Finite Group Databasebeta

← #30603825

Subgroups — #30603825

Order 15616.

686 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 90 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#21461051C212yesmaximal, complemented
2#21460744C212yesmaximal, complemented
3#21460743C212yesmaximal, complemented
4#21448400C211yesmaximal, characteristic
5#16840779—22no—
6#16840608—22no—
7#16840605C2214yes—
8C23×C22⋊D61C411yescharacteristic
9C2×#14519733C411yescharacteristic
10C2×#14519733C2211yescharacteristic
11#16839662—22no—
12#16839662—22no—
13#16839618C2211yescharacteristic
14#16839617C411yescharacteristic
15#16839617C2211yescharacteristic
16C61×#379C411yescharacteristic
17#14519844—42no—
18C22⋊#92420—24no—
19C4×#92420—42no—
20C24×D61C2×C411yescharacteristic
21C22×C22⋊D61—21no—
22C22×C22⋊D61—21no—
23C22×C22⋊D61—22no—
24C22×C22⋊D61—42no—
25C22×C22⋊D61—42no—
26C22×C22⋊D61C814yescomplemented
27C23×Dic61C2×C411yescharacteristic
28#14519733—42no—
29#14519733—42no—
30C2×#11758196—21no—
31C2×#11758196D412yes—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61C2×C411yescharacteristic
34C2×C22⋊Dic61C2311yescharacteristic
35C2×#11758198—22no—
36C2×#11758198—22no—
37C2×C4×Dic61—21no—
38C2×C4×Dic61D412yes—
39C25×C61C2×C411yescharacteristic
40C2×C22⋊C4×C61—22no—
41C2×C22⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61C2×C411yescharacteristic
43C24×C61—21no—
44C24×C61—21no—
45C24×C61—41no—
46C24×C61—41no—
47C24×C61—82no—
48C24×C61C2×C812yes—
49C24×C61C22×C411yescharacteristic
50C23×D61—21no—
51C23×D61—21no—
52C23×D61—41no—
53C23×D61—41no—
54C23×D61—82no—
55C23×D61C2×C812yes—
56C22×C4×C61—21no—
57C22×C4×C61C22⋊C412yes—
58C2×C22⋊D61—24no—
59C2×C22⋊D61—24no—
60C2×C22⋊D61—42no—
61C2×C22⋊D61—42no—
62C2×C22⋊D61—48no—
63C2×C22⋊D61—48no—
64C2×C22⋊D61—42no—
65C2×C22⋊D61—42no—
66C22×Dic61—22no—
67C22×Dic61—22no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C2×C812yes—
72C22×Dic61M1612yes—
73C22×Dic61C2×D412yes—
74C2×#92420—44no—
75C22⋊C4×C61—41no—
76C22⋊C4×C61—41no—
77C22⋊C4×C61—82no—
78C22⋊Dic61—41no—
79C22⋊Dic61—41no—
80C22⋊Dic61—82no—
81#11758198—82no—
82#11758198—82no—
83#11758196—42no—
84#11758196—42no—
85C4×Dic61—22no—
86C4×Dic61—22no—
87C4×Dic61—42no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—22no—
91C23×C61—42no—
92C23×C61—42no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61—42no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61—82no—
99C23×C61—82no—
100C23×C61—84no—
101C23×C61—84no—
102C23×C61C2×C22⋊C411yescharacteristic
103C23×C61C22×C811yescharacteristic
104C23×C61C2×M1611yescharacteristic
105C22×D61—22no—
106C22×D61—42no—
107C22×D61—42no—
108C22×D61—42no—
109C22×D61—41no—
110C22×D61—41no—
111C22×D61—42no—
112C22×D61—82no—
113C22×D61—82no—
114C22×D61—84no—
115C22×D61—84no—
116C2×C4×C61—22no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—41no—
120C22⋊D61—48no—
121C22⋊D61—48no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—24no—
125C2×Dic61—24no—
126C2×Dic61—22no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—41no—
130C2×Dic61C22⋊C812yes—
131C2×Dic61C22⋊C812yes—
132C2×Dic61#9912yes—
133C2×Dic61#10012yes—
134#92420—84no—
135#11427—611nomaximal, Sylow
136C22×C61—22no—
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—42no—
142C22×C61—42no—
143C22×C61—42no—
144C22×C61—42no—
145C22×C61—42no—
146C22×C61—82no—
147C22×C61—84no—
148C22×C61—84no—
149C22×C61—82no—
150C22×C61C2×C22⋊C811yescharacteristic
151C22×C61C2×#9911yescharacteristic
152C22×C61C2×#10011yescharacteristic
153C2×D61—42no—
154C2×D61—42no—
155C2×D61—42no—
156C2×D61—82no—
157C2×D61—84no—
158C2×D61—82no—
159C2×D61—84no—
160C4×C61—42no—
