Finite Group Databasebeta

← #86027

Subgroups — #86027

Order 384.

1142 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#525C211yesmaximal, characteristic, complemented
2#5830C211yesmaximal, characteristic, complemented
3#5827C211yesmaximal, characteristic, complemented
4C2×#930C211yesmaximal, characteristic, complemented
5C2×#926C211yesmaximal, characteristic, complemented
6#5658C214yesmaximal, complemented
7#5643C212yesmaximal, complemented
8#5624C212yesmaximal, complemented
9#5619C212yesmaximal, complemented
10#5596C212yesmaximal, complemented
11#5591C214yesmaximal, complemented
12#5565C212yesmaximal, complemented
13#5565C212yesmaximal, complemented
14#5563C212yesmaximal, complemented
15#5555C212yesmaximal, complemented
16#5521C211yesmaximal, characteristic, complemented
17#5519C211yesmaximal, characteristic, complemented
18#3494—31nomaximal, Sylow
19C2×C3×#41C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C22×#67C2212yes—
22C2×#237C2214yescomplemented
23C22×D12C2211yescharacteristic
24C3×#123C2212yes—
25C3×C22⋊Q8C2211yescharacteristic, complemented
26C3×C22⋊Q8C2211yescharacteristic, complemented
27C3×#121C2211yescharacteristic, complemented
28C3×#121C2211yescharacteristic, complemented
29C3×#120C2212yes—
30C3×C4×D4C2212yescomplemented
31C3×C4×D4C2212yescomplemented
32C3×#117C2211yescharacteristic
33C2×#217C2212yes—
34#947C2212yescomplemented
35Q8×Dic3C2212yescomplemented
36#940C2212yescomplemented
37#939C2214yescomplemented
38#939C2214yescomplemented
39#938C2214yescomplemented
40#937C2214yes—
41D4×Dic3C2214yes—
42D4×Dic3C2212yescomplemented
43#930C2212yes—
44#930C2212yes—
45#930C2212yes—
46#930C2212yes—
47#930C2212yescomplemented
48#930C2212yescomplemented
49C2×(C2×D3)⋊C4C2212yes—
50#926C2212yes—
51#926C2211yescharacteristic
52#926C2211yescharacteristic
53#926C2212yescomplemented
54#926C2211yescharacteristic, complemented
55#926C2211yescharacteristic, complemented
56#926C2212yescomplemented
57C2×C4⋊Dic3C2211yescharacteristic
58C2×C4⋊Dic3C2211yescharacteristic, complemented
59C2×C4⋊Dic3C2211yescharacteristic, complemented
60C2×C4⋊Dic3C2212yescomplemented
61C2×C4×Dic3C2212yescomplemented
62#898C2214yescomplemented
63#898C2214yescomplemented
64#897C2214yescomplemented
65#893C2212yescomplemented
66#893C2212yescomplemented
67#892C2214yescomplemented
68#890C2212yescomplemented
69#888C2212yescomplemented
70#886C2214yescomplemented
71#886C2214yescomplemented
72#885C2214yescomplemented
73#885C2214yescomplemented
74#883C2214yescomplemented
75#883C2214yescomplemented
76#67⋊C4C2214yescomplemented
77#67⋊C4C2214yescomplemented
78D3×C22⋊C4C2214yes—
79#878C2214yes—
80C4×D12C2212yescomplemented
81C4×D12C2212yescomplemented
82#870C2212yescomplemented
83#869C2212yescomplemented
84#555—32no—
85#553—32no—
86#553—32no—
87#551—34no—
88#547—32no—
89#546—34no—
90#544—31no—
91#542—32no—
92#540—32no—
93#535—31no—
94#532—32no—
95#529—32no—
96#525D311yescharacteristic, complemented
97C2×#121—31no—
98#520—31no—
99#518—31no—
100C2×#117—31no—
101C24×C3C2311yescharacteristic
