Finite Group Databasebeta

← #10698678

Subgroups — #10698678

Order 768.

386 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 93 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×#2822C211yesmaximal, characteristic
2#71741C212yesmaximal
3C2×#4543C212yesmaximal
4#69548C211yesmaximal, characteristic
5#67630C211yesmaximal, characteristic
6#10428—31nomaximal, Sylow
7C2×C3×#125C2212yes—
8C3×#400—22no—
9C3×#398—22no—
10C3×#398—22no—
11C3×#382C2211yescharacteristic
12#4935C2211yescharacteristic
13C2×#803C2212yes—
14C2×#803C2211yescharacteristic
15#4543—24no—
16#4529—22no—
17#4529—22no—
18#4529—22no—
19#2822D311yescharacteristic
20#1992—32no—
21#1836—31no—
22C2×#330—32no—
23#1295—31no—
24C3×#125—22no—
25C3×#125—22no—
26C3×#125—22no—
27C3×#125—22no—
28C2×C3×C4⋊C4—22no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4D412yes—
32C2×C3×C4⋊C4D412yes—
33C2×C3×C4⋊C4D412yes—
34C2×C3×C42C2311yescharacteristic
35C2×C4⋊Dic3—22no—
36C2×C4⋊Dic3—21no—
37C22×C3⋊C8—22no—
38C22×C3⋊C8—21no—
39C3×#95—24no—
40C3×#95—22no—
41C3×#95—22no—
42#803—21no—
43#803—22no—
44#803—22no—
45#803D412yes—
46C2×#125C2×D312yes—
47C2×#97—31no—
48C2×#97—32no—
49#400—22no—
50#398—22no—
51#398—22no—
52#384—31no—
53#382C2×D311yescharacteristic
54#340—62no—
55#340—62no—
56#340—62no—
57#330—64no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4C2×D412yes—
62C22×C3×C4C2×D411yescharacteristic
63C22×Dic3—41no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—42no—
69C3×C4⋊C4—42no—
70C3×C4⋊C4—42no—
71C3×C4⋊C4—42no—
72C3×C4⋊C4—42no—
73C3×C4⋊C4—82no—
74C3×C42C2×D411yescharacteristic
75C3×C42#4111yescharacteristic
76C3×C42#4112yes—
77C4⋊Dic3—42no—
78C4⋊Dic3—44no—
79C2×C3⋊C8—24no—
80C2×C3⋊C8—22no—
81C2×C3⋊C8—42no—
82C2×C3⋊C8—41no—
83C2×C3⋊C8—41no—
84C2×C3⋊C8—42no—
85C22×C8—62no—
86C22×C8—61no—
87#125—22no—
88#125—22no—
89#125—22no—
90#125—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—62no—
95C2×C4⋊C4—61no—
96C2×C4⋊C4#6712yes—
97C2×C4⋊C4#6712yes—
98C2×C4⋊C4#6712yes—
99C2×C42C22×D311yescharacteristic
100#97—32no—
101#97—62no—
102#97—61no—
103#97—62no—
104#95—24no—
105#95—22no—
106#95—22no—
107C23×C3#12011yescharacteristic
108C2×C3×C4—22no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—42no—
113C2×C3×C4—42no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—42no—
118C2×C3×C4—42no—
119C2×C3×C4—42no—
120C2×C3×C4—42no—
121C2×C3×C4#12111yescharacteristic
122C2×C3×C4#12112yes—
123C2×C3×C4#12312yes—
124C2×C3×C4#12411yescharacteristic
125C2×Dic3—81no—
126C2×Dic3—82no—
127C3⋊C8—44no—
128C3⋊C8—42no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—121no—
133C22×C4C2×#6712yes—
134C22×C4C2×#6711yescharacteristic
135C2×C8—64no—
136C2×C8—62no—
137C2×C8—121no—
138C2×C8—122no—
139C2×C8—121no—
140C2×C8—122no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—82no—
151C4⋊C4—124no—
152C4⋊C4—122no—
153C42D3×D411yescharacteristic
154C42#23711yescharacteristic
155C42#23712yes—
156C22×C3#39411yescharacteristic
157C22×C3#45411yescharacteristic
158C22×C3#45412yes—
159C22×C3#45511yescharacteristic
160C22×C3#45612yes—
161C3×C4—22no—
162C3×C4—22no—
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—82no—
166C3×C4—82no—
167C3×C4—82no—
168Dic3—161no—
169C23#95311yescharacteristic
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—242no—
184C2×C4—241no—
185C2×C4#93512yes—
186C2×C4#93611yescharacteristic
187C2×C4#93811yescharacteristic
188C2×C4#93912yes—
189C8—124no—
190C8—122no—
191C2×C3#165811yescharacteristic
192C2×C3#166311yescharacteristic
193C2×C3#166412yes—
194C2×C3#203812yes—
195C2×C3#204011yescharacteristic
196C22#522211yescharacteristic
197C22#524211yescharacteristic
198C22#524512yes—
199C22#524511yescharacteristic
200C22#524612yes—
201C4—22no—
202C4—22no—
203C4—21no—
204C4—21no—
205C4—82no—
206C4—82no—
207C4—82no—
208C4—481no—
209C3#1042811yescharacteristic, complemented, Sylow
210C2#7186312yes—
211C2#7187711yescharacteristic
212C2#7195712yes—
213C2#7196011yescharacteristic
214C2#7196111yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2822C2
socle, prime radicalC23×C3#120
hypercenter#2822D3
hypocenterC3#10428
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
209. C3#10428semidirect

Maximal subgroups

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

subgroupindex
C3×#28222
#717412
C2×#45432
#695482
#676302
#104283