Finite Group Databasebeta

← #86322

Subgroups — #86322

Order 384.

1058 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#535C211yesmaximal, characteristic, complemented
2#5828C211yesmaximal, characteristic, complemented
3#5825C211yesmaximal, characteristic, complemented
4#5806C212yesmaximal, complemented
5#5720C212yesmaximal, complemented
6#5713C212yesmaximal, complemented
7#5669C214yesmaximal, complemented
8#5616C212yesmaximal, complemented
9#5606C212yesmaximal, complemented
10#5603C212yesmaximal, complemented
11#5598C212yesmaximal, complemented
12#5594C212yesmaximal, complemented
13#5593C214yesmaximal, complemented
14#5587C212yesmaximal, complemented
15#5533C212yesmaximal, complemented
16#3540—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic, complemented
19C2×#237C2212yes—
20C22×C4×D3C2211yescharacteristic, complemented
21C22×Dic6C2211yescharacteristic, complemented
22C3×#127C2212yes—
23C3×#124C2212yescomplemented
24C3×#121C2212yes—
25C3×#121C2212yescomplemented
26C3×#120C2212yes—
27C3×#120C2212yescomplemented
28C3×#117C2211yescharacteristic, complemented
29#953C2214yes—
30C2×#217C2214yescomplemented
31#946C2212yescomplemented
32#944C2212yescomplemented
33#939C2214yes—
34#939C2214yescomplemented
35#939C2214yescomplemented
36#938C2212yes—
37#938C2212yes—
38#937C2211yescharacteristic
39#937C2211yescharacteristic, complemented
40#937C2214yescomplemented
41#936C2212yes—
42#936C2214yescomplemented
43#936C2214yescomplemented
44#936C2212yescomplemented
45#935C2211yescharacteristic
46#935C2214yescomplemented
47#935C2211yescharacteristic, complemented
48#935C2212yescomplemented
49#935C2214yescomplemented
50#935C2212yescomplemented
51D4×Dic3C2214yescomplemented
52D4×Dic3C2214yescomplemented
53D4×Dic3C2212yescomplemented
54C4×#67C2212yes—
55C4×#67C2212yescomplemented
56#926C2211yescharacteristic, complemented
57#926C2211yescharacteristic, complemented
58#924C2212yescomplemented
59#924C2212yescomplemented
60#897C2212yescomplemented
61#892C2212yescomplemented
62#888C2212yescomplemented
63#887C2212yescomplemented
64#885C2214yes—
65#885C2214yescomplemented
66#885C2214yescomplemented
67#883C2214yescomplemented
68#67⋊C4C2214yescomplemented
69#879C2214yescomplemented
70#879C2214yescomplemented
71#878C2214yescomplemented
72#878C2214yescomplemented
73#877C2214yescomplemented
74#876C2214yescomplemented
75C4×Dic6C2214yescomplemented
76H(3,𝔽4)—32no—
77#560—32no—
78#559—32no—
79#556—32no—
80#555—32no—
81#551—32no—
82#548—32no—
83#546—32no—
84#542—34no—
85#542—32no—
86#538—34no—
87#536—31no—
88#535D311yescharacteristic, complemented
89#525—31no—
90#525—32no—
91C24×C3C2312yes—
92C23×D3C2311yescharacteristic
93C3×#41—42no—
94C2×C3×Q8C2311yescharacteristic
95C2×C3×D4—42no—
96C2×C3×D4C2312yes—
97C2×C3×D4C2314yes—
98C2×C3×D4C2314yes—
99C2×C3×D4C2311yescharacteristic
100C2×C3×D4C2311yescharacteristic
101C2×C3×D4C2311yescharacteristic
102C2×C3×D4C2312yes—
103C22×C3×C4C2311yescharacteristic
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C2×#67C2314yes—
107C2×#67C2314yes—
108C2×#67C2312yes—
109C2×#67C2312yes—
110C22×Dic3C2314yes—
111C22×Dic3C2314yes—
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2312yes—
114#237—44no—
115C2×C4×D3—42no—
116C2×C4×D3C2312yes—
117C2×C4×D3C2312yes—
118C2×Dic6—42no—
119C2×Dic6C2312yes—
120C2×Dic6C2314yes—
121C2×Dic6C2312yes—
122C3×C4⋊C4C2312yes—
123C3×C22⋊C4C2314yes—
124C3×C22⋊C4C2314yes—
125C3×C22⋊C4C2312yes—
126C3×C42C2312yes—
127#217C2314yes—
128#217C2314yes—
129#217C2314yes—
130#217C2312yes—
131#217C2314yes—
132#217C2314yes—
133#217C2314yes—
