Finite Group Databasebeta

← #29575521

Subgroups — #29575521

Order 15616.

546 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 135 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#21454900C211yesmaximal, characteristic, complemented
2#21448526C211yesmaximal, characteristic
3#21444209C212yesmaximal
4#21443647C212yesmaximal, complemented
5C61×#1461C211yesmaximal, characteristic, complemented
6C2×#14519766C411yescharacteristic
7C2×C4×D244C2211yescharacteristic
8Q8×#92418—24no—
9#16839815—22no—
10#16839815C2212yes—
11#16839669C411yescharacteristic
12#16839663C411yescharacteristic
13#16839653—22no—
14#16839633—22no—
15#16839631C411yescharacteristic
16#16839608C2211yescharacteristic
17#16839489—24no—
18C2×C4×Q8×C61C2211yescharacteristic
19C61×#431C2212yes—
20C61×#324—22no—
21C22×D244C2×C411yescharacteristic
22C22×C4×D61—21no—
23#14519766—22no—
24#14519766C814yescomplemented
25C4×D244—22no—
26C4×D244—22no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758196—22no—
29C2×#11758198—22no—
30C2×#11758198—21no—
31C2×#11758198—22no—
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61C2×C411yescharacteristic
34C4⋊#92418—24no—
35C4⋊#92418—22no—
36C22⋊#92418—22no—
37C4×#92418—24no—
38C4×#92418D414yes—
39#14519684—22no—
40#14519684—21no—
41#14519684—21no—
42C22×Q8×C61C2×C411yescharacteristic
43C4×Q8×C61—22no—
44C4×Q8×C61—22no—
45C2×C4⋊C4×C61C2×C411yescharacteristic
46C2×C4⋊C4×C61D412yes—
47C2×C42×C61D412yes—
48C2×C42×C61C2311yescharacteristic
49C4⋊C8×C61—22no—
50C22⋊C8×C61—22no—
51C4×C8×C61—22no—
52C23×D61—21no—
53C2×Q8×C61—22no—
54C2×Q8×C61—22no—
55C2×Q8×C61C2×C812yes—
56C2×Q8×C61M1612yes—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×Dic61—21no—
63C22×Dic61—21no—
64C22×Dic61—21no—
65C2×D244—22no—
66C2×D244—22no—
67C2×D244—21no—
68C2×D244C2×C812yes—
69C2×C4×D61—42no—
70C2×C4×D61—82no—
71C2×C8×C61—42no—
72C4⋊C4×C61—22no—
73C4⋊C4×C61—24no—
74C42×C61—22no—
75C42×C61—22no—
76C42×C61C2×D412yes—
77C42×C61#4112yes—
78#11758198—28no—
79#11758198—41no—
80#11758198—41no—
81#11758198—82no—
82C4⋊Dic61—22no—
83#11758196—82no—
84C4×Dic61—21no—
85C4×Dic61—22no—
86C4×Dic61C2×C812yes—
87C2×#92418—24no—
88C23×C61C2×C22⋊C411yescharacteristic
89C22×D61—22no—
90C22×D61—42no—
91C22×D61—42no—
92C22×D61—41no—
93C22×D61—41no—
94Q8×C61—24no—
95Q8×C61—24no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61C22⋊C814yes—
103C2×C4×C61C4×D412yes—
104C2×C4×C61#12011yescharacteristic
105C2×C4×C61#12112yes—
106C2×C4×C61#12311yescharacteristic
107C2×C4×C61C22×C811yescharacteristic
108C2×C4×C61C2×M1611yescharacteristic
109C2×C4×C61#13112yes—
110C2×Dic61—22no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118D244—24no—
119D244—42no—
120C4×D61—84no—
121C8×C61—82no—
122#92418—44no—
123#9102—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61#38611yescharacteristic
127C22×C61C2×C22⋊C811yescharacteristic
128C22×C61#40811yescharacteristic
129C2×D61—44no—
130C2×D61—42no—
131C2×D61—82no—
132C2×D61—81no—
133C2×D61—81no—
134C4×C61—24no—
135C4×C61—22no—
136C4×C61—21no—
137C4×C61—21no—
138C4×C61—42no—
139C4×C61C8×D412yes—
140C4×C61#43512yes—
141Dic61—42no—
142Dic61—81no—
143Dic61—81no—
144#2982—611no—
145#2326—611no—
146#1478—612no—
147#1461D6111yescharacteristic, complemented
148#1274—612no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61#186911yescharacteristic
