Finite Group Databasebeta

← #10594635

Subgroups — #10594635

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#64105C211yesmaximal, characteristic, complemented
2C3×#897C211yesmaximal, characteristic, complemented
3C3×#896C211yesmaximal, characteristic, complemented
4#63921C211yesmaximal, characteristic
5D3×(C2×D3)⋊C4C211yesmaximal, characteristic
6D3×(C2×D3)⋊C4C211yesmaximal, characteristic
7#63899C211yesmaximal, characteristic, complemented
8#63896C211yesmaximal, characteristic, complemented
9#63895C211yesmaximal, characteristic, complemented
10#63888C211yesmaximal, characteristic, complemented
11#63887C211yesmaximal, characteristic, complemented
12#63887C211yesmaximal, characteristic, complemented
13#63871C211yesmaximal, characteristic, complemented
14#63857C211yesmaximal, characteristic, complemented
15#63851C211yesmaximal, characteristic, complemented
16#5634—31nomaximal
17#5634—31nomaximal
18C22×D32C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×#634C2211yescharacteristic, complemented
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic, complemented
24C2×C3×Dic6C2211yescharacteristic, complemented
25C2×#627C2211yescharacteristic, complemented
26C2×#624C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic
28C2×D3×Dic3C2211yescharacteristic
29C32×C4⋊C4C2211yescharacteristic, complemented
30#3752C2211yescharacteristic
31#3752C2211yescharacteristic
32#3750C2211yescharacteristic, complemented
33#3750C2211yescharacteristic, complemented
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×C4⋊Dic3C2211yescharacteristic, complemented
39C3×C4⋊Dic3C2211yescharacteristic, complemented
40C3×Dic3⋊C4C2211yescharacteristic, complemented
41C3×Dic3⋊C4C2211yescharacteristic, complemented
42#3724C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45Dic3⋊Dic3C2211yescharacteristic, complemented
46Dic3⋊Dic3C2211yescharacteristic, complemented
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52(C2×D3)⋊Dic3C2211yescharacteristic
53C3×C22⋊Q8—31no—
54C3×C22⋊Q8—31no—
55#946—31no—
56#946—31no—
57#946—31no—
58#946—31no—
59C2×(C2×D3)⋊C4—31no—
60C2×(C2×D3)⋊C4—31no—
61#924—31no—
62#924—31no—
63#897—31no—
64#897—31no—
65#897—31no—
66#897D311yescharacteristic, complemented
67#896—31no—
68#896—31no—
69#896—31no—
70#896—31no—
71#896—31no—
72#896—31no—
73#896—31no—
74#896—61no—
75#896D311yescharacteristic, complemented
76D3×C22⋊C4—31no—
77D3×C22⋊C4—31no—
78D3×C22⋊C4—31no—
79D3×C22⋊C4—31no—
80#878—31no—
81#878—31no—
82#878—31no—
83#878—31no—
84C22×C3⋊D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C2×D32—41no—
88C2×D32—41no—
89C2×D32Q814yescomplemented
90C2×C32×C4C2311yescharacteristic
91C2×C32×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C4×C3⋊D3—41no—
97#634—41no—
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C3×C4×D3—41no—
105C3×C4×D3—41no—
106C3×Dic6—41no—
107C3×Dic6—41no—
108#627—41no—
109#624—41no—
110D3×Dic3—41no—
111D3×Dic3—41no—
112#543—91noSylow
113C23×D3—31no—
114C23×D3—31no—
115C2×C3×Q8—31no—
116C2×C3×Q8—31no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×Dic3—31no—
120C22×Dic3—31no—
121C2×C4×D3—31no—
122C2×C4×D3—31no—
123C2×C4×D3—31no—
124C2×C4×D3—31no—
125C2×C4×D3—31no—
126C2×C4×D3—31no—
