Finite Group Databasebeta

← #10792525

Subgroups — #10792525

Order 768.

1526 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4708C212yesmaximal
2C2×#4707C211yesmaximal, characteristic
3C2×#4689C212yesmaximal
4C2×C4×D3×C8C211yesmaximal, characteristic
5#73382C211yesmaximal, characteristic
6C3×#1726C211yesmaximal, characteristic, complemented
7#69804C211yesmaximal, characteristic, complemented
8Dic3×#97C212yesmaximal, complemented
9#69749C211yesmaximal, characteristic, complemented
10C4×C8×Dic3C212yesmaximal, complemented
11#69434C211yesmaximal, characteristic, complemented
12C4×#405—31nomaximal, Sylow
13C22×ΓL(1,𝔽25)C2212yes—
14C22×D3×C8C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16C2×#872C2212yes—
17C2×C42×D3C2211yescharacteristic
18C2×C3×#97C2211yescharacteristic
19C2×C3×#97C2212yes—
20C2×C3×C4×C8C2211yescharacteristic
21C2×C3×C4×C8C2212yes—
22C3×#376C2211yescharacteristic
23C2×#815C2212yes—
24C2×#815C2212yes—
25C2×C8×Dic3C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27C2×C8×Dic3C2212yes—
28C2×C8×Dic3C2212yes—
29C2×C8×Dic3C2212yes—
30#4932C2211yescharacteristic
31C42×Dic3C2212yes—
32C2×#803C2211yescharacteristic
33C2×#803C2212yes—
34C2×C4×C3⋊C8C2211yescharacteristic
35C2×C4×C3⋊C8C2212yes—
36#4708C4116yescomplemented
37#4707C418yescomplemented
38#4689C4116yescomplemented
39C4×D3×C8C418yescomplemented
40#4666C414yes—
41#4649C414yes—
42C2×#405—32no—
43C2×#405—32no—
44C2×C4×M16—31no—
45C22×C4×C8—31no—
46C4×#117—31no—
47#1726—31no—
48#1726—31no—
49#1726—31no—
50#1726D311yescharacteristic, complemented
51C4×#97—32no—
52C42×C8—32no—
53C22×C4×D3C2×C412yes—
54C22×C4×D3C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2312yes—
57C22×C3×C8C2311yescharacteristic
58C2×C3×C42C2312yes—
59C2×C3×C42C2311yescharacteristic
60C2×(C2×D3)⋊C4C2×C414yes—
61C2×Dic3⋊C4C2×C414yes—
62C2×C4×Dic3C2×C418yes—
63C2×C4×Dic3C2×C412yes—
64C2×C4×Dic3C2311yescharacteristic
65C2×C4×Dic3C2312yes—
66C22×C3⋊C8C2311yescharacteristic
67C22×C3⋊C8C2311yescharacteristic
68C22×C3⋊C8C2312yes—
69C2×ΓL(1,𝔽25)C2×C418yes—
70C2×ΓL(1,𝔽25)C2×C418yes—
71C2×ΓL(1,𝔽25)C2×C418yes—
72C2×D3×C8C2×C414yes—
73C2×D3×C8C2×C414yes—
74C2×D3×C8C2×C418yes—
75C2×D3×C8C2×C418yes—
76#872C2×C418yes—
77#872C2×C418yes—
78C42×D3C2×C414yes—
79C42×D3C2×C414yes—
80C3×#97C2×C414yes—
81C3×#97C2×C418yescomplemented
82C3×C4×C8C2×C414yes—
83C3×C4×C8C2×C418yescomplemented
84C3×#95C2×C414yes—
85C3⋊#95C2×C414yes—
86#815C2×C418yescomplemented
87#815C2×C418yescomplemented
88C8×Dic3C2×C414yes—
89C8×Dic3C2×C414yes—
90C8×Dic3C2×C418yes—
91C8×Dic3C2×C418yescomplemented
92C8×Dic3C2×C418yescomplemented
93#803C2×C414yes—
94#803C2×C418yescomplemented
95C4×C3⋊C8C2×C414yes—
96C4×C3⋊C8C2×C418yescomplemented
97C22×M16—32no—
98C23×C8—31no—
99C23×C8—31no—
100C2×#117—32no—
101C22×C42—31no—
102#405—316no—
103#405—316no—
104C4×M16—38no—
105C2×#97—32no—
106C2×#97—32no—
107C2×#97—31no—
108C2×#97—32no—
109C2×#97C2×D311yescharacteristic
110C2×#97C2×D312yes—
111C2×C4×C8—31no—
112C2×C4×C8—32no—
113C2×C4×C8—32no—
114C2×C4×C8—31no—
115C2×C4×C8—32no—
116C2×C4×C8—31no—
117C2×C4×C8—32no—
118C2×C4×C8—38no—
119C2×C4×C8C2×D311yescharacteristic
120C2×C4×C8C2×D312yes—
121C4×C4⋊C4—34no—
122C4×C22⋊C4—34no—
123#376—31no—
124#376C2×D311yescharacteristic
125C43—32no—
126C23×D3C22×C411yescharacteristic
127C22×C3×C4C22×C412yes—
128C22×C3×C4C22×C414yes—
129C22×C3×C4C2411yescharacteristic
130C22×Dic3C22×C414yes—
131C22×Dic3C22×C411yescharacteristic
132C22×Dic3C22×C412yes—
133C2×C4×D3—28no—
134C2×C4×D3C4218yes—
135C2×C4×D3C22×C412yes—
136C2×C4×D3C22×C412yes—
137C2×C4×D3C22×C414yes—
138C2×C4×D3C22×C414yes—
139C2×C3×C8C22×C412yes—
140C2×C3×C8C22×C412yes—
141C2×C3×C8C22×C414yes—
142C2×C3×C8C22×C414yes—
143C2×C3×C8C22×C414yes—
144C2×C3×C8C22×C414yes—
145C2×C3×C8C22×C414yes—
146C3×C42C22×C412yes—
147C3×C42C22×C414yes—
148C3×C42C22×C412yes—
149C3×C42C22×C414yes—
150(C2×D3)⋊C4C42116yes—
151Dic3⋊C4C42116yes—
152C4×Dic3—216no—
153C4×Dic3C4218yes—
154C4×Dic3C22×C412yes—
155C4×Dic3C22×C412yes—
156C4×Dic3C22×C414yes—
157C4×Dic3C22×C414yes—
