Finite Group Databasebeta

← #10594662

Subgroups — #10594662

Order 576.

534 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64107C211yesmaximal, characteristic, complemented
2C3×#895C211yesmaximal, characteristic, complemented
3C3×#894C211yesmaximal, characteristic, complemented
4#63922C211yesmaximal, characteristic, complemented
5#63922C211yesmaximal, characteristic, complemented
6#63922C211yesmaximal, characteristic, complemented
7#63914C211yesmaximal, characteristic, complemented
8#63908C211yesmaximal, characteristic, complemented
9#63906C211yesmaximal, characteristic, complemented
10#63906C211yesmaximal, characteristic, complemented
11#63905C211yesmaximal, characteristic, complemented
12#63891C211yesmaximal, characteristic, complemented
13#63886C211yesmaximal, characteristic, complemented
14#63885C211yesmaximal, characteristic, complemented
15#63862C211yesmaximal, characteristic, complemented
16#5666—31nomaximal
17#5620—31nomaximal
18C22×D32C2211yescharacteristic
19C2×C3×D12C2211yescharacteristic
20C2×C3×D12C2211yescharacteristic
21C2×C3×D12C2211yescharacteristic
22C2×C3×D12C2211yescharacteristic, complemented
23C2×C3×C4×D3C2211yescharacteristic, complemented
24C2×C3×C4×D3C2211yescharacteristic, complemented
25C2×#626C2211yescharacteristic
26C2×#626C2211yescharacteristic
27C2×#626C2211yescharacteristic
28C2×#625C2211yescharacteristic
29C2×#625C2211yescharacteristic
30C2×#625C2211yescharacteristic
31C2×#625C2211yescharacteristic, complemented
32C2×#625C2211yescharacteristic, complemented
33C2×D3×Dic3C2211yescharacteristic, complemented
34C32×C4⋊C4C2211yescharacteristic, complemented
35#3752C2211yescharacteristic
36#3752C2211yescharacteristic
37#3752C2211yescharacteristic
38C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
39#3750C2211yescharacteristic, complemented
40C4×C3⋊Dic3C2211yescharacteristic, complemented
41C3×(C2×D3)⋊C4C2211yescharacteristic
42C3×(C2×D3)⋊C4C2211yescharacteristic
43C3×(C2×D3)⋊C4C2211yescharacteristic
44C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C3×Dic3⋊C4C2211yescharacteristic, complemented
47Dic3⋊Dic3C2211yescharacteristic, complemented
48Dic3⋊Dic3C2211yescharacteristic, complemented
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52(C2×D3)⋊Dic3C2211yescharacteristic, complemented
53C3×#123—31no—
54C3×#121—31no—
55#940—31no—
56#940—31no—
57#939—31no—
58#939—31no—
59#938—31no—
60#937—31no—
61#937—31no—
62#937—31no—
63#930—31no—
64#930—31no—
65#930—31no—
66#929—31no—
67#898—31no—
68#898—31no—
69#898—31no—
70#898—61no—
71#895—31no—
72#895D311yescharacteristic, complemented
73#894D311yescharacteristic, complemented
74#886—31no—
75#884—31no—
76#884—31no—
77#884—31no—
78#883—31no—
79#883—31no—
80#882—31no—
81#882—31no—
82#882—31no—
83#879—31no—
84C22×C3⋊D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C22×C3×D3C2311yescharacteristic
89C22×C3×D3C2311yescharacteristic
90C2×D32—41no—
91C2×D32—41no—
92C2×D32—41no—
93C2×C32×C4C2311yescharacteristic
94C2×C32×C4C2311yescharacteristic
