Finite Group Databasebeta

← #10732476

Subgroups — #10732476

Order 768.

1498 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4972C213yesmaximal
2C2×#4647C213yesmaximal
3C3×#1444C211yesmaximal, characteristic
4#71568C213yesmaximal
5#69420C213yesmaximal
6#69419C211yesmaximal, characteristic
7Dic3×#95C211yesmaximal, characteristic
8#15211—31nomaximal, Sylow
9C22×C3×C4⋊C4C2213yes—
10C22×C4⋊Dic3C2213yes—
11C22×C4⋊Dic3C2211yescharacteristic
12C22×Dic3⋊C4C2213yes—
13C22×Dic3⋊C4C2213yes—
14C22×Dic3⋊C4C2213yes—
15C22×C4×Dic3C2213yes—
16C2×C3×#95C2213yes—
17C2×C3×#95C2211yescharacteristic
18C2×C3⋊#95C2213yes—
19C2×C3⋊#95C2216yes—
20C2×C3⋊#95C2213yes—
21#4972—212no—
22#4647—212no—
23C2×#389—33no—
24C2×#389—33no—
25#1444—31no—
26#1444—33no—
27#1444—33no—
28#1444D311yescharacteristic
29C4×#95—31no—
30C23×C3×C4C2313yes—
31C23×C3×C4C2311yescharacteristic
32C23×C3×C4C2313yes—
33C23×Dic3C2311yescharacteristic
34C23×Dic3C2313yes—
35C23×Dic3C2311yescharacteristic
36C23×Dic3C2313yes—
37C2×C3×C4⋊C4—26no—
38C2×C3×C4⋊C4—46no—
39C2×C4⋊Dic3—26no—
40C2×C4⋊Dic3—212no—
41C2×C4⋊Dic3—26no—
42C2×Dic3⋊C4—26no—
43C2×Dic3⋊C4—26no—
44C2×Dic3⋊C4—26no—
45C2×Dic3⋊C4—26no—
46C2×Dic3⋊C4—26no—
47C2×Dic3⋊C4—212no—
48C2×Dic3⋊C4—46no—
49C2×C4×Dic3—46no—
50C2×C4×Dic3D4112yes—
51C3×#95—26no—
52C3×#95Q814yes—
53C3⋊#95—26no—
54C3⋊#95—212no—
55C3⋊#95Q8112yes—
56C22×C4⋊C4—33no—
57C22×C4⋊C4—33no—
58C22×C4⋊C4—33no—
59C22×C4⋊C4—31no—
60C22×C4⋊C4—33no—
61C22×C4⋊C4C2×D313yes—
62C22×C42—33no—
63#389—612no—
64#389—612no—
65C2×#95—33no—
66C2×#95—36no—
67C2×#95—33no—
68C2×#95C2×D313yes—
69C2×#95C2×D311yescharacteristic
70C24×C3C2411yescharacteristic
71C22×C3×C4—26no—
72C22×C3×C4—26no—
73C22×C3×C4—23no—
74C22×C3×C4—23no—
75C22×C3×C4—26no—
76C22×C3×C4—23no—
77C22×C3×C4—26no—
78C22×C3×C4—23no—
79C22×C3×C4—23no—
80C22×C3×C4C2×D416yes—
81C22×C3×C4C2×Q816yes—
82C22×C3×C4C2×Q816yes—
83C22×C3×C4#4112yes—
84C22×Dic3—23no—
85C22×Dic3—23no—
86C22×Dic3—23no—
87C22×Dic3—26no—
88C22×Dic3—21no—
89C22×Dic3—23no—
90C22×Dic3—26no—
91C22×Dic3—23no—
92C22×Dic3—26no—
93C22×Dic3—23no—
94C22×Dic3—23no—
95C22×Dic3—21no—
96C22×Dic3C2×D416yes—
97C22×Dic3C2×D416yes—
98C22×Dic3C2×Q8112yes—
99C22×Dic3#4116yes—
100C3×C4⋊C4—412no—
101C4⋊Dic3—224no—
102C4⋊Dic3—44no—
103Dic3⋊C4—224no—
104Dic3⋊C4—412no—
105Dic3⋊C4—412no—
106C4×Dic3—412no—
107C23×C4—31no—
108C23×C4—33no—
109C23×C4—33no—
110C23×C4—31no—
111C23×C4C22×D313yes—
112C23×C4C22×D311yescharacteristic
113C23×C4C22×D313yes—
114C2×C4⋊C4—26no—
115C2×C4⋊C4—46no—
116C2×C4⋊C4—66no—
117C2×C4⋊C4—66no—
118C2×C4⋊C4—612no—
119C2×C4⋊C4—66no—
120C2×C4⋊C4—66no—
121C2×C4⋊C4—66no—
122C2×C4⋊C4—612no—
123C2×C4⋊C4—66no—
124C2×C4⋊C4—66no—
125C2×C4⋊C4—126no—
126C2×C42—312no—
127C2×C42—126no—
128#95—26no—
129#95—312no—
130#95—612no—
131#95—66no—
132#95Dic614yes—
133C23×C3C22×D413yes—
134C23×C3C22×Q811yescharacteristic
135C23×C3C22×Q813yes—
136C23×C3C2×#4111yescharacteristic
137C23×C3C2×#4113yes—
138C23×C3H(5,𝔽2)13yes—
139C23×C32−1+411yescharacteristic
140C2×C3×C4—26no—
141C2×C3×C4—26no—
142C2×C3×C4—212no—
143C2×C3×C4—212no—
144C2×C3×C4—26no—
145C2×C3×C4—43no—
146C2×C3×C4—41no—
