Finite Group Databasebeta

← #10704581

Subgroups — #10704581

Order 768.

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

normal subgroups only

This group has 510 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×#2310C211yesmaximal, characteristic
2#71572C212yesmaximal
3#67643C214yesmaximal, complemented
4#11510—31nomaximal, Sylow
5C23×C3×Q8C411yescharacteristic
6C2×C3×#117C411yescharacteristic
7C2×C3×#117C2211yescharacteristic
8C3×#391—22no—
9C3×#391C412yes—
10C3×#391C2212yes—
11C3×#381—22no—
12#4926—22no—
13#4528C2214yes—
14#4525—22no—
15#2310D311yescharacteristic
16#1464—32no—
17#1319—34no—
18C22×C3×Q8—24no—
19C22×C3×Q8—21no—
20C22×C3×Q8—42no—
21C22×C3×Q8—42no—
22C22×C3×Q8C2×C412yes—
23C23×C3×C4—21no—
24C23×C3×C4C2×C411yescharacteristic
25C3×#117—42no—
26C3×#117—42no—
27C2×C3×C4⋊C4—21no—
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—
35C3×#95—24no—
36#830—24no—
37C4⋊(C3⋊C8)—42no—
38#803—42no—
39C23×Q8Dic311yescharacteristic
40C2×#117Dic311yescharacteristic
41C2×#117C2×D311yescharacteristic
42#432—62no—
43#391—22no—
44#391Dic312yes—
45#391C2×D312yes—
46#381—22no—
47#324—34no—
48#323—62no—
49C24×C3C22⋊C411yescharacteristic
50C2×C3×Q8—24no—
51C2×C3×Q8—22no—
52C2×C3×Q8—416no—
53C2×C3×Q8—42no—
54C2×C3×Q8—42no—
55C2×C3×Q8—42no—
56C2×C3×Q8—42no—
57C2×C3×Q8M1614yes—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4—24no—
62C22×C3×C4—22no—
63C22×C3×C4—22no—
64C22×C3×C4—42no—
65C22×C3×C4—42no—
66C22×C3×C4C22⋊C412yes—
67C22×C3×C4C22⋊C411yescharacteristic
68C22×C3×C4C22×C411yescharacteristic
69C22×C3×C4C2×D412yes—
70C3×C4⋊C4—41no—
71C3×C4⋊C4—41no—
72C3×C4⋊C4—42no—
73C3×C22⋊C4—42no—
74C3×C22⋊C4—42no—
75C3×C42—22no—
76C3×C42—22no—
77C3×C42—42no—
78C2×C3⋊C8—44no—
79C22×Q8—24no—
80C22×Q8—21no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8C2×Dic312yes—
84C23×C4—21no—
85C23×C4C2×Dic311yescharacteristic
86#117—42no—
87#117—42no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2×Dic311yescharacteristic
90C2×C4⋊C4C22×D311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4#6712yes—
93C2×C42C2×Dic312yes—
94C2×C42#6712yes—
95C4⋊C8—122no—
96C22⋊C8—64no—
97#97—122no—
98#95—24no—
99#95—24no—
100C23×C3—21no—
101C23×C3—21no—
102C23×C3—41no—
103C23×C3—41no—
104C23×C3#10012yes—
105C23×C3C2×C22⋊C411yescharacteristic
106C3×Q8—44no—
107C3×Q8—48no—
108C3×Q8—44no—
109C2×C3×C4—28no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—24no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—48no—
118C2×C3×C4—42no—
119C2×C3×C4—42no—
120C2×C3×C4—42no—
121C2×C3×C4—42no—
122C2×C3×C4—42no—
123C2×C3×C4—42no—
124C2×C3×C4—42no—
125C2×C3×C4#9912yes—
126C2×C3×C4C2×M1612yes—
127C3⋊C8—84no—
128C24#21711yescharacteristic
129C2×Q8—24no—
130C2×Q8—22no—
131C2×Q8—416no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8#20814yes—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4#21712yes—
146C22×C4#21711yescharacteristic
147C22×C4C22×Dic311yescharacteristic
148C22×C4C2×#6712yes—
149C2×C8—124no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C42—22no—
156C42—22no—
157C42—42no—
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—22no—
161C22×C3—42no—
162C22×C3—42no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3#40711yescharacteristic
166C22×C3C2×#9911yescharacteristic
167C22×C3C2×#10011yescharacteristic
168C3×C4—22no—
169C3×C4—22no—
170C3×C4—42no—
171C3×C4—42no—
172C3×C4—44no—
173C3×C4—82no—
174C23—21no—
175C23—21no—
176C23—41no—
177C23—41no—
178C23#83312yes—
179C23C2×#21711yescharacteristic
180Q8—44no—
181Q8—48no—
182Q8—44no—
183C2×C4—28no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—48no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4#83412yes—
200C2×C4C2×#20812yes—
201C8—244no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—42no—
205C2×C3#146211yescharacteristic
206C2×C3#212911yescharacteristic
207C2×C3#213511yescharacteristic
208C22—21no—
209C22—21no—
210C22—22no—
211C22—42no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22#520711yescharacteristic
216C22C2×#83311yescharacteristic
217C22C2×#83411yescharacteristic
218C4—22no—
219C4—22no—
220C4—44no—
221C4—42no—
222C4—42no—
223C4—82no—
224C3#1151011yescharacteristic, complemented, Sylow
225C2—21no—
226C2—21no—
227C2—42no—
228C2#7153911yescharacteristic
229C2#7187111yescharacteristic
230C2#7187311yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2310C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2310D3
hypocenterC3#11510
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
3. #67643C2semidirect
224. C3#11510semidirect

Maximal subgroups

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

subgroupindex
C3×#23102
#715722
#676432
#115103