Finite Group Databasebeta

← #10594115

Subgroups — #10594115

Order 576.

526 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64088C211yesmaximal, characteristic, complemented
2C3×#882C211yesmaximal, characteristic
3C3×#879C211yesmaximal, characteristic, complemented
4#63969C211yesmaximal, characteristic, complemented
5D3×#217C211yesmaximal, characteristic, complemented
6D3×#217C211yesmaximal, characteristic, complemented
7#63959C211yesmaximal, characteristic, complemented
8#63899C211yesmaximal, characteristic, complemented
9#63898C211yesmaximal, characteristic, complemented
10#63893C211yesmaximal, characteristic, complemented
11Dic3×D12C211yesmaximal, characteristic, complemented
12#63884C211yesmaximal, characteristic, complemented
13#63884C211yesmaximal, characteristic, complemented
14#63878C211yesmaximal, characteristic, complemented
15#63877C211yesmaximal, characteristic, complemented
16#5705—31nomaximal
17#5587—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C2×C3×#67C2211yescharacteristic
20C2×C3×D12C2211yescharacteristic
21C2×C3×D12C2211yescharacteristic
22C2×#627C2211yescharacteristic, complemented
23C2×D3×Dic3C2211yescharacteristic
24C2×D3×Dic3C2211yescharacteristic
25C2×D3×Dic3C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic, complemented
28C32×C22⋊C4C2211yescharacteristic
29#3757C2211yescharacteristic
30#3757C2211yescharacteristic
31C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
32#3750C2211yescharacteristic, complemented
33#3750C2211yescharacteristic, complemented
34C4×C3⋊Dic3C2211yescharacteristic, complemented
35C3×#217C2211yescharacteristic
36C3×#217C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×C4⋊Dic3C2211yescharacteristic, complemented
40C3×Dic3⋊C4C2211yescharacteristic, complemented
41C3×Dic3⋊C4C2211yescharacteristic, complemented
42C3×C4×Dic3C2211yescharacteristic, complemented
43#3724C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45(C2×D3)⋊Dic3C2211yescharacteristic
46(C2×D3)⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic, complemented
50(C2×D3)⋊Dic3C2211yescharacteristic, complemented
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#126—31no—
54C3×#120—31no—
55C2×#217—31no—
56C2×#217—31no—
57#936—31no—
58#935—31no—
59#935—31no—
60#935—31no—
61D4×Dic3—31no—
62D4×Dic3—31no—
63#897—31no—
64#896—31no—
65#894—31no—
66#894—31no—
67#893—31no—
68#892—31no—
69#885—31no—
70#883—31no—
71#882D311yescharacteristic
72D3×C22⋊C4—31no—
73D3×C22⋊C4—31no—
74#879—31no—
75#879—31no—
76#879—61no—
77#879D311yescharacteristic, complemented
78#878—31no—
79#878—31no—
80#877—31no—
81#877—31no—
82C4×D12—31no—
83#872—31no—
84C23×C32C2311yescharacteristic
85C22×C3×D3—22no—
86C22×C3×D3—22no—
87C22×C3×D3—41no—
88C22×C3×D3C2311yescharacteristic
89C22×C3×D3C2311yescharacteristic
90C22×C3×D3C2311yescharacteristic
91C2×C32×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C3×#67—41no—
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C3×D12—22no—
104C3×D12—22no—
105#627—41no—
106D3×Dic3—22no—
107D3×Dic3—22no—
108D3×Dic3—41no—
109D3×Dic3—41no—
110D3×Dic3—41no—
111#551—91noSylow
112C24×C3—31no—
113C23×D3C2×D311yescharacteristic
114C2×C3×D4—31no—
115C2×C3×D4—31no—
116C2×C3×D4—31no—
117C2×#67C2×D311yescharacteristic
118C22×Dic3—31no—
119C22×Dic3—31no—
120C22×Dic3—31no—
121C22×Dic3—31no—
122C22×Dic3—31no—
123C2×D12C2×D311yescharacteristic
124C2×D12C2×D311yescharacteristic
