Finite Group Databasebeta

← #10594170

Subgroups — #10594170

Order 576.

521 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64087C211yesmaximal, characteristic, complemented
2C3×D3×C22⋊C4C211yesmaximal, characteristic
3C3×#878C211yesmaximal, characteristic, complemented
4#63982C211yesmaximal, characteristic, complemented
5D3×#217C211yesmaximal, characteristic
6#63964C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic
8#63899C211yesmaximal, characteristic, complemented
9#63896C211yesmaximal, characteristic, complemented
10#63893C211yesmaximal, characteristic, complemented
11#63887C211yesmaximal, characteristic, complemented
12#63857C211yesmaximal, characteristic, complemented
13#63856C211yesmaximal, characteristic, complemented
14#63851C211yesmaximal, characteristic, complemented
15#63851C211yesmaximal, characteristic, complemented
16#5634—31nomaximal
17#5483—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic
20C2×#634C2211yescharacteristic, complemented
21C22×C3×Dic3C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×Dic6C2211yescharacteristic, complemented
25C2×#627C2211yescharacteristic, complemented
26C2×#627C2211yescharacteristic, complemented
27C2×D3×Dic3C2211yescharacteristic
28C2×D3×Dic3C2211yescharacteristic
29C32×C22⋊C4C2211yescharacteristic
30#3757C2211yescharacteristic
31C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
32#3750C2211yescharacteristic, complemented
33#3750C2211yescharacteristic, complemented
34C3×#217C2211yescharacteristic
35C3×#217C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×C4⋊Dic3C2211yescharacteristic, complemented
39C3×Dic3⋊C4C2211yescharacteristic, complemented
40C3×Dic3⋊C4C2211yescharacteristic, complemented
41#3724C2211yescharacteristic, complemented
42#3724C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45Dic3⋊Dic3C2211yescharacteristic, complemented
46Dic3⋊Dic3C2211yescharacteristic, complemented
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52(C2×D3)⋊Dic3C2211yescharacteristic
53C3×C22⋊Q8—31no—
54C2×C3×C22⋊C4—31no—
55C2×#217—31no—
56C2×#217—31no—
57#946—31no—
58#946—31no—
59C2×(C2×D3)⋊C4—31no—
60#924—31no—
61#924—31no—
62#924—31no—
63#924—31no—
64#924—31no—
65#897—31no—
66#897—31no—
67#896—31no—
68#896—31no—
69#896—31no—
70#896—31no—
71D3×C22⋊C4—31no—
72D3×C22⋊C4D311yescharacteristic
73#878—31no—
74#878—31no—
75#878—31no—
76#878—31no—
77#878—31no—
78#878—31no—
79#878—31no—
80#878—31no—
81#878—31no—
82#878—61no—
83#878D311yescharacteristic, complemented
84C23×C32C2311yescharacteristic
85C22×C3×D3—41no—
86C22×C3×D3—41no—
87C22×C3×D3Q814yescomplemented
88C22×C3×D3C2311yescharacteristic
89C22×C3×D3C2311yescharacteristic
90C2×C32×C4C2311yescharacteristic
91C2×C32×C4C2311yescharacteristic
92C2×C3⋊Dic3—41no—
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97#634—41no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C3×C4×D3—41no—
106C3×C4×D3—41no—
107C3×Dic6—41no—
108#627—41no—
109#627—41no—
110D3×Dic3—41no—
111D3×Dic3—41no—
112#543—91noSylow
113C24×C3—31no—
114C23×D3C2×D311yescharacteristic
115C2×C3×Q8—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×Dic3—31no—
120C22×Dic3—31no—
121C22×Dic3—31no—
122C22×Dic3—31no—
123C22×Dic3—61no—
124C22×Dic3C2×D311yescharacteristic
125C2×C4×D3—31no—
126C2×C4×D3—31no—
127C2×C4×D3C2×D311yescharacteristic
128C2×C4×D3C2×D311yescharacteristic
129C2×Dic6—31no—
130C2×Dic6—31no—
131C2×Dic6—31no—
132C2×Dic6—31no—
133C2×Dic6—31no—
