Finite Group Databasebeta

← #10690131

Subgroups — #10690131

Order 768.

458 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×#2382C211yesmaximal, characteristic
2C2×#5069C211yesmaximal, characteristic
3C2×#4543C211yesmaximal, characteristic
4#69555C212yesmaximal
5#67630C212yesmaximal
6#9004—31nomaximal, Sylow
7C2×C3×#125C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×#389—22no—
10C3×#387—22no—
11C3×#387—22no—
12C3×#382C2212yes—
13#5069—22no—
14#5069C414yescomplemented
15C2×#810—22no—
16C2×#803C2212yes—
17C2×#803C2211yescharacteristic
18#4543—22no—
19#4529—22no—
20#4529—22no—
21C2×#491—31no—
22#2382D311yescharacteristic
23#1993—32no—
24C2×#330—31no—
25#1295—32no—
26C22×Dic6—21no—
27C3×#125—22no—
28C3×#125C2×C412yes—
29C3×#125D412yes—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4D412yes—
33C2×C3×C4⋊C4D412yes—
34C2×C3×C4⋊C4D412yes—
35C2×C3×C42—22no—
36C2×C3×C42C2311yescharacteristic
37C2×C4⋊Dic3—22no—
38C22×C3⋊C8—21no—
39C22×C3⋊C8—22no—
40#869—22no—
41#869—21no—
42#869C2×C412yes—
43C3×#95—22no—
44C3×#95—22no—
45C3×#95—22no—
46C3×#95—22no—
47#810—28no—
48#803—21no—
49#803—22no—
50#803—22no—
51#803C2×C412yes—
52C2×C4⋊Q8—31no—
53C2×#125C2×D311yescharacteristic
54#491—34no—
55#491—62no—
56C2×#103—62no—
57C2×#97—31no—
58C2×#97—32no—
59#389—22no—
60#387—22no—
61#387—22no—
62#382C2×D312yes—
63#340—62no—
64#340—62no—
65#330—62no—
66C22×C3×C4—22no—
67C22×C3×C4—22no—
68C22×C3×C4—22no—
69C22×C3×C4C2×D411yescharacteristic
70C22×C3×C4C2×D412yes—
71C22×Dic3—41no—
72C2×Dic6—22no—
73C2×Dic6—42no—
74C2×Dic6—42no—
75C2×Dic6—41no—
76C3×C4⋊C4—24no—
77C3×C4⋊C4—24no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—42no—
80C3×C4⋊C4—42no—
81C3×C4⋊C4—82no—
82C3×C4⋊C4C22⋊C414yes—
83C3×C42—82no—
84C3×C42C22×C411yescharacteristic
85C3×C42C2×D411yescharacteristic
86C3×C42#4112yes—
87C4⋊Dic3—24no—
88C4⋊Dic3—42no—
89C2×C3⋊C8—24no—
90C2×C3⋊C8—22no—
91C2×C3⋊C8—41no—
92C2×C3⋊C8—41no—
93C2×C3⋊C8—42no—
94C2×C3⋊C8—42no—
95C22×Q8—61no—
96C22×C8—62no—
97C22×C8—61no—
98C4⋊Q8—32no—
99C4⋊Q8—61no—
100C4⋊Q8—62no—
101#125—22no—
102#125C4×D312yes—
103#125D1212yes—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4#6712yes—
108C2×C4⋊C4#6712yes—
109C2×C4⋊C4#6712yes—
110C2×C42—22no—
111C2×C42C22×D311yescharacteristic
112#103—68no—
113#97—32no—
114#97—61no—
115#97—62no—
116#97—62no—
117#95—22no—
118#95—22no—
119#95—22no—
120#95—22no—
121C23×C3#12011yescharacteristic
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—24no—
127C2×C3×C4—42no—
128C2×C3×C4—42no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4C2×C22⋊C411yescharacteristic
136C2×C3×C4C4×D412yes—
137C2×C3×C4#12112yes—
138C2×C3×C4#12311yescharacteristic
139C2×Dic3—42no—
140C2×Dic3—81no—
141C2×Dic3—81no—
142Dic6—44no—
143Dic6—82no—
144C3⋊C8—44no—
145C3⋊C8—42no—
146C2×Q8—62no—
147C2×Q8—122no—
148C2×Q8—122no—
149C2×Q8—121no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—121no—
154C22×C4C2×#6711yescharacteristic
155C22×C4C2×#6712yes—
156C2×C8—64no—
157C2×C8—62no—
158C2×C8—122no—
159C2×C8—121no—
160C2×C8—121no—
161C2×C8—122no—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—24no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—64no—
168C4⋊C4—82no—
169C4⋊C4—122no—
170C4⋊C4(C2×D3)⋊C414yes—
171C42—82no—
172C42C2×C4×D311yescharacteristic
173C42C2×D1211yescharacteristic
174C42#23512yes—
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—
186C3×C4—82no—
187Dic3—82no—
188C23#95311yescharacteristic
189Q8—124no—
190Q8—242no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—122no—
205C2×C4—241no—
206C2×C4—241no—
207C2×C4C2×(C2×D3)⋊C411yescharacteristic
208C2×C4C4×#6712yes—
209C2×C4#92911yescharacteristic
210C2×C4#93012yes—
211C8—122no—
212C8—124no—
213C2×C3#166211yescharacteristic
214C2×C3#166311yescharacteristic
215C2×C3#166412yes—
216C2×C3#188212yes—
217C2×C3#188811yescharacteristic
218C22#521311yescharacteristic
219C22#524212yes—
220C22#524511yescharacteristic
221C22#524511yescharacteristic
222C22#524612yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—82no—
229C4—82no—
230C4—242no—
231C3#900411yescharacteristic, complemented, Sylow
232C2#7166111yescharacteristic
233C2#7166612yes—
234C2#7195411yescharacteristic
235C2#7195712yes—
236C2#7196111yescharacteristic
subgroupG/H
centerC23#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2382C2
socle, prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2382D3
hypocenterC3#9004
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2382

Complements

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

NH ≅ G/Ntype
14. #5069C4semidirect
231. C3#9004semidirect

Maximal subgroups

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

subgroupindex
C3×#23822
C2×#50692
C2×#45432
#695552
#676302
#90043