Finite Group Databasebeta

← #10595255

Subgroups — #10595255

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#64146C211yesmaximal, characteristic, complemented
2C3×#938C211yesmaximal, characteristic, complemented
3C3×#935C211yesmaximal, characteristic, complemented
4#63981C211yesmaximal, characteristic, complemented
5#63978C211yesmaximal, characteristic, complemented
6#63974C211yesmaximal, characteristic, complemented
7#63959C211yesmaximal, characteristic, complemented
8#63958C211yesmaximal, characteristic, complemented
9#63956C211yesmaximal, characteristic, complemented
10#63953C211yesmaximal, characteristic, complemented
11#63952C211yesmaximal, characteristic, complemented
12#63904C211yesmaximal, characteristic, complemented
13#63895C211yesmaximal, characteristic, complemented
14#63850C211yesmaximal, characteristic, complemented
15#63848C211yesmaximal, characteristic, complemented
16#5657—31nomaximal
17#5613—31nomaximal
18C23×C3⋊D3C2211yescharacteristic
19C2×C32×D4C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×#638C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic, complemented
24C22×C3×Dic3C2211yescharacteristic, complemented
25C2×C3×C4×D3C2211yescharacteristic, complemented
26C2×#627C2211yescharacteristic, complemented
27C2×#626C2211yescharacteristic
28C2×#626C2211yescharacteristic
29C2×#624C2211yescharacteristic
30C2×#624C2211yescharacteristic
31#3757C2211yescharacteristic
32#3752C2211yescharacteristic
33#3752C2211yescharacteristic
34C3×#217C2211yescharacteristic
35C3×#217C2211yescharacteristic
36C3×#217C2211yescharacteristic
37C3×#217C2211yescharacteristic, complemented
38C3×#217C2211yescharacteristic, complemented
39C3×C4⋊Dic3C2211yescharacteristic, complemented
40C3×Dic3⋊C4C2211yescharacteristic, complemented
41C3×Dic3⋊C4C2211yescharacteristic, complemented
42#3724C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic, complemented
50(C2×D3)⋊Dic3C2211yescharacteristic, complemented
51Dic32C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#123—31no—
54C3×#121—31no—
55#939—31no—
56#938—31no—
57#938D311yescharacteristic, complemented
58#937—31no—
59#937—31no—
60#937—31no—
61#937—61no—
62#936—31no—
63#935D311yescharacteristic, complemented
64#930—31no—
65#929—31no—
66C4×#67—31no—
67C4×#67—31no—
68#898—31no—
69#898—31no—
70#897—31no—
71#896—31no—
72#894—31no—
73#893—31no—
74#885—31no—
75#885—31no—
76#885—31no—
77#884—31no—
78#882—31no—
79#67⋊C4—31no—
80D3×C22⋊C4—31no—
81D3×C22⋊C4—31no—
82#879—31no—
83#879—31no—
84C23×C32C2311yescharacteristic
85C23×C32C2311yescharacteristic
86C22×C3⋊D3—22no—
87C22×C3⋊D3—41no—
88C22×C3⋊D3—41no—
89C22×C3⋊D3C2311yescharacteristic
90C22×C3⋊D3C2311yescharacteristic
91C22×C3×D3C2311yescharacteristic
92C32×D4—41no—
93C2×C32×C4C2311yescharacteristic
94#638—22no—
95#638—41no—
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C3×#67—41no—
99C3×#67—41no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C3×C4×D3—41no—
109#627—41no—
110#626—22no—
111#626—41no—
112#624—41no—
113#624—41no—
114#538—91noSylow
115C23×D3—31no—
116C23×D3—31no—
117C23×D3—61no—
118C2×C3×D4—21no—
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C2×C3×D4C2×D311yescharacteristic
122C2×C3×D4C2×D311yescharacteristic
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C2×#67—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67—31no—
129C2×#67—31no—
130C2×#67—31no—
131C2×#67—61no—
132C2×#67—61no—
133C2×#67C2×D311yescharacteristic
134C2×#67C2×D311yescharacteristic, complemented
135C22×Dic3C2×D311yescharacteristic, complemented
136C2×D12—31no—
137C2×D12—31no—
138C2×C4×D3—31no—
139C2×C4×D3—31no—
140C2×C4×D3—31no—
