Finite Group Databasebeta

← #10692545

Subgroups — #10692545

Order 768.

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

normal subgroups only

This group has 486 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
1#82478C211yesmaximal, characteristic
2#82435C211yesmaximal, characteristic
3C3×#1801C211yesmaximal, characteristic
4#70507C212yesmaximal
5#67715C212yesmaximal
6#9392—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×#423C2212yes—
10C2×C3×#101—21no—
11#5229—21no—
12Q8×C3⋊C8—24no—
13#4979C414yes—
14#4975—21no—
15C4×C4⋊Dic3C2211yescharacteristic
16#4928—22no—
17#4925—22no—
18#4925C2212yes—
19C3×#343—21no—
20#4481—24no—
21#2982—31no—
22#2567—31no—
23#1801D311yescharacteristic
24#1548—32no—
25#1303—32no—
26C22×C3×Q8D411yescharacteristic
27C22×Dic6C2×C412yes—
28C2×C3×M16—22no—
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⋊Dic3—21no—
37C2×C4⋊Dic3—21no—
38C2×Dic3⋊C4—24no—
39C2×C4×Dic3—21no—
40#869—22no—
41#869—22no—
42C3×C4⋊C8—22no—
43C3×C4⋊C8—22no—
44C3×#101—42no—
45C3⋊#95—24no—
46C3⋊#95C2×C412yes—
47#830—22no—
48C4⋊(C3⋊C8)—24no—
49C4⋊(C3⋊C8)D414yes—
50C4×C3⋊C8—24no—
51C4×C3⋊C8—22no—
52C2×C4⋊Q8—31no—
53C2×C4×Q8C2×D311yescharacteristic
54C8×Q8—64no—
55#431—32no—
56#431—62no—
57#423C2×D312yes—
58C2×#101—21no—
59#395—34no—
60#391—61no—
61#389—62no—
62#384—61no—
63C4×C4⋊C4—31no—
64#343—21no—
65#333—64no—
66C2×C3×Q8—22no—
67C2×C3×Q8—22no—
68C2×C3×Q8SD1612yes—
69C2×C3×Q8Q1612yes—
70C22×C3×C4C22⋊C412yes—
71C22×C3×C4C22⋊C412yes—
72C22×C3×C4C22×C411yescharacteristic
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D411yescharacteristic
75C22×Dic3—22no—
76C22×Dic3—22no—
77C2×Dic6—24no—
78C2×Dic6—82no—
79C3×M16—42no—
80C3×M16—42no—
81C2×C3×C8—42no—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—22no—
84C3×C42—22no—
85C3×C42—22no—
86C3×C42C2×D412yes—
87C3×C42#4112yes—
88C4⋊Dic3—24no—
89C4⋊Dic3—41no—
90C4⋊Dic3—42no—
91C4⋊Dic3—41no—
92Dic3⋊C4—84no—
93C4×Dic3—42no—
94C2×C3⋊C8—24no—
95C22×Q8—32no—
96C22×Q8#6711yescharacteristic
97C2×M16—22no—
98C4⋊Q8—62no—
99C4⋊Q8—62no—
100C4×Q8—22no—
101C4×Q8—22no—
102C2×C4⋊C4—61no—
103C2×C4⋊C4—61no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4—64no—
106C2×C4⋊C4C4×D312yes—
107C2×C4⋊C4#6711yescharacteristic
108C2×C42—61no—
109C2×C42D1212yes—
110C2×C42C22×D311yescharacteristic
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—34no—
114C4⋊C8—64no—
115#101—42no—
116C22⋊C8—62no—
117C4×C8—64no—
118C4×C8—62no—
119#95—32no—
120#95—64no—
121C23×C3C2×C22⋊C411yescharacteristic
122C3×Q8—24no—
123C3×Q8—24no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—41no—
129C2×C3×C4—42no—
130C2×C3×C4#10314yes—
131C2×C3×C4C4×D412yes—
132C2×C3×C4#12011yescharacteristic
133C2×C3×C4#12112yes—
134C2×C3×C4#12311yescharacteristic
135C2×C3×C4C2×SD1611yescharacteristic
136C2×C3×C4C2×Q1611yescharacteristic
137C2×C3×C4#13512yes—
138C2×Dic3—24no—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—44no—
142C2×Dic3—42no—
143Dic6—84no—
144C3×C8—82no—
145C3⋊C8—44no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—64no—
149C2×Q8—242no—
150C2×Q8#21412yes—
151C2×Q8#21612yes—
152C22×C4—62no—
153C22×C4—62no—
154C22×C4(C2×D3)⋊C412yes—
155C22×C4(C2×D3)⋊C412yes—
156C22×C4C2×C4×D311yescharacteristic
157C22×C4C2×D1211yescharacteristic
158C22×C4C2×#6711yescharacteristic
159M16—42no—
160M16—42no—
161C2×C8—42no—
162C2×C8—64no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—64no—
166C4⋊C4—121no—
167C4⋊C4—121no—
168C4⋊C4—122no—
169C4⋊C4—244no—
170C42—22no—
171C42—22no—
172C42—122no—
173C42D3×D412yes—
174C42#23712yes—
175C22×C3—21no—
176C22×C3—21no—
177C22×C3#38611yescharacteristic
178C22×C3C2×#10311yescharacteristic
179C22×C3#41611yescharacteristic
180C3×C4—21no—
181C3×C4—24no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4#45612yes—
185C3×C4#46512yes—
186C3×C4#46712yes—
187Dic3—44no—
188Dic3—82no—
189C23C2×(C2×D3)⋊C411yescharacteristic
190Q8—24no—
191Q8—24no—
192Q8—244no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—64no—
200C2×C4—122no—
201C2×C4—122no—
202C2×C4—124no—
203C2×C4—122no—
204C2×C4#81014yes—
205C2×C4#67⋊C412yes—
206C2×C4#88211yescharacteristic
207C2×C4#88611yescharacteristic
208C2×C4C2×#21411yescharacteristic
209C2×C4#93912yes—
210C2×C4C2×#21611yescharacteristic
211C2×C4#95012yes—
212C8—82no—
213C8—124no—
214C2×C3—21no—
215C2×C3—21no—
216C2×C3#188611yescharacteristic
217C2×C3#189211yescharacteristic
218C2×C3#191411yescharacteristic
219C22—21no—
220C22—21no—
221C22#495811yescharacteristic
222C22C2×#81011yescharacteristic
223C22#500211yescharacteristic
224C4—24no—
225C4—21no—
226C4—22no—
227C4—42no—
228C4—124no—
229C4—242no—
230C4#475212yes—
231C4#504112yes—
232C4#505112yes—
233C3#939211yescharacteristic, complemented, Sylow
234C2—21no—
235C2—21no—
236C2#6951411yescharacteristic
237C2#7010311yescharacteristic
238C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1801C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1801D3
hypocenterC3#9392
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
233. C3#9392semidirect

Maximal subgroups

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

subgroupindex
#824782
#824352
C3×#18012
#705072
#677152
#93923