Finite Group Databasebeta

← #30604743

Subgroups — #30604743

Order 15616.

468 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461518C211yesmaximal, characteristic
2#21461052C211yesmaximal, characteristic
3#21460799C214yesmaximal
4#21451586C211yesmaximal, characteristic
5C2×#14519856—21no—
6#16840778—21no—
7#16840640—21no—
8#16840640C2212yes—
9#16840608—21no—
10#16840606C2212yes—
11C2×#14519753—21no—
12C2×#14519734C2211yescharacteristic
13#16839693—21no—
14#16839693C412yescomplemented
15#16839688—21no—
16#16839688C2212yes—
17#16839669—21no—
18#16839628C411yescharacteristic, complemented
19C2×C4⋊Q8×C61C411yescharacteristic, complemented
20#14519856—24no—
21#14519856—42no—
22C2×C4⋊#6923—41no—
23C2×C22⋊#6923—22no—
24C2×C4×#6923—41no—
25C4⋊#92420—41no—
26C22⋊#92420—22no—
27C4×#92420—41no—
28C22×C4×D61D411yescharacteristic
29#14519753—42no—
30#14519753—42no—
31#14519734—41no—
32#14519734—41no—
33C2×C4⋊Dic61—21no—
34C2×#11758196—22no—
35C2×#11758196C2×C411yescharacteristic
36C2×#11758196D411yescharacteristic
37C2×#11758198—22no—
38C2×#11758198D411yescharacteristic
39C2×#11758198C2311yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41#14519684—21no—
42#14519684—22no—
43#14519684—22no—
44#14519684C2×C412yes—
45C22×Q8×C61—21no—
46C4⋊Q8×C61—41no—
47C4⋊Q8×C61—41no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61C2×C412yes—
50C2×C42×C61C2×C411yescharacteristic
51C23×D61C2×D411yescharacteristic
52C22×#6923—42no—
53C2×Q8×C61—41no—
54C2×Q8×C61—41no—
55C2×Q8×C61—42no—
56C2×Q8×C61—42no—
57C22×C4×C61—22no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22×C411yescharacteristic
61C22×Dic61—22no—
62C22×Dic61C22⋊C411yescharacteristic
63C22×Dic61C22⋊C411yescharacteristic
64C22×Dic61C2×D411yescharacteristic
65C2×C4×D61—21no—
66C2×C4×D61—21no—
67C2×C4×D61—22no—
68C2×C4×D61D812yes—
69C2×C4×D61SD1612yes—
70C22⋊#6923—24no—
71C22⋊#6923—42no—
72C2×#92420—42no—
73C4⋊#6923—44no—
74C4×#6923—44no—
75C4⋊C4×C61—42no—
76C4⋊C4×C61—42no—
77C42×C61—21no—
78C42×C61—21no—
79#11758198—41no—
80#11758198—41no—
81#11758198—42no—
82#11758198—42no—
83C4⋊Dic61—42no—
84#11758196—41no—
85#11758196—41no—
86#11758196—44no—
87C4×Dic61—21no—
88C4×Dic61—21no—
89C23×C61C2×C22⋊C411yescharacteristic
90C22×D61—21no—
91C22×D61—21no—
92C22×D61—41no—
93C22×D61—41no—
94C22×D61C2×D811yescharacteristic
95C22×D61C2×SD1611yescharacteristic
96C2×#6923—44no—
97C2×#6923—82no—
98C2×#6923—82no—
99Q8×C61—82no—
100Q8×C61—82no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61#9911yescharacteristic
109C2×C4×C61#9911yescharacteristic
110C2×C4×C61#10212yes—
111C2×C4×C61#10412yes—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61#10212yes—
122C4×D61—22no—
123C4×D61—22no—
124C4×D61—22no—
125C4×D61—22no—
126#92420—82no—
127#11694—611nomaximal, Sylow
128C22×C61—21no—
129C22×C61—21no—
130C22×C61C2×#9911yescharacteristic
131C22×C61C2×#10211yescharacteristic
132C22×C61C2×#10411yescharacteristic
133C2×D61—21no—
134C2×D61—21no—
135C2×D61—41no—
136C2×D61—41no—
137C2×D61—41no—
138C2×D61—41no—
139C2×D61—81no—
140#6923—84no—
141C4×C61—41no—
142C4×C61—41no—
143C4×C61—82no—
144C4×C61#32712yes—
145C4×C61#32712yes—
146Dic61—22no—
147Dic61—41no—
148Dic61—41no—
149Dic61—82no—
150#2621—611no—
151#1740—611no—
152#1458—611no—
153#1344—614no—
154C2×C61—21no—
155C2×C61—21no—
156C2×C61C2×#32711yescharacteristic
157C2×C61#212311yescharacteristic
