Finite Group Databasebeta

← #10693031

Subgroups — #10693031

Order 768.

474 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 131 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×#2982C211yesmaximal, characteristic
2#82481C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#70770C212yesmaximal
5#67738C212yesmaximal
6#9484—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C4×Dic6C2211yescharacteristic
9C3×C8×Q8—24no—
10C3×#431—22no—
11C3×#431C2212yes—
12#5229—21no—
13C2×#836—21no—
14#4986C412yes—
15#4979C412yes—
16#4972—21no—
17#4936C2211yescharacteristic
18#4933—22no—
19#4924C2212yes—
20#4530—21no—
21#4456—24no—
22#2982D311yescharacteristic
23#2581—31no—
24#1801—31no—
25#1487—32no—
26#1286—32no—
27C22×C3×Q8D411yescharacteristic
28C22×Dic6C2×C411yescharacteristic
29C3×C4×Q8—22no—
30C3×C4×Q8—22no—
31C2×C3×C4⋊C4C2×C412yes—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C42D412yes—
34C2×C3×C42C2311yescharacteristic
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3C2×C411yescharacteristic
37C2×Dic3⋊C4—22no—
38C2×Dic3⋊C4—21no—
39C2×C4×Dic3—21no—
40C2×#208—22no—
41C4×Dic6—22no—
42C4×Dic6—22no—
43C3×C4⋊C8—24no—
44C3×C4⋊C8—22no—
45C3×C22⋊C8—22no—
46C3×C4×C8—24no—
47C3×C4×C8D414yes—
48#836—42no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52C3⋊#95—21no—
53C3⋊#95C2×C412yes—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C2×C4×Q8—31no—
57C2×C4×Q8C2×D311yescharacteristic
58C8×Q8—24no—
59#431—22no—
60#431C2×D312yes—
61#423—32no—
62C2×#101—61no—
63#400—32no—
64#395—32no—
65#391—61no—
66#389—61no—
67#387—62no—
68#382—31no—
69#343—61no—
70Q8⋊C8—64no—
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C3×Q8SD1612yes—
74C2×C3×Q8Q1612yes—
75C22×C3×C4C22⋊C412yes—
76C22×C3×C4C22⋊C412yes—
77C22×C3×C4C22×C411yescharacteristic
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×D411yescharacteristic
80C22×Dic3—21no—
81C22×Dic3—22no—
82C22×Dic3—21no—
83C2×Dic6—22no—
84C2×Dic6—42no—
85C2×C3×C8—24no—
86C3×C4⋊C4—24no—
87C3×C4⋊C4—22no—
88C3×C42—22no—
89C3×C42—22no—
90C3×C42C2×D412yes—
91C3×C42#4112yes—
92C4⋊Dic3—22no—
93C4⋊Dic3—42no—
94Dic3⋊C4—44no—
95Dic3⋊C4—41no—
96Dic3⋊C4—41no—
97C4×Dic3—42no—
98#208—42no—
99#208—42no—
100C2×C3⋊C8—42no—
101C22×Q8—31no—
102C22×Q8D1211yescharacteristic
103C2×M16—62no—
104C4×Q8—22no—
105C4×Q8—22no—
106C4×Q8—62no—
107C4×Q8—62no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—61no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4C4×D312yes—
113C2×C4⋊C4D1211yescharacteristic
114C2×C42—61no—
115C2×C42#6712yes—
116C2×C42C22×D311yescharacteristic
117C4⋊C8—24no—
118C4⋊C8—22no—
119C4⋊C8—62no—
120C4⋊C8—62no—
121#101—122no—
122C22⋊C8—22no—
123C4×C8—24no—
124C4×C8#6714yes—
125#95—32no—
126#95—62no—
127#95—62no—
128#95—62no—
129#95—61no—
130C23×C3C2×C22⋊C411yescharacteristic
131C3×Q8—24no—
132C3×Q8—24no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—41no—
137C2×C3×C4—41no—
138C2×C3×C4—42no—
139C2×C3×C4#10314yes—
140C2×C3×C4C4×D412yes—
141C2×C3×C4#12011yescharacteristic
142C2×C3×C4#12112yes—
143C2×C3×C4#12311yescharacteristic
144C2×C3×C4C2×SD1611yescharacteristic
145C2×C3×C4C2×Q1611yescharacteristic
146C2×C3×C4#13512yes—
147C2×Dic3—22no—
148C2×Dic3—41no—
149C2×Dic3—41no—
150C2×Dic3—42no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C2×Dic3—41no—
154C2×Dic3—42no—
155Dic6—24no—
156Dic6—42no—
157C3×C8—44no—
158C3⋊C8—82no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—62no—
162C2×Q8—122no—
163C2×Q8#20412yes—
164C2×Q8Dic1212yes—
165C22×C4—61no—
166C22×C4—62no—
167C22×C4—61no—
168C22×C4(C2×D3)⋊C412yes—
169C22×C4(C2×D3)⋊C412yes—
170C22×C4C2×C4×D311yescharacteristic
171C22×C4C2×D1211yescharacteristic
172C22×C4C2×#6711yescharacteristic
173M16—122no—
174M16—122no—
175C2×C8—24no—
176C2×C8—122no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—62no—
180C4⋊C4—124no—
181C4⋊C4—122no—
182C4⋊C4—121no—
183C4⋊C4—121no—
184C42—22no—
185C42—22no—
186C42—122no—
187C42#23512yes—
188C42C2×#6712yes—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3#38611yescharacteristic
192C22×C3C2×#10311yescharacteristic
193C22×C3#41611yescharacteristic
194C3×C4—22no—
195C3×C4—24no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—42no—
199C3×C4#46512yes—
200C3×C4#46712yes—
201Dic3—42no—
202Dic3—82no—
203Dic3—81no—
204C23C2×(C2×D3)⋊C411yescharacteristic
205Q8—24no—
206Q8—24no—
207Q8—64no—
208Q8—122no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—62no—
216C2×C4—121no—
217C2×C4—121no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4—121no—
222C2×C4—122no—
223C2×C4#81614yes—
224C2×C4C2×#20411yescharacteristic
225C2×C4#90412yes—
226C2×C4C2×Dic1211yescharacteristic
227C2×C4C4×#6712yes—
228C2×C4#92911yescharacteristic
229C2×C4#93012yes—
230C2×C4#95311yescharacteristic
231C8—44no—
232C8—242no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3#189111yescharacteristic
236C2×C3#189211yescharacteristic
237C2×C3#191211yescharacteristic
238C22—21no—
239C22—21no—
240C22C2×#81611yescharacteristic
241C22#511311yescharacteristic
242C22#521311yescharacteristic
243C4—24no—
244C4—22no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—122no—
249C4—242no—
250C4—241no—
251C4#511412yes—
252C4#511612yes—
253C3#948411yescharacteristic, complemented, Sylow
254C2—21no—
255C2—21no—
256C2#7166311yescharacteristic
257C2#7197211yescharacteristic
258C2#7202911yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2982C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2982D3
hypocenterC3#9484
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
253. C3#9484semidirect

Maximal subgroups

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

subgroupindex
C3×#29822
#824812
#716372
#707702
#677382
#94843