Finite Group Databasebeta

← #10692583

Subgroups — #10692583

Order 768.

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

normal subgroups only

This group has 514 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×#2563C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic
3#71774C212yesmaximal, complemented
4#71617C211yesmaximal, characteristic
5#67620C212yesmaximal, complemented
6#9399—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#393C414yes—
10C3×#390—21no—
11C3×#382—21no—
12C3×#380—22no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#833—21no—
15#5207—22no—
16#5012—24no—
17#4925C2212yes—
18#4924C2212yes—
19#4546—24no—
20#4530—21no—
21#2975—31no—
22#2563D311yescharacteristic
23#1800—31no—
24#1548—32no—
25#1303—32no—
26C22×C3×Q8C2×C412yes—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4D412yes—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—22no—
31C3×C4×D4—22no—
32C3×C4×D4—22no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C22⋊C4—24no—
38C2×C3×C22⋊C4C2×C412yes—
39C2×C3×C42—21no—
40C2×C3×C42C2311yescharacteristic
41C2×#208—24no—
42C2×#208—22no—
43C3×#95—24no—
44C3×#95C2×C412yes—
45#833—42no—
46#830—24no—
47#830—22no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51C4×C3⋊C8—22no—
52C2×C4⋊Q8C2×D311yescharacteristic
53C2×C4×D4C2×D311yescharacteristic
54#436—64no—
55#431—32no—
56#423—32no—
57C2×#100—61no—
58#407—62no—
59#393Dic314yes—
60#390—21no—
61#382—21no—
62#380—22no—
63C4×C4⋊C4C2×D311yescharacteristic
64#343—61no—
65#333—64no—
66C24×C3C22⋊C412yes—
67C2×C3×Q8—24no—
68C2×C3×Q8—82no—
69C2×C3×D4—22no—
70C2×C3×D4—21no—
71C2×C3×D4—22no—
72C2×C3×D4—21no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—44no—
78C22×C3×C4C22⋊C412yes—
79C22×C3×C4C22×C411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C3×C4⋊C4—22no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4—41no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—41no—
87C3×C22⋊C4—24no—
88C3×C22⋊C4—84no—
89C3×C42—41no—
90C3×C42—41no—
91C3×C42C2×D412yes—
92C3×C42#4112yes—
93#208—42no—
94#208—42no—
95#208—84no—
96C2×C3⋊C8—24no—
97C2×C3⋊C8—42no—
98C22×Q8C2×Dic312yes—
99C22×D4#6711yescharacteristic
100C23×C4#6712yes—
101C2×M16—64no—
102C2×M16—62no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4×D4—22no—
106C4×D4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4#6711yescharacteristic
111C2×C22⋊C4—24no—
112C2×C22⋊C4C2×Dic312yes—
113C2×C42—21no—
114C2×C42C22×D311yescharacteristic
115C4⋊C8—34no—
116C4⋊C8—62no—
117C4⋊C8—62no—
118#100—122no—
119C22⋊C8—64no—
120C22⋊C8—62no—
121C4×C8—62no—
122#95—24no—
123#95C2×Dic312yes—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—42no—
128C23×C3—42no—
129C23×C3C2×C22⋊C411yescharacteristic
130C3×Q8—84no—
131C3×D4—42no—
132C3×D4—42no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—41no—
136C2×C3×C4—44no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4—44no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4—82no—
144C2×C3×C4C4×D412yes—
145C2×C3×C4#12011yescharacteristic
146C2×C3×C4#12112yes—
147C2×C3×C4#12311yescharacteristic
148C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
149C2×C3×C4#13712yes—
150C2×C3×C4#13711yescharacteristic
151C3⋊C8—44no—
152C3⋊C8—82no—
153C24#21712yes—
154C2×Q8—24no—
155C2×Q8—82no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—21no—
159C2×D4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—44no—
165C22×C4#21712yes—
166C22×C4C22×Dic311yescharacteristic
167C22×C4C2×#6711yescharacteristic
168C22×C4C2×#6711yescharacteristic
169M16—122no—
170M16—122no—
171M16—244no—
172C2×C8—64no—
173C2×C8—122no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C22⋊C4—24no—
180C22⋊C4—84no—
181C42—41no—
182C42—41no—
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—82no—
199C3×C4—82no—
200C3×C4#45512yes—
201C3×C4#46812yes—
202C3×C4#47012yes—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—42no—
207C23—42no—
208C23C2×#21711yescharacteristic
209Q8—84no—
210D4—42no—
211D4—42no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4D4×Dic312yes—
224C2×C4#93511yescharacteristic
225C2×C4#93711yescharacteristic
226C2×C4#93912yes—
227C2×C4#94212yes—
228C2×C4#94911yescharacteristic
229C2×C4#95111yescharacteristic
230C8—124no—
231C8—242no—
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#522011yescharacteristic
246C22#522511yescharacteristic
247C22#523211yescharacteristic
248C4—21no—
249C4—22no—
250C4—42no—
251C4—82no—
252C4—82no—
253C4#504912yes—
254C4#505212yes—
255C4#506112yes—
256C3#939911yescharacteristic, complemented, Sylow
257C2—21no—
258C2—21no—
259C2—82no—
260C2#7169711yescharacteristic
261C2#7172311yescharacteristic
262C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2563C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2563D3
hypocenterC3#9399
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #71774C2semidirect
5. #67620C2semidirect
256. C3#9399semidirect

Maximal subgroups

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

subgroupindex
C3×#25632
#823042
#717742
#716172
#676202
#93993