Finite Group Databasebeta

← #10696059

Subgroups — #10696059

Order 768.

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

normal subgroups only

This group has 528 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#79310C212yesmaximal
2#73397C211yesmaximal, characteristic
3#70770C212yesmaximal
4#69441C211yesmaximal, characteristic
5C3×#1274C211yesmaximal, characteristic
6#10017—31nomaximal, Sylow
7#5470C2211yescharacteristic
8C2×C4×Dic6C2212yes—
9C3×#431C2212yes—
10C3×#376C2211yescharacteristic
11C4×C4⋊Dic3—22no—
12#4924C2211yescharacteristic
13#4850—22no—
14#4849—22no—
15#4739—22no—
16#4722—22no—
17C4×Dic12—24no—
18#4661—22no—
19#4656—22no—
20#4523—22no—
21#4456—24no—
22#3104—32no—
23#2399—31no—
24#1995—31no—
25#1487—32no—
26#1274D311yescharacteristic
27C22×Dic6D412yes—
28C2×C3×C42D412yes—
29C2×C3×C42C2311yescharacteristic
30#926—21no—
31C2×C4⋊Dic3—24no—
32C2×C4⋊Dic3D412yes—
33#924—22no—
34C2×Dic3⋊C4—22no—
35C2×C4×Dic3—22no—
36C2×#208—21no—
37C2×Dic12—24no—
38#870—21no—
39#869D412yes—
40C4×Dic6—22no—
41C4×Dic6—22no—
42C4×Dic6—22no—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C3×C4⋊C8—22no—
46C3×C22⋊C8—22no—
47C3×C4×C8D414yes—
48C3×#95—22no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51#818—24no—
52#816—24no—
53#816—24no—
54C4⋊(C3⋊C8)—22no—
55#533—31no—
56C2×C4×Q8—32no—
57#484—62no—
58#477—62no—
59#462—62no—
60#451—62no—
61C4×Q16—64no—
62#431C2×D312yes—
63#423—31no—
64#398—62no—
65#395—62no—
66C4×C4⋊C4—62no—
67#376C2×D311yescharacteristic
68#339—62no—
69Q8⋊C8—64no—
70C22×C3×C4—22no—
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4C2×D412yes—
73C22×Dic3—22no—
74C22×Dic3—22no—
75C2×Dic6—22no—
76C2×Dic6—22no—
77C2×Dic6—22no—
78C2×Dic6—44no—
79C2×C3×C8—24no—
80C3×C42—22no—
81C3×C42Q1614yescomplemented
82C3×C42C2×D412yes—
83C3×C42C2×D411yescharacteristic
84C3×C42#4111yescharacteristic
85#217—41no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90C4⋊Dic3—48no—
91Dic3⋊C4—42no—
92Dic3⋊C4—42no—
93Dic3⋊C4—42no—
94C4×Dic3—42no—
95C4×Dic3—42no—
96C4×Dic3—41no—
97#208—42no—
98C2×C3⋊C8—41no—
99Dic12—48no—
100C22×Q8—32no—
101C2×Q16—64no—
102C2×M16—61no—
103C4⋊Q8—32no—
104#125—61no—
105C22⋊Q8—62no—
106C4×Q8—62no—
107C4×Q8—62no—
108C4×Q8—62no—
109C4×Q8—62no—
110C4×Q8—62no—
111#117—61no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—64no—
114C2×C4⋊C4—62no—
115C2×C42—62no—
116C2×C42D1212yes—
117C2×C42C22×D311yescharacteristic
118#107—64no—
119C4⋊C8—22no—
120C4⋊C8—62no—
121#103—64no—
122#103—64no—
123C22⋊C8—22no—
124C4×C8D1214yes—
125#95—22no—
126#95—62no—
127#95—62no—
128C23×C3#12011yescharacteristic
129C2×C3×C4—24no—
130C2×C3×C4—42no—
131C2×C3×C4—44no—
132C2×C3×C4#12012yes—
133C2×C3×C4#12112yes—
134C2×C3×C4#12111yescharacteristic
135C2×C3×C4#12711yescharacteristic
136C2×C3×C4C2×Q1612yes—
137C2×C3×C4#13712yes—
138C2×Dic3—24no—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—44no—
142C2×Dic3—42no—
143C2×Dic3—42no—
144Dic6—24no—
145Dic6—24no—
146Dic6—42no—
147Dic6—42no—
148Dic6—44no—
149C3×C8—44no—
150C3⋊C8—81no—
151C2×Q8—62no—
152C2×Q8—62no—
153C2×Q8—62no—
154C2×Q8—124no—
155C22×C4—22no—
156C22×C4—62no—
157C22×C4—62no—
158C22×C4C2×D1211yescharacteristic
159C22×C4D3×D412yes—
160Q16—128no—
161M16—122no—
162C2×C8—24no—
163C2×C8—121no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—128no—
169C4⋊C4—122no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C22⋊C4—121no—
173C42—22no—
174C42—122no—
175C42—121no—
176C42—122no—
177C42Dic1214yescomplemented
178C42C2×D1212yes—
179C42D3×D411yescharacteristic
180C42#23911yescharacteristic
181C22×C3—22no—
182C22×C3#39211yescharacteristic
183C22×C3#45112yes—
184C3×C4—22no—
185C3×C4—22no—
186C3×C4—44no—
187C3×C4#46212yes—
188C3×C4#49411yescharacteristic
189C3×C4#49711yescharacteristic
190Dic3—44no—
191Dic3—82no—
192Dic3—82no—
193C23#88211yescharacteristic
194Q8—64no—
195Q8—64no—
196Q8—122no—
197Q8—122no—
198Q8—124no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—64no—
203C2×C4—122no—
204C2×C4—122no—
205C2×C4—124no—
206C2×C4—122no—
207C2×C4—122no—
208C2×C4#87411yescharacteristic
209C2×C4#88212yes—
210C2×C4#89512yes—
211C2×C4#89511yescharacteristic
212C2×C4C2×Dic1212yes—
213C2×C4#90912yes—
214C8—44no—
215C8—241no—
216C2×C3—22no—
217C2×C3#201012yes—
218C2×C3#201611yescharacteristic
219C22—22no—
220C22#467111yescharacteristic
221C22#473912yes—
222C4—22no—
223C4—22no—
224C4—44no—
225C4—124no—
226C4—242no—
227C4—242no—
228C4#469911yescharacteristic
229C4#471511yescharacteristic
230C4#484912yes—
231C3#1001711yescharacteristic, complemented, Sylow
232C2—22no—
233C2#6868811yescharacteristic
234C2#6870312yes—
subgroupG/H
centerC2×C4#874
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1274D3
hypocenterC3#10017
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
81. C3×C42Q16semidirect
177. C42Dic12semidirect
231. C3#10017semidirect

Maximal subgroups

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

subgroupindex
#793102
#733972
#707702
#694412
C3×#12742
#100173