Finite Group Databasebeta

← #10711744

Subgroups — #10711744

Order 768.

458 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C22×C3×#106C211yesmaximal, characteristic
2#71977C211yesmaximal, characteristic
3#71963C211yesmaximal, characteristic
4C2×#4487C212yesmaximal, complemented
5#67720C212yesmaximal, complemented
6#12929—31nomaximal, Sylow
7C23×C3×C8C2211yescharacteristic
8C22×C3×C4⋊C4—21no—
9C2×C3×#106—21no—
10C2×C3×#106—24no—
11C2×C3×#106C2212yes—
12#5207—21no—
13C2×#819C2211yescharacteristic
14C2×#819C2211yescharacteristic
15C2×#812C2212yes—
16#4552—21no—
17#4549C414yescomplemented
18#4487—24no—
19C22×#106D311yescharacteristic
20C2×#365—32no—
21#2112—31no—
22#1810—31no—
23#1383—32no—
24C23×C3×C4—41no—
25C23×C3×C4D411yescharacteristic
26C22×C3×C8—22no—
27C22×C3×C8—22no—
28C22×C3×C8C2×C412yes—
29C22×C3×C8D411yescharacteristic
30C22×C3×C8Q812yes—
31C22×C3×C8C2311yescharacteristic
32C2×C3×C4⋊C4—24no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—42no—
35C2×C3×C4⋊C4—41no—
36C2×#208—21no—
37C2×#208—21no—
38C3×#106—24no—
39C3×#106—22no—
40C3×#106D414yes—
41#830—42no—
42#819—22no—
43#819—22no—
44#812—22no—
45#812—22no—
46C2×C3⋊C16C2×C414yescomplemented
47C23×C8C2×D311yescharacteristic
48C22×C4⋊C4—21no—
49C2×M32—32no—
50C2×#108—31no—
51C2×#108—31no—
52C2×#106—21no—
53C2×#106—24no—
54C2×#106C2×D312yes—
55#407—61no—
56#365—64no—
57C22⋊C16—34no—
58#341—61no—
59C24×C3SD1611yescharacteristic
60C22×C3×C4—21no—
61C22×C3×C4—21no—
62C22×C3×C4—21no—
63C22×C3×C4—21no—
64C22×C3×C4—44no—
65C22×C3×C4—42no—
66C22×C3×C4—84no—
67C22×C3×C4C4⋊C412yes—
68C22×C3×C4SD1611yescharacteristic
69C22×C3×C4SD1612yes—
70C22×C3×C4C2×D411yescharacteristic
71C2×C3×C8—21no—
72C2×C3×C8—21no—
73C2×C3×C8—24no—
74C2×C3×C8—24no—
75C2×C3×C8C4⋊C412yes—
76C2×C3×C8C22×C411yescharacteristic
77C2×C3×C8C2×D412yes—
78C2×C3×C8C2×Q811yescharacteristic
79C2×C3×C8#4112yes—
80C3×C4⋊C4—24no—
81C3×C4⋊C4—44no—
82C3×C4⋊C4—42no—
83#208—41no—
84#208—41no—
85#208—41no—
86#208—41no—
87C2×C3⋊C8—41no—
88C2×C3⋊C8—41no—
89C3⋊C16—24no—
90C23×C4—41no—
91C23×C4#6711yescharacteristic
92C2×M16—61no—
93C2×M16—61no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8Dic612yes—
97C22×C8C4×D312yes—
98C22×C8#6711yescharacteristic
99C22×C8C22×D311yescharacteristic
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104M32—62no—
105M32—62no—
106C2×C16—34no—
107#108—62no—
108#108—62no—
109#106—24no—
110#106—22no—
111#106#6714yes—
112C22⋊C8—122no—
113C23×C3—24no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3#10612yes—
117C23×C3C2×SD1611yescharacteristic
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—44no—
125C2×C3×C4—88no—
126C2×C3×C4—84no—
127C2×C3×C4#10612yes—
128C2×C3×C4C2×C4⋊C411yescharacteristic
129C2×C3×C4#12011yescharacteristic
130C2×C3×C4#12311yescharacteristic
131C2×C3×C4C2×SD1611yescharacteristic
132C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
133C2×C3×C4#13711yescharacteristic
134C3×C8—22no—
135C3×C8C4×D412yes—
136C3×C8C22⋊Q812yes—
137C3⋊C8—81no—
138C3⋊C8—81no—
139C24#21411yescharacteristic
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—84no—
147C22×C4Dic3⋊C412yes—
148C22×C4#21411yescharacteristic
149C22×C4#21512yes—
150C22×C4C2×#6711yescharacteristic
151M16—121no—
152M16—121no—
153M16—121no—
154M16—121no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—24no—
158C2×C8—24no—
159C2×C8—121no—
160C2×C8—121no—
161C2×C8Dic3⋊C412yes—
162C2×C8C2×Dic611yescharacteristic
163C2×C8C2×C4×D311yescharacteristic
164C2×C8#23512yes—
165C2×C8C2×#6712yes—
166C4⋊C4—24no—
167C4⋊C4—44no—
168C4⋊C4—42no—
169C16—64no—
170C22×C3—24no—
171C22×C3—24no—
172C22×C3—22no—
173C22×C3—22no—
174C22×C3—22no—
175C22×C3#36514yescomplemented
176C22×C3C2×#10611yescharacteristic
177C22×C3#45011yescharacteristic
178C22×C3#48311yescharacteristic
179C3×C4—22no—
180C3×C4—84no—
181C3×C4#38511yescharacteristic
182C3×C4#42811yescharacteristic
183C3×C4#44811yescharacteristic
184C3×C4#48111yescharacteristic
185C23—21no—
186C23—21no—
187C23—24no—
188C23#80812yes—
189C23C2×#21411yescharacteristic
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—44no—
197C2×C4—88no—
198C2×C4—84no—
199C2×C4#80812yes—
200C2×C4C2×Dic3⋊C411yescharacteristic
201C2×C4#92911yescharacteristic
202C2×C4#93211yescharacteristic
203C2×C4C2×#21511yescharacteristic
204C2×C4#94211yescharacteristic
205C2×C4#95311yescharacteristic
206C8—22no—
207C8—241no—
208C8—241no—
209C8#92412yes—
210C8C4×#6712yes—
211C2×C3—21no—
212C2×C3—24no—
213C2×C3—21no—
214C2×C3#185311yescharacteristic
215C2×C3C2×#36512yes—
216C22—24no—
217C22—24no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22#448714yescomplemented
222C22C2×#80811yescharacteristic
223C22#497111yescharacteristic
224C22#521811yescharacteristic
225C4—22no—
226C4—84no—
227C4#496411yescharacteristic
228C4#496811yescharacteristic
229C4#521011yescharacteristic
230C4#522711yescharacteristic
231C3#1292911yescharacteristic, complemented, Sylow
232C2—21no—
233C2—21no—
234C2—24no—
235C2C2×#448712yes—
236C2#7155611yescharacteristic
subgroupG/H
centerC22C2×#808
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC22×C3×#106C2
socleC22×C3C2×#106
prime radicalC24×C3SD16
cosocleC22×C3×C8C23
hypercenterC22×#106D3
hypocenterC3#12929
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
4. C2×#4487C2semidirect
5. #67720C2semidirect
17. #4549C4semidirect
46. C2×C3⋊C16C2×C4semidirect
175. C22×C3#365semidirect
221. C22#4487semidirect
231. C3#12929semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1062
#719772
#719632
C2×#44872
#677202
#129293