Finite Group Databasebeta

← #10950788

Subgroups — #10950788

Order 768.

532 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#80699C212yesmaximal, complemented
2C3×#2218C211yesmaximal, characteristic, complemented
3#71464C211yesmaximal, characteristic, complemented
4#71461C211yesmaximal, characteristic, complemented
5#71443C211yesmaximal, characteristic, complemented
6#71419C211yesmaximal, characteristic, complemented
7#71071C212yesmaximal, complemented
8#69343C211yesmaximal, characteristic, complemented
9#69339C211yesmaximal, characteristic, complemented
10#69186C212yesmaximal, complemented
11#69174C212yesmaximal, complemented
12#33740—31nomaximal, Sylow
13#5619—22no—
14C2×C3×D16C2211yescharacteristic
15C2×C3×M32C2211yescharacteristic
16C3×#466C2212yes—
17#5151C2212yes—
18Dic3×D8C2212yes—
19C2×#826C2212yes—
20#5115C2212yes—
21#5101C2211yescharacteristic
22C22×C3⋊C16C2211yescharacteristic
23#5039—22no—
24C2×D48C2211yescharacteristic
25#4878C2212yes—
26#4872C2212yes—
27#4785—22no—
28#4764—22no—
29C3×#367C2211yescharacteristic
30C3×#357C2212yes—
31#4562C2212yes—
32#4509C2212yes—
33#4505C2211yescharacteristic
34#4502C2211yescharacteristic
35#4502C2211yescharacteristic
36Dic3×C16C2211yescharacteristic
37Dic3×C16C2211yescharacteristic
38#4491C2212yes—
39#4489C2212yes—
40#3176—32no—
41#2253—31no—
42#2246—31no—
43#2244—32no—
44#2218—31no—
45#2218D311yescharacteristic, complemented
46#2215—32no—
47#2177—31no—
48#2177—31no—
49C4×D16—32no—
50#2108—31no—
51C2×C3×D8C2312yes—
52C22×C3×C8C2311yescharacteristic
53C3×#121—22no—
54#940—22no—
55#939—42no—
56#938—42no—
57D4×Dic3—22no—
58C2×#213—22no—
59#930—22no—
60#926D411yescharacteristic
61C2×#208D411yescharacteristic
62C2×D24C2312yes—
63#893—22no—
64#890—22no—
65#886—42no—
66#883—42no—
67C3×D16—22no—
68C3×M32—22no—
69C2×C3×C16C2311yescharacteristic
70C2×C3×C16C2311yescharacteristic
71C3×#107C2312yes—
72C3×#102—22no—
73#832—22no—
74#826—24no—
75#821—22no—
76#818C2312yes—
77#815D411yescharacteristic
78#815D411yescharacteristic
79C8×Dic3C2311yescharacteristic
80C8×Dic3C2311yescharacteristic
81C2×C3⋊C16—22no—
82C2×C3⋊C16C2311yescharacteristic
83C2×C3⋊C16C2311yescharacteristic
84#809—22no—
85#808—22no—
86D48—22no—
87#540—62no—
88C2×D16—32no—
89C2×D16—31no—
90C2×D16C2×D311yescharacteristic
91C2×M32C2×D311yescharacteristic
92C22×C16—31no—
93#499—32no—
94#493—32no—
95#490—62no—
96#469—62no—
97#466—32no—
98#466C2×D312yes—
99#442—62no—
100C4×D8—32no—
101C4×D8—32no—
102#405—31no—
103#367—31no—
104#367C2×D311yescharacteristic
105#366—32no—
106#357—32no—
107#357—32no—
108#357—32no—
109#357C2×D312yes—
110#346—31no—
111#346—31no—
112C4×C16—31no—
113C4×C16—31no—
114C2×C3×D4—22no—
115C2×C3×D4—42no—
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—42no—
118C2×#67—42no—
119C2×#67—42no—
120C22×Dic3—42no—
121C2×D12—22no—
122C2×C4×D3—42no—
123C3×D8—22no—
124C3×D8—22no—
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8C2×D411yescharacteristic
128C2×C3×C8C2411yescharacteristic
129C3×C4⋊C4—22no—
130C3×C22⋊C4—42no—
131#217—42no—
132#217D812yes—
133#213—44no—
134(C2×D3)⋊C4—42no—
135(C2×D3)⋊C4—42no—
136C4⋊Dic3—22no—
137C4⋊Dic3—42no—
138C4⋊Dic3D812yes—
139Dic3⋊C4—42no—
140C4×Dic3C2×D411yescharacteristic
141C4×Dic3C2×D411yescharacteristic
142#208—22no—
143C2×C3⋊C8C2×D411yescharacteristic
144C2×C3⋊C8C2×D411yescharacteristic
145D24—22no—
146D24—22no—
147C3×C16—21no—
148C3×C16#4112yes—
149C3⋊C16—21no—
150C3⋊C16#4112yes—
151C2×D8—32no—
152C2×D8—62no—
153C2×D8C22×D312yes—
154C2×M16—31no—
155C22×C8C22×D311yescharacteristic
156#127—62no—
157#125—62no—