127C2×C4×D3—61no—
128C2×C4×D3C2×D311yescharacteristic
129C2×C4×D3C2×D311yescharacteristic
130C2×Dic6—31no—
131C2×Dic6—31no—
132C2×Dic6—31no—
133C2×Dic6—31no—
134C2×Dic6—61no—
135C2×Dic6C2×D311yescharacteristic, complemented
136C2×Dic6C2×D311yescharacteristic, complemented
137C3×C4⋊C4—21no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4C2×D311yescharacteristic, complemented
143C3×C4⋊C4C2×D311yescharacteristic, complemented
144C3×C22⋊C4—31no—
145C3×C22⋊C4—31no—
146C3×C22⋊C4—31no—
147C3×C22⋊C4—31no—
148#217—31no—
149#217—31no—
150#217—31no—
151#217—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—31no—
164(C2×D3)⋊C4—61no—
165(C2×D3)⋊C4—61no—
166(C2×D3)⋊C4C2×D311yescharacteristic
167(C2×D3)⋊C4C2×D311yescharacteristic
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
175C4⋊Dic3C2×D311yescharacteristic, complemented
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×C32C2411yescharacteristic
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3C2×Q812yes—
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3C2×Q812yes—
199C2×C3×D3C2×Q812yes—
200D32—44no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—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—31no—
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—91no—
226C2×C22⋊C4—91no—
227C2×C22⋊C4—91no—
228C23×C3—31no—
229C23×C3—31no—
230C22×D3—31no—
231C22×D3—31no—
232C22×D3—34no—
233C22×D3—34no—
234C22×D3—61no—
235C22×D3—121no—
236C22×D3—121no—
237C22×D3—121no—
238C22×D3—121no—
239C22×D3C22×D311yescharacteristic
240C22×D3C22×D311yescharacteristic
241C3×Q8—121no—
242C3×Q8—121no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—121no—
253C2×C3×C4—121no—
254C2×C3×C4C22×D311yescharacteristic
255C2×C3×C4C22×D311yescharacteristic
256C2×C3×C4C22×D311yescharacteristic
257C2×C3×C4C22×D311yescharacteristic
258C2×C3×C4C22×D311yescharacteristic
259C2×C3×C4C22×D311yescharacteristic
260C2×Dic3—31no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—121no—
270C2×Dic3—121no—
271C2×Dic3C22×D311yescharacteristic
272C2×Dic3C22×D311yescharacteristic
273C2×Dic3C22×D311yescharacteristic
274C2×Dic3C22×D311yescharacteristic
275C2×Dic3C22×D311yescharacteristic
276C2×Dic3C22×D311yescharacteristic
277C4×D3—41no—
278C4×D3—41no—
279C4×D3—121no—
280C4×D3—121no—
281C4×D3—121no—
282C4×D3—121no—
283C4×D3—122no—
284C4×D3—121no—
285C4×D3—121no—
286Dic6—41no—
287Dic6—41no—
288Dic6—121no—
289Dic6—121no—
290Dic6—121no—
291Dic6—121no—
292Dic6—122no—
293C2×C32C22×Q811yescharacteristic
294C2×C32H(5,𝔽2)11yescharacteristic
295C2×C32H(5,𝔽2)11yescharacteristic
296C3⋊D3—22no—
297C3×D3—22no—
298C3×D3—22no—
299C24—91no—
300C2×Q8—31no—
301C2×Q8—31no—
302C2×Q8—91no—
303C2×Q8—91no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4—91no—
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—31no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C4⋊C4—91no—
318C4⋊C4—91no—
319C4⋊C4—91no—
320C4⋊C4—91no—
321C4⋊C4D3211yescharacteristic, complemented
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⋊C4—91no—
334C22×C3—21no—
335C22×C3—32no—
336C22×C3—32no—
337C22×C3—61no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3C23×D311yescharacteristic
342C22×C3C23×D311yescharacteristic
