Finite Group Databasebeta

← #10697433

Subgroups — #10697433

Order 768.

542 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 103 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#83169C212yesmaximal
2#78479C211yesmaximal, characteristic, complemented
3C3×#1961C211yesmaximal, characteristic
4#69546C212yesmaximal, complemented
5#67661C211yesmaximal, characteristic, complemented
6#10220—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8#5470C2212yes—
9C2×C3×#102—22no—
10C2×C3×#97C2211yescharacteristic
11C3×#382C2211yescharacteristic
12#5218—24no—
13#5106—22no—
14#5076—22no—
15#5075—24no—
16#5070—22no—
17#4925C2212yes—
18#4702—22no—
19#4702—22no—
20#4452—24no—
21#3208—32no—
22#3162—31no—
23#1961D311yescharacteristic
24#1518—32no—
25#1281—31no—
26C22×C3×D4—21no—
27C22×C3×D4—22no—
28C22×C3×C8D412yes—
29C3×#127—22no—
30C3×#127—21no—
31C3×#127D412yes—
32C2×C3×C4⋊C4—21no—
33C2×C3×C42C2311yescharacteristic
34C2×#214—28no—
35#926—22no—
36#924—22no—
37#924—22no—
38#870—22no—
39#869D412yes—
40#869D412yes—
41C4×Dic6—22no—
42C4×Dic6—22no—
43C3×#102—44no—
44C3×#97—22no—
45C3×#95—22no—
46#832—28no—
47#830D414yes—
48#818—24no—
49#817—22no—
50#817—22no—
51C4⋊(C3⋊C8)—22no—
52C4×C3⋊C8—22no—
53#533—32no—
54C2×#127C2×D311yescharacteristic
55#501—62no—
56#501—62no—
57#492—62no—
58#486—62no—
59#460—64no—
60#450—64no—
61#431—32no—
62#427—62no—
63C2×#102—22no—
64C2×#97C2×D311yescharacteristic
65#382C2×D311yescharacteristic
66#328—64no—
67C24×C3—41no—
68C2×C3×D4—24no—
69C2×C3×D4—22no—
70C2×C3×D4—41no—
71C2×C3×D4—42no—
72C2×C3×D4—42no—
73C2×C3×D4—42no—
74C2×C3×D4—84no—
75C22×C3×C4—41no—
76C22×C3×C4C2×D412yes—
77C22×C3×C4C2×D411yescharacteristic
78C2×Dic6—22no—
79C2×Dic6—22no—
80C2×C3×C8—22no—
81C2×C3×C8—22no—
82C2×C3×C8SD1614yescomplemented
83C3×C4⋊C4—42no—
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D412yes—
86C3×C42#4111yescharacteristic
87#217—42no—
88#214—88no—
89C4⋊Dic3—22no—
90C4⋊Dic3—22no—
91C4⋊Dic3—22no—
92C4⋊Dic3—22no—
93Dic3⋊C4—42no—
94Dic3⋊C4—42no—
95C4×Dic3—42no—
96C2×C3⋊C8—24no—
97C22×D4—22no—
98C22×D4—21no—
99C2×SD16—68no—
100C22×C8#6712yes—
101C4⋊Q8—32no—
102C4⋊Q8—32no—
103#127—21no—
104#127—22no—
105#127D1212yes—
106#125—62no—
107C22⋊Q8—62no—
108C22⋊Q8—62no—
109C4×Q8—62no—
110C4×Q8—62no—
111#117—62no—
112C2×C4⋊C4—21no—
113C2×C42C22×D311yescharacteristic
114#107—64no—
115#106—62no—
116#106—62no—
117C4⋊C8—62no—
118#102—44no—
119#102—68no—
120C22⋊C8—34no—
121#97—22no—
122C4×C8—62no—
123#95—22no—
124C23×C3—42no—
125C23×C3—84no—
126C23×C3—81no—
127C23×C3—81no—
128C23×C3#12711yescharacteristic
129C3×D4—48no—
130C3×D4—44no—
131C3×D4—84no—
132C3×D4—82no—
133C2×C3×C4—42no—
134C2×C3×C4—82no—
135C2×C3×C4—81no—
136C2×C3×C4#12012yes—
137C2×C3×C4#12011yescharacteristic
138C2×C3×C4#12112yes—
139C2×C3×C4#12111yescharacteristic
140C2×C3×C4C2×SD1612yes—
141C2×C3×C4#13712yes—
142C2×Dic3—42no—
143C2×Dic3—42no—
144Dic6—44no—
145Dic6—82no—
146C3×C8—24no—
147C3⋊C8—44no—
148C24—41no—
149C2×Q8—62no—
150C2×Q8—62no—
151C2×D4—24no—
152C2×D4—22no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—41no—
156C2×D4—42no—
157C2×D4—84no—
158C22×C4—41no—
159C22×C4D3×D412yes—
160C22×C4C2×#6711yescharacteristic
161SD16—248no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—64no—
165C2×C8#21414yescomplemented
166C4⋊C4—42no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—122no—
172C4⋊C4—122no—
173C22⋊C4—122no—
174C42—122no—
175C42C2×D1211yescharacteristic
176C42#23511yescharacteristic
177C42D3×D412yes—
178C22×C3—22no—
179C22×C3—84no—
180C22×C3—82no—
181C22×C3—162no—
182C22×C3—162no—
183C22×C3#39211yescharacteristic
184C22×C3#49211yescharacteristic
185C22×C3#49711yescharacteristic
186C3×C4—22no—
187C3×C4—22no—
188C3×C4—161no—
189C3×C4#44812yes—
190C3×C4#46112yes—
191Dic3—82no—
192Dic3—82no—
193C23—42no—
194C23—84no—
195C23—81no—
196C23—81no—
197C23#94011yescharacteristic
198Q8—124no—
199Q8—242no—
200D4—48no—
201D4—44no—
202D4—84no—
203D4—82no—
204C2×C4—42no—
205C2×C4—82no—
206C2×C4—81no—
207C2×C4—122no—
208C2×C4—122no—
209C2×C4#88212yes—
210C2×C4#88412yes—
211C2×C4#93011yescharacteristic
212C2×C4C2×#21412yes—
213C2×C4#93711yescharacteristic
214C2×C4#95112yes—
215C8—24no—
216C8—124no—
217C2×C3—21no—
218C2×C3—21no—
219C2×C3—162no—
220C2×C3#200812yes—
221C2×C3#201311yescharacteristic
222C22—22no—
223C22—82no—
224C22—84no—
225C22—162no—
226C22—162no—
227C22#496011yescharacteristic
228C22#506011yescharacteristic
229C22#507611yescharacteristic
230C4—22no—
231C4—22no—
232C4—161no—
233C4—242no—
234C4—242no—
235C4#501912yes—
236C4#504012yes—
237C3#1022011yescharacteristic, complemented, Sylow
238C2—21no—
239C2—21no—
240C2—162no—
241C2#6966111yescharacteristic
242C2#7033112yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1961C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenter#1961D3
hypocenterC3#10220
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. #78479C2semidirect
4. #69546C2semidirect
5. #67661C2semidirect
82. C2×C3×C8SD16semidirect
165. C2×C8#214semidirect
237. C3#10220semidirect

Maximal subgroups

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

subgroupindex
#831692
#784792
C3×#19612
#695462
#676612
#102203