102C23×D3C2312yes—
103C3×#41—24no—
104C2×C3×Q8C2311yescharacteristic
105C2×C3×D4C2312yes—
106C2×C3×D4C2311yescharacteristic
107C2×C3×D4C2312yes—
108C2×C3×D4C2312yes—
109C22×C3×C4—24no—
110C22×C3×C4C2311yescharacteristic
111C22×C3×C4C2311yescharacteristic
112C22×C3×C4C2312yes—
113C22×C3×C4C2311yescharacteristic
114C22×C3×C4C2311yescharacteristic
115C22×C3×C4C2311yescharacteristic
116C22×C3×C4C2311yescharacteristic
117C2×#67—44no—
118C2×#67C2314yes—
119C2×#67C2314yes—
120C2×#67C2314yes—
121C2×#67C2314yes—
122C22×Dic3C2312yes—
123C22×Dic3C2312yes—
124C22×Dic3C2312yes—
125C22×Dic3C2314yes—
126#237—48no—
127C2×D12—24no—
128C2×D12C2312yes—
129C2×D12C2312yes—
130C2×C4×D3C2314yes—
131C2×C4×D3C2314yes—
132C2×Dic6C2314yes—
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2311yescharacteristic
135C3×C4⋊C4C2312yes—
136C3×C4⋊C4C2312yes—
137C3×C22⋊C4C2312yes—
138C3×C22⋊C4C2312yes—
139C3×C22⋊C4C2312yes—
140C3×C22⋊C4C2312yes—
141C3×C22⋊C4C2312yes—
142C3×C42C2312yes—
143#217C2312yes—
144#217C2312yes—
145#217C2312yes—
146#217C2314yes—
147#217C2312yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2314yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2314yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2314yes—
154C4⋊Dic3C2311yescharacteristic
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2314yes—
157C4⋊Dic3C2311yescharacteristic
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2311yescharacteristic
160C4⋊Dic3C2312yes—
161C4⋊Dic3C2314yes—
162C4⋊Dic3C2312yes—
163C4⋊Dic3C2312yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166C4×Dic3C2312yes—
167C4×Dic3C2314yes—
168C4×Dic3C2312yes—
169C4×Dic3C2312yes—
170C4×Dic3C2312yes—
171C2×#41—34no—
172C2×#41C2×D311yescharacteristic
173C22×D4—32no—
174C22×D4—31no—
175C23×C4C2×D311yescharacteristic
176C4⋊Q8—32no—
177C4⋊Q8—32no—
178#127—32no—
179#126—34no—
180#126—34no—
181#125—32no—
182#125—32no—
183#124—32no—
184#124—34no—
185#124—34no—
186#123—34no—
187#123—34no—
188#123—34no—
189#123—34no—
190#123C2×D312yes—
191C22⋊Q8—34no—
192C22⋊Q8—34no—
193C22⋊Q8C2×D311yescharacteristic, complemented
194C22⋊Q8C2×D311yescharacteristic, complemented
195#121—34no—
196#121—34no—
197#121—34no—
198#121—32no—
199#121—32no—
200#121—32no—
201#121—32no—
202#121—32no—
203#121—32no—
204#121C2×D311yescharacteristic, complemented
205#121C2×D311yescharacteristic, complemented
206#120—34no—
207#120C2×D312yes—
208C4×Q8—32no—
209C4×D4—32no—
210C4×D4—34no—
211C4×D4—32no—
212C4×D4—32no—
213C4×D4—34no—
214C4×D4—34no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4C2×D312yescomplemented
218C4×D4C2×D312yescomplemented
219#117—31no—
220#117—31no—
221#117—32no—
222#117—32no—
223#117—31no—
224#117—31no—
225#117—32no—
226#117—34no—
227#117C2×D311yescharacteristic
228C2×C4⋊C4—31no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—32no—
232C2×C22⋊C4—32no—
233C2×C22⋊C4—32no—
234C2×C22⋊C4—34no—
235C2×C42—32no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—41no—
239C23×C3C2411yescharacteristic
240C23×C3C2412yes—
