Finite Group Databasebeta

← #10692563

Subgroups — #10692563

Order 768.

594 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 139 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×#2542C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic
3#71775C212yesmaximal, complemented
4#71617C211yesmaximal, characteristic
5#67620C212yesmaximal, complemented
6#9395—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C22⋊Q8—22no—
9C2×C3×C4×D4C2211yescharacteristic
10C3×#399C414yes—
11C3×#390—21no—
12C3×#384—21no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#833—21no—
15C2×#830—22no—
16D4×C3⋊C8—24no—
17#4925C2212yes—
18#4924C2212yes—
19#4547—24no—
20#4530—21no—
21#2972—31no—
22#2542D311yescharacteristic
23#1800—31no—
24#1549—32no—
25#1303—32no—
26C22×C3×Q8—21no—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4D412yes—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—22no—
31C3×C22⋊Q8—48no—
32C3×C4×D4—22no—
33C3×C4×D4—22no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—24no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4C2×C412yes—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43C2×#208—22no—
44C22×C3⋊C8—24no—
45C3×#95C2×C412yes—
46#833—42no—
47#830—24no—
48#830—22no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)—22no—
51C4⋊(C3⋊C8)D414yes—
52C4×C3⋊C8—22no—
53C2×C4⋊Q8C2×D311yescharacteristic
54C2×C22⋊Q8—22no—
55C2×C4×D4C2×D311yescharacteristic
56C8×D4—64no—
57#431—32no—
58#423—32no—
59C2×#100—61no—
60C2×C22⋊C8—62no—
61#399Dic314yes—
62#390—21no—
63#384—21no—
64C4×C4⋊C4C2×D311yescharacteristic
65#343—61no—
66#332—64no—
67C24×C3C22⋊C412yes—
68C2×C3×Q8—44no—
69C2×C3×Q8—41no—
70C2×C3×Q8—41no—
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4SD1612yes—
74C2×C3×D4Q1612yes—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—44no—
80C22×C3×C4C22⋊C412yes—
81C22×C3×C4C22×C411yescharacteristic
82C22×C3×C4C2×D411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C3×C4⋊C4—24no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—24no—
87C3×C4⋊C4—48no—
88C3×C4⋊C4—42no—
89C3×C22⋊C4—24no—
90C3×C22⋊C4—48no—
91C3×C42—42no—
92C3×C42C2×D412yes—
93C3×C42#4112yes—
94#208—42no—
95#208—42no—
96C2×C3⋊C8—24no—
97C2×C3⋊C8—42no—
98C2×C3⋊C8—84no—
99C22×Q8—21no—
100C22×D4#6711yescharacteristic
101C23×C4#6712yes—
102C2×M16—62no—
103C22×C8—64no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C22⋊Q8—48no—
107C4×D4—22no—
108C4×D4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4C2×Dic312yes—
113C2×C4⋊C4#6711yescharacteristic
114C2×C22⋊C4—24no—
115C2×C22⋊C4C2×Dic312yes—
116C2×C42—21no—
117C2×C42C22×D311yescharacteristic
118C4⋊C8—34no—
119C4⋊C8—62no—
120C4⋊C8—62no—
121#100—122no—
122C22⋊C8—64no—
123C22⋊C8—62no—
124C4×C8—62no—
125#95C2×Dic312yes—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—42no—
129C23×C3—42no—
130C23×C3#10314yes—
131C23×C3C2×C22⋊C411yescharacteristic
132C3×Q8—82no—
133C3×Q8—82no—
134C3×D4—24no—
135C3×D4—24no—
136C2×C3×C4—22no—
137C2×C3×C4—24no—
138C2×C3×C4—22no—
139C2×C3×C4—41no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—44no—
143C2×C3×C4—42no—
144C2×C3×C4—44no—
145C2×C3×C4—42no—
146C2×C3×C4—82no—
147C2×C3×C4C4×D412yes—
148C2×C3×C4#12011yescharacteristic
149C2×C3×C4#12112yes—
150C2×C3×C4#12311yescharacteristic
151C2×C3×C4C2×SD1611yescharacteristic
152C2×C3×C4C2×Q1611yescharacteristic
153C2×C3×C4#13512yes—
154C3⋊C8—44no—
155C3⋊C8—82no—
156C24#21712yes—
157C2×Q8—44no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4#21512yes—
163C2×D4#21612yes—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—44no—
169C22×C4#21712yes—
170C22×C4C22×Dic311yescharacteristic
171C22×C4C2×#6711yescharacteristic
172C22×C4C2×#6711yescharacteristic
173M16—122no—
174M16—122no—
175C2×C8—64no—
176C2×C8—122no—
177C2×C8—244no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—48no—
182C4⋊C4—42no—
183C22⋊C4—24no—
184C22⋊C4—48no—
185C42—42no—
186C42D3×D412yes—
187C42#23712yes—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—24no—
191C22×C3—44no—
192C22×C3—42no—
193C22×C3—42no—
194C22×C3—82no—
195C22×C3#38611yescharacteristic
196C22×C3C2×#10311yescharacteristic
197C22×C3#41611yescharacteristic
198C3×C4—21no—
199C3×C4—22no—
200C3×C4—44no—
201C3×C4—42no—
202C3×C4—82no—
203C3×C4#45312yes—
204C3×C4#46512yes—
205C3×C4#46712yes—
206C23—22no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23#83514yes—
211C23C2×#21711yescharacteristic
212Q8—82no—
213Q8—82no—
214D4—24no—
215D4—24no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4D4×Dic312yes—
228C2×C4#93511yescharacteristic
229C2×C4#93711yescharacteristic
230C2×C4#93912yes—
231C2×C4C2×#21511yescharacteristic
232C2×C4C2×#21611yescharacteristic
233C2×C4#95012yes—
234C8—124no—
235C8—242no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—44no—
239C2×C3#188711yescharacteristic
240C2×C3#189211yescharacteristic
241C2×C3#191511yescharacteristic
242C22—21no—
243C22—21no—
244C22—24no—
245C22—44no—
246C22—42no—
247C22—42no—
248C22—82no—
249C22#522011yescharacteristic
250C22C2×#83511yescharacteristic
251C22#523311yescharacteristic
252C4—21no—
253C4—22no—
254C4—44no—
255C4—42no—
256C4—82no—
257C4#504812yes—
258C4#505112yes—
259C4#507712yes—
260C3#939511yescharacteristic, complemented, Sylow
261C2—21no—
262C2—21no—
263C2—44no—
264C2#7169611yescharacteristic
265C2#7176311yescharacteristic
266C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2542C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2542D3
hypocenterC3#9395
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #71775C2semidirect
5. #67620C2semidirect
260. C3#9395semidirect

Maximal subgroups

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

subgroupindex
C3×#25422
#823012
#717752
#716172
#676202
#93953