Finite Group Databasebeta

← #10595286

Subgroups — #10595286

Order 576.

715 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#67C211yesmaximal, characteristic, complemented
2C2×#3805C211yesmaximal, characteristic, complemented
3#64148C211yesmaximal, characteristic, complemented
4C3×#940C211yesmaximal, characteristic, complemented
5C3×#937C211yesmaximal, characteristic, complemented
6#63983C211yesmaximal, characteristic, complemented
7#63980C211yesmaximal, characteristic, complemented
8#63979C211yesmaximal, characteristic, complemented
9#63976C211yesmaximal, characteristic, complemented
10#63971C211yesmaximal, characteristic, complemented
11#63959C211yesmaximal, characteristic, complemented
12#63922C211yesmaximal, characteristic, complemented
13#63885C211yesmaximal, characteristic, complemented
14#63877C211yesmaximal, characteristic, complemented
15#63852C211yesmaximal, characteristic, complemented
16#5715—31nomaximal
17#5592—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C22×D32C2211yescharacteristic
20C2×C32×D4C2211yescharacteristic, complemented
21C2×#638C2211yescharacteristic
22C2×#638C2211yescharacteristic, complemented
23C22×C3⋊Dic3C2211yescharacteristic, complemented
24C2×C3×#67C2211yescharacteristic
25C2×C3×#67C2211yescharacteristic
26C2×C3×#67C2211yescharacteristic, complemented
27C2×C3×#67C2211yescharacteristic, complemented
28C2×C3×#67C2211yescharacteristic, complemented
29C2×C3×#67C2211yescharacteristic, complemented
30C2×C3×D12C2211yescharacteristic, complemented
31D3×#67—24no—
32C2×#627C2211yescharacteristic, complemented
33C2×#626C2211yescharacteristic
34C2×#626C2211yescharacteristic, complemented
35C2×#625C2211yescharacteristic
36C2×#625C2211yescharacteristic
37C2×#625C2211yescharacteristic, complemented
38#3805—24no—
39C2×D3×Dic3C2211yescharacteristic
40C2×D3×Dic3C2211yescharacteristic, complemented
41#3757C2211yescharacteristic, complemented
42#3752C2211yescharacteristic, complemented
43#3750C2211yescharacteristic, complemented
44C3×#217C2211yescharacteristic, complemented
45C3×(C2×D3)⋊C4C2211yescharacteristic
46C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C3×C4×Dic3C2211yescharacteristic, complemented
48#3724C2211yescharacteristic, complemented
49#3722C2211yescharacteristic, complemented
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52(C2×D3)⋊Dic3C2211yescharacteristic, complemented
53(C2×D3)⋊Dic3C2211yescharacteristic, complemented
54Dic32C2211yescharacteristic, complemented
55C22×#67—31no—
56C2×#237—31no—
57C2×#237—31no—
58C2×D3×D4—31no—
59C3×#127—31no—
60C3×#120—31no—
61#953—31no—
62#940—31no—
63#940—31no—
64#940D311yescharacteristic, complemented
65#939—31no—
66#939—31no—
67#939—31no—
68#939—31no—
69#939—31no—
70#939—31no—
71#939—61no—
72#937—31no—
73#937—31no—
74#937—31no—
75#937—31no—
76#937D311yescharacteristic, complemented
77#936—31no—
78#885—31no—
79#885—31no—
80#884—31no—
81#884—31no—
82#882—31no—
83#877—31no—
84#875—31no—
85#872—31no—
86C23×C32C2311yescharacteristic
87C23×C32C2311yescharacteristic
88C22×C3⋊D3C2311yescharacteristic
89C22×C3×D3—22no—
90C22×C3×D3—41no—
91C22×C3×D3—41no—
92C22×C3×D3C2311yescharacteristic
93C22×C3×D3C2311yescharacteristic
94C22×C3×D3C2311yescharacteristic
95C22×C3×D3C2311yescharacteristic
96C2×D32—22no—
97C2×D32—41no—
98C2×D32—41no—
99C32×D4—41no—
100C2×C32×C4C2311yescharacteristic
101#638—22no—
