Finite Group Databasebeta

← #10792530

Subgroups — #10792530

Order 768.

1470 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4708C212yesmaximal
2C2×#4707C211yesmaximal, characteristic
3C2×D3×#97C211yesmaximal, characteristic
4C2×#4689C212yesmaximal
5C4×#872C211yesmaximal, characteristic
6C3×C4×#97C211yesmaximal, characteristic, complemented
7#69804C212yesmaximal, complemented
8Dic3×#97C211yesmaximal, characteristic, complemented
9Dic3×#97C211yesmaximal, characteristic, complemented
10#69749C212yesmaximal, complemented
11C4×#803C211yesmaximal, characteristic, complemented
12#20331—31nomaximal, Sylow
13C22×ΓL(1,𝔽25)C2212yes—
14C22×D3×C8C2212yes—
15C2×#872C2211yescharacteristic
16C2×#872C2211yescharacteristic
17C2×C42×D3C2211yescharacteristic
18C2×C3×#97C2211yescharacteristic
19C2×C3×#97C2211yescharacteristic
20C2×C3×#97C2212yes—
21C2×C3×C4×C8C2212yes—
22C3×C43C2211yescharacteristic
23C2×#815C2212yes—
24C2×#815C2212yes—
25C2×#815C2212yes—
26C2×C8×Dic3C2212yes—
27C2×C8×Dic3C2212yes—
28C2×C8×Dic3C2212yes—
29C4×(C2×D3)⋊C4C414yes—
30#4932C2211yescharacteristic
31#4932C2211yescharacteristic
32C4×Dic3⋊C4C414yes—
33C42×Dic3C2211yescharacteristic
34C2×#803C2211yescharacteristic
35C2×#803C2211yescharacteristic
36C2×#803C2212yes—
37C2×C4×C3⋊C8C2212yes—
38#4708C4116yescomplemented
39#4707C418yescomplemented
40D3×#97C418yescomplemented
41#4689C4116yescomplemented
42C2×#405—32no—
43C2×#405—32no—
44C2×C4×M16—31no—
45C22×#97—31no—
46C4×#117—31no—
47#1726—32no—
48#1726—32no—
49C4×#97—31no—
50C4×#97—31no—
51C4×#97—31no—
52C4×#97D311yescharacteristic, complemented
53C22×C4×D3C2×C412yes—
54C22×C4×D3C2311yescharacteristic
55C22×C3×C8C2312yes—
56C22×C3×C8C2312yes—
57C2×C3×C42C2×C414yes—
58C2×C3×C42C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×C3×C42C2311yescharacteristic
61C2×(C2×D3)⋊C4C2×C412yes—
62C2×(C2×D3)⋊C4C2×C412yes—
63C2×Dic3⋊C4C2×C412yes—
64C2×Dic3⋊C4C2×C412yes—
65C2×C4×Dic3C2×C414yes—
66C2×C4×Dic3C2×C412yes—
67C2×C4×Dic3C2311yescharacteristic
68C2×C4×Dic3C2311yescharacteristic
69C2×C4×Dic3C2311yescharacteristic
70C22×C3⋊C8C2312yes—
71C22×C3⋊C8C2312yes—
72C2×ΓL(1,𝔽25)C2×C418yes—
73C2×ΓL(1,𝔽25)C2×C418yes—
74C2×ΓL(1,𝔽25)C2×C418yes—
75C2×D3×C8C2×C418yes—
76C2×D3×C8C2×C418yes—
77C2×D3×C8C2×C418yes—
78#872C2×C418yes—
79#872C2×C418yes—
80C42×D3C2×C414yes—
81C42×D3C2×C414yes—
82C3×#97C2×C414yes—
83C3×#97C2×C414yes—
84C3×#97C2×C418yescomplemented
85C3×C4×C8C2×C418yescomplemented
86C3⋊#95C2×C414yes—
87C3⋊#95C2×C414yes—
88#815C2×C418yes—
89#815C2×C418yescomplemented
90#815C2×C418yescomplemented
91C8×Dic3C2×C418yes—
92C8×Dic3C2×C418yescomplemented
93C8×Dic3C2×C418yescomplemented
94#803C2×C414yes—
95#803C2×C414yes—
96#803C2×C418yescomplemented
97C4×C3⋊C8C2×C418yescomplemented
98C22×M16—32no—
99C23×C8—32no—
100C2×#117—31no—
101C2×#117—31no—
102C22×C42—31no—
103#405—316no—
104#405—316no—
105C4×M16—38no—
106C2×#97—32no—
107C2×#97—32no—
108C2×#97—32no—
109C2×#97—31no—
110C2×#97—31no—
111C2×#97—32no—
112C2×#97—38no—
113C2×#97C2×D311yescharacteristic
