Finite Group Databasebeta

← #86675

Subgroups — #86675

Order 384.

1132 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#547C211yesmaximal, characteristic, complemented
2#5829C212yesmaximal, complemented
3Dic3×#41C212yesmaximal, complemented
4#5732C211yesmaximal, characteristic, complemented
5#5728C211yesmaximal, characteristic, complemented
6#5658C214yesmaximal, complemented
7#5649C214yesmaximal, complemented
8#5639C212yesmaximal, complemented
9#5624C212yesmaximal, complemented
10#5618C212yesmaximal, complemented
11#5606C212yesmaximal, complemented
12#5563C211yesmaximal, characteristic, complemented
13C4×D3×D4C211yesmaximal, characteristic, complemented
14#5543C211yesmaximal, characteristic, complemented
15C4×#237C211yesmaximal, characteristic, complemented
16#5513C212yesmaximal, complemented
17C2×#892C212yesmaximal, complemented
18#3475—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×#237C2211yescharacteristic
21C2×D3×D4C2211yescharacteristic
22C22×C4×D3C2212yes—
23C3×C4⋊Q8C2211yescharacteristic, complemented
24C3×#123C2214yescomplemented
25C3×C22⋊Q8C2212yescomplemented
26C3×#121C2212yescomplemented
27C3×C4×D4C2211yescharacteristic
28C3×C4×D4C2211yescharacteristic, complemented
29C2×C3×C4⋊C4C2212yescomplemented
30#947C2212yes—
31#946C2212yes—
32Q8×Dic3C2212yescomplemented
33Q8×Dic3C2212yescomplemented
34#939C2214yes—
35#938C2212yescomplemented
36#937C2214yes—
37D4×Dic3C2212yes—
38D4×Dic3C2211yescharacteristic
39D4×Dic3C2214yescomplemented
40D4×Dic3C2214yescomplemented
41D4×Dic3C2211yescharacteristic, complemented
42#930C2212yes—
43#929C2214yes—
44C4×#67C2212yes—
45C4×#67C2212yes—
46C2×(C2×D3)⋊C4C2214yes—
47#926C2212yescomplemented
48#926C2214yescomplemented
49C2×C4⋊Dic3C2212yescomplemented
50#924C2212yescomplemented
51C2×C4×Dic3C2214yescomplemented
52C2×C4×Dic3C2212yescomplemented
53#898C2214yescomplemented
54#898C2214yescomplemented
55#897C2212yes—
56#897C2212yescomplemented
57#893C2211yescharacteristic
58#892C2214yes—
59#892C2212yes—
60#892C2211yescharacteristic
61#892C2214yescomplemented
62#892C2212yescomplemented
63#892C2212yescomplemented
64D3×C4⋊C4C2211yescharacteristic
65#890C2212yescomplemented
66#890C2212yescomplemented
67#889C2214yescomplemented
68#887C2212yescomplemented
69#887C2211yescharacteristic, complemented
70#886C2214yescomplemented
71#885C2214yescomplemented
72#883C2214yes—
73#67⋊C4C2212yes—
74#67⋊C4C2212yes—
75#67⋊C4C2214yescomplemented
76D3×C22⋊C4C2212yes—
77#879C2214yescomplemented
78#878C2214yescomplemented
79#877C2212yescomplemented
80C4×D12C2211yescharacteristic
81C42×D3C2211yescharacteristic
82#869C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#556—31no—
85#555—31no—
86#554—31no—
87#553—32no—
88#552—32no—
89#552—32no—
90#551—34no—
91#547D311yescharacteristic, complemented
92#546—32no—
93#533—31no—