125C2×C4×D3—31no—
126C2×C4×D3—31no—
127C2×C4×D3—31no—
128C2×C4×D3—31no—
129C2×C4×D3—31no—
130C2×Dic6—31no—
131C2×Dic6—31no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—31no—
135C3×C22⋊C4—21no—
136C3×C22⋊C4—31no—
137C3×C22⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4—31no—
140C3×C22⋊C4C2×D311yescharacteristic
141C3×C22⋊C4C2×D311yescharacteristic
142C3×C42—31no—
143#217—31no—
144#217—31no—
145#217—31no—
146#217—31no—
147#217—31no—
148#217—31no—
149#217—31no—
150#217—31no—
151#217—31no—
152#217—31no—
153#217—31no—
154#217—61no—
155#217—61no—
156#217C2×D311yescharacteristic
157#217C2×D311yescharacteristic
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)⋊C4C2×D311yescharacteristic
166(C2×D3)⋊C4C2×D311yescharacteristic
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3C2×D311yescharacteristic, complemented
172Dic3⋊C4—31no—
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4—31no—
177Dic3⋊C4—31no—
178Dic3⋊C4—31no—
179Dic3⋊C4—61no—
180Dic3⋊C4—61no—
181Dic3⋊C4C2×D311yescharacteristic, complemented
182Dic3⋊C4C2×D311yescharacteristic, complemented
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—31no—
187C4×Dic3—61no—
188C4×Dic3C2×D311yescharacteristic, complemented
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32C2411yescharacteristic
193C2×C3×D3—21no—
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—44no—
201C2×C3×D3#4112yes—
202C2×C3×D3#4112yes—
203C32×C4—21no—
204C32×C4—21no—
205C3⋊Dic3—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—31no—
223C4×D4—91no—
224C4×D4—91no—
225#117—91no—
226#117—91no—
227C2×C22⋊C4—91no—
228C2×C22⋊C4—91no—
229C23×C3—21no—
230C23×C3—31no—
231C23×C3—31no—
232C23×C3—31no—
233C23×C3—62no—
234C23×C3—62no—
235C23×C3—121no—
236C23×C3C22×D311yescharacteristic
237C23×C3C22×D311yescharacteristic
238C22×D3—22no—
239C22×D3—22no—
240C22×D3—41no—
241C22×D3C22×D311yescharacteristic
242C22×D3C22×D311yescharacteristic
243C22×D3C22×D311yescharacteristic
244C3×D4—62no—
245C3×D4—62no—
246C3×D4—121no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4C22×D311yescharacteristic
255C2×C3×C4C22×D311yescharacteristic
256C2×C3×C4C22×D311yescharacteristic
257C2×C3×C4C22×D311yescharacteristic
258#67—41no—
259C2×Dic3—31no—
260C2×Dic3—31no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—62no—
272C2×Dic3—62no—
273C2×Dic3—121no—
274C2×Dic3—121no—
275C2×Dic3—121no—
276C2×Dic3C22×D311yescharacteristic
277C2×Dic3C22×D311yescharacteristic
278C2×Dic3C22×D311yescharacteristic
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281D12—22no—
282D12—22no—
283C4×D3—62no—
284C4×D3—62no—
285C4×D3—121no—
286C4×D3—121no—
287C4×D3—121no—
288Dic6—121no—
289Dic6—121no—
290C2×C32—41no—
291C2×C32C2×#4111yescharacteristic
292C2×C32C2×#4111yescharacteristic
293C2×C32H(5,𝔽2)11yescharacteristic
294C3×D3—22no—
295C3×D3—22no—
296C3×D3—41no—
297C24—31no—
298C2×Q8—91no—
299C2×D4—31no—
300C2×D4—31no—
301C2×D4—31no—
302C22×C4—91no—
303C22×C4—91no—
304C22×C4—91no—
305C22×C4—91no—
306C22×C4—91no—
307C4⋊C4—31no—
308C4⋊C4—31no—
309C4⋊C4—31no—
310C4⋊C4—91no—
311C4⋊C4—91no—
312C4⋊C4—91no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C22⋊C4—31no—
316C22⋊C4—31no—
317C22⋊C4—31no—
318C22⋊C4—31no—
319C22⋊C4—91no—
320C22⋊C4—91no—
321C22⋊C4—91no—
322C22⋊C4—91no—
323C22⋊C4—91no—
324C22⋊C4—91no—
325C22⋊C4—91no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4D3211yescharacteristic
