Finite Group Databasebeta

← #30604918

Subgroups — #30604918

Order 15616.

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

normal subgroups only

This group has 350 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#21462179C211yesmaximal, characteristic
2C4×#14519846C211yesmaximal, characteristic
3#21460807C214yesmaximal
4#21456403C211yesmaximal, characteristic
5C2×#14519846—21no—
6C4×C4⋊#6923—21no—
7C4⋊#11758187—21no—
8#16840657C2212yes—
9#16840648—21no—
10#16840636C2212yes—
11#16840317—21no—
12#16840167C411yescharacteristic, complemented
13#16840112—21no—
14#16840000—21no—
15#16840000C2212yes—
16Q16×Dic61—21no—
17Q16×Dic61C412yescomplemented
18C4×C8×D61C2211yescharacteristic
19C61×#494C411yescharacteristic, complemented
20C2×C4⋊#6923—22no—
21C2×C4×#6923—41no—
22C2×#11758187—22no—
23#14519846—24no—
24(C2×#6923)⋊C4—41no—
25C2×C8×D61D411yescharacteristic
26C2×C8×D61C2311yescharacteristic
27#14519777—21no—
28#14519768—41no—
29#14519757—41no—
30#14519753—22no—
31Q8×Dic61—22no—
32#14519742—41no—
33#14519735—41no—
34C42×D61D411yescharacteristic
35#14519711—21no—
36#14519711C2×C412yes—
37#14519701—22no—
38#14519701—22no—
39C4×#92418D411yescharacteristic
40C8×Dic61C2×C411yescharacteristic
41C8×Dic61D411yescharacteristic
42C2×Q16×C61—21no—
43C2×Q16×C61C2×C412yes—
44C4⋊Q8×C61—21no—
45C4×C8×C61C2×C411yescharacteristic
46C22×#6923—42no—
47C2×Q8×C61—22no—
48C2×C4×D61D811yescharacteristic
49C2×C4×D61D811yescharacteristic
50C2×C4×D61C2×D411yescharacteristic
51C2×Dic122—41no—
52C4⋊#6923—24no—
53C4⋊#6923—42no—
54C4×#6923—82no—
55Q16×C61—42no—
56Q16×C61—42no—
57C2×C8×C61C22⋊C411yescharacteristic
58C2×C8×C61C22×C411yescharacteristic
59C4⋊C4×C61—41no—
60C4⋊C4×C61—41no—
61C42×C61C22⋊C411yescharacteristic
62#11758198—42no—
63C4⋊Dic61—22no—
64C4⋊Dic61—41no—
65C4⋊Dic61—42no—
66#11758196—42no—
67C4×Dic61—42no—
68C4×Dic61C22⋊C411yescharacteristic
69C4×Dic61SD1611yescharacteristic
70C4×Dic61SD1611yescharacteristic
71C2×#92418C22⋊C411yescharacteristic
72C2×#92418C2×D411yescharacteristic
73C8×D61—22no—
74C8×D61—41no—
75#11758187—42no—
76C22×D61C2×D811yescharacteristic
77C2×#6923—44no—
78C2×#6923—82no—
79C2×#6923—82no—
80Q8×C61—42no—
81Q8×C61—42no—
82C2×C4×C61—42no—
83C2×C4×C61#10211yescharacteristic
84C2×C4×C61#10211yescharacteristic
85C2×C4×C61C2×C22⋊C411yescharacteristic
86C2×Dic61—42no—
87C2×Dic61#10211yescharacteristic
88C2×Dic61#10211yescharacteristic
89C2×Dic61C2×SD1611yescharacteristic
90C4×D61—21no—
91C4×D61—21no—
92C4×D61—41no—
93C4×D61SD3212yes—
94C4×D61Q3212yes—
95Dic122—82no—
96C8×C61—21no—
97C8×C61#10412yes—
98#92418—21no—
99#92418—21no—
100#11743—611nomaximal, Sylow
101C22×C61C2×#10211yescharacteristic
102C2×D61—21no—
103C2×D61—21no—
104C2×D61C2×SD3211yescharacteristic
105C2×D61C2×Q3211yescharacteristic
106#6923—84no—
107C4×C61—21no—
108C4×C61—82no—
109C4×C61#35812yes—
110C4×C61#41011yescharacteristic
111C4×C61C2×#10411yescharacteristic
112Dic61—21no—
113Dic61—21no—
114Dic61—82no—
115Dic61#35812yes—
