Finite Group Databasebeta

← #10594140

Subgroups — #10594140

Order 576.

471 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64092C211yesmaximal, characteristic, complemented
2C3×#885C211yesmaximal, characteristic, complemented
3C3×#878C211yesmaximal, characteristic, complemented
4#63978C211yesmaximal, characteristic
5#63969C211yesmaximal, characteristic, complemented
6#63955C211yesmaximal, characteristic, complemented
7#63953C211yesmaximal, characteristic, complemented
8#63907C211yesmaximal, characteristic, complemented
9#63893C211yesmaximal, characteristic, complemented
10#63881C211yesmaximal, characteristic, complemented
11#63871C211yesmaximal, characteristic, complemented
12#63868C211yesmaximal, characteristic, complemented
13#63858C211yesmaximal, characteristic, complemented
14Dic3×Dic6C211yesmaximal, characteristic, complemented
15#63839C211yesmaximal, characteristic, complemented
16#5669—31nomaximal
17#5643—31nomaximal
18C2×#638C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C22×C3×Dic3C2211yescharacteristic
22C2×C3×Dic6C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic
24C2×#627C2211yescharacteristic
25C2×#626C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C32×C22⋊C4C2211yescharacteristic
28#3757C2211yescharacteristic
29#3752C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2211yescharacteristic
31#3750C2211yescharacteristic
32C3×#217C2211yescharacteristic
33C3×#217C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×C4⋊Dic3C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×Dic3⋊C4C2211yescharacteristic
39C3×C4×Dic3C2211yescharacteristic
40#3724C2211yescharacteristic
41Dic3⋊Dic3C2211yescharacteristic
42Dic3⋊Dic3C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47#3722C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50Dic32C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#124—31no—
54C3×C22⋊Q8—31no—
55#947—31no—
56#946—31no—
57Q8×Dic3—31no—
58#944—31no—
59#939—31no—
60#935—31no—
61#930—31no—
62#926—31no—
63#898—31no—
64#898—31no—
65#897—31no—
66#897—31no—
67#893—31no—
68#888—31no—
69#885—31no—
70#885—31no—
71#885—31no—
72#885D311yescharacteristic, complemented
73#883—31no—
74#883—31no—
75#883—31no—
76#883—61no—
77#67⋊C4—31no—
78#879—31no—
79#878—31no—
80#878D311yescharacteristic, complemented
81#877—31no—
82#876—31no—
83C4×Dic6—31no—
84C23×C32C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89#638—41no—
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C4×C3⋊D3—41no—
94C3×#67—41no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C3×Dic6—22no—
104C3×Dic6—41no—
105#627—41no—
106#626—22no—
107D3×Dic3—41no—
108#542—91noSylow
109C2×C3×Q8—31no—
110C2×C3×Q8—31no—
111C2×C3×D4—31no—
112C22×C3×C4—31no—
113C2×#67—31no—
114C2×#67—31no—
115C2×#67—31no—
116C2×#67—61no—
117C2×#67C2×D311yescharacteristic
118C22×Dic3—31no—
119C22×Dic3C2×D311yescharacteristic
120C2×D12—31no—
121C2×C4×D3—31no—
122C2×C4×D3—31no—
123C2×C4×D3—31no—
124C2×C4×D3—61no—
125C2×Dic6—31no—
126C2×Dic6—31no—
127C2×Dic6C2×D311yescharacteristic
128C2×Dic6C2×D311yescharacteristic
129C3×C4⋊C4—31no—
130C3×C4⋊C4—31no—
131C3×C4⋊C4—31no—
132C3×C22⋊C4—21no—
133C3×C22⋊C4—31no—
134C3×C22⋊C4—31no—
135C3×C22⋊C4—31no—
136C3×C22⋊C4—31no—
137C3×C22⋊C4C2×D311yescharacteristic
138C3×C22⋊C4C2×D311yescharacteristic
139C3×C42—31no—
140#217—31no—
141#217—31no—
142#217—31no—
143#217—31no—
144#217—61no—
145#217C2×D311yescharacteristic
146#217C2×D311yescharacteristic
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—31no—
157(C2×D3)⋊C4—61no—
158(C2×D3)⋊C4C2×D311yescharacteristic
159(C2×D3)⋊C4C2×D311yescharacteristic
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4⋊Dic3—31no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—31no—
166C4⋊Dic3—61no—
167C4⋊Dic3C2×D311yescharacteristic
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
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—31no—
182C4×Dic3—31no—
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3C2×D311yescharacteristic
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
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#4112yes—
208C4⋊Q8—91no—
209#126—91no—
210#126—91no—
211#124—31no—
212#124—91no—
213#124—91no—
214#123—91no—
215#123—91no—
216C22⋊Q8—31no—
217C22⋊Q8—91no—
218C22⋊Q8—91no—
219#121—91no—
220C4×Q8—91no—
221C4×D4—91no—
222#117—91no—
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×Q8—62no—
