Finite Group Databasebeta

← #86332

Subgroups — #86332

Order 384.

1142 conjugacy classes of subgroups.

The subgroup lattice has 480 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#5827C211yesmaximal, characteristic, complemented
4C2×#938C211yesmaximal, characteristic, complemented
5#5798C211yesmaximal, characteristic, complemented
6#5719C212yesmaximal, complemented
7#5716C212yesmaximal, complemented
8#5682C212yesmaximal, complemented
9#5680C212yesmaximal, complemented
10#5618C212yesmaximal, complemented
11#5616C212yesmaximal, complemented
12#5614C212yesmaximal, complemented
13#5611C212yesmaximal, complemented
14#5591C214yesmaximal, complemented
15#5587C214yesmaximal, complemented
16D3×#117C211yesmaximal, characteristic, complemented
17#5519C211yesmaximal, characteristic, complemented
18#3494—31nomaximal, Sylow
19C2×C3×#41C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic
21C22×#67C2212yes—
22C2×#237C2214yescomplemented
23C22×C4×D3C2211yescharacteristic
24C3×#127C2212yescomplemented
25C3×#124C2212yescomplemented
26C3×#121C2212yescomplemented
27C3×#121C2212yescomplemented
28C3×#120C2214yes—
29C3×#117C2211yescharacteristic
30C2×#217C2214yes—
31C2×#217C2212yes—
32#946C2211yescharacteristic, complemented
33#946C2211yescharacteristic, complemented
34Q8×Dic3C2212yescomplemented
35#939C2214yescomplemented
36#939C2214yescomplemented
37#938C2212yes—
38#938C2212yes—
39#938C2212yes—
40#938C2212yes—
41#938C2211yescharacteristic, complemented
42#938C2214yescomplemented
43#938C2211yescharacteristic, complemented
44#937C2212yes—
45#936C2214yescomplemented
46#935C2212yes—
47#935C2214yescomplemented
48#935C2214yescomplemented
49D4×Dic3C2214yes—
50D4×Dic3C2212yescomplemented
51D4×Dic3C2212yescomplemented
52D4×Dic3C2212yescomplemented
53D4×Dic3C2214yescomplemented
54D4×Dic3C2212yescomplemented
55D4×Dic3C2212yescomplemented
56C4×#67C2212yescomplemented
57C4×#67C2212yescomplemented
58#926C2211yescharacteristic
59#926C2211yescharacteristic
60#926C2212yescomplemented
61C2×C4⋊Dic3C2211yescharacteristic
62C2×C4⋊Dic3C2212yescomplemented
63#892C2211yescharacteristic, complemented
64#892C2211yescharacteristic, complemented
65D3×C4⋊C4C2211yescharacteristic, complemented
66D3×C4⋊C4C2211yescharacteristic, complemented
67#890C2212yescomplemented
68#888C2212yescomplemented
69#886C2214yescomplemented
70#885C2214yescomplemented
71#883C2214yescomplemented
72#67⋊C4C2214yescomplemented
73D3×C22⋊C4C2212yes—
74#879C2214yescomplemented
75#879C2214yescomplemented
76#878C2214yes—
77#878C2214yescomplemented
78#877C2212yes—
79#877C2214yescomplemented
80#872C2212yescomplemented
81C42×D3C2212yescomplemented
82#870C2212yescomplemented
83#869C2212yescomplemented
84#555—32no—
85#553—32no—
86#553—32no—
87#551—34no—
88#547—32no—
89#546—34no—
90#544—31no—
91#542—32no—
92#540—32no—
93#535D311yescharacteristic, complemented
94#532—32no—
95#529—32no—
96#525—31no—
97C2×#121—31no—