158#123—122no—
159#123—122no—
160#121—22no—
161#121—62no—
162#121—122no—
163#121—122no—
164C4×D4—62no—
165C4×D4—62no—
166#117—31no—
167D16—22no—
168D16—64no—
169D16—62no—
170M32—22no—
171C2×C16—31no—
172C2×C16—31no—
173C2×C16—62no—
174C2×C16C22×D311yescharacteristic
175C2×C16C22×D311yescharacteristic
176#107—32no—
177#107C22×D312yes—
178#106—62no—
179#102—22no—
180#102—62no—
181#102—62no—
182#102—62no—
183#97—31no—
184#97—31no—
185C4×C8—31no—
186C4×C8—31no—
187C23×C3—42no—
188C23×C3C2×D811yescharacteristic
189C22×D3—42no—
190C3×D4—42no—
191C3×D4—42no—
192C3×D4—162no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—42no—
196C2×C3×C4C2×D811yescharacteristic
197C2×C3×C4C22×D411yescharacteristic
198#67—162no—
199#67—162no—
200#67—162no—
201C2×Dic3—42no—
202C2×Dic3—162no—
203C2×Dic3C2×D812yes—
204C2×Dic3C2×D812yes—
205D12—42no—
206D12—42no—
207C4×D3—162no—
208C3×C8—21no—
209C3×C8C2×#4111yescharacteristic
210C3×C8C2×#4111yescharacteristic
211C3⋊C8—21no—
212C3⋊C8#13512yes—
213C2×D4—22no—
214C2×D4—42no—
215C2×D4—62no—
216C2×D4—122no—
217C2×D4—122no—
218C2×D4—122no—
219C22×C4—122no—
220C22×C4—122no—
221C22×C4D3×D411yescharacteristic
222D8—22no—
223D8—22no—
224D8—62no—
225D8—62no—
226D8—124no—
227M16—62no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8D3×D411yescharacteristic
233C2×C8C23×D311yescharacteristic
234C4⋊C4—22no—
235C4⋊C4—32no—
236C4⋊C4—62no—
237C4⋊C4—122no—
238C4⋊C4—122no—
239C22⋊C4—32no—
240C22⋊C4—42no—
241C22⋊C4—122no—
242C22⋊C4—122no—
243C22⋊C4—122no—
244C42—31no—
245C42—31no—
246C16—21no—
247C16—32no—
248C16—61no—
249C16#23712yes—
250C22×C3—22no—
251C22×C3—21no—
252C22×C3—82no—
253C22×C3—82no—
254C22×C3—82no—
255C22×C3C22×D811yescharacteristic
256C2×D3—82no—
257C2×D3—82no—
258C2×D3—82no—
259C3×C4—21no—
260C3×C4—82no—
261C3×C4#53211yescharacteristic
262C3×C4C2×#13511yescharacteristic
263Dic3—22no—
264Dic3—22no—
265Dic3—82no—
266C23—42no—
267C23—122no—
268C23D3×D811yescharacteristic
269D4—42no—
270D4—42no—
271D4—122no—
272D4—122no—
273D4—162no—
274D4—482no—
275D4—482no—
276D4—482no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—42no—
282C2×C4—122no—
283C2×C4—482no—
284C2×C4—482no—
285C2×C4D3×D811yescharacteristic
286C2×C4C2×D3×D411yescharacteristic
287C8—21no—
288C8—32no—
289C8—61no—
290C8C2×#23711yescharacteristic
291C8D3×#4111yescharacteristic
292C2×C3—41no—
293C2×C3—162no—
294C2×C3#316911yescharacteristic
295C2×C3#341512yes—
296D3—162no—
297C22—21no—
298C22—22no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—242no—
303C22—242no—
304C22—242no—
305C22C2×D3×D811yescharacteristic
306C4—21no—
307C4—62no—
308C4—62no—
309C4—82no—
310C4—242no—
311C4#559611yescharacteristic
312C4D3×#13511yescharacteristic
313C3#3374011yescharacteristic, complemented, Sylow
314C2—41no—
315C2—162no—
316C2—482no—
317C2#8069711yescharacteristic
318C2#8203612yes—
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2218C2
socleC22×C3C22×D8
hypercenter#2218D3
hypocenterC3#33740
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2218

Complements

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

NH ≅ G/Ntype
1. #80699C2semidirect
2. C3×#2218C2semidirect
3. #71464C2semidirect
4. #71461C2semidirect
5. #71443C2semidirect
6. #71419C2semidirect
7. #71071C2semidirect
8. #69343C2semidirect
9. #69339C2semidirect
10. #69186C2semidirect
11. #69174C2semidirect
45. #2218D3semidirect
313. C3#33740semidirect

Maximal subgroups

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

subgroupindex
#806992
C3×#22182
#714642
#714612
#714432
#714192
#710712
#693432
#693392
#691862
#691742
#337403