134#217C2311yescharacteristic
135#217C2312yes—
136#217C2311yescharacteristic
137#217C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2312yes—
142C4⋊Dic3C2314yes—
143C4⋊Dic3C2312yes—
144C4⋊Dic3C2312yes—
145Dic3⋊C4C2312yes—
146Dic3⋊C4C2314yes—
147Dic3⋊C4C2314yes—
148Dic3⋊C4C2314yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2314yes—
151C4×Dic3C2314yes—
152C4×Dic3C2314yes—
153C4×Dic3C2312yes—
154C4×Dic3C2312yes—
155C2×#41—32no—
156C2×#41C2×D311yescharacteristic
157C22×Q8—31no—
158C22×D4C2×D311yescharacteristic, complemented
159C23×C4—31no—
160C4⋊Q8—32no—
161C4⋊Q8—32no—
162#127C2×D312yes—
163#126—34no—
164#126—34no—
165#126—34no—
166#124—32no—
167#124—34no—
168#124—32no—
169#124—34no—
170#124—34no—
171#124—34no—
172#124—34no—
173#124C2×D312yescomplemented
174#123—31no—
175#123—31no—
176#123—32no—
177#123—32no—
178#123—34no—
179#123—34no—
180#123—34no—
181C22⋊Q8—32no—
182C22⋊Q8—32no—
183C22⋊Q8—32no—
184C22⋊Q8—32no—
185C22⋊Q8—34no—
186C22⋊Q8—34no—
187#121—32no—
188#121—32no—
189#121—34no—
190#121—34no—
191#121—34no—
192#121C2×D312yes—
193#121C2×D312yescomplemented
194#120—34no—
195#120—31no—
196#120—34no—
197#120—31no—
198#120C2×D312yes—
199#120C2×D312yescomplemented
200C4×Q8—32no—
201C4×Q8—34no—
202C4×D4—34no—
203C4×D4—34no—
204C4×D4—32no—
205C4×D4—32no—
206C4×D4—32no—
207C4×D4—34no—
208#117—31no—
209#117—31no—
210#117—32no—
211#117—34no—
212#117C2×D311yescharacteristic, complemented
213C2×C22⋊C4—34no—
214C23×C3—42no—
215C23×C3—44no—
216C23×C3C2411yescharacteristic
217C23×C3C2411yescharacteristic
218C23×C3C2414yes—
219C23×C3C2411yescharacteristic
220C22×D3—41no—
221C22×D3—42no—
222C22×D3C2412yes—
223C3×Q8—41no—
224C3×D4—24no—
225C3×D4—42no—
226C3×D4—44no—
227C3×D4—44no—
228C3×D4—41no—
229C3×D4—41no—
230C3×D4—41no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4C2414yes—
235C2×C3×C4C2411yescharacteristic
236C2×C3×C4C2411yescharacteristic
237C2×C3×C4C2411yescharacteristic
238C2×C3×C4C2411yescharacteristic
239#67—44no—
240#67—42no—
241#67—44no—
242#67—42no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—44no—
246C2×Dic3—44no—
247C2×Dic3C2412yes—
248C2×Dic3C2414yes—
249C2×Dic3C2412yes—
250C2×Dic3C2412yes—
251C2×Dic3C2414yes—
252C4×D3—24no—
253C4×D3—42no—
254Dic6—24no—
255Dic6—42no—
256Dic6—44no—
257C24—31no—
258C24C22×D312yes—
259#41—42no—
260#41—124no—
261C2×Q8—32no—
262C2×Q8—34no—
263C2×Q8—32no—
264C2×Q8—122no—
265C2×Q8C22×D311yescharacteristic
266C2×D4—32no—
267C2×D4—34no—
268C2×D4—32no—
269C2×D4—34no—
270C2×D4—42no—
271C2×D4C22×D312yes—
272C2×D4C22×D314yes—
273C2×D4C22×D314yes—
274C2×D4C22×D311yescharacteristic
275C2×D4C22×D311yescharacteristic
276C2×D4C22×D311yescharacteristic
277C2×D4C22×D312yes—
278C22×C4—34no—
279C22×C4—34no—
280C22×C4—31no—
281C22×C4—32no—
282C22×C4—32no—
283C22×C4—32no—
284C22×C4—122no—
285C22×C4C22×D311yescharacteristic
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D311yescharacteristic
288C4⋊C4—32no—
289C4⋊C4—32no—
290C4⋊C4—32no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4—32no—
294C4⋊C4—34no—
295C4⋊C4—34no—
296C4⋊C4—34no—
297C4⋊C4C22×D312yes—
298C22⋊C4—32no—
299C22⋊C4—34no—
300C22⋊C4—34no—
301C22⋊C4—34no—
302C22⋊C4—31no—
303C22⋊C4—31no—
304C22⋊C4—32no—
305C22⋊C4—32no—
306C22⋊C4—34no—
307C22⋊C4—34no—
308C22⋊C4—34no—
309C22⋊C4—34no—
310C22⋊C4—32no—
311C22⋊C4—34no—
312C22⋊C4—32no—
313C22⋊C4C22×D314yes—
314C22⋊C4C22×D312yes—
315C22⋊C4C22×D314yes—
316C42—32no—
317C42—34no—
318C42—32no—
319C42—34no—
320C42C22×D312yes—
321C22×C3—21no—
322C22×C3—24no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—24no—
326C22×C3—22no—
