Finite Group Databasebeta

← #83504

Subgroups — #83504

Order 384.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#546C211yesmaximal, characteristic, complemented
2C2×#949C211yesmaximal, characteristic, complemented
3C2×#932C211yesmaximal, characteristic, complemented
4#5550C211yesmaximal, characteristic, complemented
5#5239C211yesmaximal, characteristic, complemented
6#5217C211yesmaximal, characteristic, complemented
7#5059C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5049C211yesmaximal, characteristic, complemented
10#5047C211yesmaximal, characteristic, complemented
11#5042C211yesmaximal, characteristic, complemented
12#5036C211yesmaximal, characteristic, complemented
13#5016C211yesmaximal, characteristic, complemented
14C4×#213C211yesmaximal, characteristic, complemented
15#5012C211yesmaximal, characteristic, complemented
16#3273—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic, complemented
19C2×#235C2211yescharacteristic
20C22×D12C2211yescharacteristic, complemented
21C3×#124C2211yescharacteristic, complemented
22C3×#123—21no—
23C3×C22⋊Q8C2211yescharacteristic, complemented
24C3×#121—21no—
25C3×#121C2211yescharacteristic
26C3×#120—21no—
27C3×C4×D4C2211yescharacteristic
28C3×C4×D4C2211yescharacteristic, complemented
29C2×C3×C22⋊C4—21no—
30#949—24no—
31C2×#216C2211yescharacteristic, complemented
32C2×#215C2211yescharacteristic, complemented
33C2×#215C2211yescharacteristic, complemented
34#937—21no—
35C2×#214C2211yescharacteristic, complemented
36C2×#214C2211yescharacteristic, complemented
37#932—24no—
38C2×#213C2211yescharacteristic
39C2×#213C2211yescharacteristic
40C2×#213C2211yescharacteristic
41C2×#213C2211yescharacteristic
42#930C2211yescharacteristic
43#924C2211yescharacteristic, complemented
44C2×#208C2211yescharacteristic, complemented
45C2×#208C2211yescharacteristic, complemented
46#894—21no—
47#884—21no—
48D3×C22⋊C4—21no—
49#875C2211yescharacteristic, complemented
50C4×D12C2211yescharacteristic, complemented
51#835C2211yescharacteristic, complemented
52#832C2211yescharacteristic, complemented
53#832C2211yescharacteristic, complemented
54#830C2211yescharacteristic, complemented
55#830C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#809C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60C4⋊(C3⋊C8)C2211yescharacteristic, complemented
61C4×C3⋊C8C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×AGL(1,ℤ/8)—31no—
64#546—31no—
65#546D311yescharacteristic, complemented
66#495—31no—
67#468—31no—
68#468—31no—
69#463—31no—
70#463—31no—
71#452—31no—
72#452—31no—
73#447—31no—
74#447—31no—
75C4×D8—31no—
76#436—31no—
77C24×C3—21no—
78C23×D3—21no—
79C3×#41—22no—
80C3×#41—41no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×D4—22no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4—41no—
87C2×C3×D4D411yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C2×#67—21no—
95C2×#67—21no—
96#235—22no—
97#235—41no—
98C2×D12—22no—
99C2×D12—21no—
100C2×D12—41no—
101C2×D12C2311yescharacteristic
102C2×D12C2311yescharacteristic
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×Dic6C2311yescharacteristic
106C3×C4⋊C4—21no—
107C3×C4⋊C4D411yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113C3×C22⋊C4—21no—
114C3×C22⋊C4D411yescharacteristic
115C3×C22⋊C4D411yescharacteristic
116C3×C42C2311yescharacteristic
117#217—21no—
118#216—41no—
119#215—22no—
120#215—22no—
121#214—22no—
122#214—22no—
123#213—22no—
124#213—22no—
125#213—41no—
126#213D414yescomplemented
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4—21no—
129(C2×D3)⋊C4—21no—
130(C2×D3)⋊C4—21no—
131C4⋊Dic3C2311yescharacteristic
132Dic3⋊C4—21no—
133#208—22no—
134#208—22no—
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137C2×C3⋊C8C2311yescharacteristic
138C2×C3⋊C8C2311yescharacteristic
139C2×#41—31no—
140C2×#41C2×D311yescharacteristic
141C22×D4—31no—
142C22×D4C2×D311yescharacteristic, complemented
143AGL(1,ℤ/8)—64no—
144AGL(1,ℤ/8)—64no—
145C2×Q16—31no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×M16—31no—
155C2×M16—31no—
156#124—31no—
157#124C2×D311yescharacteristic, complemented
158#123—21no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8C2×D311yescharacteristic, complemented
162#121—21no—
163#121—31no—
164#121—61no—
165#121C2×D311yescharacteristic
166#120—21no—
167#120—61no—
168C4×D4—31no—
169C4×D4C2×D311yescharacteristic
170C4×D4C2×D311yescharacteristic, complemented
171C2×C22⋊C4—21no—
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#102—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183C4×C8—31no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—81no—
189C23×C3C2×D411yescharacteristic
190C23×C3C2×D411yescharacteristic
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—81no—
196C3×Q8—21no—
197C3×Q8—21no—