141C2×C4×D3—31no—
142C2×C4×D3C2×D311yescharacteristic, complemented
143C2×Dic6—31no—
144C2×Dic6—31no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—31no—
148C3×C22⋊C4—31no—
149C3×C22⋊C4—31no—
150C3×C22⋊C4—31no—
151C3×C22⋊C4—31no—
152C3×C22⋊C4—31no—
153#217—31no—
154#217—31no—
155#217—31no—
156#217—31no—
157#217—61no—
158#217C2×D311yescharacteristic
159#217C2×D311yescharacteristic
160#217C2×D311yescharacteristic
161#217C2×D311yescharacteristic, complemented
162#217C2×D311yescharacteristic, complemented
163(C2×D3)⋊C4—31no—
164(C2×D3)⋊C4—31no—
165(C2×D3)⋊C4—31no—
166(C2×D3)⋊C4—31no—
167(C2×D3)⋊C4—31no—
168(C2×D3)⋊C4—31no—
169(C2×D3)⋊C4—31no—
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—
176(C2×D3)⋊C4—31no—
177(C2×D3)⋊C4—61no—
178(C2×D3)⋊C4—61no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3C2×D311yescharacteristic, complemented
182Dic3⋊C4—31no—
183Dic3⋊C4—31no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4C2×D311yescharacteristic, complemented
187Dic3⋊C4C2×D311yescharacteristic, complemented
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×C32C2411yescharacteristic
199C2×C3⋊D3—21no—
200C2×C3⋊D3—21no—
201C2×C3⋊D3—21no—
202C2×C3⋊D3—21no—
203C2×C3⋊D3—21no—
204C2×C3⋊D3—44no—
205C2×C3⋊D3#4112yes—
206C2×C3×D3—21no—
207C2×C3×D3—21no—
208C2×C3×D3—21no—
209C32×C4—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—31no—
224#123—91no—
225C22⋊Q8—91no—
226#121—31no—
227#121—91no—
228#121—91no—
229#120—91no—
230#120—91no—
231C4×D4—91no—
232C4×D4—91no—
233C2×C22⋊C4—91no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—31no—
237C23×C3C22×D311yescharacteristic
238C23×C3C22×D311yescharacteristic
239C23×C3C22×D311yescharacteristic
240C23×C3C22×D311yescharacteristic
241C22×D3—31no—
242C22×D3—31no—
243C22×D3—31no—
244C22×D3—31no—
245C22×D3—61no—
246C22×D3—61no—
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—122no—
250C22×D3—121no—
251C22×D3—121no—
252C22×D3—121no—
253C22×D3—121no—
254C22×D3—122no—
255C22×D3—122no—
256C22×D3C22×D311yescharacteristic
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—42no—
260C3×D4—121no—
261C3×D4—121no—
262C2×C3×C4—21no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—121no—
271C2×C3×C4—121no—
272C2×C3×C4C22×D311yescharacteristic
273C2×C3×C4C22×D311yescharacteristic
274#67—41no—
275#67—41no—
276#67—62no—
277#67—62no—
278#67—62no—
279#67—122no—
280#67—121no—
281#67—121no—
282#67—121no—
283#67—122no—
284C2×Dic3—31no—
285C2×Dic3—31no—
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—41no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3C22×D311yescharacteristic
292C2×Dic3C22×D311yescharacteristic
293C2×Dic3C22×D311yescharacteristic
294C2×Dic3C22×D311yescharacteristic
295C2×Dic3C22×D311yescharacteristic
296C2×Dic3C22×D311yescharacteristic
297C2×Dic3C22×D311yescharacteristic
298D12—62no—
299D12—121no—
300C4×D3—41no—
301C4×D3—121no—
302C4×D3—121no—
303C4×D3—121no—
304C4×D3—121no—
305Dic6—121no—
306Dic6—121no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32C2×#4111yescharacteristic
310C2×C32H(5,𝔽2)11yescharacteristic
311C2×C32H(5,𝔽2)11yescharacteristic
312C3⋊D3—22no—
313C3⋊D3—41no—
314C3×D3—41no—
315C24—91no—
316C2×Q8—91no—
317C2×D4—31no—
318C2×D4—31no—
319C2×D4—91no—
320C2×D4—91no—
321C2×D4—91no—
322C2×D4—91no—
323C2×D4D3211yescharacteristic
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—91no—
327C22×C4—91no—
328C4⋊C4—31no—
329C4⋊C4—31no—
330C4⋊C4—31no—
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—91no—
340C22⋊C4—91no—
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—22no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
