Finite Group Databasebeta

← #10594117

Subgroups — #10594117

Order 576.

533 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64087C211yesmaximal, characteristic, complemented
2C3×#882C211yesmaximal, characteristic
3C3×#877C211yesmaximal, characteristic, complemented
4#67⋊Dic3C211yesmaximal, characteristic, complemented
5D3×#217C211yesmaximal, characteristic, complemented
6#63961C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
8#63896C211yesmaximal, characteristic, complemented
9Dic3×D12C211yesmaximal, characteristic, complemented
10#63884C211yesmaximal, characteristic, complemented
11#63884C211yesmaximal, characteristic, complemented
12#63877C211yesmaximal, characteristic, complemented
13#63861C211yesmaximal, characteristic, complemented
14#63860C211yesmaximal, characteristic, complemented
15#63859C211yesmaximal, characteristic, complemented
16#5587—31nomaximal
17#5527—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic
20C2×#634C2211yescharacteristic, complemented
21C2×C3×#67C2211yescharacteristic
22C22×C3×Dic3C2211yescharacteristic
23C2×C3×D12C2211yescharacteristic
24C2×C3×D12C2211yescharacteristic
25C2×D3×Dic3C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic, complemented
28C32×C22⋊C4C2211yescharacteristic
29#3757C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
31#3750C2211yescharacteristic, complemented
32#3750C2211yescharacteristic, complemented
33C3×#217C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic, complemented
37C3×Dic3⋊C4C2211yescharacteristic, complemented
38C3×C4×Dic3C2211yescharacteristic, complemented
39C3×C4×Dic3C2211yescharacteristic, complemented
40#3724C2211yescharacteristic, complemented
41#3724C2211yescharacteristic, complemented
42Dic3⋊Dic3C2211yescharacteristic, complemented
43(C2×D3)⋊Dic3C2211yescharacteristic
44(C2×D3)⋊Dic3C2211yescharacteristic
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×#120—31no—
54C3×#117—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#929—31no—
64C2×(C2×D3)⋊C4—31no—
65#898—31no—
66#896—31no—
67#894—31no—
68#894—31no—
69#882D311yescharacteristic
70#67⋊C4—31no—
71D3×C22⋊C4—31no—
72#879—31no—
73#879—31no—
74#878—31no—
75#878—31no—
76#878—31no—
77#878—61no—
78#877—31no—
79#877D311yescharacteristic, complemented
80#876—31no—
81#875—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⋊Dic3—41no—
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98#634—41no—
99C3×#67—22no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C3×D12—22no—
107C3×D12—41no—
108D3×Dic3—22no—
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—
117C22×C3×C4—31no—
118C2×#67C2×D311yescharacteristic
119C22×Dic3—31no—
120C22×Dic3—31no—
121C22×Dic3—31no—
122C22×Dic3—31no—
123C22×Dic3—31no—
124C22×Dic3—61no—
125C22×Dic3C2×D311yescharacteristic
126C2×D12C2×D311yescharacteristic
127C2×D12C2×D311yescharacteristic
128C2×C4×D3—31no—
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×Dic6—31no—
132C2×Dic6—31no—
133C2×Dic6—61no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C22⋊C4—21no—
137C3×C22⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4—31no—
140C3×C22⋊C4C2×D311yescharacteristic
141C3×C22⋊C4C2×D311yescharacteristic
142C3×C42—31no—
143C3×C42—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—31no—
155#217—61no—
156#217C2×D311yescharacteristic
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)⋊C4C2×D311yescharacteristic
167(C2×D3)⋊C4C2×D311yescharacteristic
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—61no—
172Dic3⋊C4—31no—
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4—31no—
177Dic3⋊C4—31no—
178Dic3⋊C4—31no—
179Dic3⋊C4—31no—
180Dic3⋊C4—31no—
181Dic3⋊C4—61no—
182Dic3⋊C4—61no—
183Dic3⋊C4C2×D311yescharacteristic, complemented
184Dic3⋊C4C2×D311yescharacteristic, complemented
185C4×Dic3—31no—
186C4×Dic3—31no—
187C4×Dic3C2×D311yescharacteristic, complemented
188C4×Dic3C2×D311yescharacteristic, complemented
189C22×C32—21no—
190C22×C32—21no—
191C22×C32#4112yes—
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—21no—
201C2×C3×D3—44no—
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—91no—
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—31no—
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×C4—62no—
255C2×C3×C4C22×D311yescharacteristic
256C2×C3×C4C22×D311yescharacteristic
257C2×C3×C4C22×D311yescharacteristic
258C2×C3×C4C22×D311yescharacteristic
259#67—22no—