102#638—41no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3C2311yescharacteristic
105C2×C3⋊Dic3C2311yescharacteristic
106C2×C3⋊Dic3C2311yescharacteristic
107C3×#67—22no—
108C3×#67—22no—
109C3×#67—22no—
110C3×#67—41no—
111C3×#67—41no—
112C3×#67—41no—
113C2×C3×Dic3C2311yescharacteristic
114C2×C3×Dic3C2311yescharacteristic
115C2×C3×Dic3C2311yescharacteristic
116C2×C3×Dic3C2311yescharacteristic
117C3×D12—41no—
118#627—22no—
119#626—22no—
120#626—41no—
121#625—22no—
122#625—22no—
123#625—41no—
124D3×Dic3—22no—
125D3×Dic3D414yescomplemented
126#535—91noSylow
127C24×C3—31no—
128C23×D3—31no—
129C23×D3—31no—
130C23×D3C2×D311yescharacteristic
131C2×C3×D4—21no—
132C2×C3×D4—31no—
133C2×C3×D4—31no—
134C2×C3×D4—31no—
135C2×C3×D4—31no—
136C2×C3×D4—31no—
137C2×C3×D4—31no—
138C2×C3×D4—31no—
139C2×C3×D4C2×D311yescharacteristic, complemented
140C2×C3×D4C2×D311yescharacteristic, complemented
141C2×#67—31no—
142C2×#67—31no—
143C2×#67—31no—
144C2×#67—31no—
145C2×#67—31no—
146C2×#67—31no—
147C2×#67—31no—
148C2×#67—31no—
149C2×#67—31no—
150C2×#67—31no—
151C2×#67—31no—
152C2×#67—31no—
153C2×#67—61no—
154C2×#67—61no—
155C2×#67—64no—
156C2×#67C2×D311yescharacteristic
157C2×#67C2×D311yescharacteristic
158C2×#67C2×D311yescharacteristic, complemented
159C2×#67C2×D311yescharacteristic, complemented
160C2×#67C2×D311yescharacteristic, complemented
161C2×#67C2×D311yescharacteristic, complemented
162C22×Dic3—31no—
163C22×Dic3—31no—
164C22×Dic3—31no—
165C22×Dic3—31no—
166C22×Dic3—61no—
167#237—64no—
168#237—64no—
169D3×D4—64no—
170C2×D12—31no—
171C2×D12—31no—
172C2×D12C2×D311yescharacteristic, complemented
173C2×C4×D3—31no—
174C2×C4×D3—31no—
175C2×Dic6—31no—
176C2×Dic6—31no—
177C3×C22⋊C4—31no—
178C3×C22⋊C4—31no—
179C3×C22⋊C4—31no—
180C3×C42—31no—
181#217—31no—
182#217—31no—
183#217—31no—
184#217—31no—
185#217—31no—
186#217—31no—
187#217—61no—
188#217C2×D311yescharacteristic, complemented
189(C2×D3)⋊C4—31no—
190(C2×D3)⋊C4—31no—
191(C2×D3)⋊C4—31no—
192(C2×D3)⋊C4—31no—
193(C2×D3)⋊C4—31no—
194(C2×D3)⋊C4—31no—
195(C2×D3)⋊C4—31no—
196(C2×D3)⋊C4—31no—
197(C2×D3)⋊C4—61no—
198(C2×D3)⋊C4C2×D311yescharacteristic
199(C2×D3)⋊C4C2×D311yescharacteristic, complemented
200Dic3⋊C4—31no—
201Dic3⋊C4—31no—
202Dic3⋊C4—31no—
203Dic3⋊C4—31no—
204Dic3⋊C4—61no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3C2×D311yescharacteristic, complemented
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32C2411yescharacteristic
215C2×C3⋊D3—21no—
216C2×C3⋊D3—21no—
217C2×C3⋊D3—21no—
218C2×C3×D3—21no—
219C2×C3×D3—21no—
220C2×C3×D3—21no—
221C2×C3×D3—21no—
222C2×C3×D3—21no—
223C2×C3×D3—21no—
224C2×C3×D3—21no—
225C2×C3×D3—21no—
226C2×C3×D3—21no—
227C2×C3×D3—21no—
228C2×C3×D3—21no—
229C2×C3×D3—44no—
230C2×C3×D3C2×D412yes—
231D32—44no—
232C32×C4—21no—
233C3⋊Dic3—21no—
234C3⋊Dic3—21no—
235C3⋊Dic3C2×D412yes—
236C3×Dic3—21no—
237C3×Dic3—21no—
238C3×Dic3—21no—
239C3×Dic3C2×D412yes—
240C2×#41—91no—
241C22×D4—91no—
242#127—31no—
243#127—91no—
244#124—91no—
245#124—91no—
246#121—91no—
247#121—91no—
248#121—91no—
249#121—91no—
250#120—31no—
251#120—91no—
252#120—91no—
253#120—91no—
254#117—91no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—31no—
258C23×C3—31no—
259C23×C3—31no—
260C23×C3—31no—
261C23×C3—62no—
262C23×C3—121no—
263C23×C3—121no—
264C23×C3C22×D311yescharacteristic
265C23×C3C22×D311yescharacteristic
266C23×C3C22×D311yescharacteristic
267C23×C3C22×D311yescharacteristic
