Finite Group Databasebeta

← #10594299

Subgroups — #10594299

Order 576.

534 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64094C211yesmaximal, characteristic, complemented
2C3×#883C211yesmaximal, characteristic, complemented
3C3×#67⋊C4C211yesmaximal, characteristic, complemented
4#63983C211yesmaximal, characteristic
5Dic3×#67C211yesmaximal, characteristic, complemented
6#63969C211yesmaximal, characteristic, complemented
7#63966C211yesmaximal, characteristic, complemented
8D3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
9D3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
10#63896C211yesmaximal, characteristic, complemented
11#63878C211yesmaximal, characteristic, complemented
12#63868C211yesmaximal, characteristic, complemented
13#63857C211yesmaximal, characteristic, complemented
14#63855C211yesmaximal, characteristic, complemented
15#63848C211yesmaximal, characteristic, complemented
16#5658—31nomaximal
17#5556—31nomaximal
18C22×D32C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×#634C2211yescharacteristic, complemented
21C2×C3×#67C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C22×C3×Dic3C2211yescharacteristic, complemented
24C2×C3×C4×D3C2211yescharacteristic
25C2×C3×C4×D3C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic
28C32×C22⋊C4C2211yescharacteristic, complemented
29#3757C2211yescharacteristic, complemented
30#3752C2211yescharacteristic
31#3752C2211yescharacteristic
32C4×C3⋊Dic3C2211yescharacteristic, complemented
33C3×#217C2211yescharacteristic, complemented
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×C4⋊Dic3C2211yescharacteristic, complemented
37C3×Dic3⋊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#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47#3722C2211yescharacteristic
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#935—31no—
58D4×Dic3—31no—
59#929—31no—
60C4×#67—31no—
61C2×(C2×D3)⋊C4—31no—
62C2×(C2×D3)⋊C4—31no—
63#926—31no—
64#924—31no—
65#898—31no—
66#898—31no—
67#898—31no—
68#897—31no—
69#896—31no—
70#892—31no—
71#886—31no—
72#885—31no—
73#885—31no—
74#885—61no—
75#883—31no—
76#883D311yescharacteristic, complemented
77#67⋊C4D311yescharacteristic, complemented
78D3×C22⋊C4—31no—
79D3×C22⋊C4—31no—
80#878—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×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93#638—41no—
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97#634—41no—
98C3×#67—22no—
99C3×#67—22no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C3×C4×D3—22no—
108C3×C4×D3—41no—
109D3×Dic3—22no—
110D3×Dic3—41no—
111#551—91noSylow
112C23×D3—31no—
113C23×D3—31no—
114C2×C3×D4—31no—
115C2×C3×D4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—61no—
122C2×#67C2×D311yescharacteristic
123C2×#67C2×D311yescharacteristic
124C22×Dic3—31no—
125C22×Dic3—31no—
126C22×Dic3C2×D311yescharacteristic, complemented
127C2×C4×D3—31no—
128C2×C4×D3—31no—
129C2×C4×D3C2×D311yescharacteristic
130C2×C4×D3C2×D311yescharacteristic
131C2×Dic6—31no—
132C2×Dic6—31no—
133C2×Dic6—61no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C22⋊C4—21no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4—31no—
140C3×C22⋊C4—31no—
141C3×C22⋊C4C2×D311yescharacteristic, complemented
142C3×C22⋊C4C2×D311yescharacteristic, complemented
143C3×C42—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, complemented
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—31no—
164(C2×D3)⋊C4—31no—
165(C2×D3)⋊C4—31no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4C2×D311yescharacteristic
169(C2×D3)⋊C4C2×D311yescharacteristic
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3C2×D311yescharacteristic, complemented
175Dic3⋊C4—31no—
176Dic3⋊C4—31no—
177Dic3⋊C4—31no—
178Dic3⋊C4—31no—
179Dic3⋊C4—31no—
180Dic3⋊C4—31no—
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—21no—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3#4112yes—
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—62no—
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—62no—
256C2×C3×C4—121no—
257C2×C3×C4—121no—
258C2×C3×C4C22×D311yescharacteristic
259C2×C3×C4C22×D311yescharacteristic
260C2×C3×C4C22×D311yescharacteristic
261C2×C3×C4C22×D311yescharacteristic
262#67—22no—
263#67—22no—
264#67—121no—
265#67—121no—
266#67—122no—
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—62no—
278C2×Dic3—121no—
