Finite Group Databasebeta

← #30601914

Subgroups — #30601914

Order 15616.

370 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 84 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#21461741C211yesmaximal, characteristic
2#21461253C212yesmaximal
3#21461251C212yesmaximal
4#21461033C211yesmaximal, characteristic
5#21456328C211yesmaximal, characteristic
6#16840780C2212yes—
7#16840706C2211yescharacteristic
8#16840706C2211yescharacteristic
9#16840636—22no—
10#16840616—24no—
11#16840615—24no—
12#92420⋊C8—22no—
13#16840317C2212yes—
14#16840165C412yescomplemented
15#16840162—22no—
16#16840160C411yescharacteristic, complemented
17#16839797—22no—
18C4×C8×D61C2211yescharacteristic
19C61×#498C411yescharacteristic, complemented
20C2×C4⋊#6923—21no—
21C2×C4⋊#6923—21no—
22#14519844—22no—
23#14519834D414yes—
24C4×#92420—22no—
25(C2×#6923)⋊C4—22no—
26(C2×#6923)⋊C4—22no—
27C2×C8×D61D412yes—
28#14519777C2×C412yes—
29#14519768—22no—
30#14519757—22no—
31#14519753—22no—
32Q8×Dic61—22no—
33#14519742—22no—
34#14519735—22no—
35C42×D61C2311yescharacteristic
36#14519710—24no—
37#14519709—22no—
38C4×#92418C2×C411yescharacteristic
39C8×Dic61D412yes—
40C4⋊Q8×C61C2×C412yes—
41#106×C61—22no—
42C4×C8×C61C2×C411yescharacteristic
43C22×#6923—41no—
44C22×#6923—41no—
45C2×Q8×C61—22no—
46C2×C4×D61C2×D412yes—
47C2×C4×D61C2×D411yescharacteristic
48C2×Dic122—22no—
49C2×#92420—24no—
50C4⋊#6923—42no—
51C4⋊#6923—42no—
52C2×C8×C61C22⋊C412yes—
53C4⋊C4×C61—22no—
54C4⋊C4×C61—22no—
55C42×C61C22×C411yescharacteristic
56#11758198—42no—
57C4⋊Dic61—22no—
58C4⋊Dic61—22no—
59C4⋊Dic61—42no—
60#11758196—42no—
61C4×Dic61—42no—
62C4×Dic61C2×D411yescharacteristic
63C4×Dic61#4112yes—
64C2×#92418C22⋊C412yes—
65C8×D61—22no—
66C8×D61—22no—
67C22×D61#12711yescharacteristic
68C2×#6923—82no—
69C2×#6923—82no—
70C2×#6923—81no—
71C2×#6923—81no—
72Q8×C61—82no—
73C2×C4×C61—42no—
74C2×C4×C61C2×C22⋊C411yescharacteristic
75C2×C4×C61C4×D412yes—
76C2×Dic61—42no—
77C2×Dic61#12112yes—
78C2×Dic61#12411yescharacteristic
79C4×D61—42no—
80C4×D61#13712yes—
81C4×D61#13712yes—
82Dic122—82no—
83#92420—44no—
84C8×C61—22no—
85#92418—22no—
86#9759—611nomaximal, Sylow
87C22×C61#39011yescharacteristic
88C2×D61—22no—
89C2×D61#49711yescharacteristic
90C2×D61#49711yescharacteristic
91#6923—161no—
92#6923—161no—
93C4×C61—22no—
94C4×C61—82no—
95C4×C61#41912yes—
96Dic61—22no—
97Dic61—82no—
98Dic61#49112yes—
99#3160—611no—
100#1835—611no—
101#1518—612no—
102#1518—612no—
103#1452—611no—
104C2×C61#196011yescharacteristic
105C2×C61#197612yes—
106D61—21no—
107D61—21no—
108#533—612no—
109#500—612no—
110#499—1222no—
111#498—611no—
112#498#692311yescharacteristic, complemented
113#431—612no—
114#427—1222no—
115C2×C4×C8—611no—
116#382—611no—
117#382—611no—
118#340—1222no—
119#328—1224no—
120#328—1224no—
121#322—1222no—
122C61#975911yescharacteristic, complemented, Sylow
123C22×C8—612no—
124C4⋊Q8—612no—
125C4⋊Q8C2×#692312yes—
126#125—1222no—
127C22⋊Q8—1222no—
128C22⋊Q8—1222no—
129C4×Q8—1222no—
130C4×Q8—1222no—
131#117—1222no—
132C2×C4⋊C4—1221no—
133C2×C4⋊C4—1221no—
134C2×C42—611no—
135#107—1224no—
136#106—22no—
137#106—1222no—
138C4⋊C8—1222no—
139C22⋊C8—614no—
140C4×C8—612no—
141C4×C8—611no—
142C4×C8—1222no—
143C4×C8C2×#692311yescharacteristic
144#95—1222no—
145#95—1222no—
146C2×Q8—22no—
147C2×Q8—1222no—
148C22×C4—611no—
149C22×C4—612no—
150C22×C4—2441no—
151C22×C4—2441no—
152C2×C8—612no—
153C2×C8—1222no—
154C2×C8—1222no—
155C2×C8—1224no—
156C2×C8C22⋊#692312yes—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—1222no—
160C4⋊C4—1222no—
161C4⋊C4—2442no—
162C4⋊C4—2442no—
163C4⋊C4—2442no—
164C4⋊C4—2442no—
165C22⋊C4—2442no—
166C42—611no—
167C42—612no—
168C42—2442no—
169C42C22×#692311yescharacteristic
170C23—611no—
171Q8—82no—
172Q8—4882no—
173C2×C4—42no—
174C2×C4—611no—
175C2×C4—612no—
176C2×C4—612no—
177C2×C4—612no—
178C2×C4—4881no—
179C2×C4—4881no—
180C2×C4—4882no—
181C2×C4—4882no—
182C2×C4—2442no—
183C2×C4—2442no—
184C2×C4C2×C22⋊#692311yescharacteristic
185C2×C4D4×#692312yes—
186C8—22no—
187C8—1222no—
188C8—2444no—
189C22—611no—
190C22—611no—
191C22—1222no—
192C22#1684071411yescharacteristic
193C4—22no—
194C4—82no—
195C4—612no—
196C4—1222no—
197C4—9761no—
198C4—9761no—
199C4—4882no—
200C4#1684075512yes—
201C2—1221no—
202C2—1221no—
203C2#2146195011yescharacteristic
204C2#2146197712yes—
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#498C4
socleC22×C61#390
prime radicalC22×D61#127
cosocleC42×D61C23
hypercenter#498#6923
hypocenterC61#9759
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
14. #16840165C4semidirect
16. #16840160C4semidirect
19. C61×#498C4semidirect
112. #498#6923semidirect
122. C61#9759semidirect

Maximal subgroups

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

subgroupindex
#214617412
#214612532
#214612512
#214610332
#214563282
#975961