Finite Group Databasebeta

← #86625

Subgroups — #86625

Order 384.

1150 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D42C211yesmaximal, characteristic, complemented
2#5806C212yesmaximal, complemented
3#5806C212yesmaximal, complemented
4#5805C212yesmaximal, complemented
5#5805C212yesmaximal, complemented
6#5718C211yesmaximal, characteristic, complemented
7#5716C211yesmaximal, characteristic, complemented
8#5614C212yesmaximal, complemented
9#5608C212yesmaximal, complemented
10#5603C212yesmaximal, complemented
11#5599C212yesmaximal, complemented
12#5595C214yesmaximal, complemented
13#5593C214yesmaximal, complemented
14#5557C211yesmaximal, characteristic, complemented
15#5551C211yesmaximal, characteristic, complemented
16#5548C211yesmaximal, characteristic, complemented
17#5542C211yesmaximal, characteristic, complemented
18#3546—31nomaximal, Sylow
19C22×C3×D4C2212yes—
20C22×C3×D4C2212yes—
21C2×#237C2212yescomplemented
22C2×#237C2212yescomplemented
23C2×#235C2212yescomplemented
24C2×#235C2212yescomplemented
25C3×#127C2211yescharacteristic
26C3×#121C2212yes—
27C3×#121C2212yes—
28C3×#120C2214yes—
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2211yescharacteristic
31#953C2214yescomplemented
32#953C2214yescomplemented
33#953C2214yescomplemented
34C2×#217C2214yescomplemented
35#940C2211yescharacteristic
36#940C2212yescomplemented
37#940C2214yescomplemented
38#939C2214yescomplemented
39#939C2214yescomplemented
40#939C2214yescomplemented
41#938C2211yescharacteristic
42#938C2211yescharacteristic
43#938C2212yes—
44#938C2212yescomplemented
45#938C2212yescomplemented
46#938C2212yescomplemented
47#938C2214yescomplemented
48#938C2214yescomplemented
49#936C2212yes—
50#936C2211yescharacteristic
51#936C2212yescomplemented
52#936C2214yescomplemented
53#935C2214yescomplemented
54#935C2214yescomplemented
55D4×Dic3C2212yescomplemented
56D4×Dic3C2212yescomplemented
57D4×Dic3C2212yescomplemented
58C4×#67C2212yescomplemented
59C4×#67C2212yescomplemented
60#926C2212yescomplemented
61#926C2212yescomplemented
62#924C2212yescomplemented
63#924C2212yescomplemented
64#897C2211yescharacteristic, complemented
65#895C2211yescharacteristic, complemented
66D3×C4⋊C4C2212yescomplemented
67#890C2211yescharacteristic, complemented
68#888C2211yescharacteristic, complemented
69#887C2212yescomplemented
70#885C2212yescomplemented
71#884C2212yescomplemented
72#884C2214yescomplemented
73#883C2214yescomplemented
74#883C2212yescomplemented
75#67⋊C4C2214yescomplemented
76#879C2214yescomplemented
77#879C2212yescomplemented
78#879C2214yescomplemented
79#878C2214yescomplemented
80C4×D12C2211yescharacteristic, complemented
81C42×D3C2211yescharacteristic, complemented
82#869C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#560—34no—
85#556—31no—
86#553—32no—
87#550—31no—
88#550—32no—
89#547—31no—
90#546—32no—
91#546—32no—
92D42D311yescharacteristic, complemented
93#542—32no—