232C3×Q8—121no—
233C3×D4—121no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—62no—
244C2×C3×C4C22×D311yescharacteristic
245C2×C3×C4C22×D311yescharacteristic
246C2×C3×C4C22×D311yescharacteristic
247C2×C3×C4C22×D311yescharacteristic
248#67—41no—
249#67—62no—
250#67—121no—
251#67—121no—
252#67—122no—
253C2×Dic3—22no—
254C2×Dic3—31no—
255C2×Dic3—31no—
256C2×Dic3—31no—
257C2×Dic3—31no—
258C2×Dic3—31no—
259C2×Dic3—31no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—121no—
264C2×Dic3C22×D311yescharacteristic
265C2×Dic3C22×D311yescharacteristic
266C2×Dic3C22×D311yescharacteristic
267C2×Dic3C22×D311yescharacteristic
268C2×Dic3C22×D311yescharacteristic
269C2×Dic3C22×D311yescharacteristic
270C2×Dic3C22×D311yescharacteristic
271D12—62no—
272C4×D3—121no—
273C4×D3—121no—
274C4×D3—121no—
275C4×D3—122no—
276Dic6—22no—
277Dic6—41no—
278Dic6—121no—
279Dic6—121no—
280C2×C32—41no—
281C2×C32C2×#4111yescharacteristic
282C2×C32H(5,𝔽2)11yescharacteristic
283C2×C322−1+411yescharacteristic
284C3⋊D3—41no—
285C3×D3—41no—
286C2×Q8—31no—
287C2×Q8—31no—
288C2×Q8—91no—
289C2×D4—31no—
290C2×D4—91no—
291C2×D4—91no—
292C22×C4—31no—
293C22×C4—91no—
294C22×C4—91no—
295C4⋊C4—31no—
296C4⋊C4—31no—
297C4⋊C4—31no—
298C4⋊C4—91no—
299C4⋊C4—91no—
300C4⋊C4—91no—
301C4⋊C4—91no—
302C4⋊C4—91no—
303C4⋊C4—91no—
304C4⋊C4—91no—
305C22⋊C4—31no—
306C22⋊C4—31no—
307C22⋊C4—31no—
308C22⋊C4—31no—
309C22⋊C4—91no—
310C22⋊C4—91no—
311C22⋊C4—91no—
312C22⋊C4—91no—
313C22⋊C4—91no—
314C22⋊C4—91no—
315C22⋊C4—91no—
316C22⋊C4D3211yescharacteristic
317C42—31no—
318C42—91no—
319C42—91no—
320C42—91no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—61no—
332C22×C3—61no—
333C22×C3—61no—
334C22×C3C23×D311yescharacteristic
335C22×C3C23×D311yescharacteristic
336C2×D3—21no—
337C2×D3—21no—
338C2×D3—21no—
339C2×D3—62no—
340C2×D3—61no—
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—61no—
346C2×D3—121no—
347C2×D3—121no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—32no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4—61no—
356C3×C4—61no—
357C3×C4—61no—
358C3×C4—61no—
359C3×C4—61no—
360C3×C4—61no—
361Dic3—21no—
362Dic3—21no—
363Dic3—21no—
364Dic3—21no—
365Dic3—21no—
366Dic3—21no—
367Dic3—61no—
368Dic3—61no—
369Dic3—61no—
370Dic3—61no—
371Dic3—61no—
372Dic3—61no—
373Dic3—121no—
374Dic3—121no—
375Dic3—121no—
376Dic3#23712yes—
377C32#54211yescharacteristic, complemented, Sylow
378C23—31no—
379C23—91no—
380C23C2×D3211yescharacteristic
381Q8—62no—
382Q8—121no—
383Q8—361no—
384D4—121no—
385D4—182no—
386D4—361no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—31no—
394C2×C4—62no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—361no—
399C2×C4—361no—
400C2×C4C2×D3211yescharacteristic
401C2×C4C2×D3211yescharacteristic
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—81no—
408C2×C3—121no—
409C2×C3C2×#23711yescharacteristic
410C2×C3#100411yescharacteristic
411C2×C3#100711yescharacteristic
412C2×C3D3×#4111yescharacteristic
413C2×C3#100911yescharacteristic
414C2×C3#101011yescharacteristic
415D3—41no—
416D3—122no—
417D3—121no—
418D3—121no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—61no—
423C22—61no—
424C22—61no—
425C22—181no—
426C22—181no—
427C22—181no—
428C22C22×D3211yescharacteristic
429C4—21no—
430C4—21no—
431C4—32no—
432C4—61no—
433C4—61no—
434C4—61no—
435C4—61no—
436C4—61no—
437C4—61no—
438C4—181no—
439C4—181no—
440C4—181no—
441C3—21no—
442C3#564311yescharacteristic, complemented
443C3#566911yescharacteristic, complemented
444C2—41no—
445C2—121no—
446C2—361no—
447C2#6429711yescharacteristic
448C2D3×#23711yescharacteristic
449C2#6431411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radical#63978C2
hypercenterC22⋊C4D32
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64092C2semidirect
2. C3×#885C2semidirect
3. C3×#878C2semidirect
5. #63969C2semidirect
6. #63955C2semidirect
7. #63953C2semidirect
8. #63907C2semidirect
9. #63893C2semidirect
10. #63881C2semidirect
11. #63871C2semidirect
12. #63868C2semidirect
13. #63858C2semidirect
14. Dic3×Dic6C2semidirect
15. #63839C2semidirect
72. #885D3semidirect
80. #878D3semidirect
377. C32#542semidirect
442. C3#5643semidirect
443. C3#5669semidirect

Maximal subgroups

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

subgroupindex
#640922
C3×#8852
C3×#8782
#639782
#639692
#639552
#639532
#639072
#638932
#638812
#638712
#638682
#638582
Dic3×Dic62
#638392
#56693
#56433