Finite Group Databasebeta

← #86973

Subgroups — #86973

Order 384.

1140 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#555C211yesmaximal, characteristic, complemented
2#5734C211yesmaximal, characteristic, complemented
3#5728C211yesmaximal, characteristic, complemented
4#5681C214yesmaximal, complemented
5#5678C214yesmaximal, complemented
6#5618C212yesmaximal, complemented
7#5607C212yesmaximal, complemented
8#5606C212yesmaximal, complemented
9#5599C212yesmaximal, complemented
10#5555C211yesmaximal, characteristic, complemented
11#5551C211yesmaximal, characteristic, complemented
12#5548C211yesmaximal, characteristic, complemented
13C4×D3×D4C211yesmaximal, characteristic, complemented
14#5533C212yesmaximal, complemented
15#5530C212yesmaximal, complemented
16#5524C212yesmaximal, complemented
17#5523C212yesmaximal, complemented
18#3506—31nomaximal, Sylow
19C2×#237C2212yescomplemented
20C2×D3×D4C2211yescharacteristic
21C2×#235C2212yes—
22C2×#235C2212yescomplemented
23C22×C4×D3C2212yes—
24C3×C4⋊Q8C2211yescharacteristic, complemented
25C3×#124C2214yescomplemented
26C3×#121C2212yes—
27C3×#121C2212yescomplemented
28C3×C4×D4C2211yescharacteristic
29C3×C4×D4C2211yescharacteristic, complemented
30C3×#117C2212yescomplemented
31C3×#117C2212yescomplemented
32#947C2214yescomplemented
33#946C2212yes—
34Q8×Dic3C2212yescomplemented
35#940C2214yes—
36#940C2211yescharacteristic, complemented
37#939C2214yescomplemented
38#938C2211yescharacteristic
39#938C2212yes—
40#937C2214yes—
41D4×Dic3C2211yescharacteristic
42D4×Dic3C2214yescomplemented
43#930C2212yes—
44C4×#67C2212yes—
45C4×#67C2212yescomplemented
46C2×C4⋊Dic3C2212yescomplemented
47#924C2212yescomplemented
48#897C2212yes—
49#897C2212yescomplemented
50#895C2212yes—
51#895C2211yescharacteristic
52#895C2212yescomplemented
53#893C2211yescharacteristic
54#892C2212yes—
55#892C2212yescomplemented
56#892C2212yescomplemented
57D3×C4⋊C4C2211yescharacteristic
58D3×C4⋊C4C2212yescomplemented
59#890C2212yescomplemented
60#888C2211yescharacteristic, complemented
61#887C2212yescomplemented
62#886C2214yescomplemented
63#886C2214yescomplemented
64#885C2214yes—
65#885C2212yescomplemented
66#884C2212yes—
67#884C2214yescomplemented
68#883C2214yescomplemented
69#883C2214yescomplemented
70#882C2214yes—
71#67⋊C4C2214yes—
72#67⋊C4C2212yes—
73#67⋊C4C2214yescomplemented
74D3×C22⋊C4C2212yes—
75#879C2214yescomplemented
76#876C2214yescomplemented
77C4×D12C2214yes—
78C4×D12C2211yescharacteristic
79C4×D12C2214yescomplemented
80C4×D12C2211yescharacteristic, complemented
81C42×D3C2211yescharacteristic
82#869C2211yescharacteristic, complemented
83C4×Dic6C2214yescomplemented
84#559—34no—
85#556—32no—
86#555—31no—
87#555D311yescharacteristic, complemented
88#553—32no—
89#553—32no—
90#550—32no—
91#547—31no—
92#547—31no—
93#538—34no—
94#535—32no—
95#533—31no—
96#532—31no—
97#525—32no—
98#525—32no—
99#520—32no—
100C2×C4×D4—31no—
101C23×D3C2312yes—
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2314yes—
106C22×C3×C4C2312yes—
107C22×C3×C4C2312yes—
108C2×#67C2312yes—
109C2×#67C2312yes—
110C2×#67C2312yes—
111C2×#67C2314yes—
112C2×#67C2314yes—
113C2×#67C2312yes—
114C22×Dic3C2314yes—
115C22×Dic3C2312yes—
116#237—44no—
117D3×D4—24no—
118#235—28no—
119#235—44no—
120C2×D12C2312yes—
121C2×D12C2311yescharacteristic
