Finite Group Databasebeta

← #10595221

Subgroups — #10595221

Order 576.

565 conjugacy classes of subgroups.

The subgroup lattice has 514 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×D4×Dic3C211yesmaximal, characteristic, complemented
4#63981C211yesmaximal, characteristic, complemented
5#63978C211yesmaximal, characteristic, complemented
6Dic3×#67C211yesmaximal, characteristic, complemented
7#63971C211yesmaximal, characteristic, complemented
8#63966C211yesmaximal, characteristic, complemented
9#63964C211yesmaximal, characteristic, complemented
10C2×#3722C211yesmaximal, characteristic, complemented
11#63953C211yesmaximal, characteristic, complemented
12#63890C211yesmaximal, characteristic, complemented
13#63868C211yesmaximal, characteristic, complemented
14#63852C211yesmaximal, characteristic, complemented
15#63848C211yesmaximal, characteristic, complemented
16#5617—31nomaximal
17#5560—31nomaximal
18C23×C3⋊D3C2211yescharacteristic
19C2×C32×D4C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×#638C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic, complemented
24C22×C3×Dic3C2211yescharacteristic
25C22×C3×Dic3C2211yescharacteristic
26C22×C3×Dic3C2211yescharacteristic, complemented
27C2×#627C2211yescharacteristic, complemented
28C2×#626C2211yescharacteristic
29C2×#626C2211yescharacteristic
30C2×D3×Dic3C2211yescharacteristic, complemented
31#3757C2211yescharacteristic
32#3752C2211yescharacteristic
33#3752C2211yescharacteristic
34C3×#217C2211yescharacteristic
35C3×#217C2211yescharacteristic, complemented
36C3×#217C2211yescharacteristic, complemented
37C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
38C3×C4⋊Dic3C2211yescharacteristic, complemented
39C3×Dic3⋊C4C2211yescharacteristic, complemented
40C3×C4×Dic3C2211yescharacteristic, complemented
41#3724C2211yescharacteristic, complemented
42Dic3⋊Dic3C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic
49#3722C2211yescharacteristic
50#3722C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#121—31no—
54C3×C4×D4—31no—
55#939—31no—
56#939—31no—
57#939D311yescharacteristic, complemented
58#937—31no—
59#937—31no—
60#937—31no—
61#937—61no—
62D4×Dic3—31no—
63D4×Dic3D311yescharacteristic, complemented
64#930—31no—
65#929—31no—
66C4×#67—31no—
67C2×(C2×D3)⋊C4—31no—
68C2×(C2×D3)⋊C4—31no—
69#924—31no—
70#898—31no—
71#898—31no—
72#898—31no—
73#894—31no—
74#886—31no—
75#886—31no—
76#885—31no—
77#885—31no—
78#885—31no—
79#884—31no—
80#882—31no—
81#878—31no—
82#876—31no—
83#875—31no—
84C23×C32C2311yescharacteristic
85C23×C32C2311yescharacteristic
86C22×C3⋊D3—22no—
87C22×C3⋊D3—41no—
88C22×C3⋊D3—41no—
89C22×C3⋊D3C2311yescharacteristic
90C22×C3⋊D3C2311yescharacteristic
91C22×C3×D3C2311yescharacteristic
92C32×D4—22no—
93C2×C32×C4C2311yescharacteristic
94#638—41no—
95#638—41no—
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C3×#67—22no—
99C3×#67—41no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—41no—
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C2×C3×Dic3C2311yescharacteristic
109C2×C3×Dic3C2311yescharacteristic
110#627—41no—
111#626—41no—
112#626—41no—
113D3×Dic3—41no—
114#538—91noSylow
115C23×D3—31no—
116C23×D3—31no—
117C23×D3—61no—
118C2×C3×D4—21no—
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C2×C3×D4C2×D311yescharacteristic
122C2×C3×D4C2×D311yescharacteristic
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67—31no—
129C2×#67—31no—
130C2×#67—31no—
131C2×#67—31no—
132C2×#67—61no—
133C2×#67—61no—
134C2×#67C2×D311yescharacteristic
135C2×#67C2×D311yescharacteristic, complemented
136C22×Dic3—31no—
137C22×Dic3C2×D311yescharacteristic
138C22×Dic3C2×D311yescharacteristic
139C22×Dic3C2×D311yescharacteristic, complemented
