Finite Group Databasebeta

← #79865

Subgroups — #79865

Order 384.

548 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#911C211yesmaximal, characteristic
2C4×D3×D4C211yesmaximal, characteristic, complemented
3C4×#237C211yesmaximal, characteristic, complemented
4C3×#442C211yesmaximal, characteristic, complemented
5Dic3×D8C211yesmaximal, characteristic, complemented
6#5018C211yesmaximal, characteristic, complemented
7#5014C211yesmaximal, characteristic, complemented
8#4861C211yesmaximal, characteristic, complemented
9#4852C211yesmaximal, characteristic, complemented
10#4771C211yesmaximal, characteristic, complemented
11#4770C211yesmaximal, characteristic, complemented
12#4758C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14#4710C211yesmaximal, characteristic, complemented
15#4707C211yesmaximal, characteristic, complemented
16C4×AGL(1,ℤ/8)—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C22×C4×D3—21no—
20C2×C3×D8C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic, complemented
23D4×Dic3C2211yescharacteristic
24D4×Dic3C2211yescharacteristic, complemented
25C2×#214C2211yescharacteristic
26C2×#213C2211yescharacteristic
27C4×#67—21no—
28C4×#67—21no—
29C2×C4×Dic3—21no—
30#911C418yescomplemented
31C2×#204C2211yescharacteristic
32C2×ΓL(1,𝔽25)C2211yescharacteristic
33C2×ΓL(1,𝔽25)C2211yescharacteristic
34#893C2211yescharacteristic
35#892C2211yescharacteristic
36D3×C4⋊C4C2211yescharacteristic
37#887C2211yescharacteristic, complemented
38#67⋊C4—21no—
39#67⋊C4—21no—
40D3×C22⋊C4—21no—
41#877—21no—
42C4×D12C2211yescharacteristic
43C42×D3C2211yescharacteristic
44C4×Dic6C2211yescharacteristic, complemented
45C3×#106C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic
47C3×#102C2211yescharacteristic, complemented
48C3×#97C2211yescharacteristic, complemented
49#832C2211yescharacteristic
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic
52#818C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54C8×Dic3C2211yescharacteristic, complemented
55C8×Dic3C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#803C2211yescharacteristic, complemented
61C2×AGL(1,ℤ/8)—31no—
62C4×#41—31no—
63C2×C4×D4—31no—
64#442—31no—
65#442D311yescharacteristic, complemented
66#440—31no—
67#440—31no—
68C4×SD16—31no—
69C4×SD16—31no—
70C4×D8—31no—
71C4×D8—31no—
72#428—31no—
73#418—31no—
74#417—31no—
75C4×M16—31no—
76C23×D3—21no—
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84C22×Dic3—21no—
85C22×Dic3—21no—
86#237—22no—
87#237C2×C414yescomplemented
88D3×D4—22no—
89D3×D4C2×C414yescomplemented
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3—44no—
95C2×C4×D3D411yescharacteristic
96C2×C4×D3D411yescharacteristic
97C2×C4×D3C2311yescharacteristic
98C2×Dic6C2311yescharacteristic
99C3×D8C2×C414yescomplemented
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C22⋊C4—21no—
105C3×C22⋊C4—21no—
106C3×C42C2311yescharacteristic
107#217—21no—
108#217—21no—
109#214C2×C414yescomplemented
110#213C2×C414yescomplemented
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3D411yescharacteristic
123C4×Dic3D411yescharacteristic
124C4×Dic3C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127#204C2×C414yescomplemented
128ΓL(1,𝔽25)—22no—
129ΓL(1,𝔽25)C2×C414yescomplemented
130C2×#41—31no—
131C22×D4—31no—
132C23×C4—61no—
133AGL(1,ℤ/8)—38no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×D8—31no—
137C2×D8C2×D311yescharacteristic
138C2×M16—31no—
139C2×M16—31no—
140C4×Q8—31no—
141C4×Q8—31no—
142C4×D4—31no—
143C4×D4—31no—
144C4×D4—31no—
145C4×D4—31no—
146C4×D4—61no—
147C4×D4—61no—
148C4×D4—61no—
149C4×D4—61no—
150C4×D4C2×D311yescharacteristic
151C4×D4C2×D311yescharacteristic, complemented
152#117—31no—
153#117—61no—
154C2×C4⋊C4—31no—
155C2×C22⋊C4—61no—
156C2×C42—31no—
157C2×C42—61no—
158#107—31no—
159#107—31no—
160#106—31no—
161#106C2×D311yescharacteristic, complemented
162#103—31no—
163#103—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169#102C2×D311yescharacteristic, complemented
170#97—31no—
171#97C2×D311yescharacteristic, complemented
172C4×C8—31no—
173C4×C8—31no—
174C23×C3—21no—
175C23×C3—21no—
176C22×D3—21no—
177C22×D3—24no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3C2×D411yescharacteristic
181C3×D4—21no—
182C3×D4—21no—
183C3×D4C22×C412yes—
184C3×D4C22×C412yescomplemented
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—24no—
193#67—24no—
194C2×Dic3—21no—
