Finite Group Databasebeta

← #86345

Subgroups — #86345

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×#536C211yesmaximal, characteristic, complemented
2#5830C211yesmaximal, characteristic, complemented
3#5828C211yesmaximal, characteristic, complemented
4#5817C212yesmaximal, complemented
5#5733C214yesmaximal, complemented
6#5683C212yesmaximal, complemented
7#5679C212yesmaximal, complemented
8#5666C212yesmaximal, complemented
9#5664C212yesmaximal, complemented
10#5657C214yesmaximal, complemented
11#5643C212yesmaximal, complemented
12#5641C212yesmaximal, complemented
13#5630C212yesmaximal, complemented
14#5627C212yesmaximal, complemented
15#5532C212yesmaximal, complemented
16#3540—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic, complemented
19C2×#239C2212yes—
20C22×D12C2211yescharacteristic, complemented
21C22×C4×D3C2211yescharacteristic, complemented
22C3×C4⋊Q8C2212yescomplemented
23C3×#124C2212yes—
24C3×#123C2212yescomplemented
25C3×#123C2212yescomplemented
26C3×C22⋊Q8C2212yescomplemented
27C3×C22⋊Q8C2212yescomplemented
28C3×#117C2211yescharacteristic, complemented
29#947C2212yescomplemented
30#947C2214yescomplemented
31#947C2212yescomplemented
32#946C2212yescomplemented
33#946C2214yescomplemented
34#946C2212yescomplemented
35Q8×Dic3C2212yescomplemented
36#940C2212yes—
37#938C2212yes—
38#937C2212yes—
39#937C2211yescharacteristic
40#937C2212yescomplemented
41#937C2211yescharacteristic, complemented
42#935C2211yescharacteristic
43#935C2211yescharacteristic, complemented
44#930C2212yes—
45#930C2214yescomplemented
46#930C2212yescomplemented
47C4×#67C2212yes—
48C4×#67C2214yescomplemented
49C4×#67C2212yescomplemented
50#926C2211yescharacteristic, complemented
51#926C2211yescharacteristic, complemented
52#898C2214yescomplemented
53#898C2214yescomplemented
54#897C2214yescomplemented
55#895C2214yes—
56#895C2212yes—
57#894C2214yescomplemented
58#894C2214yescomplemented
59#893C2212yescomplemented
60#893C2214yescomplemented
61#892C2212yescomplemented
62#888C2212yescomplemented
63#887C2214yescomplemented
64#885C2214yes—
65#885C2214yescomplemented
66#885C2214yescomplemented
67#884C2214yescomplemented
68#883C2214yescomplemented
69#882C2214yes—
70#882C2214yescomplemented
71#67⋊C4C2214yescomplemented
72D3×C22⋊C4C2214yescomplemented
73#879C2214yescomplemented
74#875C2214yescomplemented
75#872C2214yescomplemented
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#536D311yescharacteristic, complemented
88#535—31no—
89#525—31no—
90#525—32no—
91C23×D3C2312yes—
92C23×D3C2311yescharacteristic
93C3×#41—42no—
94C2×C3×Q8—42no—
95C2×C3×Q8C2312yes—
96C2×C3×Q8C2312yes—
97C2×C3×Q8C2311yescharacteristic
98C2×C3×D4C2311yescharacteristic
99C2×C3×D4C2311yescharacteristic
100C2×C3×D4C2311yescharacteristic
101C22×C3×C4C2311yescharacteristic
102C22×C3×C4C2311yescharacteristic
103C22×C3×C4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C2×#67C2314yes—
106C2×#67C2312yes—
107C2×#67C2312yes—
108C2×#67C2314yes—
109C22×Dic3C2311yescharacteristic
110#239—44no—
111C2×D12—42no—
112C2×D12C2312yes—
113C2×D12C2314yes—
114C2×D12C2314yes—
115C2×D12C2312yes—
116C2×C4×D3—42no—
117C2×C4×D3C2312yes—
118C2×C4×D3C2314yes—
119C2×C4×D3C2314yes—
120C2×C4×D3C2312yes—
121C2×Dic6C2314yes—
122C3×C4⋊C4C2314yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2314yes—
125C3×C22⋊C4C2314yes—
126C3×C22⋊C4C2314yes—
127C3×C22⋊C4C2312yes—
128C3×C42C2312yes—
129#217C2312yes—
