Finite Group Databasebeta

← #10690504

Subgroups — #10690504

Order 768.

404 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 111 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×#3160C211yesmaximal, characteristic
2#83332C211yesmaximal, characteristic
3#71825C212yesmaximal
4#71821C211yesmaximal, characteristic
5#67656C212yesmaximal
6#9068—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C3×#533C2211yescharacteristic
9C3×#500C412yes—
10C3×#499—22no—
11C3×#498C412yes—
12C3×#427—22no—
13C2×C3×C4×C8C2211yescharacteristic
14#5024—22no—
15Q8×C3⋊C8—22no—
16#5000—22no—
17#4925C2212yes—
18C4×#208C2212yes—
19#4543—22no—
20#4535—22no—
21#4535—22no—
22#3160D311yescharacteristic
23#2986—31no—
24#1526—31no—
25#1485—32no—
26#1278—32no—
27C22×C3×C8—21no—
28C3×C4⋊Q8C2×C412yes—
29C3×C4⋊Q8C2×C412yes—
30C3×#125—21no—
31C3×#125D412yes—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C3×#117—21no—
37C3×#117D412yes—
38C2×C3×C42C2311yescharacteristic
39C2×#208—22no—
40C3×#107—24no—
41C3×#106—24no—
42C3×C4×C8—21no—
43C3×C4×C8C2×C411yescharacteristic
44C3×C4×C8C2×C411yescharacteristic
45#830—22no—
46C4⋊(C3⋊C8)—24no—
47C4⋊(C3⋊C8)D412yes—
48C4⋊(C3⋊C8)D412yes—
49#803—22no—
50#803—22no—
51C4×C3⋊C8—22no—
52C4×C3⋊C8—22no—
53C4×C3⋊C8—22no—
54#533C2×D311yescharacteristic
55#533C2×D311yescharacteristic
56#500Dic312yes—
57#499—22no—
58#498Dic312yes—
59#446—62no—
60C8×Q8—62no—
61#433—62no—
62#431—32no—
63#427—22no—
64C4×M16—32no—
65C2×C4×C8C2×D311yescharacteristic
66#330—62no—
67Q8⋊C8—62no—
68Q8⋊C8—62no—
69C2×C3×Q8—22no—
70C2×C3×Q8—22no—
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4#4112yes—
73C2×C3×C8—24no—
74C2×C3×C8—21no—
75C2×C3×C8—21no—
76C3×C4⋊C4—22no—
77C3×C4⋊C4—24no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—42no—
81C3×C4⋊C4C22⋊C414yes—
82C3×C22⋊C4—22no—
83C3×C22⋊C4—41no—
84C3×C42—22no—
85C3×C42—41no—
86C3×C42C22×C411yescharacteristic
87C3×C42C2×D412yes—
88C3×C42C2×D411yescharacteristic
89#208—24no—
90#208—42no—
91C2×C3⋊C8—22no—
92C2×C3⋊C8—22no—
93C2×C3⋊C8—42no—
94C2×M16—62no—
95C22×C8—21no—
96C4⋊Q8C2×Dic312yes—
97C4⋊Q8C2×Dic312yes—
98#125—21no—
99#125#6712yes—
100C22⋊Q8—22no—
101C22⋊Q8—22no—
102C4×Q8—22no—
103C4×Q8—22no—
104#117—21no—
105#117#6712yes—
106C2×C42C22×D311yescharacteristic
107#107—24no—
108#106—24no—
109C4⋊C8—32no—
110C4⋊C8—32no—
111C4⋊C8—64no—
112C22⋊C8—62no—
113#97—62no—
114#97—62no—
115C4×C8—21no—
116C4×C8—62no—
117C4×C8—62no—
118C4×C8—62no—
119C4×C8C2×Dic311yescharacteristic
120C4×C8C2×Dic311yescharacteristic
121C23×C3#12311yescharacteristic
122C3×Q8—42no—
123C3×Q8—42no—
124C3×Q8—42no—
125C3×Q8—42no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—24no—
130C2×C3×C4—42no—
131C2×C3×C4C2×C22⋊C411yescharacteristic
132C2×C3×C4C4×D412yes—
133C2×C3×C4#12011yescharacteristic
134C2×C3×C4#12112yes—
135C3×C8—22no—
136C3×C8—22no—
137C3⋊C8—44no—
138C3⋊C8—42no—
139C3⋊C8—42no—
140C2×Q8—22no—
141C2×Q8—22no—
142C22×C4#23712yes—
143C22×C4C2×#6711yescharacteristic
144M16—64no—
145M16—122no—
146C2×C8—24no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—62no—
150C2×C8—62no—
151C2×C8—122no—
152C4⋊C4—22no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4#21714yes—
158C22⋊C4—22no—
159C22⋊C4—41no—
160C42—22no—
161C42—41no—
162C42D3×D412yes—
163C42C22×Dic311yescharacteristic
164C42C2×#6711yescharacteristic
165C22×C3—22no—
166C22×C3—21no—
167C22×C3#38611yescharacteristic
168C3×C4—22no—
169C3×C4—21no—
170C3×C4—44no—
171C3×C4—82no—
172C3×C4#44312yes—
173C3×C4#44412yes—
174C3×C4#45612yes—
175C23#93511yescharacteristic
176Q8—42no—
177Q8—42no—
178Q8—42no—
179Q8—42no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—42no—
185C2×C4D4×Dic312yes—
186C2×C4#93711yescharacteristic
187C2×C4#93912yes—
188C2×C4C2×#21711yescharacteristic
189C8—22no—
190C8—22no—
191C8—124no—
192C8—122no—
193C8—122no—
194C2×C3—41no—
195C2×C3#187612yes—
196C2×C3#188611yescharacteristic
197C22—21no—
198C22—22no—
199C22#522011yescharacteristic
200C4—21no—
201C4—22no—
202C4—44no—
203C4—82no—
204C4#509512yes—
205C4#519612yes—
206C4#519712yes—
207C3#906811yescharacteristic, complemented, Sylow
208C2—41no—
209C2#7178411yescharacteristic
210C2#7208712yes—
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3160C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#3160D3
hypocenterC3#9068
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
207. C3#9068semidirect

Maximal subgroups

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

subgroupindex
C3×#31602
#833322
#718252
#718212
#676562
#90683