Finite Group Databasebeta

← #30603841

Subgroups — #30603841

Order 15616.

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

normal subgroups only

This group has 594 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
1D61⋊#324C212yesmaximal, complemented
2#21460750C212yesmaximal, complemented
3#21460747C212yesmaximal, complemented
4#21448409C211yesmaximal, characteristic
5D61⋊(C22⋊C8)—22no—
6#16840608—22no—
7#16840607C2214yes—
8C22×#11758224C411yescharacteristic
9C2×#14519734C411yescharacteristic
10C2×C4⋊C4×D61C2211yescharacteristic
11#16839668C2211yescharacteristic
12#16839667C411yescharacteristic
13#16839665C2211yescharacteristic
14#16839638—22no—
15#16839620—22no—
16C61×#390C411yescharacteristic
17C2×D61⋊C8—42no—
18#14519834—42no—
19C22⋊#92420—24no—
20C2×#11758224—22no—
21C2×#11758224—42no—
22C2×#11758224—42no—
23C2×#11758224C814yescomplemented
24C22×Dic122C2×C411yescharacteristic
25C22×C22⋊D61—21no—
26C23×Dic61—21no—
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D412yes—
29#14519734—42no—
30C2×C4⋊Dic61C2311yescharacteristic
31C2×#11758196—21no—
32C2×#11758196D412yes—
33C4⋊C4×D61—42no—
34C2×C22⋊Dic61—22no—
35C2×#11758198—21no—
36C2×C4×Dic61C2×C411yescharacteristic
37#14519684—22no—
38#14519684—22no—
39C22×D4×C61C2×C411yescharacteristic
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C22⋊C4×C61—22no—
42C2×C42×C61C2×C411yescharacteristic
43C24×C61—21no—
44C23×D61C22⋊C411yescharacteristic
45C2×D4×C61—21no—
46C2×D4×C61—41no—
47C2×D4×C61—41no—
48C2×D4×C61C2×C812yescomplemented
49C22×C4×C61—21no—
50C22×C4×C61C22⋊C412yes—
51C22×C4×C61C22×C411yescharacteristic
52C2×C22⋊D61—24no—
53C2×C22⋊D61—42no—
54C2×C22⋊D61—42no—
55C22×Dic61—21no—
56C22×Dic61—24no—
57C22×Dic61—21no—
58C22×Dic61—42no—
59C22×Dic61—42no—
60C22×Dic61C22⋊C411yescharacteristic
61C22×Dic61C2×D412yes—
62#11758224—48no—
63#11758224—48no—
64C2×C4×D61—22no—
65C2×C4×D61—22no—
66C2×C4×D61—22no—
67C2×C4×D61—22no—
68C2×C4×D61—82no—
69C2×C4×D61C2×C812yes—
70C2×C4×D61M1612yes—
71C2×Dic122—21no—
72C2×Dic122—41no—
73C2×Dic122—41no—
74C2×Dic122C2×C812yescomplemented
75C2×#92420—44no—
76D61⋊C8—84no—
77C4⋊C4×C61—41no—
78C22⋊C4×C61—82no—
79C42×C61—41no—
80C22⋊Dic61—82no—
81#11758198—42no—
82C4⋊Dic61—41no—
83#11758196—42no—
84#11758196—42no—
85C4×Dic61—41no—
86C23×C61—22no—
87C23×C61—42no—
88C23×C61—42no—
89C23×C61—41no—
90C23×C61—41no—
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—41no—
93C22×D61—41no—
94C22×D61C22⋊C812yes—
95C22×D61#9912yes—
96C22×D61#10012yes—
97D4×C61—42no—
98D4×C61—42no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—22no—
102C2×C4×C61—42no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61C22×C811yescharacteristic
107C2×C4×C61C2×M1611yescharacteristic
108C22⋊D61—48no—
109C2×Dic61—21no—
110C2×Dic61—21no—
111C2×Dic61—22no—
112C2×Dic61—22no—
113C2×Dic61—48no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61—41no—
118C2×Dic61—41no—
119C2×Dic61—41no—
120C2×Dic61C22⋊C812yes—
121C4×D61—24no—
122C4×D61—24no—
123C4×D61—44no—
124C4×D61—82no—
125Dic122—42no—
126Dic122—42no—
127#92420—84no—
128#11432—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—44no—
132C22×C61—42no—
133C22×C61—82no—
134C22×C61—81no—
135C22×C61—81no—
136C22×C61C2×C22⋊C811yescharacteristic
137C22×C61C2×#9911yescharacteristic
138C22×C61C2×#10011yescharacteristic
139C2×D61—22no—
140C2×D61—22no—
141C2×D61—42no—
142C2×D61—42no—
143C2×D61#32314yes—
144C4×C61—21no—
145C4×C61—21no—
146C4×C61—42no—
147C4×C61—81no—
148Dic61—22no—
149Dic61—42no—