114C2×#97C2×D311yescharacteristic
115C2×#97C2×D312yes—
116C2×C4×C8—32no—
117C2×C4×C8—32no—
118C2×C4×C8—32no—
119C2×C4×C8—32no—
120C2×C4×C8C2×D312yes—
121C4×C4⋊C4—34no—
122C4×C22⋊C4—34no—
123#376—31no—
124#376—31no—
125C43—31no—
126C43C2×D311yescharacteristic
127C23×D3C22×C411yescharacteristic
128C22×C3×C4C22×C412yes—
129C22×C3×C4C22×C412yes—
130C22×C3×C4C22×C412yes—
131C22×C3×C4C2411yescharacteristic
132C22×Dic3C22×C412yes—
133C22×Dic3C22×C412yes—
134C22×Dic3C22×C412yes—
135C22×Dic3C22×C411yescharacteristic
136C2×C4×D3—44no—
137C2×C4×D3C4218yes—
138C2×C4×D3C22×C412yes—
139C2×C4×D3C22×C412yes—
140C2×C4×D3C22×C414yes—
141C2×C4×D3C22×C414yes—
142C2×C3×C8C22×C414yes—
143C2×C3×C8C22×C414yes—
144C2×C3×C8C22×C414yes—
145C2×C3×C8C22×C414yes—
146C2×C3×C8C22×C414yes—
147C2×C3×C8C22×C414yes—
148C3×C42—44no—
149C3×C42C22×C414yes—
150C3×C42C22×C412yes—
151C3×C42C22×C412yes—
152C3×C42C22×C414yes—
153(C2×D3)⋊C4C4218yes—
154(C2×D3)⋊C4C4218yes—
155Dic3⋊C4C4218yes—
156Dic3⋊C4C4218yes—
157C4×Dic3—44no—
158C4×Dic3C4218yes—
159C4×Dic3C22×C412yes—
160C4×Dic3C22×C414yes—
161C4×Dic3C22×C414yes—
162C4×Dic3C22×C412yes—
163C2×C3⋊C8C22×C414yes—
164C2×C3⋊C8C22×C414yes—
165C2×C3⋊C8C22×C414yes—
166C2×C3⋊C8C22×C414yes—
167C2×C3⋊C8C22×C414yes—
168C2×C3⋊C8C22×C414yes—
169ΓL(1,𝔽25)C42132yescomplemented
170D3×C8C42132yescomplemented
171C23×C4—31no—
172C23×C4—32no—
173C2×M16—38no—
174C2×M16—38no—
175C2×M16—38no—
176C22×C8—32no—
177C22×C8—32no—
178C22×C8—38no—
179C22×C8—38no—
180C22×C8—38no—
181C22×C8C22×D312yes—
182C22×C8C22×D312yes—
183#117—38no—
184#117—38no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—32no—
187C2×C22⋊C4—32no—
188C2×C22⋊C4—32no—
189C2×C42—32no—
190C2×C42—31no—
191C2×C42—31no—
192C2×C42—31no—
193C2×C42—34no—
194C2×C42—34no—
195C2×C42—34no—
196C2×C42C4×D314yes—
197C2×C42C22×D311yescharacteristic
198C2×C42C22×D311yescharacteristic
199C2×C42C22×D311yescharacteristic
200#97—38no—
201#97—38no—
202#97—38no—
203#97—34no—
204#97—34no—
205#97—38no—
206#97C4×D314yes—
207#97C4×D314yes—
208#97C4×D318yescomplemented
209C4×C8—38no—
210C4×C8—38no—
211C4×C8—38no—
212C4×C8—38no—
213C4×C8C4×D318yescomplemented
214#95—34no—
215#95—34no—
216C23×C3C23×C411yescharacteristic
217C22×D3—24no—
218C22×D3C2×C4212yes—
219C22×D3C2×C4212yes—
220C22×D3C2×C4212yes—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4C2×C4214yes—
225C2×C3×C4C2×C4214yes—
226C2×C3×C4C2×C4214yes—
227C2×C3×C4C23×C411yescharacteristic
228C2×C3×C4C23×C411yescharacteristic
229C2×C3×C4C23×C412yes—
230C2×C3×C4C23×C412yes—
231C2×Dic3—24no—
232C2×Dic3—24no—
233C2×Dic3—24no—
234C2×Dic3C2×C4214yes—
235C2×Dic3C2×C4214yes—
236C2×Dic3C2×C4214yes—
237C2×Dic3C2×C4212yes—
238C2×Dic3C2×C4212yes—
239C2×Dic3C2×C4212yes—
240C4×D3—48no—
241C4×D3C2×C4218yes—
242C4×D3C2×C4218yes—
243C3×C8C2×C4218yes—
244C3×C8C2×C4218yes—
245C3⋊C8C2×C4218yes—
246C3⋊C8C2×C4218yes—
247C24—31no—
248C22×C4—31no—
249C22×C4—32no—
250C22×C4—32no—
251C22×C4—34no—
252C22×C4—34no—
253C22×C4—32no—
254C22×C4—32no—
255C22×C4—32no—
256C22×C4—38no—
257C22×C4—124no—
258C22×C4C2×C4×D312yes—