158C2×C3⋊C8C22×C412yes—
159C2×C3⋊C8C22×C412yes—
160C2×C3⋊C8C22×C414yes—
161C2×C3⋊C8C22×C414yes—
162C2×C3⋊C8C22×C414yes—
163C2×C3⋊C8C22×C414yes—
164C2×C3⋊C8C22×C414yes—
165ΓL(1,𝔽25)C42132yescomplemented
166D3×C8C42116yescomplemented
167D3×C8C42116yescomplemented
168C23×C4—31no—
169C23×C4—32no—
170C2×M16—38no—
171C2×M16—38no—
172C2×M16—38no—
173C22×C8—31no—
174C22×C8—31no—
175C22×C8—32no—
176C22×C8—34no—
177C22×C8—34no—
178C22×C8—38no—
179C22×C8—38no—
180C22×C8C22×D311yescharacteristic
181C22×C8C22×D312yes—
182C22×C8C22×D311yescharacteristic
183#117—38no—
184#117—38no—
185C2×C4⋊C4—34no—
186C2×C22⋊C4—34no—
187C2×C42—32no—
188C2×C42—32no—
189C2×C42—31no—
190C2×C42—34no—
191C2×C42—34no—
192C2×C42—38no—
193C2×C42C22×D312yes—
194C2×C42C22×D311yescharacteristic
195#97—38no—
196#97—38no—
197#97—34no—
198#97—38no—
199#97C4×D314yes—
200#97C4×D318yescomplemented
201C4×C8—34no—
202C4×C8—34no—
203C4×C8—38no—
204C4×C8—38no—
205C4×C8—38no—
206C4×C8—34no—
207C4×C8—38no—
208C4×C8C4×D314yes—
209C4×C8C4×D318yescomplemented
210#95—34no—
211#95C4×D314yes—
212C23×C3C23×C411yescharacteristic
213C22×D3—24no—
214C22×D3C2×C4214yes—
215C22×D3C2×C4212yes—
216C2×C3×C4—28no—
217C2×C3×C4—24no—
218C2×C3×C4C2×C4214yes—
219C2×C3×C4C2×C4218yes—
220C2×C3×C4C23×C411yescharacteristic
221C2×C3×C4C23×C411yescharacteristic
222C2×C3×C4C23×C412yes—
223C2×C3×C4C23×C412yes—
224C2×Dic3—28no—
225C2×Dic3C2×C4218yes—
226C2×Dic3C2×C4212yes—
227C2×Dic3C2×C4214yes—
228C2×Dic3C2×C4214yes—
229C2×Dic3#13118yes—
230C4×D3—216no—
231C4×D3C2×C4214yes—
232C4×D3C2×C4214yes—
233C4×D3C2×C4218yes—
234C3×C8C2×C4214yes—
235C3×C8C2×C4214yes—
236C3×C8C2×C4218yes—
237C3⋊C8C2×C4214yes—
238C3⋊C8C2×C4214yes—
239C3⋊C8C2×C4218yes—
240C24—31no—
241C22×C4—32no—
242C22×C4—31no—
243C22×C4—32no—
244C22×C4—32no—
245C22×C4—34no—
246C22×C4—34no—
247C22×C4—34no—
248C22×C4—38no—
249C22×C4—68no—
250C22×C4C2×C4×D312yes—
251C22×C4C2×C4×D314yes—
252C22×C4C23×D311yescharacteristic
253M16—332no—
254C2×C8—32no—
255C2×C8—32no—
256C2×C8—34no—
257C2×C8—34no—
258C2×C8—34no—
259C2×C8—34no—
260C2×C8—34no—
261C2×C8—316no—
262C2×C8—316no—
263C2×C8C2×C4×D312yes—
264C2×C8C2×C4×D312yes—
265C2×C8C2×C4×D314yes—
266C2×C8C2×C4×D314yes—
267C2×C8C2×C4×D314yes—
268C2×C8C2×C4×D314yes—
269C2×C8C2×C4×D314yes—
270C4⋊C4—316no—
271C22⋊C4—316no—
272C42—38no—
273C42—32no—
274C42—32no—
275C42—34no—
276C42—34no—
277C42—616no—
278C42C2×C4×D312yes—
279C42C2×C4×D314yes—
280C42C2×C4×D314yes—
281C42C2×C4×D312yes—
282C22×C3C22×C4211yescharacteristic
283C22×C3C22×C4212yes—
284C22×C3C2×#13114yes—
285C2×D3—24no—
286C2×D3—28no—
287C2×D3C4314yes—
288C3×C4—24no—
289C3×C4—28no—
290C3×C4C22×C4211yescharacteristic
291C3×C4C22×C4211yescharacteristic
292C3×C4C22×C4212yes—
293Dic3—28no—
294Dic3C4314yes—
295Dic3#40518yescomplemented
296C23—34no—
297C23—32no—
298C23—64no—
299C23C22×C4×D311yescharacteristic
300C2×C4—28no—
301C2×C4—24no—
302C2×C4—32no—
303C2×C4—34no—
304C2×C4—34no—
305C2×C4—38no—
306C2×C4—34no—
307C2×C4—34no—
308C2×C4—38no—
309C2×C4—38no—
310C2×C4—616no—
311C2×C4—68no—
312C2×C4C42×D314yes—
313C2×C4C42×D318yes—
314C2×C4C22×C4×D311yescharacteristic
315C2×C4C22×C4×D311yescharacteristic
316C2×C4C22×C4×D312yes—
317C2×C4C22×C4×D312yes—
318C8—34no—
319C8—34no—
320C8—38no—
321C8C42×D314yes—
322C8C42×D314yes—
323C8C42×D318yes—
324C2×C3C2×C4311yescharacteristic
325C2×C3C2×#40512yes—
326C2×C3C4×#13114yes—
327D3—24no—
328C22—34no—
329C22—64no—
330C22—68no—
331C22C2×C42×D311yescharacteristic
332C22C2×C42×D312yes—
333C22D3×#13114yes—
334C4—24no—
335C4—28no—
336C4—34no—
337C4—38no—
338C4—68no—
339C4C2×C42×D311yescharacteristic
340C4C2×C42×D311yescharacteristic
341C4C2×C42×D312yes—
342C3C4×#40511yescharacteristic, complemented, Sylow
343C2—64no—
344C2C43×D311yescharacteristic
345C2D3×#40512yes—
346C2#7854914yes—
subgroupG/H
centerC22×C8C22×D3
commutatorC2×C3C2×C43
FrattiniC22×C4C23×D3
FittingC3×#1726C2
socleC23×C3C23×C4
prime radicalC23×D3C22×C4
cosocleC22×C3×C4C24
hypercenter#1726D3
hypocenterC3C4×#405
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C8 ◁ #1726

