Finite Group Databasebeta

← #10704608

Subgroups — #10704608

Order 768.

446 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2296C211yesmaximal, characteristic
2#71572C212yesmaximal
3#67636C214yesmaximal
4#11518—31nomaximal, Sylow
5C2×C3×#117C411yescharacteristic
6C2×C3×#117C2211yescharacteristic
7C22×C3×C42C411yescharacteristic
8C3×#389—22no—
9C3×#385C412yes—
10C3×#385C2212yes—
11C3×C4×C4⋊C4—22no—
12#4926—22no—
13#4528—22no—
14#4525C2214yes—
15#2296D311yescharacteristic
16#1464—32no—
17#1326—34no—
18C23×C3×C4C2×C411yescharacteristic
19C23×C3×C4C2×C412yes—
20C3×#117—24no—
21C3×#117—42no—
22C2×C3×C4⋊C4—21no—
23C2×C3×C4⋊C4—22no—
24C2×C3×C4⋊C4—22no—
25C2×C3×C4⋊C4D412yes—
26C2×C3×C22⋊C4C2×C412yes—
27C2×C3×C22⋊C4C2×C411yescharacteristic
28C2×C3×C22⋊C4C2311yescharacteristic
29C2×C3×C42—21no—
30C2×C3×C42—21no—
31C2×C3×C42—21no—
32C2×C3×C42—42no—
33C2×C3×C42D412yes—
34C3×#95—22no—
35C3×#95—22no—
36#830—24no—
37C4⋊(C3⋊C8)—42no—
38#803—42no—
39C2×#117Dic311yescharacteristic
40C2×#117C2×D311yescharacteristic
41C22×C42Dic311yescharacteristic
42#432—62no—
43#389—22no—
44#385Dic312yes—
45#385C2×D312yes—
46C4×C4⋊C4—22no—
47#324—62no—
48#323—34no—
49C24×C3C22×C411yescharacteristic
50C22×C3×C4—22no—
51C22×C3×C4—22no—
52C22×C3×C4—22no—
53C22×C3×C4—22no—
54C22×C3×C4—22no—
55C22×C3×C4—44no—
56C22×C3×C4—42no—
57C22×C3×C4C22⋊C411yescharacteristic
58C22×C3×C4C22⋊C412yes—
59C22×C3×C4C22⋊C411yescharacteristic
60C22×C3×C4C2×D412yes—
61C3×C4⋊C4—24no—
62C3×C4⋊C4—42no—
63C3×C4⋊C4—44no—
64C3×C4⋊C4—44no—
65C3×C22⋊C4—22no—
66C3×C22⋊C4—41no—
67C3×C22⋊C4—41no—
68C3×C22⋊C4M1614yes—
69C3×C42—24no—
70C3×C42—22no—
71C3×C42—22no—
72C3×C42—42no—
73C3×C42—42no—
74C2×C3⋊C8—44no—
75C23×C4C2×Dic311yescharacteristic
76C23×C4C2×Dic312yes—
77#117—24no—
78#117—42no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4#6712yes—
83C2×C22⋊C4C2×Dic312yes—
84C2×C22⋊C4C2×Dic311yescharacteristic
85C2×C22⋊C4C22×D311yescharacteristic
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—42no—
90C2×C42#6712yes—
91C4⋊C8—122no—
92C22⋊C8—64no—
93#97—122no—
94#95—22no—
95#95—22no—
96C23×C3—21no—
97C23×C3—21no—
98C23×C3—42no—
99C23×C3C2×C22⋊C411yescharacteristic
100C23×C3C2×M1612yes—
101C2×C3×C4—28no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—24no—
107C2×C3×C4—24no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—42no—
111C2×C3×C4—42no—
112C2×C3×C4—42no—
113C2×C3×C4—44no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—44no—
118C2×C3×C4#9912yes—
119C2×C3×C4#10112yes—
120C3⋊C8—84no—
121C24C22×Dic311yescharacteristic
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4#21711yescharacteristic
130C22×C4#21712yes—
131C22×C4#21711yescharacteristic
132C22×C4C2×#6712yes—
133C2×C8—124no—
134C4⋊C4—24no—
135C4⋊C4—42no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C22⋊C4—22no—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C22⋊C4#20814yes—
142C42—24no—
143C42—22no—
144C42—22no—
145C42—42no—
146C42—42no—
147C22×C3—22no—
148C22×C3—22no—
149C22×C3—42no—
150C22×C3—42no—
151C22×C3—42no—
152C22×C3#40711yescharacteristic
153C22×C3C2×#9911yescharacteristic
154C22×C3C2×#10111yescharacteristic
155C3×C4—22no—
156C3×C4—22no—
157C3×C4—44no—
158C3×C4—42no—
159C3×C4—42no—
160C3×C4—44no—
161C23—21no—
162C23—21no—
163C23—42no—
164C23C2×#20812yes—
165C23C2×#21711yescharacteristic
166C2×C4—28no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4#83412yes—
184C2×C4#83612yes—
185C8—244no—
186C2×C3—21no—
187C2×C3—21no—
188C2×C3—42no—
189C2×C3#146311yescharacteristic
190C2×C3#212012yes—
191C22—22no—
192C22—22no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22#520711yescharacteristic
197C22C2×#83411yescharacteristic
198C22C2×#83611yescharacteristic
199C4—22no—
200C4—22no—
201C4—44no—
202C4—42no—
203C4—44no—
204C4—42no—
205C3#1151811yescharacteristic, complemented, Sylow
206C2—21no—
207C2—21no—
208C2—42no—
209C2#7155211yescharacteristic
210C2#7171612yes—
subgroupG/H
centerC22C2×#834
commutator, prime radicalC24×C3C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2296C2
socleC22×C3C2×#99
cosocleC2×C3×C22⋊C4C23
hypercenter#2296D3
hypocenterC3#11518
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2296

Complements

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

NH ≅ G/Ntype
205. C3#11518semidirect

Maximal subgroups

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

subgroupindex
C3×#22962
#715722
#676362
#115183