Finite Group Databasebeta

← #10690503

Subgroups — #10690503

Order 768.

404 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2986C211yesmaximal, characteristic
2#83339C211yesmaximal, characteristic
3#71821C211yesmaximal, characteristic
4#70769C212yesmaximal
5#67739C212yesmaximal
6#9068—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#446—22no—
10C3×C8×Q8—22no—
11C3×#433—22no—
12C3×#431C2212yes—
13#5080C412yes—
14#5078C412yes—
15#5065—22no—
16#4998—22no—
17C4×#208C2212yes—
18C2×C4×C3⋊C8C2211yescharacteristic
19#4543—22no—
20#4484—22no—
21#4484—22no—
22#3160—31no—
23#2986D311yescharacteristic
24#1526—31no—
25#1485—32no—
26#1278—32no—
27C3×C4⋊Q8C2×C412yes—
28C3×#125D412yes—
29C3×C22⋊Q8—22no—
30C3×C4×Q8—22no—
31C3×#117D412yes—
32C2×C3×C42C2311yescharacteristic
33#926—21no—
34#924—22no—
35C2×#208—22no—
36C22×C3⋊C8—21no—
37#870—21no—
38#869C2×C412yes—
39C4×Dic6—22no—
40C3×C4⋊C8—24no—
41C3×C4⋊C8D412yes—
42C3×C4⋊C8D412yes—
43C3×C22⋊C8—22no—
44C3×#97—22no—
45C3×C4×C8—22no—
46C3×C4×C8—22no—
47#808—24no—
48#807—24no—
49#803—22no—
50C4×C3⋊C8—21no—
51C4×C3⋊C8—22no—
52C4×C3⋊C8C2×C411yescharacteristic
53C4×C3⋊C8C2×C411yescharacteristic
54#533—31no—
55#533C2×D311yescharacteristic
56#500—32no—
57#499—62no—
58#498—32no—
59#446—22no—
60C8×Q8—22no—
61#433—22no—
62#431C2×D312yes—
63#427—62no—
64C4×M16—32no—
65C2×C4×C8—31no—
66#330—62no—
67Q8⋊C8—62no—
68Q8⋊C8—62no—
69C2×C3×Q8—22no—
70C22×C3×C4C2×D411yescharacteristic
71C22×C3×C4#4112yes—
72C2×Dic6—22no—
73C2×C3×C8—22no—
74C2×C3×C8—22no—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—24no—
77C3×C4⋊C4C22⋊C414yes—
78C3×C22⋊C4—22no—
79C3×C42—22no—
80C3×C42C22×C411yescharacteristic
81C3×C42C2×D411yescharacteristic
82C3×C42C2×D412yes—
83#217—41no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86Dic3⋊C4—42no—
87C4×Dic3—41no—
88#208—24no—
89#208—42no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8—21no—
92C2×C3⋊C8—21no—
93C2×C3⋊C8—42no—
94C2×M16—62no—
95C22×C8—61no—
96C4⋊Q8—32no—
97C4⋊Q8C4×D312yes—
98#125—61no—
99#125D1212yes—
100C22⋊Q8—22no—
101C22⋊Q8—62no—
102C4×Q8—22no—
103C4×Q8—62no—
104#117—61no—
105#117#6712yes—
106C2×C42C22×D311yescharacteristic
107#107—64no—
108#106—64no—
109C4⋊C8—24no—
110C4⋊C8#6712yes—
111C4⋊C8#6712yes—
112C22⋊C8—22no—
113#97—22no—
114#97—62no—
115C4×C8—22no—
116C4×C8—22no—
117C4×C8—31no—
118C4×C8—31no—
119C4×C8—61no—
120C4×C8—62no—
121C23×C3#12311yescharacteristic
122C3×Q8—42no—
123C3×Q8—42no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—24no—
128C2×C3×C4C2×C22⋊C411yescharacteristic
129C2×C3×C4C4×D412yes—
130C2×C3×C4#12011yescharacteristic
131C2×C3×C4#12112yes—
132C2×Dic3—42no—
133Dic6—42no—
134Dic6—42no—
135C3×C8—42no—
136C3×C8—42no—
137C3⋊C8—22no—
138C3⋊C8—22no—
139C3⋊C8—44no—
140C2×Q8—22no—
141C2×Q8—62no—
142C22×C4#23512yes—
143C22×C4C2×#6711yescharacteristic
144M16—64no—
145M16—122no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—64no—
149C2×C8—61no—
150C2×C8—61no—
151C2×C8—122no—
152C4⋊C4—22no—
153C4⋊C4—24no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—122no—
157C4⋊C4(C2×D3)⋊C414yes—
158C22⋊C4—22no—
159C22⋊C4—121no—
160C42—22no—
161C42—121no—
162C42C2×C4×D311yescharacteristic
163C42C2×D1211yescharacteristic
164C42C2×#6712yes—
165C22×C3—22no—
166C22×C3—21no—
167C22×C3#38611yescharacteristic
168C3×C4—22no—
169C3×C4—21no—
170C3×C4—44no—
171C3×C4#44312yes—
172C3×C4#44412yes—
173C3×C4#45612yes—
174Dic3—82no—
175C23#92911yescharacteristic
176Q8—42no—
177Q8—42no—
178Q8—122no—
179Q8—122no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—122no—
185C2×C4C2×(C2×D3)⋊C411yescharacteristic
186C2×C4C4×#6712yes—
187C2×C4#93012yes—
188C2×C4#95311yescharacteristic
189C8—42no—
190C8—42no—
191C8—62no—
192C8—62no—
193C8—124no—
194C2×C3—41no—
195C2×C3#187612yes—
196C2×C3#188611yescharacteristic
197C22—22no—
198C22—21no—
199C22#521311yescharacteristic
200C4—22no—
201C4—21no—
202C4—44no—
203C4—242no—
204C4#514412yes—
205C4#514512yes—
206C4#524612yes—
207C3#906811yescharacteristic, complemented, Sylow
208C2—41no—
209C2#7166011yescharacteristic
210C2#7201012yes—
subgroupG/H
centerC2×C4#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2986C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#2986D3
hypocenterC3#9068
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2986

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×#29862
#833392
#718212
#707692
#677392
#90683