Finite Group Databasebeta

← #10695814

Subgroups — #10695814

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
1C3×#3189C211yesmaximal, characteristic
2#82727C212yesmaximal
3#71691C212yesmaximal
4#69911C211yesmaximal, characteristic
5#67679C211yesmaximal, characteristic
6#9979—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2212yes—
8C2×#926C2211yescharacteristic
9C3×#484—22no—
10C3×#467—22no—
11C3×#451—22no—
12C2×C3×C22⋊C8C2211yescharacteristic
13#5225—22no—
14#5207C2212yes—
15C2×#830C2211yescharacteristic
16#5045—24no—
17#5044—22no—
18#5043—22no—
19#4964—22no—
20#4550—22no—
21#4528—24no—
22#3189D311yescharacteristic
23#3095—32no—
24#1818—31no—
25#1472—32no—
26#1293—31no—
27C22×C3×Q8D412yes—
28C23×C3×C4C2311yescharacteristic
29C2×C3×Q16—22no—
30C22×C3×C8D412yes—
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C3×C22⋊Q8—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4D412yes—
37C2×C3×C22⋊C4—22no—
38C2×#217—21no—
39#926—22no—
40#926—22no—
41C2×C4⋊Dic3D411yescharacteristic
42C2×C4⋊Dic3D411yescharacteristic
43C2×C4×Dic3—21no—
44C2×#208—22no—
45C22×C3⋊C8—21no—
46C3×#107—22no—
47C3×#103—24no—
48C3×C22⋊C8—22no—
49#835—24no—
50#835—24no—
51#830—22no—
52#830—22no—
53#830D414yes—
54#808—24no—
55#807—24no—
56C2×C22⋊Q8C2×D312yes—
57C2×#117—31no—
58#485—64no—
59#484—22no—
60#484—62no—
61#483—62no—
62#467—22no—
63#451—22no—
64#428—62no—
65#419—62no—
66#407—32no—
67C2×C22⋊C8—31no—
68C2×C22⋊C8C2×D311yescharacteristic
69#336—62no—
70#324—64no—
71C24×C3#4111yescharacteristic
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C2×C3×Q8—44no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—42no—
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×D412yes—
80C22×C3×C4C2×D412yes—
81C22×C3×C4C2×D411yescharacteristic
82C22×Dic3—21no—
83C22×Dic3—21no—
84C3×Q16—82no—
85C2×C3×C8—22no—
86C2×C3×C8—42no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—42no—
90C3×C4⋊C4—42no—
91C3×C22⋊C4—42no—
92C3×C22⋊C4—42no—
93#217—42no—
94C4⋊Dic3—22no—
95C4⋊Dic3—22no—
96C4×Dic3—41no—
97C4×Dic3—41no—
98#208—44no—
99C2×C3⋊C8—24no—
100C2×C3⋊C8—42no—
101C2×C3⋊C8—41no—
102C22×Q8#6712yes—
103C23×C4C22×D311yescharacteristic
104C2×Q16—22no—
105C2×M16—62no—
106C22×C8—61no—
107C22×C8#6712yes—
108C22⋊Q8—22no—
109C22⋊Q8—22no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112#117—62no—
113#117—62no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4#6712yes—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—61no—
120C2×C42—61no—
121#107—22no—
122#107—64no—
123#106—64no—
124#103—24no—
125#103—64no—
126#103—64no—
127C22⋊C8—22no—
128C22⋊C8—34no—
129C22⋊C8—62no—
130C22⋊C8—62no—
131C23×C3—22no—
132C23×C3—41no—
133C23×C3—41no—
134C23×C3#12111yescharacteristic
135C23×C3#12112yes—
136C3×Q8—44no—
137C3×Q8—44no—
138C2×C3×C4—22no—
139C2×C3×C4—24no—
140C2×C3×C4—41no—
141C2×C3×C4—42no—
142C2×C3×C4—44no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4—82no—
146C2×C3×C4#12012yes—
147C2×C3×C4#12011yescharacteristic
148C2×C3×C4#12711yescharacteristic
149C2×Dic3—42no—
150C2×Dic3—41no—
151C2×Dic3—41no—
152C2×Dic3—42no—
153C3×C8—42no—
154C3⋊C8—44no—
155C3⋊C8—81no—
156C24#23711yescharacteristic
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—44no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—61no—
164C22×C4—61no—
165C22×C4D3×D411yescharacteristic
166C22×C4D3×D412yes—
167C22×C4C2×#6712yes—
168C22×C4C2×#6711yescharacteristic
169Q16—82no—
170M16—124no—
171C2×C8—22no—
172C2×C8—42no—
173C2×C8—64no—
174C2×C8—122no—
175C2×C8—121no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C22⋊C4—42no—
183C22⋊C4—42no—
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—22no—
196C3×C4—21no—
197C3×C4—82no—
198C3×C4—82no—
199C3×C4#45812yes—
200Dic3—81no—
201Dic3—81no—
202C23—22no—
203C23—41no—
204C23—41no—
205C23#93811yescharacteristic
206C23#93912yes—
207Q8—44no—
208Q8—44no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4—121no—
220C2×C4—121no—
221C2×C4#93712yes—
222C2×C4#94011yescharacteristic
223C2×C4#95311yescharacteristic
224C8—42no—
225C8—124no—
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#519212yes—
239C22#522311yescharacteristic
240C4—22no—
241C4—21no—
242C4—82no—
243C4—82no—
244C4—241no—
245C4—241no—
246C4#524312yes—
247C3#997911yescharacteristic, complemented, Sylow
248C2—22no—
249C2—42no—
250C2#7184911yescharacteristic
251C2#7207411yescharacteristic
252C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3189C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenter#3189D3
hypocenterC3#9979
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

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
C3×#31892
#827272
#716912
#699112
#676792
#99793