Finite Group Databasebeta

← #10688784

Subgroups — #10688784

Order 768.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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#82338C211yesmaximal, characteristic
2C3×#1460C211yesmaximal, characteristic
3#71617C211yesmaximal, characteristic
4#67746C212yesmaximal, complemented
5#67738C212yesmaximal, complemented
6#8770—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C3×#431C2212yes—
9#5221—21no—
10C2×#833—21no—
11#5211C412yes—
12C4×#217C412yes—
13Dic3×C4⋊C4—21no—
14#4951—22no—
15#4936C2211yescharacteristic
16C4×C4⋊Dic3C2211yescharacteristic
17#4924C2212yes—
18C3×#323—22no—
19#4530—21no—
20#2357—31no—
21#1800—31no—
22#1460D311yescharacteristic
23#1303—32no—
24#1286—32no—
25C22×C3×D4D411yescharacteristic
26C23×C3×C4C2×C412yes—
27C3×C4×D4—22no—
28C3×C4×D4—22no—
29C2×C3×C4⋊C4D411yescharacteristic
30C2×C3×C22⋊C4D412yes—
31C2×C3×C42C2311yescharacteristic
32C2×#217—22no—
33C2×#217—22no—
34C2×C4⋊Dic3—21no—
35C2×C4⋊Dic3—21no—
36C2×C4×Dic3—21no—
37C2×C4×Dic3—22no—
38C2×#208—22no—
39C3×C4⋊C8—22no—
40C3×C22⋊C8—22no—
41C3×C4×C8—22no—
42#833—42no—
43C3⋊#95—22no—
44C3⋊#95—21no—
45C3⋊#95C2×C411yescharacteristic
46C3⋊#95C2×C411yescharacteristic
47C3⋊#95C2×C412yes—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)—22no—
50C2×C4×D4C2×D311yescharacteristic
51#431C2×D312yes—
52#423—32no—
53C2×#100—61no—
54#390—61no—
55#388—62no—
56#382—31no—
57#381—32no—
58C4×C4⋊C4—31no—
59C4×C4⋊C4—61no—
60C4×C22⋊C4—32no—
61#343—61no—
62#323—22no—
63C24×C3C22⋊C412yes—
64C2×C3×D4—22no—
65C2×C3×D4—22no—
66C2×C3×D4—21no—
67C2×C3×D4—21no—
68C22×C3×C4—22no—
69C22×C3×C4—22no—
70C22×C3×C4—82no—
71C22×C3×C4C22⋊C412yes—
72C22×C3×C4C22×C411yescharacteristic
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D411yescharacteristic
75C22×Dic3—21no—
76C22×Dic3—22no—
77C22×Dic3—21no—
78C2×C3×C8—42no—
79C3×C4⋊C4—22no—
80C3×C22⋊C4—24no—
81C3×C42#4112yes—
82C3×C42#4112yes—
83#217—44no—
84#217—82no—
85C4⋊Dic3—42no—
86C4⋊Dic3—42no—
87C4×Dic3—41no—
88C4×Dic3—41no—
89C4×Dic3—82no—
90#208—42no—
91#208—42no—
92C2×C3⋊C8—42no—
93C22×D4D1211yescharacteristic
94C23×C4C4×D312yes—
95C2×M16—62no—
96C4×D4—22no—
97C4×D4—22no—
98C2×C4⋊C4—61no—
99C2×C4⋊C4—61no—
100C2×C4⋊C4D1211yescharacteristic
101C2×C22⋊C4—62no—
102C2×C22⋊C4—62no—
103C2×C22⋊C4#6712yes—
104C2×C42—61no—
105C2×C42—62no—
106C2×C42C22×D311yescharacteristic
107C4⋊C8—22no—
108C4⋊C8—62no—
109C4⋊C8—62no—
110#100—122no—
111C22⋊C8—22no—
112C4×C8—22no—
113#95—31no—
114#95—32no—
115#95—31no—
116#95—62no—
117#95—61no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—42no—
122C23×C3—42no—
123C23×C3C2×C22⋊C411yescharacteristic
124C3×D4—42no—
125C3×D4—42no—
126C2×C3×C4—22no—
127C2×C3×C4—41no—
128C2×C3×C4—41no—
129C2×C3×C4—42no—
130C2×C3×C4—82no—
131C2×C3×C4—82no—
132C2×C3×C4#11711yescharacteristic
133C2×C3×C4#11711yescharacteristic
134C2×C3×C4#12312yes—