134C2×Dic6—31no—
135C2×Dic6—61no—
136C2×Dic6C2×D311yescharacteristic, complemented
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C22⋊C4—21no—
141C3×C22⋊C4—31no—
142C3×C22⋊C4—31no—
143C3×C22⋊C4—31no—
144C3×C22⋊C4—31no—
145C3×C22⋊C4C2×D311yescharacteristic
146C3×C22⋊C4C2×D311yescharacteristic
147#217—31no—
148#217—31no—
149#217—31no—
150#217—31no—
151#217—31no—
152#217—31no—
153#217—31no—
154#217—31no—
155#217—61no—
156#217C2×D311yescharacteristic
157#217C2×D311yescharacteristic
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)⋊C4C2×D311yescharacteristic
165(C2×D3)⋊C4C2×D311yescharacteristic
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—61no—
173C4⋊Dic3C2×D311yescharacteristic, complemented
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4—31no—
177Dic3⋊C4—31no—
178Dic3⋊C4—31no—
179Dic3⋊C4—31no—
180Dic3⋊C4—31no—
181Dic3⋊C4—31no—
182Dic3⋊C4—31no—
183Dic3⋊C4—31no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4—61no—
187Dic3⋊C4—61no—
188Dic3⋊C4C2×D311yescharacteristic, complemented
189Dic3⋊C4C2×D311yescharacteristic, complemented
190C22×C32—21no—
191C22×C32—21no—
192C22×C32C2×Q812yes—
193C22×C32C2411yescharacteristic
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—44no—
199C2×C3×D3C2×Q812yes—
200C2×C3×D3C2×Q812yes—
201C32×C4—21no—
202C32×C4—21no—
203C3⋊Dic3—21no—
204C3⋊Dic3—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—
213C22⋊Q8—31no—
214C22⋊Q8—91no—
215C22⋊Q8—91no—
216C22⋊Q8—91no—
217C22⋊Q8—91no—
218C22⋊Q8—91no—
219C22⋊Q8—91no—
220C22⋊Q8—91no—
221C22⋊Q8—91no—
222C22⋊Q8—91no—
223C22⋊Q8—91no—
224C22⋊Q8—91no—
225C2×C22⋊C4—31no—
226C2×C22⋊C4—91no—
227C2×C22⋊C4—91no—
228C23×C3—21no—
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—34no—
232C23×C3—121no—
233C23×C3—121no—
234C23×C3C22×D311yescharacteristic
235C23×C3C22×D311yescharacteristic
236C22×D3—41no—
237C22×D3—41no—
238C22×D3Dic614yescomplemented
239C22×D3C22×D311yescharacteristic
240C22×D3C22×D311yescharacteristic
241C3×Q8—121no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—121no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4C22×D311yescharacteristic
254C2×C3×C4C22×D311yescharacteristic
255C2×C3×C4C22×D311yescharacteristic
256C2×C3×C4C22×D311yescharacteristic
257C2×Dic3—31no—
258C2×Dic3—31no—
259C2×Dic3—31no—
260C2×Dic3—31no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—41no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—121no—
271C2×Dic3—121no—
272C2×Dic3—121no—
273C2×Dic3—121no—
274C2×Dic3—122no—
275C2×Dic3C22×D311yescharacteristic
276C2×Dic3C22×D311yescharacteristic
277C2×Dic3C22×D311yescharacteristic
278C2×Dic3C22×D311yescharacteristic
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C4×D3—41no—
282C4×D3—41no—
283C4×D3—121no—
284C4×D3—121no—
285Dic6—41no—
286Dic6—121no—
287Dic6—121no—
288Dic6—121no—
289Dic6—121no—
290Dic6—121no—
291Dic6—121no—
292Dic6—122no—
293C2×C32—22no—
294C2×C32C22×Q811yescharacteristic
295C2×C32H(5,𝔽2)11yescharacteristic
296C2×C32H(5,𝔽2)11yescharacteristic
297C3×D3—22no—
298C3×D3—22no—
299C24—31no—
300C2×Q8—31no—
301C2×Q8—91no—
302C2×Q8—91no—
303C2×Q8—91no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4—91no—
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C4⋊C4—91no—
318C4⋊C4—91no—
319C4⋊C4—91no—
320C4⋊C4—91no—
321C4⋊C4—91no—
322C22⋊C4—31no—
323C22⋊C4—31no—
324C22⋊C4—31no—
325C22⋊C4—31no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4—91no—
329C22⋊C4—91no—
330C22⋊C4—91no—
