Finite Group Databasebeta

← #10594209

Subgroups — #10594209

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#64086C211yesmaximal, characteristic, complemented
2C3×#67⋊C4C211yesmaximal, characteristic, complemented
3C3×D3×C22⋊C4C211yesmaximal, characteristic, complemented
4#63976C211yesmaximal, characteristic
5#63964C211yesmaximal, characteristic, complemented
6#63962C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
8#63915C211yesmaximal, characteristic, complemented
9#63905C211yesmaximal, characteristic, complemented
10#63905C211yesmaximal, characteristic, complemented
11C4×#625C211yesmaximal, characteristic, complemented
12#63862C211yesmaximal, characteristic, complemented
13#63859C211yesmaximal, characteristic, complemented
14#63855C211yesmaximal, characteristic, complemented
15#63851C211yesmaximal, characteristic, complemented
16#5556—31nomaximal
17#5495—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C22×C3×Dic3C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×C4×D3C2211yescharacteristic, complemented
25C2×#627C2211yescharacteristic, complemented
26C2×#625C2211yescharacteristic
27C2×#625C2211yescharacteristic
28C32×C22⋊C4C2211yescharacteristic
29#3757C2211yescharacteristic
30#3750C2211yescharacteristic, complemented
31#3750C2211yescharacteristic, complemented
32C4×C3⋊Dic3C2211yescharacteristic, complemented
33C4×C3⋊Dic3C2211yescharacteristic, complemented
34C3×#217C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
38C3×Dic3⋊C4C2211yescharacteristic, complemented
39C3×C4×Dic3C2211yescharacteristic, complemented
40#3724C2211yescharacteristic, complemented
41Dic3⋊Dic3C2211yescharacteristic, complemented
42Dic3⋊Dic3C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45(C2×D3)⋊Dic3C2211yescharacteristic
46(C2×D3)⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52(C2×D3)⋊Dic3C2211yescharacteristic, complemented
53C3×C4×D4—31no—
54C2×C3×C22⋊C4—31no—
55#953—31no—
56C2×#217—31no—
57#937—31no—
58#935—31no—
59#929—31no—
60#929—31no—
61C4×#67—31no—
62C4×#67—31no—
63C2×(C2×D3)⋊C4—31no—
64#926—31no—
65#924—31no—
66#924—31no—
67#898—31no—
68#896—31no—
69#886—31no—
70#885—31no—
71#885—31no—
72#883—31no—
73#883—31no—
74#67⋊C4D311yescharacteristic, complemented
75D3×C22⋊C4D311yescharacteristic, complemented
76#879—31no—
77#879—31no—
78#878—31no—
79#877—31no—
80#877—31no—
81#877—61no—
82#876—31no—
83#872—31no—
84C23×C32C2311yescharacteristic
85C22×C3×D3—22no—
86C22×C3×D3—22no—
87C22×C3×D3—41no—
88C22×C3×D3C2311yescharacteristic
89C22×C3×D3C2311yescharacteristic
90C22×C3×D3C2311yescharacteristic
91C2×C32×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C3×#67—22no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C3×C4×D3—22no—
106C3×C4×D3—41no—
107C3×C4×D3—41no—
108#627—41no—
109#625—22no—
110#625—41no—
111#551—91noSylow
112C24×C3—31no—
113C23×D3C2×D311yescharacteristic
114C2×C3×D4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—31no—
123C2×#67C2×D311yescharacteristic
124C22×Dic3—31no—
125C22×Dic3—31no—
126C22×Dic3—61no—
127C22×Dic3C2×D311yescharacteristic
128C2×C4×D3C2×D311yescharacteristic
129C2×C4×D3C2×D311yescharacteristic
130C2×C4×D3C2×D311yescharacteristic, complemented
131C2×Dic6—31no—
132C2×Dic6—31no—
133C3×C4⋊C4—31no—
134C3×C22⋊C4—21no—
135C3×C22⋊C4—31no—
136C3×C22⋊C4—31no—
137C3×C22⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4C2×D311yescharacteristic
140C3×C22⋊C4C2×D311yescharacteristic
141C3×C42—31no—
142#217—31no—
143#217—31no—
144#217—31no—
145#217—31no—
146#217—31no—
147#217—31no—
148#217—31no—
149#217—31no—
150#217—31no—
151#217—31no—
152#217—61no—
153#217C2×D311yescharacteristic
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)⋊C4C2×D311yescharacteristic
163(C2×D3)⋊C4C2×D311yescharacteristic
164(C2×D3)⋊C4C2×D311yescharacteristic, complemented
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169Dic3⋊C4—31no—
170Dic3⋊C4—31no—
171Dic3⋊C4—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⋊C4—61no—
181Dic3⋊C4C2×D311yescharacteristic, complemented
182C4×Dic3—31no—
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—61no—
187C4×Dic3—61no—
188C4×Dic3C2×D311yescharacteristic, complemented
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—91no—
221C22⋊Q8—91no—
222#120—91no—
223C4×D4—31no—
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—31no—
233C23×C3—62no—
234C23×C3—62no—
235C23×C3—121no—
236C23×C3C22×D311yescharacteristic
237C23×C3C22×D311yescharacteristic
238C22×D3—22no—
239C22×D3—22no—
240C22×D3—41no—
241C22×D3C22×D311yescharacteristic
242C22×D3C22×D311yescharacteristic
