Finite Group Databasebeta

← #86686

Subgroups — #86686

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×#547C211yesmaximal, characteristic, complemented
2#5831C212yesmaximal, complemented
3#5828C212yesmaximal, complemented
4#5728C211yesmaximal, characteristic, complemented
5#5727C211yesmaximal, characteristic, complemented
6#5660C214yesmaximal, complemented
7#5659C214yesmaximal, complemented
8#5639C212yesmaximal, complemented
9D3×C22⋊Q8C212yesmaximal, complemented
10#5612C212yesmaximal, complemented
11#5603C212yesmaximal, complemented
12#5557C211yesmaximal, characteristic, complemented
13#5551C211yesmaximal, characteristic, complemented
14C4×D3×D4C211yesmaximal, characteristic, complemented
15#5542C211yesmaximal, characteristic, complemented
16#5511C212yesmaximal, complemented
17#5504C212yesmaximal, complemented
18#3520—31nomaximal, Sylow
19C2×C3×#41C2212yescomplemented
20C2×D3×Q8C2212yescomplemented
21C2×D3×D4C2211yescharacteristic
22C2×#235C2212yes—
23C22×C4×D3C2212yes—
24C3×C4⋊Q8C2211yescharacteristic, complemented
25C3×#123C2214yescomplemented
26C3×C22⋊Q8C2212yescomplemented
27C3×#121C2212yes—
28C3×C4×D4C2211yescharacteristic
29C3×C4×D4C2211yescharacteristic, complemented
30C2×C3×C4⋊C4C2212yescomplemented
31#946C2212yes—
32#946C2212yescomplemented
33Q8×Dic3C2212yescomplemented
34#938C2214yes—
35#938C2211yescharacteristic
36#938C2212yescomplemented
37#937C2214yes—
38#936C2212yes—
39#936C2214yescomplemented
40#936C2211yescharacteristic, complemented
41#935C2214yescomplemented
42D4×Dic3C2211yescharacteristic
43C4×#67C2212yes—
44C4×#67C2214yes—
45C4×#67C2212yes—
46C4×#67C2214yescomplemented
47C4×#67C2214yescomplemented
48#926C2212yescomplemented
49#926C2212yescomplemented
50#924C2214yescomplemented
51#924C2212yescomplemented
52#924C2212yescomplemented
53#897C2212yes—
54#897C2211yescharacteristic
55#896C2214yes—
56#896C2214yescomplemented
57#893C2212yes—
58#893C2211yescharacteristic
59#892C2212yes—
60#892C2214yescomplemented
61D3×C4⋊C4C2211yescharacteristic
62D3×C4⋊C4C2212yescomplemented
63D3×C4⋊C4C2214yescomplemented
64#890C2211yescharacteristic, complemented
65#890C2212yescomplemented
66#889C2214yescomplemented
67#888C2212yescomplemented
68#887C2212yescomplemented
69#887C2212yescomplemented
70#883C2214yes—
71#883C2212yes—
72#883C2214yescomplemented
73#67⋊C4C2212yes—
74D3×C22⋊C4C2212yes—
75D3×C22⋊C4C2214yes—
76#879C2214yescomplemented
77#879C2214yescomplemented
78#879C2212yescomplemented
79#878C2214yescomplemented
80C4×D12C2211yescharacteristic
81C42×D3C2211yescharacteristic
82#869C2211yescharacteristic, 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#547D311yescharacteristic, complemented
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—
102C3×#41—44no—
103C2×C3×Q8C2312yes—
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2311yescharacteristic
106C2×C3×D4C2314yes—
107C22×C3×C4C2312yes—
108C22×C3×C4C2312yes—
109C22×C3×C4C2314yes—
110C2×#67C2312yes—
111C2×#67C2312yes—
112C2×#67C2314yes—
113C22×Dic3C2312yes—
114D3×Q8—44no—
115D3×D4—24no—
116#235—28no—
117C2×D12C2311yescharacteristic
118C2×C4×D3—44no—
119C2×C4×D3C2311yescharacteristic
120C2×C4×D3C2311yescharacteristic
121C2×C4×D3C2311yescharacteristic
122C2×C4×D3C2312yes—
123C2×C4×D3C2314yes—
124C2×C4×D3C2312yes—
125C2×C4×D3C2314yes—
126C2×C4×D3C2312yes—
127C2×Dic6C2311yescharacteristic
128C2×Dic6C2312yes—
129C2×Dic6C2314yes—
130C3×C4⋊C4C2311yescharacteristic
131C3×C4⋊C4C2312yes—
132C3×C4⋊C4C2314yes—
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2312yes—
135C3×C22⋊C4C2314yes—
136C3×C22⋊C4C2312yes—
137C3×C22⋊C4C2312yes—
138C3×C42C2311yescharacteristic
