Finite Group Databasebeta

← #10595264

Subgroups — #10595264

Order 576.

569 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64146C211yesmaximal, characteristic, complemented
2C3×#939C211yesmaximal, characteristic, complemented
3C3×#938C211yesmaximal, characteristic, complemented
4#63981C211yesmaximal, characteristic, complemented
5#63981C211yesmaximal, characteristic, complemented
6#63979C211yesmaximal, characteristic, complemented
7#63978C211yesmaximal, characteristic, complemented
8#63971C211yesmaximal, characteristic, complemented
9#63970C211yesmaximal, characteristic, complemented
10#63966C211yesmaximal, characteristic, complemented
11#63956C211yesmaximal, characteristic, complemented
12#63906C211yesmaximal, characteristic, complemented
13#63890C211yesmaximal, characteristic, complemented
14#63850C211yesmaximal, characteristic, complemented
15#63848C211yesmaximal, characteristic, complemented
16#5620—31nomaximal
17#5620—31nomaximal
18C23×C3⋊D3C2211yescharacteristic
19C2×C32×D4C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×#638C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic
24C2×C3×#67C2211yescharacteristic, complemented
25C2×C3×#67C2211yescharacteristic, complemented
26C22×C3×Dic3C2211yescharacteristic, complemented
27C2×C3×C4×D3C2211yescharacteristic, complemented
28C2×#626C2211yescharacteristic
29C2×#626C2211yescharacteristic
30C2×#626C2211yescharacteristic
31C2×#626C2211yescharacteristic
32C2×#625C2211yescharacteristic, complemented
33C2×D3×Dic3C2211yescharacteristic, complemented
34#3757C2211yescharacteristic
35#3752C2211yescharacteristic
36#3752C2211yescharacteristic
37C3×#217C2211yescharacteristic
38C3×#217C2211yescharacteristic
39C3×#217C2211yescharacteristic, complemented
40C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C3×C4⋊Dic3C2211yescharacteristic, complemented
42C3×Dic3⋊C4C2211yescharacteristic, complemented
43#3724C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45Dic3⋊Dic3C2211yescharacteristic, complemented
46Dic3⋊Dic3C2211yescharacteristic, complemented
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic
49#3722C2211yescharacteristic
50#3722C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#121—31no—
54C3×#121—31no—
55#940—31no—
56#940—31no—
57#939—31no—
58#939—31no—
59#939—31no—
60#939D311yescharacteristic, complemented
61#938—31no—
62#938D311yescharacteristic, complemented
63#937—31no—
64#937—31no—
65#937—31no—
66#937—31no—
67#937—61no—
68#930—31no—
69#930—31no—
70#929—31no—
71#929—31no—
72#898—31no—
73#898—31no—
74#894—31no—
75#894—31no—
76#886—31no—
77#886—31no—
78#884—31no—
79#884—31no—
80#882—31no—
81#882—31no—
82#879—31no—
83#879—31no—
84C23×C32C2311yescharacteristic
85C23×C32C2311yescharacteristic
86C22×C3⋊D3—41no—
87C22×C3⋊D3—41no—
88C22×C3⋊D3—41no—
89C22×C3⋊D3C2311yescharacteristic
90C22×C3⋊D3C2311yescharacteristic
91C22×C3×D3C2311yescharacteristic
92C22×C3×D3C2311yescharacteristic
93C32×D4—41no—
94C2×C32×C4C2311yescharacteristic
95#638—41no—
96#638—41no—
97C2×C3⋊Dic3C2311yescharacteristic
98C2×C3⋊Dic3C2311yescharacteristic
99C3×#67—41no—
100C3×#67—41no—
101C3×#67—41no—
102C3×#67—41no—
103C2×C3×Dic3—41no—
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C2×C3×Dic3C2311yescharacteristic
109C2×C3×Dic3C2311yescharacteristic
110C3×C4×D3—41no—
111#626—41no—
112#626—41no—
113#626—41no—
114#626—41no—
115#625—41no—
116D3×Dic3—41no—
117#560—91noSylow
118C23×D3—31no—
119C23×D3—31no—
120C23×D3—61no—
121C2×C3×D4—21no—
122C2×C3×D4—31no—
123C2×C3×D4—31no—
124C2×C3×D4—31no—
125C2×C3×D4—31no—
126C2×C3×D4C2×D311yescharacteristic
127C2×C3×D4C2×D311yescharacteristic
128C22×C3×C4—31no—
129C22×C3×C4—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—
139C2×#67—31no—
140C2×#67—61no—
141C2×#67—61no—
142C2×#67C2×D311yescharacteristic
143C2×#67C2×D311yescharacteristic
144C2×#67C2×D311yescharacteristic, complemented