268C22×D3—22no—
269C22×D3—31no—
270C22×D3—31no—
271C22×D3—41no—
272C22×D3—41no—
273C22×D3—61no—
274C22×D3—62no—
275C22×D3—62no—
276C22×D3—121no—
277C22×D3—121no—
278C22×D3—121no—
279C22×D3—121no—
280C22×D3C22×D311yescharacteristic
281C22×D3C22×D311yescharacteristic
282C22×D3C22×D311yescharacteristic
283C22×D3C22×D311yescharacteristic
284C3×D4—41no—
285C3×D4—41no—
286C3×D4—42no—
287C3×D4—62no—
288C3×D4—62no—
289C3×D4—62no—
290C3×D4—121no—
291C3×D4—121no—
292C3×D4—121no—
293C3×D4—121no—
294C2×C3×C4—21no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4C22×D311yescharacteristic
300C2×C3×C4C22×D311yescharacteristic
301#67—22no—
302#67—22no—
303#67—22no—
304#67—41no—
305#67—41no—
306#67—41no—
307#67—62no—
308#67—62no—
309#67—62no—
310#67—62no—
311#67—62no—
312#67—62no—
313#67—62no—
314#67—122no—
315#67—121no—
316#67—121no—
317#67—121no—
318#67—121no—
319#67—121no—
320#67—122no—
321C2×Dic3—31no—
322C2×Dic3—31no—
323C2×Dic3—34no—
324C2×Dic3—31no—
325C2×Dic3—31no—
326C2×Dic3—31no—
327C2×Dic3—31no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—62no—
332C2×Dic3—62no—
333C2×Dic3—62no—
334C2×Dic3—122no—
335C2×Dic3C22×D311yescharacteristic
336C2×Dic3C22×D311yescharacteristic
337C2×Dic3C22×D311yescharacteristic
338C2×Dic3C22×D311yescharacteristic
339D12—41no—
340D12—62no—
341D12—121no—
342C4×D3—34no—
343C4×D3—62no—
344Dic6—62no—
345Dic6—62no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32C22×D411yescharacteristic
349C2×C32H(5,𝔽2)11yescharacteristic
350C2×C32H(5,𝔽2)11yescharacteristic
351C3⋊D3—41no—
352C3×D3—22no—
353C3×D3—41no—
354C3×D3—41no—
355C3×D3—41no—
356C24—31no—
357C24—91no—
358#41—184no—
359C2×Q8—91no—
360C2×D4—31no—
361C2×D4—31no—
362C2×D4—31no—
363C2×D4—31no—
364C2×D4—31no—
365C2×D4—31no—
366C2×D4—31no—
367C2×D4—91no—
368C2×D4—91no—
369C2×D4—91no—
370C2×D4—91no—
371C2×D4—91no—
372C2×D4—91no—
373C2×D4—91no—
374C2×D4—184no—
375C2×D4D3211yescharacteristic, complemented
376C22×C4—91no—
377C22×C4—91no—
378C22×C4—91no—
379C4⋊C4—91no—
380C4⋊C4—91no—
381C22⋊C4—31no—
382C22⋊C4—31no—
383C22⋊C4—31no—
384C22⋊C4—91no—
385C22⋊C4—91no—
386C22⋊C4—91no—
387C22⋊C4—91no—
388C22⋊C4—91no—
389C22⋊C4—91no—
390C22⋊C4—91no—
391C42—31no—
392C42—91no—
393C22×C3—22no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—32no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—61no—
414C22×C3—61no—
415C22×C3—61no—
416C22×C3—61no—
417C22×C3—61no—
418C22×C3—61no—
419C22×C3—61no—
420C22×C3—61no—
421C22×C3—61no—
422C22×C3—61no—
423C22×C3—61no—
424C22×C3—124no—
425C22×C3C23×D311yescharacteristic
426C22×C3C23×D311yescharacteristic
427C2×D3—21no—
428C2×D3—21no—
429C2×D3—21no—
430C2×D3—21no—
431C2×D3—21no—
432C2×D3—21no—
433C2×D3—21no—
434C2×D3—21no—
435C2×D3—21no—
436C2×D3—21no—
437C2×D3—21no—
438C2×D3—44no—
439C2×D3—61no—
440C2×D3—61no—
441C2×D3—61no—
442C2×D3—61no—
443C2×D3—61no—
444C2×D3—61no—
445C2×D3—121no—
446C2×D3—121no—
447C2×D3—121no—
448C2×D3—124no—
449C2×D3—124no—
450C2×D3D3×D412yes—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—32no—
454C3×C4—41no—
455C3×C4—61no—
456C3×C4—61no—
457C3×C4—61no—
458Dic3—21no—
459Dic3—21no—
460Dic3—21no—
461Dic3—32no—
462Dic3—32no—
463Dic3—62no—
464Dic3—61no—
465Dic3—61no—
466Dic3—61no—
467Dic3—61no—
468Dic3—121no—
469Dic3—121no—
470Dic3D3×D412yes—
471C32#53511yescharacteristic, complemented, Sylow
472C23—31no—
473C23—31no—
474C23—31no—
475C23—31no—
476C23—62no—
477C23—91no—
