Finite Group Databasebeta

← #86990

Subgroups — #86990

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×#556C211yesmaximal, characteristic, complemented
2#5728C211yesmaximal, characteristic, complemented
3#5726C211yesmaximal, characteristic, complemented
4#5707C214yesmaximal, complemented
5#5705C214yesmaximal, complemented
6#5682C212yesmaximal, complemented
7#5676C212yesmaximal, complemented
8#5633C212yesmaximal, complemented
9D3×C22⋊Q8C212yesmaximal, complemented
10#5580C211yesmaximal, characteristic, complemented
11#5579C212yesmaximal, complemented
12Q8×D12C211yesmaximal, characteristic, complemented
13#5574C212yesmaximal, complemented
14#5551C211yesmaximal, characteristic, complemented
15C4×D3×D4C211yesmaximal, characteristic, complemented
16#5529C212yesmaximal, complemented
17#5524C212yesmaximal, complemented
18#3520—31nomaximal, Sylow
19C2×#239C2212yes—
20C2×D3×Q8C2212yescomplemented
21C2×D3×D4C2211yescharacteristic
22C2×#235C2212yescomplemented
23C22×C4×D3C2212yes—
24C3×C4⋊Q8C2211yescharacteristic, complemented
25C3×#126C2214yescomplemented
26C3×#124C2212yes—
27C3×C22⋊Q8C2212yescomplemented
28C3×C4×Q8C2212yescomplemented
29C3×C4×Q8C2211yescharacteristic, complemented
30C3×C4×D4C2211yescharacteristic
31C3×#117C2212yescomplemented
32#947C2211yescharacteristic, complemented
33#946C2212yes—
34#946C2211yescharacteristic
35#946C2212yescomplemented
36Q8×Dic3C2212yescomplemented
37#938C2212yes—
38D4×Dic3C2211yescharacteristic
39C4×#67C2212yes—
40#924C2212yescomplemented
41#898C2212yescomplemented
42#897C2212yes—
43#897C2214yescomplemented
44#897C2212yescomplemented
45#896C2214yes—
46#896C2214yescomplemented
47#895C2211yescharacteristic
48#895C2212yescomplemented
49#894C2212yes—
50#894C2214yescomplemented
51#893C2211yescharacteristic
52#893C2214yescomplemented
53#892C2212yes—
54D3×C4⋊C4C2211yescharacteristic
55D3×C4⋊C4C2214yescomplemented
56D3×C4⋊C4C2212yescomplemented
57D3×C4⋊C4C2212yescomplemented
58#890C2212yescomplemented
59#890C2211yescharacteristic, complemented
60#887C2212yescomplemented
61#886C2214yescomplemented
62#885C2214yescomplemented
63#884C2214yes—
64#883C2214yes—
65#883C2214yescomplemented
66#882C2214yes—
67#67⋊C4C2212yes—
68D3×C22⋊C4C2214yes—
69D3×C22⋊C4C2212yes—
70#879C2214yescomplemented
71#878C2214yescomplemented
72C4×D12C2212yes—
73C4×D12C2212yes—
74C4×D12C2214yes—
75C4×D12C2211yescharacteristic
76C4×D12C2214yescomplemented
77C4×D12C2211yescharacteristic, complemented
78#872C2212yescomplemented
79#872C2214yescomplemented
80C42×D3C2211yescharacteristic
81#870C2214yescomplemented
82#869C2212yescomplemented
83#869C2211yescharacteristic, complemented
84#556D311yescharacteristic, complemented
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#533—31no—
96#529—31no—
97#525—32no—
98C2×C22⋊Q8—32no—
99#520—32no—
100C2×C4×D4—31no—
101C23×D3C2312yes—
102C2×C3×Q8C2312yes—
103C2×C3×Q8C2311yescharacteristic
104C2×C3×D4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C2×#67C2312yes—
107C2×#67C2314yes—
108C22×Dic3C2312yes—
109#239—28no—
110D3×Q8—44no—
111D3×D4—24no—
112#235—44no—
113C2×D12C2312yes—
114C2×D12C2314yes—
115C2×D12C2311yescharacteristic
116C2×D12C2311yescharacteristic
117C2×C4×D3—44no—
118C2×C4×D3C2311yescharacteristic
119C2×C4×D3C2311yescharacteristic
120C2×C4×D3C2311yescharacteristic
121C2×C4×D3C2312yes—
122C2×C4×D3C2314yes—
123C2×C4×D3C2312yes—
124C2×C4×D3C2312yes—
125C2×C4×D3C2314yes—
126C2×C4×D3C2314yes—
127C2×C4×D3C2312yes—
128C2×Dic6C2312yes—
129C2×Dic6C2314yes—
130C3×C4⋊C4C2312yes—
131C3×C4⋊C4C2312yes—
132C3×C4⋊C4C2311yescharacteristic
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2314yes—
135C3×C4⋊C4C2312yes—
136C3×C22⋊C4C2314yes—
