Finite Group Databasebeta

← #10594187

Subgroups — #10594187

Order 576.

733 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#626C211yesmaximal, characteristic, complemented
2C2×#3799C211yesmaximal, characteristic, complemented
3#64091C211yesmaximal, characteristic, complemented
4C3×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#878C211yesmaximal, characteristic, complemented
6#63977C211yesmaximal, characteristic, complemented
7#63974C211yesmaximal, characteristic, complemented
8#63952C211yesmaximal, characteristic, complemented
9#63910C211yesmaximal, characteristic, complemented
10#63910C211yesmaximal, characteristic, complemented
11#63909C211yesmaximal, characteristic, complemented
12#63907C211yesmaximal, characteristic, complemented
13#63904C211yesmaximal, characteristic, complemented
14#63854C211yesmaximal, characteristic, complemented
15#63850C211yesmaximal, characteristic, complemented
16#5630—31nomaximal
17#5490—31nomaximal
18C23×C3⋊D3C2211yescharacteristic
19C23×C3×D3C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
22C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
23C22×C3×Dic3C2211yescharacteristic
24C2×C3×C4×D3C2211yescharacteristic, complemented
25C2×C3×C4×D3C2211yescharacteristic, complemented
26C2×C3×Dic6C2211yescharacteristic, complemented
27C2×#626—24no—
28C2×#626C2211yescharacteristic
29C2×#626C2211yescharacteristic
30C2×#626C2211yescharacteristic
31C2×#626C2211yescharacteristic
32C2×#626C2211yescharacteristic, complemented
33C2×#626C2211yescharacteristic, complemented
34C2×#624C2211yescharacteristic, complemented
35C2×#624C2211yescharacteristic, complemented
36C2×#624C2211yescharacteristic, complemented
37#3799—24no—
38C32×C22⋊C4C2211yescharacteristic
39#3752C2211yescharacteristic, complemented
40#3752C2211yescharacteristic, complemented
41C3×#217C2211yescharacteristic
42C3×#217C2211yescharacteristic
43C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C3×C4⋊Dic3C2211yescharacteristic, complemented
46C3×Dic3⋊C4C2211yescharacteristic, complemented
47C3×Dic3⋊C4C2211yescharacteristic, complemented
48#3724C2211yescharacteristic, complemented
49#3722C2211yescharacteristic, complemented
50#3722C2211yescharacteristic, complemented
51#3722C2211yescharacteristic, complemented
52(C2×D3)⋊Dic3C2211yescharacteristic, complemented
53(C2×D3)⋊Dic3C2211yescharacteristic, complemented
54Dic32C2211yescharacteristic, complemented
55C22×#67—31no—
56C2×#239—31no—
57C2×#235—31no—
58C22×D12—31no—
59C3×C22⋊Q8—31no—
60C2×C3×C22⋊C4—31no—
61#953—31no—
62#947—31no—
63#930—31no—
64#929—31no—
65C4×#67—31no—
66C4×#67—31no—
67#895—31no—
68#895—31no—
69#895—31no—
70#894—31no—
71#894—31no—
72#893—31no—
73#885—31no—
74#884—31no—
75#884—31no—
76#883—31no—
77#882—31no—
78#882—31no—
79#882—31no—
80#882—61no—
81#67⋊C4—31no—
82D3×C22⋊C4—31no—
83D3×C22⋊C4—31no—
84D3×C22⋊C4D311yescharacteristic, complemented
85#878D311yescharacteristic, complemented
86C23×C32C2311yescharacteristic
87C22×C3⋊D3—22no—
88C22×C3⋊D3—22no—
89C22×C3⋊D3—41no—
90C22×C3⋊D3C2311yescharacteristic
91C22×C3⋊D3C2311yescharacteristic
92C22×C3⋊D3C2311yescharacteristic
93C22×C3×D3—22no—
94C22×C3×D3—22no—
95C22×C3×D3—41no—
96C22×C3×D3C2311yescharacteristic
97C22×C3×D3C2311yescharacteristic
98C2×C32×C4C2311yescharacteristic
99C2×C32×C4C2311yescharacteristic
100#638—22no—
