Finite Group Databasebeta

← #10697411

Subgroups — #10697411

Order 768.

468 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3255C211yesmaximal, characteristic, complemented
2#83361C212yesmaximal
3#71836C212yesmaximal, complemented
4#70513C211yesmaximal, characteristic
5#67716C211yesmaximal, characteristic, complemented
6#10216—31nomaximal, Sylow
7C3×#533C2212yes—
8#5477C2211yescharacteristic
9C3×#475—22no—
10C3×#470—22no—
11C3×#462—22no—
12C3×#423C2211yescharacteristic
13C2×#825—22no—
14#5087—22no—
15#5084—22no—
16#5067—22no—
17#5064—22no—
18#5003—24no—
19#4932C2211yescharacteristic
20#4926C2212yes—
21#4538—24no—
22#3255D311yescharacteristic, complemented
23#3235—32no—
24#1963—31no—
25#1506—32no—
26#1307—31no—
27C2×#235—21no—
28C2×C3×Q16—22no—
29C2×C3×M16D412yes—
30C3×C4⋊Q8D412yes—
31C3×C4⋊Q8D412yes—
32C3×#125—22no—
33C3×C22⋊Q8—22no—
34C3×C22⋊Q8—22no—
35C3×C4×Q8—22no—
36C3×C4×Q8—22no—
37C3×#117—22no—
38C2×C3×C42C2311yescharacteristic
39#930—22no—
40C2×C4×Dic3—21no—
41#875—21no—
42#874D411yescharacteristic
43C4×D12—22no—
44#869D411yescharacteristic
45C3×#106—22no—
46C3×C4⋊C8—22no—
47C3×#103—24no—
48C3⋊#95—22no—
49#830D414yes—
50#825—44no—
51#809—24no—
52#809—24no—
53#808—24no—
54#807—24no—
55C4⋊(C3⋊C8)—22no—
56#803—22no—
57#533C2×D312yes—
58#532—31no—
59#501—62no—
60#490—62no—
61#482—64no—
62#476—62no—
63#475—22no—
64#474—62no—
65#470—22no—
66#462—22no—
67#432—32no—
68#423C2×D311yescharacteristic
69C2×#104—62no—
70#376—31no—
71#328—64no—
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C22×C3×C4C2×D412yes—
75C22×C3×C4C2×D411yescharacteristic
76C2×#67—41no—
77C22×Dic3—41no—
78#235—42no—
79#235—42no—
80C2×D12—22no—
81C2×D12—21no—
82C2×C4×D3—41no—
83C2×Dic6—21no—
84C3×Q16—82no—
85C3×M16—42no—
86C2×C3×C8—22no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—42no—
92C3×C4⋊C4—42no—
93C3×C22⋊C4—42no—
94C3×C42—42no—
95C3×C42C2×D411yescharacteristic
96C3×C42C2×D412yes—
97C3×C42#4111yescharacteristic
98(C2×D3)⋊C4—42no—
99C4⋊Dic3—22no—
100C4×Dic3—22no—
101C4×Dic3—41no—
102C2×C3⋊C8—24no—
103C2×#41—61no—
104C2×Q16—22no—
105C2×M16#6712yes—
106C4⋊Q8—31no—
107C4⋊Q8#6712yes—
108C4⋊Q8#6712yes—
109#127—31no—
110#125—22no—
111#124—61no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114#121—62no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×D4—62no—
118#117—22no—
119C2×C42—61no—
120C2×C42C22×D311yescharacteristic
121#107—64no—
122#106—22no—
123#106—64no—
124C4⋊C8—22no—
125C4⋊C8—62no—
126#104—124no—
127#103—24no—
128#102—64no—
129#102—64no—
130C22⋊C8—34no—
131#97—62no—
132#95—62no—
133C23×C3#12711yescharacteristic
134C22×D3—41no—
135C3×Q8—42no—
136C3×Q8—42no—
137C3×Q8—82no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4#12012yes—
144C2×C3×C4#12011yescharacteristic
145C2×C3×C4#12111yescharacteristic
146C2×C3×C4#12112yes—
147#67—82no—
148C2×Dic3—42no—
149C2×Dic3—41no—
150C2×Dic3—82no—
151D12—44no—
152D12—42no—
153C4×D3—81no—
154C4×D3—81no—
155Dic6—42no—
156C3×C8—42no—
157C3⋊C8—44no—
158#41—122no—
159#41—122no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—61no—
163C2×D4—62no—
164C2×D4—61no—
165C2×D4—121no—
166C22×C4—121no—
167C22×C4—121no—
168C22×C4D3×D412yes—
169C22×C4C2×#6711yescharacteristic
170Q16—82no—
171M16—42no—
172C2×C8—22no—
173C2×C8—64no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—62no—
181C22⋊C4—42no—
182C22⋊C4—122no—
183C42—42no—
184C42—62no—
185C42—121no—
186C42D3×D411yescharacteristic
187C42#23711yescharacteristic
188C42C2×#6712yes—
189C22×C3—21no—
190C22×C3—22no—
191C22×C3#39211yescharacteristic
192C2×D3—81no—
193C2×D3—82no—
194C3×C4—21no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—82no—
198C3×C4—82no—
199C3×C4#45311yescharacteristic
200C3×C4#45411yescharacteristic
201Dic3—82no—
202Dic3—81no—
203C23—121no—
204C23#94011yescharacteristic
205Q8—42no—
206Q8—42no—
207Q8—82no—
208Q8—122no—
209D4—122no—
210D4—124no—
211D4—242no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—122no—
218C2×C4—121no—
219C2×C4—242no—
220C2×C4—241no—
221C2×C4—241no—
222C2×C4#93712yes—
223C2×C4#93811yescharacteristic
224C2×C4#93912yes—
225C2×C4#95311yescharacteristic
226C8—42no—
227C8—124no—
228C2×C3—41no—
229C2×C3#200611yescharacteristic
230C2×C3#200912yes—
231D3—161no—
232C22—22no—
233C22—21no—
234C22—242no—
235C22—241no—
236C22#522311yescharacteristic
237C4—22no—
238C4—22no—
239C4—21no—
240C4—82no—
241C4—82no—
242C4—242no—
243C4—241no—
244C4#524111yescharacteristic
245C4#524511yescharacteristic
246C3#1021611yescharacteristic, complemented, Sylow
247C2—41no—
248C2—481no—
249C2#7187911yescharacteristic
250C2#7211112yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3255C2
socleC22×C3#392
prime radical#5477C22
hypercenter#3255D3
hypocenterC3#10216
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

Complements

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

NH ≅ G/Ntype
1. C3×#3255C2semidirect
3. #71836C2semidirect
5. #67716C2semidirect
22. #3255D3semidirect
246. C3#10216semidirect

Maximal subgroups

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

subgroupindex
C3×#32552
#833612
#718362
#705132
#677162
#102163