Finite Group Databasebeta

← #86771

Subgroups — #86771

Order 384.

1232 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#550C211yesmaximal, characteristic, complemented
2#5832C212yesmaximal, complemented
3#5829C212yesmaximal, complemented
4#5718C212yesmaximal, complemented
5#5715C212yesmaximal, complemented
6D3×#127C211yesmaximal, characteristic, complemented
7#5713C211yesmaximal, characteristic, complemented
8#5615C214yesmaximal, complemented
9#5612C212yesmaximal, complemented
10#5610C214yesmaximal, complemented
11#5596C212yesmaximal, complemented
12#5585C211yesmaximal, characteristic, complemented
13#5578C211yesmaximal, characteristic, complemented
14#5559C212yesmaximal, complemented
15#5553C211yesmaximal, characteristic, complemented
16#5545C212yesmaximal, complemented
17C4×#237C211yesmaximal, characteristic, complemented
18#3508—31nomaximal, Sylow
19C2×C3×#41C2212yescomplemented
20C2×#237C2211yescharacteristic
21C2×#237C2214yescomplemented
22C2×D3×D4C2212yes—
23C2×D3×D4C2212yes—
24C2×D3×D4C2212yes—
25C2×#235C2212yescomplemented
26C3×#127C2211yescharacteristic
27C3×#127C2212yescomplemented
28C3×#121C2212yescomplemented
29C3×#121C2214yescomplemented
30C3×C4×Q8C2211yescharacteristic, complemented
31C3×C4×D4C2211yescharacteristic, complemented
32C3×C4×D4C2212yescomplemented
33#947C2212yescomplemented
34#946C2211yescharacteristic
35#944C2211yescharacteristic, complemented
36#940C2212yes—
37#940C2212yes—
38#940C2212yes—
39#940C2212yescomplemented
40#939C2214yescomplemented
41#939C2214yescomplemented
42#939C2214yescomplemented
43#939C2214yescomplemented
44#938C2214yescomplemented
45#937C2214yes—
46#937C2214yes—
47#936C2212yescomplemented
48#936C2212yescomplemented
49#935C2214yescomplemented
50D4×Dic3C2212yescomplemented
51D4×Dic3C2212yescomplemented
52D4×Dic3C2211yescharacteristic, complemented
53#930C2212yescomplemented
54#929C2214yescomplemented
55C4×#67C2212yes—
56C4×#67C2214yescomplemented
57C4×#67C2214yescomplemented
58C2×(C2×D3)⋊C4C2214yes—
59C2×C4×Dic3C2212yescomplemented
60#897C2211yescharacteristic
61#896C2212yes—
62#894C2214yescomplemented
63#893C2212yescomplemented
64#893C2212yescomplemented
65#892C2211yescharacteristic
66#889C2212yescomplemented
67#888C2211yescharacteristic, complemented
68#887C2211yescharacteristic, complemented
69#885C2214yescomplemented
70#884C2214yescomplemented
71#883C2214yescomplemented
72#67⋊C4C2212yes—
73#67⋊C4C2214yescomplemented
74D3×C22⋊C4C2214yes—
75#877C2212yescomplemented
76#875C2212yescomplemented
77#874C2211yescharacteristic
78C4×D12C2212yescomplemented
79#872C2212yes—
80C42×D3C2211yescharacteristic
81C4×Dic6C2211yescharacteristic, complemented
82C4×Dic6C2211yescharacteristic, complemented
83C4×Dic6C2212yescomplemented
84#559—31no—
85#559—31no—
86#559—32no—
87#550—32no—
88#550—31no—
89#550D311yescharacteristic, complemented
90#546—32no—
91#546—34no—
92#546—32no—
93#540—34no—
94#535—32no—
95#533—31no—
96#532—32no—
97#532—32no—
98C2×#127—31no—