195C2×Dic3—24no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12C22×C412yes—
205C4×D3—22no—
206C4×D3—41no—
207C4×D3—42no—
208C4×D3—42no—
209C4×D3—42no—
210C4×D3C22×C412yes—
211C4×D3#4112yes—
212Dic6—21no—
213Dic6C22×C412yescomplemented
214C3×C8—21no—
215C3×C8C22×C412yescomplemented
216C3⋊C8—21no—
217C3⋊C8C22×C412yescomplemented
218C24—61no—
219#41—34no—
220#41—62no—
221C2×Q8—31no—
222C2×D4—34no—
223C2×D4—31no—
224C2×D4—62no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4C22×D311yescharacteristic
228C2×D4C22×D311yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—124no—
241SD16—34no—
242SD16—34no—
243D8—34no—
244D8C4×D314yescomplemented
245M16—34no—
246M16—62no—
247C2×C8—31no—
248C2×C8—31no—
249C2×C8C22×D311yescharacteristic
250C2×C8C22×D311yescharacteristic
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4C22×D311yescharacteristic
257C4⋊C4C22×D311yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C42—31no—
266C42—31no—
267C42—31no—
268C42—61no—
269C42—61no—
270C42—61no—
271C42C22×D311yescharacteristic
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3C22×D411yescharacteristic
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—22no—
282C2×D3—48no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3C4×D412yes—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4C23×C411yescharacteristic
291C3×C4C2×#4111yescharacteristic
292Dic3—22no—
293Dic3—21no—
294Dic3—41no—
295Dic3—41no—
296Dic3C4×D412yes—
297Dic3AGL(1,ℤ/8)12yescomplemented
298C23—21no—
299C23—21no—
300C23—31no—
301C23—64no—
302C23—61no—
303C23—122no—
304C23—122no—
305Q8—32no—
306Q8—61no—
307D4—21no—
308D4—21no—
309D4—32no—
310D4—64no—
311D4—64no—
312D4—61no—
313D4C2×C4×D312yes—
314D4C2×C4×D312yescomplemented
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—31no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—62no—
325C2×C4—64no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4D3×D411yescharacteristic
336C2×C4D3×D411yescharacteristic
337C2×C4C23×D311yescharacteristic
338C8—21no—
339C8—32no—
340C8—61no—
341C8C2×C4×D312yescomplemented
342C2×C3—42no—
343C2×C3—42no—
344C2×C3C2×C4×D411yescharacteristic
345C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
346C2×C3#57511yescharacteristic
347D3—22no—
348D3—42no—
349C22—22no—
350C22—22no—
351C22—32no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—61no—
357C22—61no—
358C22—62no—
359C22—128no—
360C22—121no—
361C22—121no—
362C22C2×D3×D411yescharacteristic
363C4—21no—
364C4—21no—
365C4—32no—
366C4—32no—
367C4—41no—
368C4—41no—
369C4—61no—
370C4—62no—
371C4—121no—
372C4—121no—
373C4C22×C4×D311yescharacteristic
374C4D3×#4111yescharacteristic
375C3C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
376C2—42no—
377C2—42no—
378C2—62no—
379C2—122no—
380C2C4×D3×D411yescharacteristic
381C2D3×AGL(1,ℤ/8)11yescharacteristic
382C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×#442C2
socleC22×C3C22×D4
prime radicalC2×#911C2
cosocleC2×C3×C4C24
hypercenter#442D3
hypocenterC3C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. C4×D3×D4C2semidirect
3. C4×#237C2semidirect
4. C3×#442C2semidirect
5. Dic3×D8C2semidirect
6. #5018C2semidirect
7. #5014C2semidirect
8. #4861C2semidirect
9. #4852C2semidirect
10. #4771C2semidirect
11. #4770C2semidirect
12. #4758C2semidirect
13. #4756C2semidirect
14. #4710C2semidirect
15. #4707C2semidirect
22. C3×C4×D4C22semidirect
24. D4×Dic3C22semidirect
30. #911C4semidirect
37. #887C22semidirect
44. C4×Dic6C22semidirect
45. C3×#106C22semidirect
47. C3×#102C22semidirect
48. C3×#97C22semidirect
50. #832C22semidirect
52. #818C22semidirect
53. #816C22semidirect
54. C8×Dic3C22semidirect
55. C8×Dic3C22semidirect
56. #810C22semidirect
58. #808C22semidirect
59. #807C22semidirect
60. #803C22semidirect
65. #442D3semidirect
87. #237C2×C4semidirect
89. D3×D4C2×C4semidirect
99. C3×D8C2×C4semidirect
109. #214C2×C4semidirect
110. #213C2×C4semidirect
127. #204C2×C4semidirect
129. ΓL(1,𝔽25)C2×C4semidirect
151. C4×D4C2×D3semidirect
161. #106C2×D3semidirect
169. #102C2×D3semidirect
171. #97C2×D3semidirect
184. C3×D4C22×C4semidirect
213. Dic6C22×C4semidirect
215. C3×C8C22×C4semidirect
217. C3⋊C8C22×C4semidirect
244. D8C4×D3semidirect
297. Dic3AGL(1,ℤ/8)semidirect
314. D4C2×C4×D3semidirect
341. C8C2×C4×D3semidirect
375. C3C4×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
C4×D3×D42
C4×#2372
C3×#4422
Dic3×D82
#50182
#50142
#48612
#48522
#47712
#47702
#47582
#47562
#47102
#47072
C4×AGL(1,ℤ/8)3