Finite Group Databasebeta

← #10595083

Subgroups — #10595083

Order 576.

715 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#67C211yesmaximal, characteristic, complemented
2C2×#3804C211yesmaximal, characteristic, complemented
3C4×#638C211yesmaximal, characteristic, complemented
4C3×#930C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6#63976C211yesmaximal, characteristic, complemented
7#63961C211yesmaximal, characteristic, complemented
8#63922C211yesmaximal, characteristic, complemented
9#63921C211yesmaximal, characteristic, complemented
10#63915C211yesmaximal, characteristic, complemented
11#63912C211yesmaximal, characteristic, complemented
12#63902C211yesmaximal, characteristic, complemented
13#63899C211yesmaximal, characteristic, complemented
14#63885C211yesmaximal, characteristic, complemented
15#63854C211yesmaximal, characteristic, complemented
16#5610—31nomaximal
17#5490—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C22×D32C2211yescharacteristic
20C22×C32×C4C2211yescharacteristic, complemented
21C2×#638C2211yescharacteristic
22C2×C4×C3⋊D3C2211yescharacteristic, complemented
23C2×C3×#67C2211yescharacteristic
24C2×C3×#67C2211yescharacteristic, complemented
25C22×C3×Dic3C2211yescharacteristic, complemented
26C2×C3×D12C2211yescharacteristic, complemented
27D3×#67—24no—
28C2×#627C2211yescharacteristic, complemented
29C2×#626C2211yescharacteristic
30C2×#626C2211yescharacteristic, complemented
31C2×#625C2211yescharacteristic
32C2×#625C2211yescharacteristic
33C2×#625C2211yescharacteristic, complemented
34C2×#624C2211yescharacteristic, complemented
35#3804—24no—
36C2×D3×Dic3C2211yescharacteristic
37C2×D3×Dic3C2211yescharacteristic, complemented
38#3757C2211yescharacteristic, complemented
39#3752C2211yescharacteristic, complemented
40#3750C2211yescharacteristic, complemented
41C4×C3⋊Dic3C2211yescharacteristic, complemented
42C3×(C2×D3)⋊C4C2211yescharacteristic
43C3×(C2×D3)⋊C4C2211yescharacteristic
44C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
48C3×C4⋊Dic3C2211yescharacteristic, complemented
49#3724C2211yescharacteristic, complemented
50#3724C2211yescharacteristic, complemented
51(C2×D3)⋊Dic3C2211yescharacteristic
52(C2×D3)⋊Dic3C2211yescharacteristic, complemented
53(C2×D3)⋊Dic3C2211yescharacteristic, complemented
54(C2×D3)⋊Dic3C2211yescharacteristic, complemented
55C22×#67—31no—
56C2×#237—31no—
57C2×D3×D4—31no—
58C2×#235—31no—
59C3×#121—31no—
60C2×C3×C22⋊C4—31no—
61#953—31no—
62#939—31no—
63#937—31no—
64#937—31no—
65#935—31no—
66#930D311yescharacteristic, complemented
67#929—31no—
68C4×#67—31no—
69C4×#67—31no—
70C4×#67—61no—
71C2×(C2×D3)⋊C4D311yescharacteristic, complemented
72#896—31no—
73#894—31no—
74#885—31no—
75#885—31no—
76#884—31no—
77#884—31no—
78#884—31no—
79#883—31no—
80#882—31no—
81#67⋊C4—31no—
82#67⋊C4—31no—
83D3×C22⋊C4—31no—
84D3×C22⋊C4—31no—
85#878—31no—
86C23×C32C2311yescharacteristic
87C22×C3⋊D3C2311yescharacteristic
88C22×C3×D3—22no—
89C22×C3×D3—22no—
90C22×C3×D3—41no—
91C22×C3×D3C2311yescharacteristic
92C22×C3×D3C2311yescharacteristic
93C22×C3×D3C2311yescharacteristic
94C22×C3×D3C2311yescharacteristic
95C2×D32—22no—
96C2×D32—22no—
97C2×D32—41no—
98C2×C32×C4—41no—
99C2×C32×C4C2311yescharacteristic
100C2×C32×C4C2311yescharacteristic
101#638D414yescomplemented
102C2×C3⋊Dic3C2311yescharacteristic