99#518—32no—
100C4×#41—31no—
101C23×D3C2314yes—
102C3×#41—44no—
103C2×C3×Q8C2311yescharacteristic
104C2×C3×D4C2312yes—
105C2×C3×D4C2312yes—
106C2×C3×D4C2311yescharacteristic
107C2×C3×D4C2312yes—
108C2×C3×D4C2314yes—
109C2×C3×D4C2314yes—
110C22×C3×C4C2312yes—
111C22×C3×C4C2314yes—
112C2×#67C2312yes—
113C2×#67C2314yes—
114C2×#67C2314yes—
115C2×#67C2314yes—
116C2×#67C2314yes—
117C2×#67C2314yes—
118C22×Dic3C2312yes—
119C22×Dic3C2312yes—
120C22×Dic3C2314yes—
121#237—24no—
122#237—48no—
123D3×D4—28no—
124D3×D4—44no—
125D3×D4—44no—
126#235—44no—
127C2×D12C2312yes—
128C2×D12C2312yes—
129C2×D12C2312yes—
130C2×C4×D3C2311yescharacteristic
131C2×C4×D3C2311yescharacteristic
132C2×C4×D3C2314yes—
133C2×C4×D3C2314yes—
134C2×C4×D3C2311yescharacteristic
135C2×Dic6C2311yescharacteristic
136C2×Dic6C2312yes—
137C2×Dic6C2311yescharacteristic
138C3×C4⋊C4C2311yescharacteristic
139C3×C4⋊C4C2311yescharacteristic
140C3×C4⋊C4C2312yes—
141C3×C22⋊C4C2312yes—
142C3×C22⋊C4C2314yes—
143C3×C42C2311yescharacteristic
144C3×C42C2312yes—
145#217C2312yes—
146#217C2314yes—
147#217C2314yes—
148#217C2314yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2314yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2314yes—
154(C2×D3)⋊C4C2312yes—
155(C2×D3)⋊C4C2314yes—
156C4⋊Dic3C2311yescharacteristic
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2311yescharacteristic
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2312yes—
165Dic3⋊C4C2312yes—
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2312yes—
168C4×Dic3C2312yes—
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
171C4×Dic3C2312yes—
172C4×Dic3C2312yes—
173C2×#41—31no—
174C2×#41—34no—
175C2×#41—32no—
176C2×#41C2×D312yescomplemented
177C22×D4—32no—
178C22×D4—32no—
179C22×D4—32no—
180C4⋊Q8—31no—
181C4⋊Q8—31no—
182#127—32no—
183#127—32no—
184#127—32no—
185#127—32no—
186#127—31no—
187#127C2×D311yescharacteristic
188#127C2×D312yescomplemented
189#125—32no—
190#124—32no—
191#124—32no—
192#124—32no—
193#124—34no—
194#124—32no—
195#123—34no—
196#123—34no—
197#123—34no—
198#123—34no—
199C22⋊Q8—31no—
200C22⋊Q8—32no—
201C22⋊Q8—31no—
202#121—34no—
203#121—34no—
204#121—34no—
205#121—34no—
206#121—34no—
207#121—32no—
208#121—34no—
209#121C2×D312yescomplemented
210#121C2×D314yescomplemented
211#120—34no—
212#120—34no—
213C4×Q8—31no—
214C4×Q8—31no—
215C4×Q8—32no—
216C4×Q8—31no—
217C4×Q8C2×D311yescharacteristic, complemented
218C4×D4—31no—
219C4×D4—32no—
220C4×D4—32no—
221C4×D4—32no—
222C4×D4—34no—
223C4×D4—34no—
224C4×D4—32no—
225C4×D4—32no—
226C4×D4—32no—
227C4×D4—34no—
228C4×D4—32no—
229C4×D4C2×D311yescharacteristic, complemented
230C4×D4C2×D312yescomplemented
231#117—31no—
232#117—32no—
233#117—32no—
234C2×C22⋊C4—34no—
235C2×C22⋊C4—34no—
236C2×C42—32no—
237C2×C42—31no—
238C23×C3C2414yes—
239C23×C3C2412yes—
240C22×D3—28no—
241C22×D3—44no—
242C22×D3—44no—
243C22×D3C2414yes—
244C22×D3C2411yescharacteristic
245C3×Q8—22no—
246C3×D4—24no—
247C3×D4—22no—
248C3×D4—24no—
249C3×D4—42no—