101C2×C3⋊Dic3C2311yescharacteristic
102C4⋊(C3⋊D3)—22no—
103C4⋊(C3⋊D3)—41no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C2×C3×Dic3C2311yescharacteristic
109C2×C3×Dic3C2311yescharacteristic
110C2×C3×Dic3C2311yescharacteristic
111C3×C4×D3—22no—
112C3×C4×D3—41no—
113C3×Dic6—22no—
114#626—22no—
115#626—22no—
116#626—22no—
117#626—22no—
118#626—41no—
119#626D414yescomplemented
120#624—22no—
121#624—22no—
122#624—41no—
123#546—91noSylow
124C24×C3—31no—
125C23×D3—31no—
126C23×D3—31no—
127C23×D3—61no—
128C23×D3C2×D311yescharacteristic
129C2×C3×Q8—31no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—31no—
133C2×#67—31no—
134C2×#67—31no—
135C2×#67—31no—
136C2×#67—31no—
137C2×#67—31no—
138C2×#67—31no—
139C2×#67—31no—
140C2×#67—31no—
141C2×#67—61no—
142C2×#67—64no—
143C22×Dic3C2×D311yescharacteristic
144#239—64no—
145#235—64no—
146C2×D12—31no—
147C2×D12—31no—
148C2×D12—31no—
149C2×D12—31no—
150C2×D12—31no—
151C2×D12—31no—
152C2×D12—31no—
153C2×D12—31no—
154C2×D12—31no—
155C2×D12—31no—
156C2×D12—61no—
157C2×D12—61no—
158C2×D12—64no—
159C2×C4×D3—31no—
160C2×C4×D3—31no—
161C2×C4×D3—31no—
162C2×C4×D3—31no—
163C2×C4×D3—31no—
164C2×C4×D3—31no—
165C2×C4×D3C2×D311yescharacteristic, complemented
166C2×C4×D3C2×D311yescharacteristic, complemented
167C2×Dic6C2×D311yescharacteristic, complemented
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C22⋊C4—21no—
172C3×C22⋊C4—31no—
173C3×C22⋊C4—31no—
174C3×C22⋊C4—31no—
175C3×C22⋊C4—31no—
176C3×C22⋊C4C2×D311yescharacteristic
177C3×C22⋊C4C2×D311yescharacteristic
178#217—31no—
179#217—31no—
180#217C2×D311yescharacteristic
181#217C2×D311yescharacteristic
182(C2×D3)⋊C4—31no—
183(C2×D3)⋊C4—31no—
184(C2×D3)⋊C4—31no—
185(C2×D3)⋊C4—31no—
186(C2×D3)⋊C4—31no—
187(C2×D3)⋊C4—31no—
188(C2×D3)⋊C4—31no—
189(C2×D3)⋊C4—31no—
190(C2×D3)⋊C4—31no—
191(C2×D3)⋊C4—31no—
192(C2×D3)⋊C4—31no—
193(C2×D3)⋊C4—31no—
194(C2×D3)⋊C4—61no—
195(C2×D3)⋊C4—61no—
196(C2×D3)⋊C4C2×D311yescharacteristic, complemented
197(C2×D3)⋊C4C2×D311yescharacteristic, complemented
198C4⋊Dic3C2×D311yescharacteristic, complemented
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4C2×D311yescharacteristic, complemented
202Dic3⋊C4C2×D311yescharacteristic, complemented
203C4×Dic3—31no—
204C4×Dic3—31no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32#4112yescomplemented
208C22×C32C2411yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3—21no—
214C2×C3⋊D3—21no—
215C2×C3⋊D3—21no—
216C2×C3⋊D3—21no—
217C2×C3⋊D3—44no—
218C2×C3⋊D3C2×D412yes—
219C2×C3×D3—21no—
220C2×C3×D3—21no—
221C2×C3×D3—21no—
222C2×C3×D3—21no—
223C2×C3×D3—21no—
224C2×C3×D3—44no—
225C2×C3×D3C2×D412yes—
226C32×C4—21no—
227C32×C4—21no—
228C3⋊Dic3—21no—
229C3×Dic3—21no—
230C3×Dic3—21no—
231C3×Dic3—21no—
232C3×Dic3—21no—
233C3×Dic3—21no—
234C3×Dic3C2×D412yes—
235C2×#41—91no—
236C22×D4—91no—
237#124—91no—
238#123—91no—
239#123—91no—
240C22⋊Q8—31no—
241#121—91no—
242#121—91no—
243#121—91no—
244#120—91no—
245#120—91no—
246C4×D4—91no—
247C4×D4—91no—
248C2×C22⋊C4—31no—
249C2×C22⋊C4—91no—
250C23×C3—21no—
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—62no—
254C23×C3—62no—
255C23×C3—121no—
256C23×C3C22×D311yescharacteristic
257C23×C3C22×D311yescharacteristic
258C22×D3—22no—
259C22×D3—22no—
260C22×D3—31no—
261C22×D3—31no—
262C22×D3—31no—
263C22×D3—31no—
264C22×D3—31no—
265C22×D3—31no—
266C22×D3—41no—
267C22×D3—61no—