198C3×D4—21no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—41no—
202C3×D4—42no—
203C3×D4—42no—
204C3×D4—81no—
205C3×D4C2×D412yes—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—42no—
213C2×C3×C4—81no—
214C2×C3×C4C2×D411yescharacteristic
215C2×C3×C4C2×D411yescharacteristic
216C2×C3×C4C2×D411yescharacteristic
217C2×C3×C4C2×D411yescharacteristic
218C2×C3×C4C2411yescharacteristic
219#67—42no—
220#67—81no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223D12—21no—
224D12—21no—
225D12—21no—
226D12—42no—
227D12C2×D412yes—
228C4×D3—42no—
229C4×D3—81no—
230Dic6—21no—
231Dic6—21no—
232C3⋊C8—21no—
233C3⋊C8—21no—
234C3⋊C8—21no—
235C3⋊C8C2×D412yes—
236C24—21no—
237C24—61no—
238#41—22no—
239#41—41no—
240#41—62no—
241#41—121no—
242C2×Q8—31no—
243C2×Q8C22×D311yescharacteristic
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—31no—
249C2×D4—31no—
250C2×D4—41no—
251C2×D4—62no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C2×D4#6711yescharacteristic
257C2×D4C22×D311yescharacteristic
258C2×D4C22×D311yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4C22×D311yescharacteristic
264C22×C4C22×D311yescharacteristic
265Q16—121no—
266SD16—62no—
267SD16—62no—
268SD16—62no—
269SD16—62no—
270D8—34no—
271D8—62no—
272D8—62no—
273D8—121no—
274M16—62no—
275M16—62no—
276C2×C8—31no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—31no—
280C4⋊C4—21no—
281C4⋊C4—31no—
282C4⋊C4—61no—
283C4⋊C4#6711yescharacteristic
284C4⋊C4C22×D311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4#6711yescharacteristic
296C22⋊C4#6711yescharacteristic
297C42C22×D311yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—84no—
311C22×C3C22×D411yescharacteristic
312C2×D3—22no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—84no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4C22×D411yescharacteristic
326C3×C4H(5,𝔽2)11yescharacteristic
327Dic3—41no—
328Dic3—41no—
329C23—21no—
330C23—21no—
331C23—42no—
332C23—42no—
333C23—61no—
334C23—61no—
335C23—81no—
336C23—122no—
337C23—122no—
338C23—241no—
339C23C2×#6711yescharacteristic
340C23C2×#6711yescharacteristic
341Q8—21no—
342Q8—21no—
343Q8—61no—
344Q8—61no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—32no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—61no—
353D4—61no—
354D4—61no—
355D4—81no—
356D4—122no—
357D4—122no—
358D4—241no—
359D4D3×D412yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—81no—
370C2×C4—122no—
371C2×C4—241no—
372C2×C4C2×#6711yescharacteristic
373C2×C4C2×#6711yescharacteristic
374C2×C4C2×#6711yescharacteristic
375C2×C4C2×#6711yescharacteristic
376C2×C4C23×D311yescharacteristic
377C8—32no—
378C8—61no—
379C8—61no—
380C8—61no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—42no—
384C2×C3—81no—
385C2×C3D4211yescharacteristic
386C2×C3#57511yescharacteristic
387C2×C3#57611yescharacteristic
388D3—42no—
389D3—81no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—62no—
403C22—84no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22—244no—
410C22C22×#6711yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—41no—
416C4—41no—
417C4—121no—
418C4—121no—
419C4C2×D3×D411yescharacteristic
420C4#100411yescharacteristic
421C3#327311yescharacteristic, complemented, Sylow
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—122no—
427C2—241no—
428C2D4×#6711yescharacteristic
429C2#582211yescharacteristic
430C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#546C2
socleC22×C3C22×D4
hypercenter#546D3
hypocenterC3#3273
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. C2×#932C2semidirect
4. #5550C2semidirect
5. #5239C2semidirect
6. #5217C2semidirect
7. #5059C2semidirect
8. #5057C2semidirect
9. #5049C2semidirect
10. #5047C2semidirect
11. #5042C2semidirect
12. #5036C2semidirect
13. #5016C2semidirect
14. C4×#213C2semidirect
15. #5012C2semidirect
18. C22×C3×D4C22semidirect
20. C22×D12C22semidirect
21. C3×#124C22semidirect
23. C3×C22⋊Q8C22semidirect
28. C3×C4×D4C22semidirect
31. C2×#216C22semidirect
32. C2×#215C22semidirect
33. C2×#215C22semidirect
35. C2×#214C22semidirect
36. C2×#214C22semidirect
43. #924C22semidirect
44. C2×#208C22semidirect
45. C2×#208C22semidirect
49. #875C22semidirect
50. C4×D12C22semidirect
51. #835C22semidirect
52. #832C22semidirect
53. #832C22semidirect
54. #830C22semidirect
55. #830C22semidirect
56. #810C22semidirect
57. #809C22semidirect
58. #809C22semidirect
59. #807C22semidirect
60. C4⋊(C3⋊C8)C22semidirect
61. C4×C3⋊C8C22semidirect
65. #546D3semidirect
126. #213D4semidirect
142. C22×D4C2×D3semidirect
157. #124C2×D3semidirect
161. C22⋊Q8C2×D3semidirect
170. C4×D4C2×D3semidirect
421. C3#3273semidirect

Maximal subgroups

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

subgroupindex
C3×#5462
C2×#9492
C2×#9322
#55502
#52392
#52172
#50592
#50572
#50492
#50472
#50422
#50362
#50162
C4×#2132
#50122
#32733