137C3×C22⋊C4C2314yes—
138C3×C22⋊C4C2312yes—
139C3×C42C2311yescharacteristic
140C3×C42C2312yes—
141C3×C42C2314yes—
142#217C2314yes—
143#217C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2312yes—
154C4⋊Dic3C2311yescharacteristic
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2314yes—
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2312yes—
159C4⋊Dic3C2314yes—
160C4⋊Dic3C2311yescharacteristic
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2312yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2314yes—
171C2×#41—32no—
172C2×#41—32no—
173C22×Q8—32no—
174C22×D4—31no—
175C23×C4—32no—
176C4⋊Q8—31no—
177C4⋊Q8—32no—
178C4⋊Q8C2×D311yescharacteristic, complemented
179#126—32no—
180#126—34no—
181#126C2×D314yescomplemented
182#125—31no—
183#125—32no—
184#125—34no—
185#124—31no—
186#124—34no—
187#124C2×D312yes—
188#123—32no—
189#123—34no—
190#123—34no—
191#123—34no—
192#123—34no—
193C22⋊Q8—31no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198C22⋊Q8—34no—
199C22⋊Q8—32no—
200C22⋊Q8—34no—
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203C22⋊Q8C2×D312yescomplemented
204#121—32no—
205#121—31no—
206#121—32no—
207#121—34no—
208#120—34no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8C2×D312yescomplemented
212C4×Q8C2×D311yescharacteristic, complemented
213C4×D4—31no—
214C4×D4—32no—
215C4×D4—31no—
216C4×D4—34no—
217C4×D4—32no—
218C4×D4—31no—
219C4×D4—34no—
220C4×D4—32no—
221C4×D4—34no—
222C4×D4—32no—
223C4×D4—31no—
224C4×D4C2×D311yescharacteristic
225#117—32no—
226#117—32no—
227#117—34no—
228#117C2×D312yescomplemented
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×C42—31no—
236C23×C3C2412yes—
237C22×D3—24no—
238C22×D3—24no—
239C22×D3—42no—
240C22×D3C2412yes—
241C22×D3C2412yes—
242C22×D3C2411yescharacteristic
243C3×Q8—24no—
244C3×Q8—22no—
245C3×D4—22no—
246C2×C3×C4—24no—
247C2×C3×C4C2414yes—
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2412yes—
250C2×C3×C4C2412yes—
251C2×C3×C4C2412yes—
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254#67—24no—
255#67—44no—
256C2×Dic3—42no—
257C2×Dic3C2411yescharacteristic
258C2×Dic3C2411yescharacteristic
259C2×Dic3C2411yescharacteristic
260C2×Dic3C2414yes—
261C2×Dic3C2412yes—
262C2×Dic3C2412yes—
263D12—28no—
264D12—24no—
265D12—22no—
266D12#4114yescomplemented
267C4×D3—28no—
268C4×D3—22no—
269C4×D3—22no—
270C4×D3—24no—
271C4×D3—22no—
272C4×D3—24no—
273C4×D3—42no—
274C4×D3—42no—
275C4×D3—44no—
276C4×D3—44no—
277Dic6—24no—
278Dic6—44no—
279C24—32no—
280#41—68no—
281#41—124no—
282C2×Q8—32no—
283C2×Q8—34no—
284C2×Q8—124no—
285C2×Q8C22×D312yes—
286C2×Q8C22×D311yescharacteristic
287C2×D4—32no—
288C2×D4—34no—
289C2×D4—31no—
290C2×D4—32no—
291C2×D4—34no—
292C2×D4—31no—
293C2×D4—64no—
294C2×D4C22×D311yescharacteristic
295C22×C4—32no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—34no—
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—34no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—34no—
306C22×C4—124no—
307C22×C4C22×D312yes—
308C4⋊C4—32no—
309C4⋊C4—34no—
310C4⋊C4—31no—
311C4⋊C4—32no—
312C4⋊C4—31no—
313C4⋊C4—32no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4—34no—
319C4⋊C4—32no—
320C4⋊C4—34no—
321C4⋊C4C22×D311yescharacteristic
322C4⋊C4C22×D312yes—
323C4⋊C4C22×D314yes—
324C4⋊C4C22×D312yes—
325C4⋊C4C22×D311yescharacteristic
326C4⋊C4C22×D312yes—
327C22⋊C4—32no—
328C22⋊C4—34no—
329C22⋊C4—32no—
330C22⋊C4—32no—
331C22⋊C4—34no—
332C22⋊C4—32no—
333C22⋊C4—34no—
334C22⋊C4—32no—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—34no—
338C22⋊C4—32no—
339C22⋊C4C22×D312yes—
340C22⋊C4C22×D314yes—
341C22⋊C4C22×D314yes—
342C42—31no—
343C42—31no—
344C42—31no—
345C42—34no—
346C42C22×D311yescharacteristic
347C42C22×D314yes—