130#217C2312yes—
131#217C2311yescharacteristic
132#217C2311yescharacteristic
133#217C2312yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2314yes—
136(C2×D3)⋊C4C2312yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2314yes—
143(C2×D3)⋊C4C2312yes—
144C4⋊Dic3C2314yes—
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2312yes—
147Dic3⋊C4C2312yes—
148Dic3⋊C4C2314yes—
149Dic3⋊C4C2314yes—
150Dic3⋊C4C2312yes—
151C4×Dic3C2314yes—
152C4×Dic3C2314yes—
153C4×Dic3C2312yes—
154C4×Dic3C2312yes—
155C2×#41—32no—
156C2×#41C2×D311yescharacteristic
157C22×Q8C2×D311yescharacteristic, complemented
158C22×D4—31no—
159C23×C4—31no—
160C4⋊Q8—32no—
161C4⋊Q8C2×D312yescomplemented
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#124C2×D312yes—
174#123—31no—
175#123—31no—
176#123—34no—
177#123—34no—
178#123—34no—
179#123C2×D312yescomplemented
180#123C2×D312yescomplemented
181C22⋊Q8—32no—
182C22⋊Q8—32no—
183C22⋊Q8—34no—
184C22⋊Q8—34no—
185C22⋊Q8C2×D312yescomplemented
186C22⋊Q8C2×D312yescomplemented
187#121—32no—
188#121—32no—
189#121—32no—
190#121—34no—
191#121—32no—
192#121—34no—
193#121—34no—
194#120—31no—
195#120—31no—
196#120—32no—
197#120—32no—
198#120—34no—
199#120—34no—
200C4×Q8—32no—
201C4×Q8—34no—
202C4×D4—32no—
203C4×D4—32no—
204C4×D4—34no—
205C4×D4—34no—
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×C3C2411yescharacteristic
215C23×C3C2411yescharacteristic
216C23×C3C2411yescharacteristic
217C22×D3—41no—
218C22×D3—42no—
219C22×D3—42no—
220C22×D3—44no—
221C22×D3C2412yes—
222C22×D3C2414yes—
223C3×Q8—24no—
224C3×Q8—42no—
225C3×Q8—41no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—41no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—42no—
233C2×C3×C4C2411yescharacteristic
234C2×C3×C4C2414yes—
235C2×C3×C4C2414yes—
236C2×C3×C4C2411yescharacteristic
237C2×C3×C4C2411yescharacteristic
238C2×C3×C4C2411yescharacteristic
239#67—42no—
240#67—44no—
241#67—44no—
242#67—42no—
243C2×Dic3—41no—
244C2×Dic3C2414yes—
245C2×Dic3C2412yes—
246C2×Dic3C2412yes—
247C2×Dic3C2412yes—
248D12—24no—
249D12—42no—
250D12—44no—
251D12—44no—
252C4×D3—24no—
253C4×D3—42no—
254C4×D3—44no—
255C4×D3—44no—
256Dic6—44no—
257C24—32no—
258C24—31no—
259#41—42no—
260#41—124no—
261C2×Q8—34no—
262C2×Q8—42no—
263C2×Q8C22×D312yes—
264C2×Q8C22×D312yes—
265C2×Q8C22×D311yescharacteristic
266C2×D4—32no—
267C2×D4—34no—
268C2×D4—32no—
269C2×D4—34no—
270C2×D4—32no—
271C2×D4—34no—
272C2×D4—34no—
273C2×D4—32no—
274C2×D4—122no—
275C2×D4C22×D311yescharacteristic
276C2×D4C22×D311yescharacteristic
277C2×D4C22×D311yescharacteristic
278C22×C4—31no—
279C22×C4—32no—
280C22×C4—34no—
281C22×C4—34no—
282C22×C4—32no—
283C22×C4—122no—
284C22×C4C22×D311yescharacteristic
285C22×C4C22×D311yescharacteristic
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D312yes—
288C4⋊C4—32no—
289C4⋊C4—32no—
290C4⋊C4—32no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4—32no—
294C4⋊C4—34no—
295C4⋊C4C22×D312yes—
296C4⋊C4C22×D314yes—
297C4⋊C4C22×D314yes—
298C22⋊C4—32no—
299C22⋊C4—32no—
300C22⋊C4—31no—
301C22⋊C4—31no—
302C22⋊C4—32no—
303C22⋊C4—34no—
304C22⋊C4—34no—
305C22⋊C4—34no—
306C22⋊C4—34no—
307C22⋊C4—32no—
308C22⋊C4—34no—
309C22⋊C4—34no—
310C22⋊C4—34no—
311C22⋊C4—32no—
312C22⋊C4—34no—
313C22⋊C4C22×D314yes—
314C22⋊C4C22×D312yes—
315C22⋊C4C22×D314yes—
316C42—34no—
317C42—32no—
318C42—32no—
319C42—34no—
320C42C22×D312yes—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3C2511yescharacteristic