329C42—31no—
330C42—91no—
331C42—91no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—32no—
340C22×C3—32no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3—124no—
352C22×C3C23×D311yescharacteristic
353C22×C3C23×D311yescharacteristic
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—21no—
360C2×D3—21no—
361C2×D3—44no—
362C2×D3#23712yes—
363C2×D3#23712yes—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—41no—
369C3×C4—41no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—61no—
375Dic3—21no—
376Dic3—21no—
377Dic3—21no—
378Dic3—21no—
379Dic3—21no—
380Dic3—61no—
381Dic3—61no—
382Dic3—61no—
383Dic3—61no—
384Dic3—61no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—121no—
389Dic3—121no—
390Dic3—121no—
391Dic3—121no—
392C32#55111yescharacteristic, complemented, Sylow
393C23—31no—
394C23—31no—
395C23—31no—
396C23—62no—
397C23—62no—
398C23—121no—
399C23C2×D3211yescharacteristic
400Q8—361no—
401D4—62no—
402D4—62no—
403D4—121no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—182no—
414C2×C4—182no—
415C2×C4—361no—
416C2×C4—361no—
417C2×C4—361no—
418C2×C4C2×D3211yescharacteristic
419C2×C4C2×D3211yescharacteristic
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—62no—
426C2×C3—62no—
427C2×C3—81no—
428C2×C3—121no—
429C2×C3C2×#23711yescharacteristic
430C2×C3C2×#23711yescharacteristic
431C2×C3#100411yescharacteristic
432C2×C3D3×#4111yescharacteristic
433C2×C3D3×#4111yescharacteristic
434C2×C3#100911yescharacteristic
435D3—22no—
436D3—22no—
437D3—41no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—32no—
442C22—32no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—61no—
449C22—61no—
450C22—124no—
451C22C22×D3211yescharacteristic
452C4—21no—
453C4—21no—
454C4—61no—
455C4—61no—
456C4—61no—
457C4—61no—
458C4—61no—
459C4—181no—
460C4—181no—
461C4—181no—
462C4—181no—
463C3—21no—
464C3#558711yescharacteristic, complemented
465C3#570511yescharacteristic, complemented
466C2—41no—
467C2—62no—
468C2—62no—
469C2—121no—
470C2#6430711yescharacteristic
471C2D3×#23711yescharacteristic
472C2D3×#23711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC3×#882C2
hypercenterC22⋊C4D32
hypocenterC32#551
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64088C2semidirect
3. C3×#879C2semidirect
4. #63969C2semidirect
5. D3×#217C2semidirect
6. D3×#217C2semidirect
7. #63959C2semidirect
8. #63899C2semidirect
9. #63898C2semidirect
10. #63893C2semidirect
11. Dic3×D12C2semidirect
12. #63884C2semidirect
13. #63884C2semidirect
14. #63878C2semidirect
15. #63877C2semidirect
22. C2×#627C22semidirect
27. C2×D3×Dic3C22semidirect
31. C4⋊(C3⋊Dic3)C22semidirect
32. #3750C22semidirect
33. #3750C22semidirect
34. C4×C3⋊Dic3C22semidirect
39. C3×C4⋊Dic3C22semidirect
40. C3×Dic3⋊C4C22semidirect
41. C3×Dic3⋊C4C22semidirect
42. C3×C4×Dic3C22semidirect
43. #3724C22semidirect
44. Dic3⋊Dic3C22semidirect
49. (C2×D3)⋊Dic3C22semidirect
50. (C2×D3)⋊Dic3C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. Dic32C22semidirect
77. #879D3semidirect
171. C4⋊Dic3C2×D3semidirect
181. Dic3⋊C4C2×D3semidirect
182. Dic3⋊C4C2×D3semidirect
188. C4×Dic3C2×D3semidirect
392. C32#551semidirect
464. C3#5587semidirect
465. C3#5705semidirect

Maximal subgroups

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

subgroupindex
#640882
C3×#8822
C3×#8792
#639692
D3×#2172
#639592
#638992
#638982
#638932
Dic3×D122
#638842
#638782
#638772
#57053
#55873