Finite Group Databasebeta

← #10696040

Subgroups — #10696040

Order 768.

566 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2399C211yesmaximal, characteristic, complemented
2#82442C212yesmaximal
3#71622C212yesmaximal, complemented
4#69440C211yesmaximal, characteristic
5#67614C211yesmaximal, characteristic, complemented
6#10014—31nomaximal, Sylow
7C2×C3×C4×Q8C2212yes—
8#5477C2211yescharacteristic
9C3×#398—22no—
10C3×#395—22no—
11C3×C4×C4⋊C4—22no—
12C3×#376C2211yescharacteristic
13#5227—22no—
14#5027—22no—
15C4×#215—24no—
16#5020—22no—
17#4968—22no—
18C2×#805—22no—
19#4925C2212yes—
20#4924C2211yescharacteristic
21#4535—24no—
22#3103—32no—
23#2399D311yescharacteristic, complemented
24#1963—31no—
25#1487—32no—
26#1274—31no—
27C22×C3×Q8D412yes—
28C2×#235—21no—
29C3×C4×Q8—22no—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C2×C3×C4⋊C4—24no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C42—22no—
37C2×C3×C42D412yes—
38C2×C3×C42C2311yescharacteristic
39C2×#215—24no—
40#930—22no—
41C2×#208—21no—
42#875—21no—
43#874D411yescharacteristic
44C4×D12—22no—
45#869D411yescharacteristic
46C3×#95—22no—
47C3×#95—22no—
48C3×#95—22no—
49#835—24no—
50#830—22no—
51#809—24no—
52#808—24no—
53#805—44no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C4×C3⋊C8D414yes—
57#532—31no—
58C2×C4×Q8C2×D312yes—
59#481—62no—
60#475—62no—
61#460—62no—
62#448—62no—
63C4×SD16—64no—
64#431—32no—
65#423—31no—
66C2×#104—62no—
67#398—22no—
68#395—22no—
69C4×C4⋊C4—22no—
70#376C2×D311yescharacteristic
71Q8⋊C8—64no—
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C2×C3×Q8—44no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4C2×D412yes—
79C22×C3×C4C2×D411yescharacteristic
80C2×#67—41no—
81#235—42no—
82#235—42no—
83C2×D12—22no—
84C2×D12—21no—
85C2×C4×D3—41no—
86C2×Dic6—21no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—48no—
90C3×C4⋊C4—42no—
91C3×C4⋊C4—42no—
92C3×C42—22no—
93C3×C42—42no—
94C3×C42—42no—
95C3×C42SD1614yescomplemented
96C3×C42C2×D411yescharacteristic
97C3×C42C2×D412yes—
98C3×C42#4111yescharacteristic
99#215—48no—
100(C2×D3)⋊C4—42no—
101C4⋊Dic3—22no—
102#208—42no—
103C2×C3⋊C8—24no—
104C2×C3⋊C8—41no—
105C2×#41—61no—
106C22×Q8#6712yes—
107C2×SD16—64no—
108C2×M16—61no—
109C4⋊Q8—31no—
110#127—31no—
111#124—61no—
112#121—62no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×D4—62no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4#6712yes—
121C2×C42—22no—
122C2×C42#6712yes—
123C2×C42C22×D311yescharacteristic
124#106—64no—
125C4⋊C8—62no—
126C4⋊C8—62no—
127#104—124no—
128#103—64no—
129#102—64no—
130C22⋊C8—62no—
131C4×C8—34no—
132#95—22no—
133#95—22no—
134#95—22no—
135C23×C3#12011yescharacteristic
136C22×D3—41no—
137C3×Q8—24no—
138C3×Q8—24no—
139C3×Q8—42no—
140C3×Q8—42no—
141C2×C3×C4—24no—
142C2×C3×C4—24no—
143C2×C3×C4—42no—
144C2×C3×C4—44no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4—44no—
148C2×C3×C4—42no—
149C2×C3×C4#12012yes—
150C2×C3×C4#12111yescharacteristic
151C2×C3×C4#12112yes—
152C2×C3×C4#12711yescharacteristic
153C2×C3×C4C2×SD1612yes—
154C2×C3×C4#13712yes—
155#67—82no—
156C2×Dic3—41no—
157D12—44no—
158D12—42no—
159C4×D3—81no—
160C4×D3—81no—
161Dic6—42no—
162C3⋊C8—44no—
163C3⋊C8—81no—
164#41—122no—
165#41—122no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—44no—
169C2×Q8—61no—
170C2×D4—62no—
171C2×D4—61no—
172C2×D4—121no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—121no—
177C22×C4C2×#6712yes—
178C22×C4C2×#6711yescharacteristic
179SD16—128no—
180M16—122no—
181C2×C8—64no—
182C2×C8—121no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—48no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—62no—
189C22⋊C4—122no—
190C42—22no—
191C42—42no—
192C42—42no—
193C42#21514yescomplemented
194C42D3×D411yescharacteristic
195C42D3×D412yes—
196C42#23711yescharacteristic
197C22×C3—22no—
198C22×C3#39211yescharacteristic
199C22×C3#44812yes—
200C2×D3—82no—
201C2×D3—81no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—44no—
205C3×C4—44no—
206C3×C4—82no—
207C3×C4#46112yes—
208C3×C4#49211yescharacteristic
209C3×C4#49711yescharacteristic
210Dic3—81no—
211C23—121no—
212C23#95311yescharacteristic
213Q8—24no—
214Q8—24no—
215Q8—42no—
216Q8—42no—
217Q8—122no—
218D4—124no—
219D4—122no—
220D4—242no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—121no—
230C2×C4—241no—
231C2×C4—241no—
232C2×C4#93712yes—
233C2×C4#93811yescharacteristic
234C2×C4#93912yes—
235C2×C4#94011yescharacteristic
236C2×C4C2×#21512yes—
237C2×C4#94212yes—
238C8—124no—
239C8—241no—
240C2×C3—22no—
241C2×C3#201012yes—
242C2×C3#201311yescharacteristic
243D3—161no—
244C22—22no—
245C22—242no—
246C22—241no—
247C22#522311yescharacteristic
248C22#522712yes—
249C4—22no—
250C4—22no—
251C4—44no—
252C4—44no—
253C4—82no—
254C4—241no—
255C4#508312yes—
256C4#508511yescharacteristic
257C4#508611yescharacteristic
258C3#1001411yescharacteristic, complemented, Sylow
259C2—22no—
260C2—481no—
261C2#7171512yes—
262C2#7179111yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2399C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2399D3
hypocenterC3#10014
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
1. C3×#2399C2semidirect
3. #71622C2semidirect
5. #67614C2semidirect
23. #2399D3semidirect
95. C3×C42SD16semidirect
193. C42#215semidirect
258. C3#10014semidirect

Maximal subgroups

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

subgroupindex
C3×#23992
#824422
#716222
#694402
#676142
#100143