Finite Group Databasebeta

← #10595088

Subgroups — #10595088

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
2C3×#929C211yesmaximal, characteristic, complemented
3C3×C4×#67C211yesmaximal, characteristic, complemented
4#63983C211yesmaximal, characteristic
5#63975C211yesmaximal, characteristic, complemented
6#63969C211yesmaximal, characteristic, complemented
7#63959C211yesmaximal, characteristic, complemented
8#63921C211yesmaximal, characteristic, complemented
9D3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
10#63878C211yesmaximal, characteristic, complemented
11#63865C211yesmaximal, characteristic, complemented
12#63862C211yesmaximal, characteristic, complemented
13#63859C211yesmaximal, characteristic, complemented
14#63851C211yesmaximal, characteristic, complemented
15#63836C211yesmaximal, characteristic, complemented
16#5659—31nomaximal
17#5556—31nomaximal
18C22×D32C2211yescharacteristic
19C22×C32×C4C2211yescharacteristic, complemented
20C2×#638C2211yescharacteristic
21C2×C4×C3⋊D3C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic
24C2×C3×C4×D3C2211yescharacteristic
25C2×#627C2211yescharacteristic, complemented
26C2×#624C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic
28#3757C2211yescharacteristic, complemented
29#3752C2211yescharacteristic
30#3750C2211yescharacteristic, complemented
31C4×C3⋊Dic3C2211yescharacteristic, complemented
32C3×#217C2211yescharacteristic, complemented
33C3×#217C2211yescharacteristic, complemented
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic, complemented
38C3×Dic3⋊C4C2211yescharacteristic, complemented
39C3×Dic3⋊C4C2211yescharacteristic, complemented
40C3×C4×Dic3C2211yescharacteristic, complemented
41#3724C2211yescharacteristic, complemented
42Dic3⋊Dic3C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic, complemented
53C3×#123—31no—
54C3×C4×D4—31no—
55#937—31no—
56#937—31no—
57#936—31no—
58#935—31no—
59#929D311yescharacteristic, complemented
60C4×#67—31no—
61C4×#67—31no—
62C4×#67—61no—
63C4×#67D311yescharacteristic, complemented
64C2×(C2×D3)⋊C4—31no—
65#924—31no—
66#898—31no—
67#896—31no—
68#896—31no—
69#896—31no—
70#892—31no—
71#892—31no—
72#885—31no—
73#883—31no—
74#883—31no—
75#883—31no—
76D3×C22⋊C4—31no—
77D3×C22⋊C4—31no—
78D3×C22⋊C4—31no—
79#879—31no—
80#879—31no—
81#877—31no—
82#876—31no—
83#872—31no—
84C23×C32C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×D32—22no—
89C2×D32—22no—
90C2×D32—41no—
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—
99C3×#67—22no—
100C3×#67—41no—
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C3×C4×D3—41no—
108#627—41no—
109#624—22no—
110D3×Dic3—22no—
111#551—91noSylow
112C23×D3—31no—
113C23×D3—31no—
114C2×C3×D4—31no—
115C2×C3×D4—31no—
116C22×C3×C4—21no—
117C22×C3×C4—31no—
118C22×C3×C4C2×D311yescharacteristic, complemented
119C22×C3×C4C2×D311yescharacteristic, complemented
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—61no—
123C2×#67C2×D311yescharacteristic
124C2×#67C2×D311yescharacteristic
125C22×Dic3—31no—
126C2×C4×D3—31no—
127C2×C4×D3—31no—
128C2×C4×D3—31no—
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×C4×D3—61no—
132C2×C4×D3C2×D311yescharacteristic
133C2×Dic6—31no—
134C2×Dic6—31no—
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×C42—31no—
144#217—31no—
145#217—31no—
146#217—31no—
147#217—31no—
148#217—31no—
149#217—31no—
150#217—31no—
151#217—61no—
152#217C2×D311yescharacteristic, complemented
153#217C2×D311yescharacteristic, complemented
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—31no—
164(C2×D3)⋊C4—31no—
165(C2×D3)⋊C4—61no—
166(C2×D3)⋊C4C2×D311yescharacteristic
167(C2×D3)⋊C4C2×D311yescharacteristic
168(C2×D3)⋊C4C2×D311yescharacteristic
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172Dic3⋊C4—31no—
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
182Dic3⋊C4C2×D311yescharacteristic, complemented
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—31no—
187C4×Dic3—61no—
188C4×Dic3C2×D311yescharacteristic, complemented
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32C2411yescharacteristic
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3⋊D3#4112yes—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3#4112yes—
202D32—44no—
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—31no—
218#123—91no—
219#123—91no—
220C22⋊Q8—91no—
221C22⋊Q8—91no—
222#120—91no—
223C4×D4—31no—
224C4×D4—91no—
225#117—91no—
226#117—91no—
227C2×C22⋊C4—91no—
228C2×C22⋊C4—91no—
229C23×C3—21no—
230C23×C3—31no—
231C23×C3—31no—
