Finite Group Databasebeta

← #10594017

Subgroups — #10594017

Order 576.

543 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#876C211yesmaximal, characteristic, complemented
3C3×#872C211yesmaximal, characteristic, complemented
4#63923C211yesmaximal, characteristic
5#63921C211yesmaximal, characteristic, complemented
6D3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7#63897C211yesmaximal, characteristic, complemented
8#63893C211yesmaximal, characteristic, complemented
9#63881C211yesmaximal, characteristic, complemented
10#63865C211yesmaximal, characteristic, complemented
11#63861C211yesmaximal, characteristic, complemented
12#63853C211yesmaximal, characteristic, complemented
13#63852C211yesmaximal, characteristic, complemented
14#63850C211yesmaximal, characteristic, complemented
15#63836C211yesmaximal, characteristic, complemented
16#5705—31nomaximal
17#5527—31nomaximal
18C22×D32C2211yescharacteristic
19C2×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C4×C3⋊D3C2211yescharacteristic
21C2×C4×C3⋊D3C2211yescharacteristic, complemented
22C2×C3×C4×D3C2211yescharacteristic
23C2×#627C2211yescharacteristic, complemented
24C2×#626C2211yescharacteristic
25C2×#626C2211yescharacteristic
26C2×#624C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic
28C32×C42C2211yescharacteristic, complemented
29#3752C2211yescharacteristic
30#3752C2211yescharacteristic, complemented
31C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
32C3×(C2×D3)⋊C4C2211yescharacteristic
33C3×(C2×D3)⋊C4C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic, complemented
38C3×Dic3⋊C4C2211yescharacteristic, complemented
39C3×Dic3⋊C4C2211yescharacteristic, complemented
40C3×Dic3⋊C4C2211yescharacteristic, complemented
41C3×Dic3⋊C4C2211yescharacteristic, complemented
42C3×C4×Dic3C2211yescharacteristic, complemented
43#3724C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47#3722C2211yescharacteristic, complemented
48#3722C2211yescharacteristic, complemented
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic, complemented
53C3×#126—31no—
54C3×#117—31no—
55#929—31no—
56C2×(C2×D3)⋊C4—31no—
57#898—31no—
58#897—31no—
59#896—31no—
60#896—31no—
61#894—31no—
62#894—31no—
63#894—31no—
64#894—31no—
65#893—31no—
66#892—31no—
67#885—31no—
68#883—31no—
69#882—31no—
70#882—31no—
71#67⋊C4—31no—
72D3×C22⋊C4—31no—
73D3×C22⋊C4—31no—
74D3×C22⋊C4—31no—
75#879—31no—
76#878—31no—
77#876D311yescharacteristic, complemented
78#875—31no—
79C4×D12—31no—
80C4×D12—31no—
81C4×D12—61no—
82#872—31no—
83#872D311yescharacteristic, complemented
84C22×C3⋊D3C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×D32—22no—
89C2×D32—22no—
90C2×D32—41no—
91C2×C32×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C4⋊(C3⋊D3)—22no—
97C4×C3⋊D3—41no—
98C4×C3⋊D3—41no—
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C3×C4×D3—22no—
106#627—41no—
107#626—22no—
108#626—41no—
109#624—22no—
110D3×Dic3—41no—
111#551—91noSylow
112C23×D3—31no—
113C23×D3—31no—
114C22×C3×C4—31no—
115C2×#67—31no—
116C2×#67—31no—
117C22×Dic3—31no—
118C2×D12—31no—
119C2×D12—31no—
120C2×D12—31no—
121C2×D12—31no—
122C2×D12—61no—
123C2×C4×D3—31no—
124C2×C4×D3—31no—
125C2×C4×D3—31no—
126C2×C4×D3—31no—
127C2×C4×D3—31no—
128C2×C4×D3—31no—
129C2×C4×D3—31no—
130C2×C4×D3—61no—
131C2×C4×D3—61no—
132C2×C4×D3C2×D311yescharacteristic
133C2×Dic6—31no—
134C2×Dic6—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C22⋊C4—31no—
141C3×C22⋊C4—31no—
142C3×C22⋊C4—31no—
143C3×C22⋊C4—31no—
144C3×C22⋊C4—31no—
145C3×C42—21no—
146C3×C42—31no—
147C3×C42C2×D311yescharacteristic, complemented