152C2×C61#191011yescharacteristic
153C2×C61#191211yescharacteristic
154D61—82no—
155C2×#124—611no—
156C2×C4×Q8C2×D6111yescharacteristic
157C2×C4×D4—611no—
158C8×Q8—1224no—
159#431—612no—
160#431—1222no—
161#431C2×D6112yes—
162#391—611no—
163#390—611no—
164#388—1222no—
165#385—1222no—
166#384—611no—
167#376—611no—
168D4⋊C8—1224no—
169#324—22no—
170C61#910211yescharacteristic, complemented, Sylow
171C22×Q8C4×D6111yescharacteristic
172C22×D4—611no—
173C23×C4—1221no—
174#124—614no—
175#124—1222no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1222no—
179C4×D4—1222no—
180C2×C4⋊C4—611no—
181C2×C4⋊C4—1222no—
182C2×C4⋊C4C4×D6111yescharacteristic
183C2×C4⋊C4C22⋊D6112yes—
184C2×C22⋊C4—1222no—
185C2×C22⋊C4—1222no—
186C2×C22⋊C4—1221no—
187C2×C42—611no—
188C2×C42—611no—
189C2×C42D24412yes—
190C2×C42C22×D6111yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—1224no—
193C4⋊C8—1222no—
194C22⋊C8—22no—
195C22⋊C8—1222no—
196C4×C8—22no—
197C4×C8—614no—
198C4×C8—1224no—
199#95—1222no—
200#95—1221no—
201#95—1221no—
202C24—1221no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C8×D6112yes—
206C2×Q8#1175818912yes—
207C2×D4—612no—
208C2×D4—1222no—
209C2×D4—1221no—
210C2×D4—1222no—
211C22×C4—1221no—
212C22×C4—1221no—
213C22×C4—1221no—
214C22×C4—4882no—
215C22×C4—2442no—
216C22×C4#1175819812yes—
217C22×C4#1175819812yes—
218C22×C4C2×C4×D6111yescharacteristic
219C22×C4C2×D24411yescharacteristic
220C22×C4C2×C22⋊D6111yescharacteristic
221C2×C8—42no—
222C2×C8—1224no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—1222no—
226C4⋊C4—4882no—
227C22⋊C4—1228no—
228C22⋊C4—4882no—
229C22⋊C4—2441no—
230C22⋊C4—2441no—
231C42—22no—
232C42—22no—
233C42—612no—
234C42—1221no—
235C42—1222no—
236C42D4×D6112yes—
237C42#1175822412yes—
238C23—1222no—
239C23—2442no—
240C23—2442no—
241C23—2441no—
242C23—2441no—
243C23C2×#1175819811yescharacteristic
244Q8—24no—
245Q8—24no—
246D4—1224no—
247D4—2442no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—1222no—
255C2×C4—4884no—
256C2×C4—2441no—
257C2×C4—2442no—
258C2×C4—2441no—
259C2×C4—2441no—
260C2×C4—2441no—
261C2×C4—2441no—
262C2×C4—2441no—
263C2×C4#1451968914yes—
264C2×C4#1451973712yes—
265C2×C4#1451974711yescharacteristic
266C2×C4#1451974912yes—
267C2×C4#1451976111yescharacteristic
268C2×C4C2×C8×D6111yescharacteristic
269C2×C4C2×#1175818911yescharacteristic
270C2×C4#1451978512yes—
271C8—82no—
272C8—2444no—
273C22—21no—
274C22—21no—
275C22—4882no—
276C22—4881no—
277C22—4881no—
278C22—2444no—
279C22—2442no—
280C22#1683963911yescharacteristic
281C22C2×#1451968911yescharacteristic
282C22#1683973011yescharacteristic
283C4—22no—
284C4—21no—
285C4—21no—
286C4—24no—
287C4—42no—
288C4—4881no—
289C4—4881no—
290C4—2442no—
291C4#1683982512yes—
292C4#1683983012yes—
293C2—21no—
294C2—21no—
295C2—4882no—
296C2#2144610911yescharacteristic
297C2#2144634411yescharacteristic
298C2#2144635411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC61×#1461C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1461D61
hypocenterC61#9102
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #21454900C2semidirect
4. #21443647C2semidirect
5. C61×#1461C2semidirect
24. #14519766C8semidirect
147. #1461D61semidirect
170. C61#9102semidirect

Maximal subgroups

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

subgroupindex
#214549002
#214485262
#214442092
#214436472
C61×#14612
#910261