Finite Group Databasebeta

← #10795166

Subgroups — #10795166

Order 768.

1166 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 267 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×#5812C211yesmaximal, characteristic
2#78620C212yesmaximal
3C3×#1784C211yesmaximal, characteristic
4#72133C211yesmaximal, characteristic
5#70097C214yesmaximal, complemented
6Dic3×#101C214yesmaximal, complemented
7#69974C212yesmaximal
8#20730—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10#5812—28no—
11C2×Q8×Dic3C414yes—
12C2×Q8×Dic3C2212yes—
13C2×Q8×Dic3C2212yes—
14C2×#926C414yes—
15C2×#926C2211yescharacteristic
16C2×#926C2211yescharacteristic
17C2×C3×#101C2214yes—
18C3×#407C2212yes—
19C2×#836C2214yes—
20#5207C2212yes—
21#5118C2214yes—
22Dic3×M16C2214yes—
23#4720—28no—
24#4554C2218yes—
25C2×#519—31no—
26#2921—32no—
27#1822—32no—
28#1801—34no—
29#1784—31no—
30#1784D311yescharacteristic
31C4×#101—34no—
32C22×C3×Q8—28no—
33C22×C3×Q8—24no—
34C22×C3×Q8C2×C412yes—
35C22×C3×Q8C2312yes—
36C23×C3×C4C2×C412yes—
37C23×C3×C4C2311yescharacteristic
38C2×C3×M16C2314yes—
39C2×#217C2×C412yes—
40C2×#217C2×C412yes—
41Q8×Dic3—216no—
42Q8×Dic3—28no—
43Q8×Dic3—28no—
44#926—24no—
45#926—216no—
46#926D418yes—
47C2×C4⋊Dic3C2×C414yes—
48C2×C4⋊Dic3C2×C414yes—
49C2×C4⋊Dic3C2×C412yes—
50C2×C4⋊Dic3C2312yes—
51C2×C4×Dic3C2×C414yes—
52C2×C4×Dic3C2×C414yes—
53C2×C4×Dic3C2×C412yes—
54C2×C4×Dic3C2311yescharacteristic
55C2×C4×Dic3C2311yescharacteristic
56C2×#208C2314yes—
57C3×#101—28no—
58C3×C22⋊C8—24no—
59#836—28no—
60#830—24no—
61#815—24no—
62Dic3⋊C8—28no—
63C8×Dic3—24no—
64C23×Q8C2×D311yescharacteristic
65#519—68no—
66C2×C4×Q8—34no—
67C2×C4×Q8—32no—
68C2×C4×Q8—32no—
69C2×#117—31no—
70C2×#117—34no—
71C2×#117—31no—
72#433—68no—
73#423—34no—
74C2×#101—34no—
75C2×#101C2×D314yes—
76#407—32no—
77#407C2×D312yes—
78C4×M16—34no—
79#343—38no—
80C24×C3C22×C411yescharacteristic
81C2×C3×Q8—232no—
82C2×C3×Q8—24no—
83C2×C3×Q8—28no—
84C2×C3×Q8—24no—
85C2×C3×Q8—24no—
86C2×C3×Q8#4118yes—
87C22×C3×C4—24no—
88C22×C3×C4—28no—
89C22×C3×C4—24no—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C22×C414yes—
92C22×C3×C4C2×D414yes—
93C22×C3×C4C2411yescharacteristic
94C22×Dic3C22×C414yes—
95C22×Dic3C22×C412yes—
96C22×Dic3C22×C412yes—
97C3×M16—28no—
98C2×C3×C8—24no—
99#217—24no—
100#217C22⋊C418yes—
101C4⋊Dic3—24no—
102C4⋊Dic3—28no—
103C4⋊Dic3—28no—
104C4⋊Dic3C22⋊C418yes—
105C4×Dic3—24no—
106C4×Dic3—28no—
107C4×Dic3—28no—
108C4×Dic3C2×D414yes—
109C4×Dic3C2×D414yes—
110#208—28no—
111C2×C3⋊C8—24no—
112C22×Q8—28no—
113C22×Q8—24no—
114C22×Q8C4×D312yes—
115C22×Q8C22×D312yes—
116C23×C4C4×D312yes—
117C23×C4C22×D311yescharacteristic
118C2×M16—34no—
119C2×M16C22×D314yes—
120C4×Q8—68no—
121C4×Q8—68no—
122C4×Q8—616no—
123#117—38no—
124#117—616no—
125#117—64no—
126C2×C4⋊C4—32no—