331C22⋊C4—91no—
332C22⋊C4—91no—
333C22⋊C4D3211yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—32no—
341C22×C3—32no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—124no—
349C22×C3C2×Dic612yes—
350C22×C3D3×Q812yes—
351C22×C3C23×D311yescharacteristic
352C22×C3C23×D311yescharacteristic
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—44no—
358C2×D3C2×Dic612yes—
359C2×D3C2×Dic612yes—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—61no—
367C3×C4—61no—
368C3×C4—61no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—61no—
372Dic3—21no—
373Dic3—21no—
374Dic3—21no—
375Dic3—21no—
376Dic3—21no—
377Dic3—21no—
378Dic3—62no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382Dic3—61no—
383Dic3—61no—
384Dic3—61no—
385Dic3—61no—
386Dic3—61no—
387Dic3—121no—
388Dic3—121no—
389Dic3—121no—
390C32#54311yescharacteristic, complemented, Sylow
391C23—31no—
392C23—31no—
393C23—34no—
394C23—121no—
395C23—121no—
396C23C2×D3211yescharacteristic
397Q8—121no—
398Q8—361no—
399Q8—361no—
400Q8—361no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—91no—
408C2×C4—91no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—361no—
415C2×C4—361no—
416C2×C4—361no—
417C2×C4C2×D3211yescharacteristic
418C2×C4C2×D3211yescharacteristic
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—42no—
425C2×C3—62no—
426C2×C3—62no—
427C2×C3C22×Dic611yescharacteristic
428C2×C3#100411yescharacteristic
429C2×C3#100411yescharacteristic
430C2×C3#100411yescharacteristic
431C2×C3C2×D3×Q811yescharacteristic
432C2×C3#100911yescharacteristic
433D3—22no—
434D3—22no—
435C22—21no—
436C22—21no—
437C22—32no—
438C22—32no—
439C22—61no—
440C22—61no—
441C22—61no—
442C22—61no—
443C22—124no—
444C22D3×Dic612yes—
445C22C22×D3211yescharacteristic
446C4—21no—
447C4—21no—
448C4—61no—
449C4—61no—
450C4—61no—
451C4—61no—
452C4—61no—
453C4—61no—
454C4—181no—
455C4—181no—
456C4—181no—
457C4—181no—
458C3—21no—
459C3#548311yescharacteristic, complemented
460C3#563411yescharacteristic, complemented
461C2—22no—
462C2—62no—
463C2—62no—
464C2C2×D3×Dic611yescharacteristic
465C2#6431311yescharacteristic
466C2#6431411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC23×C3×D3C22
hypercenterC22⋊C4D32
hypocenterC32#543
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64087C2semidirect
3. C3×#878C2semidirect
4. #63982C2semidirect
6. #63964C2semidirect
8. #63899C2semidirect
9. #63896C2semidirect
10. #63893C2semidirect
11. #63887C2semidirect
12. #63857C2semidirect
13. #63856C2semidirect
14. #63851C2semidirect
15. #63851C2semidirect
20. C2×#634C22semidirect
24. C2×C3×Dic6C22semidirect
25. C2×#627C22semidirect
26. C2×#627C22semidirect
31. C4⋊(C3⋊Dic3)C22semidirect
32. #3750C22semidirect
33. #3750C22semidirect
38. C3×C4⋊Dic3C22semidirect
39. C3×Dic3⋊C4C22semidirect
40. C3×Dic3⋊C4C22semidirect
41. #3724C22semidirect
42. #3724C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
45. Dic3⋊Dic3C22semidirect
46. Dic3⋊Dic3C22semidirect
83. #878D3semidirect
87. C22×C3×D3Q8semidirect
136. C2×Dic6C2×D3semidirect
173. C4⋊Dic3C2×D3semidirect
188. Dic3⋊C4C2×D3semidirect
189. Dic3⋊C4C2×D3semidirect
238. C22×D3Dic6semidirect
390. C32#543semidirect
459. C3#5483semidirect
460. C3#5634semidirect

Maximal subgroups

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

subgroupindex
#640872
C3×D3×C22⋊C42
C3×#8782
#639822
D3×#2172
#639642
C2×(C2×D3)⋊Dic32
#638992
#638962
#638932
#638872
#638572
#638562
#638512
#56343
#54833