Finite Group Databasebeta

← #30606298

Subgroups — #30606298

Order 15616.

854 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C2×#16840741C212yesmaximal
2#21460913C214yesmaximal, complemented
3#21448309C211yesmaximal, characteristic
4C2×#14519856—22no—
5#16840741—24no—
6#16840741C418yescomplemented
7C2×#14519836C2214yes—
8C2×#14519835—22no—
9C22×#11758222C2211yescharacteristic
10C2×#14519736C411yescharacteristic, complemented
11#16839668C412yescomplemented
12#16839664C2212yes—
13#16839641—22no—
14C4×#11758198—22no—
15C22×C42×C61C411yescharacteristic, complemented
16#14519856—28no—
17#14519856—44no—
18C2×C4⋊#6923—42no—
19C2×C22⋊#6923—24no—
20C2×C4×#6923—42no—
21#14519836—24no—
22#14519836C2×C418yescomplemented
23#14519835—28no—
24C2×#11758222—41no—
25C2×#11758222—41no—
26C2×#11758222—42no—
27C2×#11758222C2×C414yes—
28C2×#11758222D412yes—
29C2×#11758222Q812yes—
30C22×Dic122C2×C411yescharacteristic
31C22×D244C2311yescharacteristic
32C22×C22⋊D61D412yes—
33C22×C4×D61D412yes—
34#14519736—41no—
35#14519736—41no—
36C2×C4⋊Dic61C2×C412yes—
37C2×C22⋊Dic61—21no—
38C2×#11758198—24no—
39C2×C4×Dic61—21no—
40#14519684—24no—
41C23×C4×C61—21no—
42C23×C4×C61C2×C411yescharacteristic
43C2×C42×C61—21no—
44C2×C42×C61—42no—
45C2×C42×C61C2×C412yes—
46C24×C61C22⋊C411yescharacteristic
47C23×D61C2×D412yes—
48C22×#6923—44no—
49C22×C4×C61—22no—
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—22no—
53C22×C4×C61—44no—
54C22×C4×C61—42no—
55C22×C4×C61C4212yes—
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C4⋊C412yes—
58C22×C4×C61C22×C411yescharacteristic
59C2×C22⋊D61—42no—
60C2×C22⋊D61—42no—
61C2×C22⋊D61C22⋊C414yes—
62C2×C22⋊D61C4⋊C414yes—
63C22×Dic61—22no—
64C22×Dic61C22⋊C412yes—
65#11758222—44no—
66#11758222—44no—
67#11758222—42no—
68#11758222—42no—
69#11758222—42no—
70#11758222—42no—
71C2×D244—41no—
72C2×D244—41no—
73C2×D244C22×C412yes—
74C2×D244C2×D411yescharacteristic
75C2×D244C2×Q811yescharacteristic
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—24no—
79C2×C4×D61C22⋊C414yes—
80C2×C4×D61C4⋊C414yes—
81C2×Dic122—41no—
82C2×Dic122—41no—
83C2×Dic122C4212yes—
84C2×Dic122C4⋊C412yes—
85C22⋊#6923—28no—
86C22⋊#6923—44no—
87C4⋊#6923—48no—
88C4×#6923—48no—
89C42×C61—42no—
90C42×C61—42no—
91C22⋊Dic61—42no—
92#11758198—48no—
93C4⋊Dic61—42no—
94C4×Dic61—41no—
95C4×Dic61—41no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—42no—
99C23×C61#9512yes—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—22no—
102C22×D61—22no—
103C22×D61—44no—
104C22×D61C2×C22⋊C412yes—
105C22×D61C2×C4⋊C412yes—
106C2×#6923—48no—
107C2×#6923—88no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—24no—
115C2×C4×C61—42no—
116C2×C4×C61—48no—
117C2×C4×C61—42no—
118C2×C4×C61#9512yes—
119C2×C4×C61#9912yes—
120C2×C4×C61#9911yescharacteristic
121C2×C4×C61#9911yescharacteristic
122C2×C4×C61C2×C4211yescharacteristic
123C2×C4×C61C2×C4⋊C411yescharacteristic
124C22⋊D61—44no—
125C22⋊D61—44no—
126C2×Dic61—22no—
127C2×Dic61—22no—
128C2×Dic61—44no—
129C2×Dic61—42no—
130C2×Dic61#9514yes—
131D244—42no—
132D244—41no—
133D244—41no—
134C4×D61—24no—
135C4×D61—24no—
136C4×D61—24no—
137C4×D61—24no—
138Dic122—42no—
139Dic122—42no—
140#12581—611nomaximal, Sylow
141C22×C61—22no—
142C22×C61—22no—
143C22×C61—42no—
144C22×C61—42no—
145C22×C61—42no—
146C22×C61C2×#9511yescharacteristic
147C22×C61C2×#9911yescharacteristic
148C22×C61C2×#9911yescharacteristic
149C2×D61—22no—
150C2×D61—22no—
151C2×D61—44no—
152C2×D61—42no—
153C2×D61—42no—
154C2×D61—82no—
155#6923—88no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—44no—
159C4×C61#34212yes—
160C4×C61#34212yes—
161Dic61—24no—
162Dic61—82no—
163C2×#410—612no—
164#2297—611no—
165#1393—614no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—42no—