478C23—121no—
479C23—121no—
480C23—182no—
481C23—361no—
482C23—361no—
483C23C2×D3211yescharacteristic
484C23C2×D3211yescharacteristic
485Q8—182no—
486D4—41no—
487D4—62no—
488D4—62no—
489D4—62no—
490D4—121no—
491D4—121no—
492D4—121no—
493D4—121no—
494D4—182no—
495D4—182no—
496D4—182no—
497D4—182no—
498D4—361no—
499D4—361no—
500D4—361no—
501C2×C4—31no—
502C2×C4—31no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—94no—
506C2×C4—91no—
507C2×C4—91no—
508C2×C4—91no—
509C2×C4—182no—
510C2×C4—182no—
511C2×C4C2×D3211yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—42no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—41no—
520C2×C3—62no—
521C2×C3—81no—
522C2×C3—121no—
523C2×C3—121no—
524C2×C3—121no—
525C2×C3C2×D3×D411yescharacteristic
526C2×C3C2×D3×D411yescharacteristic
527C2×C3#100411yescharacteristic
528C2×C3#100411yescharacteristic
529C2×C3#100411yescharacteristic
530C2×C3#100411yescharacteristic
531D3—22no—
532D3—41no—
533D3—41no—
534D3—41no—
535D3—121no—
536D3—121no—
537D3—241no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—32no—
545C22—61no—
546C22—61no—
547C22—61no—
548C22—61no—
549C22—61no—
550C22—61no—
551C22—61no—
552C22—61no—
553C22—61no—
554C22—61no—
555C22—61no—
556C22—124no—
557C22—181no—
558C22—181no—
559C22—181no—
560C22—364no—
561C22C22×D3211yescharacteristic
562C4—21no—
563C4—32no—
564C4—61no—
565C4—61no—
566C4—61no—
567C4—92no—
568C4—181no—
569C4—181no—
570C3—21no—
571C3#559211yescharacteristic, complemented
572C3#571511yescharacteristic, complemented
573C2—41no—
574C2—41no—
575C2—62no—
576C2—121no—
577C2—121no—
578C2—121no—
579C2—361no—
580C2D32×D411yescharacteristic
581C2#6431311yescharacteristic
582C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#535
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D3×#67C2semidirect
2. C2×#3805C2semidirect
3. #64148C2semidirect
4. C3×#940C2semidirect
5. C3×#937C2semidirect
6. #63983C2semidirect
7. #63980C2semidirect
8. #63979C2semidirect
9. #63976C2semidirect
10. #63971C2semidirect
11. #63959C2semidirect
12. #63922C2semidirect
13. #63885C2semidirect
14. #63877C2semidirect
15. #63852C2semidirect
20. C2×C32×D4C22semidirect
22. C2×#638C22semidirect
23. C22×C3⋊Dic3C22semidirect
26. C2×C3×#67C22semidirect
27. C2×C3×#67C22semidirect
28. C2×C3×#67C22semidirect
29. C2×C3×#67C22semidirect
30. C2×C3×D12C22semidirect
32. C2×#627C22semidirect
34. C2×#626C22semidirect
37. C2×#625C22semidirect
40. C2×D3×Dic3C22semidirect
41. #3757C22semidirect
42. #3752C22semidirect
43. #3750C22semidirect
44. C3×#217C22semidirect
46. C3×(C2×D3)⋊C4C22semidirect
47. C3×C4×Dic3C22semidirect
48. #3724C22semidirect
49. #3722C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. (C2×D3)⋊Dic3C22semidirect
53. (C2×D3)⋊Dic3C22semidirect
54. Dic32C22semidirect
64. #940D3semidirect
76. #937D3semidirect
125. D3×Dic3D4semidirect
139. C2×C3×D4C2×D3semidirect
140. C2×C3×D4C2×D3semidirect
158. C2×#67C2×D3semidirect
159. C2×#67C2×D3semidirect
160. C2×#67C2×D3semidirect
161. C2×#67C2×D3semidirect
172. C2×D12C2×D3semidirect
188. #217C2×D3semidirect
199. (C2×D3)⋊C4C2×D3semidirect
207. C4×Dic3C2×D3semidirect
375. C2×D4D32semidirect
471. C32#535semidirect
571. C3#5592semidirect
572. C3#5715semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#672
C2×#38052
#641482
C3×#9402
C3×#9372
#639832
#639802
#639792
#639762
#639712
#639592
#639222
#638852
#638772
#638522
#57153
#55923