268C22×D3—61no—
269C22×D3—61no—
270C22×D3—62no—
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—62no—
274C22×D3—122no—
275C22×D3—122no—
276C22×D3—121no—
277C22×D3—121no—
278C22×D3—122no—
279C22×D3C22×D311yescharacteristic
280C22×D3C22×D311yescharacteristic
281C3×Q8—62no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—121no—
293C2×C3×C4C22×D311yescharacteristic
294C2×C3×C4C22×D311yescharacteristic
295C2×C3×C4C22×D311yescharacteristic
296C2×C3×C4C22×D311yescharacteristic
297#67—34no—
298#67—62no—
299#67—62no—
300#67—62no—
301#67—62no—
302#67—62no—
303#67—62no—
304#67—122no—
305#67—121no—
306C2×Dic3—22no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—61no—
310C2×Dic3C22×D311yescharacteristic
311C2×Dic3C22×D311yescharacteristic
312C2×Dic3C22×D311yescharacteristic
313C2×Dic3C22×D311yescharacteristic
314C2×Dic3C22×D311yescharacteristic
315C2×Dic3C22×D311yescharacteristic
316D12—34no—
317D12—62no—
318D12—62no—
319D12—62no—
320D12—62no—
321D12—62no—
322D12—62no—
323D12—122no—
324D12—121no—
325D12—121no—
326D12—122no—
327D12—121no—
328C4×D3—22no—
329C4×D3—41no—
330C4×D3—62no—
331C4×D3—62no—
332C4×D3—62no—
333C4×D3—62no—
334C4×D3—121no—
335C4×D3—121no—
336Dic6—22no—
337C2×C32—22no—
338C2×C32C22×D411yescharacteristic
339C2×C32C2×#4111yescharacteristic
340C2×C32H(5,𝔽2)11yescharacteristic
341C3⋊D3—22no—
342C3⋊D3—41no—
343C3⋊D3—41no—
344C3×D3—22no—
345C3×D3—41no—
346C24—31no—
347C24—91no—
348#41—184no—
349C2×Q8—31no—
350C2×D4—91no—
351C2×D4—91no—
352C2×D4—91no—
353C2×D4—91no—
354C2×D4—91no—
355C2×D4—91no—
356C2×D4—91no—
357C2×D4—91no—
358C2×D4—91no—
359C2×D4—184no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—31no—
363C22×C4—91no—
364C22×C4—91no—
365C22×C4—91no—
366C4⋊C4—31no—
367C4⋊C4—31no—
368C4⋊C4—31no—
369C4⋊C4—91no—
370C22⋊C4—31no—
371C22⋊C4—31no—
372C22⋊C4—31no—
373C22⋊C4—31no—
374C22⋊C4—91no—
375C22⋊C4—91no—
376C22⋊C4—91no—
377C22⋊C4—91no—
378C22⋊C4—91no—
379C22⋊C4—91no—
380C22⋊C4—91no—
381C22⋊C4D3211yescharacteristic
382C42—91no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—32no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—124no—
398C22×C3#23512yescomplemented
399C22×C3#23912yescomplemented
400C22×C3C23×D311yescharacteristic
401C22×C3C23×D311yescharacteristic
402C2×D3—21no—
403C2×D3—21no—
404C2×D3—21no—
405C2×D3—21no—
406C2×D3—21no—
407C2×D3—32no—
408C2×D3—32no—
409C2×D3—44no—
410C2×D3—62no—
411C2×D3—61no—
412C2×D3—61no—
413C2×D3—61no—
414C2×D3—61no—
415C2×D3—62no—
416C2×D3—61no—
417C2×D3—61no—
418C2×D3—61no—
419C2×D3—61no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3—61no—
423C2×D3—61no—
424C2×D3—61no—
425C2×D3—61no—
426C2×D3—61no—
427C2×D3—61no—
428C2×D3—121no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—124no—
436C2×D3—124no—
437C2×D3—128no—
438C2×D3D3×D412yes—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—32no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—61no—
450C3×C4—61no—
451Dic3—21no—
452Dic3—21no—
453Dic3—21no—
454Dic3—21no—
455Dic3—21no—
456Dic3—61no—
457Dic3—61no—
458Dic3—121no—
459Dic3D3×D412yes—
460C32#54611yescharacteristic, complemented, Sylow
461C23—31no—
462C23—31no—
463C23—62no—
464C23—62no—
465C23—91no—
466C23—91no—
467C23—91no—
468C23—121no—
469C23—182no—
470C23—182no—
471C23—361no—
472C23C2×D3211yescharacteristic
473Q8—62no—
474D4—94no—
475D4—182no—
476D4—182no—