169C2×C61C2×#34211yescharacteristic
170C2×C61#212612yes—
171D61—42no—
172D61—42no—
173C22×#41—611no—
174C2×#117—611no—
175C2×#117—1222no—
176C22×C42#692311yescharacteristic, complemented
177#410—618no—
178#410—1224no—
179C2×#99—614no—
180C2×#99—1222no—
181#391—612no—
182#390—612no—
183#386—1222no—
184C4×C22⋊C4—1222no—
185C61#1258111yescharacteristic, complemented, Sylow
186C2×#41—614no—
187C2×#41—612no—
188C2×#41—612no—
189C2×#41—2442no—
190C2×#41—2441no—
191C2×#41—2441no—
192C22×Q8—611no—
193C22×D4—612no—
194C22×D4—611no—
195C23×C4—21no—
196C23×C4—612no—
197C23×C4C2×#692311yescharacteristic
198#117—1228no—
199#117—2444no—
200#117—2441no—
201#117—2441no—
202C2×C4⋊C4—612no—
203C2×C4⋊C4—2442no—
204C2×C22⋊C4—1224no—
205C2×C22⋊C4—1224no—
206C2×C22⋊C4—1221no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42—1221no—
210C2×C42—2442no—
211C2×C42C2×#692312yes—
212#99—618no—
213#99—1228no—
214#99—1224no—
215#95—1224no—
216C24—612no—
217C24C22⋊#692311yescharacteristic
218#41—2444no—
219#41—2444no—
220#41—2442no—
221#41—2442no—
222#41—2442no—
223#41—2442no—
224C2×Q8—612no—
225C2×Q8—612no—
226C2×Q8—2441no—
227C2×Q8—2441no—
228C2×D4—614no—
229C2×D4—614no—
230C2×D4—612no—
231C2×D4—611no—
232C2×D4—611no—
233C2×D4—2441no—
234C2×D4—2441no—
235C2×D4—2442no—
236C2×D4—2442no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—612no—
244C22×C4—614no—
245C22×C4—614no—
246C22×C4—1224no—
247C22×C4—1222no—
248C22×C4—1222no—
249C22×C4—1222no—
250C22×C4—2444no—
251C22×C4C4×#692312yes—
252C22×C4C4⋊#692312yes—
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22×#692311yescharacteristic
255C4⋊C4—2442no—
256C4⋊C4—2448no—
257C22⋊C4—1228no—
258C22⋊C4—2448no—
259C22⋊C4—2442no—
260C22⋊C4—2444no—
261C42—42no—
262C42—42no—
263C42—2448no—
264C42—2441no—
265C42—2441no—
266C23—21no—
267C23—21no—
268C23—42no—
269C23—612no—
270C23—612no—
271C23—1222no—
272C23—1222no—
273C23—2444no—
274C23(C2×#6923)⋊C412yes—
275C23C2×C22⋊#692311yescharacteristic
276Q8—2442no—
277Q8—2442no—
278D4—2442no—
279D4—2441no—
280D4—2441no—
281D4—2444no—
282D4—2444no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—42no—
291C2×C4—48no—
292C2×C4—42no—
293C2×C4—614no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—1224no—
297C2×C4—1224no—
298C2×C4—1224no—
299C2×C4—1224no—
300C2×C4—4888no—
301C2×C4—2444no—
302C2×C4—2448no—
303C2×C4—2442no—
304C2×C4(C2×#6923)⋊C412yes—
305C2×C4#1451983512yes—
306C2×C4#1451983611yescharacteristic
307C2×C4#1451983611yescharacteristic
308C2×C4C2×C4×#692311yescharacteristic
309C2×C4C2×C4⋊#692311yescharacteristic
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—1222no—
316C22—1222no—
317C22—4882no—
318C22—2442no—
319C22—2442no—
320C22—2444no—
321C22C2×(C2×#6923)⋊C411yescharacteristic
322C22C2×#1451983511yescharacteristic
323C22C2×#1451983611yescharacteristic
324C4—21no—
325C4—21no—
326C4—44no—
327C4—1224no—
328C4—4888no—
329C4—4882no—
330C4#1684064012yes—
331C4#1684064012yes—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—2442no—
336C2—2442no—
337C2C2×#1684064011yescharacteristic
338C2#2146209212yes—
subgroupG/H
centerC2×C4#14519836
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC22×C42×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC22×D244C23
hypercenterC22×C42#6923
hypocenterC61#12581
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #21460913C2semidirect
6. #16840741C4semidirect
10. C2×#14519736C4semidirect
11. #16839668C4semidirect
15. C22×C42×C61C4semidirect
22. #14519836C2×C4semidirect
176. C22×C42#6923semidirect
185. C61#12581semidirect

Maximal subgroups

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

subgroupindex
C2×#168407412
#214609132
#214483092
#1258161