Finite Group Databasebeta

← #10696510

Subgroups — #10696510

Order 768.

600 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#5089C212yesmaximal, complemented
2#74254C211yesmaximal, characteristic
3C3×#1960C211yesmaximal, characteristic, complemented
4C2×#4464C212yesmaximal, complemented
5#67630C211yesmaximal, characteristic, complemented
6#10085—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×#875C2212yes—
9C2×C3×#103—22no—
10C2×C3×#97C2211yescharacteristic
11C3×#382C2211yescharacteristic
12#5089—24no—
13#5089—24no—
14#4991—22no—
15C2×#810—22no—
16C2×#809—22no—
17C2×#803C2212yes—
18#4673—22no—
19#4671—22no—
20#4464—24no—
21C2×#489—32no—
22#2819—31no—
23#1960D311yescharacteristic, complemented
24C2×#329—32no—
25#1295—31no—
26C22×C3×Q8—21no—
27C22×D12—22no—
28C22×D12D412yes—
29C22×Dic6D412yes—
30C22×C3×C8—21no—
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—21no—
33C3×C4⋊Q8D412yescomplemented
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C42C2311yescharacteristic
37C2×(C2×D3)⋊C4—22no—
38C2×(C2×D3)⋊C4—22no—
39C2×(C2×D3)⋊C4—24no—
40C22×C3⋊C8—22no—
41#875—22no—
42#875—22no—
43#875—22no—
44#875—22no—
45C3×#103—44no—
46C3×#97—22no—
47C3×#95—22no—
48C3⋊#95—22no—
49#810—28no—
50#809—28no—
51#803—22no—
52#803D414yescomplemented
53C2×C4⋊Q8C2×D311yescharacteristic
54C2×#124—32no—
55#489—64no—
56#489—64no—
57C2×#103—22no—
58C2×#103—62no—
59C2×#102—62no—
60C2×#97—32no—
61C2×#97C2×D311yescharacteristic
62#394—62no—
63#393—62no—
64#392—62no—
65#382C2×D311yescharacteristic
66#329—64no—
67C23×D3—22no—
68C2×C3×Q8—22no—
69C2×C3×Q8—42no—
70C2×C3×Q8—42no—
71C2×C3×Q8—41no—
72C22×C3×C4—22no—
73C22×C3×C4—41no—
74C22×C3×C4C2×D412yes—
75C22×C3×C4C2×D411yescharacteristic
76C22×Dic3—22no—
77C2×D12—22no—
78C2×D12—22no—
79C2×D12—44no—
80C2×D12—82no—
81C2×D12—84no—
82C2×Dic6—22no—
83C2×Dic6—22no—
84C2×Dic6—44no—
85C2×C3×C8—22no—
86C2×C3×C8—41no—
87C2×C3×C8—41no—
88C3×C4⋊C4—24no—
89C3×C4⋊C4—42no—
90C3×C4⋊C4—42no—
91C3×C42C2×D411yescharacteristic
92C3×C42C2×D412yes—
93C3×C42#4111yescharacteristic
94(C2×D3)⋊C4—42no—
95(C2×D3)⋊C4—42no—
96(C2×D3)⋊C4—42no—
97(C2×D3)⋊C4—42no—
98(C2×D3)⋊C4—84no—
99C2×C3⋊C8—24no—
100C2×C3⋊C8—44no—
101C22×Q8—21no—
102C22×Q8—32no—
103C22×D4—32no—
104C22×D4—62no—
105C22×C8—21no—
106C22×C8—62no—
107C4⋊Q8—21no—
108C4⋊Q8—22no—
109C4⋊Q8D1212yescomplemented
110#124—62no—
111#124—62no—
112#124—62no—
113#124—62no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4#6712yes—
116C2×C22⋊C4—62no—
117C2×C22⋊C4—62no—
118C2×C22⋊C4—64no—
119C2×C42C22×D311yescharacteristic
120#103—44no—
121#103—68no—
122#102—68no—
123#97—22no—
124#97—34no—
125#97—62no—
126#95—22no—
127#95—62no—
128C23×C3#12011yescharacteristic
129C22×D3—42no—
130C22×D3—42no—
131C22×D3—42no—
132C22×D3—44no—
133C22×D3—44no—
134C3×Q8—44no—
