Finite Group Databasebeta

← #10697390

Subgroups — #10697390

Order 768.

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

normal subgroups only

This group has 536 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#79350C212yesmaximal, complemented
2#79064C211yesmaximal, characteristic, complemented
3#70832C212yesmaximal, complemented
4#70513C211yesmaximal, characteristic, complemented
5C3×#1307C211yesmaximal, characteristic, complemented
6#10212—31nomaximal, Sylow
7#5477C2211yescharacteristic
8#5477C2212yes—
9C3×#432C2212yes—
10C3×#423C2211yescharacteristic
11C2×#825—22no—
12#5135—22no—
13#4932C2211yescharacteristic
14#4843—22no—
15#4843—22no—
16#4842—22no—
17#4841—22no—
18#4734—24no—
19#4714—22no—
20#4712—22no—
21#4470—24no—
22#3253—31no—
23#3214—32no—
24#1963—31no—
25#1505—32no—
26#1307D311yescharacteristic, complemented
27C2×#235—21no—
28C2×#235—22no—
29C2×C3×M16D412yes—
30C2×C3×C42C2311yescharacteristic
31#930—22no—
32#930—22no—
33#930—22no—
34C2×C4×Dic3—21no—
35C2×D24—22no—
36C2×D24—24no—
37C2×#204—24no—
38#875—22no—
39#875—21no—
40#874D411yescharacteristic
41#874D412yes—
42C4×D12—22no—
43C4×D12—22no—
44C4×D12—22no—
45#869D411yescharacteristic
46#869D412yes—
47C3×C4⋊C8—22no—
48C3×C4⋊C8—22no—
49C3×C22⋊C8D414yes—
50C3×#97—22no—
51C3⋊#95—22no—
52#825—44no—
53#821—24no—
54#821—24no—
55#817—22no—
56#816—24no—
57#532—31no—
58#532—32no—
59#496—62no—
60#489—62no—
61#476—62no—
62#469—62no—
63#461—62no—
64#459—62no—
65#459—62no—
66#449—64no—
67#432C2×D312yes—
68#423C2×D311yescharacteristic
69C2×#104—62no—
70#376—31no—
71#327—64no—
72C22×C3×C4C2×D411yescharacteristic
73C22×C3×C4C2×D412yes—
74C2×#67—41no—
75C2×#67—42no—
76C22×Dic3—41no—
77#235—42no—
78#235—42no—
79#235—84no—
80C2×D12—22no—
81C2×D12—22no—
82C2×D12—22no—
83C2×D12—21no—
84C2×D12—22no—
85C2×C4×D3—41no—
86C2×C4×D3—42no—
87C2×Dic6—22no—
88C2×Dic6—21no—
89C3×M16—42no—
90C2×C3×C8—24no—
91C2×C3×C8—22no—
92C3×C42C2×D411yescharacteristic
93C3×C42C2×D412yes—
94C3×C42#4111yescharacteristic
95(C2×D3)⋊C4—42no—
96(C2×D3)⋊C4—42no—
97(C2×D3)⋊C4—42no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101C4×Dic3—22no—
102C4×Dic3—41no—
103D24—84no—
104D24—82no—
105#204—84no—
106C2×#41—61no—
107C2×#41—62no—
108C2×SD16—64no—
109C2×D8—64no—
110C2×D8—62no—
111C2×M16D1212yes—
112C4⋊Q8—32no—
113C4⋊Q8—31no—
114#127—32no—
115#127—31no—
116#124—62no—
117#124—61no—
118#121—62no—
119#121—62no—
120#121—62no—
121C4×D4—62no—
122C4×D4—62no—
123C4×D4—62no—
124C2×C42—61no—
125C2×C42C22×D311yescharacteristic
126#106—62no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129#104—124no—
130#103—64no—
131#102—64no—
132#102—64no—
133C22⋊C8D1214yes—
134#97—22no—
135#95—62no—
136C23×C3#12711yescharacteristic
137C22×D3—41no—
138C22×D3—42no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—42no—
