Finite Group Databasebeta

← #83502

Subgroups — #83502

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
1C3×#546C211yesmaximal, characteristic, complemented
2C2×#949C211yesmaximal, characteristic, complemented
3C22×#213C211yesmaximal, characteristic, complemented
4D4×D12C211yesmaximal, characteristic, complemented
5#5238C211yesmaximal, characteristic, complemented
6#5217C211yesmaximal, characteristic, complemented
7#5058C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5048C211yesmaximal, characteristic, complemented
10#5046C211yesmaximal, characteristic, complemented
11#5039C211yesmaximal, characteristic, complemented
12#5036C211yesmaximal, characteristic, complemented
13#5015C211yesmaximal, characteristic, complemented
14#5014C211yesmaximal, characteristic, complemented
15#5011C211yesmaximal, characteristic, complemented
16#3264—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18C22×C3×D4C2211yescharacteristic, complemented
19C2×D3×D4—21no—
20C22×D12C2211yescharacteristic
21C22×D12C2211yescharacteristic, complemented
22C3×#124C2211yescharacteristic, complemented
23C3×#123—21no—
24C3×C22⋊Q8C2211yescharacteristic, complemented
25C3×#121—21no—
26C3×#121C2211yescharacteristic, complemented
27C3×#120—21no—
28C3×C4×D4C2211yescharacteristic
29C3×C4×D4C2211yescharacteristic, complemented
30C2×C3×C22⋊C4—21no—
31#949—24no—
32C2×#215C2211yescharacteristic, complemented
33C2×#215C2211yescharacteristic, complemented
34C2×#215C2211yescharacteristic, complemented
35C2×#213—24no—
36C2×#213C2211yescharacteristic
37C2×#213C2211yescharacteristic
38C2×#213C2211yescharacteristic, complemented
39C2×#213C2211yescharacteristic, complemented
40C2×#213C2211yescharacteristic, complemented
41C2×#213C2211yescharacteristic, complemented
42#930C2211yescharacteristic
43#930C2211yescharacteristic, complemented
44C2×#208C2211yescharacteristic, complemented
45C22×C3⋊C8C2211yescharacteristic, complemented
46#895—21no—
47#882—21no—
48#882—21no—
49#874C2211yescharacteristic, complemented
50C4×D12C2211yescharacteristic, complemented
51#835C2211yescharacteristic, complemented
52#832C2211yescharacteristic, complemented
53#832C2211yescharacteristic, complemented
54#830C2211yescharacteristic, complemented
55#830C2211yescharacteristic, complemented
56#809C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#809C2211yescharacteristic, complemented
59#808C2211yescharacteristic, complemented
60C4⋊(C3⋊C8)C2211yescharacteristic, complemented
61#803C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×D8—31no—
64#546D311yescharacteristic, complemented
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#442—31no—
76#435—31no—
77C24×C3—21no—
78C23×D3—21no—
79C23×D3—21no—
80C3×#41—22no—
81C3×#41—41no—
82C2×C3×Q8C2311yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4—21no—
87C2×C3×D4—41no—
88C2×C3×D4D411yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C2×C3×D4C2311yescharacteristic
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C2×#67—21no—
96C2×#67—21no—
97D3×D4—44no—
98C2×D12—22no—
99C2×D12—22no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12—41no—
105C2×D12—41no—
106C2×D12C2311yescharacteristic
107C2×D12C2311yescharacteristic
108C2×D12C2311yescharacteristic
109C2×C4×D3—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4D411yescharacteristic
112C3×C4⋊C4C2311yescharacteristic
113C3×C22⋊C4—21no—
114C3×C22⋊C4—21no—
115C3×C22⋊C4—21no—
116C3×C22⋊C4—21no—
117C3×C22⋊C4—21no—
118C3×C22⋊C4D411yescharacteristic
119C3×C22⋊C4D411yescharacteristic
