Finite Group Databasebeta

← #10594941

Subgroups — #10594941

Order 576.

543 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#638C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
3C3×#924C211yesmaximal, characteristic, complemented
4#63977C211yesmaximal, characteristic
5D3×#217C211yesmaximal, characteristic, complemented
6#63966C211yesmaximal, characteristic, complemented
7#63955C211yesmaximal, characteristic, complemented
8#63907C211yesmaximal, characteristic, complemented
9#63890C211yesmaximal, characteristic, complemented
10#63887C211yesmaximal, characteristic, complemented
11#63881C211yesmaximal, characteristic, complemented
12#63878C211yesmaximal, characteristic, complemented
13#63862C211yesmaximal, characteristic, complemented
14#63858C211yesmaximal, characteristic, complemented
15#63853C211yesmaximal, characteristic, complemented
16#5637—31nomaximal
17#5495—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C22×C32×C4C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×C4×C3⋊D3C2211yescharacteristic, complemented
22C22×C3×Dic3C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic, complemented
24C2×#626C2211yescharacteristic
25C2×#626C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic
28#3757C2211yescharacteristic
29#3752C2211yescharacteristic, complemented
30#3750C2211yescharacteristic, complemented
31C4×C3⋊Dic3C2211yescharacteristic, complemented
32C3×#217C2211yescharacteristic
33C3×#217C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×C4⋊Dic3C2211yescharacteristic, complemented
39C3×Dic3⋊C4C2211yescharacteristic, complemented
40C3×Dic3⋊C4C2211yescharacteristic, complemented
41Dic3⋊Dic3C2211yescharacteristic, complemented
42Dic3⋊Dic3C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45#3722C2211yescharacteristic, complemented
46#3722C2211yescharacteristic, complemented
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51Dic32C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×C22⋊Q8—31no—
54C2×C3×C22⋊C4—31no—
55#953—31no—
56C2×#217—31no—
57#947—31no—
58#946—31no—
59#929—31no—
60C4×#67—31no—
61C4×#67—31no—
62C4×#67—61no—
63C2×(C2×D3)⋊C4D311yescharacteristic, complemented
64#926—31no—
65#924D311yescharacteristic, complemented
66#898—31no—
67#898—31no—
68#894—31no—
69#894—31no—
70#893—31no—
71#892—31no—
72#886—31no—
73#886—31no—
74#885—31no—
75#883—31no—
76#882—31no—
77#67⋊C4—31no—
78D3×C22⋊C4—31no—
79#879—31no—
80#879—31no—
81#878—31no—
82#877—31no—
83#877—31no—
84C23×C32C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3×D3—22no—
87C22×C3×D3—22no—
88C22×C3×D3—41no—
89C22×C3×D3C2311yescharacteristic
90C22×C3×D3C2311yescharacteristic
91C2×C32×C4—41no—
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94#638—22no—
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C4×C3⋊D3—41no—
99C2×C3×Dic3—22no—
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C3×Dic6—41no—
107#626—22no—
108#626—41no—
109D3×Dic3—22no—
110D3×Dic3—41no—
111#551—91noSylow
112C24×C3—31no—
113C23×D3C2×D311yescharacteristic
114C2×C3×Q8—31no—
115C22×C3×C4—21no—
116C22×C3×C4—31no—
117C22×C3×C4C2×D311yescharacteristic
118C22×C3×C4C2×D311yescharacteristic
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—31no—
123C2×#67—61no—
124C22×Dic3—31no—
125C22×Dic3—31no—
126C22×Dic3C2×D311yescharacteristic
127C2×D12—31no—
128C2×D12—31no—
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×C4×D3—31no—
132C2×C4×D3—31no—
133C2×C4×D3—61no—
134C2×Dic6C2×D311yescharacteristic, complemented
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4—31no—
