Finite Group Databasebeta

← #10698749

Subgroups — #10698749

Order 768.

478 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 125 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#74343C211yesmaximal, characteristic
2#70485C212yesmaximal
3C2×#4484C212yesmaximal
4#69577C211yesmaximal, characteristic
5C3×#1296C211yesmaximal, characteristic
6#10439—31nomaximal, Sylow
7C2×C4×Dic6C2212yes—
8C2×C3×C4⋊C8C2212yes—
9C3×#384C2211yescharacteristic
10C2×#816—22no—
11#4984—22no—
12#4981—22no—
13#4973—22no—
14C2×#807—22no—
15#4935C2211yescharacteristic
16C2×C4×C3⋊C8C2211yescharacteristic
17#4551—22no—
18#4484—24no—
19#2582—31no—
20#1845—31no—
21#1803—32no—
22C2×Q8⋊C8—32no—
23#1296D311yescharacteristic
24C22×Dic6D412yes—
25C22×C3×C8—22no—
26C2×C3×C4⋊C4—22no—
27C2×C3×C4⋊C4D412yes—
28C2×C3×C42C2311yescharacteristic
29C2×C4⋊Dic3—21no—
30C2×C4⋊Dic3—22no—
31C2×C4⋊Dic3—22no—
32C2×C4⋊Dic3D412yes—
33C2×Dic3⋊C4—22no—
34C2×C4×Dic3—22no—
35C22×C3⋊C8—21no—
36C4×Dic6—22no—
37C4×Dic6—22no—
38C4×Dic6—22no—
39C4×Dic6—22no—
40C3×C4⋊C8—22no—
41C3×C4⋊C8—22no—
42C3⋊#95—22no—
43C3⋊#95—24no—
44#816—44no—
45#807—44no—
46C4×C3⋊C8—21no—
47C4×C3⋊C8D412yes—
48C2×C4×Q8—32no—
49C2×#107—62no—
50C2×C4⋊C8C2×D312yes—
51C2×#103—62no—
52C2×C4×C8—31no—
53#398—62no—
54#391—62no—
55#389—62no—
56#384—31no—
57#384C2×D311yescharacteristic
58#340—62no—
59Q8⋊C8—64no—
60C22×C3×C4—22no—
61C22×C3×C4C2×D411yescharacteristic
62C22×C3×C4C2×D412yes—
63C22×Dic3—22no—
64C22×Dic3—22no—
65C22×Dic3—41no—
66C2×Dic6—22no—
67C2×Dic6—22no—
68C2×Dic6—22no—
69C2×Dic6—22no—
70C2×Dic6—22no—
71C2×Dic6Q1614yes—
72C2×C3×C8—42no—
73C2×C3×C8—42no—
74C2×C3×C8—42no—
75C3×C4⋊C4—42no—
76C3×C4⋊C4—42no—
77C3×C4⋊C4—42no—
78C3×C42C2×D411yescharacteristic
79C3×C42#4112yes—
80C3×C42#4111yescharacteristic
81C4⋊Dic3—22no—
82C4⋊Dic3—22no—
83C4⋊Dic3—42no—
84C4⋊Dic3—42no—
85C4⋊Dic3—42no—
86C4⋊Dic3—82no—
87Dic3⋊C4—42no—
88Dic3⋊C4—42no—
89C4×Dic3—42no—
90C4×Dic3—42no—
91C2×C3⋊C8—22no—
92C2×C3⋊C8—22no—
93C2×C3⋊C8—22no—
94C22×Q8—32no—
95C22×C8—22no—
96C22×C8—61no—
97C4×Q8—62no—
98C4×Q8—62no—
99C4×Q8—62no—
100C4×Q8—62no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—62no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4—62no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4D1212yes—
108C2×C42—62no—
109C2×C42C22×D311yescharacteristic
110#107—124no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113#103—124no—
114C4×C8—32no—
115C4×C8—61no—
116#95—64no—
117#95—62no—
118C23×C3#12011yescharacteristic
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—44no—
122C2×C3×C4—42no—
123C2×C3×C4#12112yes—
124C2×C3×C4#12111yescharacteristic
125C2×C3×C4#12312yes—
126C2×C3×C4#12411yescharacteristic
127C2×C3×C4C2×Q1612yes—
128C2×C3×C4#13512yes—
129C2×C3×C4AGL(1,ℤ/8)12yes—
130C2×C3×C4#13712yes—
131C2×Dic3—24no—
132C2×Dic3—42no—
133C2×Dic3—42no—
134C2×Dic3—42no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—82no—
138C2×Dic3—81no—
139Dic6—24no—
140Dic6—24no—
141Dic6—42no—
142Dic6—42no—
143C3×C8—82no—
144C3⋊C8—24no—
145C2×Q8—34no—
146C2×Q8—62no—
147C2×Q8—62no—
148C2×Q8—62no—
149C2×Q8—62no—
150C2×Q8—62no—
151C22×C4—22no—
152C22×C4—62no—
153C22×C4—62no—
154C22×C4—121no—
155C22×C4C2×D1211yescharacteristic
156C22×C4D3×D412yes—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—62no—
161C2×C8—62no—
162C2×C8—62no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—122no—
169C4⋊C4—122no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C4⋊C4—122no—
173C4⋊C4—242no—
174C42—122no—
175C42—122no—
176C42#23512yes—
177C42D3×D411yescharacteristic
178C42#23911yescharacteristic
179C22×C3#39411yescharacteristic
180C22×C3#44912yes—
181C22×C3#45112yes—
182C22×C3#45311yescharacteristic
183C22×C3#45611yescharacteristic
184C3×C4—21no—
185C3×C4—21no—
186C3×C4—82no—
187C3×C4#46712yes—
188C3×C4#46812yes—
189C3×C4#48412yes—
190C3×C4#48512yes—
191Dic3—44no—
192Dic3—82no—
193Dic3—161no—
194C23#88211yescharacteristic
195Q8—64no—
196Q8—64no—
197Q8—122no—
198Q8—122no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—64no—
204C2×C4—122no—
205C2×C4—122no—
206C2×C4—122no—
207C2×C4—122no—
208C2×C4—122no—
209C2×C4—241no—
210C2×C4—242no—
211C2×C4#87511yescharacteristic
212C2×C4#88412yes—
213C2×C4#89412yes—
214C2×C4#89511yescharacteristic
215C2×C4#91112yes—
216C2×C4#91512yes—
217C2×C4#91612yes—
218C2×C4D3×Q1612yes—
219C8—64no—
220C8—82no—
221C2×C3#162011yescharacteristic
222C2×C3#163311yescharacteristic
223C2×C3#164812yes—
224C2×C3#203612yes—
225C2×C3#203911yescharacteristic
226C22#467211yescharacteristic
227C22#474711yescharacteristic
228C22#475211yescharacteristic
229C22#478112yes—
230C22#480912yes—
231C4—21no—
232C4—21no—
233C4—82no—
234C4—124no—
235C4—242no—
236C4—481no—
237C4#478012yes—
238C4#481312yes—
239C4#486812yes—
240C4#488412yes—
241C3#1043911yescharacteristic, complemented, Sylow
242C2#6840211yescharacteristic
243C2#6846012yes—
244C2#6850511yescharacteristic
245C2#6882512yes—
246C2#6897611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socle, prime radicalC23×C3#120
hypercenter#1296D3
hypocenterC3#10439
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
241. C3#10439semidirect

Maximal subgroups

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

subgroupindex
#743432
#704852
C2×#44842
#695772
C3×#12962
#104393