148C3×C42C2×D311yescharacteristic, complemented
149#217—31no—
150#217—31no—
151#217—31no—
152(C2×D3)⋊C4—31no—
153(C2×D3)⋊C4—31no—
154(C2×D3)⋊C4—31no—
155(C2×D3)⋊C4—31no—
156(C2×D3)⋊C4—31no—
157(C2×D3)⋊C4—31no—
158(C2×D3)⋊C4—31no—
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—61no—
168(C2×D3)⋊C4—61no—
169(C2×D3)⋊C4C2×D311yescharacteristic
170(C2×D3)⋊C4C2×D311yescharacteristic
171(C2×D3)⋊C4C2×D311yescharacteristic
172(C2×D3)⋊C4C2×D311yescharacteristic
173(C2×D3)⋊C4C2×D311yescharacteristic
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—61no—
178Dic3⋊C4—31no—
179Dic3⋊C4—31no—
180Dic3⋊C4—31no—
181Dic3⋊C4C2×D311yescharacteristic, complemented
182Dic3⋊C4C2×D311yescharacteristic, complemented
183Dic3⋊C4C2×D311yescharacteristic, complemented
184Dic3⋊C4C2×D311yescharacteristic, complemented
185Dic3⋊C4C2×D311yescharacteristic, complemented
186C4×Dic3—31no—
187C4×Dic3—31no—
188C4×Dic3C2×D311yescharacteristic, complemented
189C22×C32C2411yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3⋊D3#4112yes—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3#4112yes—
202D32—44no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3—21no—
208C3×Dic3—21no—
209C3×Dic3—21no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3—21no—
213C3×Dic3—21no—
214#126—31no—
215#126—91no—
216#124—91no—
217#123—91no—
218#123—91no—
219#123—91no—
220C22⋊Q8—91no—
221C22⋊Q8—91no—
222#120—91no—
223C4×D4—91no—
224C4×D4—91no—
225#117—31no—
226#117—91no—
227C2×C22⋊C4—91no—
228C2×C22⋊C4—91no—
229C23×C3—31no—
230C23×C3—31no—
231C22×D3—31no—
232C22×D3—31no—
233C22×D3—31no—
234C22×D3—31no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—62no—
240C22×D3—62no—
241C22×D3—121no—
242C22×D3—121no—
243C22×D3C22×D311yescharacteristic
244C22×D3C22×D311yescharacteristic
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—62no—
255C2×C3×C4C22×D311yescharacteristic
256C2×C3×C4C22×D311yescharacteristic
257C2×C3×C4C22×D311yescharacteristic
258C2×C3×C4C22×D311yescharacteristic
259C2×C3×C4C22×D311yescharacteristic
260C2×C3×C4C22×D311yescharacteristic
261#67—62no—
262#67—121no—
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—121no—
270C2×Dic3C22×D311yescharacteristic
271C2×Dic3C22×D311yescharacteristic
272C2×Dic3C22×D311yescharacteristic
273C2×Dic3C22×D311yescharacteristic
274C2×Dic3C22×D311yescharacteristic
275C2×Dic3C22×D311yescharacteristic
276D12—62no—
277D12—62no—
278D12—64no—
279D12—62no—
280D12—121no—
281C4×D3—22no—
282C4×D3—62no—
283C4×D3—62no—
284C4×D3—121no—
285C4×D3—121no—
286C4×D3—121no—
287C4×D3—121no—
288C4×D3—121no—
289C4×D3—122no—
290C4×D3—122no—
291Dic6—121no—
292Dic6—121no—
293C2×C32C2×#4111yescharacteristic
294C2×C32C2×#4111yescharacteristic
295C2×C32H(5,𝔽2)11yescharacteristic
296C3⋊D3—22no—
297C3⋊D3—41no—
298C3×D3—22no—
299C3×D3—41no—
300C24—91no—
301C2×Q8—91no—
302C2×D4—91no—
303C2×D4—91no—
304C2×D4—91no—
305C22×C4—31no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4—91no—
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—31no—
314C4⋊C4—31no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C4⋊C4—91no—
318C22⋊C4—31no—
319C22⋊C4—31no—
320C22⋊C4—31no—
321C22⋊C4—31no—
322C22⋊C4—31no—
323C22⋊C4—91no—
324C22⋊C4—91no—
325C22⋊C4—91no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4—91no—
329C22⋊C4—91no—
330C22⋊C4—91no—
331C22⋊C4—91no—
332C42—31no—
333C42—91no—
334C42D3211yescharacteristic, complemented
335C22×C3—21no—
336C22×C3—32no—
337C22×C3—61no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3C23×D311yescharacteristic
