Finite Group Databasebeta

← #10594018

Subgroups — #10594018

Order 576.

534 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64081C211yesmaximal, characteristic, complemented
2C3×#876C211yesmaximal, characteristic, complemented
3C3×C4×D12C211yesmaximal, characteristic, complemented
4#63922C211yesmaximal, characteristic
5#63921C211yesmaximal, characteristic, complemented
6D3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7#63915C211yesmaximal, characteristic, complemented
8#63905C211yesmaximal, characteristic, complemented
9#63904C211yesmaximal, characteristic, complemented
10#63895C211yesmaximal, characteristic, complemented
11#63884C211yesmaximal, characteristic, complemented
12#63878C211yesmaximal, characteristic, complemented
13#63858C211yesmaximal, characteristic, complemented
14#63853C211yesmaximal, characteristic, complemented
15#63851C211yesmaximal, characteristic, complemented
16#5705—31nomaximal
17#5556—31nomaximal
18C22×D32C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×C3×D12C2211yescharacteristic
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic, complemented
23C2×#627C2211yescharacteristic, complemented
24C2×#626C2211yescharacteristic
25C2×#625C2211yescharacteristic
26C2×#624C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic
28C32×C42C2211yescharacteristic, complemented
29#3752C2211yescharacteristic
30#3752C2211yescharacteristic
31#3750C2211yescharacteristic, complemented
32#3750C2211yescharacteristic, complemented
33C4×C3⋊Dic3C2211yescharacteristic, complemented
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
39C3×C4⋊Dic3C2211yescharacteristic, complemented
40C3×Dic3⋊C4C2211yescharacteristic, complemented
41C3×Dic3⋊C4C2211yescharacteristic, complemented
42C3×Dic3⋊C4C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45#3722C2211yescharacteristic
46(C2×D3)⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic, complemented
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#126—31no—
54C3×C4×D4—31no—
55#937—31no—
56#935—31no—
57#929—31no—
58C4×#67—31no—
59C2×(C2×D3)⋊C4—31no—
60#924—31no—
61#898—31no—
62#897—31no—
63#896—31no—
64#896—31no—
65#894—31no—
66#894—31no—
67#893—31no—
68#892—31no—
69#885—31no—
70#885—31no—
71#883—31no—
72#883—31no—
73#882—31no—
74D3×C22⋊C4—31no—
75D3×C22⋊C4—31no—
76D3×C22⋊C4—31no—
77#879—31no—
78#877—31no—
79#876D311yescharacteristic, complemented
80C4×D12D311yescharacteristic, complemented
81#872—31no—
82#872—31no—
83#872—61no—
84C22×C3⋊D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×D32—22no—
89C2×D32—22no—
90C2×D32—41no—
91C2×C32×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C4×C3⋊D3—22no—
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C3×D12—22no—
104C3×C4×D3—41no—
105C3×C4×D3—41no—
106#627—41no—
107#626—22no—
108#625—41no—
109#624—41no—
110D3×Dic3—22no—
111#551—91noSylow
112C23×D3—31no—
113C23×D3—31no—
114C2×C3×D4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C2×#67—31no—
118C2×#67—31no—
119C2×#67—31no—
120C22×Dic3—31no—
121C2×D12—31no—
122C2×D12C2×D311yescharacteristic
123C2×C4×D3—31no—
124C2×C4×D3—31no—
125C2×C4×D3—31no—
126C2×C4×D3—31no—
127C2×C4×D3—31no—
128C2×C4×D3—61no—
129C2×C4×D3C2×D311yescharacteristic
130C2×C4×D3C2×D311yescharacteristic, complemented
131C2×Dic6—31no—
132C2×Dic6—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C22⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4—31no—
140C3×C22⋊C4—31no—
141C3×C22⋊C4—31no—
142C3×C42—21no—
143C3×C42C2×D311yescharacteristic, complemented
144C3×C42C2×D311yescharacteristic, complemented
145#217—31no—
146#217—31no—
147#217—31no—
148#217—31no—
149#217—31no—
150#217—31no—
151(C2×D3)⋊C4—31no—
152(C2×D3)⋊C4—31no—
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)⋊C4—61no—
165(C2×D3)⋊C4C2×D311yescharacteristic
166(C2×D3)⋊C4C2×D311yescharacteristic
167(C2×D3)⋊C4C2×D311yescharacteristic
168(C2×D3)⋊C4C2×D311yescharacteristic
169(C2×D3)⋊C4C2×D311yescharacteristic, complemented
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⋊C4—61no—
181Dic3⋊C4C2×D311yescharacteristic, complemented
182Dic3⋊C4C2×D311yescharacteristic, complemented
183Dic3⋊C4C2×D311yescharacteristic, complemented
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—31no—
187C4×Dic3—31no—
188C4×Dic3—61no—
189C22×C32C2411yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3#4112yes—
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#4112yes—
202D32—44no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—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—31no—
215#126—91no—
216#124—91no—
217#123—91no—
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—31no—
230C23×C3—31no—
231C23×C3—31no—
232C22×D3—31no—
233C22×D3—31no—
234C22×D3—61no—
235C22×D3—62no—
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—121no—
240C22×D3—121no—
241C22×D3C22×D311yescharacteristic
242C22×D3C22×D311yescharacteristic
243C22×D3C22×D311yescharacteristic
244C3×D4—62no—
245C2×C3×C4—21no—
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—121no—
254C2×C3×C4—121no—
255C2×C3×C4C22×D311yescharacteristic
