Finite Group Databasebeta

← #30599885

Subgroups — #30599885

Order 15616.

586 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 126 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#21462666C211yesmaximal, characteristic
2#21461121C212yesmaximal, complemented
3#21461059C211yesmaximal, characteristic
4#21460647C212yesmaximal, complemented
5#124×Dic61C211yesmaximal, characteristic
6D4×#92420—24no—
7#16840778C2212yes—
8#16840778C2212yes—
9C2×C22⋊#92420—22no—
10Q8⋊#92420—24no—
11#16840605—22no—
12C2×Q8×Dic61C2211yescharacteristic
13C2×D4×Dic61C2211yescharacteristic
14#16839669C411yescharacteristic
15#16839665C411yescharacteristic
16#16839655—22no—
17#16839628C411yescharacteristic
18C22⋊C4×Dic61—22no—
19C42×Dic61C2211yescharacteristic
20C2×#124×C61C411yescharacteristic
21C22×#92420—24no—
22C4⋊#92420—22no—
23C4⋊#92420D414yes—
24C22⋊#92420—24no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C4×#92420—22no—
28C4×#92420—22no—
29C23×Dic61D412yes—
30Q8×Dic61—22no—
31Q8×Dic61—22no—
32D4×Dic61—22no—
33D4×Dic61—22no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61D412yes—
39C2×C4×Dic61—21no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61—21no—
42C2×C4×Dic61C2×C411yescharacteristic
43C2×C4×Dic61C2311yescharacteristic
44#14519684—22no—
45#14519684—22no—
46C22×Q8×C61C2×C411yescharacteristic
47C22×D4×C61C2×C411yescharacteristic
48#124×C61—22no—
49#124×C61C814yescomplemented
50C2×C22⋊C4×C61—22no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61C22⋊C412yes—
53C2×Q8×C61—21no—
54C2×Q8×C61—22no—
55C2×Q8×C61—22no—
56C2×Q8×C61C2×C812yescomplemented
57C2×D4×C61—22no—
58C2×D4×C61—22no—
59C2×D4×C61C2×C812yes—
60C2×D4×C61M1612yes—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61C22×C411yescharacteristic
64C22×Dic61—21no—
65C22×Dic61—22no—
66C22×Dic61—22no—
67C22×Dic61—21no—
68C22×Dic61—44no—
69C22×Dic61C22⋊C412yes—
70C22×Dic61C2×D411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C2×#92420—24no—
73C2×#92420—42no—
74C2×#92420—84no—
75C22⋊C4×C61—28no—
76C42×C61—21no—
77C42×C61C2×C812yescomplemented
78C22⋊Dic61—24no—
79C22⋊Dic61—44no—
80C4⋊Dic61—22no—
81C4⋊Dic61—22no—
82C4⋊Dic61—42no—
83C4×Dic61—22no—
84C4×Dic61—42no—
85C4×Dic61—42no—
86C4×Dic61—41no—
87C4×Dic61—41no—
88C4×Dic61C2×D412yes—
89C4×Dic61#4112yes—
90C23×C61—22no—
91C23×C61—22no—
92C23×C61—42no—
93C23×C61—42no—
94C23×C61C22⋊C814yes—
95C23×C61C2×C22⋊C411yescharacteristic
96Q8×C61—24no—
97Q8×C61—42no—
98D4×C61—24no—
99D4×C61—24no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61C4×D412yes—
107C2×C4×C61C22×C811yescharacteristic
108C2×C4×C61C2×M1611yescharacteristic
109C2×C4×C61#13112yes—
110C2×Dic61—22no—
111C2×Dic61—22no—
112C2×Dic61—22no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—44no—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—41no—
120C2×Dic61—82no—
121C2×Dic61#12011yescharacteristic
122C2×Dic61#12112yes—
123C2×Dic61#12311yescharacteristic
124#92420—44no—
125#92420—82no—
126#8957—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—24no—
130C22×C61—44no—
131C22×C61—42no—
132C22×C61—42no—
133C22×C61—82no—
134C22×C61#38611yescharacteristic
135C22×C61C2×C22⋊C811yescharacteristic
136C22×C61#40811yescharacteristic
