Finite Group Databasebeta

← #10594213

Subgroups — #10594213

Order 576.

530 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64088C211yesmaximal, characteristic, complemented
2C3×#883C211yesmaximal, characteristic, complemented
3C3×#882C211yesmaximal, characteristic, complemented
4#63976C211yesmaximal, characteristic, complemented
5#63970C211yesmaximal, characteristic, complemented
6D3×#217C211yesmaximal, characteristic, complemented
7#63959C211yesmaximal, characteristic, complemented
8#63915C211yesmaximal, characteristic, complemented
9#63908C211yesmaximal, characteristic, complemented
10#63898C211yesmaximal, characteristic, complemented
11#63893C211yesmaximal, characteristic, complemented
12#63885C211yesmaximal, characteristic, complemented
13#63884C211yesmaximal, characteristic, complemented
14#63879C211yesmaximal, characteristic, complemented
15#63858C211yesmaximal, characteristic, complemented
16#5657—31nomaximal
17#5593—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C2×C3×#67C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×D12C2211yescharacteristic
22C2×C3×D12C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic, complemented
24C2×#627C2211yescharacteristic, complemented
25C2×#625C2211yescharacteristic
26C2×#625C2211yescharacteristic
27C2×#625C2211yescharacteristic, complemented
28C2×D3×Dic3C2211yescharacteristic
29C2×D3×Dic3C2211yescharacteristic
30C2×D3×Dic3C2211yescharacteristic, complemented
31C32×C22⋊C4C2211yescharacteristic
32#3757C2211yescharacteristic
33#3757C2211yescharacteristic
34C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
35#3750C2211yescharacteristic, complemented
36#3750C2211yescharacteristic, complemented
37C4×C3⋊Dic3C2211yescharacteristic, complemented
38C3×#217C2211yescharacteristic
39C3×(C2×D3)⋊C4C2211yescharacteristic
40C3×(C2×D3)⋊C4C2211yescharacteristic
41C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
42C3×C4⋊Dic3C2211yescharacteristic, complemented
43C3×Dic3⋊C4C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45(C2×D3)⋊Dic3C2211yescharacteristic
46(C2×D3)⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic, complemented
50(C2×D3)⋊Dic3C2211yescharacteristic, complemented
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#123—31no—
54C3×#120—31no—
55#953—31no—
56C2×#217—31no—
57#939—31no—
58#939—31no—
59#938—31no—
60#938—31no—
61#937—31no—
62#936—31no—
63#935—31no—
64D4×Dic3—31no—
65#930—31no—
66C4×#67—31no—
67#898—31no—
68#897—31no—
69#894—31no—
70#893—31no—
71#885—31no—
72#885—31no—
73#885—31no—
74#884—31no—
75#883D311yescharacteristic, complemented
76#882D311yescharacteristic, complemented
77#67⋊C4—31no—
78D3×C22⋊C4—31no—
79#879—31no—
80#879—31no—
81#879—31no—
82#879—61no—
83#878—31no—
84C23×C32C2311yescharacteristic
85C22×C3×D3—22no—
86C22×C3×D3—41no—
87C22×C3×D3—41no—
88C22×C3×D3C2311yescharacteristic
89C22×C3×D3C2311yescharacteristic
90C22×C3×D3C2311yescharacteristic
91C22×C3×D3C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C3×#67—41no—
99C3×#67—41no—
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C3×D12—22no—
105C3×D12—41no—
106C3×C4×D3—41no—
107#627—41no—
108#625—22no—
109#625—41no—
110#625—41no—
111D3×Dic3—41no—
112D3×Dic3—41no—
113D3×Dic3—41no—
114#538—91noSylow
115C24×C3—31no—
116C23×D3C2×D311yescharacteristic
117C2×C3×D4—31no—
118C2×C3×D4—31no—
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C22×C3×C4—31no—
122C2×#67—31no—
123C2×#67—31no—
124C2×#67—31no—
125C2×#67—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67C2×D311yescharacteristic
129C2×#67C2×D311yescharacteristic
130C22×Dic3—31no—
131C22×Dic3—31no—
132C22×Dic3—31no—
133C2×D12C2×D311yescharacteristic