241C23×C3C2411yescharacteristic
242C23×C3C2411yescharacteristic
243C22×D3—24no—
244C22×D3—42no—
245C22×D3—42no—
246C22×D3C2414yes—
247C3×Q8—22no—
248C3×D4—24no—
249C3×D4—24no—
250C3×D4—22no—
251C3×D4—42no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—24no—
258C2×C3×C4#4114yescomplemented
259C2×C3×C4C2411yescharacteristic
260C2×C3×C4C2412yes—
261C2×C3×C4C2412yes—
262C2×C3×C4C2412yes—
263C2×C3×C4C2411yescharacteristic
264C2×C3×C4C2411yescharacteristic
265C2×C3×C4C2411yescharacteristic
266#67—28no—
267#67—44no—
268#67—44no—
269#67—44no—
270C2×Dic3—24no—
271C2×Dic3—44no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3C2412yes—
275C2×Dic3C2412yes—
276C2×Dic3C2412yes—
277C2×Dic3C2412yes—
278C2×Dic3C2414yes—
279D12—24no—
280D12—24no—
281C4×D3—44no—
282C4×D3—44no—
283Dic6—44no—
284C24—32no—
285C24C22×D311yescharacteristic
286#41—24no—
287#41—128no—
288C2×Q8—34no—
289C2×Q8C22×D311yescharacteristic
290C2×D4—34no—
291C2×D4—34no—
292C2×D4—34no—
293C2×D4—34no—
294C2×D4—32no—
295C2×D4—32no—
296C2×D4—64no—
297C2×D4—124no—
298C2×D4C22×D312yes—
299C2×D4C22×D311yescharacteristic
300C2×D4C22×D312yes—
301C2×D4C22×D312yes—
302C22×C4—24no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—32no—
306C22×C4—34no—
307C22×C4—34no—
308C22×C4—34no—
309C22×C4C22×D311yescharacteristic
310C22×C4C22×D311yescharacteristic
311C22×C4C22×D311yescharacteristic
312C22×C4C22×D312yes—
313C22×C4C22×D311yescharacteristic
314C22×C4C22×D311yescharacteristic
315C22×C4C22×D311yescharacteristic
316C4⋊C4—31no—
317C4⋊C4—32no—
318C4⋊C4—31no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4—32no—
322C4⋊C4—32no—
323C4⋊C4—32no—
324C4⋊C4—34no—
325C4⋊C4—34no—
326C4⋊C4—34no—
327C4⋊C4—34no—
328C4⋊C4C22×D312yes—
329C4⋊C4C22×D311yescharacteristic
330C4⋊C4C22×D311yescharacteristic
331C4⋊C4C22×D312yes—
332C22⋊C4—32no—
333C22⋊C4—32no—
334C22⋊C4—32no—
335C22⋊C4—32no—
336C22⋊C4—34no—
337C22⋊C4—34no—
338C22⋊C4—34no—
339C22⋊C4—34no—
340C22⋊C4—34no—
341C22⋊C4—34no—
342C22⋊C4—34no—
343C22⋊C4C22×D312yes—
344C22⋊C4C22×D312yes—
345C22⋊C4C22×D312yes—
346C22⋊C4C22×D312yes—
347C22⋊C4C22×D312yes—
348C42—32no—
349C42—32no—
350C42—32no—
351C42—32no—
352C42—34no—
353C42C22×D312yes—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—44no—
365C22×C3C2×#4112yes—
366C22×C3H(5,𝔽2)12yes—
367C22×C3C2511yescharacteristic
368C2×D3—24no—
369C2×D3—24no—
370C2×D3—24no—
371C2×D3—84no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4C2×#4112yes—
378C3×C4C2×#4112yes—
379Dic3—22no—
380Dic3—24no—
381Dic3—22no—
382Dic3—22no—
383Dic3—22no—
384C23—22no—
385C23—22no—
386C23—34no—
387C23—41no—
388C23—64no—
389C23—122no—
390C23—122no—
391C23C23×D311yescharacteristic
392C23C23×D312yes—
393C23C23×D311yescharacteristic
394C23C23×D311yescharacteristic
395Q8—22no—
396Q8—124no—
397D4—24no—
398D4—24no—
399D4—22no—
400D4—42no—
401D4—68no—
402D4—64no—
403D4—64no—
404D4—124no—
405D4—124no—
406D4—124no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—32no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—34no—
418C2×C4—64no—
419C2×C4—124no—
420C2×C4—124no—