135C2×C3×C4#12312yes—
136C2×C3×C4#13512yes—
137C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
138C2×C3×C4#13711yescharacteristic
139C2×Dic3—22no—
140C2×Dic3—41no—
141C2×Dic3—41no—
142C2×Dic3—42no—
143C2×Dic3—41no—
144C2×Dic3—42no—
145C2×Dic3—41no—
146C2×Dic3—42no—
147C2×Dic3—41no—
148C3×C8—82no—
149C3⋊C8—82no—
150C24(C2×D3)⋊C412yes—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—61no—
158C22×C4—62no—
159C22×C4—61no—
160C22×C4—82no—
161C22×C4(C2×D3)⋊C412yes—
162C22×C4C2×C4×D311yescharacteristic
163C22×C4C2×D1211yescharacteristic
164C22×C4C2×#6711yescharacteristic
165M16—122no—
166M16—122no—
167C2×C8—42no—
168C2×C8—122no—
169C4⋊C4—22no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C22⋊C4—24no—
173C22⋊C4—124no—
174C22⋊C4—242no—
175C42—121no—
176C42—121no—
177C42—242no—
178C42#23712yes—
179C42#23712yes—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—42no—
183C22×C3—42no—
184C22×C3—42no—
185C22×C3—82no—
186C22×C3—82no—
187C22×C3#38111yescharacteristic
188C22×C3#41611yescharacteristic
189C22×C3#41711yescharacteristic
190C3×C4—22no—
191C3×C4—21no—
192C3×C4—21no—
193C3×C4—42no—
194C3×C4#48212yes—
195C3×C4#48512yes—
196Dic3—42no—
197Dic3—81no—
198Dic3—82no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23C2×(C2×D3)⋊C411yescharacteristic
205D4—42no—
206D4—42no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—62no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—121no—
215C2×C4—121no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—121no—
219C2×C4—121no—
220C2×C4—121no—
221C2×C4—122no—
222C2×C4#87711yescharacteristic
223C2×C4#87711yescharacteristic
224C2×C4#88612yes—
225C2×C4#90412yes—
226C2×C4#90811yescharacteristic
227C2×C4#90911yescharacteristic
228C2×C4#93512yes—
229C8—82no—
230C8—242no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—82no—
234C2×C3#182511yescharacteristic
235C2×C3#182711yescharacteristic
236C2×C3#182911yescharacteristic
237C22—21no—
238C22—21no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—82no—
243C22—82no—
244C22#494311yescharacteristic
245C22#511311yescharacteristic
246C22#513311yescharacteristic
247C4—22no—
248C4—21no—
249C4—21no—
250C4—42no—
251C4—122no—
252C4—241no—
253C4—242no—
254C4#472312yes—
255C4#473712yes—
256C3#877011yescharacteristic, complemented, Sylow
257C2—21no—
258C2—21no—
259C2—82no—
260C2#6992611yescharacteristic
261C2#7003911yescharacteristic
262C2#7006711yescharacteristic
subgroupG/H
centerC22#4943
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1460C2
socleC22×C3#381
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1460D3
hypocenterC3#8770
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
4. #67746C2semidirect
5. #67738C2semidirect
256. C3#8770semidirect

Maximal subgroups

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

subgroupindex
#823382
C3×#14602
#716172
#677462
#677382
#87703