Finite Group Databasebeta

← #10704352

Subgroups — #10704352

Order 768.

434 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 131 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×#2317C211yesmaximal, characteristic
2#71571C212yesmaximal
3#67632C214yesmaximal, complemented
4#11449—31nomaximal, Sylow
5C2×C3×#117C411yescharacteristic
6C2×C3×#117C2211yescharacteristic
7C22×C3×C4⋊C4C411yescharacteristic
8C3×#389—22no—
9C3×#385—22no—
10C3×#384C412yes—
11C3×#382C2212yes—
12#4925—22no—
13#4528C2214yes—
14#4525—22no—
15#2317D311yescharacteristic
16#1458—32no—
17#1324—34no—
18C23×C3×C4—21no—
19C23×C3×C4C2×C411yescharacteristic
20C3×#117—22no—
21C3×#117—42no—
22C3×#117C814yescomplemented
23C2×C3×C4⋊C4—21no—
24C2×C3×C4⋊C4—21no—
25C2×C3×C4⋊C4—24no—
26C2×C3×C4⋊C4—42no—
27C2×C3×C4⋊C4C2×C412yes—
28C2×C3×C4⋊C4C2×C411yescharacteristic
29C2×C3×C4⋊C4C2311yescharacteristic
30C2×C3×C22⋊C4—21no—
31C2×C3×C22⋊C4D412yes—
32C2×C3×C42C2×C412yes—
33C2×C3×C42D412yes—
34C3×#95—24no—
35#830—24no—
36C4⋊(C3⋊C8)—42no—
37C4×C3⋊C8—42no—
38C2×#117Dic311yescharacteristic
39C2×#117C2×D311yescharacteristic
40C22×C4⋊C4Dic311yescharacteristic
41#431—62no—
42#389—22no—
43#385—22no—
44#384Dic312yes—
45#382C2×D312yes—
46#324—34no—
47#323—62no—
48C24×C3C22⋊C411yescharacteristic
49C22×C3×C4—22no—
50C22×C3×C4—22no—
51C22×C3×C4—22no—
52C22×C3×C4—22no—
53C22×C3×C4—42no—
54C22×C3×C4—82no—
55C22×C3×C4C22⋊C412yes—
56C22×C3×C4C22⋊C411yescharacteristic
57C22×C3×C4C2×C812yes—
58C22×C3×C4M1612yes—
59C22×C3×C4C22×C411yescharacteristic
60C22×C3×C4C2×D412yes—
61C3×C4⋊C4—24no—
62C3×C4⋊C4—41no—
63C3×C4⋊C4—41no—
64C3×C4⋊C4—42no—
65C3×C4⋊C4—42no—
66C3×C4⋊C4—84no—
67C3×C22⋊C4—24no—
68C3×C22⋊C4—42no—
69C3×C42—22no—
70C3×C42—22no—
71C3×C42—22no—
72C3×C42C2×C814yes—
73C2×C3⋊C8—44no—
74C23×C4—21no—
75C23×C4C2×Dic311yescharacteristic
76#117—22no—
77#117—42no—
78#117C3⋊C814yescomplemented
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4C2×Dic312yes—
84C2×C4⋊C4C2×Dic311yescharacteristic
85C2×C4⋊C4C22×D311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4#6712yes—
88C2×C42C2×Dic312yes—
89C2×C42#6712yes—
90C4⋊C8—122no—
91C22⋊C8—64no—
92C4×C8—122no—
93#95—24no—
94C23×C3—21no—
95C23×C3—21no—
96C23×C3—41no—
97C23×C3—41no—
98C23×C3C22⋊C812yes—
99C23×C3C2×C22⋊C411yescharacteristic
100C2×C3×C4—21no—
101C2×C3×C4—21no—
102C2×C3×C4—24no—
103C2×C3×C4—24no—
104C2×C3×C4—24no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—42no—
108C2×C3×C4—42no—
109C2×C3×C4—42no—
110C2×C3×C4—42no—
111C2×C3×C4—44no—
112C2×C3×C4—42no—
113C2×C3×C4—84no—
114C2×C3×C4C22⋊C812yes—
115C2×C3×C4#9912yes—
116C2×C3×C4#10112yes—
117C2×C3×C4C22×C811yescharacteristic
118C2×C3×C4C2×M1611yescharacteristic
119C3⋊C8—84no—
120C24#21711yescharacteristic
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—42no—
126C22×C4—82no—
127C22×C4C2×C3⋊C812yes—
128C22×C4#20812yes—
129C22×C4#21712yes—
130C22×C4#21711yescharacteristic
131C22×C4C22×Dic311yescharacteristic
132C22×C4C2×#6712yes—
133C2×C8—124no—
134C4⋊C4—24no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—84no—
140C22⋊C4—24no—
141C22⋊C4—42no—
142C42—22no—
143C42—22no—
144C42—22no—
145C42C2×C3⋊C814yes—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—22no—
149C22×C3—42no—
150C22×C3—42no—
151C22×C3—41no—
152C22×C3—41no—
153C22×C3C2×C22⋊C811yescharacteristic
154C22×C3C2×#9911yescharacteristic
155C22×C3C2×#10111yescharacteristic
156C3×C4—21no—
157C3×C4—21no—
158C3×C4—44no—
159C3×C4—42no—
160C3×C4—42no—
161C3×C4—82no—
162C3×C4#32414yes—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23#83012yes—
168C23C2×#21711yescharacteristic
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4#83012yes—
184C2×C4#83412yes—
185C2×C4#83612yes—
186C2×C4C22×C3⋊C811yescharacteristic
187C2×C4C2×#20811yescharacteristic
188C8—244no—
189C2×C3—21no—
190C2×C3—21no—
191C2×C3—42no—
192C2×C3C2×#32411yescharacteristic
193C2×C3#212911yescharacteristic
194C2×C3#213411yescharacteristic
195C22—21no—
196C22—21no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22C2×#83011yescharacteristic
203C22C2×#83411yescharacteristic
204C22C2×#83611yescharacteristic
205C4—21no—
206C4—21no—
207C4—44no—
208C4—42no—
209C4—42no—
210C4—82no—
211C4#452814yes—
212C3#1144911yescharacteristic, complemented, Sylow
213C2—21no—
214C2—21no—
215C2—42no—
216C2C2×#452811yescharacteristic
217C2#7186211yescharacteristic
218C2#7187111yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC2×C3×C4C22×C8
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2317C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2317D3
hypocenterC3#11449
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. #67632C2semidirect
22. C3×#117C8semidirect
78. #117C3⋊C8semidirect
212. C3#11449semidirect

Maximal subgroups

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

subgroupindex
C3×#23172
#715712
#676322
#114493