Finite Group Databasebeta

← #10594274

Subgroups — #10594274

Order 576.

706 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#67C211yesmaximal, characteristic, complemented
2C2×#3795C211yesmaximal, characteristic, complemented
3#64094C211yesmaximal, characteristic, complemented
4C3×#883C211yesmaximal, characteristic, complemented
5C3×#882C211yesmaximal, characteristic, complemented
6#63971C211yesmaximal, characteristic, complemented
7#63970C211yesmaximal, characteristic, complemented
8D3×#217C211yesmaximal, characteristic, complemented
9#63922C211yesmaximal, characteristic, complemented
10D3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
11#63902C211yesmaximal, characteristic, complemented
12#63898C211yesmaximal, characteristic, complemented
13#63896C211yesmaximal, characteristic, complemented
14#63890C211yesmaximal, characteristic, complemented
15#63886C211yesmaximal, characteristic, complemented
16#5654—31nomaximal
17#5591—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C22×D32C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×#634C2211yescharacteristic, complemented
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic, complemented
24C2×C3×D12C2211yescharacteristic
25C2×C3×D12C2211yescharacteristic, complemented
26C2×C3×C4×D3C2211yescharacteristic, complemented
27D3×#67—24no—
28C2×#626C2211yescharacteristic
29C2×#626C2211yescharacteristic, complemented
30C2×#625C2211yescharacteristic
31C2×#625C2211yescharacteristic, complemented
32C2×D3×Dic3C2211yescharacteristic
33C2×D3×Dic3C2211yescharacteristic, complemented
34C2×D3×Dic3C2211yescharacteristic, complemented
35C2×D3×Dic3C2211yescharacteristic, complemented
36C2×D3×Dic3C2211yescharacteristic, complemented
37#3795—24no—
38C32×C22⋊C4C2211yescharacteristic, complemented
39#3757C2211yescharacteristic, complemented
40#3752C2211yescharacteristic, complemented
41#3752C2211yescharacteristic, complemented
42C4×C3⋊Dic3C2211yescharacteristic, complemented
43C3×#217C2211yescharacteristic, complemented
44C3×(C2×D3)⋊C4C2211yescharacteristic
45C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C3×C4⋊Dic3C2211yescharacteristic, complemented
48C3×Dic3⋊C4C2211yescharacteristic, complemented
49#3724C2211yescharacteristic, complemented
50Dic3⋊Dic3C2211yescharacteristic, complemented
51#3722C2211yescharacteristic, complemented
52(C2×D3)⋊Dic3C2211yescharacteristic
53(C2×D3)⋊Dic3C2211yescharacteristic
54(C2×D3)⋊Dic3C2211yescharacteristic, complemented
55C22×#67—31no—
56C2×#237—31no—
57C2×D3×D4—31no—
58C2×#235—31no—
59C3×#123—31no—
60C3×#120—31no—
61C2×#217—31no—
62#939—31no—
63#939—31no—
64#939—31no—
65#938—31no—
66#937—31no—
67D4×Dic3—31no—
68C2×(C2×D3)⋊C4—31no—
69#896—31no—
70#895—31no—
71#894—31no—
72#893—31no—
73#886—31no—
74#885—31no—
75#885—31no—
76#885—61no—
77#884—31no—
78#883D311yescharacteristic, complemented
79#882—31no—
80#882D311yescharacteristic, complemented
81#67⋊C4—31no—
82#67⋊C4—31no—
83D3×C22⋊C4—31no—
84D3×C22⋊C4—31no—
85#878—31no—
86C23×C32C2311yescharacteristic
87C22×C3⋊D3C2311yescharacteristic
88C22×C3×D3—22no—
89C22×C3×D3—22no—
90C22×C3×D3—41no—
91C22×C3×D3C2311yescharacteristic
92C22×C3×D3C2311yescharacteristic
93C22×C3×D3C2311yescharacteristic