421C2×C4—122no—
422C2×C4—124no—
423C2×C4—122no—
424C2×C4#23714yescomplemented
425C2×C4C23×D311yescharacteristic
426C2×C4C23×D312yes—
427C2×C4C23×D312yes—
428C2×C4C23×D312yes—
429C2×C4C23×D311yescharacteristic
430C2×C4C23×D311yescharacteristic
431C2×C4C23×D311yescharacteristic
432C2×C3—22no—
433C2×C3—22no—
434C2×C3—42no—
435C2×C3—41no—
436C2×C3C22×#4111yescharacteristic
437C2×C3C2×H(5,𝔽2)11yescharacteristic
438C2×C3#58511yescharacteristic
439D3—44no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—44no—
451C22—64no—
452C22—64no—
453C22—64no—
454C22—244no—
455C22C2×#23712yes—
456C22#100912yes—
457C22C24×D311yescharacteristic
458C4—21no—
459C4—21no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—62no—
464C4—62no—
465C4—64no—
466C4—62no—
467C4—62no—
468C4C2×#23712yes—
469C4C2×#23712yes—
470C3#349411yescharacteristic, complemented, Sylow
471C2—22no—
472C2—22no—
473C2—41no—
474C2—42no—
475C2—124no—
476C2C22×#23711yescharacteristic
477C2C2×#100911yescharacteristic
478C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#525C2
hypercenter#525D3
hypocenterC3#3494
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C3×#525C2semidirect
2. #5830C2semidirect
3. #5827C2semidirect
4. C2×#930C2semidirect
5. C2×#926C2semidirect
6. #5658C2semidirect
7. #5643C2semidirect
8. #5624C2semidirect
9. #5619C2semidirect
10. #5596C2semidirect
11. #5591C2semidirect
12. #5565C2semidirect
13. #5565C2semidirect
14. #5563C2semidirect
15. #5555C2semidirect
16. #5521C2semidirect
17. #5519C2semidirect
22. C2×#237C22semidirect
25. C3×C22⋊Q8C22semidirect
26. C3×C22⋊Q8C22semidirect
27. C3×#121C22semidirect
28. C3×#121C22semidirect
30. C3×C4×D4C22semidirect
31. C3×C4×D4C22semidirect
34. #947C22semidirect
35. Q8×Dic3C22semidirect
36. #940C22semidirect
37. #939C22semidirect
38. #939C22semidirect
39. #938C22semidirect
42. D4×Dic3C22semidirect
47. #930C22semidirect
48. #930C22semidirect
53. #926C22semidirect
54. #926C22semidirect
55. #926C22semidirect
56. #926C22semidirect
58. C2×C4⋊Dic3C22semidirect
59. C2×C4⋊Dic3C22semidirect
60. C2×C4⋊Dic3C22semidirect
61. C2×C4×Dic3C22semidirect
62. #898C22semidirect
63. #898C22semidirect
64. #897C22semidirect
65. #893C22semidirect
66. #893C22semidirect
67. #892C22semidirect
68. #890C22semidirect
69. #888C22semidirect
70. #886C22semidirect
71. #886C22semidirect
72. #885C22semidirect
73. #885C22semidirect
74. #883C22semidirect
75. #883C22semidirect
76. #67⋊C4C22semidirect
77. #67⋊C4C22semidirect
80. C4×D12C22semidirect
81. C4×D12C22semidirect
82. #870C22semidirect
83. #869C22semidirect
96. #525D3semidirect
193. C22⋊Q8C2×D3semidirect
194. C22⋊Q8C2×D3semidirect
204. #121C2×D3semidirect
205. #121C2×D3semidirect
217. C4×D4C2×D3semidirect
218. C4×D4C2×D3semidirect
258. C2×C3×C4#41semidirect
424. C2×C4#237semidirect
470. C3#3494semidirect

Maximal subgroups

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

subgroupindex
C3×#5252
#58302
#58272
C2×#9302
C2×#9262
#56582
#56432
#56242
#56192
#55962
#55912
#55652
#55632
#55552
#55212
#55192
#34943