327C22×C3—24no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—84no—
331C22×C3C2511yescharacteristic
332C2×D3—22no—
333C2×D3—22no—
334C2×D3—22no—
335C2×D3—44no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—24no—
340C3×C4H(5,𝔽2)12yes—
341Dic3—24no—
342Dic3—22no—
343Dic3—22no—
344Dic3—24no—
345Dic3—22no—
346C23—32no—
347C23—42no—
348C23—44no—
349C23—121no—
350C23—122no—
351C23C23×D311yescharacteristic
352C23C23×D311yescharacteristic
353C23C23×D314yes—
354C23C23×D311yescharacteristic
355Q8—41no—
356Q8—64no—
357Q8—122no—
358Q8—124no—
359D4—24no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—44no—
365D4—44no—
366D4—124no—
367D4—122no—
368D4—124no—
369D4—122no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—32no—
374C2×C4—34no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—34no—
378C2×C4—64no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—122no—
382C2×C4—124no—
383C2×C4—124no—
384C2×C4C23×D314yes—
385C2×C4C23×D311yescharacteristic
386C2×C4C23×D311yescharacteristic
387C2×C4C23×D311yescharacteristic
388C2×C4C23×D311yescharacteristic
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—44no—
392C2×C3—41no—
393C2×C3C2×H(5,𝔽2)11yescharacteristic
394C2×C3#58512yes—
395D3—42no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—24no—
400C22—24no—
401C22—22no—
402C22—22no—
403C22—24no—
404C22—22no—
405C22—62no—
406C22—62no—
407C22—62no—
408C22—84no—
409C22—124no—
410C22C24×D311yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—24no—
415C4—62no—
416C4—64no—
417C4—62no—
418C4—64no—
419C4—62no—
420C4#100412yes—
421C3#354011yescharacteristic, complemented, Sylow
422C2—41no—
423C2—44no—
424C2—41no—
425C2—41no—
426C2—122no—
427C2C2×#100411yescharacteristic
428C2#596612yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#535C2
hypercenter#535D3
hypocenterC3#3540
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C3×#535C2semidirect
2. #5828C2semidirect
3. #5825C2semidirect
4. #5806C2semidirect
5. #5720C2semidirect
6. #5713C2semidirect
7. #5669C2semidirect
8. #5616C2semidirect
9. #5606C2semidirect
10. #5603C2semidirect
11. #5598C2semidirect
12. #5594C2semidirect
13. #5593C2semidirect
14. #5587C2semidirect
15. #5533C2semidirect
18. C22×C3×D4C22semidirect
20. C22×C4×D3C22semidirect
21. C22×Dic6C22semidirect
23. C3×#124C22semidirect
25. C3×#121C22semidirect
27. C3×#120C22semidirect
28. C3×#117C22semidirect
30. C2×#217C22semidirect
31. #946C22semidirect
32. #944C22semidirect
34. #939C22semidirect
35. #939C22semidirect
39. #937C22semidirect
40. #937C22semidirect
42. #936C22semidirect
43. #936C22semidirect
44. #936C22semidirect
46. #935C22semidirect
47. #935C22semidirect
48. #935C22semidirect
49. #935C22semidirect
50. #935C22semidirect
51. D4×Dic3C22semidirect
52. D4×Dic3C22semidirect
53. D4×Dic3C22semidirect
55. C4×#67C22semidirect
56. #926C22semidirect
57. #926C22semidirect
58. #924C22semidirect
59. #924C22semidirect
60. #897C22semidirect
61. #892C22semidirect
62. #888C22semidirect
63. #887C22semidirect
65. #885C22semidirect
66. #885C22semidirect
67. #883C22semidirect
68. #67⋊C4C22semidirect
69. #879C22semidirect
70. #879C22semidirect
71. #878C22semidirect
72. #878C22semidirect
73. #877C22semidirect
74. #876C22semidirect
75. C4×Dic6C22semidirect
88. #535D3semidirect
158. C22×D4C2×D3semidirect
173. #124C2×D3semidirect
193. #121C2×D3semidirect
199. #120C2×D3semidirect
212. #117C2×D3semidirect
421. C3#3540semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#58282
#58252
#58062
#57202
#57132
#56692
#56162
#56062
#56032
#55982
#55942
#55932
#55872
#55332
#35403