Finite Group Databasebeta

← #10688113

Subgroups — #10688113

Order 768.

486 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×D4×C3⋊C8C211yesmaximal, characteristic
2C3×#1800C211yesmaximal, characteristic
3#71617C211yesmaximal, characteristic
4#70491C212yesmaximal, complemented
5#67711C212yesmaximal, complemented
6#8616—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C23×C3⋊C8C412yes—
9C3×#423C2212yes—
10C2×C3×#100—21no—
11C2×#833—21no—
12C2×#830C412yes—
13D4×C3⋊C8—22no—
14D4×C3⋊C8—22no—
15#4924C2212yes—
16C2×C4⋊(C3⋊C8)C2211yescharacteristic
17C2×C4×C3⋊C8C2211yescharacteristic
18C3×#343—21no—
19#4530—21no—
20#4479—22no—
21#4479—22no—
22C2×C8×D4—31no—
23#1800—31no—
24#1800D311yescharacteristic
25#1567—32no—
26#1292—32no—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4C2×C412yes—
29C2×C3×M16—22no—
30C3×C4×D4—22no—
31C3×C4×D4—22no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C22⋊C4Q812yes—
34C2×C3×C42C2311yescharacteristic
35C2×#208—22no—
36C22×C3⋊C8—22no—
37C22×C3⋊C8—22no—
38C22×C3⋊C8—44no—
39C22×C3⋊C8C2×C411yescharacteristic
40C22×C3⋊C8C2×C412yes—
41C22×C3⋊C8C2×C411yescharacteristic
42C3×C4⋊C8—22no—
43C3×C4⋊C8—22no—
44C3×#100—42no—
45#833—42no—
46#830—24no—
47C4⋊(C3⋊C8)—22no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)—22no—
50C4×C3⋊C8D412yes—
51C4×C3⋊C8D412yes—
52C23×C8—32no—
53C2×C4×D4C2×D311yescharacteristic
54C8×D4—62no—
55C8×D4—62no—
56#423—32no—
57#423C2×D312yes—
58C2×C4⋊C8—31no—
59C2×#100—21no—
60C2×#100—61no—
61C2×C22⋊C8—32no—
62C2×C4×C8—31no—
63#343—21no—
64#343—61no—
65#335—62no—
66#335—62no—
67C24×C3C4⋊C412yes—
68C2×C3×D4—22no—
69C2×C3×D4—22no—
70C2×C3×D4D812yes—
71C2×C3×D4Q1612yes—
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—42no—
75C22×C3×C4—42no—
76C22×C3×C4C4⋊C412yes—
77C22×C3×C4C22×C411yescharacteristic
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×Q811yescharacteristic
80C3×M16—42no—
81C3×M16—42no—
82C2×C3×C8—42no—
83C3×C4⋊C4—22no—
84C3×C22⋊C4—24no—
85C3×C42C2×D412yes—
86C3×C42#4112yes—
87#208—42no—
88#208—42no—
89C2×C3⋊C8—22no—
90C2×C3⋊C8—22no—
91C2×C3⋊C8—41no—
92C2×C3⋊C8—44no—
93C2×C3⋊C8—44no—
94C2×C3⋊C8—42no—
95C2×C3⋊C8—42no—
96C2×C3⋊C8C22⋊C412yes—
97C2×C3⋊C8C22⋊C411yescharacteristic
98C2×C3⋊C8C22⋊C411yescharacteristic
99C22×D4#6711yescharacteristic
100C23×C4C4×D312yes—
101C2×M16—22no—
102C2×M16—62no—
103C22×C8—31no—
104C22×C8—31no—
105C22×C8—32no—
106C22×C8—62no—
107C22×C8—62no—
108C22×C8—124no—
109C4×D4—22no—
110C4×D4—22no—
111C2×C4⋊C4#6711yescharacteristic
112C2×C22⋊C4Dic612yes—
113C2×C42C22×D311yescharacteristic
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—62no—
117C4⋊C8—62no—
118C4⋊C8—62no—
119#100—42no—
120#100—122no—
121C22⋊C8—64no—
122C4×C8—32no—
123C4×C8—32no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—42no—
127C23×C3—42no—
128C23×C3#10714yes—
129C23×C3C2×C4⋊C411yescharacteristic
130C3×D4—24no—
131C3×D4—24no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—41no—
