Finite Group Databasebeta

← #79878

Subgroups — #79878

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#911C211yesmaximal, characteristic, complemented
2C2×D3×D8C211yesmaximal, characteristic, complemented
3#5554C211yesmaximal, characteristic, complemented
4D4×D12C211yesmaximal, characteristic, complemented
5C3×#442C211yesmaximal, characteristic, complemented
6#5016C211yesmaximal, characteristic, complemented
7#5015C211yesmaximal, characteristic, complemented
8#4865C211yesmaximal, characteristic, complemented
9#4864C211yesmaximal, characteristic, complemented
10#4781C211yesmaximal, characteristic, complemented
11#4776C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13#4773C211yesmaximal, characteristic, complemented
14#4712C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3264—31nomaximal, Sylow
17C2×#237C2211yescharacteristic, complemented
18C2×D3×D4C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic, complemented
21C22×D12—21no—
22C2×C3×D8C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic, complemented
24C3×C4×D4C2211yescharacteristic, complemented
25C2×#214C2211yescharacteristic, complemented
26C2×#213C2211yescharacteristic
27C2×#213C2211yescharacteristic, complemented
28C2×#213C2211yescharacteristic, complemented
29#930—21no—
30#930—21no—
31C2×(C2×D3)⋊C4—21no—
32#911—24no—
33D3×D8—24no—
34C2×D24C2211yescharacteristic, complemented
35C2×D24C2211yescharacteristic, complemented
36C2×#204C2211yescharacteristic, complemented
37C2×#204C2211yescharacteristic, complemented
38C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
39C2×D3×C8C2211yescharacteristic, complemented
40#897C2211yescharacteristic, complemented
41#895C2211yescharacteristic
42#895C2211yescharacteristic, complemented
43#895C2211yescharacteristic, complemented
44#886—21no—
45#882—21no—
46#882—21no—
47#882—21no—
48#875C2211yescharacteristic, complemented
49#874C2211yescharacteristic, complemented
50C4×D12C2211yescharacteristic
51C3×#106C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×#97C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic, complemented
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#809C2211yescharacteristic, complemented
59#809C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61C4⋊(C3⋊C8)C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×D8—31no—
64#546—31no—
65D42—31no—
66#496—31no—
67#469—31no—
68#465—31no—
69#463—31no—
70#459—31no—
71#450—31no—
72#449—31no—
73#447—31no—
74#447—31no—
75#442D311yescharacteristic, complemented
76#435—31no—
77C23×D3—21no—
78C23×D3—21no—
79C23×D3—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C22×Dic3—21no—
89#237—22no—
90#237—41no—
91D3×D4—22no—
92D3×D4—22no—
93D3×D4—22no—
94D3×D4—41no—
95D3×D4—41no—
96D3×D4—41no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—44no—
103C2×D12D411yescharacteristic
104C2×D12C2311yescharacteristic
105C2×D12C2311yescharacteristic
106C2×D12C2311yescharacteristic
107C2×C4×D3C2311yescharacteristic
108C2×C4×D3C2311yescharacteristic
109C2×Dic6C2311yescharacteristic
110C3×D8D414yescomplemented
111C2×C3×C8C2311yescharacteristic
112C2×C3×C8C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C4⋊C4C2311yescharacteristic
115C3×C22⋊C4—21no—
116C3×C22⋊C4—21no—
117C3×C42C2311yescharacteristic
118#214—22no—
119#213—22no—
120#213—22no—
121#213—22no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4D411yescharacteristic
