Finite Group Databasebeta

← #10697984

Subgroups — #10697984

Order 768.

618 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 159 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×#5078C212yesmaximal
2C2×#4699C211yesmaximal, characteristic
3C3×#1956C211yesmaximal, characteristic
4#69582C212yesmaximal
5C3⋊#1273C211yesmaximal, characteristic
6#10309—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C22×Dic12—22no—
9C2×#869C2212yes—
10C2×C3×#103—22no—
11C2×C3×C4×C8C2211yescharacteristic
12C3×#384C2211yescharacteristic
13#5078—24no—
14#5078—24no—
15C2×#807—24no—
16C2×C4×C3⋊C8C2212yes—
17#4699—24no—
18#4529—24no—
19C2×#500—32no—
20C2×#494—31no—
21#1990—32no—
22#1956D311yescharacteristic
23#1273—31no—
24C22×C3×Q8—21no—
25C22×Dic6—22no—
26C22×C3×C8D412yes—
27C3×C4⋊Q8—22no—
28C3×C4⋊Q8—21no—
29C3×C4⋊Q8D412yes—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—21no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C42C2311yescharacteristic
34C2×C4⋊Dic3—22no—
35C2×C4⋊Dic3—22no—
36C22×C3⋊C8D414yes—
37C2×Dic12—28no—
38C2×Dic12—44no—
39C2×Dic12—44no—
40#869—22no—
41#869—22no—
42#869—22no—
43#869D414yes—
44C3×#103—44no—
45C3×C4×C8—22no—
46#807—216no—
47C4×C3⋊C8—22no—
48C4×C3⋊C8—22no—
49C22×Q16—62no—
50C2×C4⋊Q8—32no—
51C2×C4⋊Q8C2×D311yescharacteristic
52#500—64no—
53#500—64no—
54#494—64no—
55C2×#107—64no—
56C2×#103—22no—
57C2×C4×C8—32no—
58C2×C4×C8C2×D311yescharacteristic
59#384C2×D311yescharacteristic
60#340—64no—
61C2×C3×Q8—22no—
62C2×C3×Q8—41no—
63C2×C3×Q8—42no—
64C2×C3×Q8—42no—
65C22×C3×C4—41no—
66C22×C3×C4—41no—
67C22×C3×C4C2×D412yes—
68C22×C3×C4C2×D411yescharacteristic
69C22×Dic3—42no—
70C2×Dic6—24no—
71C2×Dic6—44no—
72C2×Dic6—44no—
73C2×Dic6—42no—
74C2×C3×C8—22no—
75C2×C3×C8—22no—
76C2×C3×C8D814yes—
77C3×C4⋊C4—24no—
78C3×C4⋊C4—42no—
79C3×C4⋊C4—42no—
80C3×C4⋊C4—44no—
81C3×C42C2×D411yescharacteristic
82C3×C42C2×D412yes—
83C3×C42#4111yescharacteristic
84C4⋊Dic3—24no—
85C4⋊Dic3—24no—
86C4⋊Dic3—42no—
87C4⋊Dic3—42no—
88C2×C3⋊C8—24no—
89C2×C3⋊C8—24no—
90C2×C3⋊C8Q1618yescomplemented
91Dic12—416no—
92C22×Q8—21no—
93C22×Q8—62no—
94C2×Q16—68no—
95C2×Q16—124no—
96C2×Q16—124no—
97C22×C8—34no—
98C22×C8#6712yes—
99C4⋊Q8—21no—
100C4⋊Q8—22no—
101C4⋊Q8—34no—
102C4⋊Q8—62no—
103C4⋊Q8—62no—
104C4⋊Q8—62no—
105C4⋊Q8D1212yes—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—62no—
111C2×C42C22×D311yescharacteristic
112#107—616no—
113#103—44no—
114C4×C8—22no—
115C4×C8—62no—
116C4×C8—62no—
117C23×C3#12711yescharacteristic
118C3×Q8—44no—
119C3×Q8—82no—
120C2×C3×C4—42no—
121C2×C3×C4—81no—
122C2×C3×C4—81no—
123C2×C3×C4—81no—
124C2×C3×C4—82no—
125C2×C3×C4#12012yes—
126C2×C3×C4#12011yescharacteristic
127C2×C3×C4#12112yes—
128C2×C3×C4#12111yescharacteristic
129C2×C3×C4C2×D812yes—
130C2×C3×C4C2×Q1614yes—
131C2×C3×C4AGL(1,ℤ/8)12yes—
132C2×C3×C4#13714yes—
133C2×Dic3—44no—
134C2×Dic3—82no—
135C2×Dic3—82no—
136Dic6—48no—
137Dic6—84no—
138C3×C8—24no—
139C3⋊C8—28no—
140C2×Q8—22no—
141C2×Q8—41no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—64no—
145C2×Q8—122no—
146C2×Q8—124no—
147C2×Q8—124no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—122no—
151C22×C4D3×D412yes—
152C22×C4C2×#6711yescharacteristic
153Q16—1216no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—38no—
157C2×C8—64no—
158C2×C8—64no—
159C2×C8#21314yes—
160C4⋊C4—24no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4—42no—
164C4⋊C4—64no—
165C4⋊C4—64no—
166C4⋊C4—122no—
167C4⋊C4—122no—
168C42C2×D1211yescharacteristic
169C42#23511yescharacteristic
170C42D3×D412yes—
171C22×C3#39211yescharacteristic
172C22×C3#49311yescharacteristic
173C22×C3#49412yes—
174C22×C3#49611yescharacteristic
175C22×C3#49712yes—
176C3×C4—82no—
177C3×C4—161no—
178C3×C4#44712yes—
179C3×C4#45114yes—
180C3×C4#45912yes—
181C3×C4#46214yes—
182Dic3—84no—
183C23#94011yescharacteristic
184Q8—44no—
185Q8—82no—
186Q8—128no—
187Q8—244no—
188C2×C4—42no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4—81no—
193C2×C4—124no—
194C2×C4—242no—
195C2×C4—242no—
196C2×C4#88212yes—
197C2×C4#88412yes—
198C2×C4#91514yes—
199C2×C4D3×Q1614yes—
200C2×C4#93011yescharacteristic
201C2×C4C2×#21312yes—
202C2×C4#93711yescharacteristic
203C2×C4#94912yes—
204C8—24no—
205C8—68no—
206C2×C3#171311yescharacteristic
207C2×C3#172211yescharacteristic
208C2×C3#172312yes—
209C2×C3#201111yescharacteristic
210C2×C3#201612yes—
211C22#496011yescharacteristic
212C22#505811yescharacteristic
213C22#507311yescharacteristic
214C22#516712yes—
215C22#518312yes—
216C4—82no—
217C4—161no—
218C4—244no—
219C4#479514yes—
220C4#480914yes—
221C4#501512yes—
222C4#503612yes—
223C3#1030911yescharacteristic, complemented, Sylow
224C2#6965811yescharacteristic
225C2#7028312yes—
226C2#7122612yes—
227C2#7126311yescharacteristic
228C2#7129811yescharacteristic
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1956C2
socle, prime radicalC23×C3#127
hypercenter#1956D3
hypocenterC3#10309
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
90. 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×#50782
C2×#46992
C3×#19562
#695822
C3⋊#12732
#103093