136C2×C3×C4—44no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
140C2×C3×C4C2×C22⋊C411yescharacteristic
141C2×C3×C4#11711yescharacteristic
142C2×C3×C4#12112yes—
143C2×C3×C4C22⋊Q812yes—
144C2×C3×C4C2×D811yescharacteristic
145C2×C3×C4C2×Q1611yescharacteristic
146C2×C3×C4#13512yes—
147C3×C8—82no—
148C3⋊C8—21no—
149C3⋊C8—22no—
150C3⋊C8—42no—
151C3⋊C8—82no—
152C3⋊C8#10012yescomplemented
153C24Dic3⋊C412yes—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4#21312yes—
157C2×D4#21612yes—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4Dic3⋊C412yes—
163C22×C4C2×Dic611yescharacteristic
164C22×C4C2×C4×D311yescharacteristic
165C22×C4C2×#6711yescharacteristic
166M16—42no—
167M16—42no—
168M16—122no—
169M16—122no—
170C2×C8—31no—
171C2×C8—32no—
172C2×C8—31no—
173C2×C8—42no—
174C2×C8—62no—
175C2×C8—62no—
176C2×C8—124no—
177C2×C8—121no—
178C2×C8—124no—
179C2×C8—122no—
180C2×C8—122no—
181C4⋊C4—22no—
182C22⋊C4—24no—
183C42D3×D412yes—
184C42#23712yes—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—24no—
188C22×C3—44no—
189C22×C3—44no—
190C22×C3—42no—
191C22×C3—42no—
192C22×C3#38011yescharacteristic
193C22×C3C2×#10711yescharacteristic
194C22×C3#42711yescharacteristic
195C3×C4—21no—
196C3×C4—22no—
197C3×C4—42no—
198C3×C4C2×#10011yescharacteristic
199C3×C4#41311yescharacteristic
200C3×C4#46612yes—
201C3×C4#46712yes—
202C23—22no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23#80714yes—
207C23C2×Dic3⋊C411yescharacteristic
208D4—24no—
209D4—24no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4#87711yescharacteristic
219C2×C4#87812yes—
220C2×C4D3×C22⋊C411yescharacteristic
221C2×C4C2×#21311yescharacteristic
222C2×C4#93912yes—
223C2×C4C2×#21611yescharacteristic
224C2×C4#95012yes—
225C8—32no—
226C8—61no—
227C8—62no—
228C8—82no—
229C8—122no—
230C8—242no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—44no—
234C2×C3#180011yescharacteristic
235C2×C3#180811yescharacteristic
236C2×C3#181311yescharacteristic
237C22—21no—
238C22—21no—
239C22—24no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22#494211yescharacteristic
245C22C2×#80711yescharacteristic
246C22#499811yescharacteristic
247C4—21no—
248C4—22no—
249C4—42no—
250C4D3×#10011yescharacteristic
251C4#474511yescharacteristic
252C4#503812yes—
253C4#505112yes—
254C3#861611yescharacteristic, complemented, Sylow
255C2—21no—
256C2—21no—
257C2—44no—
258C2#6951211yescharacteristic
259C2#7006311yescharacteristic
260C2#7013411yescharacteristic
subgroupG/H
centerC22#4942
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1800C2
socleC22×C3#380
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1800D3
hypocenterC3#8616
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
4. #70491C2semidirect
5. #67711C2semidirect
152. C3⋊C8#100semidirect
254. C3#8616semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊C82
C3×#18002
#716172
#704912
#677112
#86163