140C3×C22⋊C4—31no—
141C3×C22⋊C4—31no—
142C3×C22⋊C4—31no—
143C3×C22⋊C4—31no—
144#217—31no—
145#217—31no—
146#217—31no—
147#217—31no—
148#217—31no—
149#217—31no—
150#217—61no—
151#217C2×D311yescharacteristic
152#217C2×D311yescharacteristic
153(C2×D3)⋊C4—31no—
154(C2×D3)⋊C4—31no—
155(C2×D3)⋊C4—31no—
156(C2×D3)⋊C4—31no—
157(C2×D3)⋊C4—31no—
158(C2×D3)⋊C4—31no—
159(C2×D3)⋊C4—31no—
160(C2×D3)⋊C4—31no—
161(C2×D3)⋊C4—31no—
162(C2×D3)⋊C4—31no—
163(C2×D3)⋊C4—61no—
164(C2×D3)⋊C4C2×D311yescharacteristic
165(C2×D3)⋊C4C2×D311yescharacteristic
166(C2×D3)⋊C4C2×D311yescharacteristic
167(C2×D3)⋊C4C2×D311yescharacteristic
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3C2×D311yescharacteristic, complemented
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4—31no—
177Dic3⋊C4—31no—
178Dic3⋊C4—31no—
179Dic3⋊C4—61no—
180Dic3⋊C4C2×D311yescharacteristic, complemented
181Dic3⋊C4C2×D311yescharacteristic, complemented
182C4×Dic3—31no—
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—31no—
187C4×Dic3—31no—
188C4×Dic3—61no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32#4112yes—
192C22×C32C2411yescharacteristic
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3⋊D3—21no—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—44no—
202C2×C3×D3#4112yes—
203C32×C4—21no—
204C32×C4—21no—
205C3⋊Dic3—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3—21no—
208C3×Dic3—21no—
209C3×Dic3—21no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3—21no—
213C3×Dic3—21no—
214#126—91no—
215#126—91no—
216#124—91no—
217#123—91no—
218#123—91no—
219#123—91no—
220C22⋊Q8—31no—
221C22⋊Q8—91no—
222#120—91no—
223C4×D4—91no—
224C4×D4—91no—
225#117—91no—
226#117—91no—
227C2×C22⋊C4—31no—
228C2×C22⋊C4—91no—
229C23×C3—21no—
230C23×C3—31no—
231C23×C3—31no—
232C23×C3—62no—
233C23×C3—62no—
234C23×C3—121no—
235C23×C3C22×D311yescharacteristic
236C23×C3C22×D311yescharacteristic
237C22×D3—22no—
238C22×D3—22no—
239C22×D3—31no—
240C22×D3—31no—
241C22×D3—41no—
242C22×D3—61no—
243C22×D3C22×D311yescharacteristic
244C22×D3C22×D311yescharacteristic
245C3×Q8—121no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—62no—
258C2×C3×C4C22×D311yescharacteristic
259C2×C3×C4C22×D311yescharacteristic
260C2×C3×C4C22×D311yescharacteristic
261C2×C3×C4C22×D311yescharacteristic
262#67—62no—
263#67—64no—
264#67—62no—
265#67—62no—
266#67—121no—
267C2×Dic3—22no—
268C2×Dic3—31no—
269C2×Dic3—31no—
270C2×Dic3—31no—
271C2×Dic3—31no—
272C2×Dic3—31no—
273C2×Dic3—31no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—62no—
278C2×Dic3—121no—
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282C2×Dic3C22×D311yescharacteristic
283C2×Dic3C22×D311yescharacteristic
284C2×Dic3C22×D311yescharacteristic
285D12—62no—
286D12—121no—
287C4×D3—62no—
288C4×D3—121no—
289C4×D3—121no—
290C4×D3—121no—
291C4×D3—122no—
292Dic6—41no—
293C2×C32—22no—
294C2×C32C2×#4111yescharacteristic
295C2×C32C2×#4111yescharacteristic
296C2×C32H(5,𝔽2)11yescharacteristic
297C3⋊D3—41no—
298C3×D3—22no—
299C3×D3—41no—
300C24—31no—
301C2×Q8—31no—
302C2×D4—91no—
303C2×D4—91no—
304C2×D4—91no—
305C22×C4—31no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4D3211yescharacteristic
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C4⋊C4—91no—
318C22⋊C4—31no—
319C22⋊C4—31no—
320C22⋊C4—31no—
321C22⋊C4—31no—
322C22⋊C4—31no—
323C22⋊C4—31no—
324C22⋊C4—91no—
325C22⋊C4—91no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4—91no—
329C22⋊C4—91no—
330C22⋊C4—91no—
331C22⋊C4—91no—
332C42—91no—
333C42—91no—
334C42—91no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—32no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—124no—