116#3170—611no—
117#2162—611no—
118C4×#107—611no—
119#1348—614no—
120C2×C61#150211yescharacteristic
121C2×C61C2×#35811yescharacteristic
122C2×C61#214511yescharacteristic
123D61—21no—
124D61—21no—
125#533—1221no—
126#500—611no—
127#494#692311yescharacteristic, complemented
128#487—1221no—
129#467—612no—
130#467—1221no—
131C4×Q16—612no—
132C4×Q16—1221no—
133C2×#107—1221no—
134C2×C4×C8—611no—
135C4×C4⋊C4—1221no—
136C4⋊C16—1221no—
137C22⋊C16—612no—
138C4×C16—1221no—
139#340—612no—
140C61#1174311yescharacteristic, complemented, Sylow
141C2×Q16—21no—
142C2×Q16C2×#692312yes—
143C22×C8—611no—
144C22×C8—611no—
145C4⋊Q8—21no—
146C4⋊Q8—1221no—
147#125—2441no—
148C22⋊Q8—1222no—
149C22⋊Q8—2441no—
150C4×Q8—1222no—
151C4×Q8—2441no—
152#117—2441no—
153C2×C4⋊C4—1222no—
154C2×C42—611no—
155C2×C42—2441no—
156C2×C16—1222no—
157#107—612no—
158#107—1224no—
159#107—1221no—
160#103—1222no—
161#103—1222no—
162C4×C8—611no—
163C4×C8—611no—
164C4×C8—611no—
165C4×C8C2×#692311yescharacteristic
166#95—2441no—
167C2×Q8—22no—
168C2×Q8—2441no—
169C22×C4—611no—
170C22×C4—611no—
171C22×C4—611no—
172C22×C4—2442no—
173Q16—42no—
174Q16—42no—
175C2×C8—611no—
176C2×C8—611no—
177C2×C8—1222no—
178C2×C8—2441no—
179C2×C8C22⋊#692311yescharacteristic
180C2×C8C22×#692311yescharacteristic
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—1224no—
184C4⋊C4—1222no—
185C4⋊C4—2442no—
186C4⋊C4—2442no—
187C4⋊C4—2442no—
188C4⋊C4—2441no—
189C22⋊C4—2442no—
190C42—611no—
191C42—611no—
192C42—611no—
193C42—4882no—
194C42—2442no—
195C42C22⋊#692311yescharacteristic
196C16—2442no—
197C23—611no—
198Q8—42no—
199Q8—42no—
200Q8—4882no—
201C2×C4—42no—
202C2×C4—611no—
203C2×C4—612no—
204C2×C4—612no—
205C2×C4—611no—
206C2×C4—611no—
207C2×C4—1221no—
208C2×C4—1221no—
209C2×C4—4882no—
210C2×C4—4882no—
211C2×C4—2441no—
212C2×C4—2444no—
213C2×C4—2442no—
214C2×C4#1451983911yescharacteristic
215C2×C4#1451983911yescharacteristic
216C2×C4C2×C22⋊#692311yescharacteristic
217C8—21no—
218C8—1221no—
219C8—1221no—
220C8#1451984112yes—
221C22—611no—
222C22—611no—
223C22—1221no—
224C22—1221no—
225C22C2×#1451983911yescharacteristic
226C4—21no—
227C4—82no—
228C4—612no—
229C4—1221no—
230C4—1221no—
231C4—4884no—
232C4—4882no—
233C4#1684067512yes—
234C4#1684074211yescharacteristic
235C4C2×#1451984111yescharacteristic
236C2—1221no—
237C2—1221no—
238C2#2146119911yescharacteristic
239C2C2×#1684067511yescharacteristic
240C2#2146214111yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#494C4
socleC22×C61C2×#102
prime radicalC22×D61C2×D8
cosocleC2×C8×D61C23
hypercenter#494#6923
hypocenterC61#11743
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #494

Complements

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

NH ≅ G/Ntype
12. #16840167C4semidirect
17. Q16×Dic61C4semidirect
19. C61×#494C4semidirect
127. #494#6923semidirect
140. C61#11743semidirect

Maximal subgroups

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

subgroupindex
#214621792
C4×#145198462
#214608072
#214564032
#1174361