Finite Group Databasebeta

← #10594842

Subgroups — #10594842

Order 576.

483 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#638C211yesmaximal, characteristic, complemented
2C3×C4×#67C211yesmaximal, characteristic, complemented
3C2×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
4#63978C211yesmaximal, characteristic
5#63966C211yesmaximal, characteristic, complemented
6#63964C211yesmaximal, characteristic, complemented
7#63961C211yesmaximal, characteristic, complemented
8#63870C211yesmaximal, characteristic, complemented
9#63868C211yesmaximal, characteristic, complemented
10#63867C211yesmaximal, characteristic, complemented
11#63865C211yesmaximal, characteristic, complemented
12#63862C211yesmaximal, characteristic, complemented
13#63853C211yesmaximal, characteristic, complemented
14#63851C211yesmaximal, characteristic, complemented
15#63850C211yesmaximal, characteristic, complemented
16#5564—31nomaximal
17#5515—31nomaximal
18C22×C32×C4C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×C4×C3⋊D3C2211yescharacteristic
21C2×C3×#67C2211yescharacteristic
22C22×C3×Dic3C2211yescharacteristic
23C22×C3×Dic3C2211yescharacteristic
24C2×C3×C4×D3C2211yescharacteristic
25C2×#627C2211yescharacteristic
26C2×#626C2211yescharacteristic
27#3757C2211yescharacteristic
28#3752C2211yescharacteristic
29#3750C2211yescharacteristic
30C4×C3⋊Dic3C2211yescharacteristic
31C3×#217C2211yescharacteristic
32C3×(C2×D3)⋊C4C2211yescharacteristic
33C3×Dic3⋊C4C2211yescharacteristic
34C3×Dic3⋊C4C2211yescharacteristic
35C3×Dic3⋊C4C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×C4×Dic3C2211yescharacteristic
39#3724C2211yescharacteristic
40#3724C2211yescharacteristic
41#3724C2211yescharacteristic
42Dic3⋊Dic3C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic
49#3722C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52(C2×D3)⋊Dic3C2211yescharacteristic
53C3×C4×D4—31no—
54C2×C3×C4⋊C4—31no—
55#939—31no—
56#935—31no—
57#930—31no—
58#929—31no—
59#929—31no—
60#929—31no—
61C4×#67—31no—
62C4×#67—31no—
63C4×#67—61no—
64C4×#67D311yescharacteristic, complemented
65#924—31no—
66#924—31no—
67C2×Dic3⋊C4D311yescharacteristic, complemented
68#898—31no—
69#898—31no—
70#896—31no—
71#896—31no—
72#896—31no—
73#894—31no—
74#894—31no—
75#889—31no—
76#889—31no—
77#889—31no—
78#886—31no—
79#883—31no—
80#879—31no—
81#878—31no—
82#876—31no—
83#870—31no—
84C23×C32C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C2×C32×C4—41no—
88C2×C32×C4C2311yescharacteristic
89C2×C32×C4C2311yescharacteristic
90#638—22no—
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C4×C3⋊D3—41no—
95C3×#67—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C3×C4×D3—41no—
106#627—41no—
107#626—41no—
108#539—91noSylow
109C2×C3×D4—31no—
110C22×C3×C4—21no—
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4C2×D311yescharacteristic
115C22×C3×C4C2×D311yescharacteristic
116C2×#67—31no—
117C2×#67—31no—
118C2×#67—31no—
119C2×#67—61no—
120C2×#67C2×D311yescharacteristic
121C22×Dic3C2×D311yescharacteristic
122C22×Dic3C2×D311yescharacteristic
123C2×D12—31no—
124C2×C4×D3—31no—
125C2×C4×D3—31no—
126C2×C4×D3—61no—
127C2×C4×D3C2×D311yescharacteristic
