Finite Group Databasebeta

← #30599903

Subgroups — #30599903

Order 15616.

558 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 98 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#21462681C211yesmaximal, characteristic
2#21461123C212yesmaximal, complemented
3#21461059C211yesmaximal, characteristic
4#21460647C212yesmaximal, complemented
5#21448366C211yesmaximal, characteristic
6#16840797—24no—
7#16840778C2212yes—
8#16840778C2212yes—
9#16840732—22no—
10#16840612—24no—
11#16840605—22no—
12C2×#14519764C411yescharacteristic
13C2×#14519742C2211yescharacteristic
14C2×D4×Dic61C2211yescharacteristic
15#16839668C411yescharacteristic
16#16839650—22no—
17#16839627C411yescharacteristic
18C22⋊C4×Dic61—22no—
19C42×Dic61C2211yescharacteristic
20C61×#390C411yescharacteristic
21C2×#11758217—24no—
22C4⋊#92420—22no—
23C4⋊#92420D414yes—
24C22⋊#92420—24no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C4×#92420—22no—
28C4×#92420—22no—
29C22×Dic122C2×C411yescharacteristic
30C23×Dic61D412yes—
31#14519764—22no—
32#14519764—22no—
33#14519742—22no—
34#14519742—22no—
35D4×Dic61—22no—
36D4×Dic61—22no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61D412yes—
41C2×C4×Dic61—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61C2311yescharacteristic
46#14519684—22no—
47#14519684—22no—
48C22×D4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—22no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61C22⋊C412yes—
53C2×D4×C61—22no—
54C2×D4×C61—21no—
55C2×D4×C61—21no—
56C2×D4×C61—22no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—21no—
61C22×Dic61—21no—
62C22×Dic61—22no—
63C22×Dic61—22no—
64C22×Dic61—44no—
65C22×Dic61C22⋊C412yes—
66C22×Dic61C2×D411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C2×Dic122—22no—
69C2×Dic122—22no—
70C2×Dic122—21no—
71C2×Dic122M1612yescomplemented
72#11758217—84no—
73C2×#92420—24no—
74C2×#92420—42no—
75C4⋊C4×C61—22no—
76C22⋊C4×C61—44no—
77C42×C61—22no—
78C22⋊Dic61—28no—
79C22⋊Dic61—24no—
80C4⋊Dic61—22no—
81#11758196—42no—
82C4×Dic61—21no—
83C4×Dic61—42no—
84C4×Dic61—42no—
85C4×Dic61—41no—
86C4×Dic61—41no—
87C4×Dic61M1612yescomplemented
88C4×Dic61C2×D412yes—
89C4×Dic61#4112yes—
90C23×C61—22no—
91C23×C61—22no—
92C23×C61—22no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61C2×C22⋊C411yescharacteristic
96D4×C61—42no—
97D4×C61—42no—
98C2×C4×C61—22no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61—41no—
103C2×C4×C61C4×D412yes—
104C2×C4×C61C2×M1611yescharacteristic
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—41no—
115C2×Dic61—41no—
116C2×Dic61—44no—
117C2×Dic61—42no—
118C2×Dic61—82no—
119C2×Dic61#12011yescharacteristic
120C2×Dic61#12112yes—
121C2×Dic61#12311yescharacteristic
122Dic122—24no—
123Dic122—42no—
124#92420—44no—
125#92420—82no—
126#8961—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—44no—
130C22×C61—42no—
131C22×C61—42no—
132C22×C61—42no—
133C22×C61—82no—
134C22×C61#38611yescharacteristic
135C22×C61#40711yescharacteristic
136C22×C61#40811yescharacteristic
137C4×C61—42no—
138C4×C61—81no—
139C4×C61#43512yes—
140C4×C61#43612yes—
141Dic61—22no—
142Dic61—21no—
143Dic61—42no—
144Dic61—42no—
145Dic61—42no—
146Dic61#45512yes—
147#2975—611no—