95C2×C32×C4C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C2×C3⋊Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C3×D12—41no—
103C3×D12—41no—
104C3×D12—41no—
105C3×D12—41no—
106C3×C4×D3—41no—
107C3×C4×D3—41no—
108#626—41no—
109#626—41no—
110#626—41no—
111#625—41no—
112#625—41no—
113#625—41no—
114#625—41no—
115#625—41no—
116D3×Dic3—41no—
117#560—91noSylow
118C23×D3—31no—
119C23×D3—31no—
120C2×C3×D4—31no—
121C2×C3×D4—31no—
122C2×C3×D4—31no—
123C2×C3×D4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67—31no—
129C2×#67—31no—
130C2×#67—31no—
131C2×#67—31no—
132C2×#67—31no—
133C2×#67—31no—
134C2×#67—31no—
135C2×#67—31no—
136C2×#67—31no—
137C2×#67—31no—
138C2×#67—31no—
139C22×Dic3—31no—
140C2×D12—31no—
141C2×D12—31no—
142C2×D12—31no—
143C2×D12C2×D311yescharacteristic
144C2×D12C2×D311yescharacteristic
145C2×D12C2×D311yescharacteristic
146C2×D12C2×D311yescharacteristic, complemented
147C2×C4×D3—31no—
148C2×C4×D3C2×D311yescharacteristic, complemented
149C2×C4×D3C2×D311yescharacteristic, complemented
150C3×C4⋊C4—21no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4C2×D311yescharacteristic, complemented
153C3×C4⋊C4C2×D311yescharacteristic, complemented
154C3×C22⋊C4—31no—
155C3×C22⋊C4—31no—
156C3×C22⋊C4—31no—
157C3×C22⋊C4—31no—
158C3×C22⋊C4—31no—
159#217—31no—
160#217—31no—
161#217—31no—
162#217—31no—
163(C2×D3)⋊C4—31no—
164(C2×D3)⋊C4—31no—
165(C2×D3)⋊C4—31no—
166(C2×D3)⋊C4—31no—
167(C2×D3)⋊C4—31no—
168(C2×D3)⋊C4—31no—
169(C2×D3)⋊C4—31no—
170(C2×D3)⋊C4—31no—
171(C2×D3)⋊C4—31no—
172(C2×D3)⋊C4—31no—
173(C2×D3)⋊C4—61no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—61no—
176(C2×D3)⋊C4C2×D311yescharacteristic
177(C2×D3)⋊C4C2×D311yescharacteristic
178(C2×D3)⋊C4C2×D311yescharacteristic
179(C2×D3)⋊C4C2×D311yescharacteristic, complemented
180(C2×D3)⋊C4C2×D311yescharacteristic, complemented
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—61no—
186Dic3⋊C4—31no—
187Dic3⋊C4—31no—
188Dic3⋊C4—31no—
189Dic3⋊C4—31no—
190Dic3⋊C4—61no—
191Dic3⋊C4C2×D311yescharacteristic, complemented
192C4×Dic3—31no—
193C4×Dic3—31no—
194C4×Dic3—61no—
195C22×C32C2411yescharacteristic
196C2×C3⋊D3—21no—
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—21no—
202C2×C3×D3—21no—
203C2×C3×D3—21no—
204C2×C3×D3—21no—
205C2×C3×D3—21no—
206C2×C3×D3—21no—
207C2×C3×D3—21no—
208C2×C3×D3—21no—
209C2×C3×D3—21no—
210C2×C3×D3—21no—
211C2×C3×D3—21no—
212C2×C3×D3—21no—
213C2×C3×D3—21no—
214D32—44no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—21no—
218C3⋊Dic3—21no—
219C3⋊Dic3—21no—
220C3⋊Dic3—21no—
221C3×Dic3—21no—
222C3×Dic3—21no—
223C3×Dic3—21no—
224#127—91no—
225#126—91no—
226#126—91no—
227#123—31no—
228#123—91no—
229#123—91no—
230#121—31no—
231#121—91no—
232#121—91no—
233#121—91no—
234#121—91no—
235#121—91no—
236#120—91no—
237#120—91no—
238#120—91no—
239C23×C3—31no—
240C23×C3—31no—
241C23×C3—31no—
242C23×C3—31no—
243C23×C3—31no—
244C22×D3—31no—
245C22×D3—31no—
246C22×D3—61no—
247C22×D3—121no—
248C22×D3—121no—
249C22×D3—121no—