129(C2×D3)⋊C4D411yescharacteristic
130C4⋊Dic3—21no—
131C4⋊Dic3D411yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133C2×C3⋊C8C2311yescharacteristic
134D24—22no—
135D24—41no—
136#204—22no—
137#204—41no—
138ΓL(1,𝔽25)—22no—
139D3×C8—22no—
140C2×#41—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
145AGL(1,ℤ/8)—64no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×D8—31no—
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×D8—64no—
155C2×D8C2×D311yescharacteristic
156C2×M16—31no—
157C22×C8—31no—
158#127—31no—
159#124—31no—
160#123—61no—
161C22⋊Q8—31no—
162#121—31no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168#120—61no—
169#120—61no—
170C4×D4—31no—
171C4×D4C2×D311yescharacteristic, complemented
172C4×D4C2×D311yescharacteristic, complemented
173C2×C22⋊C4—61no—
174#106C2×D311yescharacteristic, complemented
175C4⋊C8—31no—
176#103—31no—
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic, complemented
181#102C2×D311yescharacteristic, complemented
182C22⋊C8—31no—
183C22⋊C8—31no—
184#97C2×D311yescharacteristic, complemented
185C23×C3—21no—
186C23×C3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—42no—
196C22×D3—81no—
197C22×D3—81no—
198C22×D3—81no—
199C22×D3C2×D411yescharacteristic
200C22×D3C2×D411yescharacteristic
201C3×D4—21no—
202C3×D4—21no—
203C3×D4C2×D412yes—
204C3×D4C2×D412yescomplemented
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—42no—
208C2×C3×C4—81no—
209C2×C3×C4C2×D411yescharacteristic
210C2×C3×C4C2×D411yescharacteristic
211C2×C3×C4C2411yescharacteristic
212#67—42no—
213#67—42no—
214#67—42no—
215#67—42no—
216C2×Dic3—21no—
217C2×Dic3—42no—
218C2×Dic3C2×D411yescharacteristic
219C2×Dic3C2×D411yescharacteristic
220D12—22no—
221D12—21no—
222D12—21no—
223D12—21no—
224D12—21no—
225D12—21no—
226D12—21no—
227D12—42no—
228D12—42no—
229D12—42no—
230D12—81no—
231D12—81no—
232C4×D3—21no—
233C4×D3—21no—
234C4×D3—21no—
235C4×D3—21no—
236Dic6—21no—
237Dic6—21no—
238C3×C8—21no—
239C3×C8C2×D412yescomplemented
240C3⋊C8—21no—
241C3⋊C8—21no—
242C24—61no—
243C24—61no—
244C24—61no—
245#41—62no—
246#41—121no—
247C2×Q8—31no—
248C2×D4—31no—
249C2×D4—31no—
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—124no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4C22×D311yescharacteristic
269C2×D4C22×D311yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—61no—
275SD16—62no—
276SD16—62no—
277SD16—121no—
278D8—62no—
279D8—62no—
280D8—62no—
281D8—62no—
282D8—121no—
283D8D1214yescomplemented
284M16—62no—
285C2×C8—31no—
286C2×C8—31no—
287C2×C8—62no—
288C2×C8C22×D311yescharacteristic
289C2×C8C22×D311yescharacteristic
290C4⋊C4—31no—
291C4⋊C4—61no—
292C4⋊C4C22×D311yescharacteristic
293C4⋊C4C22×D311yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—31no—
297C22⋊C4—31no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C42C22×D311yescharacteristic
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3C22×D411yescharacteristic
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—21no—
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—84no—
327C2×D3—84no—
328C2×D3—84no—
329C2×D3AGL(1,ℤ/8)12yescomplemented
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4C22×D411yescharacteristic
335C3×C4H(5,𝔽2)11yescharacteristic
336Dic3—21no—
337Dic3—21no—
338Dic3—41no—
339C23—21no—
340C23—21no—
341C23—31no—
342C23—31no—
343C23—61no—
344C23—61no—
345C23—61no—