147C2×C3×C4—43no—
148C2×C3×C4—43no—
149C2×C3×C4—46no—
150C2×C3×C4—46no—
151C2×C3×C4—43no—
152C2×C3×C4C4⋊Q8112yes—
153C2×Dic3—26no—
154C2×Dic3—26no—
155C2×Dic3—26no—
156C2×Dic3—26no—
157C2×Dic3—212no—
158C2×Dic3—212no—
159C2×Dic3—26no—
160C2×Dic3—212no—
161C2×Dic3—43no—
162C2×Dic3—46no—
163C2×Dic3—43no—
164C2×Dic3—41no—
165C2×Dic3—43no—
166C2×Dic3—43no—
167C2×Dic3#12714yes—
168C2×Dic3C4⋊Q8112yes—
169C24C23×D311yescharacteristic
170C22×C4—26no—
171C22×C4—26no—
172C22×C4—23no—
173C22×C4—23no—
174C22×C4—23no—
175C22×C4—26no—
176C22×C4—23no—
177C22×C4—26no—
178C22×C4—23no—
179C22×C4—36no—
180C22×C4—36no—
181C22×C4—312no—
182C22×C4—36no—
183C22×C4—63no—
184C22×C4—66no—
185C22×C4—63no—
186C22×C4—63no—
187C22×C4—61no—
188C22×C4—66no—
189C22×C4—63no—
190C22×C4—63no—
191C22×C4—66no—
192C22×C4—63no—
193C22×C4—63no—
194C22×C4—61no—
195C22×C4C2×Dic616yes—
196C22×C4D3×D416yes—
197C22×C4D3×Q816yes—
198C22×C4#23912yes—
199C4⋊C4—412no—
200C4⋊C4—624no—
201C4⋊C4—624no—
202C4⋊C4—1212no—
203C4⋊C4—124no—
204C4⋊C4—1212no—
205C42—1212no—
206C22×C3#52513yes—
207C22×C3#52513yes—
208C22×C3C2×#12711yescharacteristic
209C22×C3C2×C4⋊Q813yes—
210C22×C3C2×C4⋊Q813yes—
211C22×C3#53413yes—
212C22×C3#53713yes—
213C22×C3#55416yes—
214C22×C3#55413yes—
215C22×C3#55413yes—
216C22×C3#55713yes—
217C22×C3#55711yescharacteristic
218C3×C4—212no—
219C3×C4—42no—
220C3×C4—83no—
221Dic3—212no—
222Dic3—24no—
223Dic3—46no—
224Dic3—81no—
225C23C22×Dic611yescharacteristic
226C23C2×D3×D413yes—
227C23#100413yes—
228C23C2×D3×Q813yes—
229C23C2×#23911yescharacteristic
230C23D3×#4113yes—
231C23#101011yescharacteristic
232C2×C4—212no—
233C2×C4—26no—
234C2×C4—212no—
235C2×C4—26no—
236C2×C4—26no—
237C2×C4—34no—
238C2×C4—312no—
239C2×C4—43no—
240C2×C4—41no—
241C2×C4—43no—
242C2×C4—46no—
243C2×C4—43no—
244C2×C4—43no—
245C2×C4—46no—
246C2×C4—612no—
247C2×C4—612no—
248C2×C4—66no—
249C2×C4—66no—
250C2×C4—66no—
251C2×C4—66no—
252C2×C4—612no—
253C2×C4—66no—
254C2×C4—123no—
255C2×C4—123no—
256C2×C4—126no—
257C2×C4—121no—
258C2×C4—123no—
259C2×C4—123no—
260C2×C4#888112yes—
261C2×C3#239711yescharacteristic
262C2×C3#258313yes—
263C2×C3#258313yes—
264C2×C3#258413yes—
265C2×C3#258413yes—
266C2×C3#265611yescharacteristic
267C2×C3#286811yescharacteristic
268C22C2×#88813yes—
269C22#553913yes—
270C22#557011yescharacteristic
271C22#558913yes—
272C22#560813yes—
273C22#562713yes—
274C22#563616yes—
275C22#565613yes—
276C22#568513yes—
277C22D3×#12711yescharacteristic
278C22D3×C4⋊Q813yes—
279C22#573013yes—
280C4—212no—
281C4—42no—
282C4—612no—
283C4—64no—
284C4—83no—
285C4—126no—
286C4—241no—
287C3#1521111yescharacteristic, complemented, Sylow
288C2#7435611yescharacteristic
289C2#7448913yes—
290C2#7607913yes—
291C2#7676211yescharacteristic
292C2#7725513yes—
293C2#7779713yes—
294C2#7801911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1444C2
hypercenter#1444D3
hypocenterC3#15211
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
287. C3#15211semidirect

Maximal subgroups

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

subgroupindex
C2×#49722
C2×#46472
C3×#14442
#715682
#694202
#694192
Dic3×#952
#152113