Finite Group Databasebeta

← #10692744

Subgroups — #10692744

Order 768.

508 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 95 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#83231C211yesmaximal, characteristic, complemented
2#83203C211yesmaximal, characteristic
3C3×#1558C211yesmaximal, characteristic, complemented
4#70835C212yesmaximal
5#67752C212yesmaximal, complemented
6#9429—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×#942—22no—
9#5477C2211yescharacteristic
10C3×#432C2212yes—
11#5229—22no—
12#5092C412yescomplemented
13#5086—22no—
14#5085C412yescomplemented
15#4980—24no—
16#4937C2212yes—
17C4×#208C2211yescharacteristic
18C3×#333—22no—
19#4470—24no—
20#3152—31no—
21#2381—31no—
22#1558D311yescharacteristic, complemented
23#1514—32no—
24#1288—32no—
25C22×C3×Q8D412yes—
26C2×#235—21no—
27C3×C4⋊Q8—22no—
28C3×C4⋊Q8C2×C412yes—
29C3×C4⋊Q8D412yes—
30C2×C3×C4⋊C4—22no—
31C2×C3×C42C2311yescharacteristic
32C2×#216—24no—
33#942—44no—
34#942—44no—
35C2×#215—24no—
36#930—22no—
37C2×C4⋊Dic3—22no—
38C2×C4×Dic3—22no—
39C2×#208—21no—
40#875—21no—
41#874C2×C411yescharacteristic
42C4×D12—22no—
43#869C2×C411yescharacteristic
44C3×C4⋊C8—22no—
45C3×C22⋊C8—22no—
46C3×#97—22no—
47C3⋊#95—24no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50C3⋊#95D414yes—
51#803—21no—
52C4×C3⋊C8C2×C412yes—
53C2×#137—62no—
54#532—31no—
55C2×C4⋊Q8C2×D311yescharacteristic
56#497—62no—
57#494—32no—
58#492—32no—
59#432C2×D312yes—
60C4×M16—31no—
61#391—62no—
62#387—64no—
63#383—32no—
64#333—22no—
65#327—64no—
66C2×C3×Q8—22no—
67C2×C3×Q8—42no—
68C2×C3×Q8—42no—
69C2×C3×Q8C22⋊C414yes—
70C22×C3×C4—22no—
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4C2×D412yes—
73C2×#67—41no—
74C22×Dic3—24no—
75#235—42no—
76#235—42no—
77C2×D12—21no—
78C2×D12—22no—
79C2×C4×D3—41no—
80C2×Dic6—21no—
81C2×C3×C8—42no—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—24no—
84C3×C42C22×C411yescharacteristic
85C3×C42C2×D411yescharacteristic
86C3×C42#4112yes—
87#216—48no—
88#215—48no—
89(C2×D3)⋊C4—42no—
90C4⋊Dic3—22no—
91C4⋊Dic3—82no—
92C4×Dic3—82no—
93#208—24no—
94C2×C3⋊C8—22no—
95C2×#41—61no—
96C22×Q8D1212yes—
97#137—124no—
98#137—124no—
99C2×Q16—64no—
100C2×SD16—64no—
101C2×M16—61no—
102C4⋊Q8—22no—
103C4⋊Q8—31no—
104C4⋊Q8C4×D312yes—
105C4⋊Q8#6712yes—
106#127—31no—
107#124—61no—
108#121—62no—
109C4×D4—62no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—62no—
112C2×C42—62no—
113C2×C42C22×D311yescharacteristic
114C4⋊C8—22no—
115C22⋊C8—22no—
116#97—22no—
117#97—61no—
118C4×C8—32no—
119#95—34no—
120#95—64no—
121#95—62no—
122#95—62no—
123C23×C3#12011yescharacteristic
124C22×D3—41no—
125C3×Q8—44no—
126C3×Q8—82no—
127C2×C3×C4—22no—
128C2×C3×C4—24no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4C2×C22⋊C411yescharacteristic
133C2×C3×C4C4×D412yes—
134C2×C3×C4#12112yes—
135C2×C3×C4#12311yescharacteristic
136#67—82no—
137C2×Dic3—48no—
138C2×Dic3—41no—
139C2×Dic3—44no—
140D12—42no—
141D12—82no—
142C4×D3—81no—
143C4×D3—81no—
144Dic6—42no—
145C3×C8—82no—
146C3⋊C8—24no—
147#41—122no—
148#41—122no—
149C2×Q8—22no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—61no—
153C2×Q8(C2×D3)⋊C414yes—
154C2×D4—62no—
155C2×D4—61no—
156C2×D4—121no—
157C22×C4—22no—
158C22×C4—64no—
159C22×C4—121no—
160C22×C4C2×D1211yescharacteristic
161C22×C4D3×D412yes—
162Q16—128no—
163SD16—128no—
164M16—64no—
165C2×C8—42no—
166C2×C8—62no—
167C4⋊C4—24no—
168C4⋊C4—24no—
169C4⋊C4—62no—
170C4⋊C4—242no—
171C22⋊C4—122no—
172C42—242no—
173C42C2×C4×D311yescharacteristic
174C42#23712yes—
175C42C2×#6711yescharacteristic
176C22×C3—22no—
177C22×C3#38611yescharacteristic
178C22×C3#45312yes—
179C2×D3—81no—
180C2×D3—82no—
181C3×C4—21no—
182C3×C4—21no—
183C3×C4—22no—
184C3×C4—22no—
185C3×C4—44no—
186Dic3—84no—
187Dic3—81no—
188C23—121no—
189C23#88211yescharacteristic
190Q8—44no—
191Q8—82no—
192Q8—122no—
193D4—122no—
194D4—242no—
195D4—242no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—128no—
202C2×C4—121no—
203C2×C4—124no—
204C2×C4—241no—
205C2×C4—241no—
206C2×C4#67⋊C412yes—
207C2×C4#88611yescharacteristic
208C2×C4C2×(C2×D3)⋊C411yescharacteristic
209C2×C4#93912yes—
210C8—64no—
211C8—82no—
212C2×C3—22no—
213C2×C3#188711yescharacteristic
214C2×C3#190512yes—
215D3—161no—
216C22—22no—
217C22—241no—
218C22—242no—
219C22#474712yes—
220C22#495811yescharacteristic
221C4—21no—
222C4—21no—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—244no—
227C4—241no—
228C3#942911yescharacteristic, complemented, Sylow
229C2—22no—
230C2—481no—
231C2#7007011yescharacteristic
232C2#7033412yes—
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1558C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1558D3
hypocenterC3#9429
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
1. #83231C2semidirect
3. C3×#1558C2semidirect
5. #67752C2semidirect
12. #5092C4semidirect
14. #5085C4semidirect
22. #1558D3semidirect
228. C3#9429semidirect

Maximal subgroups

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

subgroupindex
#832312
#832032
C3×#15582
#708352
#677522
#94293