98#520—31no—
99#518—31no—
100C2×#117—31no—
101C24×C3C2312yes—
102C23×D3C2311yescharacteristic
103C3×#41—24no—
104C2×C3×Q8C2311yescharacteristic
105C2×C3×D4—24no—
106C2×C3×D4C2312yes—
107C2×C3×D4C2312yes—
108C2×C3×D4C2312yes—
109C2×C3×D4C2311yescharacteristic
110C2×C3×D4C2314yes—
111C2×C3×D4C2314yes—
112C22×C3×C4C2312yes—
113C22×C3×C4C2311yescharacteristic
114C2×#67—44no—
115C2×#67C2314yes—
116C2×#67C2312yes—
117C2×#67C2314yes—
118C2×#67C2312yes—
119C22×Dic3C2311yescharacteristic
120C22×Dic3C2314yes—
121C22×Dic3C2312yes—
122C22×Dic3C2314yes—
123C22×Dic3C2314yes—
124#237—48no—
125C2×C4×D3—24no—
126C2×C4×D3C2311yescharacteristic
127C2×C4×D3C2312yes—
128C2×C4×D3C2311yescharacteristic
129C2×C4×D3C2311yescharacteristic
130C2×C4×D3C2311yescharacteristic
131C2×C4×D3C2312yes—
132C2×Dic6C2314yes—
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2311yescharacteristic
135C3×C22⋊C4C2312yes—
136C3×C22⋊C4C2314yes—
137C3×C22⋊C4C2314yes—
138C3×C42C2312yes—
139#217C2312yes—
140#217C2314yes—
141#217C2312yes—
142#217C2314yes—
143#217C2314yes—
144#217C2314yes—
145#217C2314yes—
146#217C2312yes—
147#217C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2312yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2314yes—
155C4⋊Dic3C2311yescharacteristic
156C4⋊Dic3C2314yes—
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2311yescharacteristic
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2312yes—
165Dic3⋊C4C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2312yes—
169C4×Dic3C2314yes—
170C4×Dic3C2312yes—
171C2×#41—34no—
172C2×#41C2×D311yescharacteristic
173C22×D4—32no—
174C22×D4C2×D311yescharacteristic
175C23×C4—31no—
176C4⋊Q8—32no—
177C4⋊Q8—32no—
178#127C2×D312yescomplemented
179#126—34no—
180#126—34no—
181#125—32no—
182#125—32no—
183#124—34no—
184#124—34no—
185#124C2×D312yescomplemented
186#123—32no—
187#123—34no—
188#123—34no—
189#123—34no—
190#123—34no—
191C22⋊Q8—31no—
192C22⋊Q8—31no—
193C22⋊Q8—34no—
194C22⋊Q8—34no—
195#121—32no—
196#121—32no—
197#121—32no—
198#121—32no—
199#121—31no—
200#121—31no—
201#121—34no—
202#121—34no—
203#121—34no—
204#121C2×D312yescomplemented
205#121C2×D312yescomplemented
206#120—32no—
207#120C2×D314yes—
208C4×Q8—32no—
209C4×D4—32no—
210C4×D4—32no—
211C4×D4—32no—
212C4×D4—32no—
213C4×D4—32no—
214C4×D4—34no—
215C4×D4—34no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—34no—
219#117—31no—
220#117—31no—
221#117—32no—
222#117—31no—
223#117—31no—
224#117—32no—
225#117—34no—
226#117—32no—
227#117C2×D311yescharacteristic
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—31no—
232C2×C22⋊C4—34no—
233C2×C22⋊C4—32no—
234C2×C22⋊C4—32no—
235C2×C42—32no—
236C23×C3—24no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3C2414yes—
240C23×C3C2412yes—
241C23×C3C2411yescharacteristic
242C22×D3—22no—
243C22×D3—22no—
244C22×D3—41no—
245C22×D3C2411yescharacteristic
246C22×D3C2411yescharacteristic
247C3×Q8—22no—
248C3×D4—24no—