250C3×D4—44no—
251C3×D4—44no—
252C2×C3×C4—24no—
253C2×C3×C4—44no—
254C2×C3×C4C2412yes—
255C2×C3×C4C2411yescharacteristic
256C2×C3×C4C2412yes—
257C2×C3×C4C2412yes—
258C2×C3×C4C2411yescharacteristic
259C2×C3×C4C2411yescharacteristic
260#67—24no—
261#67—28no—
262#67—28no—
263#67—44no—
264#67—44no—
265#67—44no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—44no—
269C2×Dic3C2411yescharacteristic
270C2×Dic3C2412yes—
271C2×Dic3C2412yes—
272C2×Dic3C2412yes—
273C2×Dic3C2412yes—
274C2×Dic3C2411yescharacteristic
275C2×Dic3C2411yescharacteristic
276D12—24no—
277D12—24no—
278D12—42no—
279C4×D3—22no—
280C4×D3—22no—
281C4×D3—44no—
282C4×D3—44no—
283C4×D3#4114yescomplemented
284Dic6—22no—
285Dic6—22no—
286Dic6—24no—
287C24—34no—
288#41—44no—
289#41—64no—
290#41—124no—
291#41—128no—
292C2×Q8—31no—
293C2×Q8—32no—
294C2×Q8—31no—
295C2×Q8C22×D311yescharacteristic
296C2×D4—32no—
297C2×D4—34no—
298C2×D4—34no—
299C2×D4—34no—
300C2×D4—34no—
301C2×D4—34no—
302C2×D4—32no—
303C2×D4—32no—
304C2×D4—32no—
305C2×D4—68no—
306C2×D4—124no—
307C2×D4—124no—
308C2×D4C22×D312yes—
309C2×D4C22×D311yescharacteristic
310C2×D4C22×D312yes—
311C2×D4C22×D314yes—
312C2×D4C22×D314yes—
313C2×D4C22×D312yes—
314C22×C4—31no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—34no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—34no—
321C22×C4—34no—
322C22×C4C22×D312yes—
323C22×C4C22×D314yes—
324C4⋊C4—31no—
325C4⋊C4—32no—
326C4⋊C4—32no—
327C4⋊C4—32no—
328C4⋊C4—32no—
329C4⋊C4—32no—
330C4⋊C4—31no—
331C4⋊C4—32no—
332C4⋊C4—32no—
333C4⋊C4—32no—
334C4⋊C4C22×D311yescharacteristic
335C4⋊C4C22×D311yescharacteristic
336C4⋊C4C22×D312yes—
337C22⋊C4—34no—
338C22⋊C4—32no—
339C22⋊C4—34no—
340C22⋊C4—34no—
341C22⋊C4—32no—
342C22⋊C4—32no—
343C22⋊C4—32no—
344C22⋊C4—34no—
345C22⋊C4—34no—
346C22⋊C4—34no—
347C22⋊C4—34no—
348C22⋊C4C22×D314yes—
349C22⋊C4C22×D312yes—
350C42—31no—
351C42—32no—
352C42—31no—
353C42—32no—
354C42—32no—
355C42—31no—
356C42—32no—
357C42C22×D311yescharacteristic
358C42C22×D312yes—
359C22×C3—22no—
360C22×C3—24no—
361C22×C3—22no—
362C22×C3—24no—
363C22×C3—22no—
364C22×C3—24no—
365C22×C3C2511yescharacteristic
366C2×D3—21no—
367C2×D3—21no—
368C2×D3—24no—
369C2×D3—24no—
370C2×D3—24no—
371C2×D3—88no—
372C2×D3C2×#4112yes—
373C3×C4—22no—
374C3×C4—21no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4C2×#4112yes—
378C3×C4H(5,𝔽2)12yes—
379Dic3—22no—
380Dic3—22no—
381Dic3—22no—
382Dic3—21no—
383Dic3—22no—
384Dic3C2×#4112yes—
385Dic3H(5,𝔽2)12yes—
386C23—31no—
387C23—34no—
388C23—68no—
389C23—124no—
390C23—124no—
391C23C23×D314yes—
392C23C23×D312yes—
393Q8—22no—
394Q8—64no—
395Q8—62no—
396Q8—62no—
397D4—24no—
398D4—24no—
399D4—22no—
400D4—42no—
401D4—44no—
402D4—44no—
403D4—68no—
404D4—68no—
405D4—64no—
406D4—64no—
407D4—64no—
408D4—122no—
409D4—124no—
410D4—124no—
411D4—124no—
412C2×C4—24no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—34no—
416C2×C4—32no—
417C2×C4—32no—
418C2×C4—31no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—44no—