134C2×D12C2×D311yescharacteristic
135C2×C4×D3—31no—
136C2×C4×D3—31no—
137C2×C4×D3—31no—
138C2×C4×D3C2×D311yescharacteristic, complemented
139C2×Dic6—31no—
140C2×Dic6—31no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—31no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—31no—
145C3×C22⋊C4—31no—
146C3×C22⋊C4—31no—
147C3×C22⋊C4—31no—
148C3×C22⋊C4C2×D311yescharacteristic
149C3×C22⋊C4C2×D311yescharacteristic
150#217—31no—
151#217—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—31no—
161#217—61no—
162#217—61no—
163#217C2×D311yescharacteristic
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)⋊C4C2×D311yescharacteristic
172(C2×D3)⋊C4C2×D311yescharacteristic
173(C2×D3)⋊C4C2×D311yescharacteristic, complemented
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—61no—
178C4⋊Dic3C2×D311yescharacteristic, complemented
179Dic3⋊C4—31no—
180Dic3⋊C4—31no—
181Dic3⋊C4—31no—
182Dic3⋊C4—31no—
183Dic3⋊C4—31no—
184Dic3⋊C4—61no—
185Dic3⋊C4—61no—
186Dic3⋊C4C2×D311yescharacteristic, complemented
187C4×Dic3—31no—
188C4×Dic3—31no—
189C4×Dic3—31no—
190C4×Dic3—31no—
191C4×Dic3—61no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32C2411yescharacteristic
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—21no—
202C2×C3×D3—21no—
203C2×C3×D3—21no—
204C2×C3×D3—21no—
205C2×C3×D3—21no—
206C2×C3×D3—21no—
207C2×C3×D3—44no—
208C2×C3×D3#4112yes—
209C32×C4—21no—
210C32×C4—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—91no—
227#121—91no—
228#121—91no—
229#120—31no—
230#120—91no—
231C4×D4—91no—
232C4×D4—91no—
233C2×C22⋊C4—91no—
234C23×C3—21no—
235C23×C3—31no—
236C23×C3—31no—
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—62no—
240C23×C3—121no—
241C23×C3—121no—
242C23×C3C22×D311yescharacteristic
243C23×C3C22×D311yescharacteristic
244C22×D3—22no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3C22×D311yescharacteristic
248C22×D3C22×D311yescharacteristic
249C22×D3C22×D311yescharacteristic
250C22×D3C22×D311yescharacteristic
251C3×D4—62no—
252C3×D4—121no—
253C3×D4—121no—
254C3×D4—121no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—121no—
262C2×C3×C4C22×D311yescharacteristic
263C2×C3×C4C22×D311yescharacteristic
264C2×C3×C4C22×D311yescharacteristic
265C2×C3×C4C22×D311yescharacteristic
266#67—41no—
267#67—41no—
268#67—62no—
269#67—62no—
270#67—121no—
271#67—121no—
272#67—121no—
273#67—121no—
274C2×Dic3—31no—
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—31no—
278C2×Dic3—31no—
279C2×Dic3—31no—
280C2×Dic3—31no—
281C2×Dic3—31no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—121no—
287C2×Dic3—121no—
288C2×Dic3—121no—
289C2×Dic3C22×D311yescharacteristic
290C2×Dic3C22×D311yescharacteristic
291C2×Dic3C22×D311yescharacteristic
292C2×Dic3C22×D311yescharacteristic
293D12—22no—
294D12—41no—
295C4×D3—41no—
296C4×D3—121no—
297C4×D3—121no—
298C4×D3—121no—
299Dic6—121no—
300Dic6—121no—
301C2×C32—41no—
302C2×C32C2×#4111yescharacteristic
303C2×C32H(5,𝔽2)11yescharacteristic
304C2×C32H(5,𝔽2)11yescharacteristic
305C3×D3—22no—
306C3×D3—41no—
307C3×D3—41no—
308C3×D3—41no—
309C24—31no—
310C2×Q8—91no—
311C2×D4—31no—
312C2×D4—31no—
313C2×D4—31no—
314C2×D4—31no—
315C2×D4—91no—
316C2×D4—91no—
317C2×D4—91no—
318C22×C4—31no—
319C22×C4—91no—
320C22×C4—91no—
321C22×C4—91no—
322C4⋊C4—31no—
323C4⋊C4—31no—
324C4⋊C4—91no—
325C4⋊C4—91no—
326C4⋊C4—91no—
327C4⋊C4—91no—
328C22⋊C4—31no—
329C22⋊C4—31no—
330C22⋊C4—31no—
331C22⋊C4—31no—
332C22⋊C4—91no—
333C22⋊C4—91no—
334C22⋊C4—91no—
335C22⋊C4—91no—
336C22⋊C4—91no—
337C22⋊C4—91no—
338C22⋊C4—91no—
339C22⋊C4—91no—
340C22⋊C4—91no—
341C22⋊C4D3211yescharacteristic
342C42—91no—