120C3×C42C2311yescharacteristic
121#215—22no—
122#215—22no—
123#215—41no—
124#213—22no—
125#213—22no—
126#213—22no—
127#213—22no—
128#213—41no—
129#213D414yescomplemented
130(C2×D3)⋊C4—21no—
131(C2×D3)⋊C4—21no—
132(C2×D3)⋊C4—21no—
133C4⋊Dic3C2311yescharacteristic
134#208—22no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8C2311yescharacteristic
137C2×C3⋊C8C2311yescharacteristic
138C2×C3⋊C8C2311yescharacteristic
139C2×C3⋊C8C2311yescharacteristic
140C2×#41C2×D311yescharacteristic, complemented
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—61no—
144C22×D4C2×D311yescharacteristic, complemented
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—31no—
155C2×D8—64no—
156C2×M16—31no—
157C22×C8—31no—
158#127—31no—
159#124C2×D311yescharacteristic, complemented
160#123—21no—
161C22⋊Q8C2×D311yescharacteristic, complemented
162#121—21no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121C2×D311yescharacteristic, complemented
167#120—21no—
168#120—61no—
169#120—61no—
170C4×D4—31no—
171C4×D4C2×D311yescharacteristic
172C4×D4C2×D311yescharacteristic, complemented
173C2×C22⋊C4—21no—
174#106—31no—
175C4⋊C8—31no—
176#103—31no—
177#102—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181#102—31no—
182C22⋊C8—31no—
183C22⋊C8—31no—
184#97—31no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—81no—
190C23×C3C2×D411yescharacteristic
191C23×C3C2×D411yescharacteristic
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—42no—
196C22×D3—42no—
197C22×D3—42no—
198C22×D3—42no—
199C22×D3—81no—
200C22×D3—81no—
201C3×Q8—21no—
202C3×Q8—21no—
203C3×D4—22no—
204C3×D4—21no—
205C3×D4—21no—
206C3×D4—21no—
207C3×D4—42no—
208C3×D4—42no—
209C3×D4—81no—
210C3×D4C2×D412yescomplemented
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—42no—
218C2×C3×C4—81no—
219C2×C3×C4C2×D411yescharacteristic
220C2×C3×C4C2×D411yescharacteristic
221C2×C3×C4C2×D411yescharacteristic
222C2×C3×C4C2×D411yescharacteristic
223C2×C3×C4C2411yescharacteristic
224#67—42no—
225#67—42no—
226C2×Dic3—21no—
227D12—21no—
228D12—21no—
229D12—21no—
230D12—21no—
231D12—21no—
232D12—42no—
233D12—42no—
234D12—42no—
235D12—81no—
236D12C2×D412yes—
237C4×D3—42no—
238C3⋊C8—21no—
239C3⋊C8—21no—
240C3⋊C8—21no—
241C3⋊C8C2×D412yescomplemented
242C24—21no—
243C24—61no—
244C24—61no—
245#41—22no—
246#41—41no—
247C2×Q8C22×D311yescharacteristic
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—31no—
253C2×D4—31no—
254C2×D4—31no—
255C2×D4—41no—
256C2×D4—62no—
257C2×D4—62no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—124no—
267C2×D4#6711yescharacteristic
268C2×D4C22×D311yescharacteristic
269C2×D4C22×D311yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—61no—
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D311yescharacteristic
275SD16—62no—
276SD16—62no—
277SD16—121no—
278D8—34no—
279D8—62no—
280D8—62no—
281D8—62no—
282D8—62no—
283D8—121no—
284M16—62no—
285C2×C8—31no—
286C2×C8—31no—
287C2×C8—31no—
288C2×C8—31no—
289C2×C8—62no—
290C4⋊C4—21no—
291C4⋊C4—31no—
292C4⋊C4#6711yescharacteristic
293C4⋊C4C22×D311yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4#6711yescharacteristic
303C22⋊C4#6711yescharacteristic
304C42C22×D311yescharacteristic
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—22no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—84no—
317C22×C3AGL(1,ℤ/8)12yescomplemented
318C22×C3C22×D411yescharacteristic
319C2×D3—22no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—84no—
329C2×D3—84no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4C22×D411yescharacteristic
