Finite Group Databasebeta

← #10697851

Subgroups — #10697851

Order 768.

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

normal subgroups only

This group has 542 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
1#79269C212yesmaximal
2#73308C211yesmaximal, characteristic, complemented
3#70780C212yesmaximal, complemented
4#69395C211yesmaximal, characteristic
5C3×#1276C211yesmaximal, characteristic, complemented
6#10285—31nomaximal, Sylow
7#5477C2211yescharacteristic
8C2×#869C2212yes—
9C3×#431C2212yes—
10C3×C43C2211yescharacteristic
11#4936—22no—
12#4929—24no—
13C2×#805—22no—
14#4924C2211yescharacteristic
15#4842—24no—
16#4735—24no—
17#4694—22no—
18#4680—22no—
19#4482—24no—
20#3227—32no—
21#2862—31no—
22#1962—31no—
23#1549—32no—
24#1276D311yescharacteristic, complemented
25C2×#235—21no—
26C22×Dic6—22no—
27C2×C3×C42—22no—
28C2×C3×C42D412yes—
29C2×C3×C42C2311yescharacteristic
30#930—22no—
31C2×C4⋊Dic3—22no—
32C2×C4⋊Dic3D414yes—
33C2×#208—21no—
34#875—21no—
35#874D411yescharacteristic
36C4×D12—22no—
37#869—22no—
38#869—22no—
39#869—22no—
40#869—22no—
41#869D411yescharacteristic
42C3×C4⋊C8D414yes—
43C3×C22⋊C8—22no—
44C3×C4×C8—22no—
45C3⋊#95—28no—
46#821—28no—
47#817—28no—
48#805—44no—
49C4⋊(C3⋊C8)—22no—
50#532—31no—
51C2×C4⋊Q8—32no—
52#498—62no—
53#486—62no—
54#481—64no—
55#476—64no—
56#431C2×D312yes—
57#423—31no—
58C2×#104—62no—
59#391—64no—
60#382—62no—
61C43C2×D311yescharacteristic
62#332—64no—
63C22×C3×C4—22no—
64C22×C3×C4C2×D411yescharacteristic
65C22×C3×C4C2×D412yes—
66C2×#67—41no—
67C22×Dic3—24no—
68#235—42no—
69#235—42no—
70C2×D12—22no—
71C2×D12—21no—
72C2×C4×D3—41no—
73C2×Dic6—21no—
74C2×Dic6—48no—
75C2×Dic6—42no—
76C2×Dic6—42no—
77C2×C3×C8—24no—
78C3×C42—22no—
79C3×C42—44no—
80C3×C42SD1614yes—
81C3×C42C2×D412yes—
82C3×C42C2×D411yescharacteristic
83C3×C42#4111yescharacteristic
84(C2×D3)⋊C4—42no—
85C4⋊Dic3—22no—
86C4⋊Dic3—24no—
87C4⋊Dic3—24no—
88C4⋊Dic3—42no—
89C4⋊Dic3—42no—
90#208—42no—
91C2×C3⋊C8—41no—
92C2×#41—61no—
93C22×Q8—62no—
94C2×M16—61no—
95C4⋊Q8—31no—
96C4⋊Q8—62no—
97C4⋊Q8—62no—
98C4⋊Q8—62no—
99C4⋊Q8—62no—
100#127—31no—
101#124—61no—
102#121—62no—
103C4×D4—62no—
104C2×C4⋊C4—34no—
105C2×C4⋊C4—62no—
106C2×C42—22no—
107C2×C42D1212yes—
108C2×C42C22×D311yescharacteristic
109#106—68no—
110C4⋊C8—62no—
111C4⋊C8D1214yes—
112#104—124no—
113#102—68no—
114C22⋊C8—22no—
115C4×C8—22no—
116#95—68no—
117C23×C3#12011yescharacteristic
118C22×D3—41no—
119C2×C3×C4—24no—
120C2×C3×C4—24no—
121C2×C3×C4—44no—
122C2×C3×C4#12012yes—
123C2×C3×C4#12112yes—
124C2×C3×C4#12111yescharacteristic
125C2×C3×C4#12711yescharacteristic
126C2×C3×C4C2×SD1612yes—
127C2×C3×C4#13712yes—
128#67—82no—
129C2×Dic3—48no—
130C2×Dic3—44no—
131C2×Dic3—41no—
132D12—44no—
133D12—42no—
134C4×D3—81no—
135C4×D3—81no—
136Dic6—42no—
137Dic6—84no—
138Dic6—84no—
139C3×C8—44no—
140C3⋊C8—81no—
141#41—122no—
142#41—122no—
143C2×Q8—61no—
144C2×Q8—128no—
145C2×Q8—122no—
146C2×Q8—122no—
147C2×D4—62no—
148C2×D4—61no—
149C2×D4—121no—
150C22×C4—22no—
151C22×C4—64no—
152C22×C4—121no—
153C22×C4C2×D1211yescharacteristic
154C22×C4D3×D412yes—
155M16—122no—
156C2×C8—24no—
157C2×C8—121no—
158C4⋊C4—64no—
159C4⋊C4—62no—
160C4⋊C4—64no—
161C4⋊C4—122no—
162C4⋊C4—122no—
163C22⋊C4—122no—
164C42—22no—
165C42—44no—
166C42#20414yes—
167C42C2×D1212yes—
168C42D3×D411yescharacteristic
169C42#23911yescharacteristic
170C22×C3—22no—
171C22×C3#39211yescharacteristic
172C22×C3#44812yes—
173C2×D3—82no—
174C2×D3—81no—
175C3×C4—22no—
176C3×C4—44no—
177C3×C4#45414yes—
178C3×C4#46112yes—
179C3×C4#49211yescharacteristic
180C3×C4#49711yescharacteristic
181Dic3—84no—
182Dic3—81no—
183C23—121no—
184C23#88211yescharacteristic
185Q8—122no—
186Q8—244no—
187Q8—244no—
188D4—124no—
189D4—122no—
190D4—242no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—128no—
195C2×C4—121no—
196C2×C4—124no—
197C2×C4—241no—
198C2×C4—241no—
199C2×C4#87411yescharacteristic
200C2×C4#88212yes—
201C2×C4#89512yes—
202C2×C4#89511yescharacteristic
203C2×C4C2×#20412yes—
204C2×C4#90912yes—
205C8—44no—
206C8—241no—
207C2×C3—22no—
208C2×C3#200312yes—
209C2×C3#201311yescharacteristic
210D3—161no—
211C22—22no—
212C22—241no—
213C22—242no—
214C22#467111yescharacteristic
215C22#473812yes—
216C4—22no—
217C4—44no—
218C4—241no—
219C4—244no—
220C4#469311yescharacteristic
221C4#471511yescharacteristic
222C4#482714yes—
223C4#484112yes—
224C3#1028511yescharacteristic, complemented, Sylow
225C2—22no—
226C2—481no—
227C2#6868711yescharacteristic
228C2#6876712yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1276D3
hypocenterC3#10285
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #73308C2semidirect
3. #70780C2semidirect
5. C3×#1276C2semidirect
24. #1276D3semidirect
224. C3#10285semidirect

Maximal subgroups

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

subgroupindex
#792692
#733082
#707802
#693952
C3×#12762
#102853