128C2×Dic6—31no—
129C2×Dic6—31no—
130C3×C4⋊C4—31no—
131C3×C4⋊C4—31no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—31no—
135C3×C22⋊C4—31no—
136C3×C22⋊C4—31no—
137C3×C42—31no—
138#217—31no—
139#217—31no—
140#217—31no—
141#217—31no—
142#217—31no—
143#217—61no—
144#217C2×D311yescharacteristic
145(C2×D3)⋊C4—31no—
146(C2×D3)⋊C4—31no—
147(C2×D3)⋊C4—31no—
148(C2×D3)⋊C4—31no—
149(C2×D3)⋊C4—31no—
150(C2×D3)⋊C4—31no—
151(C2×D3)⋊C4—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—61no—
157(C2×D3)⋊C4C2×D311yescharacteristic
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4⋊Dic3—31no—
163Dic3⋊C4—31no—
164Dic3⋊C4—31no—
165Dic3⋊C4—31no—
166Dic3⋊C4—31no—
167Dic3⋊C4—31no—
168Dic3⋊C4—31no—
169Dic3⋊C4—31no—
170Dic3⋊C4—31no—
171Dic3⋊C4—31no—
172Dic3⋊C4—31no—
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4—61no—
177Dic3⋊C4C2×D311yescharacteristic
178Dic3⋊C4C2×D311yescharacteristic
179Dic3⋊C4C2×D311yescharacteristic
180Dic3⋊C4C2×D311yescharacteristic
181Dic3⋊C4C2×D311yescharacteristic
182C4×Dic3—31no—
183C4×Dic3—31no—
184C4×Dic3—61no—
185C4×Dic3C2×D311yescharacteristic
186C22×C32—21no—
187C22×C32—21no—
188C22×C32#4112yes—
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—21no—
196C32×C4—21no—
197C32×C4—21no—
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3⋊Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—21no—
203C3×Dic3—21no—
204C3×Dic3—21no—
205C3×Dic3—21no—
206C3×Dic3—21no—
207C3×Dic3—21no—
208#126—91no—
209#126—91no—
210#125—91no—
211#125—91no—
212#123—91no—
213#123—91no—
214#123—91no—
215#123—91no—
216C22⋊Q8—91no—
217C22⋊Q8—91no—
218C22⋊Q8—91no—
219#121—91no—
220C4×D4—31no—
221C4×D4—91no—
222C2×C4⋊C4—31no—
223C23×C3—21no—
224C23×C3—31no—
225C23×C3C22×D311yescharacteristic
226C23×C3C22×D311yescharacteristic
227C22×D3—31no—
228C22×D3—31no—
229C22×D3—61no—
230C22×D3C22×D311yescharacteristic
231C3×D4—62no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—62no—
245C2×C3×C4—121no—
246C2×C3×C4—121no—
247C2×C3×C4C22×D311yescharacteristic
248C2×C3×C4C22×D311yescharacteristic
249C2×C3×C4C22×D311yescharacteristic
250C2×C3×C4C22×D311yescharacteristic
251#67—22no—
252#67—64no—
253#67—62no—
254#67—62no—
255#67—121no—
256C2×Dic3—22no—
257C2×Dic3—31no—
258C2×Dic3—31no—
259C2×Dic3—31no—
260C2×Dic3—31no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—41no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3C22×D311yescharacteristic
268C2×Dic3C22×D311yescharacteristic
269C2×Dic3C22×D311yescharacteristic
270C2×Dic3C22×D311yescharacteristic
271C2×Dic3C22×D311yescharacteristic
272C2×Dic3C22×D311yescharacteristic
273C2×Dic3C22×D311yescharacteristic
274D12—121no—
275C4×D3—41no—
276C4×D3—121no—
277C4×D3—121no—
278C4×D3—122no—
279Dic6—121no—
280Dic6—121no—
281C2×C32—22no—
282C2×C32C2×#4111yescharacteristic
283C2×C32H(5,𝔽2)11yescharacteristic
284C2×C322−1+411yescharacteristic
285C3⋊D3—41no—
286C3×D3—41no—
287C2×Q8—91no—
288C2×D4—31no—
289C2×D4—91no—
290C2×D4—91no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—91no—
295C22×C4D3211yescharacteristic
296C4⋊C4—31no—
297C4⋊C4—31no—
298C4⋊C4—31no—
299C4⋊C4—31no—
300C4⋊C4—31no—
301C4⋊C4—91no—
