Finite Group Databasebeta

← #10692792

Subgroups — #10692792

Order 768.

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

normal subgroups only

This group has 400 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×#3139C211yesmaximal, characteristic
2#83332C211yesmaximal, characteristic
3#71834C211yesmaximal, characteristic
4#71775C212yesmaximal
5#67669C212yesmaximal
6#9436—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×C3×C4⋊Q8C2211yescharacteristic
9C3×#491C414yes—
10C3×#487—22no—
11C3×#419—22no—
12C3×C4×M16C2211yescharacteristic
13#5024—22no—
14Q8×C3⋊C8—22no—
15#5000—22no—
16#4925C2212yes—
17#4924C2212yes—
18#4547—22no—
19#4547—22no—
20#4547—22no—
21#3139D311yescharacteristic
22#2986—31no—
23#1561—31no—
24#1549—32no—
25#1299—32no—
26C22×C3×Q8—21no—
27C2×C3×M16—21no—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—21no—
30C3×C4⋊Q8C2×C412yes—
31C3×C4⋊Q8D412yes—
32C3×#125C2×C412yes—
33C3×C22⋊Q8—22no—
34C3×C4×Q8—22no—
35C3×#117D412yes—
36C2×C3×C4⋊C4—22no—
37C2×C3×C42C2311yescharacteristic
38C2×#208—22no—
39C3×#103—28no—
40C3×#97C2×C412yes—
41C3×C4×C8—21no—
42#830—22no—
43C4⋊(C3⋊C8)—24no—
44C4⋊(C3⋊C8)—22no—
45C4⋊(C3⋊C8)—22no—
46C4⋊(C3⋊C8)D412yes—
47C4⋊(C3⋊C8)D412yes—
48#803—22no—
49C4×C3⋊C8—22no—
50C4×C3⋊C8—22no—
51#533C2×D311yescharacteristic
52C2×C4⋊Q8C2×D311yescharacteristic
53#491Dic314yes—
54#487—22no—
55#446—62no—
56C8×Q8—62no—
57#433—62no—
58#431—32no—
59#423—32no—
60#419—22no—
61C4×M16C2×D311yescharacteristic
62#332—62no—
63#332—62no—
64#332—62no—
65C2×C3×Q8—22no—
66C2×C3×Q8—22no—
67C2×C3×Q8—44no—
68C2×C3×Q8—41no—
69C22×C3×C4—41no—
70C22×C3×C4C2×D411yescharacteristic
71C22×C3×C4#4112yes—
72C3×M16—42no—
73C2×C3×C8—22no—
74C3×C4⋊C4—24no—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—24no—
77C3×C4⋊C4—42no—
78C3×C4⋊C4C22⋊C414yes—
79C3×C22⋊C4—22no—
80C3×C42—22no—
81C3×C42C22×C411yescharacteristic
82C3×C42C2×D412yes—
83C3×C42C2×D411yescharacteristic
84#208—82no—
85C2×C3⋊C8—22no—
86C2×C3⋊C8—22no—
87C2×C3⋊C8—42no—
88C22×Q8—21no—
89C2×M16—21no—
90C2×M16—62no—
91C4⋊Q8—22no—
92C4⋊Q8—21no—
93C4⋊Q8C2×Dic312yes—
94C4⋊Q8#6712yes—
95#125C2×Dic312yes—
96C22⋊Q8—22no—
97C4×Q8—22no—
98#117#6712yes—
99C2×C4⋊C4—22no—
100C2×C42C22×D311yescharacteristic
101C4⋊C8—32no—
102C4⋊C8—32no—
103C4⋊C8—64no—
104C4⋊C8—62no—
105C4⋊C8—62no—
106#103—28no—
107C22⋊C8—62no—
108#97—62no—
109#97C2×Dic312yes—
110C4×C8—21no—
111C4×C8—62no—
112C4×C8—62no—
113C23×C3#12311yescharacteristic
114C3×Q8—44no—
115C3×Q8—42no—
116C3×Q8—42no—
117C3×Q8—82no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—24no—
122C2×C3×C4—42no—
123C2×C3×C4—82no—
124C2×C3×C4C2×C22⋊C411yescharacteristic
125C2×C3×C4C4×D412yes—
126C2×C3×C4#12011yescharacteristic
127C2×C3×C4#12112yes—
128C3×C8—42no—
129C3⋊C8—42no—
130C3⋊C8—42no—
131C3⋊C8—82no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—44no—
135C2×Q8—41no—
136C22×C4—41no—
137C22×C4#23712yes—
138C22×C4C2×#6711yescharacteristic
139M16—42no—
140M16—242no—
141C2×C8—22no—
142C2×C8—62no—
143C2×C8—62no—
144C2×C8—122no—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—24no—
148C4⋊C4—42no—
149C4⋊C4#21714yes—
150C22⋊C4—22no—
151C42—22no—
152C42D3×D412yes—
153C42C22×Dic311yescharacteristic
154C42C2×#6711yescharacteristic
155C22×C3—21no—
156C22×C3—22no—
157C22×C3#38611yescharacteristic
158C3×C4—21no—
159C3×C4—22no—
160C3×C4—44no—
161C3×C4—82no—
162C3×C4#45412yes—
163C3×C4#47112yes—
164C3×C4#47312yes—
165C23#93511yescharacteristic
166Q8—44no—
167Q8—42no—
168Q8—42no—
169Q8—82no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4—82no—
176C2×C4D4×Dic312yes—
177C2×C4#93711yescharacteristic
178C2×C4#93912yes—
179C2×C4C2×#21711yescharacteristic
180C8—42no—
181C8—122no—
182C8—122no—
183C8—242no—
184C2×C3—41no—
185C2×C3#188811yescharacteristic
186C2×C3#190812yes—
187C22—21no—
188C22—22no—
189C22#522011yescharacteristic
190C4—21no—
191C4—22no—
192C4—44no—
193C4—82no—
194C4#506212yes—
195C4#520312yes—
196C4#520412yes—
197C3#943611yescharacteristic, complemented, Sylow
198C2—41no—
199C2#7172411yescharacteristic
200C2#7210312yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3139C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#3139D3
hypocenterC3#9436
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
197. C3#9436semidirect

Maximal subgroups

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

subgroupindex
C3×#31392
#833322
#718342
#717752
#676692
#94363