Finite Group Databasebeta

← #10697873

Subgroups — #10697873

Order 768.

614 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 113 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×#2868C211yesmaximal, characteristic, complemented
2#83215C212yesmaximal
3#71773C212yesmaximal, complemented
4#69395C211yesmaximal, characteristic
5#67611C211yesmaximal, characteristic, complemented
6#10289—31nomaximal, Sylow
7C2×C3×C4⋊Q8—22no—
8C2×C3×C4⋊Q8C2212yes—
9#5477C2211yescharacteristic
10C3×#384—24no—
11C3×C43C2211yescharacteristic
12#5088—22no—
13#5087—24no—
14#5078—22no—
15#4967—24no—
16C2×#805—22no—
17#4925C2212yes—
18#4924C2211yescharacteristic
19#4547—24no—
20#3226—32no—
21#2868D311yescharacteristic, complemented
22#1962—31no—
23#1546—32no—
24#1276—31no—
25C22×C3×Q8—22no—
26C2×#235—21no—
27C3×C4⋊Q8—22no—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—22no—
31C3×C4⋊Q8—44no—
32C3×C4⋊Q8—44no—
33C2×C3×C4⋊C4—28no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4D414yes—
36C2×C3×C42—22no—
37C2×C3×C42D412yes—
38C2×C3×C42C2311yescharacteristic
39#930—22no—
40C2×#208—21no—
41#875—21no—
42#874D411yescharacteristic
43C4×D12—22no—
44#869D411yescharacteristic
45#830—22no—
46#809—28no—
47#807—28no—
48#805—44no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51C4×C3⋊C8—22no—
52#532—31no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×D312yes—
55#500—62no—
56#486—62no—
57#480—64no—
58#474—64no—
59#431—32no—
60#423—31no—
61C2×#104—62no—
62#384—24no—
63C43C2×D311yescharacteristic
64#332—64no—
65C2×C3×Q8—42no—
66C2×C3×Q8—42no—
67C2×C3×Q8—44no—
68C2×C3×Q8—44no—
69C22×C3×C4—22no—
70C22×C3×C4—24no—
71C22×C3×C4C2×D412yes—
72C22×C3×C4C2×D411yescharacteristic
73C2×#67—41no—
74#235—42no—
75#235—42no—
76C2×D12—22no—
77C2×D12—21no—
78C2×C4×D3—41no—
79C2×Dic6—21no—
80C3×C4⋊C4—24no—
81C3×C4⋊C4—24no—
82C3×C4⋊C4—416no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4—42no—
85C3×C42—28no—
86C3×C42—22no—
87C3×C42D814yes—
88C3×C42C2×D411yescharacteristic
89C3×C42C2×D412yes—
90C3×C42#4111yescharacteristic
91(C2×D3)⋊C4—42no—
92C4⋊Dic3—22no—
93#208—42no—
94C2×C3⋊C8—24no—
95C2×C3⋊C8—41no—
96C2×#41—61no—
97C22×Q8—22no—
98C2×M16—61no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C4⋊Q8—31no—
104C4⋊Q8—44no—
105C4⋊Q8—44no—
106#127—31no—
107#124—61no—
108#121—62no—
109C4×D4—62no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4#6714yes—
113C2×C42—22no—
114C2×C42#6712yes—
115C2×C42C22×D311yescharacteristic
116#107—68no—
117C4⋊C8—34no—
118C4⋊C8—62no—
119#104—124no—
120#102—68no—
121C22⋊C8—62no—
122C4×C8—62no—
123C23×C3#12011yescharacteristic
124C22×D3—41no—
125C3×Q8—84no—
126C3×Q8—82no—
127C3×Q8—82no—
128C2×C3×C4—28no—
129C2×C3×C4—24no—
130C2×C3×C4—24no—
131C2×C3×C4—44no—
132C2×C3×C4—48no—
133C2×C3×C4#12012yes—
134C2×C3×C4#12111yescharacteristic
135C2×C3×C4#12112yes—
136C2×C3×C4#12711yescharacteristic
137C2×C3×C4C2×D812yes—
138C2×C3×C4AGL(1,ℤ/8)12yes—
139#67—82no—
140C2×Dic3—41no—
141D12—44no—
142D12—42no—
143C4×D3—81no—
144C4×D3—81no—
145Dic6—42no—
146C3⋊C8—44no—
147C3⋊C8—81no—
148#41—122no—
149#41—122no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—61no—
155C2×D4—62no—
156C2×D4—61no—
157C2×D4—121no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—121no—
161C22×C4C2×#6712yes—
162C22×C4C2×#6711yescharacteristic
163M16—122no—
164C2×C8—64no—
165C2×C8—121no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—416no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—62no—
172C22⋊C4—122no—
173C42—28no—
174C42—22no—
175C42#21314yes—
176C42D3×D411yescharacteristic
177C42D3×D412yes—
178C42#23711yescharacteristic
179C22×C3—22no—
180C22×C3#39211yescharacteristic
181C22×C3#44712yes—
182C2×D3—82no—
183C2×D3—81no—
184C3×C4—28no—
185C3×C4—22no—
186C3×C4—84no—
187C3×C4#45614yes—
188C3×C4#45912yes—
189C3×C4#49311yescharacteristic
190C3×C4#49611yescharacteristic
191Dic3—81no—
192C23—121no—
193C23#95311yescharacteristic
194Q8—84no—
195Q8—82no—
196Q8—82no—
197Q8—122no—
198D4—122no—
199D4—124no—
200D4—242no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—48no—
205C2×C4—44no—
206C2×C4—121no—
207C2×C4—241no—
208C2×C4—241no—
209C2×C4C2×#21312yes—
210C2×C4#93212yes—
211C2×C4#93712yes—
212C2×C4#93811yescharacteristic
213C2×C4#93912yes—
214C2×C4#94011yescharacteristic
215C8—124no—
216C8—241no—
217C2×C3—22no—
218C2×C3#200512yes—
219C2×C3#201111yescharacteristic
220D3—161no—
221C22—22no—
222C22—241no—
223C22—242no—
224C22#521712yes—
225C22#522311yescharacteristic
226C4—28no—
227C4—22no—
228C4—84no—
229C4—241no—
230C4#507212yes—
231C4#507311yescharacteristic
232C4#507411yescharacteristic
233C4#509514yes—
234C3#1028911yescharacteristic, complemented, Sylow
235C2—22no—
236C2—481no—
237C2#7176711yescharacteristic
238C2#7179512yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2868C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2868D3
hypocenterC3#10289
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
1. C3×#2868C2semidirect
3. #71773C2semidirect
5. #67611C2semidirect
21. #2868D3semidirect
234. C3#10289semidirect

Maximal subgroups

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

subgroupindex
C3×#28682
#832152
#717732
#693952
#676112
#102893