Finite Group Databasebeta

← #10697861

Subgroups — #10697861

Order 768.

566 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 113 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×#2862C211yesmaximal, characteristic, complemented
2#83213C212yesmaximal
3#71774C212yesmaximal, complemented
4#69395C211yesmaximal, characteristic
5#67611C211yesmaximal, characteristic, complemented
6#10287—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2212yes—
8#5477C2211yescharacteristic
9C3×#391—24no—
10C3×#382—22no—
11C3×C43C2211yescharacteristic
12#5227—24no—
13#5085—22no—
14#5083—24no—
15#5079—22no—
16C2×#805—22no—
17#4925C2212yes—
18#4924C2211yescharacteristic
19#4546—24no—
20#3206—32no—
21#2862D311yescharacteristic, complemented
22#1962—31no—
23#1548—32no—
24#1276—31no—
25C22×C3×Q8D414yes—
26C2×#235—21no—
27C3×C4⋊Q8—22no—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—22no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C42—22no—
34C2×C3×C42D412yes—
35C2×C3×C42C2311yescharacteristic
36C2×#215—28no—
37#930—22no—
38C2×#208—21no—
39#875—21no—
40#874D411yescharacteristic
41C4×D12—22no—
42#869D411yescharacteristic
43C3×#95—28no—
44#835—28no—
45#830—22no—
46#805—44no—
47C4⋊(C3⋊C8)—22no—
48C4⋊(C3⋊C8)D414yes—
49C4×C3⋊C8—22no—
50#532—31no—
51C2×C4⋊Q8C2×D312yes—
52#492—62no—
53#487—62no—
54#461—64no—
55#448—64no—
56#431—32no—
57#423—31no—
58C2×#104—62no—
59#391—24no—
60#382—22no—
61C43C2×D311yescharacteristic
62#333—64no—
63C2×C3×Q8—24no—
64C2×C3×Q8—24no—
65C2×C3×Q8—48no—
66C22×C3×C4—22no—
67C22×C3×C4—24no—
68C22×C3×C4C2×D412yes—
69C22×C3×C4C2×D411yescharacteristic
70C2×#67—41no—
71#235—42no—
72#235—42no—
73C2×D12—22no—
74C2×D12—21no—
75C2×C4×D3—41no—
76C2×Dic6—21no—
77C3×C4⋊C4—42no—
78C3×C4⋊C4—42no—
79C3×C4⋊C4—42no—
80C3×C4⋊C4—42no—
81C3×C42—22no—
82C3×C42—44no—
83C3×C42SD1614yescomplemented
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D412yes—
86C3×C42#4111yescharacteristic
87#215—88no—
88(C2×D3)⋊C4—42no—
89C4⋊Dic3—22no—
90#208—42no—
91C2×C3⋊C8—24no—
92C2×C3⋊C8—41no—
93C2×#41—61no—
94C22×Q8#6714yes—
95C2×SD16—68no—
96C2×M16—61no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—31no—
102#127—31no—
103#124—61no—
104#121—62no—
105C4×D4—62no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C42—22no—
109C2×C42#6712yes—
110C2×C42C22×D311yescharacteristic
111C4⋊C8—34no—
112C4⋊C8—62no—
113#104—124no—
114#103—68no—
115C22⋊C8—62no—
116C4×C8—62no—
117#95—28no—
118C23×C3#12011yescharacteristic
119C22×D3—41no—
120C3×Q8—48no—
121C3×Q8—84no—
122C2×C3×C4—24no—
123C2×C3×C4—24no—
124C2×C3×C4—44no—
125C2×C3×C4—48no—
126C2×C3×C4—44no—
127C2×C3×C4#12012yes—
128C2×C3×C4#12111yescharacteristic
129C2×C3×C4#12112yes—
130C2×C3×C4#12711yescharacteristic
131C2×C3×C4C2×SD1612yes—
132C2×C3×C4#13712yes—
133#67—82no—
134C2×Dic3—41no—
135D12—44no—
136D12—42no—
137C4×D3—81no—
138C4×D3—81no—
139Dic6—42no—
140C3⋊C8—44no—
141C3⋊C8—81no—
142#41—122no—
143#41—122no—
144C2×Q8—24no—
145C2×Q8—24no—
146C2×Q8—48no—
147C2×Q8—61no—
148C2×D4—62no—
149C2×D4—61no—
150C2×D4—121no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—121no—
154C22×C4C2×#6712yes—
155C22×C4C2×#6711yescharacteristic
156SD16—248no—
157M16—122no—
158C2×C8—64no—
159C2×C8—121no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—62no—
165C22⋊C4—122no—
166C42—22no—
167C42—44no—
168C42#21514yescomplemented
169C42D3×D411yescharacteristic
170C42D3×D412yes—
171C42#23711yescharacteristic
172C22×C3—22no—
173C22×C3#39211yescharacteristic
174C22×C3#44812yes—
175C2×D3—82no—
176C2×D3—81no—
177C3×C4—22no—
178C3×C4—44no—
179C3×C4—84no—
180C3×C4#45514yes—
181C3×C4#46112yes—
182C3×C4#49211yescharacteristic
183C3×C4#49711yescharacteristic
184Dic3—81no—
185C23—121no—
186C23#95311yescharacteristic
187Q8—48no—
188Q8—84no—
189Q8—122no—
190D4—124no—
191D4—122no—
192D4—242no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—44no—
196C2×C4—48no—
197C2×C4—44no—
198C2×C4—121no—
199C2×C4—241no—
200C2×C4—241no—
201C2×C4#93712yes—
202C2×C4#93811yescharacteristic
203C2×C4#93912yes—
204C2×C4#94011yescharacteristic
205C2×C4C2×#21512yes—
206C2×C4#94212yes—
207C8—124no—
208C8—241no—
209C2×C3—22no—
210C2×C3#200412yes—
211C2×C3#201311yescharacteristic
212D3—161no—
213C22—22no—
214C22—242no—
215C22—241no—
216C22#522311yescharacteristic
217C22#522712yes—
218C4—22no—
219C4—44no—
220C4—84no—
221C4—241no—
222C4#506114yes—
223C4#508312yes—
224C4#508511yescharacteristic
225C4#508611yescharacteristic
226C3#1028711yescharacteristic, complemented, Sylow
227C2—22no—
228C2—481no—
229C2#7173112yes—
230C2#7179111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2862C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2862D3
hypocenterC3#10287
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
1. C3×#2862C2semidirect
3. #71774C2semidirect
5. #67611C2semidirect
21. #2862D3semidirect
83. C3×C42SD16semidirect
168. C42#215semidirect
226. C3#10287semidirect

Maximal subgroups

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

subgroupindex
C3×#28622
#832132
#717742
#693952
#676112
#102873