Finite Group Databasebeta

← #86285

Subgroups — #86285

Order 384.

1124 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#533C211yesmaximal, characteristic, complemented
2#5732C211yesmaximal, characteristic, complemented
3#5727C211yesmaximal, characteristic, complemented
4#5726C211yesmaximal, characteristic, complemented
5#5726C211yesmaximal, characteristic, complemented
6#5692C214yesmaximal, complemented
7#5635C212yesmaximal, complemented
8#5630C212yesmaximal, complemented
9#5627C212yesmaximal, complemented
10D3×C22⋊Q8C212yesmaximal, complemented
11#5582C212yesmaximal, complemented
12#5581C212yesmaximal, complemented
13#5576C212yesmaximal, complemented
14#5574C212yesmaximal, complemented
15#5527C214yesmaximal, complemented
16#5479C211yesmaximal, characteristic, complemented
17C2×C4×D12C211yesmaximal, characteristic, complemented
18#3520—31nomaximal, Sylow
19C2×#239C2212yes—
20C2×#239C2212yescomplemented
21C2×D3×Q8C2212yescomplemented
22C22×D12C2211yescharacteristic
23C22×C4×D3C2212yes—
24C3×C4⋊Q8C2211yescharacteristic, complemented
25C3×C4⋊Q8C2211yescharacteristic, complemented
26C3×#125C2212yescomplemented
27C3×C22⋊Q8C2212yes—
28C3×C22⋊Q8C2212yes—
29C3×C4×Q8C2212yescomplemented
30C3×C4×Q8C2212yescomplemented
31C3×#117C2212yes—
32C2×C3×C42C2211yescharacteristic
33#947C2212yes—
34#946C2212yescomplemented
35#946C2212yescomplemented
36#946C2212yescomplemented
37Q8×Dic3C2212yescomplemented
38#930C2211yescharacteristic
39#929C2212yes—
40C4×#67C2212yes—
41C2×(C2×D3)⋊C4C2212yes—
42C2×C4⋊Dic3C2211yescharacteristic
43#924C2211yescharacteristic
44#898C2214yescomplemented
45#897C2214yescomplemented
46#897C2212yescomplemented
47#896C2214yescomplemented
48#896C2214yescomplemented
49#895C2214yes—
50#895C2212yes—
51#895C2212yescomplemented
52#894C2214yes—
53#894C2214yescomplemented
54#894C2214yescomplemented
55#894C2214yescomplemented
56#893C2212yes—
57#893C2214yescomplemented
58#892C2212yescomplemented
59D3×C4⋊C4C2212yescomplemented
60D3×C4⋊C4C2214yescomplemented
61D3×C4⋊C4C2212yescomplemented
62#889C2214yescomplemented
63#888C2212yescomplemented
64#882C2214yes—
65#67⋊C4C2214yes—
66D3×C22⋊C4C2214yes—
67#878C2214yes—
68#876C2212yescomplemented
69#876C2214yescomplemented
70#875C2211yescharacteristic, complemented
71#875C2214yescomplemented
72C4×D12C2212yes—
73C4×D12C2211yescharacteristic
74C4×D12C2211yescharacteristic
75C4×D12C2211yescharacteristic
76C4×D12C2212yes—
77C4×D12C2211yescharacteristic
78C4×D12C2211yescharacteristic, complemented
79C4×D12C2214yescomplemented
80#872C2212yescomplemented
81#872C2214yescomplemented
82#870C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#556—31no—
85#554—32no—
86#552—32no—
87#551—34no—
88D4×Q8—31no—
89#548—32no—
90#547—31no—
91#547—31no—
92#546—32no—
93#542—32no—
94#539—34no—
95#533D311yescharacteristic, complemented
96#529—31no—
97#525—32no—
98C2×C22⋊Q8—32no—
99#520—32no—
100C2×C4×D4—31no—
101C23×D3C2312yes—
102C2×C3×Q8C2312yes—
103C2×C3×Q8C2312yes—
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C2×#67C2312yes—
108C22×Dic3C2312yes—
109#239—28no—
110#239—44no—
111D3×Q8—44no—
112C2×D12—24no—
113C2×D12C2312yes—
114C2×D12C2311yescharacteristic
115C2×D12C2311yescharacteristic
116C2×D12C2314yes—
117C2×D12C2314yes—
118C2×D12C2311yescharacteristic
119C2×C4×D3—44no—
120C2×C4×D3C2312yes—
121C2×C4×D3C2314yes—
122C2×C4×D3C2312yes—
123C2×C4×D3C2312yes—
124C2×C4×D3C2312yes—
125C2×C4×D3C2312yes—
126C2×C4×D3C2314yes—
127C2×C4×D3C2314yes—
128C2×Dic6C2311yescharacteristic
129C2×Dic6C2314yes—
130C3×C4⋊C4C2312yes—
131C3×C4⋊C4C2314yes—
132C3×C4⋊C4C2312yes—
133C3×C4⋊C4C2314yes—
134C3×C4⋊C4C2312yes—
135C3×C4⋊C4C2312yes—
136C3×C22⋊C4C2314yes—
137C3×C42C2311yescharacteristic
