Finite Group Databasebeta

← #86055

Subgroups — #86055

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×#525C211yesmaximal, characteristic, complemented
2#5830C211yesmaximal, characteristic, complemented
3#5825C211yesmaximal, characteristic, complemented
4#5791C212yesmaximal, complemented
5#5664C212yesmaximal, complemented
6#5659C212yesmaximal, complemented
7#5640C214yesmaximal, complemented
8#5613C214yesmaximal, complemented
9#5595C212yesmaximal, complemented
10#5594C212yesmaximal, complemented
11#5563C212yesmaximal, complemented
12#5552C212yesmaximal, complemented
13#5550C212yesmaximal, complemented
14#5542C212yesmaximal, complemented
15#5534C212yesmaximal, complemented
16#3540—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic, complemented
19C2×#235C2212yes—
20C22×D12C2211yescharacteristic, complemented
21C22×Dic6C2211yescharacteristic, complemented
22C3×#123C2211yescharacteristic
23C3×#123C2211yescharacteristic, complemented
24C3×C22⋊Q8C2212yescomplemented
25C3×#121C2212yes—
26C3×#120C2211yescharacteristic
27C3×#120C2211yescharacteristic, complemented
28C3×C4×D4C2212yes—
29C3×C4×D4C2212yescomplemented
30C3×#117C2211yescharacteristic, complemented
31#953C2214yescomplemented
32#947C2212yescomplemented
33#944C2212yescomplemented
34#940C2212yes—
35#939C2214yescomplemented
36#938C2214yescomplemented
37#937C2212yes—
38#937C2214yes—
39#937C2212yescomplemented
40#936C2212yes—
41#936C2214yes—
42#935C2212yescomplemented
43#935C2212yescomplemented
44#930C2212yes—
45#930C2212yes—
46#930C2212yescomplemented
47#929C2214yescomplemented
48C4×#67C2214yescomplemented
49C4×#67C2214yescomplemented
50#926C2212yescomplemented
51#926C2211yescharacteristic, complemented
52#926C2211yescharacteristic, complemented
53#924C2212yescomplemented
54#924C2212yescomplemented
55#924C2212yescomplemented
56#898C2214yescomplemented
57#896C2214yescomplemented
58#896C2214yescomplemented
59#894C2214yescomplemented
60#892C2214yescomplemented
61#887C2214yescomplemented
62#885C2214yescomplemented
63#885C2214yescomplemented
64#885C2214yescomplemented
65#885C2214yescomplemented
66#884C2214yes—
67#883C2214yescomplemented
68#67⋊C4C2214yescomplemented
69D3×C22⋊C4C2214yescomplemented
70#879C2214yescomplemented
71#879C2214yescomplemented
72#875C2212yescomplemented
73C4×D12C2212yescomplemented
74#869C2212yescomplemented
75C4×Dic6C2212yescomplemented
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#535—31no—
89#525—32no—
90#525D311yescharacteristic, complemented
91C24×C3C2311yescharacteristic
92C23×D3C2312yes—
93C3×#41—42no—
94C2×C3×Q8C2311yescharacteristic
95C2×C3×D4C2311yescharacteristic
96C2×C3×D4C2311yescharacteristic
97C2×C3×D4C2312yes—
98C2×C3×D4C2311yescharacteristic
99C2×C3×D4C2312yes—
100C22×C3×C4—42no—
101C22×C3×C4C2311yescharacteristic
102C22×C3×C4C2311yescharacteristic
103C22×C3×C4C2312yes—
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2312yes—
107C2×#67C2314yes—
108C2×#67C2314yes—
109C2×#67C2314yes—
110C2×#67C2314yes—
111C22×Dic3C2312yes—
112#235—44no—
113C2×D12—42no—
114C2×D12C2312yes—
115C2×D12C2312yes—
116C2×C4×D3C2314yes—
117C2×C4×D3C2314yes—
118C2×Dic6—42no—
119C2×Dic6C2312yes—
120C2×Dic6C2314yes—
121C2×Dic6C2312yes—
122C3×C4⋊C4C2312yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C22⋊C4C2312yes—
126C3×C22⋊C4C2312yes—
127C3×C22⋊C4C2312yes—
128C3×C22⋊C4C2311yescharacteristic
129C3×C22⋊C4C2311yescharacteristic
130C3×C22⋊C4C2312yes—
131C3×C42C2312yes—
132#217C2312yes—
133#217C2314yes—
134#217C2314yes—
135#217C2314yes—
