Finite Group Databasebeta

← #10694260

Subgroups — #10694260

Order 768.

478 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 159 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
2C2×#4547C212yesmaximal
3C2×#4547C212yesmaximal
4#69392C211yesmaximal, characteristic
5C4×C4⋊(C3⋊C8)C211yesmaximal, characteristic
6#9700—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C4⋊Q8C2211yescharacteristic
9C3×#391—21no—
10C3×#391—21no—
11C3×#391—22no—
12C3×#382C414yes—
13C3×C43C2211yescharacteristic
14C2×C4⋊(C3⋊C8)C2212yes—
15C2×C4⋊(C3⋊C8)C2212yes—
16C2×C4×C3⋊C8—21no—
17#4547—24no—
18#4547—24no—
19#4524—21no—
20#4524—22no—
21#2862D311yescharacteristic
22#1831—31no—
23C4×C4⋊C8—31no—
24C2×#332—32no—
25C2×#332—32no—
26C22×C3×Q8—21no—
27C22×C3×Q8—21no—
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×C42C2×C412yes—
37C2×C3×C42D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D412yes—
40C2×C3×C42C2311yescharacteristic
41C22×C3⋊C8—22no—
42C22×C3⋊C8—22no—
43C3×#95—24no—
44C3×#95—24no—
45C4⋊(C3⋊C8)—22no—
46C4⋊(C3⋊C8)—22no—
47C4⋊(C3⋊C8)—22no—
48C4⋊(C3⋊C8)D414yes—
49C4×C3⋊C8—42no—
50C2×C4⋊Q8C2×D311yescharacteristic
51C2×C4⋊Q8C2×D311yescharacteristic
52C2×C4⋊C8—32no—
53C2×C4⋊C8—32no—
54C2×C4×C8—61no—
55#391—21no—
56#391—22no—
57#391—21no—
58#382Dic314yes—
59C43C2×D311yescharacteristic
60#336—62no—
61#336—61no—
62#332—64no—
63#332—64no—
64C2×C3×Q8—42no—
65C2×C3×Q8—41no—
66C2×C3×Q8—41no—
67C2×C3×Q8—42no—
68C2×C3×Q8—44no—
69C2×C3×Q8—41no—
70C2×C3×Q8—41no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4C22⋊C412yes—
74C22×C3×C4C22⋊C412yes—
75C22×C3×C4C22×C411yescharacteristic
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4C2×D411yescharacteristic
78C3×C4⋊C4—24no—
79C3×C4⋊C4—24no—
80C3×C4⋊C4—42no—
81C3×C4⋊C4—41no—
82C3×C4⋊C4—41no—
83C3×C42—24no—
84C3×C42—24no—
85C3×C42—22no—
86C3×C42—21no—
87C3×C42—21no—
88C3×C42C2×D412yes—
89C3×C42#4112yes—
90C2×C3⋊C8—24no—
91C2×C3⋊C8—42no—
92C2×C3⋊C8—42no—
93C2×C3⋊C8—42no—
94C2×C3⋊C8—44no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×C8—62no—
98C22×C8—62no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4C2×Dic312yes—
106C2×C4⋊C4C2×Dic312yes—
107C2×C42C2×Dic312yes—
108C2×C42#6711yescharacteristic
109C2×C42#6712yes—
110C2×C42#6711yescharacteristic
111C2×C42C22×D311yescharacteristic
112C4⋊C8—34no—
113C4⋊C8—62no—
114C4⋊C8—62no—
115C4⋊C8—62no—
116C4×C8—122no—
117#95—24no—
118#95—24no—
119C23×C3C2×C22⋊C411yescharacteristic
120C3×Q8—82no—
121C3×Q8—82no—
122C3×Q8—82no—
123C3×Q8—82no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—44no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4#10114yes—
136C2×C3×C4C4×D412yes—
137C2×C3×C4#12112yes—
138C2×C3×C4#12411yescharacteristic
139C2×C3×C4#12711yescharacteristic
140C2×C3×C4#13512yes—
141C2×C3×C4#13514yes—
142C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
143C2×C3×C4#13711yescharacteristic
144C3⋊C8—44no—
145C3⋊C8—82no—
146C2×Q8—41no—
147C2×Q8—42no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—42no—
151C2×Q8—44no—
152C2×Q8—41no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4#21712yes—
156C22×C4#21712yes—
157C22×C4C22×Dic311yescharacteristic
158C22×C4C2×#6711yescharacteristic
159C22×C4C2×#6711yescharacteristic
160C2×C8—64no—
161C2×C8—122no—
162C2×C8—122no—
163C2×C8—122no—
164C2×C8—124no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C42—24no—
171C42—21no—
172C42—24no—
173C42—22no—
174C42—21no—
175C42D3×D412yes—
176C42#23712yes—
177C22×C3#39011yescharacteristic
178C22×C3C2×#10112yes—
179C22×C3#41611yescharacteristic
180C22×C3#41612yes—
181C22×C3#41711yescharacteristic
182C3×C4—24no—
183C3×C4—22no—
184C3×C4—42no—
185C3×C4—82no—
186C3×C4—82no—
187C3×C4#46312yes—
188C3×C4#46412yes—
189C3×C4#48812yes—
190C3×C4#49512yes—
191C23C2×#21711yescharacteristic
192Q8—82no—
193Q8—82no—
194Q8—82no—
195Q8—82no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4#83614yes—
208C2×C4D4×Dic312yes—
209C2×C4#93611yescharacteristic
210C2×C4#93812yes—
211C2×C4#94011yescharacteristic
212C2×C4#94911yescharacteristic
213C2×C4#95012yes—
214C2×C4#95014yes—
215C2×C4#95111yescharacteristic
216C8—124no—
217C8—242no—
218C2×C3#154412yes—
219C2×C3#155212yes—
220C2×C3#195811yescharacteristic
221C2×C3#196611yescharacteristic
222C2×C3#198311yescharacteristic
223C22#522111yescharacteristic
224C22C2×#83612yes—
225C22#523211yescharacteristic
226C22#523311yescharacteristic
227C22#523312yes—
228C4—22no—
229C4—24no—
230C4—42no—
231C4—82no—
232C4—82no—
233C4#505412yes—
234C4#505612yes—
235C4#505712yes—
236C4#505912yes—
237C3#970011yescharacteristic, complemented, Sylow
238C2#7172111yescharacteristic
239C2#7172511yescharacteristic
240C2#7178911yescharacteristic
241C2#7182912yes—
242C2#7183112yes—
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2862C2
socle, prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2862D3
hypocenterC3#9700
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ #2862

Complements

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

NH ≅ G/Ntype
237. C3#9700semidirect

Maximal subgroups

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

subgroupindex
C3×#28622
C2×#45472
#693922
C4×C4⋊(C3⋊C8)2
#97003