259C22×C4C2×C4×D312yes—
260C22×C4C2×C4×D312yes—
261C22×C4C23×D311yescharacteristic
262M16—332no—
263C2×C8—34no—
264C2×C8—34no—
265C2×C8—34no—
266C2×C8—34no—
267C2×C8—34no—
268C2×C8—34no—
269C2×C8—332no—
270C2×C8C2×C4×D314yes—
271C2×C8C2×C4×D314yes—
272C2×C8C2×C4×D314yes—
273C2×C8C2×C4×D314yes—
274C2×C8C2×C4×D314yes—
275C2×C8C2×C4×D314yes—
276C4⋊C4—38no—
277C4⋊C4—38no—
278C22⋊C4—38no—
279C22⋊C4—38no—
280C42—38no—
281C42—32no—
282C42—34no—
283C42—34no—
284C42—32no—
285C42—44no—
286C42—124no—
287C42C2×C4×D314yes—
288C42C2×C4×D314yes—
289C42C2×C4×D312yes—
290C42C2×C4×D312yes—
291C22×C3C22×C4211yescharacteristic
292C22×C3C22×C4211yescharacteristic
293C22×C3C22×C4211yescharacteristic
294C22×C3#56814yes—
295C2×D3—24no—
296C2×D3—24no—
297C2×D3—24no—
298C2×D3C4314yes—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4C22×C4212yes—
303C3×C4C22×C4212yes—
304Dic3—24no—
305Dic3—24no—
306Dic3—24no—
307Dic3C4314yes—
308C23—32no—
309C23—32no—
310C23—32no—
311C23—64no—
312C23C22×C4×D311yescharacteristic
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—32no—
317C2×C4—32no—
318C2×C4—32no—
319C2×C4—38no—
320C2×C4—38no—
321C2×C4—34no—
322C2×C4—34no—
323C2×C4—34no—
324C2×C4—64no—
325C2×C4—64no—
326C2×C4—64no—
327C2×C4—128no—
328C2×C4C42×D314yes—
329C2×C4C42×D314yes—
330C2×C4C42×D314yes—
331C2×C4C22×C4×D312yes—
332C2×C4C22×C4×D311yescharacteristic
333C2×C4C22×C4×D311yescharacteristic
334C2×C4C22×C4×D312yes—
335C8—38no—
336C8—38no—
337C8C42×D318yes—
338C8C42×D318yes—
339C2×C3C2×C4311yescharacteristic
340C2×C3#287312yes—
341C2×C3#287512yes—
342C2×C3#287512yes—
343D3—24no—
344C22—34no—
345C22—64no—
346C22—64no—
347C22—64no—
348C22C2×C42×D311yescharacteristic
349C22C2×C42×D311yescharacteristic
350C22C2×C42×D311yescharacteristic
351C22#574514yes—
352C4—24no—
353C4—24no—
354C4—24no—
355C4—34no—
356C4—64no—
357C4—64no—
358C4—64no—
359C4C2×C42×D312yes—
360C4C2×C42×D312yes—
361C3#2033111yescharacteristic, complemented, Sylow
362C2—64no—
363C2C43×D311yescharacteristic
364C2#7848612yes—
365C2#7851412yes—
366C2#7851612yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutatorC2×C3C2×C43
FittingC3×C4×#97C2
socleC23×C3C23×C4
prime radicalC23×D3C22×C4
cosocleC22×C3×C4C24
hypercenterC4×#97D3
hypocenterC3#20331
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
6. C3×C4×#97C2semidirect
7. #69804C2semidirect
8. Dic3×#97C2semidirect
9. Dic3×#97C2semidirect
10. #69749C2semidirect
11. C4×#803C2semidirect
38. #4708C4semidirect
39. #4707C4semidirect
40. D3×#97C4semidirect
41. #4689C4semidirect
52. C4×#97D3semidirect
84. C3×#97C2×C4semidirect
85. C3×C4×C8C2×C4semidirect
89. #815C2×C4semidirect
90. #815C2×C4semidirect
92. C8×Dic3C2×C4semidirect
93. C8×Dic3C2×C4semidirect
96. #803C2×C4semidirect
97. C4×C3⋊C8C2×C4semidirect
169. ΓL(1,𝔽25)C42semidirect
170. D3×C8C42semidirect
208. #97C4×D3semidirect
213. C4×C8C4×D3semidirect
361. C3#20331semidirect

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×D3×#972
C2×#46892
C4×#8722
C3×C4×#972
#698042
Dic3×#972
#697492
C4×#8032
#203313