Finite Group Databasebeta

← #10695816

Subgroups — #10695816

Order 768.

492 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 83 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#82220C212yesmaximal
2#78617C211yesmaximal, characteristic
3C3×#1818C211yesmaximal, characteristic
4#69923C212yesmaximal
5#67679C211yesmaximal, characteristic
6#9979—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#924C2212yes—
9C2×C3×C22⋊C8C2211yescharacteristic
10C2×C3×C22⋊C8C2211yescharacteristic
11#5207C2212yes—
12#5116—22no—
13#5004—22no—
14#4996—24no—
15#4995—22no—
16#4971—22no—
17#4970—22no—
18#4739—22no—
19#4722—22no—
20C3×#336—22no—
21#4469—24no—
22#3189—31no—
23#3095—32no—
24#1818D311yescharacteristic
25#1472—32no—
26#1293—31no—
27C23×C3×C4C2311yescharacteristic
28C22×Dic6D412yes—
29C22×C3×C8—21no—
30C22×C3×C8D412yes—
31C3×#117—22no—
32C3×#117—22no—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C22⋊C4—21no—
36C2×C3×C42—21no—
37C2×#217—22no—
38C2×C4⋊Dic3D412yes—
39#924—22no—
40#924—22no—
41#924—22no—
42#924—22no—
43C2×Dic3⋊C4—22no—
44C2×#208—22no—
45C2×Dic12—22no—
46C3×C22⋊C8—22no—
47C3×C22⋊C8—22no—
48#830—22no—
49#830D414yes—
50#818—22no—
51#816—24no—
52#810—24no—
53#810—24no—
54#808—24no—
55#807—24no—
56C2×C22⋊Q8—32no—
57C2×#117C2×D311yescharacteristic
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#407—32no—
67C2×C22⋊C8C2×D311yescharacteristic
68C2×C22⋊C8C2×D311yescharacteristic
69#336—22no—
70#324—64no—
71C24×C3#4111yescharacteristic
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—42no—
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4C2×D412yes—
77C22×C3×C4C2×D412yes—
78C22×C3×C4C2×D411yescharacteristic
79C22×Dic3—22no—
80C22×Dic3—22no—
81C2×Dic6—22no—
82C2×Dic6—22no—
83C2×Dic6—44no—
84C2×C3×C8—22no—
85C2×C3×C8—42no—
86C2×C3×C8—42no—
87C2×C3×C8—41no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—22no—
90C3×C22⋊C4—42no—
91C3×C42—41no—
92C3×C42—41no—
93#217—42no—
94#217—42no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97Dic3⋊C4—42no—
98Dic3⋊C4—42no—
99#208—44no—
100C2×C3⋊C8—24no—
101Dic12—82no—
102C22×Q8—32no—
103C23×C4C22×D311yescharacteristic
104C2×Q16—62no—
105C2×M16—62no—
106C22×C8—21no—
107C22×C8#6712yes—
108C22⋊Q8—62no—
109C22⋊Q8—62no—
110C22⋊Q8—62no—
111C22⋊Q8—62no—
112#117—22no—
113#117—22no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4D1211yescharacteristic
117C2×C4⋊C4D1211yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—62no—
120C2×C42—21no—
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—34no—
130C22⋊C8—62no—
131C23×C3—22no—
132C23×C3—41no—
133C23×C3—41no—
134C23×C3#12112yes—
135C23×C3#12111yescharacteristic
136C2×C3×C4—24no—
137C2×C3×C4—22no—
138C2×C3×C4—41no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—41no—
142C2×C3×C4—41no—
143C2×C3×C4—42no—
144C2×C3×C4#12012yes—
145C2×C3×C4#12011yescharacteristic
146C2×C3×C4#12711yescharacteristic
147C2×Dic3—44no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—82no—
151Dic6—44no—
152Dic6—44no—
153C3×C8—42no—
154C3×C8—81no—
155C3⋊C8—44no—
156C24#23511yescharacteristic
157C2×Q8—62no—
158C2×Q8—62no—
159C2×Q8—124no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—62no—
164C22×C4—62no—
165C22×C4C2×D1211yescharacteristic
166C22×C4D3×D412yes—
167C22×C4D3×D412yes—
168C22×C4C2×#6711yescharacteristic
169Q16—242no—
170M16—124no—
171C2×C8—22no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—41no—
175C2×C8—64no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C4⋊C4—122no—
182C22⋊C4—42no—
183C22⋊C4—122no—
184C22⋊C4—122no—
185C42—41no—
186C42—41no—
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—22no—
196C3×C4—21no—
197C3×C4—81no—
198C3×C4—81no—
199C3×C4#45812yes—
200Dic3—82no—
201Dic3—82no—
202C23—22no—
203C23—41no—
204C23—41no—
205C23#88412yes—
206C23#93011yescharacteristic
207Q8—124no—
208Q8—124no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—124no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—242no—
221C2×C4#88212yes—
222C2×C4#93711yescharacteristic
223C2×C4#94011yescharacteristic
224C8—42no—
225C8—81no—
226C8—124no—
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#496011yescharacteristic
239C22#514312yes—
240C4—21no—
241C4—22no—
242C4—81no—
243C4—81no—
244C4—242no—
245C4—242no—
246C4#505312yes—
247C3#997911yescharacteristic, complemented, Sylow
248C2—22no—
249C2—42no—
250C2#6953911yescharacteristic
251C2#7000811yescharacteristic
252C2#7001311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1818C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenter#1818D3
hypocenterC3#9979
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1818

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
#822202
#786172
C3×#18182
#699232
#676792
#99793