Finite Group Databasebeta

← #10595507

Subgroups — #10595507

Order 576.

565 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64162C211yesmaximal, characteristic, complemented
2C3×#953C211yesmaximal, characteristic, complemented
3C2×C3×#217C211yesmaximal, characteristic, complemented
4#63978C211yesmaximal, characteristic, complemented
5#63975C211yesmaximal, characteristic, complemented
6Dic3×#67C211yesmaximal, characteristic, complemented
7#63971C211yesmaximal, characteristic, complemented
8#63969C211yesmaximal, characteristic, complemented
9#63964C211yesmaximal, characteristic, complemented
10#63959C211yesmaximal, characteristic, complemented
11#63958C211yesmaximal, characteristic, complemented
12#63956C211yesmaximal, characteristic, complemented
13#63956C211yesmaximal, characteristic, complemented
14#63953C211yesmaximal, characteristic, complemented
15#63952C211yesmaximal, characteristic, complemented
16#5593—31nomaximal
17#5494—31nomaximal
18C24×C32C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×#638C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic
24C2×C3×#67C2211yescharacteristic
25C22×C3×Dic3C2211yescharacteristic
26C22×C3×Dic3C2211yescharacteristic
27C2×#627C2211yescharacteristic, complemented
28C2×#626C2211yescharacteristic, complemented
29C2×#624C2211yescharacteristic, complemented
30C2×D3×Dic3C2211yescharacteristic, complemented
31#3757C2211yescharacteristic
32#3757C2211yescharacteristic
33#3757C2211yescharacteristic
34C3×#217C2211yescharacteristic
35C3×#217C2211yescharacteristic
36C3×#217C2211yescharacteristic
37C3×#217C2211yescharacteristic
38C3×#217C2211yescharacteristic
39C3×#217C2211yescharacteristic
40C3×#217C2211yescharacteristic
41#3724C2211yescharacteristic, complemented
42#3724C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45Dic3⋊Dic3C2211yescharacteristic, complemented
46Dic3⋊Dic3C2211yescharacteristic, complemented
47#3722C2211yescharacteristic, complemented
48#3722C2211yescharacteristic, complemented
49(C2×D3)⋊Dic3C2211yescharacteristic, complemented
50(C2×D3)⋊Dic3C2211yescharacteristic, complemented
51Dic32C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#120—31no—
54C2×C3×C22⋊C4—31no—
55#953—31no—
56#953—31no—
57#953—61no—
58#953D311yescharacteristic, complemented
59C2×#217D311yescharacteristic, complemented
60#939—31no—
61#939—31no—
62#938—31no—
63#936—31no—
64#935—31no—
65D4×Dic3—31no—
66#930—31no—
67C4×#67—31no—
68C4×#67—31no—
69#924—31no—
70#885—31no—
71#885—31no—
72#885—31no—
73#884—31no—
74#884—31no—
75#883—31no—
76#883—31no—
77#883—31no—
78#67⋊C4—31no—
79#879—31no—
80#879—31no—
81#879—31no—
82#879—31no—
83#878—31no—
84C23×C32—22no—
85C23×C32—41no—
86C23×C32—41no—
87C23×C32C2311yescharacteristic
88C23×C32C2311yescharacteristic
89C23×C32C2311yescharacteristic
90C22×C3⋊D3C2311yescharacteristic
91C22×C3×D3C2311yescharacteristic
92#638—22no—
93#638—41no—
94#638—41no—
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C3×#67—22no—
99C3×#67—41no—
100C3×#67—41no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—41no—
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C2×C3×Dic3C2311yescharacteristic
109C2×C3×Dic3C2311yescharacteristic
110#627—41no—
111#626—41no—
112#624—41no—
113D3×Dic3—41no—
114#538—91noSylow
115C24×C3—21no—
116C24×C3C2×D311yescharacteristic
117C24×C3C2×D311yescharacteristic
118C2×C3×D4—31no—
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—31no—
123C2×#67—31no—
124C2×#67—31no—
125C2×#67—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67—31no—
129C2×#67—31no—
130C2×#67—61no—
131C2×#67—61no—
132C2×#67—61no—
