Finite Group Databasebeta

← #10594492

Subgroups — #10594492

Order 576.

468 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64098C211yesmaximal, characteristic, complemented
2C3×#896C211yesmaximal, characteristic, complemented
3C3×#895C211yesmaximal, characteristic, complemented
4#63915C211yesmaximal, characteristic, complemented
5#63908C211yesmaximal, characteristic, complemented
6#63905C211yesmaximal, characteristic, complemented
7#63905C211yesmaximal, characteristic, complemented
8#63893C211yesmaximal, characteristic, complemented
9#63887C211yesmaximal, characteristic, complemented
10#63885C211yesmaximal, characteristic, complemented
11#63885C211yesmaximal, characteristic, complemented
12#63884C211yesmaximal, characteristic, complemented
13#63884C211yesmaximal, characteristic, complemented
14#63863C211yesmaximal, characteristic, complemented
15#63851C211yesmaximal, characteristic, complemented
16#5647—31nomaximal
17#5621—31nomaximal
18C2×C3×D12C2211yescharacteristic
19C2×C3×D12C2211yescharacteristic
20C2×C3×D12C2211yescharacteristic
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic
24C2×#627C2211yescharacteristic
25C2×#625C2211yescharacteristic
26C2×#625C2211yescharacteristic
27C2×#625C2211yescharacteristic
28C2×D3×Dic3C2211yescharacteristic
29C2×D3×Dic3C2211yescharacteristic
30C32×C4⋊C4C2211yescharacteristic
31C4⋊(C3⋊Dic3)C2211yescharacteristic
32#3750C2211yescharacteristic
33#3750C2211yescharacteristic
34#3750C2211yescharacteristic
35#3750C2211yescharacteristic
36C4×C3⋊Dic3C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×(C2×D3)⋊C4C2211yescharacteristic
40C3×(C2×D3)⋊C4C2211yescharacteristic
41C3×Dic3⋊C4C2211yescharacteristic
42C3×Dic3⋊C4C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
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
52(C2×D3)⋊Dic3C2211yescharacteristic
53C3×C22⋊Q8—31no—
54C3×#121—31no—
55#947—31no—
56#946—31no—
57#939—31no—
58#939—31no—
59#938—31no—
60#936—31no—
61#935—31no—
62#935—31no—
63#930—31no—
64#929—31no—
65#929—31no—
66#924—31no—
67#897—31no—
68#896—31no—
69#896D311yescharacteristic, complemented
70#895D311yescharacteristic, complemented
71#894—31no—
72#894—31no—
73#889—31no—
74#889—31no—
75#889—61no—
76#885—31no—
77#885—31no—
78#884—31no—
79#884—31no—
80#883—31no—
81#883—31no—
82#878—31no—
83#878—31no—
84C22×C3×D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C32×C4C2311yescharacteristic
90C2×C32×C4C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C3×D12—41no—
100C3×D12—41no—
101C3×D12—41no—
102C3×C4×D3—41no—
103C3×C4×D3—41no—
104C3×Dic6—41no—
105#627—41no—
106#625—41no—
107#625—41no—
108#625—41no—
109D3×Dic3—41no—
110D3×Dic3—41no—
111#562—91noSylow
112C2×C3×Q8—31no—
113C2×C3×D4—31no—
114C2×C3×D4—31no—
115C2×C3×D4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C2×#67—31no—
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—31no—
123C2×#67—31no—
124C22×Dic3—31no—
125C22×Dic3—31no—
126C2×D12C2×D311yescharacteristic
127C2×D12C2×D311yescharacteristic
128C2×D12C2×D311yescharacteristic
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×C4×D3C2×D311yescharacteristic
132C2×C4×D3C2×D311yescharacteristic
133C2×Dic6—31no—
134C2×Dic6—31no—
135C2×Dic6C2×D311yescharacteristic
136C3×C4⋊C4—21no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4C2×D311yescharacteristic
140C3×C4⋊C4C2×D311yescharacteristic
141C3×C22⋊C4—31no—
142C3×C22⋊C4—31no—
143C3×C22⋊C4—31no—
144C3×C22⋊C4—31no—
145#217—31no—
146#217—31no—
147#217—31no—
148#217—31no—
149#217—31no—
150#217—31no—
151#217—31no—
152#217—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)⋊C4C2×D311yescharacteristic
162(C2×D3)⋊C4C2×D311yescharacteristic
163(C2×D3)⋊C4C2×D311yescharacteristic
164(C2×D3)⋊C4C2×D311yescharacteristic
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—61no—
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—31no—
180Dic3⋊C4—61no—
181Dic3⋊C4—61no—
182Dic3⋊C4—61no—
183Dic3⋊C4—61no—
184Dic3⋊C4C2×D311yescharacteristic
185Dic3⋊C4C2×D311yescharacteristic
186C4×Dic3—31no—
187C4×Dic3—31no—
188C4×Dic3—61no—
189C22×C32C2411yescharacteristic
190C2×C3×D3—21no—
191C2×C3×D3—21no—
192C2×C3×D3—21no—
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—21no—
202C32×C4—21no—
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—
213#125—91no—
214#124—91no—
215#124—91no—
216#123—91no—
217#123—91no—
218#123—91no—
219#123—91no—
220C22⋊Q8—31no—
221C22⋊Q8—91no—
222C22⋊Q8—91no—
223C22⋊Q8—91no—