148C4×#124—611no—
149#1479—612no—
150#1460—611no—
151#1276—612no—
152C2×C61—21no—
153C2×C61—21no—
154C2×C61—82no—
155C2×C61#187011yescharacteristic
156C2×C61#188411yescharacteristic
157C2×C61#188511yescharacteristic
158C2×#124—611no—
159C2×C4×Q8—611no—
160C2×C4×D4—611no—
161#436—1224no—
162#431—612no—
163#431—612no—
164#407—1222no—
165#391—611no—
166#390#692311yescharacteristic
167#388—1222no—
168#382—611no—
169C4×C22⋊C4—1222no—
170C43—611no—
171Q8⋊C8—1224no—
172#323—1222no—
173C61#896111yescharacteristic, complemented, Sylow
174C22×Q8—611no—
175C22×D4C2×#692311yescharacteristic
176C23×C4—612no—
177C2×M16—1224no—
178#124—1222no—
179#124—1222no—
180C4×Q8—1222no—
181C4×Q8—1222no—
182C4×D4—1222no—
183C4×D4—1222no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—1221no—
186C2×C4⋊C4C2×#692311yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—612no—
189C2×C22⋊C4—1222no—
190C2×C42—611no—
191C2×C42—611no—
192C2×C42—1221no—
193C2×C42—1221no—
194C2×C42—1221no—
195C2×C42C2×#692311yescharacteristic
196C4⋊C8—614no—
197C4⋊C8—1222no—
198C22⋊C8—1222no—
199C22⋊C8—1224no—
200C22⋊C8—1222no—
201C4×C8—1222no—
202C4×C8—1222no—
203#95—1222no—
204#95—1222no—
205C24C22⋊#692312yes—
206C2×Q8—612no—
207C2×Q8—1221no—
208C2×Q8—1222no—
209C2×Q8—1222no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—611no—
217C22×C4—611no—
218C22×C4—612no—
219C22×C4—1222no—
220C22×C4—1222no—
221C22×C4—1221no—
222C22×C4—1221no—
223C22×C4—2444no—
224C22×C4C22×#692311yescharacteristic
225M16—4884no—
226C2×C8—1224no—
227C2×C8—2442no—
228C4⋊C4—22no—
229C4⋊C4—1222no—
230C4⋊C4—2442no—
231C22⋊C4—44no—
232C22⋊C4—1224no—
233C22⋊C4—1228no—
234C42—22no—
235C42—612no—
236C42—612no—
237C42—612no—
238C42—1221no—
239C42—2442no—
240C42—2441no—
241C42—2441no—
242C42—2442no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23C2×C22⋊#692311yescharacteristic
249Q8—1224no—
250Q8—2442no—
251D4—42no—
252D4—42no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—611no—
259C2×C4—612no—
260C2×C4—611no—
261C2×C4—1222no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—4882no—
266C2×C4—2444no—
267C2×C4—2441no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2441no—
272C2×C4—2442no—
273C2×C4D4×#692312yes—
274C2×C4D61⋊M1611yescharacteristic
275C2×C4D61⋊M1611yescharacteristic
276C2×C4#1451986412yes—
277C8—4882no—
278C8—2444no—
279C22—21no—
280C22—21no—
281C22—44no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—82no—
286C22#1684071011yescharacteristic
287C22#1684073411yescharacteristic
288C22#1684073511yescharacteristic
289C4—42no—
290C4—81no—
291C4—612no—
292C4—1222no—
293C4—1221no—
294C4—2442no—
295C4—2442no—
296C4—2442no—
297C4#1684079412yes—
298C4#1684079512yes—
299C2—21no—
300C2—21no—
301C2—82no—
302C2#2146179811yescharacteristic
303C2#2146182211yescharacteristic
304C2#2146182411yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#8961
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #21461123C2semidirect
4. #21460647C2semidirect
71. C2×Dic122M16semidirect
87. C4×Dic61M16semidirect
173. C61#8961semidirect

Maximal subgroups

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

subgroupindex
#214626812
#214611232
#214610592
#214606472
#214483662
#896161