127C2×C4⋊C4—34no—
128C2×C4⋊C4—34no—
129C2×C4⋊C4—32no—
130C2×C22⋊C4—32no—
131C2×C22⋊C4—32no—
132C2×C42—31no—
133C2×C42—31no—
134C2×C42—34no—
135C2×C42—34no—
136C2×C42—32no—
137C4⋊C8—68no—
138#101—28no—
139#101—68no—
140C22⋊C8—24no—
141C22⋊C8—64no—
142#97—64no—
143C4×C8—64no—
144C23×C3—22no—
145C23×C3—42no—
146C23×C3C2×C22⋊C412yes—
147C23×C3C23×C411yescharacteristic
148C3×Q8—216no—
149C3×Q8—216no—
150C2×C3×C4—216no—
151C2×C3×C4—24no—
152C2×C3×C4—22no—
153C2×C3×C4—24no—
154C2×C3×C4—28no—
155C2×C3×C4C2×C22⋊C412yes—
156C2×C3×C4#11718yes—
157C2×C3×C4C22×D412yes—
158C2×C3×C4C2×#4114yes—
159C2×Dic3—24no—
160C2×Dic3—42no—
161C2×Dic3—42no—
162C2×Dic3—44no—
163C2×Dic3C2×C22⋊C414yes—
164C2×Dic3C2×C22⋊C414yes—
165C3×C8—44no—
166C3⋊C8—44no—
167C24C2×C4×D311yescharacteristic
168C2×Q8—232no—
169C2×Q8—28no—
170C2×Q8—24no—
171C2×Q8—24no—
172C2×Q8—24no—
173C2×Q8#23718yes—
174C22×C4—24no—
175C22×C4—28no—
176C22×C4—24no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4—34no—
180C22×C4C2×C4×D314yes—
181C22×C4C2×C4×D311yescharacteristic
182C22×C4D3×D414yes—
183C22×C4C23×D311yescharacteristic
184M16—28no—
185M16—68no—
186C2×C8—24no—
187C2×C8—64no—
188C4⋊C4—38no—
189C4⋊C4—68no—
190C4⋊C4—68no—
191C4⋊C4—64no—
192C22⋊C4—38no—
193C22⋊C4—64no—
194C42—34no—
195C42—34no—
196C42—68no—
197C42—68no—
198C42—64no—
199C22×C3—22no—
200C22×C3—24no—
201C22×C3—22no—
202C22×C3—44no—
203C22×C3C22×C22⋊C411yescharacteristic
204C22×C3C2×#11712yes—
205C3×C4—22no—
206C3×C4—28no—
207C3×C4#52514yes—
208Dic3—24no—
209Dic3—24no—
210Dic3—44no—
211C23—22no—
212C23—42no—
213C23D3×C22⋊C412yes—
214C23C22×C4×D311yescharacteristic
215Q8—216no—
216Q8—216no—
217C2×C4—216no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—28no—
222C2×C4—34no—
223C2×C4—34no—
224C2×C4—64no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—124no—
228C2×C4#87718yes—
229C2×C4D3×C22⋊C412yes—
230C2×C4C2×D3×D412yes—
231C2×C4C2×#23714yes—
232C8—44no—
233C8—124no—
234C2×C3—22no—
235C2×C3—42no—
236C2×C3#228011yescharacteristic
237C2×C3#288912yes—
238C22—22no—
239C22—22no—
240C22—24no—
241C22—44no—
242C22C2×#87712yes—
243C22C2×D3×C22⋊C411yescharacteristic
244C4—28no—
245C4—22no—
246C4—64no—
247C4—64no—
248C4—124no—
249C4#558614yes—
250C3#2073011yescharacteristic, complemented, Sylow
251C2—22no—
252C2—42no—
253C2#7381711yescharacteristic
254C2#7870612yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1784C2
socleC22×C3C22×C22⋊C4
prime radicalC24×C3C22×C4
cosocleC22×C3×C4C24
hypercenter#1784D3
hypocenterC3#20730
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
5. #70097C2semidirect
6. Dic3×#101C2semidirect
250. C3#20730semidirect

Maximal subgroups

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

subgroupindex
C2×#58122
#786202
C3×#17842
#721332
#700972
Dic3×#1012
#699742
#207303