Finite Group Databasebeta

← #10697481

Subgroups — #10697481

Order 768.

466 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 105 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#79271C212yesmaximal
2#78479C211yesmaximal, characteristic
3#70793C212yesmaximal
4#69822C211yesmaximal, characteristic
5C3×#1281C211yesmaximal, characteristic
6#10227—31nomaximal, Sylow
7#5470C2212yes—
8C2×#869C2211yescharacteristic
9C3×#431C2212yes—
10C2×C3×#97C2211yescharacteristic
11#5106—22no—
12C2×#816—22no—
13#4936C2211yescharacteristic
14#4851—24no—
15#4721—24no—
16#4702—22no—
17#4702—22no—
18#4681—22no—
19#4680—22no—
20#4467—24no—
21#3229—32no—
22#3162—31no—
23#1960—31no—
24#1518—32no—
25#1281D311yescharacteristic
26C22×Dic6—21no—
27C22×C3×C8D412yes—
28C2×C3×C42C2311yescharacteristic
29#926—22no—
30C2×C4⋊Dic3—22no—
31C2×C4⋊Dic3—21no—
32#924—22no—
33#924—22no—
34#870—22no—
35#869—22no—
36#869—21no—
37#869D412yes—
38#869D412yes—
39#869D412yes—
40C4×Dic6—22no—
41C4×Dic6—22no—
42C3×C4⋊C8—22no—
43C3×C22⋊C8D414yes—
44C3×#97—22no—
45C3×C4×C8—22no—
46C3⋊#95—22no—
47#818—24no—
48#817—28no—
49#817—22no—
50#817—22no—
51#816—28no—
52#816—44no—
53#533—32no—
54C2×C4⋊Q8—31no—
55#501—62no—
56#501—62no—
57#498—62no—
58#487—62no—
59#483—64no—
60#475—64no—
61#431C2×D312yes—
62#427—62no—
63C2×#103—62no—
64C2×#97C2×D311yescharacteristic
65#382—31no—
66#328—64no—
67C22×C3×C4C2×D412yes—
68C22×C3×C4C2×D411yescharacteristic
69C22×Dic3—41no—
70C22×Dic3—41no—
71C2×Dic6—22no—
72C2×Dic6—22no—
73C2×Dic6—22no—
74C2×Dic6—42no—
75C2×Dic6—42no—
76C2×Dic6—41no—
77C2×C3×C8—22no—
78C2×C3×C8—22no—
79C2×C3×C8—24no—
80C2×C3×C8SD1614yes—
81C3×C42C2×D412yes—
82C3×C42C2×D411yescharacteristic
83C3×C42#4111yescharacteristic
84#217—42no—
85C4⋊Dic3—24no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90C4⋊Dic3—42no—
91C4⋊Dic3—42no—
92Dic3⋊C4—42no—
93Dic3⋊C4—42no—
94C4×Dic3—42no—
95C22×Q8—61no—
96C22×C8D1212yes—
97C4⋊Q8—32no—
98C4⋊Q8—32no—
99C4⋊Q8—32no—
100C4⋊Q8—62no—
101C4⋊Q8—61no—
102#125—62no—
103C22⋊Q8—62no—
104C22⋊Q8—62no—
105C4×Q8—62no—
106C4×Q8—62no—
107#117—62no—
108C2×C4⋊C4—62no—
109C2×C4⋊C4—61no—
110C2×C42C22×D311yescharacteristic
111#107—64no—
112#106—62no—
113#106—62no—
114#106—68no—
115C4⋊C8—22no—
116#103—68no—
117#103—124no—
118C22⋊C8D1214yes—
119#97—22no—
120C4×C8—22no—
121#95—62no—
122C23×C3#12711yescharacteristic
123C2×C3×C4—42no—
124C2×C3×C4#12012yes—
125C2×C3×C4#12011yescharacteristic
126C2×C3×C4#12112yes—
127C2×C3×C4#12111yescharacteristic
128C2×C3×C4C2×SD1612yes—
129C2×C3×C4#13712yes—
130C2×Dic3—42no—
131C2×Dic3—42no—
132C2×Dic3—42no—
133C2×Dic3—81no—
134C2×Dic3—81no—
135C2×Dic3—82no—
136C2×Dic3—81no—
137Dic6—44no—
138Dic6—44no—
139Dic6—82no—
140Dic6—82no—
141C3×C8—24no—
142C3×C8—44no—
143C2×Q8—62no—
144C2×Q8—62no—
145C2×Q8—62no—
146C2×Q8—121no—
147C2×Q8—122no—
148C2×Q8—122no—
149C22×C4—121no—
150C22×C4—121no—
151C22×C4C2×D1212yes—
152C22×C4C2×D1211yescharacteristic
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—24no—
156C2×C8#20414yes—
157C4⋊C4—64no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C4⋊C4—122no—
165C4⋊C4—122no—
166C22⋊C4—122no—
167C42—122no—
168C42D3×D412yes—
169C42D3×D411yescharacteristic
170C42#23911yescharacteristic
171C22×C3—22no—
172C22×C3#39211yescharacteristic
173C22×C3#49211yescharacteristic
174C22×C3#49711yescharacteristic
175C3×C4—22no—
176C3×C4—22no—
177C3×C4#44812yes—
178C3×C4#46112yes—
179Dic3—82no—
180Dic3—82no—
181Dic3—82no—
182Dic3—161no—
183C23#87411yescharacteristic
184Q8—124no—
185Q8—124no—
186Q8—242no—
187Q8—242no—
188C2×C4—42no—
189C2×C4—122no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—241no—
193C2×C4—241no—
194C2×C4—242no—
195C2×C4—241no—
196C2×C4#88212yes—
197C2×C4#88211yescharacteristic
198C2×C4#89512yes—
199C2×C4#89511yescharacteristic
200C2×C4C2×#20412yes—
201C2×C4#90912yes—
202C8—24no—
203C8—44no—
204C2×C3—21no—
205C2×C3—21no—
206C2×C3#200912yes—
207C2×C3#201311yescharacteristic
208C22—22no—
209C22#467111yescharacteristic
210C22#469311yescharacteristic
211C22#471511yescharacteristic
212C4—22no—
213C4—22no—
214C4—242no—
215C4—242no—
216C4—242no—
217C4—481no—
218C4#473812yes—
219C4#484112yes—
220C3#1022711yescharacteristic, complemented, Sylow
221C2—21no—
222C2—21no—
223C2#6868711yescharacteristic
224C2#6903112yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1281C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1281D3
hypocenterC3#10227
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
220. C3#10227semidirect

Maximal subgroups

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

subgroupindex
#792712
#784792
#707932
#698222
C3×#12812
#102273