Finite Group Databasebeta

← #10698001

Subgroups — #10698001

Order 768.

638 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 153 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×C3×#498C211yesmaximal, characteristic, complemented
2C2×#5088C212yesmaximal
3#71785C212yesmaximal, complemented
4#69772C211yesmaximal, characteristic
5C3⋊#1273C211yesmaximal, characteristic, complemented
6#10312—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2212yes—
8C2×#874C2211yescharacteristic
9C3×#498—24no—
10C2×C3×#106—22no—
11C2×C3×C4×C8C2211yescharacteristic
12C2×#835—24no—
13C2×#821—22no—
14#5088—24no—
15#5088—24no—
16C2×#809—24no—
17#4935C2211yescharacteristic
18C2×C4×C3⋊C8C2212yes—
19C2×#498D311yescharacteristic, complemented
20C2×#486—32no—
21#1957—31no—
22#1956—32no—
23#1273—31no—
24C22×C3×Q8—22no—
25C22×D12—22no—
26C22×D12—21no—
27C22×C3×C8D412yes—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—22no—
31C3×C4⋊Q8D414yes—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C42C2311yescharacteristic
35C2×C4⋊Dic3—22no—
36C2×C4⋊Dic3—21no—
37C22×C3⋊C8D414yes—
38#874—22no—
39#874—21no—
40#874D412yes—
41C3×#106—28no—
42C3×C4×C8—22no—
43#835—48no—
44#821—44no—
45#809—216no—
46C4×C3⋊C8—22no—
47C4×C3⋊C8—22no—
48C2×C4⋊Q8C2×D312yes—
49C2×#127—31no—
50#498—24no—
51#486—64no—
52#486—64no—
53C2×#106—22no—
54C2×#103—64no—
55C2×#102—64no—
56C2×#102—62no—
57C2×C4×C8—32no—
58C2×C4×C8C2×D311yescharacteristic
59#384—31no—
60C23×D3—41no—
61C2×C3×Q8—24no—
62C2×C3×Q8—42no—
63C2×C3×Q8—44no—
64C2×C3×Q8—44no—
65C22×C3×C4—42no—
66C22×C3×C4C2×D412yes—
67C22×C3×C4C2×D411yescharacteristic
68C22×Dic3—41no—
69C2×D12—24no—
70C2×D12—22no—
71C2×D12—42no—
72C2×D12—44no—
73C2×D12—44no—
74C2×D12—42no—
75C2×D12—41no—
76C2×D12—42no—
77C2×C3×C8—22no—
78C2×C3×C8—22no—
79C2×C3×C8SD1614yes—
80C3×C4⋊C4—24no—
81C3×C4⋊C4—24no—
82C3×C4⋊C4—42no—
83C3×C4⋊C4—42no—
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D412yes—
86C3×C42#4111yescharacteristic
87C4⋊Dic3—42no—
88C4⋊Dic3—44no—
89C2×C3⋊C8—24no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8—24no—
92C22×Q8—22no—
93C22×D4—62no—
94C22×D4—61no—
95C22×C8—34no—
96C22×C8#6712yes—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8#6714yes—
101#127—32no—
102#127—61no—
103#127—62no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—61no—
108C2×C42C22×D311yescharacteristic
109#106—28no—
110#103—128no—
111#102—616no—
112#102—124no—
113C4×C8—22no—
114C4×C8—62no—
115C4×C8—62no—
116C23×C3#12711yescharacteristic
117C22×D3—42no—
118C22×D3—81no—
119C22×D3—82no—
120C22×D3—82no—
121C22×D3—81no—
122C3×Q8—48no—
123C3×Q8—84no—
124C2×C3×C4—44no—
125C2×C3×C4—82no—
126C2×C3×C4—82no—
127C2×C3×C4#12012yes—
128C2×C3×C4#12011yescharacteristic
129C2×C3×C4#12111yescharacteristic
130C2×C3×C4#12112yes—
131C2×C3×C4C2×SD1612yes—
132C2×C3×C4#13514yes—
133C2×C3×C4AGL(1,ℤ/8)14yes—
134C2×C3×C4#13712yes—
135C2×Dic3—81no—
136C2×Dic3—82no—
137D12—48no—
138D12—44no—
139D12—82no—
140D12—84no—
141C3×C8—24no—
142C3⋊C8—44no—
143C24—121no—
144C2×Q8—24no—
145C2×Q8—42no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×D4—64no—
149C2×D4—62no—
150C2×D4—124no—
151C2×D4—124no—
152C2×D4—122no—
153C2×D4—122no—
154C2×D4—122no—
155C2×D4—121no—
156C22×C4—42no—
157C22×C4—121no—
158C22×C4D3×D412yes—
159C22×C4C2×#6711yescharacteristic
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—64no—
163C2×C8—64no—
164C2×C8—64no—
165C2×C8#21514yes—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—124no—
171C4⋊C4—122no—
172C42D3×D411yescharacteristic
173C42#23711yescharacteristic
174C42C2×#6712yes—
175C22×C3#39211yescharacteristic
176C22×C3#49211yescharacteristic
177C22×C3#49512yes—
178C22×C3#49612yes—
179C22×C3#49711yescharacteristic
180C2×D3—84no—
181C2×D3—82no—
182C2×D3—161no—
183C2×D3—161no—
184C2×D3—162no—
185C3×C4—84no—
186C3×C4#44812yes—
187C3×C4#44914yes—
188C3×C4#46112yes—
189C3×C4#46314yes—
190Dic3—161no—
191C23—122no—
192C23—242no—
193C23—242no—
194C23—241no—
195C23—241no—
196C23#94011yescharacteristic
197Q8—48no—
198Q8—84no—
199D4—128no—
200D4—124no—
201D4—244no—
202D4—242no—
203C2×C4—44no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—242no—
207C2×C4—241no—
208C2×C4#91414yes—
209C2×C4#91614yes—
210C2×C4#93712yes—
211C2×C4#93811yescharacteristic
212C2×C4#93912yes—
213C2×C4C2×#21512yes—
214C2×C4#94212yes—
215C2×C4#95311yescharacteristic
216C8—24no—
217C8—124no—
218C2×C3#170811yescharacteristic
219C2×C3#171511yescharacteristic
220C2×C3#172012yes—
221C2×C3#201212yes—
222C2×C3#201311yescharacteristic
223D3—162no—
224C22—242no—
225C22—244no—
226C22—481no—
227C22—481no—
228C22—482no—
229C22#508511yescharacteristic
230C22#508611yescharacteristic
231C22#516812yes—
232C22#517612yes—
233C22#522311yescharacteristic
234C4—84no—
235C4—481no—
236C4#508312yes—
237C4#516514yes—
238C4#517214yes—
239C4#522712yes—
240C3#1031211yescharacteristic, complemented, Sylow
241C2—482no—
242C2#7133411yescharacteristic
243C2#7133611yescharacteristic
244C2#7134012yes—
245C2#7179111yescharacteristic
246C2#7205512yes—
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#498C2
socleC23×C3#127
prime radicalC2×#874C22
hypercenterC2×#498D3
hypocenterC3#10312
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
1. C2×C3×#498C2semidirect
3. #71785C2semidirect
5. C3⋊#1273C2semidirect
19. C2×#498D3semidirect
240. C3#10312semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4982
C2×#50882
#717852
#697722
C3⋊#12732
#103123