Finite Group Databasebeta

← #10698683

Subgroups — #10698683

Order 768.

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

normal subgroups only

This group has 402 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#69550C211yesmaximal, characteristic
5#67630C211yesmaximal, characteristic
6#10429—31nomaximal, Sylow
7C2×C3×#125C2212yes—
8C3×#400—22no—
9C3×#398—21no—
10C3×#398—21no—
11C3×#398—22no—
12C3×#382C2211yescharacteristic
13C2×#808—21no—
14C2×#807—21no—
15#4935C2211yescharacteristic
16C2×#803C2212yes—
17C2×#803C2211yescharacteristic
18#4543—24no—
19#4529—22no—
20#4529—22no—
21#2822D311yescharacteristic
22#1992—32no—
23#1847—31no—
24C2×#330—32no—
25#1295—31no—
26C3×#125—22no—
27C3×#125—22no—
28C3×#125—22no—
29C3×#125—22no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4D412yes—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C42C2311yescharacteristic
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3—22no—
41C22×C3⋊C8—21no—
42C22×C3⋊C8—22no—
43C3×#95—22no—
44C3×#95—22no—
45C3×#95—21no—
46C3×#95—21no—
47C3×#95—22no—
48#808—42no—
49#807—42no—
50#803—21no—
51#803—22no—
52#803—22no—
53#803D412yes—
54C2×#125C2×D312yes—
55C2×#107—61no—
56C2×#106—61no—
57C2×#97—31no—
58C2×#97—32no—
59#400—22no—
60#398—21no—
61#398—21no—
62#398—22no—
63#384—31no—
64#382C2×D311yescharacteristic
65#340—62no—
66#340—62no—
67#330—64no—
68C22×C3×C4—21no—
69C22×C3×C4—21no—
70C22×C3×C4—22no—
71C22×C3×C4—22no—
72C22×C3×C4C2×D412yes—
73C22×C3×C4C2×D411yescharacteristic
74C22×Dic3—41no—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—22no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—41no—
80C3×C4⋊C4—41no—
81C3×C4⋊C4—42no—
82C3×C4⋊C4—42no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4—42no—
85C3×C4⋊C4—81no—
86C3×C4⋊C4—81no—
87C3×C42C2×D411yescharacteristic
88C3×C42#4111yescharacteristic
89C3×C42#4112yes—
90C4⋊Dic3—41no—
91C4⋊Dic3—41no—
92C4⋊Dic3—44no—
93C2×C3⋊C8—22no—
94C2×C3⋊C8—22no—
95C2×C3⋊C8—22no—
96C2×C3⋊C8—24no—
97C2×C3⋊C8—42no—
98C2×C3⋊C8—42no—
99C22×C8—61no—
100C22×C8—62no—
101#125—22no—
102#125—22no—
103#125—22no—
104#125—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—61no—
111C2×C4⋊C4#6712yes—
112C2×C4⋊C4#6712yes—
113C2×C4⋊C4#6711yescharacteristic
114C2×C4⋊C4#6711yescharacteristic
115C2×C42C22×D311yescharacteristic
116#107—122no—
117#106—122no—
118#97—32no—
119#97—62no—
120#97—61no—
121#97—62no—
122#95—22no—
123#95—22no—
124#95—21no—
125#95—21no—
126#95—22no—
127C23×C3#12011yescharacteristic
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—41no—
135C2×C3×C4—41no—
136C2×C3×C4—42no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4#12111yescharacteristic
143C2×C3×C4#12112yes—
144C2×C3×C4#12312yes—
145C2×C3×C4#12411yescharacteristic
146C2×Dic3—81no—
147C2×Dic3—82no—
148C3⋊C8—24no—
149C3⋊C8—44no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—121no—
155C22×C4C2×#6712yes—
156C22×C4C2×#6711yescharacteristic
157C2×C8—62no—
158C2×C8—62no—
159C2×C8—62no—
160C2×C8—64no—
161C2×C8—122no—
162C2×C8—122no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—121no—
176C4⋊C4—121no—
177C4⋊C4—124no—
178C42D3×D411yescharacteristic
179C42#23711yescharacteristic
180C42#23712yes—
181C22×C3#39411yescharacteristic
182C22×C3#45311yescharacteristic
183C22×C3#45412yes—
184C22×C3#45612yes—
185C22×C3#45611yescharacteristic
186C3×C4—22no—
187C3×C4—22no—
188C3×C4—21no—
189C3×C4—21no—
190C3×C4—81no—
191C3×C4—81no—
192C3×C4—82no—
193C3×C4—82no—
194Dic3—161no—
195C23#95311yescharacteristic
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—241no—
211C2×C4—242no—
212C2×C4#93512yes—
213C2×C4#93611yescharacteristic
214C2×C4#93811yescharacteristic
215C2×C4#93912yes—
216C8—64no—
217C8—124no—
218C2×C3#165711yescharacteristic
219C2×C3#166011yescharacteristic
220C2×C3#166312yes—
221C2×C3#203812yes—
222C2×C3#203911yescharacteristic
223C22#522211yescharacteristic
224C22#524111yescharacteristic
225C22#524512yes—
226C22#524612yes—
227C22#524611yescharacteristic
228C4—22no—
229C4—22no—
230C4—21no—
231C4—21no—
232C4—81no—
233C4—81no—
234C4—82no—
235C4—82no—
236C4—481no—
237C3#1042911yescharacteristic, complemented, Sylow
238C2#7186312yes—
239C2#7187611yescharacteristic
240C2#7195511yescharacteristic
241C2#7195611yescharacteristic
242C2#7196112yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2822C2
socle, prime radicalC23×C3#120
hypercenter#2822D3
hypocenterC3#10429
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
237. C3#10429semidirect

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
#695502
#676302
#104293