145C2×#67C2×D311yescharacteristic, complemented
146C22×Dic3—31no—
147C22×Dic3C2×D311yescharacteristic, complemented
148C2×D12—31no—
149C2×D12—31no—
150C2×D12—31no—
151C2×D12—31no—
152C2×C4×D3—31no—
153C2×C4×D3C2×D311yescharacteristic, complemented
154C3×C4⋊C4—31no—
155C3×C4⋊C4—31no—
156C3×C22⋊C4—31no—
157C3×C22⋊C4—31no—
158C3×C22⋊C4—31no—
159C3×C22⋊C4—31no—
160#217—31no—
161#217—31no—
162#217—31no—
163#217—61no—
164#217C2×D311yescharacteristic
165#217C2×D311yescharacteristic
166#217C2×D311yescharacteristic, complemented
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—31no—
174(C2×D3)⋊C4—31no—
175(C2×D3)⋊C4—31no—
176(C2×D3)⋊C4—31no—
177(C2×D3)⋊C4—31no—
178(C2×D3)⋊C4—31no—
179(C2×D3)⋊C4—31no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—61no—
182(C2×D3)⋊C4C2×D311yescharacteristic, complemented
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3C2×D311yescharacteristic, complemented
187Dic3⋊C4—31no—
188Dic3⋊C4—31no—
189Dic3⋊C4—31no—
190Dic3⋊C4—31no—
191Dic3⋊C4—31no—
192Dic3⋊C4C2×D311yescharacteristic, complemented
193C4×Dic3—31no—
194C4×Dic3—31no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32C2411yescharacteristic
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—44no—
209C2×C3×D3—21no—
210C2×C3×D3—21no—
211C2×C3×D3—21no—
212C2×C3×D3—21no—
213C2×C3×D3—21no—
214C2×C3×D3—21no—
215C32×C4—21no—
216C3⋊Dic3—21no—
217C3⋊Dic3—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—91no—
228#123—91no—
229#123—91no—
230#121—31no—
231#121—31no—
232#121—91no—
233#121—91no—
234#121—91no—
235#121—91no—
236#120—91no—
237#120—91no—
238#120—91no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—31no—
242C23×C3—31no—
243C23×C3C22×D311yescharacteristic
244C23×C3C22×D311yescharacteristic
245C23×C3C22×D311yescharacteristic
246C23×C3C22×D311yescharacteristic
247C22×D3—31no—
248C22×D3—31no—
249C22×D3—31no—
250C22×D3—31no—
251C22×D3—61no—
252C22×D3—61no—
253C22×D3—121no—
254C22×D3—121no—
255C22×D3—121no—
256C22×D3—121no—
257C22×D3—121no—
258C22×D3—121no—
259C22×D3—122no—
260C22×D3—122no—
261C22×D3—122no—
262C22×D3C22×D311yescharacteristic
263C22×D3C22×D311yescharacteristic
264C3×D4—41no—
265C3×D4—41no—
266C3×D4—42no—
267C3×D4—121no—
268C3×D4—121no—
269C3×D4—121no—
270C3×D4—121no—
271C2×C3×C4—21no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—121no—
279C2×C3×C4—121no—
280C2×C3×C4C22×D311yescharacteristic
281C2×C3×C4C22×D311yescharacteristic
282#67—41no—
283#67—41no—
284#67—41no—
285#67—41no—
286#67—121no—
287#67—121no—
288#67—121no—
289#67—121no—
290#67—121no—
291#67—121no—
292#67—121no—
293#67—121no—
294#67—121no—
295#67—121no—
296#67—122no—
297#67—122no—
298C2×Dic3—31no—
299C2×Dic3—31no—
300C2×Dic3—31no—
301C2×Dic3—31no—
302C2×Dic3—41no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—121no—
306C2×Dic3C22×D311yescharacteristic
307C2×Dic3C22×D311yescharacteristic
308C2×Dic3C22×D311yescharacteristic
309C2×Dic3C22×D311yescharacteristic
310C2×Dic3C22×D311yescharacteristic
311C2×Dic3C22×D311yescharacteristic
312D12—121no—
313D12—121no—
314D12—121no—
315D12—121no—
316C4×D3—41no—
317C4×D3—121no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32H(5,𝔽2)11yescharacteristic
321C2×C32H(5,𝔽2)11yescharacteristic
322C2×C32H(5,𝔽2)11yescharacteristic
323C3⋊D3—41no—
324C3⋊D3—41no—
325C3×D3—41no—
326C3×D3—41no—
327C24—91no—
328C2×D4—31no—
329C2×D4—31no—
330C2×D4—31no—
331C2×D4—31no—
332C2×D4—91no—
333C2×D4—91no—
334C2×D4—91no—
335C2×D4—91no—
336C2×D4—91no—
337C2×D4—91no—
338C2×D4—91no—
339C2×D4D3211yescharacteristic
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—91no—
343C4⋊C4—31no—
344C4⋊C4—31no—
345C4⋊C4—91no—
346C4⋊C4—91no—
347C4⋊C4—91no—
348C4⋊C4—91no—
349C22⋊C4—31no—
350C22⋊C4—31no—
351C22⋊C4—31no—
352C22⋊C4—31no—
353C22⋊C4—91no—
354C22⋊C4—91no—
355C22⋊C4—91no—
356C22⋊C4—91no—
357C22⋊C4—91no—
358C22⋊C4—91no—
359C22⋊C4—91no—
360C22⋊C4—91no—