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—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3C23×D311yescharacteristic
373C22×C3C23×D311yescharacteristic
374C2×D3—21no—
375C2×D3—21no—
376C2×D3—21no—
377C2×D3—32no—
378C2×D3—32no—
379C2×D3—62no—
380C2×D3—61no—
381C2×D3—61no—
382C2×D3—61no—
383C2×D3—61no—
384C2×D3—62no—
385C2×D3—61no—
386C2×D3—61no—
387C2×D3—61no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—61no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—124no—
396C2×D3—124no—
397C2×D3—128no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—41no—
401C3×C4—61no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—61no—
408Dic3—21no—
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413Dic3—21no—
414Dic3—21no—
415Dic3—61no—
416Dic3—61no—
417Dic3—61no—
418Dic3—61no—
419Dic3—121no—
420Dic3—121no—
421C32#53811yescharacteristic, complemented, Sylow
422C23—31no—
423C23—91no—
424C23—91no—
425C23—182no—
426C23—361no—
427C23—361no—
428C23C2×D3211yescharacteristic
429C23C2×D3211yescharacteristic
430Q8—361no—
431D4—41no—
432D4—121no—
433D4—121no—
434D4—182no—
435D4—182no—
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—121no—
448C2×C4—121no—
449C2×C4—361no—
450C2×C4—361no—
451C2×C4C2×D3211yescharacteristic
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—42no—
456C2×C3—41no—
457C2×C3—41no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—81no—
461C2×C3—121no—
462C2×C3#100411yescharacteristic
463C2×C3#100411yescharacteristic
464C2×C3#100411yescharacteristic
465C2×C3D3×#4111yescharacteristic
466C2×C3D3×#4111yescharacteristic
467C2×C3#100911yescharacteristic
468D3—41no—
469D3—62no—
470D3—62no—
471D3—122no—
472D3—122no—
473D3—121no—
474D3—121no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—61no—
482C22—61no—
483C22—61no—
484C22—92no—
485C22—181no—
486C22—181no—
487C22—181no—
488C22—181no—
489C22—181no—
490C22—364no—
491C22C22×D3211yescharacteristic
492C4—21no—
493C4—61no—
494C4—61no—
495C4—61no—
496C4—61no—
497C4—61no—
498C4—61no—
499C4—61no—
500C4—181no—
501C4—181no—
502C3—21no—
503C3#561311yescharacteristic, complemented
504C3#565711yescharacteristic, complemented
505C2—41no—
506C2—41no—
507C2—121no—
508C2—182no—
509C2—361no—
510C2#6431211yescharacteristic
511C2#6431411yescharacteristic
512C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64146C2semidirect
2. C3×#938C2semidirect
3. C3×#935C2semidirect
4. #63981C2semidirect
5. #63978C2semidirect
6. #63974C2semidirect
7. #63959C2semidirect
8. #63958C2semidirect
9. #63956C2semidirect
10. #63953C2semidirect
11. #63952C2semidirect
12. #63904C2semidirect
13. #63895C2semidirect
14. #63850C2semidirect
15. #63848C2semidirect
23. C2×C3×#67C22semidirect
24. C22×C3×Dic3C22semidirect
25. C2×C3×C4×D3C22semidirect
26. C2×#627C22semidirect
37. C3×#217C22semidirect
38. C3×#217C22semidirect
39. C3×C4⋊Dic3C22semidirect
40. C3×Dic3⋊C4C22semidirect
41. C3×Dic3⋊C4C22semidirect
42. #3724C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
49. (C2×D3)⋊Dic3C22semidirect
50. (C2×D3)⋊Dic3C22semidirect
51. Dic32C22semidirect
52. Dic32C22semidirect
57. #938D3semidirect
63. #935D3semidirect
134. C2×#67C2×D3semidirect
135. C22×Dic3C2×D3semidirect
142. C2×C4×D3C2×D3semidirect
161. #217C2×D3semidirect
162. #217C2×D3semidirect
181. C4⋊Dic3C2×D3semidirect
186. Dic3⋊C4C2×D3semidirect
187. Dic3⋊C4C2×D3semidirect
421. C32#538semidirect
503. C3#5613semidirect
504. C3#5657semidirect

Maximal subgroups

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

subgroupindex
#641462
C3×#9382
C3×#9352
#639812
#639782
#639742
#639592
#639582
#639562
#639532
#639522
#639042
#638952
#638502
#638482
#56573
#56133