Finite Group Databasebeta

← #30595458

Subgroups — #30595458

Order 15616.

406 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 116 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#21461746C211yesmaximal, characteristic
2#21461734C211yesmaximal, characteristic
3#21460639C212yesmaximal
4#21460637C212yesmaximal
5#21448297C211yesmaximal, characteristic
6C2×C4⋊#92420—21no—
7#16840628—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12#16839669C411yescharacteristic
13#16839669C2211yescharacteristic
14#16839661C411yescharacteristic
15#16839661C2211yescharacteristic
16#16839648—21no—
17#16839648—21no—
18C4×C4⋊Dic61—21no—
19C4×C4⋊Dic61C411yescharacteristic
20#16839608—21no—
21C42×Dic61C2211yescharacteristic
22C2×C4×Q8×C61C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25C4⋊#92420—42no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×C4×Dic61—21no—
31C2×C4×Dic61C2×C411yescharacteristic
32C2×C4×Dic61D411yescharacteristic
33C2×C4×Dic61D411yescharacteristic
34C2×C4×Dic61D411yescharacteristic
35C2×C4×Dic61D411yescharacteristic
36C2×C4×Dic61C2311yescharacteristic
37#14519684—21no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44C22×Q8×C61C2×C411yescharacteristic
45C4×Q8×C61—22no—
46C4×Q8×C61C814yescomplemented
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C42×C61—21no—
50C2×C42×C61C2×C411yescharacteristic
51C2×Q8×C61—21no—
52C2×Q8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61C2×C812yes—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—21no—
61C22×Dic61—21no—
62C22×Dic61C22⋊C411yescharacteristic
63C22×Dic61C22⋊C411yescharacteristic
64C22×Dic61C2×D411yescharacteristic
65C22×Dic61C2×D411yescharacteristic
66C2×#92420—42no—
67C2×#92420—42no—
68C2×#92420—42no—
69C2×#92420—42no—
70C2×#92420—42no—
71C2×#92420—42no—
72C4⋊C4×C61—24no—
73C4⋊C4×C61—21no—
74C4⋊C4×C61C2×C812yes—
75C42×C61—24no—
76C42×C61C2×C812yes—
77C42×C61M1612yes—
78C4⋊Dic61—42no—
79C4⋊Dic61—41no—
80C4⋊Dic61—41no—
81C4⋊Dic61—41no—
82C4⋊Dic61—41no—
83C4×Dic61—21no—
84C4×Dic61—21no—
85C4×Dic61—22no—
86C4×Dic61—41no—
87C4×Dic61—41no—
88C4×Dic61—41no—
89C4×Dic61—41no—
90C4×Dic61—42no—
91C23×C61C2×C22⋊C411yescharacteristic
92Q8×C61—42no—
93Q8×C61—42no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61C22⋊C812yes—
107C2×C4×C61C22⋊C812yes—
108C2×C4×C61C22×C811yescharacteristic
109C2×C4×C61C2×M1611yescharacteristic
110C2×Dic61—21no—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61AGL(1,ℤ/8)11yescharacteristic
127C2×Dic61#13711yescharacteristic
128C2×Dic61#13712yes—
129#92420—82no—
130#92420—82no—
131#7596—611nomaximal, Sylow
132C22×C61C2×C22⋊C811yescharacteristic
133C22×C61#41011yescharacteristic
134C22×C61#41311yescharacteristic
135C22×C61#41711yescharacteristic
136C22×C61#41911yescharacteristic
137C22×C61#42111yescharacteristic
138C22×C61#42111yescharacteristic
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—22no—
142C4×C61—22no—
143C4×C61—42no—
144C4×C61—42no—
145Dic61—21no—
146Dic61—21no—
147Dic61—22no—
148Dic61—41no—
149Dic61—41no—
150Dic61—81no—
151Dic61—81no—
152#2295—611no—
153#1839—611no—
154#1831—611no—
155#1272—612no—
156#1271—612no—
157C2×C61#145811yescharacteristic
158C2×C61#147711yescharacteristic
159C2×C61#147911yescharacteristic
160C2×C61#151211yescharacteristic
161C2×C61#151511yescharacteristic
162C2×C61#153511yescharacteristic
