Finite Group Databasebeta

← #10697414

Subgroups — #10697414

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
1#79319C212yesmaximal
2#79069C211yesmaximal, characteristic, complemented
3#70836C212yesmaximal, complemented
4#70513C211yesmaximal, characteristic
5C3×#1307C211yesmaximal, characteristic, complemented
6#10216—31nomaximal, Sylow
7#5477C2211yescharacteristic
8#5470C2212yes—
9C3×#432C2212yes—
10C3×#423C2211yescharacteristic
11C2×#825—22no—
12#5137—22no—
13#4932C2211yescharacteristic
14#4851—22no—
15#4849—22no—
16#4846—22no—
17#4842—22no—
18#4737—24no—
19#4713—22no—
20#4702—22no—
21#4467—24no—
22#3255—31no—
23#3235—32no—
24#1963—31no—
25#1506—32no—
26#1307D311yescharacteristic, complemented
27C2×#235—21no—
28C2×C3×M16D412yes—
29C2×C3×C42C2311yescharacteristic
30#930—22no—
31#926—22no—
32#924—22no—
33#924—22no—
34C2×C4×Dic3—21no—
35C2×Dic12—22no—
36#875—21no—
37#874D411yescharacteristic
38C4×D12—22no—
39#870—22no—
40#869D411yescharacteristic
41#869D412yes—
42#869D412yes—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C3×C4⋊C8—22no—
46C3×C4⋊C8—22no—
47C3×C22⋊C8D414yes—
48C3×#97—22no—
49C3⋊#95—22no—
50#825—44no—
51#821—24no—
52#821—24no—
53#818—24no—
54#817—22no—
55#817—24no—
56#816—24no—
57#533—32no—
58#532—31no—
59#501—62no—
60#490—62no—
61#482—64no—
62#476—62no—
63#475—62no—
64#474—62no—
65#470—62no—
66#462—62no—
67#432C2×D312yes—
68#423C2×D311yescharacteristic
69C2×#104—62no—
70#376—31no—
71#328—64no—
72C22×C3×C4C2×D411yescharacteristic
73C22×C3×C4C2×D412yes—
74C2×#67—41no—
75C22×Dic3—41no—
76#235—42no—
77#235—42no—
78C2×D12—22no—
79C2×D12—21no—
80C2×C4×D3—41no—
81C2×Dic6—22no—
82C2×Dic6—21no—
83C2×Dic6—22no—
84C3×M16—42no—
85C2×C3×C8—22no—
86C2×C3×C8—24no—
87C3×C42C2×D411yescharacteristic
88C3×C42C2×D412yes—
89C3×C42#4111yescharacteristic
90#217—42no—
91(C2×D3)⋊C4—42no—
92C4⋊Dic3—22no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97Dic3⋊C4—42no—
98Dic3⋊C4—42no—
99C4×Dic3—22no—
100C4×Dic3—41no—
101C4×Dic3—42no—
102Dic12—82no—
103C2×#41—61no—
104C2×Q16—62no—
105C2×M16D1212yes—
106C4⋊Q8—31no—
107C4⋊Q8—32no—
108C4⋊Q8—32no—
109#127—31no—
110#125—62no—
111#124—61no—
112C22⋊Q8—62no—
113C22⋊Q8—62no—
114#121—62no—
115C4×Q8—62no—
116C4×Q8—62no—
117C4×D4—62no—
118#117—62no—
119C2×C42—61no—
120C2×C42C22×D311yescharacteristic
121#107—64no—
122#106—62no—
123#106—64no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126#104—124no—
127#103—64no—
128#102—64no—
129#102—64no—
130C22⋊C8D1214yes—
131#97—22no—
132#95—62no—
133C23×C3#12711yescharacteristic
134C22×D3—41no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—42no—
138C2×C3×C4#12012yes—
139C2×C3×C4#12011yescharacteristic
140C2×C3×C4#12112yes—
141C2×C3×C4#12111yescharacteristic
142#67—82no—
143C2×Dic3—42no—
144C2×Dic3—42no—
145C2×Dic3—42no—
146C2×Dic3—41no—
147C2×Dic3—82no—
148D12—44no—
149D12—42no—
150C4×D3—81no—
151C4×D3—81no—
152Dic6—42no—
153Dic6—42no—
154Dic6—42no—
155Dic6—82no—
156C3×C8—44no—
157C3×C8—42no—
158#41—122no—
159#41—122no—
160C2×Q8—62no—
161C2×Q8—61no—
162C2×Q8—62no—
163C2×D4—62no—
164C2×D4—61no—
165C2×D4—121no—
166C22×C4—121no—
167C22×C4—121no—
168C22×C4C2×D1211yescharacteristic
169C22×C4C2×D1212yes—
170Q16—242no—
171M16—42no—
172C2×C8—24no—
173C2×C8—22no—
174C4⋊C4—62no—
175C4⋊C4—62no—
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—122no—
180C4⋊C4—122no—
181C22⋊C4—122no—
182C22⋊C4—122no—
183C42—62no—
184C42—121no—
185C42—122no—
186C42D3×D411yescharacteristic
187C42D3×D412yes—
188C42#23911yescharacteristic
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#45311yescharacteristic
198C3×C4#45411yescharacteristic
199Dic3—82no—
200Dic3—81no—
201Dic3—82no—
202Dic3—82no—
203C23—121no—
204C23#87411yescharacteristic
205Q8—122no—
206Q8—122no—
207Q8—122no—
208Q8—242no—
209D4—122no—
210D4—124no—
211D4—242no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—121no—
219C2×C4—242no—
220C2×C4—241no—
221C2×C4—241no—
222C2×C4#88212yes—
223C2×C4#88211yescharacteristic
224C2×C4#89512yes—
225C2×C4#89511yescharacteristic
226C8—44no—
227C8—42no—
228C2×C3—41no—
229C2×C3#200611yescharacteristic
230C2×C3#200912yes—
231D3—161no—
232C22—21no—
233C22—22no—
234C22—241no—
235C22—242no—
236C22#467111yescharacteristic
237C4—21no—
238C4—22no—
239C4—22no—
240C4—242no—
241C4—241no—
242C4—242no—
243C4—242no—
244C4#474711yescharacteristic
245C4#475311yescharacteristic
246C3#1021611yescharacteristic, complemented, Sylow
247C2—41no—
248C2—481no—
249C2#6900811yescharacteristic
250C2#6903112yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1307C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1307D3
hypocenterC3#10216
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #79069C2semidirect
3. #70836C2semidirect
5. C3×#1307C2semidirect
26. #1307D3semidirect
246. C3#10216semidirect

Maximal subgroups

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

subgroupindex
#793192
#790692
#708362
#705132
C3×#13072
#102163