94#538—34no—
95#537—32no—
96#535—32no—
97#535—32no—
98#532—31no—
99#532—31no—
100#529—31no—
101C24×C3C2314yes—
102C2×C3×D4—44no—
103C2×C3×D4—44no—
104C2×C3×D4C2312yes—
105C2×C3×D4C2312yes—
106C2×C3×D4C2311yescharacteristic
107C2×C3×D4C2311yescharacteristic
108C2×C3×D4C2312yes—
109C2×C3×D4C2314yes—
110C2×C3×D4C2314yes—
111C22×C3×C4C2312yes—
112C22×C3×C4C2312yes—
113C2×#67C2314yes—
114C2×#67C2312yes—
115C2×#67C2314yes—
116C2×#67C2312yes—
117C2×#67C2312yes—
118C2×#67C2314yes—
119C2×#67C2314yes—
120C2×#67C2312yes—
121C22×Dic3C2314yes—
122C22×Dic3C2314yes—
123#237—44no—
124#237—44no—
125#235—44no—
126#235—44no—
127C2×D12C2311yescharacteristic
128C2×D12C2312yes—
129C2×C4×D3C2311yescharacteristic
130C2×C4×D3C2311yescharacteristic
131C2×C4×D3C2312yes—
132C2×C4×D3C2311yescharacteristic
133C2×C4×D3C2314yes—
134C2×Dic6C2311yescharacteristic
135C2×Dic6C2312yes—
136C2×Dic6C2312yes—
137C3×C4⋊C4C2311yescharacteristic
138C3×C4⋊C4C2311yescharacteristic
139C3×C22⋊C4C2314yes—
140C3×C22⋊C4C2312yes—
141C3×C22⋊C4C2312yes—
142C3×C42C2311yescharacteristic
143#217C2314yes—
144#217C2314yes—
145#217C2314yes—
146#217C2312yes—
147#217C2312yes—
148#217C2314yes—
149#217C2314yes—
150#217C2314yes—
151#217C2312yes—
152#217C2312yes—
153(C2×D3)⋊C4C2312yes—
154(C2×D3)⋊C4C2312yes—
155(C2×D3)⋊C4C2312yes—
156C4⋊Dic3C2311yescharacteristic
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2312yes—
159C4⋊Dic3C2311yescharacteristic
160C4⋊Dic3C2312yes—
161C4⋊Dic3C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
171C4×Dic3C2314yes—
172C2×#41—32no—
173C2×#41—32no—
174C2×#41—32no—
175C2×#41—32no—
176C22×D4C2×D312yes—
177C22×D4C2×D312yes—
178C4⋊Q8—31no—
179C4⋊Q8—31no—
180#127—31no—
181#127—32no—
182#127—34no—
183#127C2×D311yescharacteristic
184#126—32no—
185#126—34no—
186#126—34no—
187#125—31no—
188#124—31no—
189#124—32no—
190#124—34no—
191#124—32no—
192#124—32no—
193#123—34no—
194#123—34no—
195#123—32no—
196#123—34no—
197C22⋊Q8—32no—
198C22⋊Q8—32no—
199C22⋊Q8—31no—
200C22⋊Q8—34no—
201#121—31no—
202#121—32no—
203#121—31no—
204#121—32no—
205#121—34no—
206#121—32no—
207#121—32no—
208#121—34no—
209#121—34no—
210#121—34no—
211#121—34no—
212#121—31no—
213#121—32no—
214#121—34no—
215#121C2×D312yes—
216#121C2×D312yes—
217#120—34no—
218#120—34no—
219#120—34no—
220#120C2×D314yes—
221C4×Q8—32no—
222C4×Q8—31no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—32no—
226C4×D4—32no—
227C4×D4—32no—
228C4×D4—34no—
229C4×D4—31no—
230C4×D4C2×D311yescharacteristic
231C4×D4C2×D311yescharacteristic
232#117—32no—
233#117—32no—
234C2×C4⋊C4—32no—
235C2×C22⋊C4—34no—
236C2×C42—31no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3C2412yes—
241C23×C3C2412yes—
242C23×C3C2414yes—
243C22×D3C2411yescharacteristic
244C22×D3C2412yes—
245C3×D4—24no—
246C3×D4—24no—
247C3×D4—22no—
248C3×D4—22no—
249C3×D4—44no—
250C3×D4—44no—
251C3×D4—42no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4C2412yes—