94C22×C3×D3C2311yescharacteristic
95C2×D32—22no—
96C2×D32—22no—
97C2×D32—41no—
98C2×C32×C4C2311yescharacteristic
99C2×C32×C4C2311yescharacteristic
100#638—22no—
101C2×C3⋊Dic3C2311yescharacteristic
102C2×C3⋊Dic3C2311yescharacteristic
103C2×C3⋊Dic3C2311yescharacteristic
104#634—22no—
105C3×#67—22no—
106C3×#67—41no—
107C2×C3×Dic3C2311yescharacteristic
108C2×C3×Dic3C2311yescharacteristic
109C2×C3×Dic3C2311yescharacteristic
110C2×C3×Dic3C2311yescharacteristic
111C3×D12—22no—
112C3×D12—22no—
113C3×C4×D3—22no—
114#626—22no—
115#626—41no—
116#625—22no—
117#625D414yescomplemented
118D3×Dic3—22no—
119D3×Dic3—22no—
120D3×Dic3—22no—
121D3×Dic3—41no—
122D3×Dic3—41no—
123#546—91noSylow
124C24×C3—31no—
125C23×D3—31no—
126C23×D3—31no—
127C23×D3C2×D311yescharacteristic
128C2×C3×D4—31no—
129C2×C3×D4—31no—
130C2×C3×D4—31no—
131C2×C3×D4—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—
143C2×#67C2×D311yescharacteristic
144C2×#67C2×D311yescharacteristic, complemented
145C22×Dic3—31no—
146C22×Dic3—31no—
147C22×Dic3—31no—
148C22×Dic3—31no—
149C22×Dic3—31no—
150#237—64no—
151D3×D4—64no—
152#235—64no—
153C2×D12—31no—
154C2×D12—31no—
155C2×D12C2×D311yescharacteristic
156C2×D12C2×D311yescharacteristic, complemented
157C2×C4×D3—31no—
158C2×C4×D3—31no—
159C2×C4×D3—31no—
160C2×C4×D3—31no—
161C2×C4×D3—31no—
162C2×C4×D3C2×D311yescharacteristic, complemented
163C2×Dic6—31no—
164C2×Dic6—31no—
165C2×Dic6—61no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—31no—
168C3×C22⋊C4—21no—
169C3×C22⋊C4—31no—
170C3×C22⋊C4—31no—
171C3×C22⋊C4—31no—
172C3×C22⋊C4—31no—
173C3×C22⋊C4C2×D311yescharacteristic, complemented
174C3×C22⋊C4C2×D311yescharacteristic, complemented
175#217—31no—
176#217—31no—
177#217—31no—
178#217—31no—
179#217—31no—
180#217—61no—
181#217C2×D311yescharacteristic, complemented
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—61no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4C2×D311yescharacteristic
194(C2×D3)⋊C4C2×D311yescharacteristic, complemented
195(C2×D3)⋊C4C2×D311yescharacteristic, complemented
196C4⋊Dic3—31no—
197C4⋊Dic3C2×D311yescharacteristic, complemented
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4C2×D311yescharacteristic, complemented
202C4×Dic3—31no—
203C4×Dic3—31no—
204C4×Dic3—61no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—21no—
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—21no—
218C2×C3×D3—21no—
219C2×C3×D3—21no—
220C2×C3×D3—21no—
221C2×C3×D3—44no—
222C2×C3×D3C2×D412yes—
223C2×C3×D3C2×D412yes—
224C2×C3×D3#4112yescomplemented
225D32—44no—
226C32×C4—21no—
227C32×C4—21no—
228C3⋊Dic3—21no—
229C3⋊Dic3—21no—
230C3⋊Dic3C2×D412yes—
231C3×Dic3—21no—
232C3×Dic3—21no—
233C3×Dic3—21no—
234C3×Dic3—21no—
235C2×#41—91no—
236C22×D4—91no—
237#124—91no—
238#123—31no—
239#123—91no—
240C22⋊Q8—91no—
241#121—91no—
242#121—91no—
243#121—91no—
244#120—31no—
245#120—91no—
246C4×D4—91no—
247C4×D4—91no—
248C2×C22⋊C4—91no—
249C2×C22⋊C4—91no—
250C23×C3—21no—
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—31no—
254C23×C3—31no—
255C23×C3—62no—
256C23×C3—62no—
257C23×C3—121no—
258C23×C3C22×D311yescharacteristic
259C23×C3C22×D311yescharacteristic
260C22×D3—22no—
261C22×D3—22no—
262C22×D3—31no—
263C22×D3—31no—
264C22×D3—41no—
265C22×D3—61no—