250C22×D3—121no—
251C22×D3—121no—
252C22×D3—121no—
253C22×D3C22×D311yescharacteristic
254C22×D3C22×D311yescharacteristic
255C22×D3C22×D311yescharacteristic
256C22×D3C22×D311yescharacteristic
257C22×D3C22×D311yescharacteristic
258C3×D4—121no—
259C3×D4—121no—
260C3×D4—121no—
261C3×D4—121no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—121no—
269C2×C3×C4—121no—
270C2×C3×C4C22×D311yescharacteristic
271C2×C3×C4C22×D311yescharacteristic
272C2×C3×C4C22×D311yescharacteristic
273C2×C3×C4C22×D311yescharacteristic
274C2×C3×C4C22×D311yescharacteristic
275C2×C3×C4C22×D311yescharacteristic
276#67—121no—
277#67—121no—
278#67—121no—
279#67—121no—
280#67—121no—
281#67—121no—
282#67—121no—
283#67—121no—
284#67—121no—
285#67—121no—
286#67—121no—
287#67—121no—
288#67—121no—
289C2×Dic3—31no—
290C2×Dic3—31no—
291C2×Dic3—31no—
292C2×Dic3—31no—
293C2×Dic3—31no—
294C2×Dic3—31no—
295C2×Dic3—61no—
296C2×Dic3—61no—
297C2×Dic3—61no—
298C2×Dic3—121no—
299C2×Dic3C22×D311yescharacteristic
300C2×Dic3C22×D311yescharacteristic
301C2×Dic3C22×D311yescharacteristic
302D12—41no—
303D12—41no—
304D12—41no—
305D12—41no—
306D12—121no—
307D12—121no—
308D12—121no—
309C4×D3—41no—
310C4×D3—41no—
311C4×D3—121no—
312C2×C32H(5,𝔽2)11yescharacteristic
313C2×C32H(5,𝔽2)11yescharacteristic
314C2×C32H(5,𝔽2)11yescharacteristic
315C3⋊D3—41no—
316C3×D3—41no—
317C3×D3—41no—
318C3×D3—41no—
319C3×D3—41no—
320C3×D3—41no—
321C24—91no—
322C2×D4—31no—
323C2×D4—31no—
324C2×D4—31no—
325C2×D4—31no—
326C2×D4—91no—
327C2×D4—91no—
328C2×D4—91no—
329C2×D4—91no—
330C2×D4—91no—
331C2×D4—91no—
332C2×D4—91no—
333C2×D4—91no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—91no—
337C4⋊C4—31no—
338C4⋊C4—91no—
339C4⋊C4—91no—
340C4⋊C4—91no—
341C4⋊C4—91no—
342C4⋊C4D3211yescharacteristic, complemented
343C22⋊C4—31no—
344C22⋊C4—31no—
345C22⋊C4—31no—
346C22⋊C4—31no—
347C22⋊C4—31no—
348C22⋊C4—91no—
349C22⋊C4—91no—
350C22⋊C4—91no—
351C22⋊C4—91no—
352C22⋊C4—91no—
353C22⋊C4—91no—
354C22⋊C4—91no—
355C42—91no—
356C22×C3—21no—
357C22×C3—61no—
358C22×C3—61no—
359C22×C3—61no—
360C22×C3—61no—
361C22×C3—61no—
362C22×C3—61no—
363C22×C3—61no—
364C22×C3—61no—
365C22×C3—61no—
366C22×C3—61no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3C23×D311yescharacteristic
373C22×C3C23×D311yescharacteristic
374C2×D3—21no—
375C2×D3—21no—
376C2×D3—21no—
377C2×D3—21no—
378C2×D3—21no—
379C2×D3—21no—
380C2×D3—21no—
381C2×D3—21no—
382C2×D3—21no—
383C2×D3—21no—
384C2×D3—21no—
385C2×D3—21no—
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—21no—
389C2×D3—61no—
390C2×D3—61no—
391C2×D3—61no—
392C2×D3—61no—
393C2×D3—61no—
394C2×D3—61no—
395C2×D3—121no—
396C2×D3—121no—
397C2×D3—121no—
398C2×D3—124no—
399C2×D3—124no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—61no—
410C3×C4—61no—
411C3×C4—61no—
412Dic3—21no—
413Dic3—21no—
414Dic3—21no—
415Dic3—61no—
416Dic3—61no—
417Dic3—61no—
418Dic3—61no—
419Dic3—61no—
420Dic3—61no—
421Dic3—121no—
422Dic3—121no—
423Dic3—121no—
424C32#56011yescharacteristic, complemented, Sylow