94#529—34no—
95#529—32no—
96#525—32no—
97C4×#41—32no—
98C4×#41—31no—
99C2×C4×D4—31no—
100C2×#117—32no—
101C23×D3C2312yes—
102C3×#41—28no—
103C2×C3×Q8C2312yes—
104C2×C3×D4C2314yes—
105C2×C3×D4C2311yescharacteristic
106C2×C3×D4C2311yescharacteristic
107C22×C3×C4C2314yes—
108C22×C3×C4C2312yes—
109C22×C3×C4C2312yes—
110C2×#67C2312yes—
111C2×#67C2312yes—
112C2×#67C2314yes—
113C22×Dic3C2312yes—
114C22×Dic3C2312yes—
115C22×Dic3C2314yes—
116C22×Dic3C2312yes—
117#237—24no—
118D3×D4—24no—
119C2×D12C2311yescharacteristic
120C2×C4×D3—28no—
121C2×C4×D3C2311yescharacteristic
122C2×C4×D3C2312yes—
123C2×C4×D3C2312yes—
124C2×C4×D3C2312yes—
125C2×C4×D3C2311yescharacteristic
126C2×C4×D3C2311yescharacteristic
127C2×Dic6C2311yescharacteristic
128C2×Dic6C2312yes—
129C3×C4⋊C4C2311yescharacteristic
130C3×C4⋊C4C2312yes—
131C3×C4⋊C4C2314yes—
132C3×C4⋊C4C2311yescharacteristic
133C3×C4⋊C4C2312yes—
134C3×C22⋊C4C2312yes—
135C3×C22⋊C4C2312yes—
136C3×C22⋊C4C2314yes—
137C3×C42C2311yescharacteristic
138#217C2312yes—
139#217C2314yes—
140#217C2314yes—
141#217C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148(C2×D3)⋊C4C2314yes—
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2314yes—
154C4⋊Dic3C2314yes—
155C4⋊Dic3C2314yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2314yes—
161C4×Dic3C2312yes—
162C4×Dic3C2311yescharacteristic
163C4×Dic3C2311yescharacteristic
164C4×Dic3C2311yescharacteristic
165C4×Dic3C2312yes—
166C4×Dic3C2314yes—
167C4×Dic3C2312yes—
168C4×Dic3C2314yes—
169C4×Dic3C2314yes—
170C2×#41—31no—
171C2×#41C2×D312yes—
172C22×D4—31no—
173C23×C4—32no—
174C4⋊Q8—31no—
175C4⋊Q8C2×D311yescharacteristic, complemented
176#126—34no—
177#126—34no—
178#126—34no—
179#125—34no—
180#125—32no—
181#125—32no—
182#124—32no—
183#124—34no—
184#123—34no—
185#123—34no—
186#123—34no—
187#123C2×D314yescomplemented
188C22⋊Q8—32no—
189C22⋊Q8—32no—
190C22⋊Q8—32no—
191C22⋊Q8—32no—
192C22⋊Q8—34no—
193C22⋊Q8C2×D312yescomplemented
194#121—32no—
195#121—34no—
196#121—32no—
197#121C2×D312yescomplemented
198#120—34no—
199C4×Q8—32no—
200C4×Q8—32no—
201C4×Q8—31no—
202C4×Q8—32no—
203C4×Q8—31no—
204C4×D4—31no—
205C4×D4—34no—
206C4×D4—32no—
207C4×D4—34no—
208C4×D4—31no—
209C4×D4—32no—
210C4×D4—32no—
211C4×D4—31no—
212C4×D4—32no—
213C4×D4—32no—
214C4×D4—34no—
215C4×D4—31no—
216C4×D4C2×D311yescharacteristic
217C4×D4C2×D311yescharacteristic, complemented
218#117—34no—
219#117—32no—
220#117—31no—
221#117—32no—
222#117—34no—
223#117—34no—
224#117—32no—
225#117—32no—
226#117—32no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4C2×D312yescomplemented
230C2×C22⋊C4—34no—
231C2×C22⋊C4—32no—
232C2×C42—34no—
233C2×C42—32no—
234C2×C42—31no—
235C23×C3C2412yes—
236C23×C3C2412yes—
237C22×D3—24no—
238C22×D3—24no—
239C22×D3—42no—
240C22×D3C2412yes—
241C22×D3C2411yescharacteristic
242C3×Q8—24no—