142C2×C3×C4#12012yes—
143C2×C3×C4#12011yescharacteristic
144C2×C3×C4#12112yes—
145C2×C3×C4#12111yescharacteristic
146#67—82no—
147#67—84no—
148C2×Dic3—42no—
149C2×Dic3—41no—
150C2×Dic3—42no—
151C2×Dic3—82no—
152D12—42no—
153D12—42no—
154D12—44no—
155D12—42no—
156D12—42no—
157D12—42no—
158D12—82no—
159C4×D3—84no—
160C4×D3—81no—
161C4×D3—81no—
162Dic6—44no—
163Dic6—42no—
164C3×C8—44no—
165C3×C8—42no—
166#41—122no—
167#41—122no—
168#41—244no—
169C2×Q8—62no—
170C2×Q8—61no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—62no—
174C2×D4—61no—
175C2×D4—62no—
176C2×D4—121no—
177C2×D4—122no—
178C22×C4—121no—
179C22×C4—121no—
180C22×C4—122no—
181C22×C4C2×D1211yescharacteristic
182C22×C4C2×D1212yes—
183SD16—244no—
184D8—244no—
185D8—242no—
186M16—42no—
187C2×C8—24no—
188C2×C8—22no—
189C4⋊C4—62no—
190C4⋊C4—62no—
191C4⋊C4—62no—
192C22⋊C4—122no—
193C22⋊C4—122no—
194C22⋊C4—122no—
195C42—62no—
196C42—121no—
197C42D3×D411yescharacteristic
198C42D3×D412yes—
199C42#23911yescharacteristic
200C22×C3—21no—
201C22×C3—22no—
202C22×C3#39211yescharacteristic
203C2×D3—81no—
204C2×D3—82no—
205C2×D3—82no—
206C2×D3—82no—
207C2×D3—82no—
208C3×C4—21no—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4#45311yescharacteristic
212C3×C4#45411yescharacteristic
213Dic3—82no—
214Dic3—81no—
215Dic3—82no—
216C23—121no—
217C23—122no—
218C23#87411yescharacteristic
219Q8—124no—
220Q8—122no—
221D4—122no—
222D4—124no—
223D4—122no—
224D4—122no—
225D4—122no—
226D4—122no—
227D4—242no—
228D4—242no—
229D4—244no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—42no—
233C2×C4—122no—
234C2×C4—121no—
235C2×C4—122no—
236C2×C4—244no—
237C2×C4—242no—
238C2×C4—241no—
239C2×C4—241no—
240C2×C4#88212yes—
241C2×C4#88211yescharacteristic
242C2×C4#89511yescharacteristic
243C2×C4#89512yes—
244C8—44no—
245C8—42no—
246C2×C3—41no—
247C2×C3#200611yescharacteristic
248C2×C3#200812yes—
249D3—161no—
250D3—162no—
251C22—21no—
252C22—22no—
253C22—241no—
254C22—242no—
255C22—242no—
256C22—242no—
257C22—242no—
258C22#467111yescharacteristic
259C4—21no—
260C4—22no—
261C4—22no—
262C4—242no—
263C4—241no—
264C4—242no—
265C4#474711yescharacteristic
266C4#475311yescharacteristic
267C3#1021211yescharacteristic, complemented, Sylow
268C2—41no—
269C2—481no—
270C2—482no—
271C2#6900811yescharacteristic
272C2#6902812yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1307C2
socleC22×C3#392
hypercenter#1307D3
hypocenterC3#10212
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
1. #79350C2semidirect
2. #79064C2semidirect
3. #70832C2semidirect
4. #70513C2semidirect
5. C3×#1307C2semidirect
26. #1307D3semidirect
267. C3#10212semidirect

Maximal subgroups

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

subgroupindex
#793502
#790642
#708322
#705132
C3×#13072
#102123