Finite Group Databasebeta

← #29654868

Subgroups — #29654868

Order 15616.

910 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 207 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#21456764C212yesmaximal
2#21451024C211yesmaximal, characteristic
3#21444793C218yesmaximal
4#21444318C212yesmaximal
5#21444143C211yesmaximal, characteristic
6C61×#3502C211yesmaximal, characteristic
7C2×#14519777C2211yescharacteristic
8#16840141—22no—
9#16840114—22no—
10#16839972—24no—
11#16839917C2218yes—
12#16839815C2212yes—
13#16839682C2214yes—
14#16839669—21no—
15#16839660—22no—
16#16839631—21no—
17C4×C4⋊Dic61C2211yescharacteristic
18#16839499C2218yes—
19#16839488C2214yes—
20Q82×C61—22no—
21C61×#557—22no—
22C61×#557—24no—
23C61×#544—21no—
24C61×#544C2214yes—
25C61×#533—22no—
26C61×#533C2212yes—
27C61×#519—21no—
28C2×C4×Q8×C61C2211yescharacteristic
29C22×Dic122C2312yes—
30C2×#11758202—28no—
31#14519777—22no—
32#14519777—22no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61—21no—
36C2×#11758196—24no—
37C2×C4×Dic61—21no—
38C4⋊#92418—22no—
39#14519700—28no—
40C22⋊#92418C2314yes—
41C4×#92418—22no—
42#14519684—24no—
43#14519684C2312yes—
44C22×Q8×C61D411yescharacteristic
45C4⋊Q8×C61—22no—
46C4⋊Q8×C61—22no—
47C4⋊Q8×C61—24no—
48C4⋊Q8×C61—28no—
49C4⋊Q8×C61D412yes—
50C4⋊Q8×C61D412yes—
51#125×C61—24no—
52#125×C61—24no—
53#125×C61—28no—
54#125×C61—22no—
55C22⋊Q8×C61—22no—
56C22⋊Q8×C61—24no—
57C22⋊Q8×C61C2314yes—
58C4×Q8×C61—24no—
59C4×Q8×C61—24no—
60C4×Q8×C61—24no—
61C4×Q8×C61—24no—
62C4×Q8×C61—22no—
63C4×Q8×C61—22no—
64C4×Q8×C61—22no—
65C4×Q8×C61D414yes—
66#117×C61—24no—
67#117×C61—22no—
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C4⋊C4×C61C2312yes—
70C2×C42×C61D412yes—
71C2×C42×C61C2311yescharacteristic
72C2×Q8×C61—24no—
73C2×Q8×C61—22no—
74C2×Q8×C61—22no—
75C2×Q8×C61—22no—
76C2×Q8×C61Q1614yescomplemented
77C2×Q8×C61C2×D414yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D412yes—
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2411yescharacteristic
83C22×Dic61—22no—
84C22×Dic61—22no—
85C2×Dic122—24no—
86C2×Dic122—82no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61—24no—
90C4⋊C4×C61—28no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—24no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—24no—
98C4⋊C4×C61C2×D414yes—
99C22⋊C4×C61—24no—
100C42×C61—28no—
101C42×C61—22no—
102C42×C61—22no—
103C42×C61—24no—
104C42×C61C2×D412yes—
105C42×C61C2×D412yes—
106#11758202—88no—
107C4⋊Dic61—24no—
108C4⋊Dic61—41no—
109C4⋊Dic61—42no—
110C4⋊Dic61—41no—
111#11758196—84no—
112C4×Dic61—42no—
113C2×#92418—24no—
114C23×C61C22×D411yescharacteristic
115Q8×C61—28no—
116Q8×C61—44no—
117Q8×C61—44no—
118Q8×C61—84no—
119C2×C4×C61—22no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—24no—
123C2×C4×C61—24no—
124C2×C4×C61—22no—
125C2×C4×C61—41no—
126C2×C4×C61—42no—
127C2×C4×C61#12011yescharacteristic
128C2×C4×C61#12012yes—
129C2×C4×C61#12011yescharacteristic
130C2×C4×C61C2×Q1614yes—
131C2×C4×C61C2×Q1612yes—
132C2×C4×C61#13712yes—
133C2×C4×C61C22×D412yes—
134C2×Dic61—24no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—44no—
138C2×Dic61—42no—
139Dic122—48no—
140#92418—44no—
141#18404—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61C2×#12011yescharacteristic
145C22×C61C22×Q1611yescharacteristic
146C22×C61C2×#13711yescharacteristic
147C4×C61—24no—
148C4×C61—21no—
149C4×C61—22no—
150C4×C61—44no—
151C4×C61—42no—
152C4×C61—44no—
153C4×C61#45114yes—
154C4×C61#45612yes—