140C2×D12—31no—
141C2×D12—31no—
142C2×C4×D3—31no—
143C2×Dic6—31no—
144C2×Dic6—31no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C22⋊C4—31no—
148C3×C22⋊C4—31no—
149C3×C22⋊C4—31no—
150C3×C22⋊C4—31no—
151C3×C42—31no—
152#217—31no—
153#217—31no—
154#217—31no—
155#217—61no—
156#217C2×D311yescharacteristic
157#217C2×D311yescharacteristic, complemented
158#217C2×D311yescharacteristic, complemented
159(C2×D3)⋊C4—31no—
160(C2×D3)⋊C4—31no—
161(C2×D3)⋊C4—31no—
162(C2×D3)⋊C4—31no—
163(C2×D3)⋊C4—31no—
164(C2×D3)⋊C4—31no—
165(C2×D3)⋊C4—31no—
166(C2×D3)⋊C4—31no—
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—61no—
177(C2×D3)⋊C4—61no—
178(C2×D3)⋊C4C2×D311yescharacteristic, complemented
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182C4⋊Dic3C2×D311yescharacteristic, complemented
183Dic3⋊C4—31no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4—31no—
187Dic3⋊C4—31no—
188Dic3⋊C4C2×D311yescharacteristic, complemented
189C4×Dic3—31no—
190C4×Dic3—31no—
191C4×Dic3C2×D311yescharacteristic, complemented
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32#4112yes—
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—44no—
206C2×C3×D3—21no—
207C2×C3×D3—21no—
208C2×C3×D3—21no—
209C32×C4—21no—
210C3⋊Dic3—21no—
211C3⋊Dic3—21no—
212C3×Dic3—21no—
213C3×Dic3—21no—
214C3×Dic3—21no—
215C3×Dic3—21no—
216C3×Dic3—21no—
217C3×Dic3—21no—
218C3×Dic3—21no—
219#126—91no—
220#126—91no—
221#124—91no—
222#124—91no—
223#123—91no—
224#123—91no—
225C22⋊Q8—91no—
226#121—31no—
227#121—91no—
228#121—91no—
229#120—91no—
230#120—91no—
231C4×D4—31no—
232C4×D4—91no—
233C2×C22⋊C4—91no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—31no—
237C23×C3C22×D311yescharacteristic
238C23×C3C22×D311yescharacteristic
239C23×C3C22×D311yescharacteristic
240C23×C3C22×D311yescharacteristic
241C22×D3—31no—
242C22×D3—31no—
243C22×D3—31no—
244C22×D3—31no—
245C22×D3—61no—
246C22×D3—61no—
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—122no—
250C22×D3—121no—
251C22×D3—121no—
252C22×D3—121no—
253C22×D3—121no—
254C22×D3—122no—
255C22×D3—122no—
256C22×D3C22×D311yescharacteristic
257C3×D4—22no—
258C3×D4—22no—
259C3×D4—42no—
260C3×D4—62no—
261C3×D4—121no—
262C2×C3×C4—21no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—121no—
271C2×C3×C4—121no—
272C2×C3×C4—121no—
273C2×C3×C4C22×D311yescharacteristic
274C2×C3×C4C22×D311yescharacteristic
275#67—22no—
276#67—41no—
277#67—121no—
278#67—121no—
279#67—121no—
280#67—121no—
281#67—121no—
282#67—121no—
283#67—122no—
284#67—122no—
285C2×Dic3—31no—
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—31no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—121no—
295C2×Dic3C22×D311yescharacteristic
296C2×Dic3C22×D311yescharacteristic
297C2×Dic3C22×D311yescharacteristic
298C2×Dic3C22×D311yescharacteristic
299C2×Dic3C22×D311yescharacteristic
300C2×Dic3C22×D311yescharacteristic
301C2×Dic3C22×D311yescharacteristic
302D12—121no—
303D12—121no—
304C4×D3—121no—
305Dic6—121no—
306Dic6—121no—
307C2×C32—22no—
308C2×C32—41no—
309C2×C32C2×#4111yescharacteristic
310C2×C32H(5,𝔽2)11yescharacteristic
311C2×C32H(5,𝔽2)11yescharacteristic
312C3⋊D3—41no—
313C3⋊D3—41no—
314C3×D3—41no—
315C24—91no—
316C2×Q8—91no—
317C2×D4—31no—
318C2×D4—31no—
319C2×D4—91no—
320C2×D4—91no—
321C2×D4—91no—
322C2×D4—91no—
323C2×D4D3211yescharacteristic
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—91no—
328C4⋊C4—31no—
329C4⋊C4—31no—
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—91no—
333C4⋊C4—91no—
334C22⋊C4—31no—
335C22⋊C4—31no—
336C22⋊C4—31no—
337C22⋊C4—31no—
338C22⋊C4—91no—
339C22⋊C4—91no—
340C22⋊C4—91no—
341C22⋊C4—91no—
342C22⋊C4—91no—
343C22⋊C4—91no—
344C22⋊C4—91no—
345C22⋊C4—91no—
346C22⋊C4—91no—
347C22⋊C4—91no—
348C42—31no—
