Finite Group Databasebeta

← #30601614

Subgroups — #30601614

Order 15616.

618 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 142 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#21461736C211yesmaximal, characteristic, complemented
2C42×#92420C211yesmaximal, characteristic, complemented
3C2×#16840612C212yesmaximal, complemented
4C2×D4⋊#92420C212yesmaximal
5#21448366C211yesmaximal, characteristic
6C2×C4⋊#92420C2212yes—
7C2×C4×#92420—22no—
8C2×C4×#92420C2212yes—
9#16840628—22no—
10#16840612—24no—
11D4⋊#92420—24no—
12C2×#14519764C411yescharacteristic
13C2×#14519742C2211yescharacteristic
14C2×D4×Dic61C2211yescharacteristic
15#16839668C411yescharacteristic
16#16839650—22no—
17#16839627C411yescharacteristic
18C22⋊C4×Dic61—22no—
19C42×Dic61C2211yescharacteristic
20C61×#390C411yescharacteristic
21C22×#92420—22no—
22C22×#92420—22no—
23C4⋊#92420—22no—
24C4⋊#92420—22no—
25C4×#92420—28no—
26C4×#92420—22no—
27C4×#92420D414yes—
28C22×Dic122C2×C411yescharacteristic
29C23×Dic61—21no—
30#14519764—22no—
31#14519764C814yescomplemented
32#14519742—22no—
33#14519742—22no—
34D4×Dic61—22no—
35D4×Dic61—22no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×#11758196—21no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—21no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61C2×C411yescharacteristic
42C2×C4×Dic61D412yes—
43C2×C4×Dic61D412yes—
44C2×C4×Dic61C2311yescharacteristic
45#14519684—22no—
46#14519684—22no—
47C22×D4×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C22⋊C4×C61—22no—
50C2×C42×C61C2×C411yescharacteristic
51C24×C61—21no—
52C2×D4×C61—22no—
53C2×D4×C61—21no—
54C2×D4×C61—22no—
55C2×D4×C61C2×C812yes—
56C22×C4×C61—21no—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22×C411yescharacteristic
59C22×Dic61—21no—
60C22×Dic61—21no—
61C22×Dic61—41no—
62C22×Dic61—41no—
63C22×Dic61—44no—
64C22×Dic61C22⋊C412yes—
65C22×Dic61C2×D411yescharacteristic
66C22×Dic61C2×D411yescharacteristic
67C2×Dic122—22no—
68C2×Dic122—22no—
69C2×Dic122—21no—
70C2×Dic122C2×C812yes—
71C2×#92420—28no—
72C2×#92420—24no—
73C2×#92420—42no—
74C2×#92420—44no—
75C4⋊C4×C61—22no—
76C22⋊C4×C61—44no—
77C42×C61—22no—
78C22⋊Dic61—28no—
79C22⋊Dic61—42no—
80C4⋊Dic61—22no—
81#11758196—42no—
82C4×Dic61—24no—
83C4×Dic61—24no—
84C4×Dic61—41no—
85C4×Dic61—41no—
86C4×Dic61C2×C812yes—
87C4×Dic61M1612yes—
88C4×Dic61C2×D412yes—
89C4×Dic61#4112yes—
90C23×C61—22no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61C2×C22⋊C411yescharacteristic
96D4×C61—24no—
97D4×C61—42no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61#10414yes—
103C2×C4×C61C4×D412yes—
104C2×C4×C61C22×C811yescharacteristic
105C2×C4×C61C2×M1611yescharacteristic
106C2×C4×C61#13112yes—
107C2×Dic61—22no—
108C2×Dic61—22no—
109C2×Dic61—22no—
110C2×Dic61—22no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—42no—
115C2×Dic61—82no—
116C2×Dic61—82no—
117C2×Dic61C22⋊C814yes—
118C2×Dic61#12112yes—
119C2×Dic61#12411yescharacteristic
120C2×Dic61#12711yescharacteristic
121C2×Dic61#13514yes—
122Dic122—24no—
123Dic122—42no—
124#92420—28no—
125#92420—82no—
126#9639—611nomaximal, Sylow
127C22×C61—42no—
128C22×C61—44no—
129C22×C61—81no—
130C22×C61—81no—
131C22×C61—82no—
132C22×C61#39011yescharacteristic
133C22×C61C2×C22⋊C811yescharacteristic
134C22×C61#40811yescharacteristic
135C22×C61#41612yes—
136C22×C61C2×#10412yes—
137C4×C61—24no—
138C4×C61—81no—
139C4×C61C8×D412yes—
140C4×C61#43612yes—
141Dic61—22no—
142Dic61—24no—
143Dic61—42no—