243C22×D3C22×D311yescharacteristic
244C3×D4—62no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—62no—
253C2×C3×C4—121no—
254C2×C3×C4—121no—
255C2×C3×C4—121no—
256C2×C3×C4C22×D311yescharacteristic
257C2×C3×C4C22×D311yescharacteristic
258C2×C3×C4C22×D311yescharacteristic
259C2×C3×C4C22×D311yescharacteristic
260#67—22no—
261#67—62no—
262#67—62no—
263#67—121no—
264#67—121no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—31no—
270C2×Dic3—31no—
271C2×Dic3—31no—
272C2×Dic3—31no—
273C2×Dic3—41no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—64no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3C22×D311yescharacteristic
282C2×Dic3C22×D311yescharacteristic
283C2×Dic3C22×D311yescharacteristic
284C2×Dic3C22×D311yescharacteristic
285C2×Dic3C22×D311yescharacteristic
286C4×D3—22no—
287C4×D3—41no—
288C4×D3—41no—
289Dic6—121no—
290Dic6—121no—
291C2×C32—22no—
292C2×C32C2×#4111yescharacteristic
293C2×C32C2×#4111yescharacteristic
294C2×C32H(5,𝔽2)11yescharacteristic
295C3×D3—22no—
296C3×D3—41no—
297C3×D3—41no—
298C24—31no—
299C2×Q8—91no—
300C2×D4—31no—
301C2×D4—91no—
302C2×D4—91no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—91no—
308C4⋊C4—31no—
309C4⋊C4—91no—
310C4⋊C4—91no—
311C4⋊C4—91no—
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—91no—
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⋊C4D3211yescharacteristic
330C42—31no—
331C42—91no—
332C42—91no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—22no—
339C22×C3—32no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—61no—
343C22×C3—61no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—124no—
351C22×C3#23512yes—
352C22×C3#23512yes—
353C22×C3C23×D311yescharacteristic
354C22×C3C23×D311yescharacteristic
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—21no—
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—44no—
364C2×D3#23712yes—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—61no—
376Dic3—21no—
377Dic3—21no—
378Dic3—21no—
379Dic3—21no—
380Dic3—21no—
381Dic3—62no—
382Dic3—62no—
383Dic3—61no—
384Dic3—61no—
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—62no—
398C23—62no—
399C23—121no—
400C23C2×D3211yescharacteristic
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—62no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—182no—
419C2×C4C2×D3211yescharacteristic
420C2×C4C2×D3211yescharacteristic
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—42no—
427C2×C3—62no—
428C2×C3—121no—
429C2×C3—121no—
430C2×C3C2×#23511yescharacteristic
431C2×C3C2×#23511yescharacteristic
432C2×C3C2×#23711yescharacteristic
433C2×C3#100411yescharacteristic
434C2×C3#100411yescharacteristic
435C2×C3D3×#4111yescharacteristic
436D3—22no—
437D3—41no—
438D3—41no—
439C22—21no—
440C22—21no—
441C22—32no—
442C22—61no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—61no—
449C22—61no—
450C22—124no—
451C22#379812yes—
452C22C22×D3211yescharacteristic
453C4—21no—
454C4—21no—
455C4—61no—
456C4—61no—
457C4—61no—
458C4—61no—
459C4—61no—
460C4—181no—
461C4—181no—
462C4—181no—
463C4—181no—
464C3—21no—
465C3#549511yescharacteristic, complemented
466C3#555611yescharacteristic, complemented
467C2—22no—
468C2—62no—
469C2—121no—
470C2—121no—
471C2C2×#379811yescharacteristic
472C2D3×#23711yescharacteristic
473C2#6431311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radical#63976C2
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. #64086C2semidirect
2. C3×#67⋊C4C2semidirect
3. C3×D3×C22⋊C4C2semidirect
5. #63964C2semidirect
6. #63962C2semidirect
7. C2×(C2×D3)⋊Dic3C2semidirect
8. #63915C2semidirect
9. #63905C2semidirect
10. #63905C2semidirect
11. C4×#625C2semidirect
12. #63862C2semidirect
13. #63859C2semidirect
14. #63855C2semidirect
15. #63851C2semidirect
24. C2×C3×C4×D3C22semidirect
25. C2×#627C22semidirect
30. #3750C22semidirect
31. #3750C22semidirect
32. C4×C3⋊Dic3C22semidirect
33. C4×C3⋊Dic3C22semidirect
37. C3×(C2×D3)⋊C4C22semidirect
38. C3×Dic3⋊C4C22semidirect
39. C3×C4×Dic3C22semidirect
40. #3724C22semidirect
41. Dic3⋊Dic3C22semidirect
42. Dic3⋊Dic3C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. (C2×D3)⋊Dic3C22semidirect
74. #67⋊C4D3semidirect
75. D3×C22⋊C4D3semidirect
130. C2×C4×D3C2×D3semidirect
164. (C2×D3)⋊C4C2×D3semidirect
181. Dic3⋊C4C2×D3semidirect
188. C4×Dic3C2×D3semidirect
393. C32#551semidirect
465. C3#5495semidirect
466. C3#5556semidirect

Maximal subgroups

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

subgroupindex
#640862
C3×#67⋊C42
C3×D3×C22⋊C42
#639762
#639642
#639622
C2×(C2×D3)⋊Dic32
#639152
#639052
C4×#6252
#638622
#638592
#638552
#638512
#55563
#54953