158C2×C61#212911yescharacteristic
159D61—41no—
160D61—41no—
161C2×C4⋊Q8#692311yescharacteristic, complemented
162C2×C22⋊Q8—1221no—
163C2×#117—611no—
164C2×#117—1221no—
165#431—1221no—
166#398—612no—
167#398—1221no—
168#397—612no—
169#397—1221no—
170#391—1221no—
171#382—611no—
172#342—612no—
173#342—1221no—
174#324—1221no—
175#323—612no—
176C61#1169411yescharacteristic, complemented, Sylow
177C22×Q8—21no—
178C23×C4—611no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C22⋊Q8—2442no—
182C22⋊Q8—2442no—
183#117—1224no—
184#117—2441no—
185#117—2441no—
186#117—2442no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—611no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—1222no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4—2441no—
193C2×C4⋊C4C2×#692312yes—
194C2×C22⋊C4—611no—
195C2×C22⋊C4—611no—
196C2×C22⋊C4—1222no—
197C2×C22⋊C4—1222no—
198C2×C42—611no—
199C2×C42—2441no—
200C2×C42C2×#692311yescharacteristic
201C4⋊C8—2441no—
202C22⋊C8—1222no—
203C4×C8—2441no—
204#95—612no—
205#95—1222no—
206#95—1222no—
207#95—1221no—
208C24—611no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—22no—
214C22×C4—611no—
215C22×C4—612no—
216C22×C4—612no—
217C22×C4—611no—
218C22×C4—611no—
219C22×C4—1221no—
220C22×C4—1221no—
221C22×C4—1222no—
222C22×C4—1222no—
223C22×C4—2442no—
224C22×C4C22⋊#692311yescharacteristic
225C22×C4C22⋊#692311yescharacteristic
226C22×C4C22×#692311yescharacteristic
227C2×C8—2442no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—2441no—
231C4⋊C4—2441no—
232C4⋊C4—2444no—
233C4⋊C4—2444no—
234C4⋊C4—2442no—
235C22⋊C4—1224no—
236C22⋊C4—2441no—
237C22⋊C4—2441no—
238C22⋊C4—2442no—
239C22⋊C4—2442no—
240C22⋊C4—2442no—
241C42—21no—
242C42—21no—
243C42—1221no—
244C42—1221no—
245C42—2444no—
246C23—611no—
247C23—611no—
248C23—1221no—
249C23—1221no—
250C23—2441no—
251C23—2441no—
252C23C2×C22⋊#692311yescharacteristic
253Q8—82no—
254Q8—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—612no—
263C2×C4—1221no—
264C2×C4—1221no—
265C2×C4—1222no—
266C2×C4—1222no—
267C2×C4—1222no—
268C2×C4—1222no—
269C2×C4—1221no—
270C2×C4—1221no—
271C2×C4—4882no—
272C2×C4—4882no—
273C2×C4—2442no—
274C2×C4—2442no—
275C2×C4—2444no—
276C2×C4—2442no—
277C2×C4—2441no—
278C2×C4—2441no—
279C2×C4#1451983511yescharacteristic
280C2×C4#1451983511yescharacteristic
281C2×C4#1451983912yes—
282C2×C4#1451984112yes—
283C8—4882no—
284C22—21no—
285C22—21no—
286C22—1221no—
287C22—1221no—
288C22—4881no—
289C22—2441no—
290C22—2441no—
291C22—2441no—
292C22—2441no—
293C22C2×#1451983511yescharacteristic
294C22C2×#1451983911yescharacteristic
295C22C2×#1451984111yescharacteristic
296C4—41no—
297C4—41no—
298C4—82no—
299C4—1222no—
300C4—4882no—
301C4—4884no—
302C4—2441no—
303C4—2441no—
304C4#1684061312yes—
305C4#1684061312yes—
306C2—21no—
307C2—21no—
308C2—2441no—
309C2—2441no—
310C2C2×#1684061311yescharacteristic
311C2#2146208111yescharacteristic
312C2#2146210111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC23×D61C2×D4
cosocleC2×#11758198C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#11694
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
14. #16839693C4semidirect
18. #16839628C4semidirect
19. C2×C4⋊Q8×C61C4semidirect
161. C2×C4⋊Q8#6923semidirect
176. C61#11694semidirect

Maximal subgroups

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

subgroupindex
#214615182
#214610522
#214607992
#214515862
#1169461