Finite Group Databasebeta

← #10690132

Subgroups — #10690132

Order 768.

458 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 119 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×C3×#491C211yesmaximal, characteristic
2#83027C211yesmaximal, characteristic
3#71788C212yesmaximal
4C2×#4543C211yesmaximal, characteristic
5#67659C212yesmaximal
6#9004—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×#125C2211yescharacteristic
9C3×#491—22no—
10C3×#491C414yescomplemented
11C2×C3×#103—22no—
12C2×C3×#97C2211yescharacteristic
13#4984—22no—
14#4980—22no—
15#4980—22no—
16#4936C2212yes—
17C2×#803C2212yes—
18#4543—22no—
19#4529—22no—
20#4529—22no—
21C2×#491D311yescharacteristic
22#2382—31no—
23#1993—32no—
24C2×#330—31no—
25#1295—32no—
26C22×C3×Q8—21no—
27C22×C3×C8—21no—
28C3×C4⋊Q8—21no—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8C2×C412yes—
31C3×#125—22no—
32C3×#125C2×C412yes—
33C3×#125D412yes—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4D412yes—
37C2×C3×C42C2311yescharacteristic
38C2×C4⋊Dic3—22no—
39C2×C4⋊Dic3D412yes—
40C2×C4⋊Dic3D412yes—
41C2×C4×Dic3—22no—
42C22×C3⋊C8—22no—
43C3×#103—28no—
44C3×#97—21no—
45C3×#97C2×C412yes—
46C3⋊#95—22no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50#803—22no—
51#803—22no—
52C2×C4⋊Q8C2×D311yescharacteristic
53C2×#125C2×D311yescharacteristic
54#491—22no—
55#491Dic314yescomplemented
56C2×#103—22no—
57C2×#97—32no—
58C2×#97C2×D311yescharacteristic
59#389—62no—
60#387—62no—
61#387—62no—
62#382—32no—
63#340—62no—
64#340—62no—
65#330—62no—
66C2×C3×Q8—22no—
67C2×C3×Q8—42no—
68C2×C3×Q8—42no—
69C2×C3×Q8—41no—
70C22×C3×C4—22no—
71C22×C3×C4—41no—
72C22×C3×C4C2×D412yes—
73C22×C3×C4C2×D411yescharacteristic
74C22×Dic3—22no—
75C22×Dic3—22no—
76C2×C3×C8—22no—
77C2×C3×C8—41no—
78C2×C3×C8—41no—
79C3×C4⋊C4—24no—
80C3×C4⋊C4—24no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4C22⋊C414yes—
85C3×C42C22×C411yescharacteristic
86C3×C42C2×D411yescharacteristic
87C3×C42#4112yes—
88C4⋊Dic3—42no—
89C4⋊Dic3—42no—
90C4⋊Dic3—82no—
91C4×Dic3—82no—
92C2×C3⋊C8—24no—
93C2×C3⋊C8—42no—
94C2×C3⋊C8—42no—
95C22×Q8—21no—
96C22×C8—21no—
97C22×C8—62no—
98C4⋊Q8—22no—
99C4⋊Q8—21no—
100C4⋊Q8C2×Dic312yes—
101#125—22no—
102#125C2×Dic312yes—
103#125#6712yes—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—62no—
109C2×C4⋊C4#6712yes—
110C2×C42—62no—
111C2×C42C22×D311yescharacteristic
112#103—28no—
113#97—21no—
114#97—62no—
115#97—62no—
116#97C2×Dic312yes—
117#95—62no—
118#95—62no—
119#95—62no—
120#95—62no—
121C23×C3#12011yescharacteristic
122C3×Q8—44no—
123C3×Q8—82no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—24no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—81no—
133C2×C3×C4—81no—
134C2×C3×C4C2×C22⋊C411yescharacteristic
135C2×C3×C4C4×D412yes—
136C2×C3×C4#12112yes—
137C2×C3×C4#12311yescharacteristic
138C2×Dic3—42no—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—42no—
142C2×Dic3—42no—
143C2×Dic3—42no—
144C3×C8—42no—
145C3⋊C8—44no—
146C2×Q8—22no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—41no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—62no—
153C22×C4—62no—
154C22×C4D3×D412yes—
155C22×C4C2×#6711yescharacteristic
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—64no—
160C2×C8—122no—
161C2×C8—122no—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—42no—
167C4⋊C4—122no—
168C4⋊C4—122no—
169C4⋊C4—242no—
170C4⋊C4#21714yes—
171C42—242no—
172C42#23712yes—
173C42C22×Dic311yescharacteristic
174C42C2×#6711yescharacteristic
175C22×C3#38611yescharacteristic
176C22×C3#45411yescharacteristic
177C22×C3#45411yescharacteristic
178C22×C3#45512yes—
179C22×C3#45612yes—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—44no—
185C3×C4—82no—
186Dic3—82no—
187Dic3—82no—
188C23#93711yescharacteristic
189Q8—44no—
190Q8—82no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—122no—
202C2×C4—122no—
203C2×C4—122no—
204C2×C4—122no—
205C2×C4—122no—
206C2×C4—122no—
207C2×C4D4×Dic312yes—
208C2×C4#93511yescharacteristic
209C2×C4#93912yes—
210C2×C4C2×#21711yescharacteristic
211C8—42no—
212C8—124no—
213C2×C3#166211yescharacteristic
214C2×C3#166311yescharacteristic
215C2×C3#166412yes—
216C2×C3#188212yes—
217C2×C3#188811yescharacteristic
218C22#506211yescharacteristic
219C22#506211yescharacteristic
220C22#520212yes—
221C22#520512yes—
222C22#522011yescharacteristic
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—82no—
229C4—242no—
230C4—242no—
231C3#900411yescharacteristic, complemented, Sylow
232C2#7125111yescharacteristic
233C2#7125211yescharacteristic
234C2#7125312yes—
235C2#7172411yescharacteristic
236C2#7210012yes—
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#491C2
socle, prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenterC2×#491D3
hypocenterC3#9004
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
10. C3×#491C4semidirect
55. #491Dic3semidirect
231. C3#9004semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4912
#830272
#717882
C2×#45432
#676592
#90043