Finite Group Databasebeta

← #10698745

Subgroups — #10698745

Order 768.

478 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 133 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×#2582C211yesmaximal, characteristic
2#71626C212yesmaximal
3C2×#4535C212yesmaximal
4#69577C211yesmaximal, characteristic
5#67634C211yesmaximal, characteristic
6#10439—31nomaximal, Sylow
7C2×C3×C4×Q8C2212yes—
8C3×#398—22no—
9C3×#391—22no—
10C3×#389—22no—
11C3×#384C2211yescharacteristic
12C2×#835—22no—
13C2×#807—22no—
14#4935C2211yescharacteristic
15C2×C4⋊(C3⋊C8)C2212yes—
16C2×C4×C3⋊C8C2211yescharacteristic
17#4535—24no—
18#4529—22no—
19#2582D311yescharacteristic
20#1845—31no—
21#1803—32no—
22C2×Q8⋊C8—32no—
23#1296—31no—
24C22×C3×Q8D412yes—
25C3×C4×Q8—22no—
26C3×C4×Q8—22no—
27C3×C4×Q8—22no—
28C3×C4×Q8—22no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4D412yes—
33C2×C3×C4⋊C4D412yes—
34C2×C3×C42—22no—
35C2×C3×C42C2311yescharacteristic
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3—22no—
38C22×C3⋊C8—21no—
39C22×C3⋊C8—22no—
40C3×#95—24no—
41C3×#95—22no—
42#835—44no—
43#807—44no—
44C4⋊(C3⋊C8)—22no—
45C4⋊(C3⋊C8)—22no—
46C4×C3⋊C8—21no—
47C4×C3⋊C8D412yes—
48C2×C4×Q8C2×D312yes—
49C2×#107—62no—
50C2×C4⋊C8—32no—
51C2×#103—62no—
52C2×C4×C8—31no—
53#398—22no—
54#391—22no—
55#389—22no—
56#384—31no—
57#384C2×D311yescharacteristic
58#340—62no—
59Q8⋊C8—64no—
60C2×C3×Q8—22no—
61C2×C3×Q8—22no—
62C2×C3×Q8—22no—
63C2×C3×Q8—22no—
64C2×C3×Q8—22no—
65C2×C3×Q8Q1614yes—
66C22×C3×C4—22no—
67C22×C3×C4—22no—
68C22×C3×C4—22no—
69C22×C3×C4C2×D411yescharacteristic
70C22×C3×C4C2×D412yes—
71C22×Dic3—41no—
72C3×C4⋊C4—22no—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—42no—
75C3×C4⋊C4—42no—
76C3×C4⋊C4—42no—
77C3×C4⋊C4—42no—
78C3×C4⋊C4—42no—
79C3×C4⋊C4—82no—
80C3×C42—42no—
81C3×C42—42no—
82C3×C42C2×D411yescharacteristic
83C3×C42#4111yescharacteristic
84C3×C42#4112yes—
85C4⋊Dic3—42no—
86C4⋊Dic3—42no—
87C4⋊Dic3—42no—
88C2×C3⋊C8—22no—
89C2×C3⋊C8—22no—
90C2×C3⋊C8—22no—
91C2×C3⋊C8—42no—
92C2×C3⋊C8—42no—
93C2×C3⋊C8—42no—
94C22×Q8#6712yes—
95C22×C8—61no—
96C22×C8—62no—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—62no—
105C2×C4⋊C4—61no—
106C2×C4⋊C4#6712yes—
107C2×C4⋊C4#6712yes—
108C2×C42—22no—
109C2×C42C22×D311yescharacteristic
110#107—124no—
111C4⋊C8—62no—
112C4⋊C8—62no—
113#103—124no—
114C4×C8—32no—
115C4×C8—61no—
116#95—24no—
117#95—22no—
118C23×C3#12011yescharacteristic
119C3×Q8—24no—
120C3×Q8—24no—
121C3×Q8—42no—
122C3×Q8—42no—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—24no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—42no—
129C2×C3×C4—44no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4#12111yescharacteristic
134C2×C3×C4#12112yes—
135C2×C3×C4#12312yes—
136C2×C3×C4#12411yescharacteristic
137C2×C3×C4C2×Q1612yes—
138C2×C3×C4#13512yes—
139C2×C3×C4AGL(1,ℤ/8)12yes—
140C2×C3×C4#13712yes—
141C2×Dic3—82no—
142C2×Dic3—81no—
143C3⋊C8—24no—
144C3⋊C8—82no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8#21614yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—121no—
155C22×C4C2×#6711yescharacteristic
156C22×C4C2×#6712yes—
157C2×C8—62no—
158C2×C8—62no—
159C2×C8—62no—
160C2×C8—122no—
161C2×C8—122no—
162C2×C8—122no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—82no—
171C4⋊C4—122no—
172C4⋊C4—122no—
173C4⋊C4—122no—
174C42—42no—
175C42—42no—
176C42D3×D411yescharacteristic
177C42#23711yescharacteristic
178C42#23712yes—
179C22×C3#39411yescharacteristic
180C22×C3#44912yes—
181C22×C3#45112yes—
182C22×C3#45311yescharacteristic
183C22×C3#45611yescharacteristic
184C3×C4—21no—
185C3×C4—21no—
186C3×C4—44no—
187C3×C4—82no—
188C3×C4—82no—
189C3×C4#46712yes—
190C3×C4#46812yes—
191C3×C4#48412yes—
192C3×C4#48512yes—
193Dic3—161no—
194C23#95311yescharacteristic
195Q8—24no—
196Q8—24no—
197Q8—42no—
198Q8—42no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—242no—
210C2×C4—241no—
211C2×C4#93512yes—
212C2×C4#93611yescharacteristic
213C2×C4#93811yescharacteristic
214C2×C4#93912yes—
215C2×C4#94212yes—
216C2×C4C2×#21612yes—
217C2×C4#94912yes—
218C2×C4#95012yes—
219C8—64no—
220C8—242no—
221C2×C3#162011yescharacteristic
222C2×C3#163311yescharacteristic
223C2×C3#164812yes—
224C2×C3#203612yes—
225C2×C3#203911yescharacteristic
226C22#522211yescharacteristic
227C22#522812yes—
228C22#523812yes—
229C22#524111yescharacteristic
230C22#524611yescharacteristic
231C4—21no—
232C4—21no—
233C4—44no—
234C4—82no—
235C4—82no—
236C4—481no—
237C4#504412yes—
238C4#504512yes—
239C4#504912yes—
240C4#505112yes—
241C3#1043911yescharacteristic, complemented, Sylow
242C2#7169512yes—
243C2#7187611yescharacteristic
244C2#7193712yes—
245C2#7194011yescharacteristic
246C2#7194311yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2582C2
socle, prime radicalC23×C3#120
hypercenter#2582D3
hypocenterC3#10439
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2582

Complements

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

NH ≅ G/Ntype
241. C3#10439semidirect

Maximal subgroups

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

subgroupindex
C3×#25822
#716262
C2×#45352
#695772
#676342
#104393