255C2×C3×C4C2411yescharacteristic
256C2×C3×C4C2412yes—
257C2×C3×C4C2411yescharacteristic
258C2×C3×C4C2411yescharacteristic
259#67—44no—
260#67—42no—
261#67—42no—
262#67—44no—
263#67—44no—
264#67—42no—
265#67—42no—
266#67—44no—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3C2412yes—
270C2×Dic3C2411yescharacteristic
271C2×Dic3C2411yescharacteristic
272C2×Dic3C2411yescharacteristic
273C2×Dic3C2412yes—
274C2×Dic3C2414yes—
275C2×Dic3C2412yes—
276D12—22no—
277D12—42no—
278C4×D3—22no—
279C4×D3—24no—
280C4×D3—22no—
281C4×D3—41no—
282C4×D3—44no—
283Dic6—24no—
284Dic6—22no—
285Dic6—42no—
286C24C22×D314yes—
287#41—124no—
288#41—124no—
289#41—124no—
290#41—124no—
291C2×Q8—31no—
292C2×Q8—32no—
293C2×Q8—32no—
294C2×D4—32no—
295C2×D4—32no—
296C2×D4—34no—
297C2×D4—32no—
298C2×D4—34no—
299C2×D4—32no—
300C2×D4—34no—
301C2×D4—34no—
302C2×D4—31no—
303C2×D4—32no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4C22×D312yes—
307C2×D4C22×D312yes—
308C2×D4C22×D311yescharacteristic
309C2×D4C22×D311yescharacteristic
310C2×D4C22×D312yes—
311C2×D4C22×D314yes—
312C2×D4C22×D314yes—
313C22×C4—34no—
314C22×C4—34no—
315C22×C4—31no—
316C22×C4—31no—
317C22×C4—32no—
318C22×C4—31no—
319C22×C4—34no—
320C22×C4C22×D312yes—
321C22×C4C22×D312yes—
322C4⋊C4—32no—
323C4⋊C4—34no—
324C4⋊C4—31no—
325C4⋊C4—32no—
326C4⋊C4—32no—
327C4⋊C4—31no—
328C4⋊C4—32no—
329C4⋊C4—32no—
330C4⋊C4—32no—
331C4⋊C4—34no—
332C4⋊C4—32no—
333C4⋊C4C22×D311yescharacteristic
334C4⋊C4C22×D311yescharacteristic
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—34no—
338C22⋊C4—34no—
339C22⋊C4—34no—
340C22⋊C4—32no—
341C22⋊C4—32no—
342C22⋊C4—34no—
343C22⋊C4—34no—
344C22⋊C4—34no—
345C22⋊C4—32no—
346C22⋊C4—32no—
347C22⋊C4—32no—
348C22⋊C4C22×D314yes—
349C22⋊C4C22×D312yes—
350C22⋊C4C22×D312yes—
351C42—31no—
352C42—31no—
353C42—31no—
354C42—32no—
355C42—34no—
356C42C22×D311yescharacteristic
357C22×C3—24no—
358C22×C3—24no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—24no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—88no—
367C22×C3C2511yescharacteristic
368C2×D3—21no—
369C2×D3—22no—
370C2×D3—21no—
371C2×D3—22no—
372C2×D3—21no—
373C2×D3—22no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—21no—
377C3×C4H(5,𝔽2)12yes—
378C3×C4H(5,𝔽2)12yes—
379Dic3—24no—
380Dic3—22no—
381Dic3—21no—
382Dic3—22no—
383Dic3—21no—
384Dic3—21no—
385Dic3—22no—
386C23—31no—
387C23—32no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23C23×D312yes—
392C23C23×D312yes—
393C23C23×D314yes—
394Q8—62no—
395Q8—64no—
396Q8—122no—
397D4—22no—
398D4—22no—
399D4—24no—
400D4—24no—
401D4—44no—
402D4—44no—
403D4—42no—
404D4—62no—
405D4—122no—
406D4—122no—
407D4—122no—
408D4—124no—
409D4—124no—
410D4—124no—
411D4—122no—
412D4—124no—
413D4—122no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—32no—
417C2×C4—31no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—32no—
421C2×C4—34no—
422C2×C4—32no—
423C2×C4—62no—
424C2×C4—64no—
425C2×C4—62no—
