Finite Group Databasebeta

← #10697981

Subgroups — #10697981

Order 768.

618 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 161 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
2C2×#5078C212yesmaximal
3#71787C212yesmaximal
4#69771C211yesmaximal, characteristic
5C3⋊#1273C211yesmaximal, characteristic
6#10309—31nomaximal, Sylow
7C22×C3×Q16—22no—
8C2×C3×C4⋊Q8C2212yes—
9C2×#869C2211yescharacteristic
10C3×#494—24no—
11C2×C3×C4×C8C2211yescharacteristic
12C2×#816—22no—
13#5078—24no—
14#5078—24no—
15C2×#807—24no—
16#4935C2211yescharacteristic
17C2×C4×C3⋊C8C2212yes—
18#4529—24no—
19C2×#500—32no—
20C2×#494D311yescharacteristic
21#1990—32no—
22#1956—31no—
23#1273—31no—
24C22×C3×Q8—22no—
25C22×Dic6—21no—
26C2×C3×Q16—28no—
27C2×C3×Q16—44no—
28C2×C3×Q16—44no—
29C22×C3×C8D412yes—
30C3×C4⋊Q8—22no—
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×C4⋊Q8D414yes—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C42C2311yescharacteristic
37C2×C4⋊Dic3—22no—
38C2×C4⋊Dic3—22no—
39C2×C4⋊Dic3—21no—
40C22×C3⋊C8D414yes—
41#869—22no—
42#869—21no—
43#869D412yes—
44C3×C4×C8—22no—
45#816—44no—
46#807—216no—
47C4×C3⋊C8—22no—
48C4×C3⋊C8—22no—
49C22×Q16—22no—
50C2×C4⋊Q8—31no—
51C2×C4⋊Q8C2×D312yes—
52#500—64no—
53#500—64no—
54#494—24no—
55C2×#107—64no—
56C2×#103—62no—
57C2×C4×C8—32no—
58C2×C4×C8C2×D311yescharacteristic
59#384—31no—
60#340—64no—
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—
69C22×Dic3—41no—
70C2×Dic6—22no—
71C2×Dic6—42no—
72C2×Dic6—42no—
73C2×Dic6—41no—
74C3×Q16—416no—
75C2×C3×C8—22no—
76C2×C3×C8—22no—
77C2×C3×C8D814yes—
78C3×C4⋊C4—24no—
79C3×C4⋊C4—24no—
80C3×C4⋊C4—42no—
81C3×C4⋊C4—42no—
82C3×C42C2×D411yescharacteristic
83C3×C42C2×D412yes—
84C3×C42#4111yescharacteristic
85C4⋊Dic3—24no—
86C4⋊Dic3—42no—
87C4⋊Dic3—42no—
88C4⋊Dic3—44no—
89C2×C3⋊C8—24no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8Q1618yescomplemented
92C22×Q8—22no—
93C22×Q8—61no—
94C2×Q16—28no—
95C2×Q16—44no—
96C2×Q16—44no—
97C22×C8—34no—
98C22×C8#6712yes—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—32no—
103C4⋊Q8—61no—
104C4⋊Q8—62no—
105C4⋊Q8#6714yes—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—62no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—61no—
111C2×C42C22×D311yescharacteristic
112#107—616no—
113#103—124no—
114C4×C8—22no—
115C4×C8—62no—
116C4×C8—62no—
117C23×C3#12711yescharacteristic
118C3×Q8—48no—
119C3×Q8—84no—
120C2×C3×C4—44no—
121C2×C3×C4—82no—
122C2×C3×C4—82no—
123C2×C3×C4#12012yes—
124C2×C3×C4#12011yescharacteristic
125C2×C3×C4#12111yescharacteristic
126C2×C3×C4#12112yes—
127C2×C3×C4C2×D812yes—
128C2×C3×C4C2×Q1614yes—
129C2×C3×C4AGL(1,ℤ/8)12yes—
130C2×C3×C4#13714yes—
131C2×Dic3—42no—
132C2×Dic3—81no—
133C2×Dic3—81no—
134C2×Dic3—81no—
135C2×Dic3—82no—
136Dic6—44no—
137Dic6—82no—
138C3×C8—24no—
139C3⋊C8—28no—
140C2×Q8—24no—
141C2×Q8—42no—
142C2×Q8—44no—
143C2×Q8—44no—
144C2×Q8—62no—
145C2×Q8—121no—
146C2×Q8—122no—
147C2×Q8—122no—
148C22×C4—42no—
149C22×C4—121no—
150C22×C4—121no—
151C22×C4D3×D412yes—
152C22×C4C2×#6711yescharacteristic
153Q16—416no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—38no—
157C2×C8—64no—
158C2×C8—64no—
159C2×C8#21314yes—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—64no—
165C4⋊C4—122no—
166C4⋊C4—124no—
167C4⋊C4—122no—
168C42D3×D411yescharacteristic
169C42#23711yescharacteristic
170C42C2×#6712yes—
171C22×C3#39211yescharacteristic
172C22×C3#49311yescharacteristic
173C22×C3#49412yes—
174C22×C3#49611yescharacteristic
175C22×C3#49712yes—
176C3×C4—84no—
177C3×C4#44712yes—
178C3×C4#45114yes—
179C3×C4#45912yes—
180C3×C4#46214yes—
181Dic3—82no—
182Dic3—161no—
183C23#94011yescharacteristic
184Q8—48no—
185Q8—84no—
186Q8—124no—
187Q8—242no—
188C2×C4—44no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—122no—
192C2×C4—241no—
193C2×C4—241no—
194C2×C4—242no—
195C2×C4—241no—
196C2×C4D3×Q1614yes—
197C2×C4#91814yes—
198C2×C4C2×#21312yes—
199C2×C4#93212yes—
200C2×C4#93712yes—
201C2×C4#93811yescharacteristic
202C2×C4#93912yes—
203C2×C4#95311yescharacteristic
204C8—24no—
205C8—68no—
206C2×C3#171311yescharacteristic
207C2×C3#172211yescharacteristic
208C2×C3#172312yes—
209C2×C3#201111yescharacteristic
210C2×C3#201612yes—
211C22#507311yescharacteristic
212C22#507411yescharacteristic
213C22#518312yes—
214C22#519012yes—
215C22#522311yescharacteristic
216C4—84no—
217C4—242no—
218C4—481no—
219C4#507212yes—
220C4#518214yes—
221C4#518614yes—
222C4#521712yes—
223C3#1030911yescharacteristic, complemented, Sylow
224C2#7129811yescharacteristic
225C2#7130111yescharacteristic
226C2#7130212yes—
227C2#7176711yescharacteristic
228C2#7207312yes—
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#494C2
socle, prime radicalC23×C3#127
hypercenterC2×#494D3
hypocenterC3#10309
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
91. C2×C3⋊C8Q16semidirect
223. C3#10309semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4942
C2×#50782
#717872
#697712
C3⋊#12732
#103093