Finite Group Databasebeta

← #10697872

Subgroups — #10697872

Order 768.

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

normal subgroups only

This group has 614 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#83270C212yesmaximal
2#73311C211yesmaximal, characteristic, complemented
3C3×#1962C211yesmaximal, characteristic
4#70506C212yesmaximal, complemented
5#67611C211yesmaximal, characteristic, complemented
6#10289—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×#869—22no—
9C2×#869C2212yes—
10C3×#423C2211yescharacteristic
11C2×C3×#104—22no—
12C3×C43C2211yescharacteristic
13#5078—22no—
14#5070—22no—
15#5033—24no—
16#5007—24no—
17#4935—24no—
18#4925C2212yes—
19#4482—24no—
20#3226—32no—
21#2868—31no—
22#1962D311yescharacteristic
23#1546—32no—
24#1276—31no—
25C2×C3×#41—21no—
26C22×Dic6—22no—
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×C4⋊Dic3—28no—
37C2×C4⋊Dic3—22no—
38C2×C4⋊Dic3D414yes—
39#869—22no—
40#869—22no—
41#869—22no—
42#869—22no—
43#869—44no—
44#869—44no—
45C3×C4⋊C8—22no—
46C3×#104—44no—
47#832—28no—
48#830—22no—
49#807—28no—
50C4⋊(C3⋊C8)D414yes—
51C4×C3⋊C8—22no—
52#532C2×D311yescharacteristic
53C2×C4⋊Q8—32no—
54C2×C4⋊Q8—62no—
55#500—62no—
56#486—62no—
57#480—64no—
58#474—64no—
59#431—32no—
60#423C2×D311yescharacteristic
61C2×#104—22no—
62#384—64no—
63C43C2×D311yescharacteristic
64#332—64no—
65C3×#41—42no—
66C3×#41—42no—
67C2×C3×Q8—21no—
68C2×C3×D4—22no—
69C2×C3×D4—21no—
70C2×C3×D4—41no—
71C22×C3×C4—22no—
72C22×C3×C4—41no—
73C22×C3×C4C2×D412yes—
74C22×C3×C4C2×D411yescharacteristic
75C22×Dic3—24no—
76C2×Dic6—42no—
77C2×Dic6—42no—
78C2×Dic6—44no—
79C2×Dic6—44no—
80C3×M16—42no—
81C2×C3×C8—41no—
82C3×C4⋊C4—22no—
83C3×C22⋊C4—42no—
84C3×C42—28no—
85C3×C42—22no—
86C3×C42D814yes—
87C3×C42C2×D411yescharacteristic
88C3×C42C2×D412yes—
89C3×C42#4111yescharacteristic
90C4⋊Dic3—24no—
91C4⋊Dic3—24no—
92C4⋊Dic3—416no—
93C4⋊Dic3—42no—
94C4⋊Dic3—42no—
95C2×C3⋊C8—24no—
96C2×#41—21no—
97C22×Q8—62no—
98C2×M16—21no—
99C4⋊Q8—62no—
100C4⋊Q8—62no—
101C4⋊Q8—62no—
102C4⋊Q8—62no—
103C4⋊Q8—124no—
104C4⋊Q8—124no—
105C4⋊Q8D1211yescharacteristic
106#127D1211yescharacteristic
107#124—21no—
108#121—22no—
109C4×D4—22no—
110C2×C4⋊C4—34no—
111C2×C4⋊C4—68no—
112C2×C4⋊C4—62no—
113C2×C42—22no—
114C2×C42#6712yes—
115C2×C42C22×D311yescharacteristic
116#107—68no—
117C4⋊C8—22no—
118C4⋊C8—34no—
119#104—44no—
120#102—68no—
121C22⋊C8—62no—
122C4×C8—62no—
123C23×C3—41no—
124C23×C3#12011yescharacteristic
125C3×Q8—42no—
126C3×D4—44no—
127C3×D4—42no—
128C3×D4—82no—
129C2×C3×C4—28no—
130C2×C3×C4—24no—
131C2×C3×C4—24no—
132C2×C3×C4—41no—
133C2×C3×C4—81no—
134C2×C3×C4—81no—
135C2×C3×C4#12012yes—
136C2×C3×C4#12112yes—
137C2×C3×C4#12111yescharacteristic
138C2×C3×C4#12711yescharacteristic
139C2×C3×C4C2×D812yes—
140C2×C3×C4AGL(1,ℤ/8)12yes—
141C2×Dic3—44no—
142C2×Dic3—48no—
143Dic6—84no—
144Dic6—82no—
145Dic6—82no—
146C3×C8—81no—
147C3⋊C8—44no—
148#41—42no—
149#41—42no—
150C2×Q8—21no—
151C2×Q8—122no—
152C2×Q8—124no—
153C2×Q8—122no—
154C2×Q8—124no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—41no—
158C22×C4—22no—
159C22×C4—41no—
160C22×C4—64no—
161C22×C4D3×D412yes—
162C22×C4C2×#6711yescharacteristic
163M16—42no—
164C2×C8—41no—
165C2×C8—64no—
166C4⋊C4—22no—
167C4⋊C4—64no—
168C4⋊C4—64no—
169C4⋊C4—1216no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C22⋊C4—42no—
173C42—28no—
174C42—22no—
175C42#21314yes—
176C42C2×D1211yescharacteristic
177C42#23511yescharacteristic
178C42D3×D412yes—
179C22×C3—22no—
180C22×C3—82no—
181C22×C3—81no—
182C22×C3#39211yescharacteristic
183C22×C3#44712yes—
184C3×C4—28no—
185C3×C4—22no—
186C3×C4—81no—
187C3×C4#45614yes—
188C3×C4#45912yes—
189C3×C4#49311yescharacteristic
190C3×C4#49611yescharacteristic
191Dic3—84no—
192C23—41no—
193C23#93711yescharacteristic
194Q8—42no—
195Q8—244no—
196Q8—242no—
197Q8—242no—
198D4—42no—
199D4—44no—
200D4—82no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—124no—
208C2×C4—128no—
209C2×C4#88212yes—
210C2×C4#88412yes—
211C2×C4#93011yescharacteristic
212C2×C4C2×#21312yes—
213C2×C4#94011yescharacteristic
214C2×C4#94912yes—
215C8—81no—
216C8—124no—
217C2×C3—22no—
218C2×C3—161no—
219C2×C3#200512yes—
220C2×C3#201111yescharacteristic
221C22—22no—
222C22—81no—
223C22—82no—
224C22#496011yescharacteristic
225C22#503612yes—
226C4—28no—
227C4—22no—
228C4—81no—
229C4—244no—
230C4#482614yes—
231C4#501512yes—
232C4#505811yescharacteristic
233C4#507311yescharacteristic
234C3#1028911yescharacteristic, complemented, Sylow
235C2—22no—
236C2—161no—
237C2#6965811yescharacteristic
238C2#7038912yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1962D3
hypocenterC3#10289
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. #73311C2semidirect
4. #70506C2semidirect
5. #67611C2semidirect
234. C3#10289semidirect

Maximal subgroups

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

subgroupindex
#832702
#733112
C3×#19622
#705062
#676112
#102893