Finite Group Databasebeta

← #10594242

Subgroups — #10594242

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#64091C211yesmaximal, characteristic, complemented
2C3×#884C211yesmaximal, characteristic, complemented
3C3×#879C211yesmaximal, characteristic, complemented
4#63981C211yesmaximal, characteristic, complemented
5#63981C211yesmaximal, characteristic, complemented
6#63971C211yesmaximal, characteristic, complemented
7#63958C211yesmaximal, characteristic, complemented
8#63911C211yesmaximal, characteristic, complemented
9#63910C211yesmaximal, characteristic, complemented
10#63909C211yesmaximal, characteristic, complemented
11#63897C211yesmaximal, characteristic, complemented
12#63890C211yesmaximal, characteristic, complemented
13#63886C211yesmaximal, characteristic, complemented
14#63868C211yesmaximal, characteristic, complemented
15#63850C211yesmaximal, characteristic, complemented
16#5711—31nomaximal
17#5620—31nomaximal
18C23×C3⋊D3C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic
24C2×C3×D12C2211yescharacteristic, complemented
25C2×C3×C4×D3C2211yescharacteristic, complemented
26C2×#626C2211yescharacteristic
27C2×#626C2211yescharacteristic
28C2×#626C2211yescharacteristic
29C2×#626C2211yescharacteristic
30C2×#626C2211yescharacteristic, complemented
31C2×#626C2211yescharacteristic, complemented
32C2×#624C2211yescharacteristic, complemented
33C2×D3×Dic3C2211yescharacteristic, complemented
34C32×C22⋊C4C2211yescharacteristic
35#3752C2211yescharacteristic
36#3752C2211yescharacteristic
37C3×#217C2211yescharacteristic
38C3×#217C2211yescharacteristic
39C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
40C3×C4⋊Dic3C2211yescharacteristic, complemented
41C3×Dic3⋊C4C2211yescharacteristic, complemented
42C3×Dic3⋊C4C2211yescharacteristic, complemented
43C3×Dic3⋊C4C2211yescharacteristic, complemented
44C3×C4×Dic3C2211yescharacteristic, complemented
45#3724C2211yescharacteristic, complemented
46Dic3⋊Dic3C2211yescharacteristic, complemented
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic
49#3722C2211yescharacteristic
50#3722C2211yescharacteristic
51#3722C2211yescharacteristic, complemented
52(C2×D3)⋊Dic3C2211yescharacteristic, complemented
53C3×#126—31no—
54C3×#121—31no—
55#940—31no—
56#939—31no—
57#939—31no—
58#938—31no—
59#937—31no—
60#937—31no—
61#930—31no—
62#929—31no—
63#898—31no—
64#895—31no—
65#895—31no—
66#895—31no—
67#894—31no—
68#894—31no—
69#894—31no—
70#894—31no—
71#886—31no—
72#884—31no—
73#884—31no—
74#884—31no—
75#884D311yescharacteristic, complemented
76#882—31no—
77#882—31no—
78#882—31no—
79#882—31no—
80#882—61no—
81#879D311yescharacteristic, complemented
82#876—31no—
83#874—31no—
84C23×C32C2311yescharacteristic
85C22×C3⋊D3—41no—
86C22×C3⋊D3—41no—
87C22×C3⋊D3—41no—
88C22×C3⋊D3C2311yescharacteristic
89C22×C3⋊D3C2311yescharacteristic
90C22×C3⋊D3C2311yescharacteristic
91C22×C3×D3C2311yescharacteristic
92C22×C3×D3C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C2×C32×C4C2311yescharacteristic
95#638—41no—
96C2×C3⋊Dic3C2311yescharacteristic
97C4⋊(C3⋊D3)—41no—
98C4⋊(C3⋊D3)—41no—
99C3×#67—41no—
100C3×#67—41no—
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C3×D12—41no—
108C3×C4×D3—41no—
109#626—41no—
110#626—41no—
111#626—41no—
112#626—41no—
113#626—41no—
114#626—41no—
115#624—41no—
116D3×Dic3—41no—
117#560—91noSylow
118C23×D3—31no—
119C23×D3—31no—
120C23×D3—61no—
121C2×C3×D4—31no—
122C2×C3×D4—31no—
123C2×C3×D4—31no—
124C22×C3×C4—31no—
125C2×#67—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67—31no—
129C2×#67—31no—
130C2×#67—31no—
131C2×#67—31no—
132C2×#67—31no—
133C2×#67—61no—
134C2×#67C2×D311yescharacteristic
135C2×#67C2×D311yescharacteristic
