Finite Group Databasebeta

← #30602087

Subgroups — #30602087

Order 15616.

498 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461952C212yesmaximal, complemented
2#21461532C211yesmaximal, characteristic, complemented
3#21461375C211yesmaximal, characteristic, complemented
4#21461166C212yesmaximal, complemented
5#21456399C211yesmaximal, characteristic
6#16840766C2212yes—
7C2×#14519842—24no—
8C4×D61⋊C8C2211yescharacteristic
9C42×#6923C2211yescharacteristic
10(C2×#6923)⋊C8—22no—
11#16840624—22no—
12#16840610—24no—
13#16840313C2212yes—
14#16840139C411yescharacteristic, complemented
15#16840110C412yescomplemented
16#16839999—22no—
17#16839844—22no—
18C4×C8×D61C2211yescharacteristic
19C61×#500C411yescharacteristic, complemented
20C2×#11758213—22no—
21C2×D61⋊C8—21no—
22C2×C4×#6923—21no—
23C2×C4×#6923—21no—
24C2×C4×#6923D412yes—
25C4⋊#92420—22no—
26C4⋊#92420—22no—
27#14519842—48no—
28C4×#92420—21no—
29C4×#92420D412yes—
30C2×#11758226—22no—
31C2×D488—22no—
32C2×C8×D61D412yes—
33#14519774C2×C412yes—
34#14519766—22no—
35#14519752—22no—
36#14519752—22no—
37#14519739—22no—
38#14519739—22no—
39C42×D61C2311yescharacteristic
40#14519697—24no—
41C4×#92418C2×C411yescharacteristic
42C8×Dic61D412yes—
43C4⋊Q8×C61C2×C412yes—
44#107×C61—22no—
45C4×C8×C61C2×C411yescharacteristic
46C22×#6923—22no—
47C2×Q8×C61—22no—
48#11758226—44no—
49#11758226—44no—
50C2×D244—22no—
51C2×D244—22no—
52C2×D244—22no—
53C2×D244—42no—
54C2×C4×D61—42no—
55C2×C4×D61C2×D411yescharacteristic
56C2×C4×D61C2×D411yescharacteristic
57C2×C4×D61#4111yescharacteristic
58C2×#92420—22no—
59C2×#92420—42no—
60C4×#6923—24no—
61C4×#6923—24no—
62C4×#6923—42no—
63#11758213—44no—
64D61⋊C8—42no—
65C2×C8×C61C22⋊C412yes—
66C4⋊C4×C61—22no—
67C4⋊C4×C61—22no—
68C42×C61C22×C411yescharacteristic
69#11758198—42no—
70#11758198—42no—
71C4×Dic61C2×D411yescharacteristic
72C4×Dic61C2×D411yescharacteristic
73C4×Dic61#4111yescharacteristic
74C2×#92418C22⋊C412yes—
75D488—44no—
76C8×D61—42no—
77C22×D61—42no—
78C22×D61#12111yescharacteristic
79C2×#6923—24no—
80C2×#6923—42no—
81C2×#6923—42no—
82Q8×C61—44no—
83C2×C4×C61—42no—
84C2×C4×C61C2×C22⋊C411yescharacteristic
85C2×C4×C61C4×D411yescharacteristic
86C2×C4×C61C4×D411yescharacteristic
87C2×Dic61#12111yescharacteristic
88C2×Dic61#12411yescharacteristic
89C2×Dic61#12711yescharacteristic
90D244—44no—
91D244—44no—
92D244—44no—
93D244—84no—
94C4×D61—22no—
95C4×D61—22no—
96C4×D61—82no—
97C4×D61—82no—
98C4×D61AGL(1,ℤ/8)14yes—
99#92420—42no—
100#92420—82no—
101C8×C61—22no—
102#92418—22no—
103#9852—611nomaximal, Sylow
104C22×C61#39011yescharacteristic
105C2×D61—21no—
106C2×D61—21no—
107C2×D61—82no—
108C2×D61—82no—
109C2×D61—82no—
110C2×D61#46912yes—
111#6923—44no—
112C4×C61—82no—
113C4×C61#42112yes—
114C4×C61#44212yes—
115C4×C61#44312yes—
116Dic61—21no—
117Dic61—21no—
118Dic61#49612yes—
119#3169—611no—
120#1962—612no—
121#1744—611no—
122#1566—611no—