256C2×C3×C4C22×D311yescharacteristic
257C2×C3×C4C22×D311yescharacteristic
258C2×C3×C4C22×D311yescharacteristic
259C2×C3×C4C22×D311yescharacteristic
260C2×C3×C4C22×D311yescharacteristic
261#67—62no—
262#67—121no—
263#67—121no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—31no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—62no—
274C2×Dic3C22×D311yescharacteristic
275C2×Dic3C22×D311yescharacteristic
276C2×Dic3C22×D311yescharacteristic
277C2×Dic3C22×D311yescharacteristic
278C2×Dic3C22×D311yescharacteristic
279D12—22no—
280D12—62no—
281C4×D3—41no—
282C4×D3—41no—
283C4×D3—62no—
284C4×D3—62no—
285C4×D3—64no—
286C4×D3—62no—
287C4×D3—121no—
288C4×D3—121no—
289Dic6—121no—
290Dic6—121no—
291C2×C32C2×#4111yescharacteristic
292C2×C32C2×#4111yescharacteristic
293C2×C32H(5,𝔽2)11yescharacteristic
294C3⋊D3—22no—
295C3×D3—22no—
296C3×D3—41no—
297C3×D3—41no—
298C24—91no—
299C2×Q8—91no—
300C2×D4—31no—
301C2×D4—91no—
302C2×D4—91no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—91no—
306C22×C4—91no—
307C22×C4—91no—
308C4⋊C4—31no—
309C4⋊C4—31no—
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—91no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C22⋊C4—31no—
317C22⋊C4—31no—
318C22⋊C4—31no—
319C22⋊C4—31no—
320C22⋊C4—31no—
321C22⋊C4—91no—
322C22⋊C4—91no—
323C22⋊C4—91no—
324C22⋊C4—91no—
325C22⋊C4—91no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4—91no—
329C22⋊C4—91no—
330C42—91no—
331C42—91no—
332C42D3211yescharacteristic, complemented
333C22×C3—21no—
334C22×C3—32no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3C23×D311yescharacteristic
344C22×C3C23×D311yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—32no—
354C2×D3—32no—
355C2×D3—62no—
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—61no—
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—124no—
363C2×D3—124no—
364C2×D3#23512yes—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—41no—
373C3×C4—41no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—61no—
379Dic3—21no—
380Dic3—21no—
381Dic3—21no—
382Dic3—21no—
383Dic3—21no—
384Dic3—62no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—61no—
390Dic3—61no—
391Dic3—121no—
392Dic3—121no—
393C32#55111yescharacteristic, complemented, Sylow
394C23—31no—
395C23—31no—
396C23—31no—
397C23—91no—
398C23—182no—
399C23—182no—
400C23—361no—
401Q8—361no—
402D4—62no—
403D4—182no—
404D4—361no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—91no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—182no—
416C2×C4—182no—
417C2×C4—361no—
418C2×C4C2×D3211yescharacteristic
419C2×C4C2×D3211yescharacteristic
420C2×C4C2×D3211yescharacteristic
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—62no—
425C2×C3—121no—
426C2×C3—121no—
427C2×C3C2×#23511yescharacteristic
428C2×C3#100411yescharacteristic
429C2×C3D3×#4111yescharacteristic
430C2×C3D3×#4111yescharacteristic
431C2×C3D3×#4111yescharacteristic
432C2×C3#100911yescharacteristic
433D3—22no—
434D3—41no—
435D3—41no—
436D3—62no—
437D3—62no—
438D3—122no—
439C22—32no—
440C22—61no—
441C22—61no—
442C22—61no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—92no—
449C22—181no—
450C22—181no—
451C22—364no—
452C22C22×D3211yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—61no—
457C4—61no—
458C4—61no—
459C4—61no—
460C4—61no—
461C4—181no—
462C4—181no—
463C4—181no—
464C3—21no—
465C3#555611yescharacteristic, complemented
466C3#570511yescharacteristic, complemented
467C2—62no—
468C2—121no—
469C2—121no—
470C2—182no—
471C2D3×#23511yescharacteristic
472C2#6430611yescharacteristic
473C2#6430711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radical#63922C2
hypercenterC42D32
hypocenterC32#551
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64081C2semidirect
2. C3×#876C2semidirect
3. C3×C4×D12C2semidirect
5. #63921C2semidirect
6. D3×(C2×D3)⋊C4C2semidirect
7. #63915C2semidirect
8. #63905C2semidirect
9. #63904C2semidirect
10. #63895C2semidirect
11. #63884C2semidirect
12. #63878C2semidirect
13. #63858C2semidirect
14. #63853C2semidirect
15. #63851C2semidirect
22. C2×C3×C4×D3C22semidirect
23. C2×#627C22semidirect
28. C32×C42C22semidirect
31. #3750C22semidirect
32. #3750C22semidirect
33. C4×C3⋊Dic3C22semidirect
38. C3×(C2×D3)⋊C4C22semidirect
39. C3×C4⋊Dic3C22semidirect
40. C3×Dic3⋊C4C22semidirect
41. C3×Dic3⋊C4C22semidirect
42. C3×Dic3⋊C4C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
50. (C2×D3)⋊Dic3C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. Dic32C22semidirect
79. #876D3semidirect
80. C4×D12D3semidirect
130. C2×C4×D3C2×D3semidirect
143. C3×C42C2×D3semidirect
144. C3×C42C2×D3semidirect
169. (C2×D3)⋊C4C2×D3semidirect
172. C4⋊Dic3C2×D3semidirect
181. Dic3⋊C4C2×D3semidirect
182. Dic3⋊C4C2×D3semidirect
183. Dic3⋊C4C2×D3semidirect
332. C42D32semidirect
393. C32#551semidirect
465. C3#5556semidirect
466. C3#5705semidirect

Maximal subgroups

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

subgroupindex
#640812
C3×#8762
C3×C4×D122
#639222
#639212
D3×(C2×D3)⋊C42
#639152
#639052
#639042
#638952
#638842
#638782
#638582
#638532
#638512
#57053
#55563