122C2×D12C2311yescharacteristic
123C2×D12C2314yes—
124C2×C4×D3—44no—
125C2×C4×D3C2311yescharacteristic
126C2×C4×D3C2312yes—
127C2×C4×D3C2312yes—
128C2×C4×D3C2311yescharacteristic
129C2×C4×D3C2312yes—
130C2×C4×D3C2314yes—
131C2×C4×D3C2312yes—
132C2×C4×D3C2311yescharacteristic
133C2×Dic6C2312yes—
134C2×Dic6C2312yes—
135C3×C4⋊C4C2312yes—
136C3×C4⋊C4C2311yescharacteristic
137C3×C4⋊C4C2312yes—
138C3×C4⋊C4C2311yescharacteristic
139C3×C22⋊C4C2314yes—
140C3×C22⋊C4C2312yes—
141C3×C22⋊C4C2314yes—
142C3×C22⋊C4C2312yes—
143C3×C42C2314yes—
144C3×C42C2311yescharacteristic
145#217C2312yes—
146#217C2314yes—
147#217C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2314yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2312yes—
154(C2×D3)⋊C4C2312yes—
155(C2×D3)⋊C4C2314yes—
156C4⋊Dic3C2314yes—
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2312yes—
159C4⋊Dic3C2311yescharacteristic
160C4⋊Dic3C2314yes—
161C4⋊Dic3C2311yescharacteristic
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2312yes—
165Dic3⋊C4C2314yes—
166C4×Dic3C2314yes—
167C4×Dic3C2314yes—
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
171C2×#41—32no—
172C2×#41—32no—
173C2×#41—32no—
174C22×D4—31no—
175C23×C4—32no—
176C4⋊Q8—31no—
177C4⋊Q8—31no—
178C4⋊Q8C2×D311yescharacteristic, complemented
179#127—31no—
180#127—34no—
181#126—34no—
182#126—34no—
183#125—32no—
184#124—34no—
185#124—32no—
186#124—34no—
187#124C2×D314yescomplemented
188#123—34no—
189#123—34no—
190#123—34no—
191#123—34no—
192C22⋊Q8—32no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196#121—31no—
197#121—32no—
198#121—34no—
199#121—32no—
200#121—31no—
201#121—32no—
202#121—32no—
203#121—32no—
204#121—34no—
205#121C2×D312yes—
206#121C2×D312yescomplemented
207#120—34no—
208#120—34no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—34no—
212C4×D4—31no—
213C4×D4—34no—
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—31no—
217C4×D4—32no—
218C4×D4—34no—
219C4×D4—34no—
220C4×D4—31no—
221C4×D4—34no—
222C4×D4—34no—
223C4×D4—31no—
224C4×D4C2×D311yescharacteristic
225C4×D4C2×D311yescharacteristic, complemented
226#117—32no—
227#117—32no—
228#117—32no—
229#117C2×D312yescomplemented
230#117C2×D312yescomplemented
231C2×C4⋊C4—32no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—32no—
234C2×C22⋊C4—32no—
235C2×C42—31no—
236C23×C3C2412yes—
237C23×C3C2412yes—
238C22×D3—24no—
239C22×D3—42no—
240C22×D3—42no—
241C22×D3C2412yes—
242C22×D3C2412yes—
243C22×D3C2411yescharacteristic
244C3×Q8—42no—
245C3×D4—22no—
246C3×D4—22no—
247C3×D4—44no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4C2414yes—
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2412yes—
253C2×C3×C4C2412yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256#67—24no—
257#67—24no—
258#67—24no—
259#67—44no—
260#67—44no—
261#67—42no—
262C2×Dic3—42no—
263C2×Dic3—44no—
264C2×Dic3C2411yescharacteristic
265C2×Dic3C2412yes—
266C2×Dic3C2412yes—
267C2×Dic3C2411yescharacteristic
268C2×Dic3C2414yes—
269C2×Dic3C2411yescharacteristic
270D12—24no—
271D12—22no—
272D12—22no—
273D12—44no—
274C4×D3—24no—
275C4×D3—22no—
276C4×D3—24no—
277C4×D3—42no—
278C4×D3—42no—
279C4×D3—41no—
280C4×D3—44no—
281C4×D3#4114yescomplemented
282Dic6—24no—
283Dic6—24no—
284C24—32no—
285#41—68no—
286#41—124no—
287#41—124no—
288C2×Q8—32no—
289C2×Q8—32no—