349C42—91no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—22no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3#23512yes—
371C22×C3#23712yes—
372C22×C3C23×D311yescharacteristic
373C22×C3C23×D311yescharacteristic
374C2×D3—21no—
375C2×D3—21no—
376C2×D3—21no—
377C2×D3—62no—
378C2×D3—62no—
379C2×D3—61no—
380C2×D3—61no—
381C2×D3—61no—
382C2×D3—61no—
383C2×D3—61no—
384C2×D3—61no—
385C2×D3—61no—
386C2×D3—61no—
387C2×D3—61no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—61no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—124no—
396C2×D3—124no—
397C2×D3—128no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—41no—
401C3×C4—61no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—61no—
408Dic3—21no—
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413Dic3—21no—
414Dic3—21no—
415Dic3—61no—
416Dic3—61no—
417Dic3—61no—
418Dic3—61no—
419Dic3—121no—
420Dic3—121no—
421C32#53811yescharacteristic, complemented, Sylow
422C23—31no—
423C23—91no—
424C23—91no—
425C23—182no—
426C23—361no—
427C23—361no—
428C23C2×D3211yescharacteristic
429C23C2×D3211yescharacteristic
430Q8—361no—
431D4—22no—
432D4—62no—
433D4—121no—
434D4—361no—
435D4—361no—
436D4—361no—
437D4—361no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—361no—
451C2×C4C2×D3211yescharacteristic
452C2×C3—22no—
453C2×C3—22no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—42no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—81no—
461C2×C3—121no—
462C2×C3C2×#23511yescharacteristic
463C2×C3C2×#23711yescharacteristic
464C2×C3#100411yescharacteristic
465C2×C3#100411yescharacteristic
466C2×C3#100911yescharacteristic
467C2×C3#100911yescharacteristic
468D3—41no—
469D3—122no—
470D3—122no—
471D3—121no—
472D3—121no—
473D3—121no—
474D3—121no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—61no—
481C22—61no—
482C22—61no—
483C22—181no—
484C22—181no—
485C22—181no—
486C22—181no—
487C22—181no—
488C22—181no—
489C22—364no—
490C22#380412yes—
491C22C22×D3211yescharacteristic
492C4—21no—
493C4—61no—
494C4—61no—
495C4—61no—
496C4—61no—
497C4—61no—
498C4—61no—
499C4—61no—
500C4—181no—
501C4—181no—
502C3—21no—
503C3#556011yescharacteristic, complemented
504C3#561711yescharacteristic, complemented
505C2—22no—
506C2—41no—
507C2—121no—
508C2—361no—
509C2—361no—
510C2#6431411yescharacteristic
511C2#6431411yescharacteristic
512C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#538
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×D4×Dic3C2semidirect
4. #63981C2semidirect
5. #63978C2semidirect
6. Dic3×#67C2semidirect
7. #63971C2semidirect
8. #63966C2semidirect
9. #63964C2semidirect
10. C2×#3722C2semidirect
11. #63953C2semidirect
12. #63890C2semidirect
13. #63868C2semidirect
14. #63852C2semidirect
15. #63848C2semidirect
23. C2×C3×#67C22semidirect
26. C22×C3×Dic3C22semidirect
27. C2×#627C22semidirect
30. C2×D3×Dic3C22semidirect
35. C3×#217C22semidirect
36. C3×#217C22semidirect
37. C3×(C2×D3)⋊C4C22semidirect
38. C3×C4⋊Dic3C22semidirect
39. C3×Dic3⋊C4C22semidirect
40. C3×C4×Dic3C22semidirect
41. #3724C22semidirect
42. Dic3⋊Dic3C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. Dic32C22semidirect
57. #939D3semidirect
63. D4×Dic3D3semidirect
135. C2×#67C2×D3semidirect
139. C22×Dic3C2×D3semidirect
157. #217C2×D3semidirect
158. #217C2×D3semidirect
178. (C2×D3)⋊C4C2×D3semidirect
182. C4⋊Dic3C2×D3semidirect
188. Dic3⋊C4C2×D3semidirect
191. C4×Dic3C2×D3semidirect
421. C32#538semidirect
503. C3#5560semidirect
504. C3#5617semidirect

Maximal subgroups

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

subgroupindex
#641462
C3×#9392
C3×D4×Dic32
#639812
#639782
Dic3×#672
#639712
#639662
#639642
C2×#37222
#639532
#638902
#638682
#638522
#638482
#56173
#55603