138C3×C42C2311yescharacteristic
139C3×C42C2311yescharacteristic
140C3×C42C2314yes—
141C3×C42C2311yescharacteristic
142#217C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2314yes—
154(C2×D3)⋊C4C2314yes—
155(C2×D3)⋊C4C2314yes—
156C4⋊Dic3C2311yescharacteristic
157C4⋊Dic3C2311yescharacteristic
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2312yes—
160C4⋊Dic3C2311yescharacteristic
161C4⋊Dic3C2314yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2312yes—
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2314yes—
167Dic3⋊C4C2314yes—
168Dic3⋊C4C2314yes—
169C4×Dic3C2312yes—
170C4×Dic3C2314yes—
171C2×#41—32no—
172C2×#41—32no—
173C22×Q8—32no—
174C22×D4—31no—
175C23×C4—32no—
176C4⋊Q8—32no—
177C4⋊Q8C2×D311yescharacteristic, complemented
178C4⋊Q8C2×D311yescharacteristic, complemented
179#126—34no—
180#126—32no—
181#126—34no—
182#125—34no—
183#125—31no—
184#125C2×D312yescomplemented
185#124—32no—
186#124—31no—
187#124—34no—
188#123—32no—
189#123—34no—
190#123—34no—
191#123—34no—
192#123—34no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—31no—
197C22⋊Q8—34no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201C22⋊Q8—34no—
202C22⋊Q8C2×D312yes—
203C22⋊Q8C2×D312yes—
204#121—31no—
205#121—32no—
206#121—34no—
207#121—32no—
208#120—34no—
209C4×Q8—32no—
210C4×Q8—31no—
211C4×Q8C2×D312yescomplemented
212C4×Q8C2×D312yescomplemented
213C4×D4—32no—
214C4×D4—34no—
215C4×D4—32no—
216C4×D4—34no—
217C4×D4—31no—
218C4×D4—32no—
219C4×D4—31no—
220C4×D4—31no—
221C4×D4—31no—
222C4×D4—32no—
223C4×D4—34no—
224C4×D4—31no—
225#117—32no—
226#117—32no—
227#117—34no—
228#117C2×D312yes—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C22⋊C4—32no—
234C2×C22⋊C4—34no—
235C2×C42C2×D311yescharacteristic
236C23×C3C2411yescharacteristic
237C22×D3—24no—
238C22×D3—24no—
239C22×D3—42no—
240C22×D3C2412yes—
241C22×D3C2412yes—
242C22×D3C2412yes—
243C3×Q8—24no—
244C3×Q8—24no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4C2414yes—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2414yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256#67—24no—
257C2×Dic3—42no—
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260C2×Dic3C2412yes—
261C2×Dic3C2412yes—
262C2×Dic3C2412yes—
263D12—24no—
264D12—28no—
265D12—22no—
266D12—22no—
267D12—44no—
268D12#4114yescomplemented
269C4×D3—28no—
270C4×D3—24no—
271C4×D3—24no—
272C4×D3—24no—
273C4×D3—42no—
274C4×D3—42no—
275C4×D3—44no—
276C4×D3—44no—
277Dic6—22no—
278Dic6—44no—
279C24—32no—
280#41—68no—
281#41—124no—
282C2×Q8—31no—
283C2×Q8—34no—
284C2×Q8—124no—
285C2×Q8C22×D312yes—
286C2×Q8C22×D312yes—
287C2×D4—32no—
288C2×D4—31no—
289C2×D4—34no—
290C2×D4—32no—
291C2×D4—31no—
292C2×D4—31no—
293C2×D4—34no—
294C2×D4—64no—
295C22×C4—32no—
296C22×C4—34no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—34no—
301C22×C4—32no—
302C22×C4—34no—
303C22×C4—32no—
304C22×C4—124no—
305C22×C4C22×D311yescharacteristic
306C22×C4C22×D311yescharacteristic
307C22×C4C22×D311yescharacteristic
308C4⋊C4—31no—
309C4⋊C4—32no—
310C4⋊C4—34no—
311C4⋊C4—31no—
312C4⋊C4—32no—
313C4⋊C4—31no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4—31no—
318C4⋊C4—34no—
319C4⋊C4—32no—
320C4⋊C4—32no—
321C4⋊C4C22×D312yes—
322C4⋊C4C22×D312yes—
323C4⋊C4C22×D314yes—
324C4⋊C4C22×D312yes—
325C4⋊C4C22×D312yes—
326C4⋊C4C22×D314yes—
327C22⋊C4—32no—
328C22⋊C4—32no—
329C22⋊C4—32no—
330C22⋊C4—32no—
331C22⋊C4—32no—
332C22⋊C4—34no—
333C22⋊C4—34no—
334C22⋊C4—34no—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—34no—
338C22⋊C4—32no—
339C22⋊C4—34no—
340C22⋊C4—32no—
341C22⋊C4C22×D314yes—
342C42—34no—
343C42—32no—
344C42C22×D311yescharacteristic
345C42C22×D311yescharacteristic
346C42C22×D311yescharacteristic