302C4⋊C4—91no—
303C4⋊C4—91no—
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4—91no—
308C4⋊C4—91no—
309C4⋊C4—91no—
310C22⋊C4—31no—
311C22⋊C4—31no—
312C22⋊C4—91no—
313C22⋊C4—91no—
314C22⋊C4—91no—
315C22⋊C4—91no—
316C22⋊C4—91no—
317C22⋊C4—91no—
318C22⋊C4—91no—
319C22⋊C4—91no—
320C42—31no—
321C42—91no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—22no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—61no—
331C22×C3—61no—
332C22×C3—61no—
333C22×C3#23512yes—
334C22×C3#23512yes—
335C22×C3C23×D311yescharacteristic
336C22×C3C23×D311yescharacteristic
337C2×D3—21no—
338C2×D3—21no—
339C2×D3—21no—
340C2×D3—62no—
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—121no—
348C2×D3—121no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—41no—
355C3×C4—61no—
356C3×C4—61no—
357C3×C4—61no—
358C3×C4—61no—
359C3×C4—61no—
360C3×C4—61no—
361C3×C4—61no—
362Dic3—21no—
363Dic3—21no—
364Dic3—21no—
365Dic3—21no—
366Dic3—21no—
367Dic3—21no—
368Dic3—21no—
369Dic3—62no—
370Dic3—61no—
371Dic3—61no—
372Dic3—61no—
373Dic3—61no—
374Dic3—61no—
375Dic3—61no—
376Dic3—121no—
377Dic3—121no—
378C32#53911yescharacteristic, complemented, Sylow
379C23—31no—
380C23—91no—
381C23C2×D3211yescharacteristic
382Q8—361no—
383D4—62no—
384D4—182no—
385D4—361no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—41no—
394C2×C4—62no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—361no—
401C2×C4C2×D3211yescharacteristic
402C2×C4C2×D3211yescharacteristic
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—42no—
409C2×C3—121no—
410C2×C3C2×#23511yescharacteristic
411C2×C3C2×#23511yescharacteristic
412C2×C3#100411yescharacteristic
413C2×C3#100411yescharacteristic
414C2×C3#100711yescharacteristic
415C2×C3#101011yescharacteristic
416D3—41no—
417D3—122no—
418D3—121no—
419D3—121no—
420C22—21no—
421C22—21no—
422C22—61no—
423C22—61no—
424C22—61no—
425C22—181no—
426C22—181no—
427C22—181no—
428C22#379812yes—
429C22C22×D3211yescharacteristic
430C4—21no—
431C4—21no—
432C4—61no—
433C4—61no—
434C4—61no—
435C4—61no—
436C4—61no—
437C4—61no—
438C4—61no—
439C4—181no—
440C4—181no—
441C4—181no—
442C3—21no—
443C3#551511yescharacteristic, complemented
444C3#556411yescharacteristic, complemented
445C2—22no—
446C2—121no—
447C2—361no—
448C2#6429711yescharacteristic
449C2C2×#379811yescharacteristic
450C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radical#63978C2
hypercenterC22×C4D32
hypocenterC32#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C4×#638C2semidirect
2. C3×C4×#67C2semidirect
3. C2×C3×Dic3⋊C4C2semidirect
5. #63966C2semidirect
6. #63964C2semidirect
7. #63961C2semidirect
8. #63870C2semidirect
9. #63868C2semidirect
10. #63867C2semidirect
11. #63865C2semidirect
12. #63862C2semidirect
13. #63853C2semidirect
14. #63851C2semidirect
15. #63850C2semidirect
64. C4×#67D3semidirect
67. C2×Dic3⋊C4D3semidirect
378. C32#539semidirect
443. C3#5515semidirect
444. C3#5564semidirect

Maximal subgroups

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

subgroupindex
C4×#6382
C3×C4×#672
C2×C3×Dic3⋊C42
#639782
#639662
#639642
#639612
#638702
#638682
#638672
#638652
#638622
#638532
#638512
#638502
#55643
#55153