103C2×C3⋊Dic3C2311yescharacteristic
104C2×C3⋊Dic3C2311yescharacteristic
105C4×C3⋊D3—41no—
106C3×#67—22no—
107C3×#67—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3C2311yescharacteristic
110C2×C3×Dic3C2311yescharacteristic
111C2×C3×Dic3C2311yescharacteristic
112C2×C3×Dic3C2311yescharacteristic
113C3×D12—41no—
114#627—22no—
115#626—22no—
116#626—22no—
117#625—22no—
118#625—22no—
119#625—41no—
120#624—22no—
121D3×Dic3—22no—
122D3×Dic3—22no—
123#546—91noSylow
124C24×C3—31no—
125C23×D3—31no—
126C23×D3—31no—
127C23×D3C2×D311yescharacteristic
128C2×C3×D4—31no—
129C2×C3×D4—31no—
130C2×C3×D4—31no—
131C22×C3×C4—21no—
132C22×C3×C4—31no—
133C22×C3×C4C2×D311yescharacteristic, complemented
134C22×C3×C4C2×D311yescharacteristic, complemented
135C2×#67—31no—
136C2×#67—31no—
137C2×#67—31no—
138C2×#67—31no—
139C2×#67—31no—
140C2×#67—31no—
141C2×#67—31no—
142C2×#67—31no—
143C2×#67—31no—
144C2×#67—31no—
145C2×#67—61no—
146C2×#67—64no—
147C2×#67C2×D311yescharacteristic
148C2×#67C2×D311yescharacteristic, complemented
149C22×Dic3—31no—
150C22×Dic3—31no—
151C22×Dic3C2×D311yescharacteristic, complemented
152#237—64no—
153D3×D4—64no—
154#235—64no—
155C2×D12—31no—
156C2×D12—31no—
157C2×D12C2×D311yescharacteristic, complemented
158C2×C4×D3—31no—
159C2×C4×D3—31no—
160C2×C4×D3—31no—
161C2×C4×D3—31no—
162C2×C4×D3—31no—
163C2×C4×D3—31no—
164C2×C4×D3—61no—
165C2×Dic6—31no—
166C2×Dic6—31no—
167C3×C4⋊C4—31no—
168C3×C22⋊C4—31no—
169C3×C22⋊C4—31no—
170C3×C22⋊C4—31no—
171C3×C22⋊C4—31no—
172C3×C22⋊C4—31no—
173C3×C22⋊C4—31no—
174#217—31no—
175#217—31no—
176#217—31no—
177#217—31no—
178#217—31no—
179#217—31no—
180#217—61no—
181(C2×D3)⋊C4—31no—
182(C2×D3)⋊C4—31no—
183(C2×D3)⋊C4—31no—
184(C2×D3)⋊C4—31no—
185(C2×D3)⋊C4—31no—
186(C2×D3)⋊C4—31no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4C2×D311yescharacteristic
189(C2×D3)⋊C4C2×D311yescharacteristic
190(C2×D3)⋊C4C2×D311yescharacteristic, complemented
191(C2×D3)⋊C4C2×D311yescharacteristic, complemented
192(C2×D3)⋊C4C2×D311yescharacteristic, complemented
193(C2×D3)⋊C4C2×D311yescharacteristic, complemented
194C4⋊Dic3C2×D311yescharacteristic, complemented
195Dic3⋊C4—31no—
196Dic3⋊C4—31no—
197Dic3⋊C4—31no—
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4—61no—
202C4×Dic3—31no—
203C4×Dic3—31no—
204C4×Dic3—61no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32C2×D412yes—
208C22×C32C2411yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3C2×D412yes—
212C2×C3×D3—21no—
213C2×C3×D3—21no—
214C2×C3×D3—21no—
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—44no—
224C2×C3×D3#4112yescomplemented
225D32—44no—
226C32×C4—21no—
227C32×C4—21no—
228C3⋊Dic3—21no—
229C3⋊Dic3—21no—
230C3⋊Dic3C2×D412yes—
231C3×Dic3—21no—
232C3×Dic3—21no—
233C3×Dic3—21no—
234C3×Dic3—21no—
235C2×#41—91no—
236C22×D4—91no—
237#124—91no—
238#123—91no—
239#123—91no—
240C22⋊Q8—91no—
241#121—31no—
242#121—91no—
243#121—91no—
244#120—91no—
245#120—91no—
246C4×D4—91no—
247C4×D4—91no—
248C2×C22⋊C4—31no—
249C2×C22⋊C4—91no—
250C23×C3—21no—
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—31no—
254C23×C3—31no—
255C23×C3—62no—
256C23×C3—62no—
257C23×C3—121no—
258C23×C3C22×D311yescharacteristic
259C23×C3C22×D311yescharacteristic
260C22×D3—22no—
261C22×D3—22no—
262C22×D3—31no—
263C22×D3—31no—
264C22×D3—41no—
265C22×D3—61no—