249C3×D4—24no—
250C3×D4—24no—
251C3×D4—22no—
252C3×D4—44no—
253C3×D4—44no—
254C2×C3×C4—24no—
255C2×C3×C4#4114yescomplemented
256C2×C3×C4C2412yes—
257C2×C3×C4C2412yes—
258C2×C3×C4C2412yes—
259C2×C3×C4C2411yescharacteristic
260C2×C3×C4C2411yescharacteristic
261#67—24no—
262#67—28no—
263#67—44no—
264#67—42no—
265C2×Dic3—22no—
266C2×Dic3—44no—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—42no—
270C2×Dic3C2411yescharacteristic
271C2×Dic3C2412yes—
272C2×Dic3C2412yes—
273C2×Dic3C2412yes—
274C2×Dic3C2414yes—
275C2×Dic3C2412yes—
276C2×Dic3C2411yescharacteristic
277C4×D3—24no—
278C4×D3—22no—
279C4×D3—22no—
280C4×D3—22no—
281C4×D3—22no—
282C4×D3—42no—
283Dic6—44no—
284C24—31no—
285C24C22×D312yes—
286#41—24no—
287#41—128no—
288C2×Q8—34no—
289C2×Q8C22×D311yescharacteristic
290C2×D4—24no—
291C2×D4—32no—
292C2×D4—34no—
293C2×D4—32no—
294C2×D4—34no—
295C2×D4—124no—
296C2×D4C22×D312yes—
297C2×D4C22×D312yes—
298C2×D4C22×D312yes—
299C2×D4C22×D311yescharacteristic
300C2×D4C22×D314yes—
301C2×D4C22×D314yes—
302C22×C4—31no—
303C22×C4—32no—
304C22×C4—34no—
305C22×C4—34no—
306C22×C4—34no—
307C22×C4—31no—
308C22×C4—32no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—32no—
313C22×C4—64no—
314C22×C4C22×D312yes—
315C22×C4C22×D311yescharacteristic
316C4⋊C4—31no—
317C4⋊C4—32no—
318C4⋊C4—31no—
319C4⋊C4—32no—
320C4⋊C4—32no—
321C4⋊C4—32no—
322C4⋊C4—34no—
323C4⋊C4—32no—
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—34no—
327C4⋊C4—34no—
328C4⋊C4—34no—
329C4⋊C4—32no—
330C4⋊C4C22×D311yescharacteristic
331C4⋊C4C22×D311yescharacteristic
332C22⋊C4—32no—
333C22⋊C4—32no—
334C22⋊C4—34no—
335C22⋊C4—34no—
336C22⋊C4—34no—
337C22⋊C4—34no—
338C22⋊C4—34no—
339C22⋊C4—32no—
340C22⋊C4—32no—
341C22⋊C4—32no—
342C22⋊C4—32no—
343C22⋊C4—32no—
344C22⋊C4—32no—
345C22⋊C4C22×D312yes—
346C22⋊C4C22×D314yes—
347C22⋊C4C22×D314yes—
348C42—32no—
349C42—32no—
350C42—32no—
351C42—34no—
352C42—32no—
353C42C22×D312yes—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—22no—
357C22×C3—24no—
358C22×C3—22no—
359C22×C3—24no—
360C22×C3—24no—
361C22×C3—22no—
362C22×C3—84no—
363C22×C3C2×#4112yes—
364C22×C3C2511yescharacteristic
365C2×D3—21no—
366C2×D3—21no—
367C2×D3—21no—
368C2×D3—21no—
369C2×D3—21no—
370C2×D3—44no—
371C2×D3H(5,𝔽2)12yes—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4C2×#4112yes—
376C3×C4C2×#4112yes—
377Dic3—21no—
378Dic3—21no—
379Dic3—22no—
380Dic3—24no—
381Dic3—22no—
382Dic3—22no—
383Dic3—22no—
384C23—24no—
385C23—31no—
386C23—31no—
387C23—42no—
388C23—42no—
389C23—62no—
390C23—62no—
391C23—121no—
392C23C23×D314yes—
393C23C23×D312yes—
394C23C23×D311yescharacteristic
395Q8—22no—
396Q8—124no—
397D4—24no—
398D4—24no—
399D4—24no—
400D4—22no—
401D4—44no—
402D4—44no—
403D4—68no—
404D4—64no—
405D4—124no—
406D4—122no—
407C2×C4—24no—
408C2×C4—31no—
409C2×C4—32no—
410C2×C4—32no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—34no—
414C2×C4—31no—