350C22×C3#23712yes—
351C22×C3#23912yes—
352C22×C3C23×D311yescharacteristic
353C22×C3C23×D311yescharacteristic
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—44no—
360C2×D3—62no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—61no—
366C2×D3—61no—
367C2×D3—121no—
368C2×D3—121no—
369C2×D3#23512yes—
370C3×C4—22no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—41no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—61no—
379C3×C4—61no—
380C3×C4—61no—
381C3×C4—61no—
382Dic3—21no—
383Dic3—21no—
384Dic3—21no—
385Dic3—21no—
386Dic3—21no—
387Dic3—21no—
388Dic3—62no—
389Dic3—61no—
390Dic3—61no—
391Dic3—61no—
392Dic3—61no—
393Dic3—61no—
394Dic3—61no—
395Dic3—121no—
396Dic3—121no—
397C32#55111yescharacteristic, complemented, Sylow
398C23—31no—
399C23—31no—
400C23—62no—
401C23—62no—
402C23—91no—
403C23—121no—
404C23C2×D3211yescharacteristic
405Q8—121no—
406D4—182no—
407D4—182no—
408D4—361no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—41no—
416C2×C4—62no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—91no—
420C2×C4—182no—
421C2×C4—361no—
422C2×C4—361no—
423C2×C4C2×D3211yescharacteristic
424C2×C4C2×D3211yescharacteristic
425C2×C3—22no—
426C2×C3—22no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—42no—
431C2×C3—62no—
432C2×C3—121no—
433C2×C3C2×#23511yescharacteristic
434C2×C3C2×#23711yescharacteristic
435C2×C3#100411yescharacteristic
436C2×C3#100411yescharacteristic
437C2×C3C2×#23911yescharacteristic
438C2×C3D3×#4111yescharacteristic
439D3—22no—
440D3—41no—
441D3—122no—
442D3—121no—
443D3—121no—
444C22—21no—
445C22—21no—
446C22—32no—
447C22—61no—
448C22—61no—
449C22—61no—
450C22—61no—
451C22—61no—
452C22—124no—
453C22—181no—
454C22—181no—
455C22—181no—
456C22#379612yes—
457C22C22×D3211yescharacteristic
458C4—21no—
459C4—21no—
460C4—61no—
461C4—61no—
462C4—61no—
463C4—61no—
464C4—61no—
465C4—61no—
466C4—181no—
467C4—181no—
468C4—181no—
469C3—21no—
470C3#549511yescharacteristic, complemented
471C3#563711yescharacteristic, complemented
472C2—22no—
473C2—62no—
474C2—121no—
475C2—361no—
476C2C2×#379611yescharacteristic
477C2D3×#23511yescharacteristic
478C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radical#63977C2
hypercenterC22×C4D32
hypocenterC32#551
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C4×#638C2semidirect
2. C2×C3×(C2×D3)⋊C4C2semidirect
3. C3×#924C2semidirect
5. D3×#217C2semidirect
6. #63966C2semidirect
7. #63955C2semidirect
8. #63907C2semidirect
9. #63890C2semidirect
10. #63887C2semidirect
11. #63881C2semidirect
12. #63878C2semidirect
13. #63862C2semidirect
14. #63858C2semidirect
15. #63853C2semidirect
21. C2×C4×C3⋊D3C22semidirect
23. C2×C3×Dic6C22semidirect
29. #3752C22semidirect
30. #3750C22semidirect
31. C4×C3⋊Dic3C22semidirect
38. C3×C4⋊Dic3C22semidirect
39. C3×Dic3⋊C4C22semidirect
40. C3×Dic3⋊C4C22semidirect
41. Dic3⋊Dic3C22semidirect
42. Dic3⋊Dic3C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
45. #3722C22semidirect
46. #3722C22semidirect
51. Dic32C22semidirect
52. Dic32C22semidirect
63. C2×(C2×D3)⋊C4D3semidirect
65. #924D3semidirect
134. C2×Dic6C2×D3semidirect
172. C4⋊Dic3C2×D3semidirect
180. Dic3⋊C4C2×D3semidirect
181. Dic3⋊C4C2×D3semidirect
397. C32#551semidirect
470. C3#5495semidirect
471. C3#5637semidirect

Maximal subgroups

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

subgroupindex
C4×#6382
C2×C3×(C2×D3)⋊C42
C3×#9242
#639772
D3×#2172
#639662
#639552
#639072
#638902
#638872
#638812
#638782
#638622
#638582
#638532
#56373
#54953