Finite Group Databasebeta

← #10697964

Subgroups — #10697964

Order 768.

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

normal subgroups only

This group has 670 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×#494C211yesmaximal, characteristic, complemented
2C2×#5088C212yesmaximal
3#71787C212yesmaximal, complemented
4#69772C211yesmaximal, characteristic
5C3⋊#1273C211yesmaximal, characteristic, complemented
6#10306—31nomaximal, Sylow
7C22×C3×Q16—22no—
8C2×C3×C4⋊Q8C2212yes—
9C2×#874C2211yescharacteristic
10C3×#494—24no—
11C2×C3×C4×C8C2211yescharacteristic
12C2×#821—22no—
13#5088—24no—
14#5088—24no—
15C2×#809—24no—
16#4935C2211yescharacteristic
17C2×C4×C3⋊C8C2212yes—
18#4529—24no—
19C2×#494D311yescharacteristic, complemented
20C2×#486—32no—
21#1990—32no—
22#1957—31no—
23#1273—31no—
24C22×C3×Q8—22no—
25C22×D12—22no—
26C22×D12—21no—
27C2×C3×Q16—28no—
28C2×C3×Q16—44no—
29C2×C3×Q16—44no—
30C22×C3×C8D412yes—
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×C4⋊Q8—22no—
34C3×C4⋊Q8D414yes—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C42C2311yescharacteristic
38C2×C4⋊Dic3—22no—
39C2×C4⋊Dic3—21no—
40C22×C3⋊C8D414yes—
41#874—22no—
42#874—21no—
43#874D412yes—
44C3×C4×C8—22no—
45#821—44no—
46#809—216no—
47C4×C3⋊C8—22no—
48C4×C3⋊C8—22no—
49C22×Q16—22no—
50C2×C4⋊Q8C2×D312yes—
51C2×#127—31no—
52#494—24no—
53#486—64no—
54#486—64no—
55C2×#102—64no—
56C2×#102—62no—
57C2×C4×C8—32no—
58C2×C4×C8C2×D311yescharacteristic
59#384—31no—
60#340—64no—
61C23×D3—41no—
62C2×C3×Q8—24no—
63C2×C3×Q8—42no—
64C2×C3×Q8—44no—
65C2×C3×Q8—44no—
66C22×C3×C4—42no—
67C22×C3×C4C2×D412yes—
68C22×C3×C4C2×D411yescharacteristic
69C22×Dic3—41no—
70C2×D12—24no—
71C2×D12—22no—
72C2×D12—42no—
73C2×D12—44no—
74C2×D12—42no—
75C2×D12—44no—
76C2×D12—41no—
77C2×D12—42no—
78C3×Q16—416no—
79C2×C3×C8—22no—
80C2×C3×C8—22no—
81C2×C3×C8D814yes—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—24no—
84C3×C4⋊C4—42no—
85C3×C4⋊C4—42no—
86C3×C42C2×D411yescharacteristic
87C3×C42C2×D412yes—
88C3×C42#4111yescharacteristic
89C4⋊Dic3—42no—
90C4⋊Dic3—44no—
91C2×C3⋊C8—24no—
92C2×C3⋊C8—24no—
93C2×C3⋊C8—24no—
94C22×Q8—22no—
95C22×D4—62no—
96C22×D4—61no—
97C2×Q16—28no—
98C2×Q16—44no—
99C2×Q16—44no—
100C22×C8—34no—
101C22×C8#6712yes—
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8#6714yes—
106#127—32no—
107#127—61no—
108#127—62no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—61no—
113C2×C42C22×D311yescharacteristic
114#102—616no—
115#102—124no—
116C4×C8—22no—
117C4×C8—62no—
118C4×C8—62no—
119C23×C3#12711yescharacteristic
120C22×D3—42no—
121C22×D3—81no—
122C22×D3—82no—
123C22×D3—81no—
124C22×D3—82no—
125C3×Q8—48no—
126C3×Q8—84no—
127C2×C3×C4—44no—
128C2×C3×C4—82no—
129C2×C3×C4—82no—
130C2×C3×C4#12012yes—
