Finite Group Databasebeta

← #10698267

Subgroups — #10698267

Order 768.

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

normal subgroups only

This group has 392 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×#2845C211yesmaximal, characteristic
2#69555C212yesmaximal
3#69552C211yesmaximal, characteristic
4#67630C212yesmaximal
5#67630C211yesmaximal, characteristic
6#10361—31nomaximal, Sylow
7C2×#869C2211yescharacteristic
8C3×#398—22no—
9C3×#398—22no—
10C3×#398—22no—
11C3×#382C2211yescharacteristic
12C3×#382C2212yes—
13C2×#810—22no—
14C2×#803C2211yescharacteristic
15C2×#803C2212yes—
16#4529—22no—
17#4529—22no—
18#2845D311yescharacteristic
19#1993—32no—
20#1960—31no—
21#1295—31no—
22#1295—32no—
23C22×Dic6—21no—
24C2×C3×C4⋊C4—22no—
25C2×C3×C4⋊C4—22no—
26C2×C3×C4⋊C4—22no—
27C2×C3×C4⋊C4D412yes—
28C2×C3×C4⋊C4D412yes—
29C2×C3×C4⋊C4D412yes—
30C2×C3×C42C2311yescharacteristic
31C2×C4⋊Dic3—22no—
32C22×C3⋊C8—21no—
33C22×C3⋊C8—22no—
34#869—22no—
35#869—21no—
36#869Q812yes—
37C3×#95—22no—
38C3×#95—22no—
39C3×#95—22no—
40C3×#95—22no—
41#810—44no—
42#803—22no—
43#803—22no—
44#803—22no—
45C2×C4⋊Q8—31no—
46C2×#103—62no—
47C2×#97—31no—
48C2×#97—32no—
49#398—22no—
50#398—22no—
51#398—22no—
52#382C2×D311yescharacteristic
53#382C2×D312yes—
54#340—62no—
55#340—62no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4C2×D412yes—
60C22×C3×C4C2×D411yescharacteristic
61C22×Dic3—41no—
62C2×Dic6—22no—
63C2×Dic6—42no—
64C2×Dic6—42no—
65C2×Dic6—41no—
66C3×C4⋊C4—42no—
67C3×C4⋊C4—42no—
68C3×C4⋊C4—42no—
69C3×C4⋊C4—82no—
70C3×C4⋊C4—82no—
71C3×C4⋊C4—82no—
72C3×C42C2×Q811yescharacteristic
73C3×C42#4112yes—
74C3×C42#4111yescharacteristic
75C4⋊Dic3—24no—
76C4⋊Dic3—42no—
77C2×C3⋊C8—22no—
78C2×C3⋊C8—24no—
79C2×C3⋊C8—41no—
80C2×C3⋊C8—42no—
81C2×C3⋊C8—42no—
82C2×C3⋊C8—41no—
83C22×Q8—61no—
84C22×C8—62no—
85C22×C8—61no—
86C4⋊Q8—32no—
87C4⋊Q8—62no—
88C4⋊Q8—61no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—62no—
93C2×C4⋊C4#6712yes—
94C2×C4⋊C4#6712yes—
95C2×C4⋊C4#6712yes—
96C2×C42C22×D311yescharacteristic
97#103—124no—
98#97—62no—
99#97—62no—
100#97—62no—
101#95—22no—
102#95—22no—
103#95—22no—
104#95—22no—
105C23×C3#12011yescharacteristic
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—42no—
111C2×C3×C4—42no—
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×C4C22⋊Q812yes—
120C2×C3×C4C22⋊Q811yescharacteristic
121C2×C3×C4#12412yes—
122C2×C3×C4#12411yescharacteristic
123C2×Dic3—42no—
124C2×Dic3—81no—
125C2×Dic3—81no—
126Dic6—44no—
127Dic6—82no—
128C3⋊C8—42no—
129C3⋊C8—44no—
130C2×Q8—62no—
131C2×Q8—122no—
132C2×Q8—122no—
133C2×Q8—121no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—121no—
138C22×C4C2×#6712yes—
139C22×C4C2×#6711yescharacteristic
140C2×C8—62no—
141C2×C8—64no—
142C2×C8—122no—
143C2×C8—121no—
144C2×C8—122no—
145C2×C8—121no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—64no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C4⋊C4—82no—
153C4⋊C4—122no—
154C42D3×Q811yescharacteristic
155C42#23912yes—
156C42#23911yescharacteristic
157C22×C3#39311yescharacteristic
158C22×C3#45311yescharacteristic
159C22×C3#45411yescharacteristic
160C22×C3#45512yes—
161C22×C3#45612yes—
162C3×C4—21no—
163C3×C4—21no—
164C3×C4—22no—
165C3×C4—22no—
166C3×C4—82no—
167C3×C4—82no—
168C3×C4—82no—
169Dic3—82no—
170C23#95311yescharacteristic
171Q8—124no—
172Q8—242no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—122no—
187C2×C4—241no—
188C2×C4—241no—
189C2×C4#94612yes—
190C2×C4#94611yescharacteristic
191C2×C4#94712yes—
192C2×C4#94711yescharacteristic
193C8—122no—
194C8—124no—
195C2×C3#166112yes—
196C2×C3#166211yescharacteristic
197C2×C3#166311yescharacteristic
198C2×C3#202711yescharacteristic
199C2×C3#202812yes—
200C22#523111yescharacteristic
201C22#524111yescharacteristic
202C22#524212yes—
203C22#524511yescharacteristic
204C22#524612yes—
205C4—21no—
206C4—21no—
207C4—22no—
208C4—22no—
209C4—82no—
210C4—82no—
211C4—82no—
212C4—242no—
213C3#1036111yescharacteristic, complemented, Sylow
214C2#7190311yescharacteristic
215C2#7190512yes—
216C2#7195312yes—
217C2#7195411yescharacteristic
218C2#7196111yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2845C2
socle, prime radicalC23×C3#120
hypercenter#2845D3
hypocenterC3#10361
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2845

Complements

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

NH ≅ G/Ntype
213. C3#10361semidirect

Maximal subgroups

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

subgroupindex
C3×#28452
#695552
#695522
#676302
#103613