Finite Group Databasebeta

← #10695088

Subgroups — #10695088

Order 768.

422 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 131 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×#3170C211yesmaximal, characteristic
2#71839C212yesmaximal
3#71826C212yesmaximal
4#69750C211yesmaximal, characteristic
5#69731C211yesmaximal, characteristic
6#9848—31nomaximal, Sylow
7C3×#533C2212yes—
8C3×#500C411yescharacteristic, complemented
9C3×#494C411yescharacteristic, complemented
10C3×#487C412yescomplemented
11C3×#467—22no—
12C3×C4×Q16—22no—
13C2×C3×C4×C8C2211yescharacteristic
14C42×Dic3C2211yescharacteristic
15#4924C2212yes—
16C2×C4×C3⋊C8C2211yescharacteristic
17#4550—22no—
18#4535—24no—
19#4532—24no—
20#4458—22no—
21#3170D311yescharacteristic
22#1994—32no—
23#1744—31no—
24#1567—31no—
25#1493—32no—
26C2×C3×Q16—22no—
27C22×C3×C8D412yes—
28C3×C4⋊Q8C2×C412yes—
29C3×C4⋊Q8C2×C412yes—
30C3×#125—22no—
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C3×#117—22no—
36C2×C3×C42C2311yescharacteristic
37C2×C4×Dic3—21no—
38C2×C4×Dic3—21no—
39C2×C4×Dic3D412yes—
40C2×#208—22no—
41C22×C3⋊C8—21no—
42C3×#107—22no—
43C3×#103—24no—
44C3×C4×C8C2×C411yescharacteristic
45C3×C4×C8C2×C411yescharacteristic
46C3×C4×C8D412yes—
47C4⋊(C3⋊C8)—22no—
48C4⋊(C3⋊C8)—22no—
49C4×C3⋊C8—21no—
50C4×C3⋊C8D412yes—
51#533C2×D312yes—
52#500Dic311yescharacteristic, complemented
53#494Dic311yescharacteristic, complemented
54#487Dic312yescomplemented
55#467—22no—
56C4×Q16—22no—
57#423—32no—
58C2×C4×C8—31no—
59C2×C4×C8C2×D311yescharacteristic
60C43—31no—
61#339—64no—
62#336—62no—
63#335—62no—
64Q8⋊C8—64no—
65C2×C3×Q8—22no—
66C2×C3×Q8—22no—
67C22×C3×C4C2×D411yescharacteristic
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4#4111yescharacteristic
70C22×Dic3—22no—
71C3×Q16—44no—
72C2×C3×C8—42no—
73C2×C3×C8C22⋊C412yes—
74C2×C3×C8C22⋊C412yes—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—22no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—42no—
80C3×C4⋊C4—42no—
81C3×C22⋊C4—42no—
82C3×C42—42no—
83C3×C42C22×C411yescharacteristic
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D411yescharacteristic
86C3×C42#4111yescharacteristic
87C4×Dic3—22no—
88C4×Dic3—42no—
89C4×Dic3—42no—
90C4×Dic3Q1614yescomplemented
91#208—44no—
92C2×C3⋊C8—22no—
93C2×C3⋊C8—42no—
94C2×C3⋊C8—42no—
95C2×Q16—22no—
96C2×M16—62no—
97C22×C8—61no—
98C22×C8#6712yes—
99C4⋊Q8C2×Dic312yes—
100C4⋊Q8C2×Dic312yes—
101#125—22no—
102C22⋊Q8—22no—
103C22⋊Q8—22no—
104C4×Q8—22no—
105C4×Q8—22no—
106#117—22no—
107C2×C42—32no—
108C2×C42—61no—
109C2×C42—61no—
110C2×C42C22×D311yescharacteristic
111#107—22no—
112C4⋊C8—62no—
113C4⋊C8—62no—
114#103—24no—
115C4×C8—32no—
116C4×C8—61no—
117C4×C8C2×Dic311yescharacteristic
118C4×C8C2×Dic311yescharacteristic
119C4×C8#6712yes—
120C23×C3#12111yescharacteristic
121C3×Q8—44no—
122C3×Q8—44no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4C2×C22⋊C411yescharacteristic
128C2×C3×C4C4×D411yescharacteristic
129C2×C3×C4C4×D411yescharacteristic
130C2×C3×C4#12111yescharacteristic
131C2×C3×C4#12411yescharacteristic
132C2×C3×C4#12711yescharacteristic
133C2×C3×C4C2×Q1612yes—
134C2×C3×C4#13512yes—
135C2×Dic3—24no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C3×C8—22no—
139C3×C8—22no—
140C3⋊C8—42no—
141C3⋊C8—82no—
142C2×Q8—22no—
143C2×Q8—22no—
144C22×C4—62no—
145C22×C4D3×D411yescharacteristic
146C22×C4#23711yescharacteristic
147C22×C4C2×#6711yescharacteristic
148Q16—44no—
149M16—124no—
150C2×C8—42no—
151C2×C8—62no—
152C2×C8—122no—
153C2×C8—122no—
154C2×C8#21712yes—
155C2×C8#21712yes—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C22⋊C4—42no—
163C42—34no—
164C42—42no—
165C42—62no—
166C42—122no—
167C42—122no—
168C42D3×D411yescharacteristic
169C42#23711yescharacteristic
170C42C22×Dic311yescharacteristic
171C42C2×#6711yescharacteristic
172C22×C3—21no—
173C22×C3—21no—
174C22×C3#39011yescharacteristic
175C22×C3#46712yes—
176C3×C4—21no—
177C3×C4—21no—
178C3×C4—82no—
179C3×C4—82no—
180C3×C4#42112yes—
181C3×C4C4×Q1612yes—
182C3×C4#44412yes—
183C3×C4#49411yescharacteristic
184C3×C4#49511yescharacteristic
185Dic3—44no—
186C23#93811yescharacteristic
187Q8—44no—
188Q8—44no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—64no—
194C2×C4—122no—
195C2×C4—122no—
196C2×C4D3×Q1612yes—
197C2×C4#91912yes—
198C2×C4D4×Dic311yescharacteristic
199C2×C4D4×Dic311yescharacteristic
200C2×C4#93611yescharacteristic
201C2×C4#93811yescharacteristic
202C2×C4#94011yescharacteristic
203C2×C4C2×#21711yescharacteristic
204C8—22no—
205C8—22no—
206C8—122no—
207C8—242no—
208C2×C3—22no—
209C2×C3#196311yescharacteristic
210C2×C3#196911yescharacteristic
211C2×C3#197511yescharacteristic
212C22—21no—
213C22—21no—
214C22#518812yes—
215C22#522111yescharacteristic
216C4—21no—
217C4—21no—
218C4—82no—
219C4—82no—
220C4—124no—
221C4Dic3×Q1612yes—
222C4#518311yescharacteristic
223C4#519111yescharacteristic
224C4#519712yes—
225C4#523612yes—
226C3#984811yescharacteristic, complemented, Sylow
227C2—22no—
228C2#7181611yescharacteristic
229C2#7206611yescharacteristic
230C2#7208511yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3170C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#3170D3
hypocenterC3#9848
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
8. C3×#500C4semidirect
9. C3×#494C4semidirect
10. C3×#487C4semidirect
52. #500Dic3semidirect
53. #494Dic3semidirect
54. #487Dic3semidirect
90. C4×Dic3Q16semidirect
226. C3#9848semidirect

Maximal subgroups

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

subgroupindex
C3×#31702
#718392
#718262
#697502
#697312
#98483