Finite Group Databasebeta

← #30602092

Subgroups — #30602092

Order 15616.

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

normal subgroups only

This group has 422 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#21462004C212yesmaximal
2#21461532C211yesmaximal, characteristic
3#21461375C211yesmaximal, characteristic
4#21461169C212yesmaximal
5#21456403C211yesmaximal, characteristic
6#16840766C2212yes—
7C4×D61⋊C8C2211yescharacteristic
8C42×#6923C2211yescharacteristic
9#16840632—24no—
10(C2×#6923)⋊C8—22no—
11#16840624—22no—
12#16840612—24no—
13#16840317C2212yes—
14#16840167C411yescharacteristic, complemented
15#16840112C412yescomplemented
16#16840000—22no—
17Q16×Dic61—22no—
18C4×C8×D61C2211yescharacteristic
19C61×#494C411yescharacteristic, 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—
29C2×C8×D61D412yes—
30#14519777C2×C412yes—
31#14519768—22no—
32#14519757—22no—
33#14519753—22no—
34Q8×Dic61—22no—
35#14519742—22no—
36#14519735—22no—
37C42×D61C2311yescharacteristic
38#14519711—22no—
39#14519701—24no—
40C4×#92418C2×C411yescharacteristic
41C8×Dic61D412yes—
42C2×Q16×C61—22no—
43C4⋊Q8×C61C2×C412yes—
44C4×C8×C61C2×C411yescharacteristic
45C22×#6923—22no—
46C2×Q8×C61—22no—
47C2×C4×D61C2×D411yescharacteristic
48C2×C4×D61C2×D411yescharacteristic
49C2×C4×D61#4111yescharacteristic
50C2×Dic122—22no—
51C2×#92420—22no—
52C2×#92420—42no—
53C4×#6923—24no—
54C4×#6923—24no—
55C4×#6923—42no—
56#11758213—44no—
57D61⋊C8—42no—
58Q16×C61—44no—
59C2×C8×C61C22⋊C412yes—
60C4⋊C4×C61—22no—
61C4⋊C4×C61—22no—
62C42×C61C22×C411yescharacteristic
63#11758198—42no—
64C4⋊Dic61—22no—
65C4⋊Dic61—22no—
66C4⋊Dic61—42no—
67#11758196—42no—
68C4×Dic61—42no—
69C4×Dic61C2×D411yescharacteristic
70C4×Dic61C2×D411yescharacteristic
71C4×Dic61#4111yescharacteristic
72C2×#92418C22⋊C412yes—
73C8×D61—42no—
74C22×D61#12111yescharacteristic
75C2×#6923—24no—
76C2×#6923—42no—
77C2×#6923—42no—
78Q8×C61—44no—
79C2×C4×C61—42no—
80C2×C4×C61C2×C22⋊C411yescharacteristic
81C2×C4×C61C4×D411yescharacteristic
82C2×C4×C61C4×D411yescharacteristic
83C2×Dic61—42no—
84C2×Dic61#12111yescharacteristic
85C2×Dic61#12411yescharacteristic
86C2×Dic61#12711yescharacteristic
87C4×D61—22no—
88C4×D61—22no—
89C4×D61#13714yes—
90Dic122—44no—
91#92420—42no—
92#92420—82no—
93C8×C61—22no—
94#92418—22no—
95#9853—611nomaximal, Sylow
96C22×C61#39011yescharacteristic
97C2×D61—21no—
98C2×D61—21no—
99C2×D61#47012yes—
100#6923—44no—
101C4×C61—82no—
102C4×C61#42112yes—
103C4×C61#44112yes—
104C4×C61#44312yes—
105Dic61—21no—
106Dic61—21no—
107Dic61—82no—
108Dic61#49712yes—
109#3170—611no—
110#1994—612no—
111#1744—611no—
112#1566—611no—
113#1493—612no—
114C2×C61#196311yescharacteristic
115C2×C61#197111yescharacteristic
116C2×C61#197411yescharacteristic
117D61—22no—
118#533—612no—
119#500—611no—
120#494#692311yescharacteristic, complemented
121#487—612no—
122#467—1222no—
123C4×Q16—1222no—
124#423—612no—
125C2×C4×C8—611no—
126C2×C4×C8—611no—
127C43—611no—
128#339—1224no—
129#336—1222no—
130#334—1222no—
131Q8⋊C8—1224no—
132C61#985311yescharacteristic, complemented, Sylow
133C2×Q16—22no—
134C2×M16—1222no—
135C22×C8—612no—
136C22×C8—1221no—
137C4⋊Q8—612no—
138C4⋊Q8C2×#692312yes—
139#125—1222no—
140C22⋊Q8—1222no—
141C22⋊Q8—1222no—
142C4×Q8—1222no—
143C4×Q8—1222no—
144#117—1222no—
145C2×C42—611no—
146C2×C42—612no—
147C2×C42—1221no—
148C2×C42—1221no—
149#107—1222no—
150C4⋊C8—1222no—
151C4⋊C8—1222no—
152#103—1224no—
153C4×C8—612no—
154C4×C8—612no—
155C4×C8—611no—
156C4×C8—1221no—
157C4×C8C2×#692311yescharacteristic
158C2×Q8—22no—
159C2×Q8—1222no—
160C22×C4—611no—
161C22×C4—611no—
162C22×C4—611no—
163C22×C4—1222no—
164Q16—44no—
165M16—2444no—
166C2×C8—612no—
167C2×C8—1222no—
168C2×C8—2442no—
169C2×C8—2442no—
170C2×C8—2442no—
171C2×C8C22⋊#692312yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—1222no—
175C4⋊C4—1222no—
176C4⋊C4—2442no—
177C4⋊C4—2442no—
178C22⋊C4—2442no—
179C42—611no—
180C42—611no—
181C42—611no—
182C42—1224no—
183C42—1224no—
184C42—2442no—
185C42—2442no—
186C42C22×#692311yescharacteristic
187C23—611no—
188Q8—44no—
189Q8—2444no—
190C2×C4—42no—
191C2×C4—611no—
192C2×C4—614no—
193C2×C4—611no—
194C2×C4—611no—
195C2×C4—1222no—
196C2×C4—1222no—
197C2×C4—1224no—
198C2×C4—2442no—
199C2×C4—2442no—
200C2×C4—2442no—
201C2×C4C2×C22⋊#692311yescharacteristic
202C2×C4D4×#692311yescharacteristic
203C2×C4D4×#692311yescharacteristic
204C8—22no—
205C8—1222no—
206C8—4882no—
207C8—2442no—
208C22—612no—
209C22—1221no—
210C22—1221no—
211C22#1684071411yescharacteristic
212C4—82no—
213C4—612no—
214C4—1221no—
215C4—1221no—
216C4—4882no—
217C4—2444no—
218C4#1684076012yes—
219C4#1684080212yes—
220C4#1684080612yes—
221C2—1222no—
222C2#2146195711yescharacteristic
223C2#2146196711yescharacteristic
224C2#2146197011yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#494C4
socleC22×C61#390
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#494#6923
hypocenterC61#9853
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
14. #16840167C4semidirect
15. #16840112C4semidirect
19. C61×#494C4semidirect
120. #494#6923semidirect
132. C61#9853semidirect

Maximal subgroups

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

subgroupindex
#214620042
#214615322
#214613752
#214611692
#214564032
#985361