Complements

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

NH ≅ G/Ntype
6. C3×#1726C2semidirect
7. #69804C2semidirect
8. Dic3×#97C2semidirect
9. #69749C2semidirect
10. C4×C8×Dic3C2semidirect
11. #69434C2semidirect
36. #4708C4semidirect
37. #4707C4semidirect
38. #4689C4semidirect
39. C4×D3×C8C4semidirect
50. #1726D3semidirect
81. C3×#97C2×C4semidirect
83. C3×C4×C8C2×C4semidirect
86. #815C2×C4semidirect
87. #815C2×C4semidirect
91. C8×Dic3C2×C4semidirect
92. C8×Dic3C2×C4semidirect
94. #803C2×C4semidirect
96. C4×C3⋊C8C2×C4semidirect
165. ΓL(1,𝔽25)C42semidirect
166. D3×C8C42semidirect
167. D3×C8C42semidirect
200. #97C4×D3semidirect
209. C4×C8C4×D3semidirect
295. Dic3#405semidirect
342. C3C4×#405semidirect

Maximal subgroups

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

subgroupindex
C2×#47082
C2×#47072
C2×#46892
C2×C4×D3×C82
#733822
C3×#17262
#698042
Dic3×#972
#697492
C4×C8×Dic32
#694342
C4×#4053