136#217C2314yes—
137#217C2312yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2314yes—
143(C2×D3)⋊C4C2314yes—
144C4⋊Dic3C2312yes—
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2314yes—
147Dic3⋊C4C2314yes—
148Dic3⋊C4C2314yes—
149Dic3⋊C4C2314yes—
150Dic3⋊C4C2314yes—
151C4×Dic3C2314yes—
152C4×Dic3C2312yes—
153C4×Dic3C2314yes—
154C4×Dic3C2312yes—
155C2×#41—32no—
156C2×#41C2×D311yescharacteristic
157C22×Q8—31no—
158C22×D4—31no—
159C23×C4C2×D311yescharacteristic, complemented
160C4⋊Q8—32no—
161C4⋊Q8—32no—
162#127—32no—
163#126—34no—
164#126—34no—
165#126—34no—
166#124—32no—
167#124—32no—
168#124—34no—
169#124—34no—
170#124—34no—
171#124—34no—
172#124—34no—
173#124—32no—
174#123—32no—
175#123—32no—
176#123—34no—
177#123—34no—
178#123—34no—
179#123C2×D311yescharacteristic
180#123C2×D311yescharacteristic, complemented
181C22⋊Q8—32no—
182C22⋊Q8—32no—
183C22⋊Q8—32no—
184C22⋊Q8—34no—
185C22⋊Q8—34no—
186C22⋊Q8C2×D312yescomplemented
187#121—34no—
188#121—34no—
189#121—32no—
190#121—32no—
191#121—32no—
192#121—34no—
193#121C2×D312yes—
194#120—34no—
195#120—32no—
196#120—32no—
197#120—34no—
198#120C2×D311yescharacteristic
199#120C2×D311yescharacteristic, complemented
200C4×Q8—34no—
201C4×Q8—32no—
202C4×D4—34no—
203C4×D4—34no—
204C4×D4—34no—
205C4×D4—32no—
206C4×D4C2×D312yes—
207C4×D4C2×D312yescomplemented
208#117—32no—
209#117—31no—
210#117—31no—
211#117—34no—
212#117C2×D311yescharacteristic, complemented
213C2×C22⋊C4—34no—
214C23×C3—41no—
215C23×C3—42no—
216C23×C3C2411yescharacteristic
217C23×C3C2412yes—
218C23×C3C2411yescharacteristic
219C23×C3C2411yescharacteristic
220C22×D3—42no—
221C22×D3—44no—
222C22×D3C2414yes—
223C3×Q8—41no—
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—41no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—24no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4C2411yescharacteristic
236C2×C3×C4C2412yes—
237C2×C3×C4C2412yes—
238C2×C3×C4C2411yescharacteristic
239C2×C3×C4C2411yescharacteristic
240C2×C3×C4C2412yes—
241C2×C3×C4C2411yescharacteristic
242#67—44no—
243#67—44no—
244#67—44no—
245#67—44no—
246C2×Dic3—42no—
247C2×Dic3C2414yes—
248C2×Dic3C2414yes—
249C2×Dic3C2414yes—
250D12—24no—
251D12—42no—
252C4×D3—44no—
253C4×D3—44no—
254Dic6—24no—
255Dic6—42no—
256Dic6—44no—
257C24—32no—
258C24C22×D311yescharacteristic
259#41—42no—
260#41—124no—
261C2×Q8—32no—
262C2×Q8—34no—
263C2×Q8—32no—
264C2×Q8—122no—
265C2×Q8C22×D311yescharacteristic
266C2×D4—34no—
267C2×D4—34no—
268C2×D4—34no—
269C2×D4—34no—
270C2×D4—32no—
271C2×D4—32no—
272C2×D4—122no—
273C2×D4C22×D312yes—
274C2×D4C22×D312yes—
275C2×D4C22×D311yescharacteristic
276C2×D4C22×D311yescharacteristic
277C2×D4C22×D311yescharacteristic
278C22×C4—32no—
279C22×C4—34no—
280C22×C4—34no—
281C22×C4—42no—
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D312yes—
284C22×C4C22×D311yescharacteristic
285C22×C4C22×D311yescharacteristic
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D312yes—
288C4⋊C4—32no—
289C4⋊C4—32no—
290C4⋊C4—34no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4—34no—
294C4⋊C4—34no—
295C4⋊C4C22×D312yes—
296C4⋊C4C22×D312yes—
297C4⋊C4C22×D312yes—
298C22⋊C4—34no—
299C22⋊C4—32no—
300C22⋊C4—34no—
301C22⋊C4—32no—
302C22⋊C4—34no—
303C22⋊C4—34no—
304C22⋊C4—34no—
305C22⋊C4—34no—
306C22⋊C4—34no—
307C22⋊C4—34no—
308C22⋊C4—34no—
309C22⋊C4—34no—
310C22⋊C4C22×D312yes—
311C22⋊C4C22×D311yescharacteristic
312C22⋊C4C22×D311yescharacteristic
313C22⋊C4C22×D312yes—
314C22⋊C4C22×D312yes—
315C22⋊C4C22×D312yes—
316C42—32no—
317C42—34no—