266C22×D3—62no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—121no—
271C22×D3—121no—
272C22×D3C22×D311yescharacteristic
273C22×D3C22×D311yescharacteristic
274C22×D3C22×D311yescharacteristic
275C22×D3C22×D311yescharacteristic
276C3×D4—62no—
277C3×D4—62no—
278C3×D4—62no—
279C3×D4—121no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—62no—
287C2×C3×C4C22×D311yescharacteristic
288C2×C3×C4C22×D311yescharacteristic
289C2×C3×C4C22×D311yescharacteristic
290C2×C3×C4C22×D311yescharacteristic
291#67—22no—
292#67—34no—
293#67—34no—
294#67—41no—
295#67—62no—
296#67—62no—
297#67—62no—
298#67—62no—
299#67—62no—
300#67—122no—
301#67—121no—
302C2×Dic3—31no—
303C2×Dic3—31no—
304C2×Dic3—31no—
305C2×Dic3—31no—
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—62no—
312C2×Dic3—62no—
313C2×Dic3—62no—
314C2×Dic3—121no—
315C2×Dic3—121no—
316C2×Dic3C22×D311yescharacteristic
317C2×Dic3C22×D311yescharacteristic
318C2×Dic3C22×D311yescharacteristic
319C2×Dic3C22×D311yescharacteristic
320D12—22no—
321D12—22no—
322D12—62no—
323D12—121no—
324C4×D3—22no—
325C4×D3—62no—
326C4×D3—62no—
327C4×D3—62no—
328C4×D3—121no—
329C4×D3—121no—
330Dic6—62no—
331Dic6—62no—
332Dic6—122no—
333C2×C32—41no—
334C2×C32C22×D411yescharacteristic
335C2×C32C2×#4111yescharacteristic
336C2×C32H(5,𝔽2)11yescharacteristic
337C3⋊D3—41no—
338C3×D3—22no—
339C3×D3—22no—
340C3×D3—22no—
341C3×D3—41no—
342C24—31no—
343C24—91no—
344#41—184no—
345C2×Q8—91no—
346C2×D4—31no—
347C2×D4—31no—
348C2×D4—31no—
349C2×D4—31no—
350C2×D4—91no—
351C2×D4—91no—
352C2×D4—91no—
353C2×D4—91no—
354C2×D4—91no—
355C2×D4—184no—
356C22×C4—31no—
357C22×C4—91no—
358C22×C4—91no—
359C22×C4—91no—
360C22×C4—91no—
361C22×C4—91no—
362C4⋊C4—31no—
363C4⋊C4—31no—
364C4⋊C4—91no—
365C4⋊C4—91no—
366C22⋊C4—31no—
367C22⋊C4—31no—
368C22⋊C4—31no—
369C22⋊C4—31no—
370C22⋊C4—91no—
371C22⋊C4—91no—
372C22⋊C4—91no—
373C22⋊C4—91no—
374C22⋊C4—91no—
375C22⋊C4—91no—
376C22⋊C4—91no—
377C22⋊C4D3211yescharacteristic, complemented
378C42—91no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—32no—
387C22×C3—32no—
388C22×C3—32no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—61no—
398C22×C3—61no—
399C22×C3—61no—
400C22×C3—61no—
401C22×C3—124no—
402C22×C3C23×D311yescharacteristic
403C22×C3C23×D311yescharacteristic
404C2×D3—21no—
405C2×D3—21no—
406C2×D3—21no—
407C2×D3—21no—
408C2×D3—21no—
409C2×D3—21no—
410C2×D3—21no—
411C2×D3—21no—
412C2×D3—21no—
413C2×D3—44no—
414C2×D3—61no—
415C2×D3—61no—
416C2×D3—61no—
417C2×D3—61no—
418C2×D3—61no—
419C2×D3—61no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—124no—
424C2×D3—124no—
425C2×D3D3×D412yes—
426C2×D3D3×D412yes—
427C2×D3#23712yescomplemented
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—61no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—61no—
438Dic3—21no—
439Dic3—21no—
440Dic3—21no—
441Dic3—21no—
442Dic3—32no—
443Dic3—32no—
444Dic3—62no—
445Dic3—61no—
446Dic3—61no—
447Dic3—61no—
448Dic3—61no—
449Dic3—121no—
450Dic3—121no—
451C32#54611yescharacteristic, complemented, Sylow
452C23—31no—
453C23—31no—
454C23—31no—
455C23—31no—
456C23—62no—
457C23—62no—
458C23—91no—
459C23—121no—
460C23—182no—
461C23—182no—
462C23—361no—
463C23C2×D3211yescharacteristic
464Q8—182no—
465D4—62no—