224#121—31no—
225#121—91no—
226#121—91no—
227#121—91no—
228C23×C3—31no—
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—31no—
232C22×D3C22×D311yescharacteristic
233C22×D3C22×D311yescharacteristic
234C22×D3C22×D311yescharacteristic
235C22×D3C22×D311yescharacteristic
236C3×Q8—121no—
237C3×D4—121no—
238C3×D4—121no—
239C3×D4—121no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—121no—
248C2×C3×C4—121no—
249C2×C3×C4C22×D311yescharacteristic
250C2×C3×C4C22×D311yescharacteristic
251C2×C3×C4C22×D311yescharacteristic
252C2×C3×C4C22×D311yescharacteristic
253C2×C3×C4C22×D311yescharacteristic
254C2×C3×C4C22×D311yescharacteristic
255#67—121no—
256#67—121no—
257#67—121no—
258#67—121no—
259#67—121no—
260#67—121no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—121no—
274C2×Dic3—121no—
275C2×Dic3C22×D311yescharacteristic
276C2×Dic3C22×D311yescharacteristic
277C2×Dic3C22×D311yescharacteristic
278C2×Dic3C22×D311yescharacteristic
279D12—41no—
280D12—41no—
281D12—41no—
282C4×D3—41no—
283C4×D3—41no—
284C4×D3—121no—
285C4×D3—121no—
286Dic6—41no—
287Dic6—121no—
288Dic6—121no—
289C2×C32H(5,𝔽2)11yescharacteristic
290C2×C32H(5,𝔽2)11yescharacteristic
291C2×C322−1+411yescharacteristic
292C3×D3—41no—
293C3×D3—41no—
294C3×D3—41no—
295C3×D3—41no—
296C2×Q8—31no—
297C2×Q8—91no—
298C2×D4—31no—
299C2×D4—31no—
300C2×D4—31no—
301C2×D4—91no—
302C2×D4—91no—
303C2×D4—91no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—91no—
307C22×C4—91no—
308C4⋊C4—31no—
309C4⋊C4—31no—
310C4⋊C4—91no—
311C4⋊C4—91no—
312C4⋊C4—91no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C4⋊C4D3211yescharacteristic
318C22⋊C4—31no—
319C22⋊C4—31no—
320C22⋊C4—31no—
321C22⋊C4—31no—
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—
331C22×C3—21no—
332C22×C3—61no—
333C22×C3—61no—
334C22×C3—61no—
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×C3—61no—
344C22×C3C23×D311yescharacteristic
345C22×C3C23×D311yescharacteristic
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—61no—
368C3×C4—61no—
369C3×C4—61no—
370C3×C4—61no—
371Dic3—21no—
372Dic3—21no—
373Dic3—21no—
374Dic3—21no—
375Dic3—62no—
376Dic3—61no—
377Dic3—61no—
378Dic3—61no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382Dic3—61no—
383Dic3—61no—
384Dic3—121no—
385Dic3—121no—
386Dic3—121no—
387C32#56211yescharacteristic, complemented, Sylow
388C23—31no—
389C23—31no—
390C23—31no—
391C23—31no—
392Q8—121no—
393Q8—361no—
394D4—121no—
395D4—121no—
396D4—121no—
397D4—361no—
398D4—361no—
399D4—361no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—91no—
405C2×C4—91no—
406C2×C4—91no—
407C2×C4—91no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—361no—
411C2×C4—361no—
412C2×C4C2×D3211yescharacteristic
413C2×C4C2×D3211yescharacteristic
414C2×C4C2×D3211yescharacteristic
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—121no—
419C2×C3—121no—
420C2×C3—121no—
421C2×C3—121no—
422C2×C3#100411yescharacteristic
423C2×C3#100411yescharacteristic
424C2×C3#100411yescharacteristic
425C2×C3#100711yescharacteristic
426C2×C3#100911yescharacteristic
427C2×C3#101011yescharacteristic
428D3—41no—
429D3—41no—
430D3—41no—
431D3—41no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—61no—
436C22—61no—
437C22—61no—
438C22—61no—
439C22—61no—
440C22—61no—
441C22—61no—
442C22—61no—
443C22—61no—
444C22C22×D3211yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—61no—
449C4—61no—
450C4—61no—
451C4—61no—
452C4—181no—
453C4—181no—
454C4—181no—
455C4—181no—
456C3—21no—
457C3#562111yescharacteristic, complemented
458C3#564711yescharacteristic, complemented
459C2—121no—
460C2—121no—
461C2—121no—
462C2—121no—
463C2#6430711yescharacteristic
464C2#6431311yescharacteristic
465C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#562
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64098C2semidirect
2. C3×#896C2semidirect
3. C3×#895C2semidirect
4. #63915C2semidirect
5. #63908C2semidirect
6. #63905C2semidirect
7. #63905C2semidirect
8. #63893C2semidirect
9. #63887C2semidirect
10. #63885C2semidirect
11. #63885C2semidirect
12. #63884C2semidirect
13. #63884C2semidirect
14. #63863C2semidirect
15. #63851C2semidirect
69. #896D3semidirect
70. #895D3semidirect
387. C32#562semidirect
457. C3#5621semidirect
458. C3#5647semidirect

Maximal subgroups

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

subgroupindex
#640982
C3×#8962
C3×#8952
#639152
#639082
#639052
#638932
#638872
#638852
#638842
#638632
#638512
#56473
#56213