318C42—32no—
319C42—34no—
320C42C22×D312yes—
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—44no—
331C22×C3C2511yescharacteristic
332C2×D3—24no—
333C2×D3—24no—
334C2×D3—24no—
335C2×D3—84no—
336C3×C4—21no—
337C3×C4—22no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4H(5,𝔽2)12yes—
343Dic3—24no—
344Dic3—24no—
345Dic3—24no—
346C23—34no—
347C23—41no—
348C23—42no—
349C23—122no—
350C23—124no—
351C23C23×D311yescharacteristic
352C23C23×D311yescharacteristic
353C23C23×D312yes—
354C23C23×D311yescharacteristic
355Q8—41no—
356Q8—64no—
357Q8—122no—
358Q8—124no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—42no—
363D4—42no—
364D4—64no—
365D4—122no—
366D4—124no—
367D4—124no—
368D4—124no—
369D4—124no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—34no—
375C2×C4—34no—
376C2×C4—34no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—124no—
380C2×C4—124no—
381C2×C4—122no—
382C2×C4C23×D312yes—
383C2×C4C23×D311yescharacteristic
384C2×C4C23×D312yes—
385C2×C4C23×D311yescharacteristic
386C2×C4C23×D312yes—
387C2×C4C23×D311yescharacteristic
388C2×C4C23×D311yescharacteristic
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—42no—
392C2×C3—41no—
393C2×C3C2×H(5,𝔽2)11yescharacteristic
394C2×C3#58512yes—
395D3—44no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—44no—
406C22—64no—
407C22—64no—
408C22—64no—
409C22—244no—
410C22C24×D311yescharacteristic
411C4—22no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—22no—
416C4—22no—
417C4—64no—
418C4—64no—
419C4—64no—
420C4#100412yes—
421C3#354011yescharacteristic, complemented, Sylow
422C2—41no—
423C2—42no—
424C2—41no—
425C2—41no—
426C2—124no—
427C2C2×#100411yescharacteristic
428C2#596412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#525C2
hypercenter#525D3
hypocenterC3#3540
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C3×#525C2semidirect
2. #5830C2semidirect
3. #5825C2semidirect
4. #5791C2semidirect
5. #5664C2semidirect
6. #5659C2semidirect
7. #5640C2semidirect
8. #5613C2semidirect
9. #5595C2semidirect
10. #5594C2semidirect
11. #5563C2semidirect
12. #5552C2semidirect
13. #5550C2semidirect
14. #5542C2semidirect
15. #5534C2semidirect
18. C23×C3×C4C22semidirect
20. C22×D12C22semidirect
21. C22×Dic6C22semidirect
23. C3×#123C22semidirect
24. C3×C22⋊Q8C22semidirect
27. C3×#120C22semidirect
29. C3×C4×D4C22semidirect
30. C3×#117C22semidirect
31. #953C22semidirect
32. #947C22semidirect
33. #944C22semidirect
35. #939C22semidirect
36. #938C22semidirect
39. #937C22semidirect
42. #935C22semidirect
43. #935C22semidirect
46. #930C22semidirect
47. #929C22semidirect
48. C4×#67C22semidirect
49. C4×#67C22semidirect
50. #926C22semidirect
51. #926C22semidirect
52. #926C22semidirect
53. #924C22semidirect
54. #924C22semidirect
55. #924C22semidirect
56. #898C22semidirect
57. #896C22semidirect
58. #896C22semidirect
59. #894C22semidirect
60. #892C22semidirect
61. #887C22semidirect
62. #885C22semidirect
63. #885C22semidirect
64. #885C22semidirect
65. #885C22semidirect
67. #883C22semidirect
68. #67⋊C4C22semidirect
69. D3×C22⋊C4C22semidirect
70. #879C22semidirect
71. #879C22semidirect
72. #875C22semidirect
73. C4×D12C22semidirect
74. #869C22semidirect
75. C4×Dic6C22semidirect
90. #525D3semidirect
159. C23×C4C2×D3semidirect
180. #123C2×D3semidirect
186. C22⋊Q8C2×D3semidirect
199. #120C2×D3semidirect
207. C4×D4C2×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×#5252
#58302
#58252
#57912
#56642
#56592
#56402
#56132
#55952
#55942
#55632
#55522
#55502
#55422
#55342
#35403