361C42—91no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—61no—
382C22×C3—61no—
383C22×C3—61no—
384C22×C3—61no—
385C22×C3—61no—
386C22×C3—61no—
387C22×C3C23×D311yescharacteristic
388C22×C3C23×D311yescharacteristic
389C2×D3—21no—
390C2×D3—21no—
391C2×D3—21no—
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—21no—
395C2×D3—62no—
396C2×D3—62no—
397C2×D3—61no—
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—61no—
402C2×D3—61no—
403C2×D3—61no—
404C2×D3—61no—
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—121no—
410C2×D3—121no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—128no—
414C2×D3—124no—
415C2×D3—124no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—41no—
419C3×C4—61no—
420C3×C4—61no—
421C3×C4—61no—
422C3×C4—61no—
423C3×C4—61no—
424C3×C4—61no—
425Dic3—21no—
426Dic3—21no—
427Dic3—21no—
428Dic3—21no—
429Dic3—21no—
430Dic3—21no—
431Dic3—61no—
432Dic3—61no—
433Dic3—61no—
434Dic3—61no—
435Dic3—121no—
436Dic3—121no—
437C32#56011yescharacteristic, complemented, Sylow
438C23—31no—
439C23—31no—
440C23—91no—
441C23—91no—
442C23—361no—
443C23—361no—
444C23—361no—
445C23C2×D3211yescharacteristic
446C23C2×D3211yescharacteristic
447D4—41no—
448D4—121no—
449D4—121no—
450D4—121no—
451D4—121no—
452D4—361no—
453D4—361no—
454D4—361no—
455D4—361no—
456D4—361no—
457D4—361no—
458D4—361no—
459C2×C4—31no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—361no—
470C2×C4C2×D3211yescharacteristic
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—42no—
475C2×C3—41no—
476C2×C3—41no—
477C2×C3—41no—
478C2×C3—41no—
479C2×C3—81no—
480C2×C3—121no—
481C2×C3—121no—
482C2×C3#100411yescharacteristic
483C2×C3#100411yescharacteristic
484C2×C3#100411yescharacteristic
485C2×C3#100411yescharacteristic
486C2×C3#100911yescharacteristic
487C2×C3#100911yescharacteristic
488D3—41no—
489D3—41no—
490D3—122no—
491D3—122no—
492D3—121no—
493D3—121no—
494D3—121no—
495D3—121no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—61no—
503C22—61no—
504C22—61no—
505C22—61no—
506C22—61no—
507C22—61no—
508C22—181no—
509C22—181no—
510C22—181no—
511C22—181no—
512C22—181no—
513C22—181no—
514C22—364no—
515C22C22×D3211yescharacteristic
516C4—21no—
517C4—61no—
518C4—61no—
519C4—61no—
520C4—61no—
521C4—61no—
522C4—61no—
523C4—181no—
524C4—181no—
525C3—21no—
526C3#562011yescharacteristic, complemented
527C3#562011yescharacteristic, complemented
528C2—41no—
529C2—41no—
530C2—121no—
531C2—121no—
532C2—361no—
533C2—361no—
534C2#6431411yescharacteristic
535C2#6431411yescharacteristic
536C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64146C2semidirect
2. C3×#939C2semidirect
3. C3×#938C2semidirect
4. #63981C2semidirect
5. #63981C2semidirect
6. #63979C2semidirect
7. #63978C2semidirect
8. #63971C2semidirect
9. #63970C2semidirect
10. #63966C2semidirect
11. #63956C2semidirect
12. #63906C2semidirect
13. #63890C2semidirect
14. #63850C2semidirect
15. #63848C2semidirect
24. C2×C3×#67C22semidirect
25. C2×C3×#67C22semidirect
26. C22×C3×Dic3C22semidirect
27. C2×C3×C4×D3C22semidirect
32. C2×#625C22semidirect
33. C2×D3×Dic3C22semidirect
39. C3×#217C22semidirect
40. C3×(C2×D3)⋊C4C22semidirect
41. C3×C4⋊Dic3C22semidirect
42. C3×Dic3⋊C4C22semidirect
43. #3724C22semidirect
44. Dic3⋊Dic3C22semidirect
45. Dic3⋊Dic3C22semidirect
46. Dic3⋊Dic3C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. Dic32C22semidirect
60. #939D3semidirect
62. #938D3semidirect
144. C2×#67C2×D3semidirect
145. C2×#67C2×D3semidirect
147. C22×Dic3C2×D3semidirect
153. C2×C4×D3C2×D3semidirect
166. #217C2×D3semidirect
182. (C2×D3)⋊C4C2×D3semidirect
186. C4⋊Dic3C2×D3semidirect
192. Dic3⋊C4C2×D3semidirect
437. C32#560semidirect
526. C3#5620semidirect
527. C3#5620semidirect

Maximal subgroups

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

subgroupindex
#641462
C3×#9392
C3×#9382
#639812
#639792
#639782
#639712
#639702
#639662
#639562
#639062
#638902
#638502
#638482
#56203