Finite Group Databasebeta

← #10694570

Subgroups — #10694570

Order 768.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#78463C211yesmaximal, characteristic
2C3×#1450C211yesmaximal, characteristic, complemented
3#70793C212yesmaximal, complemented
4#70790C212yesmaximal
5#69763C211yesmaximal, characteristic, complemented
6#9757—31nomaximal, Sylow
7#5477C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#431C2212yes—
10C2×C3×C4×C8C2211yescharacteristic
11#5113—22no—
12#4936C2212yes—
13#4697—21no—
14#4697—21no—
15#4694C412yescomplemented
16#4681C412yescomplemented
17C3×#322—21no—
18C3×C82—21no—
19#4551—22no—
20#4470—24no—
21#4467—24no—
22#3131—31no—
23#1835—31no—
24#1518—32no—
25#1517—32no—
26#1450D311yescharacteristic, complemented
27C2×#235—21no—
28C22×C3×C8D412yes—
29C2×C3×C42C2311yescharacteristic
30#930—22no—
31#926—21no—
32C2×C4⋊Dic3—22no—
33#924—22no—
34#875—21no—
35#874C2×C411yescharacteristic
36C4×D12—22no—
37#870—21no—
38#869C2×C411yescharacteristic
39#869C2×C412yes—
40C4×Dic6—22no—
41C3×C4⋊C8—22no—
42C3×C22⋊C8D414yes—
43C3×C4×C8—22no—
44C3×C4×C8C2×C412yes—
45C3×C4×C8D411yescharacteristic
46C3×C4×C8D411yescharacteristic
47C3⋊#95—22no—
48C3⋊#95—22no—
49#821—22no—
50#821—22no—
51#816—22no—
52#816—22no—
53#533—31no—
54#532—31no—
55#488—61no—
56#488—61no—
57#487—32no—
58#486—32no—
59#431C2×D312yes—
60#416—62no—
61C2×C4×C8C2×D311yescharacteristic
62#382—32no—
63#340—62no—
64#328—64no—
65#327—64no—
66#322—21no—
67C82—21no—
68C22×C3×C4C2×D412yes—
69C22×C3×C4C2×D411yescharacteristic
70C2×#67—41no—
71C22×Dic3—42no—
72#235—44no—
73C2×D12—21no—
74C2×D12—22no—
75C2×C4×D3—41no—
76C2×Dic6—21no—
77C2×Dic6—22no—
78C2×C3×C8—24no—
79C2×C3×C8—24no—
80C2×C3×C8C22⋊C412yes—
81C2×C3×C8C22⋊C412yes—
82C3×C42C22×C411yescharacteristic
83C3×C42C2×D411yescharacteristic
84C3×C42#4112yes—
85#217—41no—
86(C2×D3)⋊C4—42no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90C4⋊Dic3—44no—
91Dic3⋊C4—42no—
92C4×Dic3—41no—
93C2×#41—61no—
94C22×C8D1212yes—
95C4⋊Q8—31no—
96C4⋊Q8—32no—
97#127—31no—
98#125—61no—
99#124—61no—
100C22⋊Q8—62no—
101#121—62no—
102C4×Q8—62no—
103C4×D4—62no—
104#117—61no—
105C2×C4⋊C4—62no—
106C2×C42C22×D311yescharacteristic
107C4⋊C8—22no—
108#103—62no—
109#103—62no—
110#102—62no—
111#102—62no—
112C22⋊C8D1214yes—
113C4×C8—22no—
114C4×C8C4×D312yes—
115C4×C8#6711yescharacteristic
116C4×C8#6711yescharacteristic
117#95—62no—
118#95—62no—
119C23×C3#12711yescharacteristic
120C22×D3—41no—
121C2×C3×C4—42no—
122C2×C3×C4C2×C22⋊C411yescharacteristic
123C2×C3×C4C4×D412yes—
124C2×C3×C4#12112yes—
125C2×C3×C4#12411yescharacteristic
126C2×C3×C4#13514yes—
127#67—82no—
128C2×Dic3—42no—
129C2×Dic3—41no—
130C2×Dic3—84no—
131C2×Dic3—82no—
132D12—42no—
133D12—82no—
134C4×D3—82no—
135Dic6—42no—
136Dic6—82no—
137C3×C8—22no—
138C3×C8—22no—
139C3×C8—44no—
140#41—124no—
141C2×Q8—62no—
142C2×Q8—61no—
143C2×D4—62no—
144C2×D4—61no—
145C2×D4—121no—
146C22×C4—121no—
147C22×C4—122no—
148C22×C4C2×D1212yes—
149C22×C4C2×D1211yescharacteristic
150C2×C8—24no—
151C2×C8—24no—
152C2×C8(C2×D3)⋊C412yes—
153C2×C8(C2×D3)⋊C412yes—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—122no—
158C4⋊C4—124no—
159C22⋊C4—121no—
160C22⋊C4—122no—
161C42—121no—
162C42C2×C4×D311yescharacteristic
163C42#23512yes—
164C42C2×#6711yescharacteristic
165C22×C3—22no—
166C22×C3#39011yescharacteristic
167C22×C3#49512yes—
168C2×D3—81no—
169C2×D3—82no—
170C3×C4—22no—
171C3×C4—22no—
172C3×C4#41612yes—
173C3×C4#48812yes—
174Dic3—81no—
175Dic3—82no—
176Dic3—162no—
177C23—121no—
178C23#87411yescharacteristic
179Q8—122no—
180Q8—242no—
181D4—122no—
182D4—242no—
183D4—242no—
184C2×C4—42no—
185C2×C4—122no—
186C2×C4—121no—
187C2×C4—242no—
188C2×C4—244no—
189C2×C4—242no—
190C2×C4C4×D1212yes—
191C2×C4#87511yescharacteristic
192C2×C4#90414yes—
193C2×C4C2×(C2×D3)⋊C411yescharacteristic
194C2×C4#93012yes—
195C8—22no—
196C8—22no—
197C8—44no—
198C2×C3—22no—
199C2×C3#195811yescharacteristic
200C2×C3#197612yes—
201D3—161no—
202C22—22no—
203C22—242no—
204C22—241no—
205C22#469612yes—
206C22#493911yescharacteristic
207C4—22no—
208C4—22no—
209C4—242no—
210C4—241no—
211C4—482no—
212C4#469712yes—
213C4#511312yes—
214C3#975711yescharacteristic, complemented, Sylow
215C2—22no—
216C2—481no—
217C2#6977311yescharacteristic
218C2#6989712yes—
subgroupG/H
centerC22#4939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1450C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1450D3
hypocenterC3#9757
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. C3×#1450C2semidirect
3. #70793C2semidirect
5. #69763C2semidirect
15. #4694C4semidirect
16. #4681C4semidirect
26. #1450D3semidirect
214. C3#9757semidirect

Maximal subgroups

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

subgroupindex
#784632
C3×#14502
#707932
#707902
#697632
#97573