348C42C22×D312yes—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3C2511yescharacteristic
353C2×D3—21no—
354C2×D3—22no—
355C2×D3—21no—
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—22no—
360C2×D3—48no—
361C2×D3C2×#4114yes—
362C2×D32−1+412yes—
363C3×C4—24no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—22no—
368C3×C4C2×#4112yes—
369C3×C4H(5,𝔽2)12yes—
370Dic3—22no—
371Dic3—21no—
372Dic3—24no—
373Dic3—22no—
374Dic3—21no—
375Dic3—21no—
376C23—31no—
377C23—32no—
378C23—32no—
379C23—64no—
380C23—64no—
381C23—122no—
382C23C23×D312yes—
383Q8—24no—
384Q8—22no—
385Q8—64no—
386Q8—124no—
387D4—22no—
388D4—34no—
389D4—68no—
390D4—64no—
391D4—64no—
392D4—62no—
393D4—124no—
394C2×C4—24no—
395C2×C4—31no—
396C2×C4—31no—
397C2×C4—31no—
398C2×C4—34no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—68no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—64no—
405C2×C4—62no—
406C2×C4—64no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—124no—
410C2×C4—124no—
411C2×C4—122no—
412C2×C4C23×D311yescharacteristic
413C2×C4C23×D314yes—
414C2×C4C23×D312yes—
415C2×C4C23×D312yes—
416C2×C4C23×D312yes—
417C2×C4C23×D311yescharacteristic
418C2×C4C23×D311yescharacteristic
419C2×C3—42no—
420C2×C3C22×#4111yescharacteristic
421C2×C3C2×H(5,𝔽2)11yescharacteristic
422C2×C3C2×2−1+411yescharacteristic
423D3—22no—
424D3—24no—
425D3—42no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—34no—
430C22—32no—
431C22—61no—
432C22—62no—
433C22—61no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—62no—
438C22—128no—
439C22C24×D311yescharacteristic
440C4—24no—
441C4—22no—
442C4—22no—
443C4—21no—
444C4—22no—
445C4—62no—
446C4—61no—
447C4—61no—
448C4—62no—
449C4—64no—
450C4—61no—
451C4D3×#4112yes—
452C4#100912yes—
453C3#352011yescharacteristic, complemented, Sylow
454C2—42no—
455C2—62no—
456C2—64no—
457C2—122no—
458C2C2×D3×#4111yescharacteristic
459C2C2×#100911yescharacteristic
460C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
hypercenter#556D3
hypocenterC3#3520
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C3×#556C2semidirect
2. #5728C2semidirect
3. #5726C2semidirect
4. #5707C2semidirect
5. #5705C2semidirect
6. #5682C2semidirect
7. #5676C2semidirect
8. #5633C2semidirect
9. D3×C22⋊Q8C2semidirect
10. #5580C2semidirect
11. #5579C2semidirect
12. Q8×D12C2semidirect
13. #5574C2semidirect
14. #5551C2semidirect
15. C4×D3×D4C2semidirect
16. #5529C2semidirect
17. #5524C2semidirect
20. C2×D3×Q8C22semidirect
22. C2×#235C22semidirect
24. C3×C4⋊Q8C22semidirect
25. C3×#126C22semidirect
27. C3×C22⋊Q8C22semidirect
28. C3×C4×Q8C22semidirect
29. C3×C4×Q8C22semidirect
31. C3×#117C22semidirect
32. #947C22semidirect
35. #946C22semidirect
36. Q8×Dic3C22semidirect
40. #924C22semidirect
41. #898C22semidirect
43. #897C22semidirect
44. #897C22semidirect
46. #896C22semidirect
48. #895C22semidirect
50. #894C22semidirect
52. #893C22semidirect
55. D3×C4⋊C4C22semidirect
56. D3×C4⋊C4C22semidirect
57. D3×C4⋊C4C22semidirect
58. #890C22semidirect
59. #890C22semidirect
60. #887C22semidirect
61. #886C22semidirect
62. #885C22semidirect
65. #883C22semidirect
70. #879C22semidirect
71. #878C22semidirect
76. C4×D12C22semidirect
77. C4×D12C22semidirect
78. #872C22semidirect
79. #872C22semidirect
81. #870C22semidirect
82. #869C22semidirect
83. #869C22semidirect
84. #556D3semidirect
178. C4⋊Q8C2×D3semidirect
181. #126C2×D3semidirect
203. C22⋊Q8C2×D3semidirect
211. C4×Q8C2×D3semidirect
212. C4×Q8C2×D3semidirect
228. #117C2×D3semidirect
266. 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×#5562
#57282
#57262
#57072
#57052
#56822
#56762
#56332
D3×C22⋊Q82
#55802
#55792
Q8×D122
#55742
#55512
C4×D3×D42
#55292
#55242
#35203