422C2×C4—62no—
423C2×C4—64no—
424C2×C4—64no—
425C2×C4—62no—
426C2×C4—124no—
427C2×C4—124no—
428C2×C4—124no—
429C2×C4C23×D312yes—
430C2×C4C23×D311yescharacteristic
431C2×C4C23×D312yes—
432C2×C4C23×D312yes—
433C2×C4C23×D311yescharacteristic
434C2×C4C23×D311yescharacteristic
435C2×C3—44no—
436C2×C3—42no—
437C2×C3C22×#4111yescharacteristic
438C2×C3C2×H(5,𝔽2)11yescharacteristic
439C2×C3C2×H(5,𝔽2)11yescharacteristic
440D3—22no—
441D3—44no—
442C22—22no—
443C22—24no—
444C22—22no—
445C22—24no—
446C22—22no—
447C22—24no—
448C22—32no—
449C22—61no—
450C22—61no—
451C22—64no—
452C22—64no—
453C22—64no—
454C22—248no—
455C22C24×D311yescharacteristic
456C4—22no—
457C4—22no—
458C4—21no—
459C4—22no—
460C4—32no—
461C4—32no—
462C4—61no—
463C4—62no—
464C4—62no—
465C4—62no—
466C4—62no—
467C4#100412yes—
468C4D3×#4112yes—
469C3#350811yescharacteristic, complemented, Sylow
470C2—44no—
471C2—42no—
472C2—62no—
473C2—124no—
474C2C2×#100411yescharacteristic
475C2C2×D3×#4111yescharacteristic
476C2D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#550C2
hypercenter#550D3
hypocenterC3#3508
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C3×#550C2semidirect
2. #5832C2semidirect
3. #5829C2semidirect
4. #5718C2semidirect
5. #5715C2semidirect
6. D3×#127C2semidirect
7. #5713C2semidirect
8. #5615C2semidirect
9. #5612C2semidirect
10. #5610C2semidirect
11. #5596C2semidirect
12. #5585C2semidirect
13. #5578C2semidirect
14. #5559C2semidirect
15. #5553C2semidirect
16. #5545C2semidirect
17. C4×#237C2semidirect
19. C2×C3×#41C22semidirect
21. C2×#237C22semidirect
25. C2×#235C22semidirect
27. C3×#127C22semidirect
28. C3×#121C22semidirect
29. C3×#121C22semidirect
30. C3×C4×Q8C22semidirect
31. C3×C4×D4C22semidirect
32. C3×C4×D4C22semidirect
33. #947C22semidirect
35. #944C22semidirect
39. #940C22semidirect
40. #939C22semidirect
41. #939C22semidirect
42. #939C22semidirect
43. #939C22semidirect
44. #938C22semidirect
47. #936C22semidirect
48. #936C22semidirect
49. #935C22semidirect
50. D4×Dic3C22semidirect
51. D4×Dic3C22semidirect
52. D4×Dic3C22semidirect
53. #930C22semidirect
54. #929C22semidirect
56. C4×#67C22semidirect
57. C4×#67C22semidirect
59. C2×C4×Dic3C22semidirect
62. #894C22semidirect
63. #893C22semidirect
64. #893C22semidirect
66. #889C22semidirect
67. #888C22semidirect
68. #887C22semidirect
69. #885C22semidirect
70. #884C22semidirect
71. #883C22semidirect
73. #67⋊C4C22semidirect
75. #877C22semidirect
76. #875C22semidirect
78. C4×D12C22semidirect
81. C4×Dic6C22semidirect
82. C4×Dic6C22semidirect
83. C4×Dic6C22semidirect
89. #550D3semidirect
176. C2×#41C2×D3semidirect
188. #127C2×D3semidirect
209. #121C2×D3semidirect
210. #121C2×D3semidirect
217. C4×Q8C2×D3semidirect
229. C4×D4C2×D3semidirect
230. C4×D4C2×D3semidirect
283. C4×D3#41semidirect
469. C3#3508semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
#58322
#58292
#57182
#57152
D3×#1272
#57132
#56152
#56122
#56102
#55962
#55852
#55782
#55592
#55532
#55452
C4×#2372
#35083