346C23—122no—
347C23—122no—
348C23—122no—
349C23—122no—
350C23—122no—
351C23—122no—
352C23—241no—
353C23—241no—
354C23—241no—
355Q8—61no—
356Q8—61no—
357D4—21no—
358D4—21no—
359D4—62no—
360D4—61no—
361D4—61no—
362D4—61no—
363D4—61no—
364D4—61no—
365D4—61no—
366D4—122no—
367D4—122no—
368D4—122no—
369D4—122no—
370D4—122no—
371D4—122no—
372D4—122no—
373D4—241no—
374D4—241no—
375D4C2×D1212yes—
376D4C2×D1212yescomplemented
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—42no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—81no—
388C2×C4—122no—
389C2×C4D3×D411yescharacteristic
390C2×C4D3×D411yescharacteristic
391C2×C4C23×D311yescharacteristic
392C8—21no—
393C8—61no—
394C8—61no—
395C8C2×D1212yescomplemented
396C2×C3—42no—
397C2×C3—42no—
398C2×C3D4211yescharacteristic
399C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
400C2×C3#57611yescharacteristic
401D3—22no—
402D3—41no—
403D3—81no—
404D3—81no—
405D3—81no—
406C22—22no—
407C22—22no—
408C22—32no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—61no—
414C22—61no—
415C22—61no—
416C22—61no—
417C22—61no—
418C22—121no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22—121no—
426C22—121no—
427C22—244no—
428C22—244no—
429C22—244no—
430C22C2×D3×D411yescharacteristic
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—61no—
436C4—61no—
437C4—121no—
438C4C22×D1211yescharacteristic
439C4#100911yescharacteristic
440C3#326411yescharacteristic, complemented, Sylow
441C2—42no—
442C2—42no—
443C2—62no—
444C2—121no—
445C2—241no—
446C2—241no—
447C2—241no—
448C2D4×D1211yescharacteristic
449C2D3×AGL(1,ℤ/8)11yescharacteristic
450C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#442C2
socleC22×C3C22×D4
hypercenter#442D3
hypocenterC3#3264
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. C2×D3×D8C2semidirect
3. #5554C2semidirect
4. D4×D12C2semidirect
5. C3×#442C2semidirect
6. #5016C2semidirect
7. #5015C2semidirect
8. #4865C2semidirect
9. #4864C2semidirect
10. #4781C2semidirect
11. #4776C2semidirect
12. #4773C2semidirect
13. #4773C2semidirect
14. #4712C2semidirect
15. #4706C2semidirect
17. C2×#237C22semidirect
19. C2×D3×D4C22semidirect
20. C2×D3×D4C22semidirect
23. C3×C4×D4C22semidirect
24. C3×C4×D4C22semidirect
25. C2×#214C22semidirect
27. C2×#213C22semidirect
28. C2×#213C22semidirect
34. C2×D24C22semidirect
35. C2×D24C22semidirect
36. C2×#204C22semidirect
37. C2×#204C22semidirect
38. C2×ΓL(1,𝔽25)C22semidirect
39. C2×D3×C8C22semidirect
40. #897C22semidirect
42. #895C22semidirect
43. #895C22semidirect
48. #875C22semidirect
49. #874C22semidirect
51. C3×#106C22semidirect
52. C3×#102C22semidirect
53. C3×#102C22semidirect
54. C3×#97C22semidirect
55. (C2×D3)⋊C8C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #810C22semidirect
58. #809C22semidirect
59. #809C22semidirect
60. #809C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
75. #442D3semidirect
110. C3×D8D4semidirect
171. C4×D4C2×D3semidirect
172. C4×D4C2×D3semidirect
174. #106C2×D3semidirect
180. #102C2×D3semidirect
181. #102C2×D3semidirect
184. #97C2×D3semidirect
204. C3×D4C2×D4semidirect
239. C3×C8C2×D4semidirect
283. D8D12semidirect
329. C2×D3AGL(1,ℤ/8)semidirect
376. D4C2×D12semidirect
395. C8C2×D12semidirect
440. C3#3264semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
C2×D3×D82
#55542
D4×D122
C3×#4422
#50162
#50152
#48652
#48642
#47812
#47762
#47732
#47122
#47062
#32643