Finite Group Databasebeta

← #30603840

Subgroups — #30603840

Order 15616.

594 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 112 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×#16840607C212yesmaximal, complemented
2#21460749C212yesmaximal, complemented
3#21460747C212yesmaximal, complemented
4#21448411C211yesmaximal, characteristic
5C2×C22⋊#92420—22no—
6#16840607—22no—
7#16840607C2214yes—
8C2×#14519736C411yescharacteristic
9C22×C4⋊Dic61C2211yescharacteristic
10#16839669C411yescharacteristic
11#16839669C2211yescharacteristic
12#16839665C411yescharacteristic
13#16839665C2211yescharacteristic
14#16839642—22no—
15#16839622—22no—
16C22×#41×C61C411yescharacteristic
17C22×#92420—42no—
18C22⋊#92420—24no—
19C22⋊#92420—42no—
20C23×Dic61D412yes—
21#14519736—42no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61—42no—
24C2×C4⋊Dic61C2×C411yescharacteristic
25C2×C4⋊Dic61D412yes—
26C2×C4⋊Dic61C2311yescharacteristic
27C2×C22⋊Dic61—21no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—22no—
30C2×C4×Dic61C2×C411yescharacteristic
31C2×C4×Dic61C2×C411yescharacteristic
32#14519684—22no—
33#14519684—22no—
34C2×#41×C61—22no—
35C2×#41×C61—42no—
36C2×#41×C61—42no—
37C2×#41×C61C814yescomplemented
38C22×Q8×C61C2×C411yescharacteristic
39C22×D4×C61—21no—
40C22×D4×C61C2×C411yescharacteristic
41C23×C4×C61—21no—
42C23×C4×C61C2×C411yescharacteristic
43C24×C61—21no—
44C24×C61C22⋊C411yescharacteristic
45#41×C61—48no—
46#41×C61—48no—
47C2×Q8×C61—21no—
48C2×Q8×C61—41no—
49C2×Q8×C61—41no—
50C2×Q8×C61C2×C812yescomplemented
51C2×D4×C61—24no—
52C2×D4×C61—21no—
53C2×D4×C61—41no—
54C2×D4×C61—41no—
55C2×D4×C61—42no—
56C2×D4×C61—42no—
57C2×D4×C61C2×C812yescomplemented
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—21no—
61C22×C4×C61—24no—
62C22×C4×C61—42no—
63C22×C4×C61—42no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C2×C812yes—
66C22×C4×C61M1612yes—
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—22no—
69C22×Dic61—22no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—82no—
73C22×Dic61C22⋊C412yes—
74C22×Dic61C2×D412yes—
75C2×#92420—44no—
76C2×#92420—84no—
77C22⋊Dic61—42no—
78C22⋊Dic61—82no—
79C22⋊Dic61—82no—
80C4⋊Dic61—41no—
81C4⋊Dic61—42no—
82C4⋊Dic61—41no—
83C4⋊Dic61—42no—
84C4×Dic61—41no—
85C4×Dic61—41no—
86C23×C61—22no—
87C23×C61—41no—
88C23×C61—41no—
89C23×C61—42no—
90C23×C61—42no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61C22⋊C812yes—
94C23×C61#9912yes—
95C23×C61#10012yes—
96C23×C61C2×C22⋊C411yescharacteristic
97Q8×C61—42no—
98Q8×C61—42no—
99D4×C61—42no—
100D4×C61—42no—
101D4×C61—48no—
102C2×C4×C61—24no—
103C2×C4×C61—24no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—22no—
109C2×C4×C61—48no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61C22⋊C812yes—
113C2×C4×C61C22×C811yescharacteristic
114C2×C4×C61C2×M1611yescharacteristic
115C2×Dic61—22no—
116C2×Dic61—22no—
117C2×Dic61—44no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—82no—
127#92420—84no—
128#11432—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—22no—
131C22×C61—21no—
132C22×C61—22no—
133C22×C61—42no—
134C22×C61—42no—
135C22×C61—44no—
136C22×C61—42no—
137C22×C61—82no—
138C22×C61—81no—
139C22×C61—81no—
140C22×C61#32314yes—
141C22×C61C2×C22⋊C811yescharacteristic
142C22×C61C2×#9911yescharacteristic
143C22×C61C2×#10011yescharacteristic
144C4×C61—22no—
145C4×C61—21no—
146C4×C61—21no—
147C4×C61—42no—
148Dic61—42no—
149Dic61—42no—
150Dic61—81no—
151Dic61—81no—
152#2311—611no—
153C2×#324—612no—
154#1319—612no—
155#1318—612no—
156C2×C61—21no—
157C2×C61—21no—