150Dic61—42no—
151Dic61—81no—
152#2311—611no—
153C2×#324—612no—
154#1319—612no—
155#1318—612no—
156C2×C61—21no—
157C2×C61—21no—
158C2×C61—82no—
159C2×C61C2×#32311yescharacteristic
160C2×C61#212711yescharacteristic
161C2×C61#213311yescharacteristic
162D61—24no—
163C22×#41—611no—
164C2×#117—611no—
165C22×C4⋊C4—611no—
166C2×C22⋊C8—1222no—
167#391—611no—
168#391—611no—
169#390—611no—
170#390#692311yescharacteristic
171#386—1222no—
172#380—1222no—
173#324—614no—
174#324—1222no—
175C61#1143211yescharacteristic, complemented, Sylow
176C2×#41—614no—
177C2×#41—1222no—
178C2×#41—2442no—
179C2×#41—2442no—
180C22×Q8—611no—
181C22×D4—1221no—
182C22×D4C2×#692311yescharacteristic
183C23×C4—611no—
184C23×C4—612no—
185C23×C4—1221no—
186C22×C8—2442no—
187#117—2442no—
188C2×C4⋊C4—612no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—1221no—
191C2×C4⋊C4—2442no—
192C2×C4⋊C4C2×#692311yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—1222no—
195C2×C22⋊C4—1221no—
196C2×C42—611no—
197C2×C42C2×#692311yescharacteristic
198C22⋊C8—1224no—
199C22⋊C8—2442no—
200#95—1222no—
201#95—1222no—
202C24—21no—
203C24—611no—
204#41—2448no—
205#41—2448no—
206C2×Q8—612no—
207C2×Q8—1221no—
208C2×Q8—2441no—
209C2×Q8—2441no—
210C2×D4—21no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—1224no—
214C2×D4—2442no—
215C2×D4—2442no—
216C2×D4D61⋊C812yescomplemented
217C22×C4—21no—
218C22×C4—611no—
219C22×C4—612no—
220C22×C4—612no—
221C22×C4—612no—
222C22×C4—1222no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—1222no—
226C22×C4—1224no—
227C22×C4—1221no—
228C22×C4—1221no—
229C22×C4—4882no—
230C22×C4—2442no—
231C22×C4—2442no—
232C22×C4C22⋊#692312yes—
233C22×C4C22×#692311yescharacteristic
234C2×C8—4884no—
235C2×C8—2444no—
236C4⋊C4—41no—
237C4⋊C4—2442no—
238C4⋊C4—2441no—
239C4⋊C4—2442no—
240C22⋊C4—82no—
241C22⋊C4—4882no—
242C22⋊C4—2442no—
243C42—41no—
244C42—2441no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—612no—
251C23—612no—
252C23—612no—
253C23—2441no—
254C23—2441no—
255C23C2×C22⋊#692311yescharacteristic
256Q8—2442no—
257Q8—2442no—
258D4—42no—
259D4—42no—
260D4—2448no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—612no—
269C2×C4—1224no—
270C2×C4—1224no—
271C2×C4—1221no—
272C2×C4—1221no—
273C2×C4—1222no—
274C2×C4—1222no—
275C2×C4—4882no—
276C2×C4—2444no—
277C2×C4—2448no—
278C2×C4—2441no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2442no—
284C2×C4C2×D61⋊C811yescharacteristic
285C2×C4D61⋊M1611yescharacteristic
286C8—4884no—
287C22—21no—
288C22—21no—
289C22—44no—
290C22—42no—
291C22—82no—
292C22—81no—
293C22—81no—
294C22—614no—
295C22—1222no—
296C22—1222no—
297C22—2442no—
298C22—2442no—
299C22D61⋊(C22⋊C8)11yescharacteristic
300C22D61⋊#9911yescharacteristic
301C22D61⋊#10011yescharacteristic
302C4—21no—
303C4—21no—
304C4—42no—
305C4—81no—
306C4—1222no—
307C4—4881no—
308C4—2442no—
309C4—2442no—
310C2—21no—
311C2—21no—
312C2—82no—
313C2—1224no—
314C2D61⋊#32311yescharacteristic
315C2#2146209511yescharacteristic
316C2#2146211511yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61C2×#99
prime radicalC22×#11758224C4
cosocleC2×C4⋊Dic61C23
hypercenter#390#6923
hypocenterC61#11432
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. D61⋊#324C2semidirect
2. #21460750C2semidirect
3. #21460747C2semidirect
23. C2×#11758224C8semidirect
48. C2×D4×C61C2×C8semidirect
74. C2×Dic122C2×C8semidirect
175. C61#11432semidirect
216. C2×D4D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
D61⋊#3242
#214607502
#214607472
#214484092
#1143261