144Dic61—81no—
145Dic61#48812yes—
146Dic61#49512yes—
147C4×#124—611no—
148#1831—611no—
149C42×C8—611no—
150C2×Q8⋊C8—612no—
151C2×D4⋊C8—612no—
152C2×C61—82no—
153C2×C61#147612yes—
154C2×C61#149412yes—
155C2×C61#195411yescharacteristic
156C2×C61#195811yescharacteristic
157C2×C61#196211yescharacteristic
158C2×#124—611no—
159C2×C4×Q8—611no—
160C2×C4×D4—611no—
161C2×C4⋊C8—612no—
162C2×C4×C8—612no—
163C2×C4×C8—1222no—
164#391—611no—
165#390#692311yescharacteristic
166#388—1222no—
167#382—611no—
168C4×C22⋊C4—1222no—
169C43—611no—
170#336—1222no—
171Q8⋊C8—1224no—
172D4⋊C8—1224no—
173C61#963911yescharacteristic, complemented, Sylow
174C22×Q8—611no—
175C22×D4C2×#692311yescharacteristic
176C23×C4—1221no—
177C22×C8—1222no—
178C22×C8—1222no—
179#124—614no—
180#124—1222no—
181C4×Q8—1222no—
182C4×Q8—1222no—
183C4×D4—1222no—
184C4×D4—1222no—
185C2×C4⋊C4—611no—
186C2×C4⋊C4—1221no—
187C2×C4⋊C4C2×#692311yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—1222no—
190C2×C22⋊C4—1221no—
191C2×C42—611no—
192C2×C42—611no—
193C2×C42—612no—
194C2×C42—612no—
195C2×C42—1221no—
196C2×C42C2×#692311yescharacteristic
197C4⋊C8—1222no—
198C4⋊C8—1222no—
199C4×C8—614no—
200C4×C8—1222no—
201C4×C8—1228no—
202#95—1222no—
203#95—1222no—
204C24—21no—
205C2×Q8—612no—
206C2×Q8—1221no—
207C2×Q8—1222no—
208C2×Q8—1222no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—22no—
212C2×D4D61⋊C812yes—
213C22×C4—21no—
214C22×C4—611no—
215C22×C4—612no—
216C22×C4—611no—
217C22×C4—1221no—
218C22×C4—1221no—
219C22×C4—2441no—
220C22×C4—2441no—
221C22×C4—2444no—
222C22×C4C22⋊#692312yes—
223C22×C4C22×#692311yescharacteristic
224C2×C8—1224no—
225C2×C8—1228no—
226C2×C8—2442no—
227C2×C8—2444no—
228C4⋊C4—22no—
229C4⋊C4—1222no—
230C4⋊C4—2442no—
231C22⋊C4—44no—
232C22⋊C4—1228no—
233C22⋊C4—2442no—
234C42—22no—
235C42—612no—
236C42—612no—
237C42—612no—
238C42—612no—
239C42—1224no—
240C42—1224no—
241C42—2441no—
242C42—2441no—
243C23—22no—
244C23—41no—
245C23—41no—
246C23—42no—
247C23—42no—
248C23C2×C22⋊#692311yescharacteristic
249Q8—1224no—
250Q8—2442no—
251D4—24no—
252D4—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—611no—
258C2×C4—611no—
259C2×C4—614no—
260C2×C4—612no—
261C2×C4—614no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—1222no—
266C2×C4—4882no—
267C2×C4—4882no—
268C2×C4—2441no—
269C2×C4—2442no—
270C2×C4—2441no—
271C2×C4—2441no—
272C2×C4#1451984114yes—
273C2×C4D4×#692312yes—
274C2×C4C2×D61⋊C811yescharacteristic
275C2×C4D61⋊M1611yescharacteristic
276C2×C4#1451986412yes—
277C8—1228no—
278C8—4882no—
279C22—44no—
280C22—42no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22#1684071411yescharacteristic
285C22D61⋊(C22⋊C8)11yescharacteristic
286C22#1684073511yescharacteristic
287C22#1684075212yes—
288C22C2×#1451984112yes—
289C4—24no—
290C4—81no—
291C4—612no—
292C4—612no—
293C4—1224no—
294C4—1222no—
295C4—4881no—
296C4—2442no—
297C4#1684078912yes—
298C4#1684079512yes—
299C2—82no—
300C2#2146111112yes—
301C2#2146117012yes—
302C2#2146194211yescharacteristic
303C2#2146194811yescharacteristic
304C2#2146195311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#9639
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461736C2semidirect
2. C42×#92420C2semidirect
3. C2×#16840612C2semidirect
31. #14519764C8semidirect
173. C61#9639semidirect

Maximal subgroups

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

subgroupindex
#214617362
C42×#924202
C2×#168406122
C2×D4⋊#924202
#214483662
#963961