Finite Group Databasebeta

← C2×#21460746 #30603815

Subgroups — C2×#21460746 #30603815

Order 15616.

726 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 120 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×#16840605C212yesmaximal
2#21460746C214yesmaximal, direct product factor
3C2×#16839623C211yesmaximal, characteristic
4C2×C22⋊#92420—22no—
5#16840605—22no—
6#16840605C2214yes—
7C24×Dic61C411yescharacteristic
8C22×C22⋊Dic61C2211yescharacteristic
9#16839623—22no—
10#16839623C412yes—
11#16839623C2212yes—
12C2×#14519684—22no—
13C22×C22⋊C4×C61C411yescharacteristic
14C22×#92420—42no—
15C22⋊#92420—24no—
16C22⋊#92420—42no—
17C23×Dic61—21no—
18C23×Dic61—21no—
19C23×Dic61—22no—
20C23×Dic61—41no—
21C23×Dic61—41no—
22C23×Dic61—42no—
23C23×Dic61C814yescomplemented
24C23×Dic61C2×C412yes—
25C23×Dic61D412yes—
26C2×C22⋊Dic61—41no—
27C2×C22⋊Dic61—41no—
28C2×C22⋊Dic61C2×C411yescharacteristic
29C2×C22⋊Dic61D412yes—
30C2×C22⋊Dic61C2311yescharacteristic
31#14519684—24no—
32#14519684—24no—
33C25×C61C2×C411yescharacteristic
34C23×C4×C61—21no—
35C2×C22⋊C4×C61—21no—
36C2×C22⋊C4×C61—42no—
37C2×C22⋊C4×C61C2×C412yes—
38C24×C61—21no—
39C24×C61—21no—
40C24×C61—22no—
41C24×C61—44no—
42C24×C61C22⋊C411yescharacteristic
43C24×C61C22⋊C411yescharacteristic
44C24×C61C2×C812yes—
45C24×C61M1612yes—
46C24×C61C22×C411yescharacteristic
47C22×C4×C61—22no—
48C22×C4×C61—44no—
49C22×C4×C61—42no—
50C22×Dic61—24no—
51C22×Dic61—24no—
52C22×Dic61—22no—
53C22×Dic61—22no—
54C22×Dic61—22no—
55C22×Dic61—22no—
56C22×Dic61—44no—
57C22×Dic61—44no—
58C22×Dic61—42no—
59C22×Dic61—42no—
60C22×Dic61—41no—
61C22×Dic61—41no—
62C22×Dic61—42no—
63C22×Dic61—44no—
64C22×Dic61—44no—
65C22×Dic61—42no—
66C22×Dic61—42no—
67C22×Dic61—44no—
68C22×Dic61C22⋊C412yes—
69C22×Dic61C2×C814yescomplemented
70C22×Dic61C2×D412yes—
71C2×#92420—44no—
72C2×#92420—84no—
73C22⋊C4×C61—42no—
74C22⋊C4×C61—42no—
75C22⋊Dic61—41no—
76C22⋊Dic61—42no—
77C22⋊Dic61—41no—
78C23×C61—24no—
79C23×C61—22no—
80C23×C61—22no—
81C23×C61—22no—
82C23×C61—22no—
83C23×C61—21no—
84C23×C61—21no—
85C23×C61—21no—
86C23×C61—21no—
87C23×C61—48no—
88C23×C61—44no—
89C23×C61—42no—
90C23×C61—44no—
91C23×C61—44no—
92C23×C61—42no—
93C23×C61—44no—
94C23×C61C22⋊C812yes—
95C23×C61C22⋊C812yes—
96C23×C61#9911yescharacteristic
97C23×C61#9911yescharacteristic
98C23×C61#10012yes—
99C23×C61C2×C22⋊C411yescharacteristic
100C23×C61C22×C811yescharacteristic
101C23×C61C2×M1611yescharacteristic
102C2×C4×C61—44no—
103C2×C4×C61—42no—
104C2×C4×C61—84no—
105C2×Dic61—24no—
106C2×Dic61—22no—
107C2×Dic61—22no—
108C2×Dic61—48no—
109C2×Dic61—44no—
110C2×Dic61—44no—
111C2×Dic61—44no—
112C2×Dic61—44no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—44no—
116C2×Dic61—42no—
117C2×Dic61—42no—
118C2×Dic61—82no—
119#92420—84no—
120C2×#1317—611nomaximal, Sylow
121C22×C61—24no—
122C22×C61—24no—
123C22×C61—22no—
124C22×C61—22no—
125C22×C61—22no—
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—42no—
129C22×C61—41no—
130C22×C61—41no—
131C22×C61—42no—
132C22×C61—42no—
133C22×C61—44no—
134C22×C61—44no—
135C22×C61—44no—
136C22×C61—44no—
137C22×C61—42no—
138C22×C61—42no—
139C22×C61#32312yes—
140C22×C61#32312yes—
141C22×C61#35214yes—
142C22×C61C2×C22⋊C811yescharacteristic
143C22×C61C2×#9911yescharacteristic
144C22×C61C2×#10011yescharacteristic
145C4×C61—82no—
146Dic61—42no—
147Dic61—42no—
148Dic61—44no—
149C2×#381—611no—
150C2×#323—612no—
151#1317—614no—
