Finite Group Databasebeta

← #10595307

Subgroups — #10595307

Order 576.

750 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3830C211yesmaximal, characteristic, complemented
2C2×D3×#67C211yesmaximal, characteristic, complemented
3C3×#939C211yesmaximal, characteristic, complemented
4C3×#937C211yesmaximal, characteristic, complemented
5#63983C211yesmaximal, characteristic, complemented
6#63980C211yesmaximal, characteristic, complemented
7#67⋊Dic3C211yesmaximal, characteristic, complemented
8#63962C211yesmaximal, characteristic, complemented
9C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
10#63958C211yesmaximal, characteristic, complemented
11#63952C211yesmaximal, characteristic, complemented
12#63921C211yesmaximal, characteristic, complemented
13#63913C211yesmaximal, characteristic, complemented
14#63896C211yesmaximal, characteristic, complemented
15#63853C211yesmaximal, characteristic, complemented
16#5610—31nomaximal
17#5591—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C22×D32C2211yescharacteristic
20C2×C32×D4C2211yescharacteristic, complemented
21C2×#638C2211yescharacteristic
22C2×#638C2211yescharacteristic, complemented
23C22×C3⋊Dic3C2211yescharacteristic, complemented
24C22×C3⋊Dic3C2211yescharacteristic, complemented
25#3830—24no—
26C2×C4×C3⋊D3C2211yescharacteristic, complemented
27C2×#634C2211yescharacteristic, complemented
28C2×C3×#67C2211yescharacteristic
29C2×C3×#67C2211yescharacteristic
30C2×C3×#67C2211yescharacteristic, complemented
31C2×C3×#67C2211yescharacteristic, complemented
32C22×C3×Dic3C2211yescharacteristic, complemented
33D3×#67—24no—
34C2×#626C2211yescharacteristic
35C2×#625C2211yescharacteristic
36C2×#624C2211yescharacteristic, complemented
37C2×D3×Dic3C2211yescharacteristic
38C3×#217C2211yescharacteristic, complemented
39C3×#217C2211yescharacteristic, complemented
40C3×(C2×D3)⋊C4C2211yescharacteristic
41C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
42C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C3×Dic3⋊C4C2211yescharacteristic, complemented
44#3724C2211yescharacteristic, complemented
45Dic3⋊Dic3C2211yescharacteristic, complemented
46Dic3⋊Dic3C2211yescharacteristic, complemented
47#3722C2211yescharacteristic, complemented
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic, complemented
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52(C2×D3)⋊Dic3C2211yescharacteristic, complemented
53(C2×D3)⋊Dic3C2211yescharacteristic, complemented
54Dic32C2211yescharacteristic, complemented
55C22×#67—31no—
56C2×#237—31no—
57C2×#237—31no—
58C2×#237—61no—
59C2×D3×D4—31no—
60C3×#121—31no—
61C3×#120—31no—
62C2×#217—31no—
63#939—31no—
64#939—31no—
65#939D311yescharacteristic, complemented
66#938—31no—
67#937—31no—
68#937—31no—
69#937D311yescharacteristic, complemented
70#935—31no—
71D4×Dic3—31no—
72C4×#67—31no—
73C2×(C2×D3)⋊C4—31no—
74#896—31no—
75#894—31no—
76#886—31no—
77#885—31no—
78#885—31no—
79#884—31no—
80#883—31no—
81#67⋊C4—31no—
82#67⋊C4—31no—
83D3×C22⋊C4—31no—
84D3×C22⋊C4—31no—
85#878—31no—
86C23×C32C2311yescharacteristic
87C23×C32C2311yescharacteristic
88C22×C3⋊D3C2311yescharacteristic
89C22×C3×D3—22no—
90C22×C3×D3—22no—
91C22×C3×D3—41no—
92C22×C3×D3C2311yescharacteristic
93C22×C3×D3C2311yescharacteristic
94C22×C3×D3C2311yescharacteristic
95C2×D32—22no—
96C2×D32—22no—
97C2×D32—41no—
98C32×D4—22no—