328C2×D3—22no—
329C2×D3—24no—
330C2×D3—24no—
331C2×D3—22no—
332C2×D3—24no—
333C2×D3—22no—
334C2×D3—44no—
335C2×D3—84no—
336C3×C4—21no—
337C3×C4—24no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—24no—
341C3×C4H(5,𝔽2)12yes—
342Dic3—22no—
343Dic3—22no—
344Dic3—24no—
345Dic3—22no—
346C23—32no—
347C23—34no—
348C23—121no—
349C23—122no—
350C23—122no—
351C23—124no—
352C23C23×D311yescharacteristic
353C23C23×D311yescharacteristic
354C23C23×D311yescharacteristic
355Q8—24no—
356Q8—42no—
357Q8—41no—
358Q8—124no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—64no—
363D4—122no—
364D4—124no—
365D4—124no—
366D4—122no—
367D4—124no—
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—42no—
378C2×C4—64no—
379C2×C4—122no—
380C2×C4—124no—
381C2×C4—124no—
382C2×C4—121no—
383C2×C4C23×D311yescharacteristic
384C2×C4C23×D314yes—
385C2×C4C23×D314yes—
386C2×C4C23×D311yescharacteristic
387C2×C4C23×D311yescharacteristic
388C2×C4C23×D311yescharacteristic
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3C2×H(5,𝔽2)11yescharacteristic
393C2×C3#58512yes—
394D3—42no—
395D3—44no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—62no—
403C22—64no—
404C22—64no—
405C22—62no—
406C22—62no—
407C22—64no—
408C22—124no—
409C22—244no—
410C22C24×D311yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—24no—
415C4—24no—
416C4—62no—
417C4—62no—
418C4—64no—
419C4—62no—
420C4#100412yes—
421C3#354011yescharacteristic, complemented, Sylow
422C2—41no—
423C2—41no—
424C2—41no—
425C2—122no—
426C2—124no—
427C2C2×#100411yescharacteristic
428C2#596812yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#536C2
hypercenter#536D3
hypocenterC3#3540
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C3×#536C2semidirect
2. #5830C2semidirect
3. #5828C2semidirect
4. #5817C2semidirect
5. #5733C2semidirect
6. #5683C2semidirect
7. #5679C2semidirect
8. #5666C2semidirect
9. #5664C2semidirect
10. #5657C2semidirect
11. #5643C2semidirect
12. #5641C2semidirect
13. #5630C2semidirect
14. #5627C2semidirect
15. #5532C2semidirect
18. C22×C3×Q8C22semidirect
20. C22×D12C22semidirect
21. C22×C4×D3C22semidirect
22. C3×C4⋊Q8C22semidirect
24. C3×#123C22semidirect
25. C3×#123C22semidirect
26. C3×C22⋊Q8C22semidirect
27. C3×C22⋊Q8C22semidirect
28. C3×#117C22semidirect
29. #947C22semidirect
30. #947C22semidirect
31. #947C22semidirect
32. #946C22semidirect
33. #946C22semidirect
34. #946C22semidirect
35. Q8×Dic3C22semidirect
40. #937C22semidirect
41. #937C22semidirect
43. #935C22semidirect
45. #930C22semidirect
46. #930C22semidirect
48. C4×#67C22semidirect
49. C4×#67C22semidirect
50. #926C22semidirect
51. #926C22semidirect
52. #898C22semidirect
53. #898C22semidirect
54. #897C22semidirect
57. #894C22semidirect
58. #894C22semidirect
59. #893C22semidirect
60. #893C22semidirect
61. #892C22semidirect
62. #888C22semidirect
63. #887C22semidirect
65. #885C22semidirect
66. #885C22semidirect
67. #884C22semidirect
68. #883C22semidirect
70. #882C22semidirect
71. #67⋊C4C22semidirect
72. D3×C22⋊C4C22semidirect
73. #879C22semidirect
74. #875C22semidirect
75. #872C22semidirect
87. #536D3semidirect
157. C22×Q8C2×D3semidirect
161. C4⋊Q8C2×D3semidirect
179. #123C2×D3semidirect
180. #123C2×D3semidirect
185. C22⋊Q8C2×D3semidirect
186. C22⋊Q8C2×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×#5362
#58302
#58282
#58172
#57332
#56832
#56792
#56662
#56642
#56572
#56432
#56412
#56302
#56272
#55322
#35403