426C2×C4—124no—
427C2×C4—121no—
428C2×C4—124no—
429C2×C4—124no—
430C2×C4C23×D312yes—
431C2×C4C23×D311yescharacteristic
432C2×C4C23×D312yes—
433C2×C4C23×D311yescharacteristic
434C2×C4C23×D311yescharacteristic
435C2×C3—44no—
436C2×C3—42no—
437C2×C3—42no—
438C2×C3C2×H(5,𝔽2)11yescharacteristic
439C2×C3C2×H(5,𝔽2)11yescharacteristic
440C2×C3#58511yescharacteristic
441D3—41no—
442D3—42no—
443C22—22no—
444C22—22no—
445C22—24no—
446C22—24no—
447C22—22no—
448C22—22no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22—61no—
453C22—62no—
454C22—61no—
455C22—62no—
456C22—61no—
457C22—62no—
458C22—88no—
459C22C24×D311yescharacteristic
460C4—22no—
461C4—22no—
462C4—21no—
463C4—61no—
464C4—62no—
465C4—61no—
466C4—64no—
467C4—62no—
468C4—61no—
469C4—62no—
470C4#100412yes—
471C4#100412yes—
472C3#354611yescharacteristic, complemented, Sylow
473C2—42no—
474C2—42no—
475C2—44no—
476C2—121no—
477C2—122no—
478C2C2×#100411yescharacteristic
479C2C2×#100411yescharacteristic
480C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×D42C2
hypercenterD42D3
hypocenterC3#3546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C3×D42C2semidirect
2. #5806C2semidirect
3. #5806C2semidirect
4. #5805C2semidirect
5. #5805C2semidirect
6. #5718C2semidirect
7. #5716C2semidirect
8. #5614C2semidirect
9. #5608C2semidirect
10. #5603C2semidirect
11. #5599C2semidirect
12. #5595C2semidirect
13. #5593C2semidirect
14. #5557C2semidirect
15. #5551C2semidirect
16. #5548C2semidirect
17. #5542C2semidirect
21. C2×#237C22semidirect
22. C2×#237C22semidirect
23. C2×#235C22semidirect
24. C2×#235C22semidirect
31. #953C22semidirect
32. #953C22semidirect
33. #953C22semidirect
34. C2×#217C22semidirect
36. #940C22semidirect
37. #940C22semidirect
38. #939C22semidirect
39. #939C22semidirect
40. #939C22semidirect
44. #938C22semidirect
45. #938C22semidirect
46. #938C22semidirect
47. #938C22semidirect
48. #938C22semidirect
51. #936C22semidirect
52. #936C22semidirect
53. #935C22semidirect
54. #935C22semidirect
55. D4×Dic3C22semidirect
56. D4×Dic3C22semidirect
57. D4×Dic3C22semidirect
58. C4×#67C22semidirect
59. C4×#67C22semidirect
60. #926C22semidirect
61. #926C22semidirect
62. #924C22semidirect
63. #924C22semidirect
64. #897C22semidirect
65. #895C22semidirect
66. D3×C4⋊C4C22semidirect
67. #890C22semidirect
68. #888C22semidirect
69. #887C22semidirect
70. #885C22semidirect
71. #884C22semidirect
72. #884C22semidirect
73. #883C22semidirect
74. #883C22semidirect
75. #67⋊C4C22semidirect
76. #879C22semidirect
77. #879C22semidirect
78. #879C22semidirect
79. #878C22semidirect
80. C4×D12C22semidirect
81. C42×D3C22semidirect
82. #869C22semidirect
83. C4×Dic6C22semidirect
92. D42D3semidirect
472. C3#3546semidirect

Maximal subgroups

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

subgroupindex
C3×D422
#58062
#58052
#57182
#57162
#56142
#56082
#56032
#55992
#55952
#55932
#55572
#55512
#55482
#55422
#35463