99C2×C32×C4C2311yescharacteristic
100#638—22no—
101#638D414yescomplemented
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3C2311yescharacteristic
105C2×C3⋊Dic3C2311yescharacteristic
106C2×C3⋊Dic3C2311yescharacteristic
107C4×C3⋊D3—22no—
108#634—22no—
109C3×#67—22no—
110C3×#67—22no—
111C3×#67—41no—
112C3×#67—41no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3C2311yescharacteristic
115C2×C3×Dic3C2311yescharacteristic
116C2×C3×Dic3C2311yescharacteristic
117C2×C3×Dic3C2311yescharacteristic
118C2×C3×Dic3C2311yescharacteristic
119#626—22no—
120#625—22no—
121#624—41no—
122D3×Dic3—22no—
123#546—91noSylow
124C24×C3—31no—
125C23×D3—31no—
126C23×D3—31no—
127C23×D3C2×D311yescharacteristic
128C2×C3×D4—21no—
129C2×C3×D4—31no—
130C2×C3×D4—31no—
131C2×C3×D4—31no—
132C2×C3×D4—31no—
133C2×C3×D4C2×D311yescharacteristic, complemented
134C2×C3×D4C2×D311yescharacteristic, complemented
135C22×C3×C4—31no—
136C2×#67—31no—
137C2×#67—31no—
138C2×#67—31no—
139C2×#67—31no—
140C2×#67—31no—
141C2×#67—31no—
142C2×#67—31no—
143C2×#67—61no—
144C2×#67—61no—
145C2×#67—64no—
146C2×#67C2×D311yescharacteristic
147C2×#67C2×D311yescharacteristic
148C2×#67C2×D311yescharacteristic, complemented
149C2×#67C2×D311yescharacteristic, complemented
150C22×Dic3—31no—
151C22×Dic3—31no—
152C22×Dic3—31no—
153C22×Dic3—31no—
154C22×Dic3—31no—
155C22×Dic3—61no—
156C22×Dic3—61no—
157C22×Dic3C2×D311yescharacteristic, complemented
158#237—64no—
159#237—64no—
160#237—68no—
161D3×D4—64no—
162C2×D12—31no—
163C2×C4×D3—31no—
164C2×C4×D3—31no—
165C2×C4×D3—31no—
166C2×C4×D3—31no—
167C2×C4×D3—31no—
168C2×C4×D3—61no—
169C2×Dic6—31no—
170C2×Dic6—31no—
171C2×Dic6—61no—
172C3×C4⋊C4—31no—
173C3×C22⋊C4—31no—
174C3×C22⋊C4—31no—
175C3×C22⋊C4—31no—
176C3×C22⋊C4—31no—
177C3×C22⋊C4—31no—
178#217—31no—
179#217—31no—
180#217—31no—
181#217—31no—
182#217—31no—
183#217—31no—
184#217C2×D311yescharacteristic, complemented
185#217C2×D311yescharacteristic, complemented
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)⋊C4C2×D311yescharacteristic
195(C2×D3)⋊C4C2×D311yescharacteristic, complemented
196(C2×D3)⋊C4C2×D311yescharacteristic, complemented
197C4⋊Dic3—31no—
198C4⋊Dic3—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4—31no—
202Dic3⋊C4—31no—
203Dic3⋊C4C2×D311yescharacteristic, complemented
204C4×Dic3—31no—
205C4×Dic3—31no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32C2×D412yes—
212C22×C32C2411yescharacteristic
213C2×C3⋊D3—21no—
214C2×C3⋊D3—21no—
215C2×C3⋊D3C2×D412yes—
216C2×C3×D3—21no—
217C2×C3×D3—21no—
218C2×C3×D3—21no—
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×D3#4112yescomplemented
226D32—44no—
227C32×C4—21no—
228C3⋊Dic3—21no—
229C3⋊Dic3—21no—
230C3⋊Dic3C2×D412yes—
231C3×Dic3—21no—
232C3×Dic3—21no—
233C3×Dic3—21no—
234C3×Dic3—21no—
235C3×Dic3—21no—
236C2×#41—91no—
237C22×D4—91no—
238#124—91no—
239#123—91no—
240#123—91no—
241C22⋊Q8—91no—
242#121—31no—
243#121—91no—
244#121—91no—
245#120—31no—
246#120—91no—
247C4×D4—91no—
248C4×D4—91no—
249C2×C22⋊C4—91no—
250C2×C22⋊C4—91no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—31no—
254C23×C3—31no—
255C23×C3—31no—
256C23×C3—62no—
257C23×C3—62no—
258C23×C3—121no—
259C23×C3C22×D311yescharacteristic
260C23×C3C22×D311yescharacteristic
261C23×C3C22×D311yescharacteristic
262C23×C3C22×D311yescharacteristic
263C22×D3—22no—
264C22×D3—22no—
265C22×D3—31no—
266C22×D3—31no—