139#217C2314yes—
140#217C2312yes—
141#217C2314yes—
142#217C2312yes—
143#217C2312yes—
144#217C2312yes—
145#217C2314yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2314yes—
151C4⋊Dic3C2311yescharacteristic
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2311yescharacteristic
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2312yes—
157C4⋊Dic3C2314yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2312yes—
165C4×Dic3C2312yes—
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2314yes—
170C4×Dic3C2314yes—
171C2×#41—32no—
172C2×#41C2×D312yescomplemented
173C22×Q8—32no—
174C22×D4—31no—
175C23×C4—32no—
176C4⋊Q8—32no—
177C4⋊Q8—31no—
178C4⋊Q8C2×D311yescharacteristic, complemented
179#126—32no—
180#126—34no—
181#126—34no—
182#125—31no—
183#125—32no—
184#125—34no—
185#124—31no—
186#124—32no—
187#124—34no—
188#123—34no—
189#123—32no—
190#123—34no—
191#123—34no—
192#123C2×D314yescomplemented
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—34no—
197C22⋊Q8—32no—
198C22⋊Q8—31no—
199C22⋊Q8—34no—
200C22⋊Q8—32no—
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203C22⋊Q8C2×D312yescomplemented
204#121—31no—
205#121—32no—
206#121—34no—
207#121C2×D312yes—
208#120—34no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—32no—
212C4×Q8—31no—
213C4×D4—31no—
214C4×D4—32no—
215C4×D4—34no—
216C4×D4—34no—
217C4×D4—34no—
218C4×D4—32no—
219C4×D4—31no—
220C4×D4—32no—
221C4×D4—32no—
222C4×D4—31no—
223C4×D4C2×D311yescharacteristic
224C4×D4C2×D311yescharacteristic, complemented
225#117—32no—
226#117—32no—
227#117—34no—
228#117—32no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4C2×D312yescomplemented
233C2×C22⋊C4—32no—
234C2×C22⋊C4—34no—
235C2×C42—31no—
236C23×C3C2412yes—
237C23×C3C2412yes—
238C22×D3—24no—
239C22×D3—24no—
240C22×D3—42no—
241C22×D3C2412yes—
242C22×D3C2411yescharacteristic
243C3×Q8—24no—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—44no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—44no—
250C2×C3×C4C2412yes—
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2414yes—
253C2×C3×C4C2412yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256#67—28no—
257#67—24no—
258#67—24no—
259C2×Dic3—42no—
260C2×Dic3C2411yescharacteristic
261C2×Dic3C2412yes—
262C2×Dic3C2411yescharacteristic
263C2×Dic3C2411yescharacteristic
264C2×Dic3C2414yes—
265C2×Dic3C2412yes—
266C2×Dic3C2412yes—
267D12#4114yescomplemented
268C4×D3—28no—
269C4×D3—22no—
270C4×D3—22no—
271C4×D3—22no—
272C4×D3—24no—
273C4×D3—42no—
274C4×D3—42no—
275C4×D3—44no—
276Dic6—24no—
277Dic6—22no—
278Dic6—44no—
279C24—32no—
280#41—44no—
281#41—68no—
282C2×Q8—32no—
283C2×Q8—31no—
284C2×Q8—34no—
285C2×Q8—124no—
286C2×Q8C22×D312yes—
287C2×D4—32no—
288C2×D4—34no—
289C2×D4—32no—
290C2×D4—31no—
291C2×D4—64no—
292C2×D4C22×D311yescharacteristic
293C2×D4C22×D311yescharacteristic
294C2×D4C22×D314yes—
295C22×C4—32no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—34no—
300C22×C4—32no—
301C22×C4—34no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—124no—
305C22×C4C22×D312yes—
306C22×C4C22×D312yes—
307C22×C4C22×D314yes—
308C4⋊C4—32no—
309C4⋊C4—32no—
310C4⋊C4—32no—
311C4⋊C4—31no—
312C4⋊C4—34no—
313C4⋊C4—34no—
314C4⋊C4—34no—
315C4⋊C4—32no—
316C4⋊C4—31no—
317C4⋊C4—32no—
318C4⋊C4—34no—
319C4⋊C4—32no—
320C4⋊C4—34no—
321C4⋊C4—32no—
322C4⋊C4C22×D311yescharacteristic
323C4⋊C4C22×D311yescharacteristic
324C4⋊C4C22×D314yes—
325C4⋊C4C22×D312yes—
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—34no—
337C22⋊C4—32no—
338C22⋊C4—34no—
339C22⋊C4C22×D312yes—
340C22⋊C4C22×D314yes—
341C22⋊C4C22×D312yes—
342C42—31no—
343C42—31no—
344C42—31no—
