Finite Group Databasebeta

← #30594168

Subgroups — #30594168

Order 15616.

368 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 76 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#21460657C212yesmaximal
2#21444086C211yesmaximal, characteristic
3C2×C22⋊#92420—22no—
4C22×C4⋊Dic61C411yescharacteristic
5C2×#14519684—22no—
6C2×#14519684C2211yescharacteristic
7C2×#95×C61C411yescharacteristic
8C22×#92420—42no—
9C22⋊#92420—24no—
10C22⋊#92420—42no—
11C23×Dic61—21no—
12C23×Dic61D412yes—
13C2×C4⋊Dic61—21no—
14C2×C4⋊Dic61—41no—
15C2×C4⋊Dic61—41no—
16C2×C4⋊Dic61C812yescomplemented
17#14519684—41no—
18#14519684—44no—
19C23×C4×C61—21no—
20C23×C4×C61C2×C411yescharacteristic
21#95×C61—41no—
22C24×C61C22⋊C411yescharacteristic
23C22×C4×C61—21no—
24C22×C4×C61—21no—
25C22×C4×C61—22no—
26C22×C4×C61—42no—
27C22×C4×C61—42no—
28C22×C4×C61—41no—
29C22×C4×C61—41no—
30C22×C4×C61—42no—
31C22×C4×C61C2×C811yescharacteristic
32C22×C4×C61M1611yescharacteristic
33C22×Dic61—22no—
34C22×Dic61—22no—
35C22×Dic61—22no—
36C22×Dic61—42no—
37C22×Dic61—42no—
38C22×Dic61—42no—
39C22×Dic61—42no—
40C22×Dic61—41no—
41C22×Dic61—41no—
42C22×Dic61SD1612yes—
43C22×Dic61Q1612yes—
44C2×#92420—44no—
45C2×#92420—82no—
46C2×#92420—82no—
47C4⋊Dic61—42no—
48C4⋊Dic61—42no—
49C23×C61—22no—
50C23×C61—21no—
51C23×C61—21no—
52C23×C61C22⋊C811yescharacteristic
53C23×C61#9911yescharacteristic
54C23×C61#10011yescharacteristic
55C23×C61#10312yes—
56C23×C61#10412yes—
57C2×C4×C61—21no—
58C2×C4×C61—21no—
59C2×C4×C61—42no—
60C2×C4×C61—42no—
61C2×C4×C61—41no—
62C2×C4×C61—41no—
63C2×C4×C61—42no—
64C2×C4×C61—44no—
65C2×C4×C61—81no—
66C2×C4×C61—82no—
67C2×C4×C61—81no—
68C2×Dic61—22no—
69C2×Dic61—22no—
70C2×Dic61—42no—
71C2×Dic61—42no—
72C2×Dic61—42no—
73C2×Dic61—42no—
74C2×Dic61—44no—
75C2×Dic61—42no—
76C2×Dic61—82no—
77C2×Dic61—82no—
78C2×Dic61—81no—
79C2×Dic61—81no—
80#92420—84no—
81#7132—611nomaximal, Sylow
82C22×C61—21no—
83C22×C61—21no—
84C22×C61—21no—
85C22×C61—21no—
86C22×C61—22no—
87C22×C61—22no—
88C22×C61—22no—
89C22×C61—22no—
90C22×C61#32311yescharacteristic
91C22×C61Q8⋊C812yes—
92C22×C61#32812yes—
93C22×C61#32911yescharacteristic
94C22×C61#33311yescharacteristic
95C22×C61#35212yes—
96C22×C61#35311yescharacteristic
97C22×C61#35511yescharacteristic
98C4×C61—41no—
99C4×C61—41no—
100C4×C61—82no—
101Dic61—42no—
102Dic61—42no—
103Dic61—82no—
104#1445—611no—
105#1280—612no—
106C2×C61—21no—
107C2×C61—22no—
108C2×C61—21no—
109C2×C61#127211yescharacteristic
110C2×C61#131711yescharacteristic
111C2×C61#131811yescharacteristic
112C2×C61#135212yes—
113C2×C61#135312yes—
114C22×C4⋊C4—611no—
115C2×C22⋊C8—1222no—
116C2×#95—611no—
117C2×#95—1222no—
118C2×#95#692311yescharacteristic
119C61#713211yescharacteristic, complemented, Sylow
120C23×C4—21no—
121C23×C4—612no—
122C23×C4—1221no—
123C23×C4C2×#692311yescharacteristic
124C22×C8—2442no—
125C2×C4⋊C4—612no—
126C2×C4⋊C4—1221no—
127C2×C4⋊C4—2441no—
128C2×C4⋊C4—2441no—
129C22⋊C8—1224no—
130C22⋊C8—2442no—
131#95—41no—
132#95—2441no—
133#95—2444no—
134C24C22⋊#692311yescharacteristic
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—612no—
144C22×C4—612no—
145C22×C4—1222no—
146C22×C4—1222no—
147C22×C4—1222no—
148C22×C4—2442no—
149C22×C4—2442no—
150C22×C4—2441no—
151C22×C4—2441no—
152C22×C4—2442no—
153C22×C4—2442no—
154C22×C4D61⋊C811yescharacteristic
155C22×C4#1175821311yescharacteristic
156C2×C8—4882no—
157C2×C8—4882no—
158C2×C8—2444no—
159C4⋊C4—2442no—
160C4⋊C4—2442no—
161C23—22no—
162C23—21no—
163C23—21no—
164C23#1451983411yescharacteristic
165C23#1451983611yescharacteristic
166C23#1451983711yescharacteristic
167C23#1451984012yes—
168C23#1451984112yes—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—81no—
178C2×C4—82no—
179C2×C4—81no—
180C2×C4—1222no—
181C2×C4—1222no—
182C2×C4—4882no—
183C2×C4—4881no—
184C2×C4—4881no—
185C2×C4—4882no—
186C2×C4—2444no—
187C2×C4—2442no—
188C2×C4—2442no—
189C2×C4—2442no—
190C2×C4—2442no—
191C2×C4—2442no—
192C8—4884no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22#1684060611yescharacteristic
202C22#1684061212yes—
203C22#1684061512yes—
204C22#1684061711yescharacteristic
205C22#1684062211yescharacteristic
206C22#1684066512yes—
207C22#1684066711yescharacteristic
208C22#1684067111yescharacteristic
209C4—41no—
210C4—41no—
211C4—82no—
212C4—4882no—
213C4—2442no—
214C4—2442no—
215C2—21no—
216C2—22no—
217C2—21no—
218C2#2146064011yescharacteristic
219C2#2146074611yescharacteristic
220C2#2146074811yescharacteristic
221C2#2146081512yes—
222C2#2146081712yes—
subgroupG/H
centerC23#14519836
commutatorC22×C4×C61C2×C8
FrattiniC23×C4C2×#6923
FittingC2×#95×C61C4
socleC23×C61#99
prime radicalC24×C61C22⋊C4
cosocleC2×#14519684C22
hypercenterC2×#95#6923
hypocenterC61#7132
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
16. C2×C4⋊Dic61C8semidirect
119. C61#7132semidirect

Maximal subgroups

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

subgroupindex
#214606572
#214440862
#713261