266C22×D3—62no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—121no—
271C22×D3—121no—
272C22×D3C22×D311yescharacteristic
273C22×D3C22×D311yescharacteristic
274C22×D3C22×D311yescharacteristic
275C22×D3C22×D311yescharacteristic
276C3×D4—62no—
277C3×D4—62no—
278C3×D4—121no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—62no—
289C2×C3×C4C22×D311yescharacteristic
290C2×C3×C4C22×D311yescharacteristic
291C2×C3×C4C22×D311yescharacteristic
292C2×C3×C4C22×D311yescharacteristic
293#67—22no—
294#67—22no—
295#67—34no—
296#67—34no—
297#67—64no—
298#67—62no—
299#67—62no—
300#67—62no—
301#67—62no—
302#67—62no—
303#67—62no—
304#67—121no—
305#67—121no—
306C2×Dic3—22no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—31no—
310C2×Dic3—31no—
311C2×Dic3—31no—
312C2×Dic3—31no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—62no—
317C2×Dic3—62no—
318C2×Dic3C22×D311yescharacteristic
319C2×Dic3C22×D311yescharacteristic
320C2×Dic3C22×D311yescharacteristic
321C2×Dic3C22×D311yescharacteristic
322D12—41no—
323D12—62no—
324D12—62no—
325C4×D3—62no—
326C4×D3—62no—
327C4×D3—62no—
328C4×D3—62no—
329C4×D3—121no—
330C4×D3—121no—
331C4×D3—122no—
332Dic6—62no—
333Dic6—62no—
334C2×C32—22no—
335C2×C32C22×D411yescharacteristic
336C2×C32C2×#4111yescharacteristic
337C2×C32H(5,𝔽2)11yescharacteristic
338C3⋊D3—22no—
339C3×D3—22no—
340C3×D3—41no—
341C3×D3—41no—
342C3×D3—41no—
343C24—31no—
344C24—91no—
345#41—184no—
346C2×Q8—91no—
347C2×D4—31no—
348C2×D4—31no—
349C2×D4—31no—
350C2×D4—91no—
351C2×D4—91no—
352C2×D4—91no—
353C2×D4—91no—
354C2×D4—91no—
355C2×D4—91no—
356C2×D4—184no—
357C22×C4—31no—
358C22×C4—91no—
359C22×C4—91no—
360C22×C4—91no—
361C22×C4—91no—
362C22×C4D3211yescharacteristic, complemented
363C4⋊C4—31no—
364C4⋊C4—91no—
365C4⋊C4—91no—
366C4⋊C4—91no—
367C22⋊C4—31no—
368C22⋊C4—31no—
369C22⋊C4—31no—
370C22⋊C4—31no—
371C22⋊C4—31no—
372C22⋊C4—31no—
373C22⋊C4—91no—
374C22⋊C4—91no—
375C22⋊C4—91no—
376C22⋊C4—91no—
377C22⋊C4—91no—
378C22⋊C4—91no—
379C42—91no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3—32no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—61no—
390C22×C3—61no—
391C22×C3—61no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—61no—
398C22×C3—61no—
399C22×C3—61no—
400C22×C3—124no—
401C22×C3D3×D412yes—
402C22×C3D3×D412yes—
403C22×C3C23×D311yescharacteristic
404C22×C3C23×D311yescharacteristic
405C2×D3—21no—
406C2×D3—21no—
407C2×D3—21no—
408C2×D3—21no—
409C2×D3—21no—
410C2×D3—21no—
411C2×D3—21no—
412C2×D3—21no—
413C2×D3—21no—
414C2×D3—21no—
415C2×D3—21no—
416C2×D3—32no—
417C2×D3—32no—
418C2×D3—44no—
419C2×D3—62no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3—61no—
423C2×D3—61no—
424C2×D3—121no—
425C2×D3—121no—
426C2×D3—124no—
427C2×D3—124no—
428C2×D3#23512yescomplemented
429C3×C4—22no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—41no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—61no—
438C3×C4—61no—
439Dic3—21no—
440Dic3—21no—
441Dic3—21no—
442Dic3—21no—
443Dic3—32no—
444Dic3—32no—
445Dic3—62no—
446Dic3—61no—
447Dic3—61no—
448Dic3—61no—
449Dic3—61no—
450Dic3—121no—
451Dic3—121no—
452C32#54611yescharacteristic, complemented, Sylow
453C23—31no—
454C23—31no—
455C23—31no—
456C23—31no—
457C23—62no—
458C23—62no—
459C23—91no—
460C23—121no—
461C23—182no—
462C23—182no—
463C23—361no—
464C23C2×D3211yescharacteristic