343C22×C3C23×D311yescharacteristic
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—32no—
350C2×D3—32no—
351C2×D3—62no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—61no—
355C2×D3—61no—
356C2×D3—62no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—121no—
364C2×D3—121no—
365C2×D3—121no—
366C2×D3—121no—
367C2×D3—124no—
368C2×D3—124no—
369C2×D3#23512yes—
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—41no—
379C3×C4—61no—
380C3×C4—61no—
381C3×C4—61no—
382C3×C4—61no—
383C3×C4—61no—
384C3×C4—61no—
385Dic3—21no—
386Dic3—21no—
387Dic3—21no—
388Dic3—21no—
389Dic3—21no—
390Dic3—21no—
391Dic3—61no—
392Dic3—61no—
393Dic3—61no—
394Dic3—61no—
395Dic3—121no—
396Dic3—121no—
397C32#55111yescharacteristic, complemented, Sylow
398C23—31no—
399C23—31no—
400C23—91no—
401C23—91no—
402C23—182no—
403C23—182no—
404C23—361no—
405Q8—361no—
406D4—182no—
407D4—182no—
408D4—361no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—62no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—182no—
419C2×C4—361no—
420C2×C4—361no—
421C2×C4—361no—
422C2×C4C2×D3211yescharacteristic
423C2×C4C2×D3211yescharacteristic
424C2×C4C2×D3211yescharacteristic
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—62no—
429C2×C3—121no—
430C2×C3C2×#23511yescharacteristic
431C2×C3D3×#4111yescharacteristic
432C2×C3D3×#4111yescharacteristic
433C2×C3D3×#4111yescharacteristic
434C2×C3#100911yescharacteristic
435C2×C3#100911yescharacteristic
436D3—22no—
437D3—41no—
438D3—62no—
439D3—62no—
440D3—122no—
441D3—122no—
442D3—121no—
443D3—121no—
444C22—32no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—61no—
449C22—61no—
450C22—92no—
451C22—181no—
452C22—181no—
453C22—181no—
454C22—181no—
455C22—181no—
456C22—364no—
457C22C22×D3211yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—61no—
462C4—61no—
463C4—61no—
464C4—61no—
465C4—61no—
466C4—61no—
467C4—181no—
468C4—181no—
469C3—21no—
470C3#552711yescharacteristic, complemented
471C3#570511yescharacteristic, complemented
472C2—62no—
473C2—121no—
474C2—182no—
475C2—361no—
476C2D3×#23511yescharacteristic
477C2#6430611yescharacteristic
478C2#6430811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radical#63923C2
hypercenterC42D32
hypocenterC32#551
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊D3)C2semidirect
2. C3×#876C2semidirect
3. C3×#872C2semidirect
5. #63921C2semidirect
6. D3×(C2×D3)⋊C4C2semidirect
7. #63897C2semidirect
8. #63893C2semidirect
9. #63881C2semidirect
10. #63865C2semidirect
11. #63861C2semidirect
12. #63853C2semidirect
13. #63852C2semidirect
14. #63850C2semidirect
15. #63836C2semidirect
21. C2×C4×C3⋊D3C22semidirect
23. C2×#627C22semidirect
28. C32×C42C22semidirect
30. #3752C22semidirect
31. C4⋊(C3⋊Dic3)C22semidirect
37. C3×Dic3⋊C4C22semidirect
38. C3×Dic3⋊C4C22semidirect
39. C3×Dic3⋊C4C22semidirect
40. C3×Dic3⋊C4C22semidirect
41. C3×Dic3⋊C4C22semidirect
42. C3×C4×Dic3C22semidirect
43. #3724C22semidirect
44. Dic3⋊Dic3C22semidirect
47. #3722C22semidirect
48. #3722C22semidirect
52. Dic32C22semidirect
77. #876D3semidirect
83. #872D3semidirect
147. C3×C42C2×D3semidirect
148. C3×C42C2×D3semidirect
181. Dic3⋊C4C2×D3semidirect
182. Dic3⋊C4C2×D3semidirect
183. Dic3⋊C4C2×D3semidirect
184. Dic3⋊C4C2×D3semidirect
185. Dic3⋊C4C2×D3semidirect
188. C4×Dic3C2×D3semidirect
334. C42D32semidirect
397. C32#551semidirect
470. C3#5527semidirect
471. C3#5705semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D3)2
C3×#8762
C3×#8722
#639232
#639212
D3×(C2×D3)⋊C42
#638972
#638932
#638812
#638652
#638612
#638532
#638522
#638502
#638362
#57053
#55273