163C2×C61#155911yescharacteristic
164C2×C4×Q8#692311yescharacteristic
165C2×C4⋊C8—1221no—
166#391—611no—
167#391—611no—
168#389—611no—
169#389—611no—
170#387—1221no—
171#387—1221no—
172C4×C4⋊C4—611no—
173C4×C4⋊C4—1221no—
174#376—1221no—
175C43—611no—
176#336—612no—
177#336—612no—
178#336—1221no—
179#336—1221no—
180#336—1221no—
181C61#759611yescharacteristic, complemented, Sylow
182C22×Q8C2×#692311yescharacteristic
183C22×C8—1222no—
184C22×C8—1222no—
185C4×Q8—22no—
186C4×Q8#9242014yescomplemented
187C2×C4⋊C4—21no—
188C2×C4⋊C4—611no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—1221no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4C2×#692311yescharacteristic
193C2×C42—21no—
194C2×C42—611no—
195C2×C42—611no—
196C2×C42—611no—
197C2×C42—611no—
198C2×C42—611no—
199C2×C42—611no—
200C2×C42—1221no—
201C2×C42C2×#692311yescharacteristic
202C4⋊C8—2442no—
203#95—1221no—
204#95—1221no—
205#95—1221no—
206#95—1221no—
207#95—1221no—
208#95—1221no—
209#95—1221no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8C2×#9242012yes—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—611no—
217C22×C4—611no—
218C22×C4—611no—
219C22×C4—611no—
220C22×C4—1221no—
221C22×C4—1221no—
222C22×C4C22⋊#692311yescharacteristic
223C22×C4C22⋊#692311yescharacteristic
224C22×C4C22×#692311yescharacteristic
225C2×C8—2442no—
226C2×C8—2442no—
227C2×C8—2442no—
228C2×C8—2442no—
229C2×C8—2442no—
230C2×C8—2442no—
231C4⋊C4—24no—
232C4⋊C4—21no—
233C4⋊C4—2441no—
234C4⋊C4—2441no—
235C4⋊C4—2441no—
236C4⋊C4—2441no—
237C4⋊C4—2442no—
238C4⋊C4C2×#9242012yes—
239C42—24no—
240C42—1221no—
241C42—1221no—
242C42—1222no—
243C42—2441no—
244C42—2441no—
245C42—2441no—
246C42—2441no—
247C42—2442no—
248C42C2×#9242012yes—
249C42#1175821712yes—
250C23C2×C22⋊#692311yescharacteristic
251Q8—42no—
252Q8—42no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—611no—
266C2×C4—611no—
267C2×C4—612no—
268C2×C4—1221no—
269C2×C4—1221no—
270C2×C4—1221no—
271C2×C4—1221no—
272C2×C4—1221no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—1221no—
278C2×C4—2441no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4C22⋊#9242012yes—
285C2×C4C22⋊#9242012yes—
286C2×C4C22×#9242011yescharacteristic
287C2×C4C2×#1175821711yescharacteristic
288C8—4882no—
289C8—4882no—
290C22C2×C22⋊#9242011yescharacteristic
291C22#1684074111yescharacteristic
292C22#1684074811yescharacteristic
293C22#1684075311yescharacteristic
294C22#1684075511yescharacteristic
295C22#1684075911yescharacteristic
296C22#1684076111yescharacteristic
297C4—21no—
298C4—21no—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—42no—
303C4—1221no—
304C4—1221no—
305C4—1222no—
306C4—4881no—
307C4—4881no—
308C4—2441no—
309C4—2441no—
310C2#2146105011yescharacteristic
311C2#2146111411yescharacteristic
312C2#2146112111yescharacteristic
313C2#2146123211yescharacteristic
314C2#2146124111yescharacteristic
315C2#2146130511yescharacteristic
316C2#2146135411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socle, prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#7596
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
46. C4×Q8×C61C8semidirect
181. C61#7596semidirect
186. C4×Q8#92420semidirect

Maximal subgroups

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

subgroupindex
#214617462
#214617342
#214606392
#214606372
#214482972
#759661