466D4—62no—
467D4—62no—
468D4—94no—
469D4—121no—
470D4—182no—
471D4—182no—
472D4—182no—
473D4—361no—
474C2×C4—31no—
475C2×C4—31no—
476C2×C4—31no—
477C2×C4—31no—
478C2×C4—62no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—182no—
483C2×C4—182no—
484C2×C4—182no—
485C2×C4—361no—
486C2×C4—361no—
487C2×C4C2×D3211yescharacteristic
488C2×C4C2×D3211yescharacteristic
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—62no—
495C2×C3—62no—
496C2×C3—62no—
497C2×C3—81no—
498C2×C3—121no—
499C2×C3C2×D3×D411yescharacteristic
500C2×C3C2×D3×D411yescharacteristic
501C2×C3C2×#23711yescharacteristic
502C2×C3#100411yescharacteristic
503C2×C3D3×#4111yescharacteristic
504C2×C3#100911yescharacteristic
505D3—22no—
506D3—22no—
507D3—22no—
508D3—41no—
509D3—121no—
510D3—121no—
511D3—241no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—32no—
516C22—32no—
517C22—32no—
518C22—61no—
519C22—61no—
520C22—61no—
521C22—61no—
522C22—61no—
523C22—61no—
524C22—61no—
525C22—61no—
526C22—61no—
527C22—124no—
528C22—181no—
529C22—181no—
530C22—181no—
531C22—364no—
532C22C22×D3211yescharacteristic
533C4—21no—
534C4—21no—
535C4—61no—
536C4—61no—
537C4—61no—
538C4—61no—
539C4—92no—
540C4—181no—
541C4—181no—
542C3—21no—
543C3#559111yescharacteristic, complemented
544C3#565411yescharacteristic, complemented
545C2—41no—
546C2—62no—
547C2—62no—
548C2—62no—
549C2—121no—
550C2—361no—
551C2#6430711yescharacteristic
552C2D32×D411yescharacteristic
553C2D3×#23711yescharacteristic
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. C2×D3×#67C2semidirect
2. C2×#3795C2semidirect
3. #64094C2semidirect
4. C3×#883C2semidirect
5. C3×#882C2semidirect
6. #63971C2semidirect
7. #63970C2semidirect
8. D3×#217C2semidirect
9. #63922C2semidirect
10. D3×(C2×D3)⋊C4C2semidirect
11. #63902C2semidirect
12. #63898C2semidirect
13. #63896C2semidirect
14. #63890C2semidirect
15. #63886C2semidirect
21. C2×#634C22semidirect
23. C2×C3×#67C22semidirect
25. C2×C3×D12C22semidirect
26. C2×C3×C4×D3C22semidirect
29. C2×#626C22semidirect
31. C2×#625C22semidirect
33. C2×D3×Dic3C22semidirect
34. C2×D3×Dic3C22semidirect
35. C2×D3×Dic3C22semidirect
36. C2×D3×Dic3C22semidirect
38. C32×C22⋊C4C22semidirect
39. #3757C22semidirect
40. #3752C22semidirect
41. #3752C22semidirect
42. C4×C3⋊Dic3C22semidirect
43. C3×#217C22semidirect
45. C3×(C2×D3)⋊C4C22semidirect
46. C3×(C2×D3)⋊C4C22semidirect
47. C3×C4⋊Dic3C22semidirect
48. C3×Dic3⋊C4C22semidirect
49. #3724C22semidirect
50. Dic3⋊Dic3C22semidirect
51. #3722C22semidirect
54. (C2×D3)⋊Dic3C22semidirect
78. #883D3semidirect
80. #882D3semidirect
117. #625D4semidirect
144. C2×#67C2×D3semidirect
156. C2×D12C2×D3semidirect
162. C2×C4×D3C2×D3semidirect
173. C3×C22⋊C4C2×D3semidirect
174. C3×C22⋊C4C2×D3semidirect
181. #217C2×D3semidirect
194. (C2×D3)⋊C4C2×D3semidirect
195. (C2×D3)⋊C4C2×D3semidirect
197. C4⋊Dic3C2×D3semidirect
201. Dic3⋊C4C2×D3semidirect
224. C2×C3×D3#41semidirect
377. C22⋊C4D32semidirect
427. C2×D3#237semidirect
451. C32#546semidirect
543. C3#5591semidirect
544. C3#5654semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#672
C2×#37952
#640942
C3×#8832
C3×#8822
#639712
#639702
D3×#2172
#639222
D3×(C2×D3)⋊C42
#639022
#638982
#638962
#638902
#638862
#56543
#55913