232C23×C3C22×D311yescharacteristic
233C23×C3C22×D311yescharacteristic
234C22×D3—31no—
235C22×D3—31no—
236C22×D3—61no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—62no—
240C22×D3—62no—
241C22×D3—121no—
242C22×D3—121no—
243C22×D3C22×D311yescharacteristic
244C22×D3C22×D311yescharacteristic
245C3×D4—62no—
246C3×D4—121no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—121no—
259C2×C3×C4C22×D311yescharacteristic
260C2×C3×C4C22×D311yescharacteristic
261C2×C3×C4C22×D311yescharacteristic
262C2×C3×C4C22×D311yescharacteristic
263#67—22no—
264#67—41no—
265#67—62no—
266#67—62no—
267#67—64no—
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×Dic3C22×D311yescharacteristic
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282C2×Dic3C22×D311yescharacteristic
283C2×Dic3C22×D311yescharacteristic
284C4×D3—41no—
285C4×D3—62no—
286C4×D3—62no—
287C4×D3—62no—
288C4×D3—121no—
289C4×D3—121no—
290C4×D3—122no—
291Dic6—121no—
292Dic6—121no—
293C2×C32—41no—
294C2×C32C2×#4111yescharacteristic
295C2×C32C2×#4111yescharacteristic
296C2×C32H(5,𝔽2)11yescharacteristic
297C3⋊D3—22no—
298C3×D3—22no—
299C3×D3—41no—
300C24—91no—
301C2×Q8—91no—
302C2×D4—31no—
303C2×D4—31no—
304C2×D4—91no—
305C22×C4—31no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4D3211yescharacteristic, complemented
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—91no—
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—31no—
333C42—91no—
334C42—91no—
335C22×C3—22no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—32no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3C23×D311yescharacteristic
352C22×C3C23×D311yescharacteristic
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—32no—
359C2×D3—32no—
360C2×D3—62no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—121no—
366C2×D3—121no—
367C2×D3—124no—
368C2×D3—124no—
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—91no—
401C23—182no—
402C23—182no—
403C23—361no—
404C23C2×D3211yescharacteristic
405Q8—361no—
406D4—62no—
407D4—121no—
408D4—182no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—41no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—121no—
420C2×C4—182no—
421C2×C4—182no—
422C2×C4—361no—
423C2×C4C2×D3211yescharacteristic
424C2×C4C2×D3211yescharacteristic
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—42no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—62no—
432C2×C3—121no—
433C2×C3C2×#23511yescharacteristic
434C2×C3#100411yescharacteristic
435C2×C3#100411yescharacteristic
436C2×C3D3×#4111yescharacteristic
437C2×C3D3×#4111yescharacteristic
438C2×C3D3×#4111yescharacteristic
439D3—22no—
440D3—41no—
441D3—62no—
442D3—62no—
443D3—122no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—32no—
448C22—61no—
449C22—61no—
450C22—61no—
451C22—61no—
452C22—61no—
453C22—92no—
454C22—181no—
455C22—181no—
456C22—364no—
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#555611yescharacteristic, complemented
471C3#565911yescharacteristic, complemented
472C2—41no—
473C2—62no—
474C2—121no—
475C2—182no—
476C2D3×#23511yescharacteristic
477C2#6430611yescharacteristic
478C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radical#63983C2
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. C3×#929C2semidirect
3. C3×C4×#67C2semidirect
5. #63975C2semidirect
6. #63969C2semidirect
7. #63959C2semidirect
8. #63921C2semidirect
9. D3×(C2×D3)⋊C4C2semidirect
10. #63878C2semidirect
11. #63865C2semidirect
12. #63862C2semidirect
13. #63859C2semidirect
14. #63851C2semidirect
15. #63836C2semidirect
19. C22×C32×C4C22semidirect
25. C2×#627C22semidirect
28. #3757C22semidirect
30. #3750C22semidirect
31. C4×C3⋊Dic3C22semidirect
32. C3×#217C22semidirect
33. C3×#217C22semidirect
37. C3×Dic3⋊C4C22semidirect
38. C3×Dic3⋊C4C22semidirect
39. C3×Dic3⋊C4C22semidirect
40. C3×C4×Dic3C22semidirect
41. #3724C22semidirect
42. Dic3⋊Dic3C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
52. Dic32C22semidirect
59. #929D3semidirect
63. C4×#67D3semidirect
118. C22×C3×C4C2×D3semidirect
119. C22×C3×C4C2×D3semidirect
152. #217C2×D3semidirect
153. #217C2×D3semidirect
180. Dic3⋊C4C2×D3semidirect
181. Dic3⋊C4C2×D3semidirect
182. Dic3⋊C4C2×D3semidirect
188. C4×Dic3C2×D3semidirect
309. C22×C4D32semidirect
397. C32#551semidirect
470. C3#5556semidirect
471. C3#5659semidirect

Maximal subgroups

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

subgroupindex
C4×#6382
C3×#9292
C3×C4×#672
#639832
#639752
#639692
#639592
#639212
D3×(C2×D3)⋊C42
#638782
#638652
#638622
#638592
#638512
#638362
#56593
#55563