Finite Group Databasebeta

← #30603817

Subgroups — #30603817

Order 15616.

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

normal subgroups only

This group has 678 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#21460746C212yesmaximal, complemented
3#21460743C212yesmaximal
4#21448197C211yesmaximal, characteristic
5#16840779—22no—
6#16840608—22no—
7#16840605C2214yes—
8C24×Dic61C411yescharacteristic
9C2×#14519733C411yescharacteristic
10C2×#14519733C2211yescharacteristic
11#16839623C411yescharacteristic
12#16839623C2211yescharacteristic
13#16839620—22no—
14#16839618C2211yescharacteristic
15C22⋊C4×Dic61—22no—
16C61×#379C411yescharacteristic
17#14519844—42no—
18C22⋊#92420—24no—
19C4×#92420—42no—
20C23×Dic61—21no—
21C23×Dic61—21no—
22C23×Dic61—22no—
23C23×Dic61—41no—
24C23×Dic61—41no—
25C23×Dic61—42no—
26C23×Dic61C814yescomplemented
27C23×Dic61C2×C411yescharacteristic
28C23×Dic61C2×C411yescharacteristic
29#14519733—41no—
30#14519733—41no—
31#14519733—42no—
32C2×#11758196—21no—
33C2×#11758196D412yes—
34C2×C22⋊Dic61—22no—
35C2×C22⋊Dic61C2×C411yescharacteristic
36C2×C22⋊Dic61C2311yescharacteristic
37C2×C4×Dic61—21no—
38C2×C4×Dic61D412yes—
39#14519684—22no—
40#14519684—22no—
41C25×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61C2×C411yescharacteristic
44C2×C22⋊C4×C61C2×C411yescharacteristic
45C24×C61—21no—
46C24×C61—21no—
47C24×C61—41no—
48C24×C61—41no—
49C24×C61—44no—
50C24×C61C2×C812yes—
51C24×C61C22×C411yescharacteristic
52C22×C4×C61—21no—
53C22×C4×C61C22⋊C412yes—
54C22×Dic61—21no—
55C22×Dic61—21no—
56C22×Dic61—22no—
57C22×Dic61—21no—
58C22×Dic61—21no—
59C22×Dic61—24no—
60C22×Dic61—24no—
61C22×Dic61—21no—
62C22×Dic61—42no—
63C22×Dic61—42no—
64C22×Dic61—44no—
65C22×Dic61—44no—
66C22×Dic61—44no—
67C22×Dic61—44no—
68C22×Dic61—41no—
69C22×Dic61—41no—
70C22×Dic61—42no—
71C22×Dic61—41no—
72C22×Dic61—41no—
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C2×C812yes—
76C22×Dic61C2×C812yes—
77C22×Dic61M1612yes—
78C22×Dic61C2×D412yes—
79C2×#92420—44no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—44no—
82C22⋊C4×C61—41no—
83C22⋊Dic61—41no—
84C22⋊Dic61—44no—
85C22⋊Dic61—41no—
86#11758196—42no—
87#11758196—42no—
88C4×Dic61—22no—
89C4×Dic61—22no—
90C4×Dic61—42no—
91C23×C61—24no—
92C23×C61—22no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—44no—
98C23×C61—42no—
99C23×C61—44no—
100C23×C61—42no—
101C23×C61—48no—
102C23×C61—44no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61—44no—
106C23×C61C2×C22⋊C411yescharacteristic
107C23×C61C22×C811yescharacteristic
108C23×C61C2×M1611yescharacteristic
109C2×C4×C61—22no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—41no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—22no—
116C2×Dic61—22no—
117C2×Dic61—22no—
118C2×Dic61—22no—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—48no—
122C2×Dic61—42no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—44no—
126C2×Dic61—44no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—41no—
131C2×Dic61C22⋊C812yes—
132C2×Dic61C22⋊C812yes—
133C2×Dic61#9911yescharacteristic
134C2×Dic61#9911yescharacteristic
135C2×Dic61#10012yes—
136#92420—84no—
137#11425—611nomaximal, Sylow
138C22×C61—22no—
139C22×C61—24no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—44no—
145C22×C61—42no—
146C22×C61—42no—
147C22×C61—44no—
148C22×C61—48no—
149C22×C61—44no—
150C22×C61—44no—
151C22×C61—42no—
152C22×C61—42no—
153C22×C61C2×C22⋊C811yescharacteristic
154C22×C61C2×#9911yescharacteristic
155C22×C61C2×#10011yescharacteristic
156C4×C61—42no—
157C4×C61—81no—
158Dic61—22no—
159Dic61—42no—
160Dic61—42no—