158C2×C61—24no—
159C2×C61—82no—
160C2×C61C2×#32311yescharacteristic
161C2×C61#212711yescharacteristic
162C2×C61#213311yescharacteristic
163C22×#41#692311yescharacteristic
164C2×#117—611no—
165C22×C4⋊C4—611no—
166C2×C22⋊C8—1222no—
167#391—611no—
168#391—611no—
169#390—611no—
170#390—611no—
171#386—1222no—
172#380—1222no—
173#324—614no—
174#324—1222no—
175C61#1143211yescharacteristic, complemented, Sylow
176C2×#41—22no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41#9242014yescomplemented
180C22×Q8C2×#692311yescharacteristic
181C22×D4—21no—
182C22×D4C2×#692311yescharacteristic
183C23×C4—21no—
184C23×C4—612no—
185C23×C4C2×#692311yescharacteristic
186C22×C8—2442no—
187#117—2442no—
188C2×C4⋊C4—612no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—611no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4—2442no—
193C2×C22⋊C4—1222no—
194C2×C22⋊C4—1222no—
195C2×C22⋊C4—1221no—
196C2×C42—611no—
197C2×C42—611no—
198C22⋊C8—1224no—
199C22⋊C8—2442no—
200#95—1222no—
201#95—1222no—
202C24—21no—
203C24C22⋊#692311yescharacteristic
204#41—48no—
205#41—48no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C2×#9242012yescomplemented
210C2×D4—24no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4C2×#9242012yescomplemented
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—612no—
224C22×C4—612no—
225C22×C4—1222no—
226C22×C4—1222no—
227C22×C4—1221no—
228C22×C4—1221no—
229C22×C4—4882no—
230C22×C4C2×#9242012yes—
231C22×C4#1175821712yes—
232C22×C4C22⋊#692311yescharacteristic
233C22×C4C22×#692311yescharacteristic
234C2×C8—4884no—
235C2×C8—2444no—
236C4⋊C4—2442no—
237C4⋊C4—2441no—
238C4⋊C4—2441no—
239C4⋊C4—2442no—
240C22⋊C4—4882no—
241C22⋊C4—4882no—
242C22⋊C4—2442no—
243C42—2441no—
244C42—2441no—
245C23—22no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23—41no—
252C23C22⋊#9242012yes—
253C23#1451983512yes—
254C23#1451983712yes—
255C23C2×C22⋊#692311yescharacteristic
256Q8—42no—
257Q8—42no—
258D4—42no—
259D4—42no—
260D4—48no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—48no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—1222no—
272C2×C4—1222no—
273C2×C4—4882no—
274C2×C4—2441no—
275C2×C4—2441no—
276C2×C4—2441no—
277C2×C4—2441no—
278C2×C4—2441no—
279C2×C4—2441no—
280C2×C4—2444no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4C22⋊#9242012yes—
284C2×C4C22×#9242011yescharacteristic
285C2×C4C2×#1175821711yescharacteristic
286C8—4884no—
287C22—22no—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—44no—
294C22—42no—
295C22—82no—
296C22—81no—
297C22—81no—
298C22#1684060514yes—
299C22C2×C22⋊#9242011yescharacteristic
300C22C2×#1451983511yescharacteristic
301C22C2×#1451983711yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—4881no—
307C4—4881no—
308C4—2442no—
309C4—2442no—
310C2—21no—
311C2—21no—
312C2—24no—
313C2—82no—
314C2C2×#1684060511yescharacteristic
315C2#2146209411yescharacteristic
316C2#2146211411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
Fitting, prime radicalC22×#41×C61C4
socleC22×C61C2×#99
cosocleC2×C4⋊Dic61C23
hypercenterC22×#41#6923
hypocenterC61#11432
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16840607C2semidirect
2. #21460749C2semidirect
3. #21460747C2semidirect
37. C2×#41×C61C8semidirect
50. C2×Q8×C61C2×C8semidirect
57. C2×D4×C61C2×C8semidirect
175. C61#11432semidirect
179. C2×#41#92420semidirect
209. C2×Q8C2×#92420semidirect
216. C2×D4C2×#92420semidirect

Maximal subgroups

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

subgroupindex
C2×#168406072
#214607492
#214607472
#214484112
#1143261