155Dic61—44no—
156Dic61—82no—
157#3502D6111yescharacteristic
158#3209—612no—
159#2567—611no—
160#1605—618no—
161#1503—612no—
162#1461—611no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61C2×#45111yescharacteristic
166C2×C61C2×#45611yescharacteristic
167C2×C61#302711yescharacteristic
168Q82—22no—
169#557—22no—
170#557—24no—
171#544—21no—
172#544C2×D6114yes—
173#533—22no—
174#533C2×D6112yes—
175C2×C4⋊Q8—611no—
176#519—21no—
177C2×C4×Q8C2×D6111yescharacteristic
178#494—1222no—
179#487—1222no—
180#462—1224no—
181#451—618no—
182#431—612no—
183#395—614no—
184#391—1221no—
185#389—1222no—
186#384—1221no—
187C4×C4⋊C4—611no—
188#328—618no—
189#324—614no—
190C61#1840411yescharacteristic, complemented, Sylow
191C22×Q8—612no—
192C22×Q8C22⋊D6111yescharacteristic
193C2×Q16—1228no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—24no—
197C4⋊Q8—28no—
198C4⋊Q8—1222no—
199C4⋊Q8—1222no—
200C4⋊Q8C22⋊D6112yes—
201C4⋊Q8C22⋊D6112yes—
202#125—24no—
203#125—24no—
204#125—28no—
205#125—22no—
206C22⋊Q8—22no—
207C22⋊Q8—24no—
208C22⋊Q8C22×D6114yes—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8C22⋊D6114yes—
217#117—24no—
218#117—22no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4—1224no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4C22⋊D6111yescharacteristic
224C2×C4⋊C4C22×D6112yes—
225C2×C42—1221no—
226C2×C42C22⋊D6112yes—
227C2×C42C22×D6111yescharacteristic
228C4⋊C8—1222no—
229#103—1228no—
230C22⋊C8—614no—
231C4×C8—1222no—
232#95—612no—
233#95—1224no—
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—1224no—
239C2×Q8—4882no—
240C2×Q8#1175820214yescomplemented
241C2×Q8C2×C22⋊D6114yes—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4C2×C22⋊D6111yescharacteristic
245C22×C4C2×C22⋊D6112yes—
246C22×C4C2×C22⋊D6112yes—
247C22×C4C2×C22⋊D6111yescharacteristic
248C22×C4C23×D6111yescharacteristic
249Q16—4888no—
250C2×C8—1224no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—1224no—
263C4⋊C4—4884no—
264C4⋊C4—2441no—
265C4⋊C4—2441no—
266C4⋊C4—2442no—
267C4⋊C4C2×C22⋊D6114yes—
268C22⋊C4—24no—
269C42—28no—
270C42—22no—
271C42—22no—
272C42—24no—
273C42—2442no—
274C42C2×C22⋊D6112yes—
275C42C2×C22⋊D6112yes—
276C23C22×C22⋊D6111yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—44no—
280Q8—84no—
281Q8—2448no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—1224no—
291C2×C4—2444no—
292C2×C4—2442no—
293C2×C4—2442no—
294C2×C4—2442no—
295C2×C4#1451974611yescharacteristic
296C2×C4#1451974611yescharacteristic
297C2×C4#1451974612yes—
298C2×C4C2×#1175820214yes—
299C2×C4C2×#1175820212yes—
300C2×C4#1451980812yes—
301C2×C4C22×C22⋊D6112yes—
302C8—2444no—
303C22—21no—
304C22—21no—
305C22C2×#1451974611yescharacteristic
306C22C22×#1175820211yescharacteristic
307C22C2×#1451980811yescharacteristic
308C4—21no—
309C4—22no—
310C4—24no—
311C4—44no—
312C4—42no—
313C4—44no—
314C4—4882no—
315C4—2444no—
316C4#1683991814yes—
317C4#1683994512yes—
318C2—21no—
319C2—21no—
320C2C2×#1683991811yescharacteristic
321C2C2×#1683994511yescharacteristic
322C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3502C2
socleC22×C61C2×#120
prime radicalC23×C61C22×D4
hypercenter#3502D61
hypocenterC61#18404
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
76. C2×Q8×C61Q16semidirect
190. C61#18404semidirect
240. C2×Q8#11758202semidirect

Maximal subgroups

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

subgroupindex
#214567642
#214510242
#214447932
#214443182
#214441432
C61×#35022
#1840461