Finite Group Databasebeta

← #30605400

Subgroups — #30605400

Order 15616.

472 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461680C211yesmaximal, characteristic
2#21461056C211yesmaximal, characteristic
3#21460799C214yesmaximal, complemented
4#21450815C211yesmaximal, characteristic
5C2×#14519836—21no—
6D61⋊(C22⋊C8)—21no—
7#16840704—21no—
8#16840640C2212yes—
9#16840607—21no—
10#16840606C2212yes—
11C2×#14519766—21no—
12C2×C4×D244C2211yescharacteristic
13#16839688C412yescomplemented
14#16839688C2212yes—
15#16839669—21no—
16#16839619—21no—
17#16839619—21no—
18C4×C4⋊Dic61C411yescharacteristic, complemented
19C2×C4⋊Q8×C61C411yescharacteristic, complemented
20C2×D61⋊C8—41no—
21C2×C4⋊#6923—41no—
22C2×C22⋊#6923—22no—
23#14519836—42no—
24#14519834—41no—
25C22⋊#92420—22no—
26(C2×#6923)⋊C4—41no—
27C22×D244D411yescharacteristic
28C22×C4×D61D411yescharacteristic
29C22×C4×D61D411yescharacteristic
30#14519766—42no—
31#14519766—42no—
32C4×D244—41no—
33C4×D244—41no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×#11758198—22no—
37C2×#11758198—22no—
38C2×#11758198C2×C411yescharacteristic
39C2×#11758198C2311yescharacteristic
40C2×C4×Dic61—21no—
41#14519684—22no—
42#14519684—22no—
43#14519684C2×C412yes—
44C22×Q8×C61—21no—
45C4⋊Q8×C61—41no—
46C4⋊Q8×C61—41no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61C2×C412yes—
49C2×C42×C61C2×C411yescharacteristic
50C23×D61C22⋊C411yescharacteristic
51C23×D61C2×D411yescharacteristic
52C22×#6923—42no—
53C2×Q8×C61—42no—
54C2×Q8×C61—42no—
55C2×Q8×C61—42no—
56C22×C4×C61—22no—
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—22no—
61C22×Dic61C22⋊C411yescharacteristic
62C22×Dic61C2×D411yescharacteristic
63C2×D244—41no—
64C2×D244—41no—
65C2×D244D812yes—
66C2×D244SD1612yes—
67C2×C4×D61—21no—
68C2×C4×D61—21no—
69C2×C4×D61—21no—
70C2×C4×D61—21no—
71C2×C4×D61—42no—
72C2×C4×D61—42no—
73C22⋊#6923—42no—
74C22⋊#6923—42no—
75C2×#92420—42no—
76C4⋊#6923—82no—
77D61⋊C8—82no—
78C4⋊C4×C61—42no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C42×C61—21no—
82C42×C61—21no—
83#11758198—41no—
84#11758198—41no—
85#11758198—44no—
86#11758198—42no—
87#11758198—42no—
88C4⋊Dic61—41no—
89C4⋊Dic61—42no—
90C4×Dic61—42no—
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—21no—
93C22×D61—21no—
94C22×D61—21no—
95C22×D61—21no—
96C22×D61—41no—
97C22×D61—41no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61#10212yes—
101C22×D61C2×D811yescharacteristic
102C22×D61C2×SD1611yescharacteristic
103C2×#6923—82no—
104C2×#6923—82no—
105C2×#6923—82no—
106Q8×C61—84no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—22no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61#9911yescharacteristic
116C2×C4×C61#9911yescharacteristic
117C2×C4×C61#10212yes—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125D244—42no—
126D244—42no—
127C4×D61—42no—
128C4×D61—42no—
129C4×D61—81no—
130C4×D61—81no—
131#92420—82no—
132#11878—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61C2×#9911yescharacteristic
136C22×C61C2×#10211yescharacteristic
137C22×C61#42111yescharacteristic
138C2×D61—21no—
139C2×D61—22no—
140C2×D61—21no—
141C2×D61—41no—
142C2×D61—41no—
143C2×D61—41no—
144C2×D61—41no—
145C2×D61—42no—
146C2×D61—41no—
147C2×D61—41no—
148C2×D61—81no—
149C2×D61—81no—
150#6923—162no—
151C4×C61—21no—
152C4×C61—22no—
153C4×C61—41no—
154C4×C61—82no—
155C4×C61#35412yes—
156Dic61—41no—
157Dic61—41no—
158Dic61—82no—
159#2547—611no—
160#1799—611no—
161#1459—611no—
162#1344—614no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61#151911yescharacteristic