161C4×C61—81no—
162Dic61—22no—
163Dic61—42no—
164Dic61—81no—
165Dic61#32314yes—
166#2309—611no—
167#1458—612no—
168#1316—612no—
169#1316—612no—
170C2×C61—21no—
171C2×C61—21no—
172C2×C61—41no—
173C2×C61—41no—
174C2×C61—82no—
175C2×C61C2×#32311yescharacteristic
176C2×C61#212212yes—
177D61—82no—
178C23×D4—611no—
179C2×#117—611no—
180C2×#117—611no—
181#431—1222no—
182#390—1222no—
183#390—1222no—
184#379—611no—
185#379—611no—
186#379—611no—
187#379#692311yescharacteristic
188#324—1222no—
189#323—614no—
190C61#1142711yescharacteristic, complemented, Sylow
191C25—611no—
192C25C2×#692311yescharacteristic
193C22×D4—614no—
194C22×D4—1222no—
195C22×D4—1221no—
196C22×D4—1221no—
197C22×D4—2442no—
198C22×D4—2442no—
199C23×C4—611no—
200#117—2442no—
201#117—2442no—
202C2×C4⋊C4—612no—
203C2×C4⋊C4—1221no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—611no—
206C2×C22⋊C4—611no—
207C2×C22⋊C4—1222no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4C2×#692311yescharacteristic
211C2×C22⋊C4C2×#692311yescharacteristic
212C2×C42—612no—
213C2×C42—1221no—
214C4⋊C8—2442no—
215C22⋊C8—1224no—
216C4×C8—2442no—
217C24—21no—
218C24—21no—
219C24—41no—
220C24—41no—
221C24—82no—
222C24—612no—
223C24—1221no—
224C24—1221no—
225C24—4882no—
226C24—2441no—
227C24—2441no—
228C24D61⋊C812yes—
229C24C22×#692311yescharacteristic
230C2×D4—1224no—
231C2×D4—1224no—
232C2×D4—2442no—
233C2×D4—2442no—
234C2×D4—2448no—
235C2×D4—2442no—
236C2×D4—2442no—
237C2×D4—2448no—
238C22×C4—21no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—612no—
242C22×C4—612no—
243C22×C4—612no—
244C22×C4—1222no—
245C22×C4—1222no—
246C22×C4—1221no—
247C22×C4C22⋊#692312yes—
248C2×C8—2444no—
249C4⋊C4—2442no—
250C4⋊C4—2442no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—82no—
254C22⋊C4—4882no—
255C22⋊C4—4882no—
256C22⋊C4—4882no—
257C22⋊C4—2441no—
258C22⋊C4—2441no—
259C42—1222no—
260C42—1222no—
261C42—2442no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—41no—
271C23—41no—
272C23—82no—
273C23—82no—
274C23—84no—
275C23—84no—
276C23—1222no—
277C23—4884no—
278C23—4884no—
279C23—4882no—
280C23—4882no—
281C23—2442no—
282C23—2442no—
283C23—2442no—
284C23—2442no—
285C23—2441no—
286C23—2441no—
287C23C2×C22⋊#692311yescharacteristic
288C23C2×D61⋊C811yescharacteristic
289C23D61⋊M1611yescharacteristic
290D4—2448no—
291D4—2448no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—612no—
297C2×C4—612no—
298C2×C4—612no—
299C2×C4—612no—
300C2×C4—1224no—
301C2×C4—1224no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1222no—
305C2×C4—2442no—
306C2×C4—2441no—
307C2×C4—2442no—
308C8—4884no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—82no—
320C22—84no—
321C22—84no—
322C22—82no—
323C22—4882no—
324C22—4884no—
325C22—4884no—
326C22—4882no—
327C22—2442no—
328C22—2442no—
329C22—2442no—
330C22D61⋊(C22⋊C8)11yescharacteristic
331C22D61⋊#9911yescharacteristic
332C22D61⋊#10011yescharacteristic
333C4—42no—
334C4—81no—
335C4—614no—
336C4—1222no—
337C4—4881no—
338C4—2442no—
339C2—21no—
340C2—21no—
341C2—41no—
342C2—41no—
343C2—82no—
344C2—4882no—
345C2D61⋊#32311yescharacteristic
346C2#2146207912yes—
subgroupG/H
centerC22D61⋊#99
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC61×#379C4
socleC22×C61C2×#99
prime radicalC23×C22⋊D61C4
cosocleC2×C22⋊Dic61C23
hypercenter#379#6923
hypocenterC61#11427
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #21461051C2semidirect
2. #21460744C2semidirect
3. #21460743C2semidirect
26. C22×C22⋊D61C8semidirect
190. C61#11427semidirect

Maximal subgroups

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

subgroupindex
#214610512
#214607442
#214607432
#214484002
#1142761