477D4—182no—
478D4—182no—
479D4—182no—
480D4—182no—
481D4—361no—
482D4—361no—
483C2×C4—31no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—31no—
487C2×C4—31no—
488C2×C4—31no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—91no—
492C2×C4—121no—
493C2×C4—182no—
494C2×C4—182no—
495C2×C4—361no—
496C2×C4C2×D3211yescharacteristic
497C2×C4C2×D3211yescharacteristic
498C2×C3—22no—
499C2×C3—22no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—42no—
504C2×C3—62no—
505C2×C3—121no—
506C2×C3C2×#23511yescharacteristic
507C2×C3C2×D3×D411yescharacteristic
508C2×C3C2×D3×D411yescharacteristic
509C2×C3#100411yescharacteristic
510C2×C3C2×#23911yescharacteristic
511C2×C3#100911yescharacteristic
512D3—22no—
513D3—41no—
514D3—62no—
515D3—62no—
516D3—122no—
517D3—122no—
518D3—121no—
519D3—121no—
520D3—121no—
521D3—121no—
522D3—241no—
523C22—21no—
524C22—21no—
525C22—32no—
526C22—61no—
527C22—61no—
528C22—61no—
529C22—61no—
530C22—61no—
531C22—92no—
532C22—124no—
533C22—181no—
534C22—181no—
535C22—181no—
536C22—181no—
537C22—181no—
538C22—181no—
539C22—181no—
540C22—181no—
541C22—364no—
542C22#379912yescomplemented
543C22C22×D3211yescharacteristic
544C4—21no—
545C4—21no—
546C4—32no—
547C4—61no—
548C4—61no—
549C4—61no—
550C4—61no—
551C4—61no—
552C4—181no—
553C3—21no—
554C3#549011yescharacteristic, complemented
555C3#563011yescharacteristic, complemented
556C2—22no—
557C2—62no—
558C2—121no—
559C2—182no—
560C2—361no—
561C2—361no—
562C2C2×#379911yescharacteristic
563C2D32×D411yescharacteristic
564C2#6431411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#546
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#626C2semidirect
2. C2×#3799C2semidirect
3. #64091C2semidirect
4. C3×D3×C22⋊C4C2semidirect
5. C3×#878C2semidirect
6. #63977C2semidirect
7. #63974C2semidirect
8. #63952C2semidirect
9. #63910C2semidirect
10. #63910C2semidirect
11. #63909C2semidirect
12. #63907C2semidirect
13. #63904C2semidirect
14. #63854C2semidirect
15. #63850C2semidirect
21. C2×C4⋊(C3⋊D3)C22semidirect
22. C2×C4⋊(C3⋊D3)C22semidirect
24. C2×C3×C4×D3C22semidirect
25. C2×C3×C4×D3C22semidirect
26. C2×C3×Dic6C22semidirect
32. C2×#626C22semidirect
33. C2×#626C22semidirect
34. C2×#624C22semidirect
35. C2×#624C22semidirect
36. C2×#624C22semidirect
39. #3752C22semidirect
40. #3752C22semidirect
43. C3×(C2×D3)⋊C4C22semidirect
44. C3×(C2×D3)⋊C4C22semidirect
45. C3×C4⋊Dic3C22semidirect
46. C3×Dic3⋊C4C22semidirect
47. C3×Dic3⋊C4C22semidirect
48. #3724C22semidirect
49. #3722C22semidirect
50. #3722C22semidirect
51. #3722C22semidirect
52. (C2×D3)⋊Dic3C22semidirect
53. (C2×D3)⋊Dic3C22semidirect
54. Dic32C22semidirect
84. D3×C22⋊C4D3semidirect
85. #878D3semidirect
119. #626D4semidirect
165. C2×C4×D3C2×D3semidirect
166. C2×C4×D3C2×D3semidirect
167. C2×Dic6C2×D3semidirect
196. (C2×D3)⋊C4C2×D3semidirect
197. (C2×D3)⋊C4C2×D3semidirect
198. C4⋊Dic3C2×D3semidirect
201. Dic3⋊C4C2×D3semidirect
202. Dic3⋊C4C2×D3semidirect
207. C22×C32#41semidirect
398. C22×C3#235semidirect
399. C22×C3#239semidirect
460. C32#546semidirect
542. C22#3799semidirect
554. C3#5490semidirect
555. C3#5630semidirect

Maximal subgroups

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

subgroupindex
C22×#6262
C2×#37992
#640912
C3×D3×C22⋊C42
C3×#8782
#639772
#639742
#639522
#639102
#639092
#639072
#639042
#638542
#638502
#56303
#54903