465Q8—182no—
466D4—62no—
467D4—62no—
468D4—94no—
469D4—121no—
470D4—182no—
471D4—182no—
472D4—182no—
473D4—182no—
474D4—361no—
475C2×C4—31no—
476C2×C4—31no—
477C2×C4—31no—
478C2×C4—31no—
479C2×C4—41no—
480C2×C4—62no—
481C2×C4—91no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—182no—
485C2×C4—182no—
486C2×C4—182no—
487C2×C4—361no—
488C2×C4C2×D3211yescharacteristic
489C2×C4C2×D3211yescharacteristic
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—42no—
496C2×C3—62no—
497C2×C3—121no—
498C2×C3—121no—
499C2×C3—121no—
500C2×C3C2×#23511yescharacteristic
501C2×C3C2×D3×D411yescharacteristic
502C2×C3C2×D3×D411yescharacteristic
503C2×C3#100411yescharacteristic
504C2×C3#100411yescharacteristic
505C2×C3D3×#4111yescharacteristic
506D3—22no—
507D3—41no—
508D3—41no—
509D3—41no—
510D3—62no—
511D3—62no—
512D3—122no—
513C22—21no—
514C22—21no—
515C22—32no—
516C22—61no—
517C22—61no—
518C22—61no—
519C22—61no—
520C22—61no—
521C22—61no—
522C22—61no—
523C22—61no—
524C22—61no—
525C22—61no—
526C22—61no—
527C22—92no—
528C22—124no—
529C22—181no—
530C22—181no—
531C22—364no—
532C22#380212yes—
533C22C22×D3211yescharacteristic
534C4—21no—
535C4—21no—
536C4—61no—
537C4—61no—
538C4—61no—
539C4—61no—
540C4—92no—
541C4—181no—
542C4—181no—
543C3—21no—
544C3#549011yescharacteristic, complemented
545C3#561011yescharacteristic, complemented
546C2—22no—
547C2—62no—
548C2—121no—
549C2—121no—
550C2—121no—
551C2—182no—
552C2C2×#380211yescharacteristic
553C2D3×#23511yescharacteristic
554C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#546
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×#67C2semidirect
2. C2×#3804C2semidirect
3. C4×#638C2semidirect
4. C3×#930C2semidirect
5. C2×C3×(C2×D3)⋊C4C2semidirect
6. #63976C2semidirect
7. #63961C2semidirect
8. #63922C2semidirect
9. #63921C2semidirect
10. #63915C2semidirect
11. #63912C2semidirect
12. #63902C2semidirect
13. #63899C2semidirect
14. #63885C2semidirect
15. #63854C2semidirect
20. C22×C32×C4C22semidirect
22. C2×C4×C3⋊D3C22semidirect
24. C2×C3×#67C22semidirect
25. C22×C3×Dic3C22semidirect
26. C2×C3×D12C22semidirect
28. C2×#627C22semidirect
30. C2×#626C22semidirect
33. C2×#625C22semidirect
34. C2×#624C22semidirect
37. C2×D3×Dic3C22semidirect
38. #3757C22semidirect
39. #3752C22semidirect
40. #3750C22semidirect
41. C4×C3⋊Dic3C22semidirect
44. C3×(C2×D3)⋊C4C22semidirect
45. C3×(C2×D3)⋊C4C22semidirect
46. C3×(C2×D3)⋊C4C22semidirect
47. C3×(C2×D3)⋊C4C22semidirect
48. C3×C4⋊Dic3C22semidirect
49. #3724C22semidirect
50. #3724C22semidirect
52. (C2×D3)⋊Dic3C22semidirect
53. (C2×D3)⋊Dic3C22semidirect
54. (C2×D3)⋊Dic3C22semidirect
66. #930D3semidirect
71. C2×(C2×D3)⋊C4D3semidirect
101. #638D4semidirect
133. C22×C3×C4C2×D3semidirect
134. C22×C3×C4C2×D3semidirect
148. C2×#67C2×D3semidirect
151. C22×Dic3C2×D3semidirect
157. C2×D12C2×D3semidirect
190. (C2×D3)⋊C4C2×D3semidirect
191. (C2×D3)⋊C4C2×D3semidirect
192. (C2×D3)⋊C4C2×D3semidirect
193. (C2×D3)⋊C4C2×D3semidirect
194. C4⋊Dic3C2×D3semidirect
224. C2×C3×D3#41semidirect
362. C22×C4D32semidirect
428. C2×D3#235semidirect
452. C32#546semidirect
544. C3#5490semidirect
545. C3#5610semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#672
C2×#38042
C4×#6382
C3×#9302
C2×C3×(C2×D3)⋊C42
#639762
#639612
#639222
#639212
#639152
#639122
#639022
#638992
#638852
#638542
#56103
#54903