131C2×C3×C4#12011yescharacteristic
132C2×C3×C4#12111yescharacteristic
133C2×C3×C4#12112yes—
134C2×C3×C4C2×D812yes—
135C2×C3×C4#13514yes—
136C2×C3×C4AGL(1,ℤ/8)12yes—
137C2×C3×C4#13714yes—
138C2×Dic3—81no—
139C2×Dic3—82no—
140D12—48no—
141D12—44no—
142D12—82no—
143D12—84no—
144C3×C8—24no—
145C3⋊C8—44no—
146C24—121no—
147C2×Q8—24no—
148C2×Q8—42no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×D4—64no—
152C2×D4—62no—
153C2×D4—124no—
154C2×D4—124no—
155C2×D4—122no—
156C2×D4—122no—
157C2×D4—122no—
158C2×D4—121no—
159C22×C4—42no—
160C22×C4—121no—
161C22×C4D3×D412yes—
162C22×C4C2×#6711yescharacteristic
163Q16—416no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—64no—
167C2×C8—64no—
168C2×C8—64no—
169C2×C8#21314yes—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—124no—
175C4⋊C4—122no—
176C42D3×D411yescharacteristic
177C42#23711yescharacteristic
178C42C2×#6712yes—
179C22×C3#39211yescharacteristic
180C22×C3#49311yescharacteristic
181C22×C3#49512yes—
182C22×C3#49611yescharacteristic
183C22×C3#49712yes—
184C2×D3—82no—
185C2×D3—84no—
186C2×D3—161no—
187C2×D3—162no—
188C2×D3—161no—
189C3×C4—84no—
190C3×C4#44712yes—
191C3×C4#45214yes—
192C3×C4#45912yes—
193C3×C4#46414yes—
194Dic3—161no—
195C23—122no—
196C23—242no—
197C23—242no—
198C23—241no—
199C23—241no—
200C23#94011yescharacteristic
201Q8—48no—
202Q8—84no—
203D4—128no—
204D4—124no—
205D4—244no—
206D4—242no—
207C2×C4—44no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—242no—
211C2×C4—241no—
212C2×C4#91814yes—
213C2×C4#91914yes—
214C2×C4C2×#21312yes—
215C2×C4#93212yes—
216C2×C4#93712yes—
217C2×C4#93811yescharacteristic
218C2×C4#93912yes—
219C2×C4#95311yescharacteristic
220C8—24no—
221C8—124no—
222C2×C3#170911yescharacteristic
223C2×C3#172012yes—
224C2×C3#172211yescharacteristic
225C2×C3#201111yescharacteristic
226C2×C3#201512yes—
227D3—162no—
228C22—242no—
229C22—244no—
230C22—481no—
231C22—482no—
232C22—481no—
233C22#507311yescharacteristic
234C22#507411yescharacteristic
235C22#519012yes—
236C22#519112yes—
237C22#522311yescharacteristic
238C4—84no—
239C4—481no—
240C4#507212yes—
241C4#518514yes—
242C4#518714yes—
243C4#521712yes—
244C3#1030611yescharacteristic, complemented, Sylow
245C2—482no—
246C2#7129711yescharacteristic
247C2#7130012yes—
248C2#7130111yescharacteristic
249C2#7176711yescharacteristic
250C2#7207212yes—
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#494C2
socleC23×C3#127
prime radicalC2×#874C22
hypercenterC2×#494D3
hypocenterC3#10306
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
1. C2×C3×#494C2semidirect
3. #71787C2semidirect
5. C3⋊#1273C2semidirect
19. C2×#494D3semidirect
244. C3#10306semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4942
C2×#50882
#717872
#697722
C3⋊#12732
#103063