133C2×#67C2×D311yescharacteristic
134C2×#67C2×D311yescharacteristic
135C2×#67C2×D311yescharacteristic
136C22×Dic3—31no—
137C22×Dic3C2×D311yescharacteristic
138C22×Dic3C2×D311yescharacteristic
139C2×D12—31no—
140C2×C4×D3—31no—
141C2×C4×D3—31no—
142C2×C4×D3—31no—
143C2×Dic6—31no—
144C2×Dic6—31no—
145C3×C22⋊C4—31no—
146C3×C22⋊C4—31no—
147C3×C22⋊C4—31no—
148C3×C22⋊C4—31no—
149C3×C22⋊C4—31no—
150C3×C22⋊C4—31no—
151C3×C22⋊C4—31no—
152#217—31no—
153#217—31no—
154#217—31no—
155#217—31no—
156#217—31no—
157#217—31no—
158#217—31no—
159#217—31no—
160#217—61no—
161#217—61no—
162#217—61no—
163#217C2×D311yescharacteristic
164#217C2×D311yescharacteristic
165#217C2×D311yescharacteristic
166#217C2×D311yescharacteristic
167#217C2×D311yescharacteristic
168#217C2×D311yescharacteristic
169#217C2×D311yescharacteristic
170(C2×D3)⋊C4—31no—
171(C2×D3)⋊C4—31no—
172(C2×D3)⋊C4—31no—
173(C2×D3)⋊C4—31no—
174(C2×D3)⋊C4—31no—
175(C2×D3)⋊C4—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180Dic3⋊C4—31no—
181Dic3⋊C4—31no—
182Dic3⋊C4—31no—
183Dic3⋊C4—31no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4—31no—
187Dic3⋊C4—31no—
188C4×Dic3—31no—
189C4×Dic3—31no—
190C4×Dic3—31no—
191C4×Dic3—31no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—44no—
201C22×C32#4112yes—
202C22×C32C2411yescharacteristic
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3×D3—21no—
207C2×C3×D3—21no—
208C2×C3×D3—21no—
209C3⋊Dic3—21no—
210C3⋊Dic3—21no—
211C3⋊Dic3—21no—
212C3×Dic3—21no—
213C3×Dic3—21no—
214C3×Dic3—21no—
215C3×Dic3—21no—
216C3×Dic3—21no—
217C3×Dic3—21no—
218C3×Dic3—21no—
219#126—91no—
220#126—91no—
221#124—91no—
222#124—91no—
223#123—91no—
224#123—91no—
225C22⋊Q8—91no—
226#121—91no—
227#121—91no—
228#121—91no—
229#120—31no—
230#120—91no—
231C4×D4—91no—
232C4×D4—91no—
233C2×C22⋊C4—31no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—31no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3C22×D311yescharacteristic
248C23×C3C22×D311yescharacteristic
249C23×C3C22×D311yescharacteristic
250C23×C3C22×D311yescharacteristic
251C23×C3C22×D311yescharacteristic
252C23×C3C22×D311yescharacteristic
253C22×D3—31no—
254C22×D3—31no—
255C22×D3—61no—
256C22×D3C22×D311yescharacteristic
257C3×D4—62no—
258C3×D4—121no—
259C3×D4—121no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—121no—
268C2×C3×C4—121no—
269#67—22no—
270#67—41no—
271#67—41no—
272#67—62no—
273#67—62no—
274#67—122no—
275#67—121no—
276#67—121no—
277#67—121no—
278#67—121no—
279#67—121no—
280#67—122no—
281#67—122no—
282C2×Dic3—31no—
283C2×Dic3—31no—
284C2×Dic3—31no—
285C2×Dic3—31no—
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—121no—
294C2×Dic3C22×D311yescharacteristic
295C2×Dic3C22×D311yescharacteristic
296C2×Dic3C22×D311yescharacteristic
297C2×Dic3C22×D311yescharacteristic
298C2×Dic3C22×D311yescharacteristic
299C2×Dic3C22×D311yescharacteristic
300C2×Dic3C22×D311yescharacteristic
301D12—121no—
302C4×D3—121no—
303C4×D3—121no—
304C4×D3—121no—
305Dic6—121no—
306Dic6—121no—
307C2×C32—22no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32C2×#4111yescharacteristic
311C2×C32H(5,𝔽2)11yescharacteristic
312C2×C32H(5,𝔽2)11yescharacteristic
313C3⋊D3—41no—
314C3×D3—41no—
315C24D3211yescharacteristic
316C2×Q8—91no—
317C2×D4—31no—
318C2×D4—31no—
319C2×D4—31no—
320C2×D4—91no—
321C2×D4—91no—
322C2×D4—91no—
323C2×D4—91no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—91no—
327C22×C4—91no—
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—91no—
333C4⋊C4—91no—
334C22⋊C4—31no—
335C22⋊C4—31no—
336C22⋊C4—31no—
337C22⋊C4—31no—
338C22⋊C4—31no—
339C22⋊C4—31no—
340C22⋊C4—31no—
341C22⋊C4—91no—
