Finite Group Databasebeta

← #30602081

Subgroups — #30602081

Order 15616.

454 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 100 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×#14519831C211yesmaximal, characteristic
3#21461374C211yesmaximal, characteristic
4#21461169C212yesmaximal
5#21456403C211yesmaximal, characteristic
6#16840766C2212yes—
7C2×#14519831—22no—
8C4×D61⋊C8C2211yescharacteristic
9C42×#6923C2211yescharacteristic
10#16840632—24no—
11#16840623—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—
27#14519831—28no—
28C4×#92420—21no—
29C4×#92420D412yes—
30C2×C8×D61D412yes—
31#14519777C2×C412yes—
32#14519768—22no—
33#14519757—22no—
34#14519753—22no—
35Q8×Dic61—22no—
36#14519742—22no—
37#14519735—22no—
38C42×D61C2311yescharacteristic
39#14519711—22no—
40#14519701—24no—
41C4×#92418C2×C411yescharacteristic
42C8×Dic61D412yes—
43C2×Q16×C61—22no—
44C4⋊Q8×C61C2×C412yes—
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—42no—
57#11758213—44no—
58D61⋊C8—24no—
59Q16×C61—44no—
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—
74C8×D61—24no—
75C22×D61#12111yescharacteristic
76C2×#6923—24no—
77C2×#6923—42no—
78C2×#6923—42no—
79Q8×C61—44no—
80C2×C4×C61—42no—
81C2×C4×C61C2×C22⋊C411yescharacteristic
82C2×C4×C61C4×D411yescharacteristic
83C2×C4×C61C4×D411yescharacteristic
84C2×Dic61—42no—
85C2×Dic61#12111yescharacteristic
86C2×Dic61#12411yescharacteristic
87C2×Dic61#12711yescharacteristic
88C4×D61—22no—
89C4×D61—22no—
90C4×D61#13714yes—
91Dic122—44no—
92#92420—24no—
93#92420—82no—
94C8×C61#10414yes—
95#92418—22no—
96#9850—611nomaximal, Sylow
97C22×C61#39011yescharacteristic
98C2×D61—21no—
99C2×D61—21no—
100C2×D61#47012yes—
101#6923—44no—
102C4×C61—82no—
103C4×C61C2×#10412yes—
104C4×C61#44112yes—
105C4×C61#44412yes—
106Dic61—21no—
107Dic61—21no—
108Dic61—82no—
109Dic61#49712yes—
110#3170—611no—
111#1994—612no—
112C4×#97—611no—
113#1566—611no—
114#1493—612no—
115C2×C61#196211yescharacteristic
116C2×C61#197111yescharacteristic
117C2×C61#197511yescharacteristic
118D61—22no—
119#533—612no—
120#500—611no—
121#494#692311yescharacteristic, complemented
122#487—612no—
123#467—1222no—
124C4×Q16—1222no—
125#423—612no—
126C2×#97—1222no—
127C2×C4×C8—611no—
128C2×C4×C8—611no—
129C43—611no—
130#339—1224no—
131#334—1222no—
132Q8⋊C8—1224no—
133C61#985011yescharacteristic, complemented, Sylow
134C2×Q16—22no—
135C2×M16—1222no—
136C22×C8—612no—
137C22×C8—1221no—
138C4⋊Q8—612no—
139C4⋊Q8C2×#692312yes—
140#125—1222no—
141C22⋊Q8—1222no—
142C22⋊Q8—1222no—
143C4×Q8—1222no—
144C4×Q8—1222no—
145#117—1222no—
146C2×C42—611no—
147C2×C42—612no—
148C2×C42—1221no—
149C2×C42—1221no—
150#107—1222no—
151C4⋊C8—1222no—
152C4⋊C8—1222no—
153#103—1224no—
154#97—1228no—
155C4×C8—612no—
156C4×C8—612no—
157C4×C8—611no—
158C4×C8—1221no—
159C4×C8C2×#692311yescharacteristic
160C2×Q8—22no—
161C2×Q8—1222no—
162C22×C4—611no—
163C22×C4—611no—
164C22×C4—611no—
165C22×C4—1222no—
166Q16—44no—
167M16—2444no—
168C2×C8—612no—
169C2×C8—1222no—
170C2×C8—1224no—
171C2×C8—1224no—
172C2×C8—2442no—
173C2×C8C22⋊#692312yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—1222no—
177C4⋊C4—1222no—
178C4⋊C4—2442no—
179C4⋊C4—2442no—
180C22⋊C4—2442no—
181C42—611no—
182C42—611no—
183C42—611no—
184C42—1224no—
185C42—1224no—
186C42—2442no—
187C42—2442no—
188C42C22×#692311yescharacteristic
189C23—611no—
190Q8—44no—
191Q8—2444no—
192C2×C4—42no—
193C2×C4—611no—
194C2×C4—614no—
195C2×C4—611no—
196C2×C4—611no—
197C2×C4—1222no—
198C2×C4—1222no—
199C2×C4—1224no—
200C2×C4—2442no—
201C2×C4—2442no—
202C2×C4—2442no—
203C2×C4C2×C22⋊#692311yescharacteristic
204C2×C4D4×#692311yescharacteristic
205C2×C4D4×#692311yescharacteristic
206C8—1224no—
207C8—1222no—
208C8—4882no—
209C8#1451984114yes—
210C22—612no—
211C22—1221no—
212C22—1221no—
213C22#1684071411yescharacteristic
214C4—82no—
215C4—612no—
216C4—1221no—
217C4—1221no—
218C4—4882no—
219C4—2444no—
220C4C2×#1451984112yes—
221C4#1684080212yes—
222C4#1684080812yes—
223C2—1222no—
224C2#2146195311yescharacteristic
225C2#2146196711yescharacteristic
226C2#2146197311yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#494C4
socleC22×C61#390
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#494#6923
hypocenterC61#9850
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
121. #494#6923semidirect
133. C61#9850semidirect

Maximal subgroups

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

subgroupindex
#214620042
C4×#145198312
#214613742
#214611692
#214564032
#985061