Finite Group Databasebeta

← #10692580

Subgroups — #10692580

Order 768.

514 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 115 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#82384C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic
3C3×#1800C211yesmaximal, characteristic
4#70507C212yesmaximal, complemented
5#67715C212yesmaximal, complemented
6#9399—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×#423C2212yes—
10C2×C3×#100—21no—
11#5221—21no—
12#5212—22no—
13#5207—22no—
14#5012—24no—
15#4977—21no—
16#4947C414yes—
17C4×C4⋊Dic3C2211yescharacteristic
18#4925C2212yes—
19C3×#343—21no—
20#4481—24no—
21#2975—31no—
22#2563—31no—
23#1800D311yescharacteristic
24#1548—32no—
25#1303—32no—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4D412yes—
28C22×Dic6C2×C412yes—
29C2×C3×M16—22no—
30C3×C4×D4—22no—
31C3×C4×D4—22no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C22⋊C4C2×C412yes—
34C2×C3×C42C2311yescharacteristic
35C2×#217—24no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3—21no—
39C2×C4×Dic3—21no—
40C2×#208—24no—
41#869—22no—
42#869—22no—
43C3×C4⋊C8—22no—
44C3×C4⋊C8—22no—
45C3×#100—42no—
46C3⋊#95—24no—
47C3⋊#95C2×C412yes—
48#830—24no—
49#830—22no—
50C4⋊(C3⋊C8)D414yes—
51C4×C3⋊C8—22no—
52C2×C4⋊Q8—31no—
53C2×C4×D4C2×D311yescharacteristic
54#436—64no—
55#431—32no—
56#423C2×D312yes—
57C2×#100—21no—
58#407—62no—
59#393—34no—
60#390—61no—
61#382—61no—
62#380—62no—
63C4×C4⋊C4—31no—
64#343—21no—
65#333—64no—
66C24×C3C22⋊C412yes—
67C2×C3×D4—22no—
68C2×C3×D4—21no—
69C2×C3×D4—21no—
70C2×C3×D4—22no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—44no—
74C22×C3×C4C22⋊C412yes—
75C22×C3×C4C22×C411yescharacteristic
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4C2×D411yescharacteristic
78C22×Dic3—22no—
79C22×Dic3—22no—
80C2×Dic6—24no—
81C2×Dic6—82no—
82C3×M16—42no—
83C3×M16—42no—
84C2×C3×C8—42no—
85C3×C4⋊C4—22no—
86C3×C22⋊C4—24no—
87C3×C42C2×D412yes—
88C3×C42#4112yes—
89#217—84no—
90C4⋊Dic3—42no—
91C4⋊Dic3—41no—
92C4⋊Dic3—42no—
93C4⋊Dic3—41no—
94C4×Dic3—41no—
95C4×Dic3—41no—
96#208—84no—
97C2×C3⋊C8—24no—
98C22×Q8—32no—
99C22×D4#6711yescharacteristic
100C23×C4D1212yes—
101C2×M16—22no—
102C2×M16—64no—
103C4⋊Q8—62no—
104C4⋊Q8—62no—
105C4×D4—22no—
106C4×D4—22no—
107C2×C4⋊C4—61no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4—61no—
110C2×C4⋊C4#6711yescharacteristic
111C2×C22⋊C4—64no—
112C2×C22⋊C4C4×D312yes—
113C2×C42—61no—
114C2×C42C22×D311yescharacteristic
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—34no—
118#100—42no—
119C22⋊C8—64no—
120C22⋊C8—62no—
121C4×C8—62no—
122#95—32no—
123#95—64no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—42no—
128C23×C3—42no—
129C23×C3C2×C22⋊C411yescharacteristic
130C3×D4—42no—
131C3×D4—42no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—41no—
135C2×C3×C4—44no—
136C2×C3×C4—42no—
137C2×C3×C4—82no—
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#13711yescharacteristic
144C2×C3×C4#13712yes—
145C2×Dic3—42no—
146C2×Dic3—42no—
147C2×Dic3—44no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150Dic6—84no—
151C3×C8—82no—
152C3⋊C8—44no—
153C24(C2×D3)⋊C412yes—
154C2×Q8—64no—
155C2×Q8—242no—
156C2×D4—21no—
157C2×D4—22no—
158C2×D4—21no—
159C2×D4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—44no—
163C22×C4—62no—
164C22×C4—62no—
165C22×C4(C2×D3)⋊C412yes—
166C22×C4C2×C4×D311yescharacteristic
167C22×C4C2×D1211yescharacteristic
168C22×C4C2×#6711yescharacteristic
169M16—42no—
170M16—42no—
171M16—244no—
172C2×C8—42no—
173C2×C8—64no—
174C4⋊C4—22no—
175C4⋊C4—122no—
176C4⋊C4—121no—
177C4⋊C4—121no—
178C4⋊C4—122no—
179C22⋊C4—24no—
180C22⋊C4—244no—
181C42—121no—
182C42—121no—
183C42D3×D412yes—
184C42#23712yes—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—44no—
188C22×C3—42no—
189C22×C3—42no—
190C22×C3—42no—
191C22×C3—82no—
192C22×C3#38611yescharacteristic
193C22×C3#41711yescharacteristic
194C22×C3#41911yescharacteristic
195C3×C4—21no—
196C3×C4—22no—
197C3×C4—42no—
198C3×C4#45512yes—
199C3×C4#46812yes—
200C3×C4#47012yes—
201Dic3—82no—
202Dic3—82no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—42no—
207C23—42no—
208C23C2×(C2×D3)⋊C411yescharacteristic
209Q8—244no—
210D4—42no—
211D4—42no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—122no—
219C2×C4—124no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4—122no—
223C2×C4#67⋊C412yes—
224C2×C4#88211yescharacteristic
225C2×C4#88611yescharacteristic
226C2×C4#93211yescharacteristic
227C2×C4#93912yes—
228C2×C4#94211yescharacteristic
229C2×C4#95112yes—
230C8—82no—
231C8—124no—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3—82no—
235C2×C3#188511yescharacteristic
236C2×C3#189411yescharacteristic
237C2×C3#191411yescharacteristic
238C22—21no—
239C22—21no—
240C22—44no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—82no—
245C22#495811yescharacteristic
246C22#496611yescharacteristic
247C22#500411yescharacteristic
248C4—21no—
249C4—22no—
250C4—42no—
251C4—242no—
252C4—242no—
253C4#474612yes—
254C4#504212yes—
255C4#505212yes—
256C3#939911yescharacteristic, complemented, Sylow
257C2—21no—
258C2—21no—
259C2—82no—
260C2#6951611yescharacteristic
261C2#7006911yescharacteristic
262C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1800C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1800D3
hypocenterC3#9399
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
4. #70507C2semidirect
5. #67715C2semidirect
256. C3#9399semidirect

Maximal subgroups

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

subgroupindex
#823842
#823042
C3×#18002
#705072
#677152
#93993