Finite Group Databasebeta

← #10698129

Subgroups — #10698129

Order 768.

754 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 87 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#84065C212yesmaximal
2#78646C211yesmaximal, characteristic
3C3×#1784C211yesmaximal, characteristic
4#69980C212yesmaximal
5#67683C211yesmaximal, characteristic
6#10334—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C2×#944—22no—
9C2×#924C2212yes—
10C2×C3×#101—22no—
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13#5230—24no—
14#5229—22no—
15#5210—24no—
16C4×#217C2212yes—
17#5137—22no—
18#4739—22no—
19#4721—22no—
20#4467—24no—
21#3195—31no—
22#2434—32no—
23#1784D311yescharacteristic
24#1508—32no—
25#1297—31no—
26C22×C3×Q8—22no—
27C22×C3×Q8—21no—
28C22×C3×Q8—22no—
29C22×C3×Q8—21no—
30C22×C3×Q8—44no—
31C23×C3×C4D412yes—
32C23×C3×C4C2311yescharacteristic
33C22×Dic6D412yes—
34C2×C3×M16—21no—
35C2×C3×M16D412yes—
36C2×#217—22no—
37C2×#217—22no—
38#944—48no—
39C2×C4⋊Dic3D412yes—
40#924—22no—
41#924—22no—
42#924—22no—
43#924—22no—
44C2×Dic3⋊C4—28no—
45C2×Dic3⋊C4—22no—
46C2×C4×Dic3—22no—
47C2×Dic12—22no—
48C3×#101—44no—
49C3×C22⋊C8—22no—
50C3×C22⋊C8—22no—
51C3⋊#95—28no—
52C3⋊#95D414yes—
53#817—22no—
54#816—24no—
55C23×Q8C2×D311yescharacteristic
56C2×C4⋊Q8—62no—
57C2×C22⋊Q8—32no—
58#483—62no—
59#470—62no—
60#451—62no—
61C2×#101—22no—
62#407C2×D311yescharacteristic
63#407C2×D311yescharacteristic
64#395—64no—
65#391—62no—
66#385—64no—
67C4×C22⋊C4—32no—
68#328—64no—
69C24×C3C2×D411yescharacteristic
70C2×C3×Q8—24no—
71C2×C3×Q8—28no—
72C2×C3×Q8—44no—
73C2×C3×Q8—42no—
74C2×C3×Q8—48no—
75C2×C3×Q8—44no—
76C2×C3×Q8—41no—
77C2×C3×Q8—42no—
78C2×C3×Q8—41no—
79C22×C3×C4—24no—
80C22×C3×C4—22no—
81C22×C3×C4—44no—
82C22×C3×C4—42no—
83C22×C3×C4C2×D412yes—
84C22×C3×C4C2×D412yes—
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4#4111yescharacteristic
87C22×Dic3—24no—
88C22×Dic3—22no—
89C22×Dic3—22no—
90C2×Dic6—22no—
91C2×Dic6—22no—
92C2×Dic6—44no—
93C3×M16—42no—
94C3×M16—42no—
95C2×C3×C8—22no—
96C2×C3×C8—41no—
97#217—28no—
98#217—42no—
99#217—42no—
100C4⋊Dic3—22no—
101C4⋊Dic3—22no—
102Dic3⋊C4—416no—
103Dic3⋊C4—42no—
104Dic3⋊C4—42no—
105C4×Dic3—44no—
106Dic12—82no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8—32no—
112C22×Q8—44no—
113C23×C4D1212yes—
114C23×C4C22×D311yescharacteristic
115C2×Q16—62no—
116C2×M16—21no—
117C2×M16#6712yes—
118C4⋊Q8—128no—
119C22⋊Q8—62no—
120C22⋊Q8—62no—
121C22⋊Q8—62no—
122C22⋊Q8—62no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—68no—
125C2×C4⋊C4—62no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—62no—
128C2×C42—62no—
129#106—62no—
130#103—64no—
131#101—44no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134#95—34no—
135#95—68no—
136C23×C3—24no—
137C23×C3—41no—
138C23×C3#12012yes—
139C23×C3#12111yescharacteristic
140C3×Q8—44no—
141C3×Q8—48no—
142C3×Q8—44no—
143C2×C3×C4—24no—
144C2×C3×C4—22no—
145C2×C3×C4—28no—
146C2×C3×C4—24no—
147C2×C3×C4—42no—
148C2×C3×C4—41no—
149C2×C3×C4—44no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4#12011yescharacteristic
153C2×C3×C4#12112yes—
154C2×C3×C4#12711yescharacteristic
155C2×Dic3—28no—
156C2×Dic3—44no—
157C2×Dic3—44no—
158C2×Dic3—44no—
159C2×Dic3—42no—
160C2×Dic3—42no—
161C2×Dic3—44no—
162Dic6—44no—
163Dic6—82no—
164C3×C8—42no—
165C3×C8—81no—
166C24C2×D1211yescharacteristic
167C2×Q8—24no—
168C2×Q8—28no—
169C2×Q8—44no—
170C2×Q8—42no—
171C2×Q8—48no—
172C2×Q8—44no—
173C2×Q8—42no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—62no—
177C2×Q8—62no—
178C2×Q8—124no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4—64no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4#23511yescharacteristic
187C22×C4D3×D412yes—
188C22×C4D3×D412yes—
189C22×C4C2×#6711yescharacteristic
190Q16—242no—
191M16—42no—
192M16—42no—
193C2×C8—22no—
194C2×C8—41no—
195C4⋊C4—62no—
196C4⋊C4—62no—
197C4⋊C4—1216no—
198C4⋊C4—122no—
199C4⋊C4—122no—
200C22⋊C4—68no—
201C22⋊C4—122no—
202C22⋊C4—122no—
203C42—124no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—21no—
208C22×C3—44no—
209C22×C3#39211yescharacteristic
210C22×C3#45614yes—
211C3×C4—21no—
212C3×C4—22no—
213C3×C4—44no—
214C3×C4B(4,𝔽2)12yes—
215Dic3—48no—
216Dic3—82no—
217Dic3—82no—
218C23—24no—
219C23—41no—
220C23#88212yes—
221C23#93011yescharacteristic
222Q8—44no—
223Q8—48no—
224Q8—44no—
225Q8—124no—
226Q8—242no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—28no—
230C2×C4—24no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—68no—
237C2×C4—124no—
238C2×C4—124no—
239C2×C4—124no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4—124no—
243C2×C4#88412yes—
244C2×C4#93711yescharacteristic
245C2×C4#94011yescharacteristic
246C8—42no—
247C8—81no—
248C2×C3—24no—
249C2×C3—41no—
250C2×C3#200512yes—
251C2×C3#202111yescharacteristic
252C22—22no—
253C22—22no—
254C22—21no—
255C22—22no—
256C22—44no—
257C22#475214yes—
258C22#496011yescharacteristic
259C4—21no—
260C4—22no—
261C4—44no—
262C4—128no—
263C4—242no—
264C4—242no—
265C4#503212yes—
266C3#1033411yescharacteristic, complemented, Sylow
267C2—24no—
268C2—41no—
269C2#6950011yescharacteristic
270C2#7011912yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1784C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1784D3
hypocenterC3#10334
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
266. C3#10334semidirect

Maximal subgroups

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

subgroupindex
#840652
#786462
C3×#17842
#699802
#676832
#103343