260C2×Dic3—22no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—62no—
274C2×Dic3—121no—
275C2×Dic3—121no—
276C2×Dic3—121no—
277C2×Dic3—121no—
278C2×Dic3—122no—
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282C2×Dic3C22×D311yescharacteristic
283C2×Dic3C22×D311yescharacteristic
284D12—22no—
285D12—41no—
286C4×D3—62no—
287C4×D3—121no—
288C4×D3—121no—
289Dic6—121no—
290Dic6—121no—
291Dic6—122no—
292C2×C32—22no—
293C2×C32C2×#4111yescharacteristic
294C2×C32C2×#4111yescharacteristic
295C2×C32H(5,𝔽2)11yescharacteristic
296C3×D3—22no—
297C3×D3—41no—
298C3×D3—41no—
299C24—31no—
300C2×Q8—91no—
301C2×D4—31no—
302C2×D4—31no—
303C2×D4—31no—
304C22×C4—31no—
305C22×C4—91no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—91no—
309C4⋊C4—31no—
310C4⋊C4—31no—
311C4⋊C4—91no—
312C4⋊C4—91no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C22⋊C4—31no—
318C22⋊C4—31no—
319C22⋊C4—31no—
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⋊C4—91no—
329C22⋊C4—91no—
330C22⋊C4D3211yescharacteristic
331C42—31no—
332C42—31no—
333C42—91no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—32no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—61no—
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×C3#23512yes—
353C22×C3#23712yes—
354C22×C3C23×D311yescharacteristic
355C22×C3C23×D311yescharacteristic
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—21no—
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—44no—
365C2×D3#23712yes—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—41no—
371C3×C4—41no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—61no—
377Dic3—21no—
378Dic3—21no—
379Dic3—21no—
380Dic3—21no—
381Dic3—21no—
382Dic3—62no—
383Dic3—61no—
384Dic3—61no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—61no—
390Dic3—61no—
391Dic3—121no—
392Dic3—121no—
393Dic3—121no—
394C32#55111yescharacteristic, complemented, Sylow
395C23—31no—
396C23—31no—
397C23—31no—
398C23—62no—
399C23—62no—
400C23—121no—
401C23C2×D3211yescharacteristic
402Q8—361no—
403D4—62no—
404D4—62no—
405D4—121no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—62no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—182no—
417C2×C4—361no—
418C2×C4—361no—
419C2×C4—361no—
420C2×C4C2×D3211yescharacteristic
421C2×C4C2×D3211yescharacteristic
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—42no—
428C2×C3—62no—
429C2×C3—121no—
430C2×C3—121no—
431C2×C3C2×#23511yescharacteristic
432C2×C3C2×#23711yescharacteristic
433C2×C3C2×#23711yescharacteristic
434C2×C3#100411yescharacteristic
435C2×C3D3×#4111yescharacteristic
436C2×C3#100911yescharacteristic
437D3—22no—
438D3—41no—
439D3—41no—
440C22—21no—
441C22—21no—
442C22—32no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—61no—
449C22—61no—
450C22—61no—
451C22—124no—
452C22#379512yes—
453C22C22×D3211yescharacteristic
454C4—21no—
455C4—21no—
456C4—61no—
457C4—61no—
458C4—61no—
459C4—61no—
460C4—61no—
461C4—181no—
462C4—181no—
463C4—181no—
464C4—181no—
465C3—21no—
466C3#552711yescharacteristic, complemented
467C3#558711yescharacteristic, complemented
468C2—22no—
469C2—62no—
470C2—121no—
471C2—121no—
472C2C2×#379511yescharacteristic
473C2D3×#23711yescharacteristic
474C2#6431411yescharacteristic
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. #64087C2semidirect
3. C3×#877C2semidirect
4. #67⋊Dic3C2semidirect
5. D3×#217C2semidirect
6. #63961C2semidirect
7. C2×(C2×D3)⋊Dic3C2semidirect
8. #63896C2semidirect
9. Dic3×D12C2semidirect
10. #63884C2semidirect
11. #63884C2semidirect
12. #63877C2semidirect
13. #63861C2semidirect
14. #63860C2semidirect
15. #63859C2semidirect
20. C2×#634C22semidirect
27. C2×D3×Dic3C22semidirect
30. C4⋊(C3⋊Dic3)C22semidirect
31. #3750C22semidirect
32. #3750C22semidirect
36. C3×Dic3⋊C4C22semidirect
37. C3×Dic3⋊C4C22semidirect
38. C3×C4×Dic3C22semidirect
39. C3×C4×Dic3C22semidirect
40. #3724C22semidirect
41. #3724C22semidirect
42. Dic3⋊Dic3C22semidirect
49. (C2×D3)⋊Dic3C22semidirect
50. (C2×D3)⋊Dic3C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. Dic32C22semidirect
79. #877D3semidirect
183. Dic3⋊C4C2×D3semidirect
184. Dic3⋊C4C2×D3semidirect
187. C4×Dic3C2×D3semidirect
188. C4×Dic3C2×D3semidirect
394. C32#551semidirect
466. C3#5527semidirect
467. C3#5587semidirect

Maximal subgroups

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

subgroupindex
#640872
C3×#8822
C3×#8772
#67⋊Dic32
D3×#2172
#639612
C2×(C2×D3)⋊Dic32
#638962
Dic3×D122
#638842
#638772
#638612
#638602
#638592
#55873
#55273