415C2×C4—64no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—124no—
424C2×C4—124no—
425C2×C4—124no—
426C2×C4#23714yescomplemented
427C2×C4C23×D312yes—
428C2×C4C23×D312yes—
429C2×C4C23×D312yes—
430C2×C4C23×D311yescharacteristic
431C2×C4C23×D311yescharacteristic
432C2×C3—22no—
433C2×C3—42no—
434C2×C3—44no—
435C2×C3C22×#4111yescharacteristic
436C2×C3C2×H(5,𝔽2)11yescharacteristic
437C2×C3#58511yescharacteristic
438D3—22no—
439D3—41no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—24no—
444C22—22no—
445C22—24no—
446C22—24no—
447C22—22no—
448C22—32no—
449C22—61no—
450C22—61no—
451C22—61no—
452C22—61no—
453C22—61no—
454C22—84no—
455C22—124no—
456C22C2×#23712yes—
457C22C24×D311yescharacteristic
458C4—22no—
459C4—22no—
460C4—22no—
461C4—61no—
462C4—61no—
463C4—62no—
464C4—64no—
465C4—62no—
466C4—62no—
467C4—62no—
468C4C2×#23712yes—
469C4C2×#23712yes—
470C3#349411yescharacteristic, complemented, Sylow
471C2—22no—
472C2—44no—
473C2—42no—
474C2—62no—
475C2—121no—
476C2C22×#23711yescharacteristic
477C2D3×H(5,𝔽2)11yescharacteristic
478C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#535C2
hypercenter#535D3
hypocenterC3#3494
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. #5827C2semidirect
4. C2×#938C2semidirect
5. #5798C2semidirect
6. #5719C2semidirect
7. #5716C2semidirect
8. #5682C2semidirect
9. #5680C2semidirect
10. #5618C2semidirect
11. #5616C2semidirect
12. #5614C2semidirect
13. #5611C2semidirect
14. #5591C2semidirect
15. #5587C2semidirect
16. D3×#117C2semidirect
17. #5519C2semidirect
22. C2×#237C22semidirect
24. C3×#127C22semidirect
25. C3×#124C22semidirect
26. C3×#121C22semidirect
27. C3×#121C22semidirect
32. #946C22semidirect
33. #946C22semidirect
34. Q8×Dic3C22semidirect
35. #939C22semidirect
36. #939C22semidirect
41. #938C22semidirect
42. #938C22semidirect
43. #938C22semidirect
45. #936C22semidirect
47. #935C22semidirect
48. #935C22semidirect
50. D4×Dic3C22semidirect
51. D4×Dic3C22semidirect
52. D4×Dic3C22semidirect
53. D4×Dic3C22semidirect
54. D4×Dic3C22semidirect
55. D4×Dic3C22semidirect
56. C4×#67C22semidirect
57. C4×#67C22semidirect
60. #926C22semidirect
62. C2×C4⋊Dic3C22semidirect
63. #892C22semidirect
64. #892C22semidirect
65. D3×C4⋊C4C22semidirect
66. D3×C4⋊C4C22semidirect
67. #890C22semidirect
68. #888C22semidirect
69. #886C22semidirect
70. #885C22semidirect
71. #883C22semidirect
72. #67⋊C4C22semidirect
74. #879C22semidirect
75. #879C22semidirect
77. #878C22semidirect
79. #877C22semidirect
80. #872C22semidirect
81. C42×D3C22semidirect
82. #870C22semidirect
83. #869C22semidirect
93. #535D3semidirect
178. #127C2×D3semidirect
185. #124C2×D3semidirect
204. #121C2×D3semidirect
205. #121C2×D3semidirect
255. C2×C3×C4#41semidirect
426. C2×C4#237semidirect
470. C3#3494semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#58282
#58272
C2×#9382
#57982
#57192
#57162
#56822
#56802
#56182
#56162
#56142
#56112
#55912
#55872
D3×#1172
#55192
#34943