342C22⋊C4—91no—
343C22⋊C4—91no—
344C22⋊C4—91no—
345C22⋊C4—91no—
346C22⋊C4—91no—
347C22⋊C4—91no—
348C42—91no—
349C42—91no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—22no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—48no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3#23512yes—
383C22×C3#23712yes—
384C22×C3C23×D311yescharacteristic
385C22×C3C23×D311yescharacteristic
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—21no—
389C2×D3—61no—
390C2×D3—61no—
391C2×D3—61no—
392C2×D3—61no—
393C2×D3—61no—
394C2×D3—61no—
395C2×D3—121no—
396C2×D3—121no—
397C2×D3—121no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—61no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—61no—
405Dic3—21no—
406Dic3—21no—
407Dic3—21no—
408Dic3—21no—
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412Dic3—61no—
413Dic3—61no—
414Dic3—61no—
415Dic3—61no—
416Dic3—61no—
417Dic3—61no—
418Dic3—121no—
419Dic3—121no—
420Dic3—121no—
421C32#53811yescharacteristic, complemented, Sylow
422C23—22no—
423C23—31no—
424C23—41no—
425C23—41no—
426C23—91no—
427C23C2×D3211yescharacteristic
428C23C2×D3211yescharacteristic
429C23C2×D3211yescharacteristic
430Q8—361no—
431D4—62no—
432D4—121no—
433D4—121no—
434D4—182no—
435D4—361no—
436D4—361no—
437D4—361no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—91no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—361no—
451C2×C4—361no—
452C2×C3—22no—
453C2×C3—22no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—42no—
458C2×C3—42no—
459C2×C3—42no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—121no—
465C2×C3C2×#23511yescharacteristic
466C2×C3C2×#23711yescharacteristic
467C2×C3#100411yescharacteristic
468C2×C3#100411yescharacteristic
469C2×C3#100411yescharacteristic
470C2×C3#100411yescharacteristic
471D3—41no—
472D3—121no—
473D3—121no—
474D3—241no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—44no—
484C22—61no—
485C22—61no—
486C22—61no—
487C22—181no—
488C22—181no—
489C22—181no—
490C22#380412yes—
491C22C22×D3211yescharacteristic
492C4—61no—
493C4—61no—
494C4—61no—
495C4—61no—
496C4—61no—
497C4—61no—
498C4—61no—
499C4—181no—
500C4—181no—
501C4—181no—
502C3—21no—
503C3#549411yescharacteristic, complemented
504C3#559311yescharacteristic, complemented
505C2—22no—
506C2—41no—
507C2—41no—
508C2—121no—
509C2—361no—
510C2C2×#380411yescharacteristic
511C2#6433011yescharacteristic
512C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC24×C32C22
hypercenterC24D32
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. #64162C2semidirect
2. C3×#953C2semidirect
3. C2×C3×#217C2semidirect
4. #63978C2semidirect
5. #63975C2semidirect
6. Dic3×#67C2semidirect
7. #63971C2semidirect
8. #63969C2semidirect
9. #63964C2semidirect
10. #63959C2semidirect
11. #63958C2semidirect
12. #63956C2semidirect
13. #63956C2semidirect
14. #63953C2semidirect
15. #63952C2semidirect
27. C2×#627C22semidirect
28. C2×#626C22semidirect
29. C2×#624C22semidirect
30. C2×D3×Dic3C22semidirect
41. #3724C22semidirect
42. #3724C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
45. Dic3⋊Dic3C22semidirect
46. Dic3⋊Dic3C22semidirect
47. #3722C22semidirect
48. #3722C22semidirect
49. (C2×D3)⋊Dic3C22semidirect
50. (C2×D3)⋊Dic3C22semidirect
51. Dic32C22semidirect
52. Dic32C22semidirect
58. #953D3semidirect
59. C2×#217D3semidirect
421. C32#538semidirect
503. C3#5494semidirect
504. C3#5593semidirect

Maximal subgroups

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

subgroupindex
#641622
C3×#9532
C2×C3×#2172
#639782
#639752
Dic3×#672
#639712
#639692
#639642
#639592
#639582
#639562
#639532
#639522
#55933
#54943