Finite Group Databasebeta

← #30602064

Subgroups — #30602064

Order 15616.

478 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 104 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#21462004C212yesmaximal
2C4×C8×#6923C211yesmaximal, characteristic
3#21461377C211yesmaximal, characteristic
4#21461168C212yesmaximal
5#21456406C211yesmaximal, characteristic
6#16840766C2212yes—
7C4×D61⋊C8C2211yescharacteristic
8C2×C8×#6923—22no—
9C42×#6923C2211yescharacteristic
10#16840632—24no—
11#16840625—22no—
12Q8⋊#92420—24no—
13#16840317C2212yes—
14#16840142C411yescharacteristic, complemented
15#16840114C412yescomplemented
16#16840004—22no—
17#16839856—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—
27C4×#92420—21no—
28C4×#92420D412yes—
29C8×#6923—28no—
30C2×Dic244—22no—
31C2×C8×D61D412yes—
32#14519777C2×C412yes—
33#14519768—22no—
34#14519757—22no—
35#14519753—22no—
36Q8×Dic61—22no—
37#14519742—22no—
38#14519735—22no—
39C42×D61C2311yescharacteristic
40#14519700—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—
48C2×C4×D61C2×D411yescharacteristic
49C2×C4×D61C2×D411yescharacteristic
50C2×C4×D61#4111yescharacteristic
51C2×Dic122—22no—
52C2×#92420—22no—
53C2×#92420—42no—
54C4×#6923—24no—
55C4×#6923—24no—
56C4×#6923—22no—
57C4×#6923Q1614yescomplemented
58#11758213—44no—
59D61⋊C8—24no—
60C2×C8×C61C22⋊C412yes—
61C4⋊C4×C61—22no—
62C4⋊C4×C61—22no—
63C42×C61C22×C411yescharacteristic
64#11758198—42no—
65C4⋊Dic61—22no—
66C4⋊Dic61—22no—
67C4⋊Dic61—42no—
68#11758196—42no—
69C4×Dic61—42no—
70C4×Dic61C2×D411yescharacteristic
71C4×Dic61C2×D411yescharacteristic
72C4×Dic61#4111yescharacteristic
73C2×#92418C22⋊C412yes—
74Dic244—44no—
75C8×D61—24no—
76C22×D61#12111yescharacteristic
77C2×#6923—24no—
78C2×#6923—24no—
79C2×#6923—24no—
80Q8×C61—44no—
81C2×C4×C61—42no—
82C2×C4×C61C2×C22⋊C411yescharacteristic
83C2×C4×C61C4×D411yescharacteristic
84C2×C4×C61C4×D411yescharacteristic
85C2×Dic61—42no—
86C2×Dic61#12111yescharacteristic
87C2×Dic61#12411yescharacteristic
88C2×Dic61#12711yescharacteristic
89C4×D61—22no—
90C4×D61—22no—
91C4×D61C2×Q1612yes—
92C4×D61#13512yes—
93Dic122—44no—
94#92420—24no—
95#92420—82no—
96C8×C61#10414yes—
97#92418—22no—
98#9845—611nomaximal, Sylow
99C22×C61#39011yescharacteristic
100C2×D61—21no—
101C2×D61—21no—
102C2×D61#46712yes—
103#6923—28no—
104C4×C61—82no—
105C4×C61C2×#10412yes—
106C4×C61C4×Q1612yes—
107C4×C61#44312yes—
108Dic61—21no—
109Dic61—21no—
110Dic61—82no—
111Dic61#49411yescharacteristic
112Dic61#49511yescharacteristic
113#3170—611no—
114#1994—612no—
115C42×C8—611no—
116#1567—611no—
117#1493—612no—
118C2×C61#196211yescharacteristic
119C2×C61#196911yescharacteristic
120C2×C61#197411yescharacteristic
121D61—22no—
122#533—612no—
123#500#692311yescharacteristic, complemented
124#494—611no—
125#487—612no—
126#467—1222no—
127C4×Q16—1222no—
128#423—612no—
129C2×C4×C8—611no—
130C2×C4×C8—611no—
131C2×C4×C8—1222no—
132C43—611no—
133#339—1224no—
134#335—1222no—
135Q8⋊C8—1224no—
136C61#984511yescharacteristic, complemented, Sylow
137C2×Q16—1222no—
138C2×M16—1222no—
139C22×C8—612no—
140C22×C8—1221no—
141C4⋊Q8—612no—
142C4⋊Q8C2×#692312yes—
143#125—1222no—
144C22⋊Q8—1222no—
145C22⋊Q8—1222no—
146C4×Q8—1222no—
147C4×Q8—1222no—
148#117—1222no—
149C2×C42—611no—
150C2×C42—612no—
151C2×C42—1221no—
152C2×C42—1221no—
153#107—22no—
154C4⋊C8—1222no—
155C4⋊C8—1222no—
156#103—1224no—
157C4×C8—612no—
158C4×C8—612no—
159C4×C8—611no—
160C4×C8—1221no—
161C4×C8—1228no—
162C4×C8C2×#692311yescharacteristic
163C2×Q8—22no—
164C2×Q8—1222no—
165C22×C4—611no—
166C22×C4—611no—
167C22×C4—611no—
168C22×C4—1222no—
169Q16—2444no—
170M16—2444no—
171C2×C8—612no—
172C2×C8—1222no—
173C2×C8—1224no—
174C2×C8—1224no—
175C2×C8—2442no—
176C2×C8C22⋊#692312yes—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—1222no—
180C4⋊C4—1222no—
181C4⋊C4—2442no—
182C4⋊C4—2442no—
183C22⋊C4—2442no—
184C42—611no—
185C42—611no—
186C42—611no—
187C42—614no—
188C42—1222no—
189C42—1224no—
190C42—1224no—
191C42—2442no—
192C42C22×#692311yescharacteristic
193C23—611no—
194Q8—44no—
195Q8—2444no—
196C2×C4—42no—
197C2×C4—611no—
198C2×C4—612no—
199C2×C4—612no—
200C2×C4—611no—
201C2×C4—611no—
202C2×C4—1224no—
203C2×C4—1224no—
204C2×C4—1224no—
205C2×C4—1222no—
206C2×C4—1222no—
207C2×C4—2442no—
208C2×C4C2×C22⋊#692311yescharacteristic
209C2×C4D4×#692311yescharacteristic
210C2×C4D4×#692311yescharacteristic
211C8—1224no—
212C8—1222no—
213C8—4882no—
214C8#1451984214yes—
215C22—612no—
216C22—1221no—
217C22—1221no—
218C22#1684071411yescharacteristic
219C4—82no—
220C4—611no—
221C4—611no—
222C4—1228no—
223C4—1221no—
224C4—1221no—
225C4—4882no—
226C4C2×#1451984212yes—
227C4Q16×#692312yes—
228C4#1684080412yes—
229C2—1222no—
230C2#2146195211yescharacteristic
231C2#2146196511yescharacteristic
232C2#2146197211yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#500C4
socleC22×C61#390
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#500#6923
hypocenterC61#9845
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
14. #16840142C4semidirect
15. #16840114C4semidirect
19. C61×#500C4semidirect
57. C4×#6923Q16semidirect
123. #500#6923semidirect
136. C61#9845semidirect

Maximal subgroups

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

subgroupindex
#214620042
C4×C8×#69232
#214613772
#214611682
#214564062
#984561