337C3×C4H(5,𝔽2)11yescharacteristic
338Dic3—41no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—61no—
344C23—61no—
345C23—61no—
346C23—81no—
347C23—122no—
348C23—122no—
349C23—122no—
350C23—122no—
351C23—241no—
352C23—241no—
353C23C2×#6711yescharacteristic
354C23C2×#6711yescharacteristic
355Q8—21no—
356Q8—21no—
357D4—22no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—32no—
362D4—42no—
363D4—42no—
364D4—61no—
365D4—61no—
366D4—61no—
367D4—61no—
368D4—61no—
369D4—81no—
370D4—122no—
371D4—122no—
372D4—122no—
373D4—122no—
374D4—122no—
375D4—241no—
376D4D3×D412yescomplemented
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—61no—
385C2×C4—81no—
386C2×C4—122no—
387C2×C4C2×#6711yescharacteristic
388C2×C4C2×#6711yescharacteristic
389C2×C4C2×#6711yescharacteristic
390C2×C4C2×#6711yescharacteristic
391C2×C4C23×D311yescharacteristic
392C8—32no—
393C8—61no—
394C8—61no—
395C8—61no—
396C2×C3—22no—
397C2×C3—41no—
398C2×C3—42no—
399C2×C3—81no—
400C2×C3D4211yescharacteristic
401C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
402C2×C3#57611yescharacteristic
403D3—42no—
404D3—81no—
405D3—81no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—62no—
418C22—84no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22—121no—
426C22—121no—
427C22—244no—
428C22—244no—
429C22#94912yescomplemented
430C22C22×#6711yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—41no—
436C4—41no—
437C4—121no—
438C4C2×D3×D411yescharacteristic
439C4#100411yescharacteristic
440C3#326411yescharacteristic, complemented, Sylow
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—122no—
446C2—241no—
447C2—241no—
448C2D4×#6711yescharacteristic
449C2C2×#94911yescharacteristic
450C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#546C2
socleC22×C3C22×D4
hypercenter#546D3
hypocenterC3#3264
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C3×#546C2semidirect
2. C2×#949C2semidirect
3. C22×#213C2semidirect
4. D4×D12C2semidirect
5. #5238C2semidirect
6. #5217C2semidirect
7. #5058C2semidirect
8. #5057C2semidirect
9. #5048C2semidirect
10. #5046C2semidirect
11. #5039C2semidirect
12. #5036C2semidirect
13. #5015C2semidirect
14. #5014C2semidirect
15. #5011C2semidirect
17. C2×C3×#41C22semidirect
18. C22×C3×D4C22semidirect
21. C22×D12C22semidirect
22. C3×#124C22semidirect
24. C3×C22⋊Q8C22semidirect
26. C3×#121C22semidirect
29. C3×C4×D4C22semidirect
32. C2×#215C22semidirect
33. C2×#215C22semidirect
34. C2×#215C22semidirect
38. C2×#213C22semidirect
39. C2×#213C22semidirect
40. C2×#213C22semidirect
41. C2×#213C22semidirect
43. #930C22semidirect
44. C2×#208C22semidirect
45. C22×C3⋊C8C22semidirect
49. #874C22semidirect
50. C4×D12C22semidirect
51. #835C22semidirect
52. #832C22semidirect
53. #832C22semidirect
54. #830C22semidirect
55. #830C22semidirect
56. #809C22semidirect
57. #809C22semidirect
58. #809C22semidirect
59. #808C22semidirect
60. C4⋊(C3⋊C8)C22semidirect
61. #803C22semidirect
64. #546D3semidirect
129. #213D4semidirect
140. C2×#41C2×D3semidirect
144. C22×D4C2×D3semidirect
159. #124C2×D3semidirect
161. C22⋊Q8C2×D3semidirect
166. #121C2×D3semidirect
172. C4×D4C2×D3semidirect
210. C3×D4C2×D4semidirect
241. C3⋊C8C2×D4semidirect
317. C22×C3AGL(1,ℤ/8)semidirect
376. D4D3×D4semidirect
429. C22#949semidirect
440. C3#3264semidirect

Maximal subgroups

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

subgroupindex
C3×#5462
C2×#9492
C22×#2132
D4×D122
#52382
#52172
#50582
#50572
#50482
#50462
#50392
#50362
#50152
#50142
#50112
#32643