343C42—91no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—32no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—61no—
359C22×C3—61no—
360C22×C3—61no—
361C22×C3—61no—
362C22×C3—61no—
363C22×C3—61no—
364C22×C3—61no—
365C22×C3—61no—
366C22×C3—124no—
367C22×C3C23×D311yescharacteristic
368C22×C3C23×D311yescharacteristic
369C2×D3—21no—
370C2×D3—21no—
371C2×D3—21no—
372C2×D3—21no—
373C2×D3—21no—
374C2×D3—21no—
375C2×D3—21no—
376C2×D3—21no—
377C2×D3—21no—
378C2×D3—21no—
379C2×D3—21no—
380C2×D3—44no—
381C2×D3#23712yes—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—41no—
387C3×C4—41no—
388C3×C4—61no—
389C3×C4—61no—
390C3×C4—61no—
391C3×C4—61no—
392Dic3—21no—
393Dic3—21no—
394Dic3—21no—
395Dic3—21no—
396Dic3—61no—
397Dic3—61no—
398Dic3—61no—
399Dic3—61no—
400Dic3—61no—
401Dic3—61no—
402Dic3—61no—
403Dic3—61no—
404Dic3—121no—
405Dic3—121no—
406Dic3—121no—
407Dic3—121no—
408C32#53811yescharacteristic, complemented, Sylow
409C23—31no—
410C23—31no—
411C23—31no—
412C23—31no—
413C23—62no—
414C23—121no—
415C23—121no—
416C23C2×D3211yescharacteristic
417Q8—361no—
418D4—62no—
419D4—121no—
420D4—121no—
421D4—121no—
422D4—182no—
423D4—361no—
424D4—361no—
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—91no—
430C2×C4—91no—
431C2×C4—91no—
432C2×C4—91no—
433C2×C4—121no—
434C2×C4—361no—
435C2×C4—361no—
436C2×C4—361no—
437C2×C4C2×D3211yescharacteristic
438C2×C4C2×D3211yescharacteristic
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—62no—
445C2×C3—81no—
446C2×C3—121no—
447C2×C3—121no—
448C2×C3—121no—
449C2×C3C2×#23711yescharacteristic
450C2×C3#100411yescharacteristic
451C2×C3#100411yescharacteristic
452C2×C3#100411yescharacteristic
453C2×C3D3×#4111yescharacteristic
454C2×C3#100911yescharacteristic
455D3—22no—
456D3—41no—
457D3—41no—
458D3—41no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—32no—
463C22—61no—
464C22—61no—
465C22—61no—
466C22—61no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22—61no—
471C22—61no—
472C22—61no—
473C22—61no—
474C22—124no—
475C22C22×D3211yescharacteristic
476C4—21no—
477C4—21no—
478C4—61no—
479C4—61no—
480C4—61no—
481C4—61no—
482C4—181no—
483C4—181no—
484C4—181no—
485C4—181no—
486C3—21no—
487C3#559311yescharacteristic, complemented
488C3#565711yescharacteristic, complemented
489C2—41no—
490C2—62no—
491C2—121no—
492C2—121no—
493C2—121no—
494C2#6430711yescharacteristic
495C2D3×#23711yescharacteristic
496C2#6431311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64088C2semidirect
2. C3×#883C2semidirect
3. C3×#882C2semidirect
4. #63976C2semidirect
5. #63970C2semidirect
6. D3×#217C2semidirect
7. #63959C2semidirect
8. #63915C2semidirect
9. #63908C2semidirect
10. #63898C2semidirect
11. #63893C2semidirect
12. #63885C2semidirect
13. #63884C2semidirect
14. #63879C2semidirect
15. #63858C2semidirect
23. C2×C3×C4×D3C22semidirect
24. C2×#627C22semidirect
27. C2×#625C22semidirect
30. C2×D3×Dic3C22semidirect
34. C4⋊(C3⋊Dic3)C22semidirect
35. #3750C22semidirect
36. #3750C22semidirect
37. C4×C3⋊Dic3C22semidirect
41. C3×(C2×D3)⋊C4C22semidirect
42. C3×C4⋊Dic3C22semidirect
43. C3×Dic3⋊C4C22semidirect
44. Dic3⋊Dic3C22semidirect
49. (C2×D3)⋊Dic3C22semidirect
50. (C2×D3)⋊Dic3C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. Dic32C22semidirect
75. #883D3semidirect
76. #882D3semidirect
138. C2×C4×D3C2×D3semidirect
173. (C2×D3)⋊C4C2×D3semidirect
178. C4⋊Dic3C2×D3semidirect
186. Dic3⋊C4C2×D3semidirect
408. C32#538semidirect
487. C3#5593semidirect
488. C3#5657semidirect

Maximal subgroups

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

subgroupindex
#640882
C3×#8832
C3×#8822
#639762
#639702
D3×#2172
#639592
#639152
#639082
#638982
#638932
#638852
#638842
#638792
#638582
#56573
#55933