136C22×Dic3—31no—
137C2×D12—31no—
138C2×D12—31no—
139C2×D12—31no—
140C2×D12—31no—
141C2×D12—31no—
142C2×D12—31no—
143C2×D12—31no—
144C2×D12—31no—
145C2×D12—31no—
146C2×D12—31no—
147C2×D12—61no—
148C2×D12—61no—
149C2×D12C2×D311yescharacteristic, complemented
150C2×C4×D3—31no—
151C2×C4×D3—31no—
152C2×C4×D3—31no—
153C2×C4×D3C2×D311yescharacteristic, complemented
154C3×C4⋊C4—31no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—31no—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—31no—
160C3×C22⋊C4—31no—
161C3×C22⋊C4—31no—
162C3×C22⋊C4C2×D311yescharacteristic
163C3×C22⋊C4C2×D311yescharacteristic
164C3×C42—31no—
165#217—31no—
166#217C2×D311yescharacteristic
167#217C2×D311yescharacteristic
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—31no—
181(C2×D3)⋊C4—31no—
182(C2×D3)⋊C4—31no—
183(C2×D3)⋊C4—61no—
184(C2×D3)⋊C4—61no—
185(C2×D3)⋊C4C2×D311yescharacteristic, complemented
186C4⋊Dic3—31no—
187C4⋊Dic3C2×D311yescharacteristic, complemented
188Dic3⋊C4—31no—
189Dic3⋊C4—31no—
190Dic3⋊C4—31no—
191Dic3⋊C4C2×D311yescharacteristic, complemented
192Dic3⋊C4C2×D311yescharacteristic, complemented
193Dic3⋊C4C2×D311yescharacteristic, complemented
194C4×Dic3C2×D311yescharacteristic, complemented
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32C2411yescharacteristic
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—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—
216C32×C4—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—31no—
226#126—91no—
227#123—91no—
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—21no—
240C23×C3—31no—
241C23×C3—31no—
242C23×C3C22×D311yescharacteristic
243C23×C3C22×D311yescharacteristic
244C22×D3—31no—
245C22×D3—31no—
246C22×D3—31no—
247C22×D3—31no—
248C22×D3—31no—
249C22×D3—31no—
250C22×D3—61no—
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—121no—
265C3×D4—121no—
266C3×D4—121no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—121no—
276C2×C3×C4C22×D311yescharacteristic
277C2×C3×C4C22×D311yescharacteristic
278C2×C3×C4C22×D311yescharacteristic
279C2×C3×C4C22×D311yescharacteristic
280#67—41no—
281#67—41no—
282#67—121no—
283#67—121no—
284#67—121no—
285#67—121no—
286#67—121no—
287#67—121no—
288#67—121no—
289#67—121no—
290#67—122no—
291C2×Dic3—31no—
292C2×Dic3—31no—
293C2×Dic3—61no—
294C2×Dic3—121no—
295C2×Dic3C22×D311yescharacteristic
296C2×Dic3C22×D311yescharacteristic
297C2×Dic3C22×D311yescharacteristic
298C2×Dic3C22×D311yescharacteristic
299C2×Dic3C22×D311yescharacteristic
300C2×Dic3C22×D311yescharacteristic
301D12—41no—
302D12—121no—
303D12—121no—
304D12—121no—
305D12—121no—
306D12—122no—
307D12—122no—
308D12—121no—
309D12—121no—
310D12—121no—
311D12—121no—
312D12—121no—
313D12—121no—
314C4×D3—41no—
315C4×D3—121no—
316C4×D3—121no—
317C4×D3—121no—
318C2×C32—41no—
319C2×C32H(5,𝔽2)11yescharacteristic
320C2×C32H(5,𝔽2)11yescharacteristic
321C2×C32H(5,𝔽2)11yescharacteristic
322C3⋊D3—41no—
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—91no—
332C2×D4—91no—
333C2×D4—91no—
334C2×D4—91no—
335C2×D4—91no—
336C2×D4—91no—
337C2×D4—91no—
338C2×D4—91no—
339C2×D4—91no—
340C22×C4—31no—
341C22×C4—91no—
342C22×C4—91no—
343C4⋊C4—31no—
344C4⋊C4—31no—
345C4⋊C4—31no—
346C4⋊C4—31no—
347C4⋊C4—91no—
348C4⋊C4—91no—
349C22⋊C4—31no—
350C22⋊C4—31no—
351C22⋊C4—31no—
352C22⋊C4—91no—
353C22⋊C4—91no—
354C22⋊C4—91no—
355C22⋊C4—91no—
356C22⋊C4—91no—
357C22⋊C4—91no—
358C22⋊C4—91no—
359C22⋊C4—91no—
360C22⋊C4D3211yescharacteristic
361C42—31no—
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—41no—
371C22×C3—41no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3—61no—
375C22×C3—61no—
376C22×C3—61no—
377C22×C3—61no—
378C22×C3C23×D311yescharacteristic
