Finite Group Databasebeta

← #10695817

Subgroups — #10695817

Order 768.

492 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 85 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#78625C212yesmaximal
2#78617C211yesmaximal, characteristic
3#69972C212yesmaximal
4#69911C211yesmaximal, characteristic
5C3×#1293C211yesmaximal, characteristic
6#9979—31nomaximal, Sylow
7C2×#926C2211yescharacteristic
8C2×#924C2212yes—
9C3×#407C2212yes—
10C2×C3×C22⋊C8C2211yescharacteristic
11C2×#830C2211yescharacteristic
12#5121—22no—
13#5120—22no—
14#5116—22no—
15#4739—22no—
16#4723—24no—
17#4722—22no—
18#4722—22no—
19#4721—22no—
20#4550—22no—
21#4469—24no—
22#3189—31no—
23#3095—32no—
24#1818—31no—
25#1472—32no—
26#1293D311yescharacteristic
27C23×C3×C4C2311yescharacteristic
28C22×Dic6D412yes—
29C2×C3×M16—22no—
30C22×C3×C8D412yes—
31C2×#217—22no—
32C2×#217—21no—
33#926—22no—
34#926—22no—
35C2×C4⋊Dic3D412yes—
36C2×C4⋊Dic3D411yescharacteristic
37C2×C4⋊Dic3D411yescharacteristic
38#924—22no—
39#924—22no—
40#924—22no—
41#924—22no—
42C2×Dic3⋊C4—22no—
43C2×C4×Dic3—21no—
44C22×C3⋊C8—21no—
45C2×Dic12—22no—
46C3×C22⋊C8—22no—
47C3×C22⋊C8—22no—
48C3×C22⋊C8D414yes—
49#830—22no—
50#818—24no—
51#818—22no—
52#817—24no—
53#816—24no—
54#816—24no—
55#816—24no—
56C2×C22⋊Q8—32no—
57C2×#117—31no—
58#485—64no—
59#484—62no—
60#484—62no—
61#483—62no—
62#467—62no—
63#451—62no—
64#428—62no—
65#419—62no—
66#407C2×D312yes—
67C2×C22⋊C8—31no—
68C2×C22⋊C8C2×D311yescharacteristic
69#336—62no—
70#324—64no—
71C24×C3#4111yescharacteristic
72C22×C3×C4—42no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D412yes—
75C22×C3×C4C2×D412yes—
76C22×C3×C4C2×D411yescharacteristic
77C22×Dic3—21no—
78C22×Dic3—21no—
79C22×Dic3—22no—
80C22×Dic3—22no—
81C2×Dic6—22no—
82C2×Dic6—22no—
83C2×Dic6—44no—
84C3×M16—44no—
85C2×C3×C8—24no—
86C2×C3×C8—22no—
87C2×C3×C8—42no—
88#217—42no—
89#217—42no—
90#217—42no—
91C4⋊Dic3—22no—
92C4⋊Dic3—22no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95Dic3⋊C4—42no—
96Dic3⋊C4—42no—
97C4×Dic3—41no—
98C4×Dic3—41no—
99C2×C3⋊C8—42no—
100C2×C3⋊C8—41no—
101Dic12—82no—
102C22×Q8—32no—
103C23×C4C22×D311yescharacteristic
104C2×Q16—62no—
105C2×M16—22no—
106C22×C8—61no—
107C22×C8D1212yes—
108C22⋊Q8—62no—
109C22⋊Q8—62no—
110C22⋊Q8—62no—
111C22⋊Q8—62no—
112#117—62no—
113#117—62no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—62no—
118C2×C22⋊C4—61no—
119C2×C22⋊C4—62no—
120C2×C42—61no—
121#107—64no—
122#107—62no—
123#106—64no—
124#103—64no—
125#103—64no—
126#103—64no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—62no—
130C22⋊C8D1214yes—
131C23×C3—22no—
132C23×C3—41no—
133C23×C3—41no—
134C23×C3#12112yes—
135C23×C3#12111yescharacteristic
136C2×C3×C4—22no—
137C2×C3×C4—24no—
138C2×C3×C4—42no—
139C2×C3×C4—41no—
140C2×C3×C4#12012yes—
141C2×C3×C4#12011yescharacteristic
142C2×C3×C4#12711yescharacteristic
143C2×Dic3—42no—
144C2×Dic3—44no—
145C2×Dic3—41no—
146C2×Dic3—41no—
147C2×Dic3—42no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—82no—
151Dic6—44no—
152Dic6—44no—
153C3×C8—44no—
154C3×C8—42no—
155C3⋊C8—81no—
156C24#23911yescharacteristic
157C2×Q8—62no—
158C2×Q8—62no—
159C2×Q8—124no—
160C22×C4—42no—
161C22×C4—61no—
162C22×C4—61no—
163C22×C4—62no—
164C22×C4—62no—
165C22×C4C2×D1211yescharacteristic
166C22×C4C2×D1212yes—
167C22×C4D3×D412yes—
168C22×C4D3×D411yescharacteristic
169Q16—242no—
170M16—44no—
171C2×C8—22no—
172C2×C8—24no—
173C2×C8—42no—
174C2×C8—122no—
175C2×C8—121no—
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C4⋊C4—122no—
182C22⋊C4—122no—
183C22⋊C4—122no—
184C22⋊C4—122no—
185C42—121no—
186C42—121no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—22no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—82no—
193C22×C3#39211yescharacteristic
194C22×C3#47312yes—
195C3×C4—21no—
196C3×C4—22no—
197C3×C4#45812yes—
198Dic3—81no—
199Dic3—82no—
200Dic3—81no—
201Dic3—82no—
202C23—22no—
203C23—41no—
204C23—41no—
205C23#89512yes—
206C23#89511yescharacteristic
207Q8—124no—
208Q8—124no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—122no—
214C2×C4—124no—
215C2×C4—122no—
216C2×C4—121no—
217C2×C4—121no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—242no—
221C2×C4#87411yescharacteristic
222C2×C4#88212yes—
223C2×C4#88211yescharacteristic
224C8—44no—
225C8—42no—
226C8—241no—
227C2×C3—22no—
228C2×C3—42no—
229C2×C3#201711yescharacteristic
230C2×C3#202011yescharacteristic
231C2×C3#202211yescharacteristic
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—82no—
238C22#467111yescharacteristic
239C22#489612yes—
240C4—22no—
241C4—21no—
242C4—241no—
243C4—242no—
244C4—241no—
245C4—242no—
246C4#474212yes—
247C3#997911yescharacteristic, complemented, Sylow
248C2—22no—
249C2—42no—
250C2#6872711yescharacteristic
251C2#6894411yescharacteristic
252C2#6896611yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1293C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenter#1293D3
hypocenterC3#9979
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
247. C3#9979semidirect

Maximal subgroups

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

subgroupindex
#786252
#786172
#699722
#699112
C3×#12932
#99793