Finite Group Databasebeta

← #10694571

Subgroups — #10694571

Order 768.

400 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 119 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×#3131C211yesmaximal, characteristic
2#71838C212yesmaximal, complemented
3#71814C212yesmaximal
4#69763C211yesmaximal, characteristic, complemented
5#69724C211yesmaximal, characteristic, complemented
6#9757—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C3×#532C2211yescharacteristic
9C3×#488—21no—
10C3×#488—21no—
11C3×#487C412yescomplemented
12C3×#486C412yescomplemented
13C3×#416—22no—
14C2×C3×C4×C8C2211yescharacteristic
15#4936C2212yes—
16#4925C2212yes—
17#4551—22no—
18#4538—24no—
19#4537—24no—
20#4455—21no—
21C8×C3⋊C8—21no—
22#3131D311yescharacteristic
23#1835—31no—
24#1518—32no—
25#1517—32no—
26#1450—31no—
27C2×C3×#41—21no—
28C22×C3×C8D412yes—
29C3×C4⋊Q8C2×C411yescharacteristic
30C3×C4⋊Q8C2×C412yes—
31C3×#127C2×C411yescharacteristic
32C3×#125—21no—
33C3×#124—21no—
34C3×C22⋊Q8—22no—
35C3×#121—22no—
36C3×C4×Q8—22no—
37C3×C4×D4—22no—
38C3×#117—21no—
39C2×C3×C42C2311yescharacteristic
40C2×C4⋊Dic3—22no—
41C3×#103—22no—
42C3×#103—22no—
43C3×#102—22no—
44C3×#102—22no—
45C3×C4×C8C2×C412yes—
46C3×C4×C8D411yescharacteristic
47C3×C4×C8D411yescharacteristic
48C3⋊#95—22no—
49C3⋊#95—22no—
50#830D414yes—
51C4⋊(C3⋊C8)—22no—
52C4×C3⋊C8—22no—
53#533C2×D311yescharacteristic
54#532C2×D311yescharacteristic
55#488—21no—
56#488—21no—
57#487Dic312yescomplemented
58#486Dic312yescomplemented
59#431—32no—
60#416—22no—
61C2×C4×C8C2×D311yescharacteristic
62#382—32no—
63#340—62no—
64#328—64no—
65#327—64no—
66#322—61no—
67C82—61no—
68C3×#41—44no—
69C2×C3×Q8—21no—
70C2×C3×Q8—22no—
71C2×C3×D4—21no—
72C2×C3×D4—22no—
73C2×C3×D4—41no—
74C22×C3×C4—41no—
75C22×C3×C4C2×D412yes—
76C22×C3×C4C2×D411yescharacteristic
77C22×Dic3—42no—
78C2×C3×C8—24no—
79C2×C3×C8C22⋊C412yes—
80C2×C3×C8C22⋊C412yes—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—22no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—42no—
85C3×C22⋊C4—41no—
86C3×C22⋊C4—42no—
87C3×C42—41no—
88C3×C42C22×C411yescharacteristic
89C3×C42C2×D411yescharacteristic
90C3×C42#4112yes—
91C4⋊Dic3—44no—
92C2×C3⋊C8—24no—
93C2×#41—21no—
94C22×C8#6712yes—
95C4⋊Q8C2×Dic311yescharacteristic
96C4⋊Q8C2×Dic312yes—
97#127C2×Dic311yescharacteristic
98#125—21no—
99#124—21no—
100C22⋊Q8—22no—
101#121—22no—
102C4×Q8—22no—
103C4×D4—22no—
104#117—21no—
105C2×C4⋊C4—62no—
106C2×C42C22×D311yescharacteristic
107C4⋊C8—62no—
108#103—22no—
109#103—22no—
110#102—22no—
111#102—22no—
112C22⋊C8—34no—
113C4×C8—62no—
114C4×C8C2×Dic312yes—
115C4×C8#6711yescharacteristic
116C4×C8#6711yescharacteristic
117#95—62no—
118#95—62no—
119C23×C3—41no—
120C23×C3#12711yescharacteristic
121C3×Q8—42no—
122C3×Q8—82no—
123C3×D4—42no—
124C3×D4—82no—
125C3×D4—82no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—41no—
129C2×C3×C4—82no—
130C2×C3×C4C2×C22⋊C411yescharacteristic
131C2×C3×C4C4×D412yes—
132C2×C3×C4#12112yes—
133C2×C3×C4#12411yescharacteristic
134C2×C3×C4#13514yes—
135C2×Dic3—84no—
136C2×Dic3—82no—
137C3×C8—22no—
138C3×C8—22no—
139C3⋊C8—44no—
140#41—44no—
141C2×Q8—22no—
142C2×Q8—21no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4—41no—
146C22×C4—41no—
147C22×C4—122no—
148C22×C4D3×D412yes—
149C22×C4C2×#6711yescharacteristic
150C2×C8—24no—
151C2×C8—64no—
152C2×C8#21712yes—
153C2×C8#21712yes—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—124no—
159C22⋊C4—41no—
160C22⋊C4—42no—
161C42—41no—
162C42#23712yes—
163C42C22×Dic311yescharacteristic
164C42C2×#6711yescharacteristic
165C22×C3—22no—
166C22×C3—81no—
167C22×C3—82no—
168C22×C3#39011yescharacteristic
169C22×C3#49512yes—
170C3×C4—22no—
171C3×C4—22no—
172C3×C4—81no—
173C3×C4—82no—
174C3×C4#41612yes—
175C3×C4#48812yes—
176Dic3—162no—
177C23—41no—
178C23#94011yescharacteristic
179Q8—42no—
180Q8—82no—
181D4—42no—
182D4—82no—
183D4—82no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—82no—
188C2×C4—244no—
189C2×C4—242no—
190C2×C4D4×Dic312yes—
191C2×C4#93611yescharacteristic
192C2×C4#93812yes—
193C2×C4#95014yes—
194C2×C4C2×#21711yescharacteristic
195C8—22no—
196C8—22no—
197C8—124no—
198C2×C3—22no—
199C2×C3—161no—
200C2×C3#195811yescharacteristic
201C2×C3#197612yes—
202C22—22no—
203C22—81no—
204C22—82no—
205C22#505912yes—
206C22#522111yescharacteristic
207C4—22no—
208C4—22no—
209C4—81no—
210C4—82no—
211C4—482no—
212C4#505612yes—
213C4#523312yes—
214C3#975711yescharacteristic, complemented, Sylow
215C2—22no—
216C2—161no—
217C2#7172511yescharacteristic
218C2#7208612yes—
subgroupG/H
centerC22#5221
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3131C2
socleC22×C3#390
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3131D3
hypocenterC3#9757
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
2. #71838C2semidirect
4. #69763C2semidirect
5. #69724C2semidirect
11. C3×#487C4semidirect
12. C3×#486C4semidirect
57. #487Dic3semidirect
58. #486Dic3semidirect
214. C3#9757semidirect

Maximal subgroups

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

subgroupindex
C3×#31312
#718382
#718142
#697632
#697242
#97573