Finite Group Databasebeta

← #10697849

Subgroups — #10697849

Order 768.

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

normal subgroups only

This group has 542 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×#2862C211yesmaximal, characteristic, complemented
2#83216C212yesmaximal
3#71775C212yesmaximal, complemented
4#69395C211yesmaximal, characteristic
5#67611C211yesmaximal, characteristic, complemented
6#10285—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2212yes—
8#5477C2211yescharacteristic
9C3×#391—24no—
10C3×#382—22no—
11C3×C43C2211yescharacteristic
12#5088—22no—
13#5084—24no—
14#5080—22no—
15#4968—24no—
16C2×#805—22no—
17#4925C2212yes—
18#4924C2211yescharacteristic
19#4547—24no—
20#3227—32no—
21#2862D311yescharacteristic, complemented
22#1962—31no—
23#1549—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—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4D414yes—
33C2×C3×C42—22no—
34C2×C3×C42D412yes—
35C2×C3×C42C2311yescharacteristic
36#930—22no—
37C2×#208—21no—
38#875—21no—
39#874D411yescharacteristic
40C4×D12—22no—
41#869D411yescharacteristic
42C3×#95—28no—
43#830—22no—
44#809—28no—
45#808—28no—
46#805—44no—
47C4⋊(C3⋊C8)—22no—
48C4⋊(C3⋊C8)D414yes—
49C4×C3⋊C8—22no—
50#532—31no—
51C2×C4⋊Q8C2×D312yes—
52#498—62no—
53#486—62no—
54#481—64no—
55#476—64no—
56#431—32no—
57#423—31no—
58C2×#104—62no—
59#391—24no—
60#382—22no—
61C43C2×D311yescharacteristic
62#332—64no—
63C2×C3×Q8—48no—
64C2×C3×Q8—42no—
65C2×C3×Q8—42no—
66C22×C3×C4—22no—
67C22×C3×C4—24no—
68C22×C3×C4C2×D412yes—
69C22×C3×C4C2×D411yescharacteristic
70C2×#67—41no—
71#235—42no—
72#235—42no—
73C2×D12—22no—
74C2×D12—21no—
75C2×C4×D3—41no—
76C2×Dic6—21no—
77C3×C4⋊C4—24no—
78C3×C4⋊C4—24no—
79C3×C4⋊C4—42no—
80C3×C4⋊C4—42no—
81C3×C42—22no—
82C3×C42—44no—
83C3×C42SD1614yes—
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D412yes—
86C3×C42#4111yescharacteristic
87(C2×D3)⋊C4—42no—
88C4⋊Dic3—22no—
89#208—42no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8—41no—
92C2×#41—61no—
93C22×Q8—22no—
94C2×M16—61no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—31no—
100#127—31no—
101#124—61no—
102#121—62no—
103C4×D4—62no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4#6714yes—
106C2×C42—22no—
107C2×C42#6712yes—
108C2×C42C22×D311yescharacteristic
109#106—68no—
110C4⋊C8—34no—
111C4⋊C8—62no—
112#104—124no—
113#102—68no—
114C22⋊C8—62no—
115C4×C8—62no—
116#95—28no—
117C23×C3#12011yescharacteristic
118C22×D3—41no—
119C3×Q8—84no—
120C3×Q8—84no—
121C2×C3×C4—24no—
122C2×C3×C4—24no—
123C2×C3×C4—44no—
124C2×C3×C4—48no—
125C2×C3×C4—44no—
126C2×C3×C4#12012yes—
127C2×C3×C4#12111yescharacteristic
128C2×C3×C4#12112yes—
129C2×C3×C4#12711yescharacteristic
130C2×C3×C4C2×SD1612yes—
131C2×C3×C4#13712yes—
132#67—82no—
133C2×Dic3—41no—
134D12—44no—
135D12—42no—
136C4×D3—81no—
137C4×D3—81no—
138Dic6—42no—
139C3⋊C8—44no—
140C3⋊C8—81no—
141#41—122no—
142#41—122no—
143C2×Q8—48no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—61no—
147C2×D4—62no—
148C2×D4—61no—
149C2×D4—121no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—121no—
153C22×C4C2×#6712yes—
154C22×C4C2×#6711yescharacteristic
155M16—122no—
156C2×C8—64no—
157C2×C8—121no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—62no—
163C22⋊C4—122no—
164C42—22no—
165C42—44no—
166C42#21514yes—
167C42D3×D411yescharacteristic
168C42D3×D412yes—
169C42#23711yescharacteristic
170C22×C3—22no—
171C22×C3#39211yescharacteristic
172C22×C3#44812yes—
173C2×D3—82no—
174C2×D3—81no—
175C3×C4—22no—
176C3×C4—44no—
177C3×C4—84no—
178C3×C4#45414yes—
179C3×C4#46112yes—
180C3×C4#49211yescharacteristic
181C3×C4#49711yescharacteristic
182Dic3—81no—
183C23—121no—
184C23#95311yescharacteristic
185Q8—84no—
186Q8—84no—
187Q8—122no—
188D4—124no—
189D4—122no—
190D4—242no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—48no—
195C2×C4—44no—
196C2×C4—121no—
197C2×C4—241no—
198C2×C4—241no—
199C2×C4#93712yes—
200C2×C4#93811yescharacteristic
201C2×C4#93912yes—
202C2×C4#94011yescharacteristic
203C2×C4C2×#21512yes—
204C2×C4#94212yes—
205C8—124no—
206C8—241no—
207C2×C3—22no—
208C2×C3#200312yes—
209C2×C3#201311yescharacteristic
210D3—161no—
211C22—22no—
212C22—242no—
213C22—241no—
214C22#522311yescharacteristic
215C22#522712yes—
216C4—22no—
217C4—44no—
218C4—84no—
219C4—241no—
220C4#506214yes—
221C4#508312yes—
222C4#508511yescharacteristic
223C4#508611yescharacteristic
224C3#1028511yescharacteristic, complemented, Sylow
225C2—22no—
226C2—481no—
227C2#7173012yes—
228C2#7179111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2862C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2862D3
hypocenterC3#10285
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
1. C3×#2862C2semidirect
3. #71775C2semidirect
5. #67611C2semidirect
21. #2862D3semidirect
224. C3#10285semidirect

Maximal subgroups

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

subgroupindex
C3×#28622
#832162
#717752
#693952
#676112
#102853