137C4×C61—42no—
138C4×C61—42no—
139C4×C61C8×D412yes—
140C4×C61#43512yes—
141Dic61—22no—
142Dic61—21no—
143Dic61—42no—
144Dic61—42no—
145Dic61—81no—
146Dic61#45512yes—
147#2972—611no—
148C4×#124—611no—
149#1479—612no—
150#1460—611no—
151#1276—612no—
152C2×C61—21no—
153C2×C61—21no—
154C2×C61—44no—
155C2×C61#186911yescharacteristic
156C2×C61#188411yescharacteristic
157C2×C61#188511yescharacteristic
158C2×#124#692311yescharacteristic
159C2×C4×Q8—611no—
160C2×C4×D4—611no—
161C8×D4—1224no—
162#431—612no—
163#431—612no—
164C2×C22⋊C8—1222no—
165#391—611no—
166#390—611no—
167#388—1222no—
168#382—611no—
169C4×C22⋊C4—1222no—
170C43—611no—
171Q8⋊C8—1224no—
172#323—1222no—
173C61#895711yescharacteristic, complemented, Sylow
174C22×Q8C2×#692311yescharacteristic
175C22×D4C2×#692311yescharacteristic
176C23×C4—612no—
177C22×C8—1224no—
178#124—22no—
179#124#9242014yescomplemented
180C4×Q8—1222no—
181C4×Q8—1222no—
182C4×D4—1222no—
183C4×D4—1222no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—611no—
186C2×C4⋊C4—1221no—
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—21no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8C2×#9242012yescomplemented
210C2×D4—22no—
211C2×D4—22no—
212C2×D4C2×#9242012yes—
213C2×D4#1175821712yes—
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
225C2×C8—1224no—
226C2×C8—4884no—
227C2×C8—2442no—
228C4⋊C4—1222no—
229C4⋊C4—1222no—
230C4⋊C4—2442no—
231C22⋊C4—28no—
232C22⋊C4—1224no—
233C22⋊C4—2444no—
234C42—21no—
235C42—612no—
236C42—612no—
237C42—1222no—
238C42—2441no—
239C42—2441no—
240C42—2442no—
241C42—2442no—
242C42C2×#9242012yescomplemented
243C23—22no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23C22⋊#9242014yes—
248C23C2×C22⋊#692311yescharacteristic
249Q8—24no—
250Q8—42no—
251D4—24no—
252D4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—611no—
260C2×C4—612no—
261C2×C4—611no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—4882no—
266C2×C4—2444no—
267C2×C4—2441no—
268C2×C4—2442no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2441no—
272C2×C4—2442no—
273C2×C4D4×#692312yes—
274C2×C4C22×#9242011yescharacteristic
275C2×C4C2×#1175821711yescharacteristic
276C2×C4#1451986412yes—
277C8—4882no—
278C8—2444no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—82no—
286C22#1684071011yescharacteristic
287C22C2×C22⋊#9242011yescharacteristic
288C22#1684073511yescharacteristic
289C4—42no—
290C4—42no—
291C4—612no—
292C4—1222no—
293C4—1221no—
294C4—4881no—
295C4—2442no—
296C4—2442no—
297C4D4×#9242012yes—
298C4#1684079312yes—
299C2—21no—
300C2—21no—
301C2—44no—
302C2#2146179211yescharacteristic
303C2#2146182011yescharacteristic
304C2#2146182411yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenterC2×#124#6923
hypocenterC61#8957
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #21461121C2semidirect
4. #21460647C2semidirect
49. #124×C61C8semidirect
56. C2×Q8×C61C2×C8semidirect
77. C42×C61C2×C8semidirect
173. C61#8957semidirect
179. #124#92420semidirect
209. C2×Q8C2×#92420semidirect
242. C42C2×#92420semidirect

Maximal subgroups

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

subgroupindex
#214626662
#214611212
#214610592
#214606472
#124×Dic612
#895761