166C2×C61#212311yescharacteristic
167C2×C61C2×#35411yescharacteristic
168D61—41no—
169D61—42no—
170D61—41no—
171C2×C4⋊Q8#692311yescharacteristic, complemented
172C2×#124—1221no—
173C2×C4×D4—611no—
174C2×#99—1221no—
175C2×C22⋊C8—1221no—
176#397—612no—
177#397—612no—
178#391—1221no—
179#380—1221no—
180#380—1221no—
181#380—1221no—
182C4×C4⋊C4—611no—
183#342—612no—
184#324—1221no—
185#323—612no—
186C61#1187811yescharacteristic, complemented, Sylow
187C22×Q8—21no—
188C22×D4—611no—
189C23×C4—611no—
190C23×C4—611no—
191C22×C8—2441no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194#124—2442no—
195#124—2442no—
196C4×D4—2441no—
197C4×D4—2441no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—611no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4—2441no—
202C2×C4⋊C4C2×#692312yes—
203C2×C22⋊C4—611no—
204C2×C22⋊C4—611no—
205C2×C22⋊C4—1222no—
206C2×C22⋊C4—1222no—
207C2×C22⋊C4—1222no—
208C2×C42—1221no—
209C2×C42C2×#692311yescharacteristic
210#99—2442no—
211C22⋊C8—1222no—
212C22⋊C8—2441no—
213#95—612no—
214#95—1222no—
215#95—1222no—
216#95—2441no—
217C24—611no—
218C24—611no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×D4—612no—
223C2×D4—612no—
224C2×D4—2441no—
225C2×D4—2441no—
226C22×C4—22no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—1221no—
230C22×C4—1221no—
231C22×C4—1221no—
232C22×C4—1221no—
233C22×C4—1222no—
234C22×C4—2442no—
235C22×C4—2442no—
236C22×C4—2442no—
237C22×C4C22⋊#692311yescharacteristic
238C22×C4C22⋊#692311yescharacteristic
239C22×C4C22×#692311yescharacteristic
240C2×C8—4882no—
241C2×C8—2442no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—4882no—
246C4⋊C4—2441no—
247C4⋊C4—2442no—
248C22⋊C4—2441no—
249C22⋊C4—2441no—
250C22⋊C4—2444no—
251C22⋊C4—2442no—
252C22⋊C4—2442no—
253C22⋊C4—2442no—
254C22⋊C4—2442no—
255C42—21no—
256C42—21no—
257C42—2442no—
258C23—612no—
259C23—611no—
260C23—611no—
261C23—1221no—
262C23—1221no—
263C23—1221no—
264C23—1221no—
265C23—2441no—
266C23—2441no—
267C23—2441no—
268C23—2441no—
269C23C2×C22⋊#692311yescharacteristic
270Q8—84no—
271D4—2442no—
272D4—2442no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—4881no—
284C2×C4—4881no—
285C2×C4—4882no—
286C2×C4—4882no—
287C2×C4—4882no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4#1451983511yescharacteristic
296C2×C4#1451983511yescharacteristic
297C2×C4#1451983912yes—
298C8—4882no—
299C22—21no—
300C22—21no—
301C22—1221no—
302C22—1222no—
303C22—1221no—
304C22—4881no—
305C22—4881no—
306C22—2441no—
307C22—2442no—
308C22—2441no—
309C22—2441no—
310C22—2441no—
311C22—2441no—
312C22—2441no—
313C22C2×#1451983511yescharacteristic
314C22C2×#1451983911yescharacteristic
315C22#1684076011yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—82no—
320C4—9762no—
321C4—4882no—
322C4—2441no—
323C4—2441no—
324C4#1684066812yes—
325C2—21no—
326C2—21no—
327C2—2441no—
328C2—2441no—
329C2—2442no—
330C2#2146125511yescharacteristic
331C2#2146208111yescharacteristic
332C2C2×#1684066811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×D244D4
cosocleC2×#11758198C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#11878
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
3. #21460799C2semidirect
13. #16839688C4semidirect
18. C4×C4⋊Dic61C4semidirect
19. C2×C4⋊Q8×C61C4semidirect
171. C2×C4⋊Q8#6923semidirect
186. C61#11878semidirect

Maximal subgroups

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

subgroupindex
#214616802
#214610562
#214607992
#214508152
#1187861