×C4⋊C4×C61—22no—
64C2×C4⋊C4×C61D412yes—
65C2×C22⋊C4×C61—22no—
66C2×C22⋊C4×C61—22no—
67C2×C22⋊C4×C61—22no—
68C2×C42×C61C2311yescharacteristic
69C24×C61—22no—
70C2×Q8×C61—22no—
71C2×Q8×C61—82no—
72C2×Q8×C61C2×D414yes—
73C2×D4×C61—28no—
74C2×D4×C61—22no—
75C2×D4×C61—82no—
76C2×D4×C61—84no—
77C2×D4×C61C2×D414yes—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61—28no—
81C22×C4×C61—84no—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D412yes—
84C2×Dic122—22no—
85C4⋊C4×C61—28no—
86C4⋊C4×C61—24no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61C2×D414yes—
90C22⋊C4×C61—28no—
91C22⋊C4×C61—24no—
92C22⋊C4×C61—24no—
93C22⋊C4×C61—24no—
94C22⋊C4×C61—24no—
95C42×C61C2×D411yescharacteristic
96C42×C61C2×D412yes—
97C42×C61C2411yescharacteristic
98C22⋊Dic61—41no—
99#11758202—84no—
100#11758200—84no—
101C4⋊Dic61—22no—
102C4⋊Dic61—22no—
103#11758196—42no—
104C4×Dic61—41no—
105#11758194—24no—
106C2×#92418—22no—
107C2×#92418—24no—
108C23×C61—24no—
109C23×C61—44no—
110C23×C61—44no—
111C23×C61—44no—
112C23×C61#12011yescharacteristic
113Q8×C61—44no—
114Q8×C61—82no—
115D4×C61—44no—
116D4×C61—82no—
117D4×C61—88no—
118C2×C4×C61—22no—
119C2×C4×C61—24no—
120C2×C4×C61—24no—
121C2×C4×C61—42no—
122C2×C4×C61—44no—
123C2×C4×C61—44no—
124C2×C4×C61—88no—
125C2×C4×C61#12011yescharacteristic
126C2×C4×C61#12012yes—
127C2×C4×C61C22×D411yescharacteristic
128C2×C4×C61C22×D412yes—
129C2×Dic61—42no—
130Dic122—44no—
131#92418—24no—
132#92418—44no—
133#19006—611nomaximal, Sylow
134C22×C61—22no—
135C22×C61—44no—
136C22×C61—44no—
137C22×C61—44no—
138C22×C61—88no—
139C22×C61#45612yes—
140C22×C61C2×#12011yescharacteristic
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—22no—
144C4×C61—22no—
145C4×C61—44no—
146C4×C61—44no—
147Dic61—82no—
148C2×#562D6111yescharacteristic
149#3215—612no—
150#3166—611no—
151#1661—618no—
152#1534—611no—
153#1527—612no—
154C2×C61—22no—
155C2×C61—84no—
156C2×C61C2×#45611yescharacteristic
157C2×C61#301912yes—
158#562—22no—
159#562—24no—
160#562C2×D6114yes—
161#533—611no—
162C2×#125C2×D6111yescharacteristic
163C2×#124C2×D6112yes—
164C2×#123—22no—
165C2×C22⋊Q8—22no—
166C2×#121—22no—
167#501—614no—
168#499—1222no—
169#497—614no—
170#489—614no—
171#464—1224no—
172#451—1222no—
173#450—1222no—
174#432—612no—
175#428—1222no—
176C4×M16—611no—
177#330—614no—
178#329—618no—
179C61#1900611yescharacteristic, complemented, Sylow
180C22×Q8C22⋊D6112yes—
181C22×D4—22no—
182C22×D4C22⋊D6112yes—
183C23×C4—22no—
184C2×Q16—1224no—
185C2×SD16—1224no—
186C2×M16—1221no—
187C4⋊Q8—612no—
188#125—22no—
189#125—1221no—
190#125C22⋊D6112yes—
191#125C22×D6112yes—
192#124—22no—
193#124—22no—
194#124C22⋊D6114yes—
195#124C22×D6114yes—
196#123—28no—
197#123—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—1222no—
201#121—28no—
202#121—28no—
203C4×Q8—1222no—
204#117—1221no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C22⋊D6112yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C42C22×D6111yescharacteristic
212#107—1224no—
213#106—1224no—
214C4⋊C8—1222no—
215#103—1224no—
216#102—1224no—
217C22⋊C8—1222no—
218#97—612no—
219#97—614no—
220C4×C8—1221no—
221C24—22no—
222C2×Q8—22no—
223C2×Q8—82no—
224C2×Q8—1222no—
225C2×Q8C2×C22⋊D6114yes—
226C2×D4—28no—
227C2×D4—22no—
228C2×D4—82no—
229C2×D4—84no—
230C2×D4C2×C22⋊D6114yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—84no—
235C22×C4C2×C22⋊D6111yescharacteristic
236C22×C4C2×C22⋊D6112yes—
237Q16—4884no—
238SD16—4884no—
239M16—1224no—
240C2×C8—1222no—
241C2×C8—1224no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—22no—
246C4⋊C4—1222no—
247C4⋊C4—1222no—
248C4⋊C4—2442no—
249C4⋊C4C2×C22⋊D6114yes—
250C22⋊C4—28no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—2441no—
256C42—2441no—
257C42C2×C22⋊D6111yescharacteristic
258C42C2×C22⋊D6112yes—
259C42C23×D6111yescharacteristic
260C23—24no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23#1451974611yescharacteristic
265Q8—44no—
266Q8—82no—
267Q8—2444no—
268D4—44no—
269D4—82no—
270D4—88no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—88no—
278C2×C4—2442no—
279C2×C4#1451974611yescharacteristic
280C2×C4#1451974612yes—
281C2×C4C22×C22⋊D6111yescharacteristic
282C2×C4C22×C22⋊D6112yes—
283C8—1224no—
284C8—2444no—
285C22—22no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—88no—
290C22#1683994512yes—
291C22C2×#1451974611yescharacteristic
292C4—21no—
293C4—21no—
294C4—22no—
295C4—22no—
296C4—44no—
297C4—44no—
298C4—4882no—
299C2—22no—
300C2—84no—
301C2C2×#1683994511yescharacteristic
302C2#2145517612yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×C61×#562C2
socleC22×C61C2×#120
hypercenterC2×#562D61
hypocenterC61#19006
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #21456822C2semidirect
2. #21456368C2semidirect
3. #21445136C2semidirect
4. #21444446C2semidirect
5. #21444417C2semidirect
8. #16840172C22semidirect
10. #16840157C22semidirect
11. #16840121C22semidirect
18. #16839504C22semidirect
19. #16839502C22semidirect
179. C61#19006semidirect

Maximal subgroups

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

subgroupindex
#214568222
#214563682
#214451362
#214444462
#214444172
C2×C61×#5622
#1900661