425C23—31no—
426C23—31no—
427C23—31no—
428C23—31no—
429C23—31no—
430C23—91no—
431C23—361no—
432C23—361no—
433C23—361no—
434D4—121no—
435D4—121no—
436D4—121no—
437D4—121no—
438D4—361no—
439D4—361no—
440D4—361no—
441D4—361no—
442D4—361no—
443D4—361no—
444D4—361no—
445D4—361no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—91no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—361no—
455C2×C4C2×D3211yescharacteristic
456C2×C4C2×D3211yescharacteristic
457C2×C4C2×D3211yescharacteristic
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—121no—
462C2×C3—121no—
463C2×C3—121no—
464C2×C3—121no—
465C2×C3—121no—
466C2×C3#100411yescharacteristic
467C2×C3#100411yescharacteristic
468C2×C3#100411yescharacteristic
469C2×C3#100911yescharacteristic
470C2×C3#100911yescharacteristic
471C2×C3#100911yescharacteristic
472D3—41no—
473D3—41no—
474D3—41no—
475D3—41no—
476D3—41no—
477D3—121no—
478D3—121no—
479D3—241no—
480C22—61no—
481C22—61no—
482C22—61no—
483C22—61no—
484C22—61no—
485C22—61no—
486C22—61no—
487C22—61no—
488C22—61no—
489C22—61no—
490C22—61no—
491C22—61no—
492C22—61no—
493C22—61no—
494C22—61no—
495C22—181no—
496C22—181no—
497C22—181no—
498C22—364no—
499C22C22×D3211yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—61no—
504C4—61no—
505C4—61no—
506C4—181no—
507C4—181no—
508C4—181no—
509C3—21no—
510C3#562011yescharacteristic, complemented
511C3#566611yescharacteristic, complemented
512C2—121no—
513C2—121no—
514C2—121no—
515C2—121no—
516C2—121no—
517C2—361no—
518C2#6430711yescharacteristic
519C2#6431311yescharacteristic
520C2#6432011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64107C2semidirect
2. C3×#895C2semidirect
3. C3×#894C2semidirect
4. #63922C2semidirect
5. #63922C2semidirect
6. #63922C2semidirect
7. #63914C2semidirect
8. #63908C2semidirect
9. #63906C2semidirect
10. #63906C2semidirect
11. #63905C2semidirect
12. #63891C2semidirect
13. #63886C2semidirect
14. #63885C2semidirect
15. #63862C2semidirect
22. C2×C3×D12C22semidirect
23. C2×C3×C4×D3C22semidirect
24. C2×C3×C4×D3C22semidirect
31. C2×#625C22semidirect
32. C2×#625C22semidirect
33. C2×D3×Dic3C22semidirect
34. C32×C4⋊C4C22semidirect
38. C4⋊(C3⋊Dic3)C22semidirect
39. #3750C22semidirect
40. C4×C3⋊Dic3C22semidirect
44. C3×(C2×D3)⋊C4C22semidirect
45. C3×(C2×D3)⋊C4C22semidirect
46. C3×Dic3⋊C4C22semidirect
47. Dic3⋊Dic3C22semidirect
48. Dic3⋊Dic3C22semidirect
52. (C2×D3)⋊Dic3C22semidirect
72. #895D3semidirect
73. #894D3semidirect
146. C2×D12C2×D3semidirect
148. C2×C4×D3C2×D3semidirect
149. C2×C4×D3C2×D3semidirect
152. C3×C4⋊C4C2×D3semidirect
153. C3×C4⋊C4C2×D3semidirect
179. (C2×D3)⋊C4C2×D3semidirect
180. (C2×D3)⋊C4C2×D3semidirect
191. Dic3⋊C4C2×D3semidirect
342. C4⋊C4D32semidirect
424. C32#560semidirect
510. C3#5620semidirect
511. C3#5666semidirect

Maximal subgroups

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

subgroupindex
#641072
C3×#8952
C3×#8942
#639222
#639142
#639082
#639062
#639052
#638912
#638862
#638852
#638622
#56663
#56203