345C42—34no—
346C42—34no—
347C42—32no—
348C42C22×D311yescharacteristic
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3C2511yescharacteristic
356C2×D3—21no—
357C2×D3—22no—
358C2×D3—21no—
359C2×D3—22no—
360C2×D3—48no—
361C2×D3C2×#4114yes—
362C2×D32−1+412yes—
363C3×C4—22no—
364C3×C4—24no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4C2×#4112yes—
368C3×C4H(5,𝔽2)12yes—
369Dic3—24no—
370Dic3—21no—
371Dic3—22no—
372Dic3—22no—
373Dic3—21no—
374Dic3—21no—
375Dic3—22no—
376C23—31no—
377C23—32no—
378C23—64no—
379C23—64no—
380C23—122no—
381C23C23×D312yes—
382C23C23×D312yes—
383Q8—24no—
384Q8—62no—
385Q8—64no—
386Q8—124no—
387D4—22no—
388D4—22no—
389D4—34no—
390D4—44no—
391D4—64no—
392D4—68no—
393D4—64no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—31no—
397C2×C4—32no—
398C2×C4—34no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—44no—
404C2×C4—68no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—62no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—124no—
412C2×C4—122no—
413C2×C4C23×D312yes—
414C2×C4C23×D311yescharacteristic
415C2×C4C23×D314yes—
416C2×C4C23×D312yes—
417C2×C4C23×D311yescharacteristic
418C2×C4C23×D311yescharacteristic
419C2×C3—42no—
420C2×C3—42no—
421C2×C3C22×#4111yescharacteristic
422C2×C3C2×H(5,𝔽2)11yescharacteristic
423C2×C3C2×2−1+411yescharacteristic
424D3—22no—
425D3—24no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—34no—
433C22—32no—
434C22—62no—
435C22—61no—
436C22—61no—
437C22—62no—
438C22—128no—
439C22C24×D311yescharacteristic
440C4—22no—
441C4—24no—
442C4—22no—
443C4—21no—
444C4—61no—
445C4—62no—
446C4—62no—
447C4—61no—
448C4—64no—
449C4—61no—
450C4—62no—
451C4#100412yes—
452C4D3×#4112yes—
453C3#352011yescharacteristic, complemented, Sylow
454C2—42no—
455C2—42no—
456C2—62no—
457C2—64no—
458C2C2×#100411yescharacteristic
459C2C2×D3×#4111yescharacteristic
460C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#547C2
hypercenter#547D3
hypocenterC3#3520
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. #5831C2semidirect
3. #5828C2semidirect
4. #5728C2semidirect
5. #5727C2semidirect
6. #5660C2semidirect
7. #5659C2semidirect
8. #5639C2semidirect
9. D3×C22⋊Q8C2semidirect
10. #5612C2semidirect
11. #5603C2semidirect
12. #5557C2semidirect
13. #5551C2semidirect
14. C4×D3×D4C2semidirect
15. #5542C2semidirect
16. #5511C2semidirect
17. #5504C2semidirect
19. C2×C3×#41C22semidirect
20. C2×D3×Q8C22semidirect
24. C3×C4⋊Q8C22semidirect
25. C3×#123C22semidirect
26. C3×C22⋊Q8C22semidirect
29. C3×C4×D4C22semidirect
30. C2×C3×C4⋊C4C22semidirect
32. #946C22semidirect
33. Q8×Dic3C22semidirect
36. #938C22semidirect
39. #936C22semidirect
40. #936C22semidirect
41. #935C22semidirect
46. C4×#67C22semidirect
47. C4×#67C22semidirect
48. #926C22semidirect
49. #926C22semidirect
50. #924C22semidirect
51. #924C22semidirect
52. #924C22semidirect
56. #896C22semidirect
60. #892C22semidirect
62. D3×C4⋊C4C22semidirect
63. D3×C4⋊C4C22semidirect
64. #890C22semidirect
65. #890C22semidirect
66. #889C22semidirect
67. #888C22semidirect
68. #887C22semidirect
69. #887C22semidirect
72. #883C22semidirect
76. #879C22semidirect
77. #879C22semidirect
78. #879C22semidirect
79. #878C22semidirect
82. #869C22semidirect
83. C4×Dic6C22semidirect
91. #547D3semidirect
172. C2×#41C2×D3semidirect
178. C4⋊Q8C2×D3semidirect
192. #123C2×D3semidirect
203. C22⋊Q8C2×D3semidirect
224. C4×D4C2×D3semidirect
232. C2×C4⋊C4C2×D3semidirect
267. 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×#5472
#58312
#58282
#57282
#57272
#56602
#56592
#56392
D3×C22⋊Q82
#56122
#56032
#55572
#55512
C4×D3×D42
#55422
#55112
#55042
#35203