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282C2×Dic3C22×D311yescharacteristic
283C2×Dic3C22×D311yescharacteristic
284C2×Dic3C22×D311yescharacteristic
285C4×D3—22no—
286C4×D3—41no—
287C4×D3—62no—
288C4×D3—121no—
289Dic6—121no—
290Dic6—121no—
291Dic6—122no—
292C2×C32—41no—
293C2×C32C2×#4111yescharacteristic
294C2×C32C2×#4111yescharacteristic
295C2×C32H(5,𝔽2)11yescharacteristic
296C3⋊D3—41no—
297C3×D3—22no—
298C3×D3—22no—
299C24—91no—
300C2×Q8—91no—
301C2×D4—31no—
302C2×D4—31no—
303C2×D4—91no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—91no—
308C22×C4—91no—
309C4⋊C4—31no—
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—91no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
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⋊C4—91no—
330C22⋊C4D3211yescharacteristic, complemented
331C42—31no—
332C42—91no—
333C42—91no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—32no—
342C22×C3—32no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3C23×D311yescharacteristic
351C22×C3C23×D311yescharacteristic
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—121no—
363C2×D3—121no—
364C2×D3—121no—
365C2×D3—124no—
366C2×D3—124no—
367C2×D3#23512yes—
368C2×D3#23712yes—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—61no—
379C3×C4—61no—
380C3×C4—61no—
381Dic3—21no—
382Dic3—21no—
383Dic3—21no—
384Dic3—21no—
385Dic3—21no—
386Dic3—21no—
387Dic3—62no—
388Dic3—61no—
389Dic3—61no—
390Dic3—61no—
391Dic3—61no—
392Dic3—61no—
393Dic3—61no—
394Dic3—121no—
395Dic3—121no—
396C32#55111yescharacteristic, complemented, Sylow
397C23—31no—
398C23—31no—
399C23—91no—
400C23—182no—
401C23—182no—
402C23—361no—
403C23C2×D3211yescharacteristic
404Q8—361no—
405D4—62no—
406D4—62no—
407D4—361no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—62no—
415C2×C4—91no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—182no—
421C2×C4—361no—
422C2×C4C2×D3211yescharacteristic
423C2×C4C2×D3211yescharacteristic
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—62no—
430C2×C3—62no—
431C2×C3—81no—
432C2×C3C2×#23511yescharacteristic
433C2×C3C2×#23711yescharacteristic
434C2×C3#100411yescharacteristic
435C2×C3D3×#4111yescharacteristic
436C2×C3D3×#4111yescharacteristic
437C2×C3#100911yescharacteristic
438D3—22no—
439D3—22no—
440D3—121no—
441D3—121no—
442D3—241no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—32no—
447C22—32no—
448C22—61no—
449C22—61no—
450C22—61no—
451C22—61no—
452C22—181no—
453C22—181no—
454C22—181no—
455C22—364no—
456C22C22×D3211yescharacteristic
457C4—21no—
458C4—21no—
459C4—61no—
460C4—61no—
461C4—61no—
462C4—61no—
463C4—61no—
464C4—61no—
465C4—181no—
466C4—181no—
467C4—181no—
468C3—21no—
469C3#555611yescharacteristic, complemented
470C3#565811yescharacteristic, complemented
471C2—41no—
472C2—62no—
473C2—62no—
474C2—361no—
475C2D3×#23511yescharacteristic
476C2D3×#23711yescharacteristic
477C2#6431411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊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. #64094C2semidirect
2. C3×#883C2semidirect
3. C3×#67⋊C4C2semidirect
5. Dic3×#67C2semidirect
6. #63969C2semidirect
7. #63966C2semidirect
8. D3×(C2×D3)⋊C4C2semidirect
9. D3×(C2×D3)⋊C4C2semidirect
10. #63896C2semidirect
11. #63878C2semidirect
12. #63868C2semidirect
13. #63857C2semidirect
14. #63855C2semidirect
15. #63848C2semidirect
20. C2×#634C22semidirect
23. C22×C3×Dic3C22semidirect
28. C32×C22⋊C4C22semidirect
29. #3757C22semidirect
32. C4×C3⋊Dic3C22semidirect
33. C3×#217C22semidirect
36. C3×C4⋊Dic3C22semidirect
37. C3×Dic3⋊C4C22semidirect
38. C3×Dic3⋊C4C22semidirect
39. C3×C4×Dic3C22semidirect
40. #3724C22semidirect
41. Dic3⋊Dic3C22semidirect
42. Dic3⋊Dic3C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
52. Dic32C22semidirect
76. #883D3semidirect
77. #67⋊C4D3semidirect
126. C22×Dic3C2×D3semidirect
141. C3×C22⋊C4C2×D3semidirect
142. C3×C22⋊C4C2×D3semidirect
151. #217C2×D3semidirect
174. C4⋊Dic3C2×D3semidirect
181. Dic3⋊C4C2×D3semidirect
182. Dic3⋊C4C2×D3semidirect
188. C4×Dic3C2×D3semidirect
330. C22⋊C4D32semidirect
396. C32#551semidirect
469. C3#5556semidirect
470. C3#5658semidirect

Maximal subgroups

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

subgroupindex
#640942
C3×#8832
C3×#67⋊C42
#639832
Dic3×#672
#639692
#639662
D3×(C2×D3)⋊C42
#638962
#638782
#638682
#638572
#638552
#638482
#56583
#55563