161Dic61—81no—
162Dic61#32312yes—
163Dic61#32312yes—
164#2280—611no—
165#1458—612no—
166#1317—612no—
167#1316—612no—
168C2×C61—21no—
169C2×C61—21no—
170C2×C61—44no—
171C2×C61—41no—
172C2×C61—41no—
173C2×C61C2×#32311yescharacteristic
174C2×C61#212012yes—
175C24×C4—611no—
176C2×#117—611no—
177C2×#117—611no—
178#431—1222no—
179#381—611no—
180#381—611no—
181#380—1222no—
182#379—611no—
183#379#692311yescharacteristic
184C4×C22⋊C4—1222no—
185#324—1222no—
186#323—614no—
187C61#1142511yescharacteristic, complemented, Sylow
188C25C2×#692311yescharacteristic
189C23×C4—611no—
190C23×C4—614no—
191C23×C4—611no—
192C23×C4—1222no—
193C23×C4—1221no—
194C23×C4—1221no—
195C23×C4—2441no—
196C23×C4—2441no—
197C23×C4—2442no—
198#117—2441no—
199#117—2441no—
200#117—2442no—
201C2×C4⋊C4—612no—
202C2×C4⋊C4—1221no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—611no—
205C2×C22⋊C4—611no—
206C2×C22⋊C4—1222no—
207C2×C22⋊C4C2×#692311yescharacteristic
208C2×C22⋊C4C2×#692311yescharacteristic
209C2×C42—612no—
210C2×C42—1221no—
211C4⋊C8—2442no—
212C22⋊C8—1224no—
213C4×C8—2442no—
214#95—1222no—
215#95—1222no—
216C24—21no—
217C24—21no—
218C24—41no—
219C24—41no—
220C24—44no—
221C24D61⋊C812yes—
222C24C22×#692311yescharacteristic
223C22×C4—21no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—612no—
227C22×C4—612no—
228C22×C4—612no—
229C22×C4—612no—
230C22×C4—1221no—
231C22×C4—1221no—
232C22×C4—1222no—
233C22×C4—1224no—
234C22×C4—1224no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—1221no—
238C22×C4—2441no—
239C22×C4—2441no—
240C22×C4—2442no—
241C22×C4—2444no—
242C22×C4—2444no—
243C22×C4—2442no—
244C22×C4—2442no—
245C22×C4—2444no—
246C22×C4—2444no—
247C22×C4—2441no—
248C22×C4—2441no—
249C22×C4C22⋊#692312yes—
250C2×C8—2444no—
251C4⋊C4—2442no—
252C4⋊C4—2442no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—41no—
256C22⋊C4—2441no—
257C22⋊C4—2444no—
258C22⋊C4—2441no—
259C42—1222no—
260C42—1222no—
261C42—2442no—
262C23—24no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—42no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23—44no—
273C23—48no—
274C23—44no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊#692311yescharacteristic
278C23C2×D61⋊C811yescharacteristic
279C23D61⋊M1611yescharacteristic
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—612no—
285C2×C4—611no—
286C2×C4—611no—
287C2×C4—612no—
288C2×C4—612no—
289C2×C4—1222no—
290C2×C4—1222no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—1222no—
296C2×C4—1222no—
297C2×C4—2444no—
298C2×C4—2444no—
299C2×C4—2448no—
300C2×C4—2442no—
301C2×C4—2441no—
302C2×C4—2442no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2442no—
306C2×C4—2442no—
307C8—4884no—
308C22—22no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22—48no—
320C22—42no—
321C22—44no—
322C22—42no—
323C22D61⋊(C22⋊C8)11yescharacteristic
324C22D61⋊#9911yescharacteristic
325C22D61⋊#10011yescharacteristic
326C4—42no—
327C4—81no—
328C4—612no—
329C4—612no—
330C4—1222no—
331C4—4881no—
332C4—2442no—
333C4—2442no—
334C2—21no—
335C2—21no—
336C2—44no—
337C2—41no—
338C2—41no—
339C2D61⋊#32311yescharacteristic
340C2#2146207112yes—
subgroupG/H
centerC22D61⋊#99
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC61×#379C4
socleC22×C61C2×#99
prime radicalC25×C61C2×C4
cosocleC2×C22⋊Dic61C23
hypercenter#379#6923
hypocenterC61#11425
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. #21460746C2semidirect
26. C23×Dic61C8semidirect
187. C61#11425semidirect

Maximal subgroups

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

subgroupindex
#214610512
#214607462
#214607432
#214481972
#1142561