123#1492—612no—
124C2×C61#196311yescharacteristic
125C2×C61#197211yescharacteristic
126C2×C61#197411yescharacteristic
127D61—22no—
128D61—162no—
129#532—612no—
130#500#692311yescharacteristic, complemented
131#493—611no—
132#486—612no—
133#466—1222no—
134C4×D8—1222no—
135#423—612no—
136C2×#104—1224no—
137C2×C4×C8—611no—
138C2×C4×C8—611no—
139C43—611no—
140#336—1222no—
141#334—1222no—
142D4⋊C8—1224no—
143C61#985211yescharacteristic, complemented, Sylow
144C2×#41—1222no—
145C2×D8—1222no—
146C2×M16—1222no—
147C22×C8—612no—
148C22×C8—1221no—
149C4⋊Q8C2×#692312yes—
150#127—612no—
151#124—1222no—
152#121—1222no—
153#121—1222no—
154C4×D4—1222no—
155C4×D4—1222no—
156C2×C42—611no—
157C2×C42—612no—
158C2×C42—1221no—
159C2×C42—1221no—
160#107—22no—
161C4⋊C8—1222no—
162C4⋊C8—1222no—
163#104—2448no—
164#102—1224no—
165C4×C8—612no—
166C4×C8—612no—
167C4×C8—611no—
168C4×C8—1221no—
169C4×C8C2×#692311yescharacteristic
170#41—2444no—
171#41—2444no—
172C2×Q8—22no—
173C2×D4—1222no—
174C2×D4—1222no—
175C2×D4—1222no—
176C2×D4—2442no—
177C22×C4—611no—
178C22×C4—611no—
179C22×C4—611no—
180C22×C4—1222no—
181C22×C4—2442no—
182D8—2444no—
183M16—2444no—
184C2×C8—612no—
185C2×C8—1222no—
186C2×C8—2442no—
187C2×C8—2442no—
188C2×C8—2442no—
189C2×C8C22⋊#692312yes—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C22⋊C4—2442no—
193C22⋊C4—2442no—
194C42—611no—
195C42—611no—
196C42—611no—
197C42—1224no—
198C42—1224no—
199C42—2442no—
200C42C22×#692311yescharacteristic
201C23—611no—
202C23—2442no—
203Q8—44no—
204D4—4884no—
205D4—2444no—
206D4—2444no—
207D4—2444no—
208C2×C4—42no—
209C2×C4—611no—
210C2×C4—614no—
211C2×C4—611no—
212C2×C4—611no—
213C2×C4—1222no—
214C2×C4—1222no—
215C2×C4—1224no—
216C2×C4—4882no—
217C2×C4—4882no—
218C2×C4—2442no—
219C2×C4—2442no—
220C2×C4C2×C22⋊#692311yescharacteristic
221C2×C4D4×#692311yescharacteristic
222C2×C4D4×#692311yescharacteristic
223C8—22no—
224C8—1222no—
225C8—4882no—
226C8—2442no—
227C22—612no—
228C22—1221no—
229C22—1221no—
230C22—4882no—
231C22—4882no—
232C22—4882no—
233C22#1684071411yescharacteristic
234C4—82no—
235C4—612no—
236C4—1221no—
237C4—1221no—
238C4—2444no—
239C4#1684076112yes—
240C4#1684080312yes—
241C4#1684080612yes—
242C2—1222no—
243C2—9762no—
244C2#2146195511yescharacteristic
245C2#2146196811yescharacteristic
246C2#2146197011yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#500C4
socleC22×C61#390
prime radical#21456399C2
cosocleC42×D61C23
hypercenter#500#6923
hypocenterC61#9852
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #21461952C2semidirect
2. #21461532C2semidirect
3. #21461375C2semidirect
4. #21461166C2semidirect
14. #16840139C4semidirect
15. #16840110C4semidirect
19. C61×#500C4semidirect
130. #500#6923semidirect
143. C61#9852semidirect

Maximal subgroups

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

subgroupindex
#214619522
#214615322
#214613752
#214611662
#214563992
#985261