290C2×Q8C22×D312yes—
291C2×D4—32no—
292C2×D4—32no—
293C2×D4—32no—
294C2×D4—32no—
295C2×D4—34no—
296C2×D4—34no—
297C2×D4—32no—
298C2×D4—31no—
299C2×D4—31no—
300C2×D4—34no—
301C2×D4—64no—
302C2×D4C22×D311yescharacteristic
303C2×D4C22×D311yescharacteristic
304C2×D4C22×D314yes—
305C22×C4—31no—
306C22×C4—34no—
307C22×C4—32no—
308C22×C4—31no—
309C22×C4—32no—
310C22×C4—32no—
311C22×C4—31no—
312C22×C4—32no—
313C22×C4—34no—
314C22×C4—32no—
315C22×C4—124no—
316C22×C4C22×D312yes—
317C22×C4C22×D312yes—
318C4⋊C4—31no—
319C4⋊C4—32no—
320C4⋊C4—32no—
321C4⋊C4—34no—
322C4⋊C4—34no—
323C4⋊C4—31no—
324C4⋊C4—34no—
325C4⋊C4—32no—
326C4⋊C4—34no—
327C4⋊C4—32no—
328C4⋊C4C22×D311yescharacteristic
329C4⋊C4C22×D312yes—
330C4⋊C4C22×D312yes—
331C4⋊C4C22×D311yescharacteristic
332C22⋊C4—32no—
333C22⋊C4—32no—
334C22⋊C4—34no—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—34no—
338C22⋊C4—34no—
339C22⋊C4—34no—
340C22⋊C4—32no—
341C22⋊C4—34no—
342C22⋊C4—32no—
343C22⋊C4C22×D312yes—
344C22⋊C4C22×D314yes—
345C22⋊C4C22×D314yes—
346C22⋊C4C22×D312yes—
347C42—31no—
348C42—31no—
349C42—31no—
350C42—34no—
351C42—34no—
352C42C22×D311yescharacteristic
353C42C22×D314yes—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3C2511yescharacteristic
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—22no—
364C2×D3—22no—
365C2×D3—22no—
366C2×D3—22no—
367C2×D3—22no—
368C2×D3—22no—
369C2×D3—48no—
370C2×D3C2×#4112yes—
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—21no—
374C3×C4—24no—
375C3×C4C2×#4112yes—
376C3×C4H(5,𝔽2)12yes—
377Dic3—22no—
378Dic3—24no—
379Dic3—21no—
380Dic3—22no—
381Dic3—21no—
382Dic3C2×#4112yes—
383C23—31no—
384C23—32no—
385C23—32no—
386C23—64no—
387C23—122no—
388C23—122no—
389C23C23×D312yes—
390C23C23×D312yes—
391Q8—42no—
392Q8—64no—
393Q8—64no—
394D4—22no—
395D4—22no—
396D4—44no—
397D4—64no—
398D4—64no—
399D4—64no—
400D4—64no—
401D4—62no—
402D4—62no—
403D4—124no—
404D4—124no—
405D4—124no—
406D4—122no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—31no—
410C2×C4—34no—
411C2×C4—32no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—32no—
415C2×C4—34no—
416C2×C4—64no—
417C2×C4—62no—
418C2×C4—64no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—124no—
422C2×C4—121no—
423C2×C4—122no—
424C2×C4—124no—
425C2×C4C23×D314yes—
426C2×C4C23×D311yescharacteristic
427C2×C4C23×D312yes—
428C2×C4C23×D312yes—
429C2×C4C23×D311yescharacteristic
430C2×C4C23×D311yescharacteristic
431C2×C3—42no—
432C2×C3—42no—
433C2×C3C22×#4111yescharacteristic
434C2×C3C2×H(5,𝔽2)11yescharacteristic
435C2×C3#58511yescharacteristic
436D3—22no—
437D3—42no—
438D3—42no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—32no—
446C22—61no—
447C22—61no—
448C22—62no—
449C22—62no—
450C22—62no—
451C22—62no—
452C22—62no—
453C22—62no—
454C22—128no—
455C22C24×D311yescharacteristic
456C4—22no—
457C4—22no—
458C4—21no—
459C4—24no—
460C4—32no—
461C4—62no—
462C4—61no—
463C4—62no—
464C4—64no—
465C4—61no—
466C4D3×#4112yes—
467C4#100912yes—
468C3#350611yescharacteristic, complemented, Sylow
469C2—42no—
470C2—42no—
471C2—62no—
472C2—122no—
473C2—122no—
474C2C2×D3×#4111yescharacteristic
475C2C2×#100911yescharacteristic
476C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#555C2
hypercenter#555D3
hypocenterC3#3506
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #555

