Finite Group Databasebeta

← #10697860

Subgroups — #10697860

Order 768.

566 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 111 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#83317C212yesmaximal
2#73308C211yesmaximal, characteristic, complemented
3C3×#1962C211yesmaximal, characteristic
4#70507C212yesmaximal, complemented
5#67611C211yesmaximal, characteristic, complemented
6#10287—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×#869C2212yes—
9C3×#423C2211yescharacteristic
10C2×C3×#104—22no—
11C3×C43C2211yescharacteristic
12#5093—22no—
13#5076—22no—
14#5040—24no—
15#5019—24no—
16#4936—22no—
17#4929—24no—
18#4925C2212yes—
19#4481—24no—
20#3206—32no—
21#2862—31no—
22#1962D311yescharacteristic
23#1548—32no—
24#1276—31no—
25C2×C3×#41—21no—
26C22×Dic6D414yes—
27C2×C3×M16—21no—
28C3×C4⋊Q8D411yescharacteristic
29C3×#127D411yescharacteristic
30C3×#124—21no—
31C3×#121—22no—
32C3×C4×D4—22no—
33C2×C3×C42—22no—
34C2×C3×C42D412yes—
35C2×C3×C42C2311yescharacteristic
36C2×#214—28no—
37C2×C4⋊Dic3—22no—
38C2×C4⋊Dic3—22no—
39#869—22no—
40#869—22no—
41#869—22no—
42#869—22no—
43C3×C4⋊C8—22no—
44C3×#104—44no—
45C3⋊#95—28no—
46#830—22no—
47#810—28no—
48C4⋊(C3⋊C8)D414yes—
49C4×C3⋊C8—22no—
50#532C2×D311yescharacteristic
51C2×C4⋊Q8—32no—
52#492—62no—
53#487—62no—
54#461—64no—
55#448—64no—
56#431—32no—
57#423C2×D311yescharacteristic
58C2×#104—22no—
59#391—64no—
60#382—62no—
61C43C2×D311yescharacteristic
62#333—64no—
63C3×#41—42no—
64C3×#41—42no—
65C2×C3×Q8—21no—
66C2×C3×D4—22no—
67C2×C3×D4—21no—
68C2×C3×D4—41no—
69C22×C3×C4—22no—
70C22×C3×C4—41no—
71C22×C3×C4C2×D412yes—
72C22×C3×C4C2×D411yescharacteristic
73C22×Dic3—24no—
74C2×Dic6—24no—
75C2×Dic6—24no—
76C2×Dic6—48no—
77C3×M16—42no—
78C2×C3×C8—41no—
79C3×C4⋊C4—22no—
80C3×C22⋊C4—42no—
81C3×C42—22no—
82C3×C42—44no—
83C3×C42SD1614yescomplemented
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D412yes—
86C3×C42#4111yescharacteristic
87#214—88no—
88C4⋊Dic3—42no—
89C4⋊Dic3—42no—
90C4⋊Dic3—42no—
91C4⋊Dic3—42no—
92C2×C3⋊C8—24no—
93C2×#41—21no—
94C22×Q8—34no—
95C2×SD16—68no—
96C2×M16—21no—
97C4⋊Q8—62no—
98C4⋊Q8—62no—
99C4⋊Q8—62no—
100C4⋊Q8—62no—
101C4⋊Q8D1211yescharacteristic
102#127D1211yescharacteristic
103#124—21no—
104#121—22no—
105C4×D4—22no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—62no—
108C2×C42—22no—
109C2×C42#6712yes—
110C2×C42C22×D311yescharacteristic
111C4⋊C8—22no—
112C4⋊C8—34no—
113#104—44no—
114#103—68no—
115C22⋊C8—62no—
116C4×C8—62no—
117#95—68no—
118C23×C3—41no—
119C23×C3#12011yescharacteristic
120C3×Q8—42no—
121C3×D4—44no—
122C3×D4—42no—
123C3×D4—82no—
124C2×C3×C4—24no—
125C2×C3×C4—24no—
126C2×C3×C4—44no—
127C2×C3×C4—41no—
128C2×C3×C4—81no—
129C2×C3×C4—81no—
130C2×C3×C4#12012yes—
131C2×C3×C4#12112yes—
132C2×C3×C4#12111yescharacteristic
133C2×C3×C4#12711yescharacteristic
134C2×C3×C4C2×SD1612yes—
135C2×C3×C4#13712yes—
136C2×Dic3—48no—
137C2×Dic3—44no—
138Dic6—48no—
139Dic6—84no—
140C3×C8—81no—
141C3⋊C8—44no—
142#41—42no—
143#41—42no—
144C2×Q8—21no—
145C2×Q8—64no—
146C2×Q8—64no—
147C2×Q8—128no—
148C2×D4—22no—
149C2×D4—21no—
150C2×D4—41no—
151C22×C4—22no—
152C22×C4—41no—
153C22×C4—64no—
154C22×C4D3×D412yes—
155C22×C4C2×#6711yescharacteristic
156SD16—248no—
157M16—42no—
158C2×C8—41no—
159C2×C8—64no—
160C4⋊C4—22no—
161C4⋊C4—122no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C4⋊C4—122no—
165C22⋊C4—42no—
166C42—22no—
167C42—44no—
168C42#21414yescomplemented
169C42C2×D1211yescharacteristic
170C42#23511yescharacteristic
171C42D3×D412yes—
172C22×C3—22no—
173C22×C3—82no—
174C22×C3—81no—
175C22×C3#39211yescharacteristic
176C22×C3#44812yes—
177C3×C4—22no—
178C3×C4—44no—
179C3×C4—81no—
180C3×C4#45514yes—
181C3×C4#46112yes—
182C3×C4#49211yescharacteristic
183C3×C4#49711yescharacteristic
184Dic3—84no—
185C23—41no—
186C23#93711yescharacteristic
187Q8—42no—
188Q8—128no—
189Q8—244no—
190D4—42no—
191D4—44no—
192D4—82no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—44no—
196C2×C4—41no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—128no—
200C2×C4—124no—
201C2×C4#88212yes—
202C2×C4#88412yes—
203C2×C4#93011yescharacteristic
204C2×C4C2×#21412yes—
205C2×C4#94011yescharacteristic
206C2×C4#95112yes—
207C8—81no—
208C8—124no—
209C2×C3—22no—
210C2×C3—161no—
211C2×C3#200412yes—
212C2×C3#201311yescharacteristic
213C22—22no—
214C22—81no—
215C22—82no—
216C22#496011yescharacteristic
217C22#504012yes—
218C4—22no—
219C4—44no—
220C4—81no—
221C4—244no—
222C4#482514yes—
223C4#501912yes—
224C4#506011yescharacteristic
225C4#507611yescharacteristic
226C3#1028711yescharacteristic, complemented, Sylow
227C2—22no—
228C2—161no—
229C2#6966111yescharacteristic
230C2#7038612yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1962D3
hypocenterC3#10287
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. #73308C2semidirect
4. #70507C2semidirect
5. #67611C2semidirect
83. C3×C42SD16semidirect
168. C42#214semidirect
226. C3#10287semidirect

Maximal subgroups

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

subgroupindex
#833172
#733082
C3×#19622
#705072
#676112
#102873