243C3×D4—28no—
244C3×D4—22no—
245C3×D4#4114yescomplemented
246C2×C3×C4—28no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4C2412yes—
250C2×C3×C4C2414yes—
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2412yes—
253C2×C3×C4C2411yescharacteristic
254C2×C3×C4C2411yescharacteristic
255#67—24no—
256#67—24no—
257#67—44no—
258C2×Dic3—24no—
259C2×Dic3—24no—
260C2×Dic3—44no—
261C2×Dic3—42no—
262C2×Dic3C2412yes—
263C2×Dic3C2412yes—
264C2×Dic3C2411yescharacteristic
265C2×Dic3C2411yescharacteristic
266C2×Dic3C2412yes—
267C2×Dic3C2414yes—
268C2×Dic3C2411yescharacteristic
269D12—22no—
270C4×D3—24no—
271C4×D3—24no—
272C4×D3—22no—
273C4×D3—24no—
274C4×D3—41no—
275C4×D3#4114yescomplemented
276Dic6—22no—
277Dic6—42no—
278C24—32no—
279#41—28no—
280#41—64no—
281C2×Q8—31no—
282C2×Q8—32no—
283C2×Q8C22×D312yes—
284C2×D4—32no—
285C2×D4—32no—
286C2×D4—34no—
287C2×D4—31no—
288C2×D4—64no—
289C2×D4C22×D314yes—
290C2×D4C22×D311yescharacteristic
291C2×D4C22×D311yescharacteristic
292C22×C4—31no—
293C22×C4—32no—
294C22×C4—32no—
295C22×C4—34no—
296C22×C4—32no—
297C22×C4—31no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—31no—
302C22×C4—68no—
303C22×C4C22×D314yes—
304C22×C4C22×D312yes—
305C22×C4C22×D312yes—
306C4⋊C4—31no—
307C4⋊C4—31no—
308C4⋊C4—32no—
309C4⋊C4—34no—
310C4⋊C4—32no—
311C4⋊C4—32no—
312C4⋊C4—32no—
313C4⋊C4—34no—
314C4⋊C4—34no—
315C4⋊C4—32no—
316C4⋊C4—34no—
317C4⋊C4—34no—
318C4⋊C4C22×D311yescharacteristic
319C4⋊C4C22×D311yescharacteristic
320C4⋊C4C22×D312yes—
321C4⋊C4C22×D314yes—
322C4⋊C4C22×D312yes—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4—32no—
326C22⋊C4—34no—
327C22⋊C4—32no—
328C22⋊C4—34no—
329C22⋊C4—34no—
330C22⋊C4—34no—
331C22⋊C4—32no—
332C22⋊C4—32no—
333C22⋊C4—34no—
334C22⋊C4C22×D312yes—
335C22⋊C4C22×D312yes—
336C22⋊C4C22×D314yes—
337C42—32no—
338C42—31no—
339C42—34no—
340C42—34no—
341C42—31no—
342C42—34no—
343C42—32no—
344C42—31no—
345C42—32no—
346C42C22×D311yescharacteristic
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3C2×#4114yes—
353C22×C3C2511yescharacteristic
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—48no—
360C2×D3C2×#4112yes—
361C3×C4—22no—
362C3×C4—22no—
363C3×C4—24no—
364C3×C4—21no—
365C3×C4C2×#4112yes—
366C3×C4C2×#4112yes—
367Dic3—22no—
368Dic3—22no—
369Dic3—21no—
370Dic3—24no—
371Dic3—21no—
372Dic3—22no—
373Dic3C2×#4112yes—
374C23—31no—
375C23—32no—
376C23—64no—
377C23—64no—
378C23—122no—
379C23C23×D312yes—
380C23C23×D312yes—
381Q8—24no—
382Q8—62no—
383Q8—122no—
384D4—28no—
385D4—22no—
386D4—64no—
387D4—64no—
388D4—62no—
389D4—124no—
390D4#23714yescomplemented
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—31no—
395C2×C4—34no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—31no—
399C2×C4—32no—
400C2×C4—34no—
401C2×C4—31no—
402C2×C4—64no—
403C2×C4—64no—
404C2×C4—64no—