267C22×D3—41no—
268C22×D3—61no—
269C22×D3—62no—
270C22×D3—62no—
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—121no—
274C22×D3—121no—
275C22×D3C22×D311yescharacteristic
276C22×D3C22×D311yescharacteristic
277C22×D3C22×D311yescharacteristic
278C3×D4—24no—
279C3×D4—22no—
280C3×D4—22no—
281C3×D4—62no—
282C3×D4—62no—
283C3×D4—121no—
284C3×D4—121no—
285C2×C3×C4—21no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—31no—
291C2×C3×C4—121no—
292C2×C3×C4C22×D311yescharacteristic
293C2×C3×C4C22×D311yescharacteristic
294#67—22no—
295#67—22no—
296#67—34no—
297#67—34no—
298#67—41no—
299#67—41no—
300#67—64no—
301#67—62no—
302#67—62no—
303#67—62no—
304#67—62no—
305#67—62no—
306#67—64no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—31no—
310C2×Dic3—31no—
311C2×Dic3—31no—
312C2×Dic3—31no—
313C2×Dic3—41no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—64no—
321C2×Dic3—64no—
322C2×Dic3—62no—
323C2×Dic3—62no—
324C2×Dic3C22×D311yescharacteristic
325C2×Dic3C22×D311yescharacteristic
326C2×Dic3C22×D311yescharacteristic
327C2×Dic3C22×D311yescharacteristic
328C2×Dic3C22×D311yescharacteristic
329D12—62no—
330C4×D3—62no—
331C4×D3—62no—
332C4×D3—62no—
333C4×D3—64no—
334C4×D3—121no—
335C4×D3—121no—
336Dic6—62no—
337Dic6—62no—
338Dic6—64no—
339C2×C32—22no—
340C2×C32—41no—
341C2×C32C22×D411yescharacteristic
342C2×C32C2×#4111yescharacteristic
343C2×C32H(5,𝔽2)11yescharacteristic
344C3⋊D3—22no—
345C3×D3—22no—
346C3×D3—41no—
347C3×D3—41no—
348C24—31no—
349C24—91no—
350#41—184no—
351C2×Q8—91no—
352C2×D4—31no—
353C2×D4—31no—
354C2×D4—31no—
355C2×D4—31no—
356C2×D4—91no—
357C2×D4—91no—
358C2×D4—91no—
359C2×D4—91no—
360C2×D4—184no—
361C2×D4D3211yescharacteristic, complemented
362C22×C4—31no—
363C22×C4—91no—
364C22×C4—91no—
365C22×C4—91no—
366C22×C4—91no—
367C22×C4—91no—
368C4⋊C4—31no—
369C4⋊C4—91no—
370C4⋊C4—91no—
371C4⋊C4—91no—
372C22⋊C4—31no—
373C22⋊C4—31no—
374C22⋊C4—31no—
375C22⋊C4—31no—
376C22⋊C4—31no—
377C22⋊C4—91no—
378C22⋊C4—91no—
379C22⋊C4—91no—
380C22⋊C4—91no—
381C22⋊C4—91no—
382C22⋊C4—91no—
383C22⋊C4—91no—
384C42—91no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—22no—
398C22×C3—32no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—61no—
404C22×C3—61no—
405C22×C3—61no—
406C22×C3—61no—
407C22×C3—61no—
408C22×C3—61no—
409C22×C3—61no—
410C22×C3—61no—
411C22×C3—124no—
412C22×C3D3×D412yes—
413C22×C3D3×D412yes—
414C22×C3C23×D311yescharacteristic
415C22×C3C23×D311yescharacteristic
416C2×D3—21no—
417C2×D3—21no—
418C2×D3—21no—
419C2×D3—21no—
420C2×D3—21no—
421C2×D3—21no—
422C2×D3—21no—
423C2×D3—21no—
424C2×D3—32no—
425C2×D3—32no—
426C2×D3—44no—
427C2×D3—62no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—61no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—124no—
435C2×D3—124no—
436C2×D3#23712yescomplemented
437C3×C4—22no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—61no—
443C3×C4—61no—
444C3×C4—61no—
445Dic3—21no—
446Dic3—21no—
447Dic3—21no—
448Dic3—21no—
449Dic3—21no—
450Dic3—32no—
451Dic3—32no—
452Dic3—62no—
453Dic3—62no—
454Dic3—62no—
455Dic3—61no—
456Dic3—61no—
457Dic3—61no—
458Dic3—61no—
459C32#54611yescharacteristic, complemented, Sylow
460C23—31no—
461C23—31no—
462C23—31no—
463C23—62no—
464C23—62no—
465C23—91no—
466C23—121no—
467C23—182no—
468C23—182no—
469C23—361no—
470C23C2×D3211yescharacteristic
471C23C2×D3211yescharacteristic
472Q8—182no—