379C22×C3C23×D311yescharacteristic
380C2×D3—21no—
381C2×D3—21no—
382C2×D3—21no—
383C2×D3—21no—
384C2×D3—21no—
385C2×D3—21no—
386C2×D3—62no—
387C2×D3—62no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—61no—
391C2×D3—61no—
392C2×D3—61no—
393C2×D3—61no—
394C2×D3—61no—
395C2×D3—61no—
396C2×D3—61no—
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—121no—
407C2×D3—121no—
408C2×D3—121no—
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—21no—
419C3×C4—21no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—61no—
423C3×C4—61no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—61no—
428Dic3—21no—
429Dic3—21no—
430Dic3—21no—
431Dic3—21no—
432Dic3—21no—
433Dic3—21no—
434Dic3—61no—
435Dic3—61no—
436Dic3—121no—
437C32#56011yescharacteristic, complemented, Sylow
438C23—31no—
439C23—31no—
440C23—91no—
441C23—91no—
442C23—91no—
443C23—361no—
444C23—361no—
445C23—361no—
446C23C2×D3211yescharacteristic
447D4—121no—
448D4—121no—
449D4—121no—
450D4—361no—
451D4—361no—
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—121no—
467C2×C4—361no—
468C2×C4—361no—
469C2×C4C2×D3211yescharacteristic
470C2×C4C2×D3211yescharacteristic
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—42no—
475C2×C3—41no—
476C2×C3—41no—
477C2×C3—121no—
478C2×C3—121no—
479C2×C3#100411yescharacteristic
480C2×C3#100411yescharacteristic
481C2×C3#100911yescharacteristic
482C2×C3#100911yescharacteristic
483C2×C3#100911yescharacteristic
484C2×C3#100911yescharacteristic
485D3—41no—
486D3—41no—
487D3—122no—
488D3—122no—
489D3—121no—
490D3—121no—
491D3—121no—
492D3—121no—
493D3—121no—
494D3—121no—
495D3—241no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—61no—
500C22—61no—
501C22—61no—
502C22—61no—
503C22—61no—
504C22—61no—
505C22—181no—
506C22—181no—
507C22—181no—
508C22—181no—
509C22—181no—
510C22—181no—
511C22—181no—
512C22—181no—
513C22—181no—
514C22—364no—
515C22C22×D3211yescharacteristic
516C4—21no—
517C4—21no—
518C4—61no—
519C4—61no—
520C4—61no—
521C4—61no—
522C4—61no—
523C4—61no—
524C4—181no—
525C3—21no—
526C3#562011yescharacteristic, complemented
527C3#571111yescharacteristic, complemented
528C2—41no—
529C2—121no—
530C2—121no—
531C2—361no—
532C2—361no—
533C2—361no—
534C2#6430811yescharacteristic
535C2#6431411yescharacteristic
536C2#6431411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64091C2semidirect
2. C3×#884C2semidirect
3. C3×#879C2semidirect
4. #63981C2semidirect
5. #63981C2semidirect
6. #63971C2semidirect
7. #63958C2semidirect
8. #63911C2semidirect
9. #63910C2semidirect
10. #63909C2semidirect
11. #63897C2semidirect
12. #63890C2semidirect
13. #63886C2semidirect
14. #63868C2semidirect
15. #63850C2semidirect
24. C2×C3×D12C22semidirect
25. C2×C3×C4×D3C22semidirect
30. C2×#626C22semidirect
31. C2×#626C22semidirect
32. C2×#624C22semidirect
33. C2×D3×Dic3C22semidirect
39. C3×(C2×D3)⋊C4C22semidirect
40. C3×C4⋊Dic3C22semidirect
41. C3×Dic3⋊C4C22semidirect
42. C3×Dic3⋊C4C22semidirect
43. C3×Dic3⋊C4C22semidirect
44. C3×C4×Dic3C22semidirect
45. #3724C22semidirect
46. Dic3⋊Dic3C22semidirect
51. #3722C22semidirect
52. (C2×D3)⋊Dic3C22semidirect
75. #884D3semidirect
81. #879D3semidirect
149. C2×D12C2×D3semidirect
153. C2×C4×D3C2×D3semidirect
185. (C2×D3)⋊C4C2×D3semidirect
187. C4⋊Dic3C2×D3semidirect
191. Dic3⋊C4C2×D3semidirect
192. Dic3⋊C4C2×D3semidirect
193. Dic3⋊C4C2×D3semidirect
194. C4×Dic3C2×D3semidirect
437. C32#560semidirect
526. C3#5620semidirect
527. C3#5711semidirect

Maximal subgroups

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

subgroupindex
#640912
C3×#8842
C3×#8792
#639812
#639712
#639582
#639112
#639102
#639092
#638972
#638902
#638862
#638682
#638502
#57113
#56203