135C3×Q8—82no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—24no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—44no—
142C2×C3×C4—81no—
143C2×C3×C4—81no—
144C2×C3×C4#12012yes—
145C2×C3×C4#12112yes—
146C2×C3×C4#12111yescharacteristic
147C2×C3×C4#12711yescharacteristic
148C2×Dic3—42no—
149C2×Dic3—44no—
150D12—44no—
151D12—82no—
152D12—88no—
153Dic6—44no—
154Dic6—82no—
155C3×C8—42no—
156C3⋊C8—44no—
157C24—62no—
158C2×Q8—22no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—41no—
162C2×Q8—62no—
163C2×Q8—62no—
164C2×Q8—124no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—124no—
168C2×D4—244no—
169C2×D4—242no—
170C22×C4—22no—
171C22×C4—41no—
172C22×C4—62no—
173C22×C4D3×D412yes—
174C22×C4C2×#6711yescharacteristic
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—64no—
179C2×C8—124no—
180C4⋊C4—24no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C22⋊C4—122no—
184C22⋊C4—122no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C22⋊C4—244no—
188C42C2×D1211yescharacteristic
189C42#23511yescharacteristic
190C42D3×D412yes—
191C22×C3#39211yescharacteristic
192C22×C3#45311yescharacteristic
193C22×C3#45411yescharacteristic
194C22×C3#45514yes—
195C2×D3—84no—
196C2×D3—84no—
197C2×D3—82no—
198C2×D3—82no—
199C2×D3—82no—
200C3×C4—22no—
201C3×C4—22no—
202C3×C4—21no—
203C3×C4—21no—
204C3×C4—82no—
205C3×C4—82no—
206Dic3—82no—
207C23—124no—
208C23—124no—
209C23—122no—
210C23—122no—
211C23—122no—
212C23#93711yescharacteristic
213Q8—44no—
214Q8—82no—
215Q8—124no—
216Q8—242no—
217D4—124no—
218D4—242no—
219D4—248no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—124no—
229C2×C4—122no—
230C2×C4#88212yes—
231C2×C4#88412yes—
232C2×C4#93011yescharacteristic
233C2×C4#94011yescharacteristic
234C8—42no—
235C8—124no—
236C2×C3#165912yes—
237C2×C3#166212yes—
238C2×C3#200412yes—
239C2×C3#200611yescharacteristic
240D3—162no—
241C22—242no—
242C22—244no—
243C22—244no—
244C22—242no—
245C22—242no—
246C22#496011yescharacteristic
247C22#506211yescharacteristic
248C22#507711yescharacteristic
249C22#520214yes—
250C4—21no—
251C4—21no—
252C4—22no—
253C4—22no—
254C4—82no—
255C4—82no—
256C4—242no—
257C3#1008511yescharacteristic, complemented, Sylow
258C2—482no—
259C2#6966411yescharacteristic
260C2#7034412yes—
261C2#7125112yes—
262C2#7132912yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1960C2
socleC23×C3#120
prime radical#74254C2
hypercenter#1960D3
hypocenterC3#10085
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. C2×#5089C2semidirect
3. C3×#1960C2semidirect
4. C2×#4464C2semidirect
5. #67630C2semidirect
23. #1960D3semidirect
33. C3×C4⋊Q8D4semidirect
52. #803D4semidirect
109. C4⋊Q8D12semidirect
257. C3#10085semidirect

Maximal subgroups

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

subgroupindex
C2×#50892
#742542
C3×#19602
C2×#44642
#676302
#100853