473D4—22no—
474D4—62no—
475D4—62no—
476D4—94no—
477D4—121no—
478D4—121no—
479D4—182no—
480D4—182no—
481D4—182no—
482C2×C4—31no—
483C2×C4—31no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—31no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—121no—
491C2×C4—182no—
492C2×C4—182no—
493C2×C4—182no—
494C2×C4—182no—
495C2×C4—361no—
496C2×C4C2×D3211yescharacteristic
497C2×C3—22no—
498C2×C3—22no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—42no—
503C2×C3—42no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—62no—
507C2×C3—121no—
508C2×C3—121no—
509C2×C3C2×D3×D411yescharacteristic
510C2×C3C2×D3×D411yescharacteristic
511C2×C3C2×#23711yescharacteristic
512C2×C3#100411yescharacteristic
513C2×C3#100411yescharacteristic
514C2×C3D3×#4111yescharacteristic
515D3—22no—
516D3—41no—
517D3—41no—
518D3—62no—
519D3—62no—
520D3—122no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—32no—
527C22—61no—
528C22—61no—
529C22—61no—
530C22—61no—
531C22—61no—
532C22—61no—
533C22—61no—
534C22—61no—
535C22—92no—
536C22—124no—
537C22—181no—
538C22—181no—
539C22—364no—
540C22#381112yes—
541C22C22×D3211yescharacteristic
542C4—21no—
543C4—61no—
544C4—61no—
545C4—61no—
546C4—61no—
547C4—61no—
548C4—92no—
549C4—181no—
550C4—181no—
551C3—21no—
552C3#559111yescharacteristic, complemented
553C3#561011yescharacteristic, complemented
554C2—22no—
555C2—41no—
556C2—62no—
557C2—121no—
558C2—121no—
559C2—182no—
560C2D3×#23711yescharacteristic
561C2#6431311yescharacteristic
562C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#546
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#3830C2semidirect
2. C2×D3×#67C2semidirect
3. C3×#939C2semidirect
4. C3×#937C2semidirect
5. #63983C2semidirect
6. #63980C2semidirect
7. #67⋊Dic3C2semidirect
8. #63962C2semidirect
9. C2×(C2×D3)⋊Dic3C2semidirect
10. #63958C2semidirect
11. #63952C2semidirect
12. #63921C2semidirect
13. #63913C2semidirect
14. #63896C2semidirect
15. #63853C2semidirect
20. C2×C32×D4C22semidirect
22. C2×#638C22semidirect
23. C22×C3⋊Dic3C22semidirect
24. C22×C3⋊Dic3C22semidirect
26. C2×C4×C3⋊D3C22semidirect
27. C2×#634C22semidirect
30. C2×C3×#67C22semidirect
31. C2×C3×#67C22semidirect
32. C22×C3×Dic3C22semidirect
36. C2×#624C22semidirect
38. C3×#217C22semidirect
39. C3×#217C22semidirect
41. C3×(C2×D3)⋊C4C22semidirect
42. C3×(C2×D3)⋊C4C22semidirect
43. C3×Dic3⋊C4C22semidirect
44. #3724C22semidirect
45. Dic3⋊Dic3C22semidirect
46. Dic3⋊Dic3C22semidirect
47. #3722C22semidirect
50. (C2×D3)⋊Dic3C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. (C2×D3)⋊Dic3C22semidirect
53. (C2×D3)⋊Dic3C22semidirect
54. Dic32C22semidirect
65. #939D3semidirect
69. #937D3semidirect
101. #638D4semidirect
133. C2×C3×D4C2×D3semidirect
134. C2×C3×D4C2×D3semidirect
148. C2×#67C2×D3semidirect
149. C2×#67C2×D3semidirect
157. C22×Dic3C2×D3semidirect
184. #217C2×D3semidirect
185. #217C2×D3semidirect
195. (C2×D3)⋊C4C2×D3semidirect
196. (C2×D3)⋊C4C2×D3semidirect
203. Dic3⋊C4C2×D3semidirect
225. C2×C3×D3#41semidirect
361. C2×D4D32semidirect
436. C2×D3#237semidirect
459. C32#546semidirect
552. C3#5591semidirect
553. C3#5610semidirect

Maximal subgroups

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

subgroupindex
C2×#38302
C2×D3×#672
C3×#9392
C3×#9372
#639832
#639802
#67⋊Dic32
#639622
C2×(C2×D3)⋊Dic32
#639582
#639522
#639212
#639132
#638962
#638532
#56103
#55913