152C2×C61—21no—
153C2×C61—22no—
154C2×C61—21no—
155C2×C61—42no—
156C2×C61—41no—
157C2×C61—41no—
158C2×C61#131714yescomplemented
159C2×C61C2×#32311yescharacteristic
160C2×C61C2×#35212yes—
161C24×C4—611no—
162C22×C22⋊C4—611no—
163C22×C22⋊C4#692311yescharacteristic
164C2×C22⋊C8—1222no—
165#381—612no—
166#381—612no—
167#381—1222no—
168C2×#95—1222no—
169#323—614no—
170#323—1222no—
171C61C2×#131711yescharacteristic, complemented, Sylow
172C25C2×#692311yescharacteristic
173C23×C4—21no—
174C23×C4—614no—
175C23×C4—612no—
176C23×C4—612no—
177C23×C4—1222no—
178C23×C4—1221no—
179C23×C4—1221no—
180C23×C4—2441no—
181C23×C4—2441no—
182C23×C4—2442no—
183C22×C8—2442no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—612no—
187C2×C22⋊C4—611no—
188C2×C22⋊C4—611no—
189C2×C22⋊C4—2441no—
190C2×C22⋊C4—2441no—
191C2×C22⋊C4C2×#692312yes—
192C22⋊C8—1224no—
193C22⋊C8—2442no—
194#95—1224no—
195#95—1224no—
196C24—21no—
197C24—21no—
198C24—22no—
199C24—44no—
200C24D61⋊C812yes—
201C24#1175821312yes—
202C24C22⋊#692311yescharacteristic
203C24C22⋊#692311yescharacteristic
204C24C22×#692311yescharacteristic
205C22×C4—22no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—612no—
209C22×C4—612no—
210C22×C4—614no—
211C22×C4—1222no—
212C22×C4—1224no—
213C22×C4—1224no—
214C22×C4—1222no—
215C22×C4—1222no—
216C22×C4—1222no—
217C22×C4—2444no—
218C22×C4—2442no—
219C22×C4—2442no—
220C22×C4—2444no—
221C22×C4—2444no—
222C22×C4—2441no—
223C22×C4—2441no—
224C22×C4—2442no—
225C22×C4—2442no—
226C22×C4—2442no—
227C22×C4—2444no—
228C22×C4—2444no—
229C2×C8—4884no—
230C2×C8—2444no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—2441no—
234C22⋊C4—2442no—
235C22⋊C4—2441no—
236C23—24no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—44no—
246C23—44no—
247C23—42no—
248C23—42no—
249C23—48no—
250C23—44no—
251C23—44no—
252C23#1451983412yes—
253C23#1451983412yes—
254C23#1451983611yescharacteristic
255C23#1451983611yescharacteristic
256C23#1451983712yes—
257C23C2×C22⋊#692311yescharacteristic
258C23C2×D61⋊C811yescharacteristic
259C23C2×#1175821311yescharacteristic
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4—1224no—
264C2×C4—1222no—
265C2×C4—1222no—
266C2×C4—4882no—
267C2×C4—2448no—
268C2×C4—2444no—
269C2×C4—2442no—
270C2×C4—2442no—
271C2×C4—2444no—
272C2×C4—2444no—
273C2×C4—2444no—
274C2×C4—2442no—
275C2×C4—2442no—
276C2×C4—2444no—
277C8—4884no—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—42no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22#1684060612yes—
297C22#1684060612yes—
298C22#1684066514yes—
299C22C2×#1451983411yescharacteristic
300C22C2×#1451983611yescharacteristic
301C22C2×#1451983711yescharacteristic
302C4—82no—
303C4—2442no—
304C4—2442no—
305C4—2444no—
306C2—21no—
307C2—21no—
308C2—22no—
309C2—42no—
310C2—41no—
311C2—41no—
312C2#2146074614yesdirect product factor
313C2C2×#1684060611yescharacteristic
314C2C2×#1684066512yes—
subgroupG/H
centerC23#14519836
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC23×C61#99
prime radicalC25×C61C2×C4
cosocleC2×C22⋊Dic61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61C2×#1317
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #21460746C2direct (N × H)
23. C23×Dic61C8semidirect
69. C22×Dic61C2×C8semidirect
158. C2×C61#1317semidirect
171. C61C2×#1317semidirect
312. C2#21460746direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168406052
#214607462
C2×#168396232
C2×#131761