Finite Group Databasebeta

← #10692608

Subgroups — #10692608

Order 768.

450 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 123 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×#2568C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3#71775C212yesmaximal
4#71637C211yesmaximal, characteristic
5#67620C212yesmaximal
6#9404—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#400C414yes—
10C3×#391—21no—
11C3×#391—22no—
12C3×#382—21no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#836—21no—
15#5024—24no—
16#4926—22no—
17#4925C2212yes—
18#4924C2212yes—
19#4547—24no—
20#4530—21no—
21#2983—31no—
22#2568D311yescharacteristic
23#1801—31no—
24#1549—32no—
25#1303—32no—
26C22×C3×Q8—21no—
27C22×C3×Q8D411yescharacteristic
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4C2×C412yes—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C42—21no—
38C2×C3×C42D412yes—
39C2×C3×C42C2311yescharacteristic
40C2×#208—22no—
41C3×#95—24no—
42C3×#95—24no—
43C3×#95C2×C412yes—
44#836—42no—
45#830—22no—
46C4⋊(C3⋊C8)—24no—
47C4⋊(C3⋊C8)—22no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)D414yes—
50#803—24no—
51C4×C3⋊C8—22no—
52C2×C4⋊Q8C2×D311yescharacteristic
53C2×C4×Q8C2×D311yescharacteristic
54#446—64no—
55#432—62no—
56#431—32no—
57#423—32no—
58C2×#101—61no—
59#400Dic314yes—
60#391—21no—
61#391—22no—
62#382—21no—
63C4×C4⋊C4C2×D311yescharacteristic
64#343—61no—
65#332—64no—
66C2×C3×Q8—22no—
67C2×C3×Q8—21no—
68C2×C3×Q8—22no—
69C2×C3×Q8—21no—
70C2×C3×Q8—44no—
71C2×C3×Q8—41no—
72C2×C3×Q8—41no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4C22⋊C412yes—
76C22×C3×C4C22⋊C412yes—
77C22×C3×C4C22×C411yescharacteristic
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×D411yescharacteristic
80C3×C4⋊C4—24no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4—42no—
85C3×C42—22no—
86C3×C42—22no—
87C3×C42—41no—
88C3×C42—41no—
89C3×C42C2×D412yes—
90C3×C42#4112yes—
91#208—42no—
92#208—42no—
93C2×C3⋊C8—24no—
94C2×C3⋊C8—42no—
95C22×Q8—21no—
96C22×Q8#6711yescharacteristic
97C2×M16—62no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4×Q8—22no—
101C4×Q8—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4C2×Dic312yes—
105C2×C4⋊C4C2×Dic312yes—
106C2×C4⋊C4#6711yescharacteristic
107C2×C42—21no—
108C2×C42#6712yes—
109C2×C42C22×D311yescharacteristic
110C4⋊C8—34no—
111C4⋊C8—64no—
112C4⋊C8—62no—
113C4⋊C8—62no—
114#101—122no—
115C22⋊C8—62no—
116#97—64no—
117C4×C8—62no—
118#95—24no—
119#95—24no—
120#95C2×Dic312yes—
121C23×C3C2×C22⋊C411yescharacteristic
122C3×Q8—42no—
123C3×Q8—42no—
124C3×Q8—82no—
125C3×Q8—82no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—41no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—44no—
136C2×C3×C4—42no—
137C2×C3×C4—42no—
138C2×C3×C4C4×D412yes—
139C2×C3×C4#12011yescharacteristic
140C2×C3×C4#12112yes—
141C2×C3×C4#12311yescharacteristic
142C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
143C2×C3×C4#13712yes—
144C2×C3×C4#13711yescharacteristic
145C3⋊C8—44no—
146C3⋊C8—82no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—44no—
152C2×Q8—41no—
153C2×Q8—41no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4#21712yes—
157C22×C4#21712yes—
158C22×C4C22×Dic311yescharacteristic
159C22×C4C2×#6711yescharacteristic
160C22×C4C2×#6711yescharacteristic
161M16—122no—
162M16—122no—
163C2×C8—64no—
164C2×C8—122no—
165C4⋊C4—24no—
166C4⋊C4—22no—
167C4⋊C4—24no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C42—22no—
171C42—22no—
172C42—41no—
173C42—41no—
174C42D3×D412yes—
175C42#23712yes—
176C22×C3—21no—
177C22×C3—21no—
178C22×C3#38611yescharacteristic
179C22×C3#41711yescharacteristic
180C22×C3#41911yescharacteristic
181C3×C4—21no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—42no—
185C3×C4—82no—
186C3×C4—82no—
187C3×C4#45412yes—
188C3×C4#46812yes—
189C3×C4#47012yes—
190C23C2×#21711yescharacteristic
191Q8—42no—
192Q8—42no—
193Q8—82no—
194Q8—82no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4D4×Dic312yes—
208C2×C4#93511yescharacteristic
209C2×C4#93711yescharacteristic
210C2×C4#93912yes—
211C2×C4#94212yes—
212C2×C4#94911yescharacteristic
213C2×C4#95111yescharacteristic
214C8—124no—
215C8—242no—
216C2×C3—21no—
217C2×C3—21no—
218C2×C3#188811yescharacteristic
219C2×C3#189411yescharacteristic
220C2×C3#191511yescharacteristic
221C22—21no—
222C22—21no—
223C22#522011yescharacteristic
224C22#522511yescharacteristic
225C22#523211yescharacteristic
226C4—21no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—82no—
231C4—82no—
232C4#504912yes—
233C4#505212yes—
234C4#506212yes—
235C3#940411yescharacteristic, complemented, Sylow
236C2—21no—
237C2—21no—
238C2#7169711yescharacteristic
239C2#7172411yescharacteristic
240C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2568C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2568D3
hypocenterC3#9404
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
235. C3#9404semidirect

Maximal subgroups

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

subgroupindex
C3×#25682
#824362
#717752
#716372
#676202
#94043