Finite Group Databasebeta

← #10692747

Subgroups — #10692747

Order 768.

508 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3152C211yesmaximal, characteristic
2#83231C211yesmaximal, characteristic, complemented
3#71813C212yesmaximal
4#71777C211yesmaximal, characteristic, complemented
5#67666C212yesmaximal, complemented
6#9429—31nomaximal, Sylow
7C2×C3×#137—22no—
8C3×#532C2211yescharacteristic
9C2×C3×C4⋊Q8C2211yescharacteristic
10C3×#497—22no—
11C3×#494C412yescomplemented
12C3×#492C412yescomplemented
13C3×C4×M16C2211yescharacteristic
14#5229—22no—
15#4980—24no—
16#4937C2212yes—
17#4926C2212yes—
18#4546—22no—
19#4537—24no—
20#3152D311yescharacteristic
21#2381—31no—
22#1558—31no—
23#1514—32no—
24#1288—32no—
25C2×C3×#41—21no—
26C22×C3×Q8D412yes—
27C3×#137—44no—
28C3×#137—44no—
29C2×C3×Q16—24no—
30C2×C3×SD16—24no—
31C2×C3×M16—21no—
32C3×C4⋊Q8—22no—
33C3×C4⋊Q8C2×C411yescharacteristic
34C3×C4⋊Q8C2×C412yes—
35C3×C4⋊Q8D412yes—
36C3×#127C2×C411yescharacteristic
37C3×#124—21no—
38C3×#121—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3—22no—
43C2×C4×Dic3—22no—
44C3×#97—21no—
45C3×C4×C8C2×C412yes—
46C3⋊#95—24no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C3⋊#95D414yes—
50#830—22no—
51C4⋊(C3⋊C8)—22no—
52#803—22no—
53C2×#137—22no—
54#532C2×D311yescharacteristic
55C2×C4⋊Q8C2×D311yescharacteristic
56#497—22no—
57#494Dic312yescomplemented
58#492Dic312yescomplemented
59#432—32no—
60C4×M16C2×D311yescharacteristic
61#391—62no—
62#387—64no—
63#383—32no—
64#333—62no—
65#327—64no—
66C3×#41—42no—
67C3×#41—42no—
68C2×C3×Q8—22no—
69C2×C3×Q8—21no—
70C2×C3×Q8—42no—
71C2×C3×Q8—42no—
72C2×C3×Q8C22⋊C414yes—
73C2×C3×D4—21no—
74C2×C3×D4—22no—
75C2×C3×D4—41no—
76C22×C3×C4—22no—
77C22×C3×C4—41no—
78C22×C3×C4C2×D412yes—
79C22×C3×C4C2×D411yescharacteristic
80C22×Dic3—24no—
81C3×Q16—48no—
82C3×SD16—48no—
83C3×M16—24no—
84C2×C3×C8—22no—
85C3×C4⋊C4—24no—
86C3×C4⋊C4—24no—
87C3×C4⋊C4—22no—
88C3×C22⋊C4—42no—
89C3×C42C22×C411yescharacteristic
90C3×C42C2×D411yescharacteristic
91C3×C42#4112yes—
92C4⋊Dic3—82no—
93C4×Dic3—82no—
94C2×C3⋊C8—42no—
95C2×#41—21no—
96C22×Q8#6712yes—
97#137—44no—
98#137—44no—
99C2×Q16—24no—
100C2×SD16—24no—
101C2×M16—21no—
102C4⋊Q8—22no—
103C4⋊Q8C2×Dic311yescharacteristic
104C4⋊Q8C2×Dic312yes—
105C4⋊Q8#6712yes—
106#127C2×Dic311yescharacteristic
107#124—21no—
108#121—22no—
109C4×D4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—62no—
112C2×C42—62no—
113C2×C42C22×D311yescharacteristic
114C4⋊C8—62no—
115C22⋊C8—62no—
116#97—21no—
117#97—62no—
118C4×C8C2×Dic312yes—
119#95—34no—
120#95—64no—
121#95—62no—
122#95—62no—
123C23×C3—41no—
124C23×C3#12011yescharacteristic
125C3×Q8—44no—
126C3×Q8—42no—
127C3×Q8—82no—
128C3×D4—42no—
129C3×D4—82no—
130C3×D4—82no—
131C2×C3×C4—22no—
132C2×C3×C4—24no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—41no—
137C2×C3×C4—81no—
138C2×C3×C4—81no—
139C2×C3×C4C2×C22⋊C411yescharacteristic
140C2×C3×C4C4×D412yes—
141C2×C3×C4#12112yes—
142C2×C3×C4#12311yescharacteristic
143C2×Dic3—48no—
144C2×Dic3—44no—
145C3×C8—24no—
146C3⋊C8—82no—
147#41—42no—
148#41—42no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8#21714yes—
154C2×D4—22no—
155C2×D4—21no—
156C2×D4—41no—
157C22×C4—22no—
158C22×C4—41no—
159C22×C4—64no—
160C22×C4D3×D412yes—
161C22×C4C2×#6711yescharacteristic
162Q16—48no—
163SD16—48no—
164M16—24no—
165C2×C8—22no—
166C2×C8—122no—
167C4⋊C4—24no—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—242no—
171C22⋊C4—42no—
172C42—242no—
173C42#23712yes—
174C42C22×Dic311yescharacteristic
175C42C2×#6711yescharacteristic
176C22×C3—22no—
177C22×C3—81no—
178C22×C3—82no—
179C22×C3#38611yescharacteristic
180C22×C3#45312yes—
181C3×C4—21no—
182C3×C4—21no—
183C3×C4—22no—
184C3×C4—22no—
185C3×C4—44no—
186C3×C4—81no—
187Dic3—84no—
188C23—41no—
189C23#93711yescharacteristic
190Q8—44no—
191Q8—42no—
192Q8—82no—
193D4—42no—
194D4—82no—
195D4—82no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—128no—
205C2×C4—124no—
206C2×C4D4×Dic312yes—
207C2×C4#93511yescharacteristic
208C2×C4#93912yes—
209C2×C4C2×#21711yescharacteristic
210C8—24no—
211C8—242no—
212C2×C3—22no—
213C2×C3—161no—
214C2×C3#188711yescharacteristic
215C2×C3#190512yes—
216C22—22no—
217C22—81no—
218C22—82no—
219C22#507712yes—
220C22#522011yescharacteristic
221C4—21no—
222C4—21no—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—81no—
227C4—244no—
228C3#942911yescharacteristic, complemented, Sylow
229C2—22no—
230C2—161no—
231C2#7176311yescharacteristic
232C2#7210112yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3152C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3152D3
hypocenterC3#9429
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. #83231C2semidirect
4. #71777C2semidirect
5. #67666C2semidirect
11. C3×#494C4semidirect
12. C3×#492C4semidirect
57. #494Dic3semidirect
58. #492Dic3semidirect
228. C3#9429semidirect

Maximal subgroups

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

subgroupindex
C3×#31522
#832312
#718132
#717772
#676662
#94293