343C2×D3—21no—
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—32no—
348C2×D3—32no—
349C2×D3—62no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—124no—
357C2×D3—124no—
358C2×D3D3×Q812yes—
359C2×D3D3×Q812yes—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—61no—
375Dic3—21no—
376Dic3—21no—
377Dic3—21no—
378Dic3—21no—
379Dic3—21no—
380Dic3—21no—
381Dic3—62no—
382Dic3—61no—
383Dic3—61no—
384Dic3—61no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—121no—
389Dic3—121no—
390C32#54311yescharacteristic, complemented, Sylow
391C23—31no—
392C23—31no—
393C23—94no—
394C23—91no—
395C23—361no—
396C23—361no—
397Q8—121no—
398Q8—121no—
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—121no—
411C2×C4—121no—
412C2×C4—361no—
413C2×C4—361no—
414C2×C4—361no—
415C2×C4—361no—
416C2×C4C2×D3211yescharacteristic
417C2×C4C2×D3211yescharacteristic
418C2×C4C2×D3211yescharacteristic
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—62no—
423C2×C3—62no—
424C2×C3#100411yescharacteristic
425C2×C3#100411yescharacteristic
426C2×C3C2×D3×Q811yescharacteristic
427C2×C3C2×D3×Q811yescharacteristic
428C2×C3#100911yescharacteristic
429C2×C3#100911yescharacteristic
430D3—22no—
431D3—22no—
432D3—62no—
433D3—62no—
434D3—122no—
435C22—32no—
436C22—32no—
437C22—61no—
438C22—61no—
439C22—61no—
440C22—61no—
441C22—92no—
442C22—181no—
443C22—181no—
444C22—364no—
445C22C22×D3211yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—61no—
450C4—61no—
451C4—61no—
452C4—61no—
453C4—61no—
454C4—61no—
455C4—181no—
456C4—181no—
457C4—181no—
458C3—21no—
459C3#563411yescharacteristic, complemented
460C3#563411yescharacteristic, complemented
461C2—62no—
462C2—62no—
463C2—182no—
464C2#6430711yescharacteristic
465C2#6431411yescharacteristic
466C2D32×Q811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC22×D32C22
hypercenterC4⋊C4D32
hypocenterC32#543
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64105C2semidirect
2. C3×#897C2semidirect
3. C3×#896C2semidirect
7. #63899C2semidirect
8. #63896C2semidirect
9. #63895C2semidirect
10. #63888C2semidirect
11. #63887C2semidirect
12. #63887C2semidirect
13. #63871C2semidirect
14. #63857C2semidirect
15. #63851C2semidirect
20. C2×#634C22semidirect
23. C2×C3×Dic6C22semidirect
24. C2×C3×Dic6C22semidirect
25. C2×#627C22semidirect
29. C32×C4⋊C4C22semidirect
32. #3750C22semidirect
33. #3750C22semidirect
38. C3×C4⋊Dic3C22semidirect
39. C3×C4⋊Dic3C22semidirect
40. C3×Dic3⋊C4C22semidirect
41. C3×Dic3⋊C4C22semidirect
42. #3724C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
45. Dic3⋊Dic3C22semidirect
46. Dic3⋊Dic3C22semidirect
66. #897D3semidirect
75. #896D3semidirect
89. C2×D32Q8semidirect
135. C2×Dic6C2×D3semidirect
136. C2×Dic6C2×D3semidirect
142. C3×C4⋊C4C2×D3semidirect
143. C3×C4⋊C4C2×D3semidirect
174. C4⋊Dic3C2×D3semidirect
175. C4⋊Dic3C2×D3semidirect
188. Dic3⋊C4C2×D3semidirect
189. Dic3⋊C4C2×D3semidirect
321. C4⋊C4D32semidirect
390. C32#543semidirect
459. C3#5634semidirect
460. C3#5634semidirect

Maximal subgroups

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

subgroupindex
#641052
C3×#8972
C3×#8962
#639212
D3×(C2×D3)⋊C42
#638992
#638962
#638952
#638882
#638872
#638712
#638572
#638512
#56343