405C2×C4—64no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—121no—
409C2×C4—122no—
410C2×C4—124no—
411C2×C4C23×D312yes—
412C2×C4C23×D314yes—
413C2×C4C23×D311yescharacteristic
414C2×C4C23×D312yes—
415C2×C4C23×D311yescharacteristic
416C2×C4C23×D311yescharacteristic
417C2×C3—24no—
418C2×C3—42no—
419C2×C3C22×#4111yescharacteristic
420C2×C3C22×#4111yescharacteristic
421C2×C3#58511yescharacteristic
422D3—22no—
423D3—42no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—32no—
430C22—61no—
431C22—61no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—128no—
436C22C2×#23714yes—
437C22C24×D311yescharacteristic
438C4—22no—
439C4—24no—
440C4—22no—
441C4—21no—
442C4—32no—
443C4—61no—
444C4—62no—
445C4—62no—
446C4—61no—
447C4—62no—
448C4—64no—
449C4C2×#23712yes—
450C4D3×#4112yes—
451C3#347511yescharacteristic, complemented, Sylow
452C2—24no—
453C2—42no—
454C2—62no—
455C2—122no—
456C2C22×#23711yescharacteristic
457C2C2×D3×#4111yescharacteristic
458C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#547C2
hypercenter#547D3
hypocenterC3#3475
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C3×#547C2semidirect
2. #5829C2semidirect
3. Dic3×#41C2semidirect
4. #5732C2semidirect
5. #5728C2semidirect
6. #5658C2semidirect
7. #5649C2semidirect
8. #5639C2semidirect
9. #5624C2semidirect
10. #5618C2semidirect
11. #5606C2semidirect
12. #5563C2semidirect
13. C4×D3×D4C2semidirect
14. #5543C2semidirect
15. C4×#237C2semidirect
16. #5513C2semidirect
17. C2×#892C2semidirect
23. C3×C4⋊Q8C22semidirect
24. C3×#123C22semidirect
25. C3×C22⋊Q8C22semidirect
26. C3×#121C22semidirect
28. C3×C4×D4C22semidirect
29. C2×C3×C4⋊C4C22semidirect
32. Q8×Dic3C22semidirect
33. Q8×Dic3C22semidirect
35. #938C22semidirect
39. D4×Dic3C22semidirect
40. D4×Dic3C22semidirect
41. D4×Dic3C22semidirect
47. #926C22semidirect
48. #926C22semidirect
49. C2×C4⋊Dic3C22semidirect
50. #924C22semidirect
51. C2×C4×Dic3C22semidirect
52. C2×C4×Dic3C22semidirect
53. #898C22semidirect
54. #898C22semidirect
56. #897C22semidirect
61. #892C22semidirect
62. #892C22semidirect
63. #892C22semidirect
65. #890C22semidirect
66. #890C22semidirect
67. #889C22semidirect
68. #887C22semidirect
69. #887C22semidirect
70. #886C22semidirect
71. #885C22semidirect
75. #67⋊C4C22semidirect
77. #879C22semidirect
78. #878C22semidirect
79. #877C22semidirect
82. #869C22semidirect
83. C4×Dic6C22semidirect
91. #547D3semidirect
175. C4⋊Q8C2×D3semidirect
187. #123C2×D3semidirect
193. C22⋊Q8C2×D3semidirect
197. #121C2×D3semidirect
217. C4×D4C2×D3semidirect
229. C2×C4⋊C4C2×D3semidirect
245. C3×D4#41semidirect
275. C4×D3#41semidirect
390. D4#237semidirect
451. C3#3475semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
#58292
Dic3×#412
#57322
#57282
#56582
#56492
#56392
#56242
#56182
#56062
#55632
C4×D3×D42
#55432
C4×#2372
#55132
C2×#8922
#34753