347C42C22×D311yescharacteristic
348C42C22×D314yes—
349C22×C3—21no—
350C22×C3—21no—
351C22×C32−1+412yes—
352C22×C3C2511yescharacteristic
353C2×D3—22no—
354C2×D3—22no—
355C2×D3—22no—
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—22no—
360C2×D3—22no—
361C2×D3—48no—
362C2×D3C2×#4114yes—
363C3×C4—24no—
364C3×C4—21no—
365C3×C4—24no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4C2×#4112yes—
370C3×C4H(5,𝔽2)12yes—
371Dic3—22no—
372Dic3—22no—
373Dic3—22no—
374Dic3—22no—
375Dic3—22no—
376C23—32no—
377C23—32no—
378C23—32no—
379C23—64no—
380C23—64no—
381C23—122no—
382C23C23×D311yescharacteristic
383Q8—24no—
384Q8—24no—
385Q8—62no—
386Q8—124no—
387D4—34no—
388D4—64no—
389D4—68no—
390D4—64no—
391D4—62no—
392D4—62no—
393D4—124no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—32no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—68no—
403C2×C4—64no—
404C2×C4—64no—
405C2×C4—64no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—124no—
409C2×C4—124no—
410C2×C4—122no—
411C2×C4C23×D314yes—
412C2×C4C23×D311yescharacteristic
413C2×C4C23×D311yescharacteristic
414C2×C4C23×D311yescharacteristic
415C2×C4C23×D311yescharacteristic
416C2×C4C23×D314yes—
417C2×C4C23×D311yescharacteristic
418C2×C4C23×D311yescharacteristic
419C2×C3—22no—
420C2×C3C22×#4111yescharacteristic
421C2×C3C2×H(5,𝔽2)11yescharacteristic
422C2×C3C2×2−1+411yescharacteristic
423D3—24no—
424D3—42no—
425D3—42no—
426C22—21no—
427C22—21no—
428C22—34no—
429C22—62no—
430C22—62no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—128no—
438C22#100712yes—
439C22C24×D311yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—24no—
444C4—21no—
445C4—24no—
446C4—62no—
447C4—62no—
448C4—62no—
449C4—62no—
450C4—62no—
451C4D3×#4112yes—
452C4#100912yes—
453C3#352011yescharacteristic, complemented, Sylow
454C2—22no—
455C2—64no—
456C2—122no—
457C2—122no—
458C2C2×#100711yescharacteristic
459C2C2×D3×#4111yescharacteristic
460C2C2×#100911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#533C2
hypercenter#533D3
hypocenterC3#3520
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
2. #5732C2semidirect
3. #5727C2semidirect
4. #5726C2semidirect
5. #5726C2semidirect
6. #5692C2semidirect
7. #5635C2semidirect
8. #5630C2semidirect
9. #5627C2semidirect
10. D3×C22⋊Q8C2semidirect
11. #5582C2semidirect
12. #5581C2semidirect
13. #5576C2semidirect
14. #5574C2semidirect
15. #5527C2semidirect
16. #5479C2semidirect
17. C2×C4×D12C2semidirect
20. C2×#239C22semidirect
21. C2×D3×Q8C22semidirect
24. C3×C4⋊Q8C22semidirect
25. C3×C4⋊Q8C22semidirect
26. C3×#125C22semidirect
29. C3×C4×Q8C22semidirect
30. C3×C4×Q8C22semidirect
34. #946C22semidirect
35. #946C22semidirect
36. #946C22semidirect
37. Q8×Dic3C22semidirect
44. #898C22semidirect
45. #897C22semidirect
46. #897C22semidirect
47. #896C22semidirect
48. #896C22semidirect
51. #895C22semidirect
53. #894C22semidirect
54. #894C22semidirect
55. #894C22semidirect
57. #893C22semidirect
58. #892C22semidirect
59. D3×C4⋊C4C22semidirect
60. D3×C4⋊C4C22semidirect
61. D3×C4⋊C4C22semidirect
62. #889C22semidirect
63. #888C22semidirect
68. #876C22semidirect
69. #876C22semidirect
70. #875C22semidirect
71. #875C22semidirect
78. C4×D12C22semidirect
79. C4×D12C22semidirect
80. #872C22semidirect
81. #872C22semidirect
82. #870C22semidirect
83. C4×Dic6C22semidirect
95. #533D3semidirect
177. C4⋊Q8C2×D3semidirect
178. C4⋊Q8C2×D3semidirect
184. #125C2×D3semidirect
211. C4×Q8C2×D3semidirect
212. C4×Q8C2×D3semidirect
268. D12#41semidirect
453. C3#3520semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#57322
#57272
#57262
#56922
#56352
#56302
#56272
D3×C22⋊Q82
#55822
#55812
#55762
#55742
#55272
#54792
C2×C4×D122
#35203