Finite Group Databasebeta

← #10692822

Subgroups — #10692822

Order 768.

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

normal subgroups only

This group has 456 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×#3142C211yesmaximal, characteristic
2#83026C211yesmaximal, characteristic, complemented
3#71812C212yesmaximal
4#71736C211yesmaximal, characteristic, complemented
5#67665C212yesmaximal, complemented
6#9442—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×C3×#125C2211yescharacteristic
9C3×#491C412yescomplemented
10C3×#490C412yescomplemented
11C3×#488—22no—
12C3×#417—22no—
13C3×C4×M16C2211yescharacteristic
14#4980—22no—
15#4980—22no—
16Dic3×C4⋊C4—22no—
17#4932C2212yes—
18#4926C2212yes—
19#4543—22no—
20#4537—22no—
21#4537—22no—
22#3142D311yescharacteristic
23#2371—31no—
24#1534—31no—
25#1505—32no—
26#1300—32no—
27C2×C3×#41—21no—
28C2×C3×M16—21no—
29C3×C4⋊Q8C2×C411yescharacteristic
30C3×#127C2×C411yescharacteristic
31C3×#125—22no—
32C3×#125C2×C412yes—
33C3×#125D412yes—
34C3×#124—21no—
35C3×#121—22no—
36C3×C4×D4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C42C2311yescharacteristic
40C2×C4⋊Dic3—22no—
41C2×C4×Dic3—22no—
42C2×C4×Dic3—22no—
43C3×#103—24no—
44C3×#102—24no—
45C3×#97C2×C412yes—
46C3×C4×C8—21no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50C3⋊#95D412yes—
51C3⋊#95D412yes—
52#830—22no—
53C4⋊(C3⋊C8)—22no—
54#803—22no—
55#532C2×D311yescharacteristic
56C2×#125C2×D311yescharacteristic
57#491Dic312yescomplemented
58#490Dic312yescomplemented
59#488—22no—
60#432—32no—
61#417—22no—
62C4×M16C2×D311yescharacteristic
63#387—62no—
64#387—62no—
65C4×C4⋊C4—62no—
66#376—32no—
67#330—62no—
68#327—62no—
69#327—62no—
70C3×#41—44no—
71C2×C3×Q8—21no—
72C2×C3×D4—21no—
73C2×C3×D4—22no—
74C2×C3×D4—41no—
75C22×C3×C4—22no—
76C22×C3×C4—41no—
77C22×C3×C4C2×D412yes—
78C22×C3×C4C2×D411yescharacteristic
79C22×Dic3—22no—
80C22×Dic3—22no—
81C3×M16—42no—
82C2×C3×C8—22no—
83C3×C4⋊C4—24no—
84C3×C4⋊C4—24no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4C22⋊C414yes—
88C3×C22⋊C4—42no—
89C3×C42C22×C411yescharacteristic
90C3×C42C2×D411yescharacteristic
91C3×C42#4112yes—
92C4⋊Dic3—44no—
93C4×Dic3—44no—
94C4×Dic3—44no—
95C2×C3⋊C8—42no—
96C2×#41—21no—
97C2×M16—21no—
98C4⋊Q8C2×Dic311yescharacteristic
99#127C2×Dic311yescharacteristic
100#125—22no—
101#125C2×Dic312yes—
102#125#6712yes—
103#124—21no—
104#121—22no—
105C4×D4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—62no—
108C2×C4⋊C4#6712yes—
109C2×C42—62no—
110C2×C42—62no—
111C2×C42C22×D311yescharacteristic
112C4⋊C8—62no—
113#103—24no—
114#102—24no—
115C22⋊C8—62no—
116#97—62no—
117#97C2×Dic312yes—
118C4×C8—21no—
119#95—32no—
120#95—32no—
121#95—62no—
122#95—62no—
123#95—62no—
124C23×C3—41no—
125C23×C3#12011yescharacteristic
126C3×Q8—42no—
127C3×D4—42no—
128C3×D4—82no—
129C3×D4—82no—
130C2×C3×C4—22no—
131C2×C3×C4—24no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—41no—
136C2×C3×C4—82no—
137C2×C3×C4C2×C22⋊C411yescharacteristic
138C2×C3×C4C4×D412yes—
139C2×C3×C4#12112yes—
140C2×C3×C4#12311yescharacteristic
141C2×Dic3—24no—
142C2×Dic3—42no—
143C2×Dic3—42no—
144C2×Dic3—42no—
145C2×Dic3—42no—
146C2×Dic3—42no—
147C3×C8—42no—
148C3⋊C8—82no—
149#41—44no—
150C2×Q8—21no—
151C2×D4—22no—
152C2×D4—21no—
153C2×D4—41no—
154C22×C4—22no—
155C22×C4—41no—
156C22×C4—62no—
157C22×C4—62no—
158C22×C4D3×D412yes—
159C22×C4C2×#6711yescharacteristic
160M16—42no—
161C2×C8—22no—
162C2×C8—122no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—24no—
166C4⋊C4—22no—
167C4⋊C4—124no—
168C4⋊C4#21714yes—
169C22⋊C4—42no—
170C42—124no—
171C42—124no—
172C42#23712yes—
173C42C22×Dic311yescharacteristic
174C42C2×#6711yescharacteristic
175C22×C3—22no—
176C22×C3—81no—
177C22×C3—82no—
178C22×C3#38611yescharacteristic
179C22×C3#45412yes—
180C3×C4—21no—
181C3×C4—21no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—44no—
185C3×C4—81no—
186Dic3—44no—
187Dic3—82no—
188C23—41no—
189C23#93711yescharacteristic
190Q8—42no—
191D4—42no—
192D4—82no—
193D4—82no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—64no—
201C2×C4—82no—
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—242no—
213C2×C3—22no—
214C2×C3—161no—
215C2×C3#188811yescharacteristic
216C2×C3#190612yes—
217C22—22no—
218C22—81no—
219C22—82no—
220C22#506212yes—
221C22#522011yescharacteristic
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—81no—
228C4—124no—
229C4—242no—
230C3#944211yescharacteristic, complemented, Sylow
231C2—22no—
232C2—161no—
233C2#7172411yescharacteristic
234C2#7211412yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3142C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3142D3
hypocenterC3#9442
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
2. #83026C2semidirect
4. #71736C2semidirect
5. #67665C2semidirect
9. C3×#491C4semidirect
10. C3×#490C4semidirect
57. #491Dic3semidirect
58. #490Dic3semidirect
230. C3#9442semidirect

Maximal subgroups

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

subgroupindex
C3×#31422
#830262
#718122
#717362
#676652
#94423