Complements

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

NH ≅ G/Ntype
1. C3×#555C2semidirect
2. #5734C2semidirect
3. #5728C2semidirect
4. #5681C2semidirect
5. #5678C2semidirect
6. #5618C2semidirect
7. #5607C2semidirect
8. #5606C2semidirect
9. #5599C2semidirect
10. #5555C2semidirect
11. #5551C2semidirect
12. #5548C2semidirect
13. C4×D3×D4C2semidirect
14. #5533C2semidirect
15. #5530C2semidirect
16. #5524C2semidirect
17. #5523C2semidirect
19. C2×#237C22semidirect
22. C2×#235C22semidirect
24. C3×C4⋊Q8C22semidirect
25. C3×#124C22semidirect
27. C3×#121C22semidirect
29. C3×C4×D4C22semidirect
30. C3×#117C22semidirect
31. C3×#117C22semidirect
32. #947C22semidirect
34. Q8×Dic3C22semidirect
36. #940C22semidirect
37. #939C22semidirect
42. D4×Dic3C22semidirect
45. C4×#67C22semidirect
46. C2×C4⋊Dic3C22semidirect
47. #924C22semidirect
49. #897C22semidirect
52. #895C22semidirect
55. #892C22semidirect
56. #892C22semidirect
58. D3×C4⋊C4C22semidirect
59. #890C22semidirect
60. #888C22semidirect
61. #887C22semidirect
62. #886C22semidirect
63. #886C22semidirect
65. #885C22semidirect
67. #884C22semidirect
68. #883C22semidirect
69. #883C22semidirect
73. #67⋊C4C22semidirect
75. #879C22semidirect
76. #876C22semidirect
79. C4×D12C22semidirect
80. C4×D12C22semidirect
82. #869C22semidirect
83. C4×Dic6C22semidirect
87. #555D3semidirect
178. C4⋊Q8C2×D3semidirect
187. #124C2×D3semidirect
206. #121C2×D3semidirect
225. C4×D4C2×D3semidirect
229. #117C2×D3semidirect
230. #117C2×D3semidirect
281. C4×D3#41semidirect
468. C3#3506semidirect

Maximal subgroups

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

subgroupindex
C3×#5552
#57342
#57282
#56812
#56782
#56182
#56072
#56062
#55992
#55552
#55512
#55482
C4×D3×D42
#55332
#55302
#55242
#55232
#35063