Finite Group Databasebeta

← #10594451

Subgroups — #10594451

Order 576.

469 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64106C211yesmaximal, characteristic, complemented
2C3×#898C211yesmaximal, characteristic, complemented
3C3×#896C211yesmaximal, characteristic, complemented
4#63913C211yesmaximal, characteristic, complemented
5#63907C211yesmaximal, characteristic, complemented
6#63906C211yesmaximal, characteristic
7#63893C211yesmaximal, characteristic, complemented
8#63891C211yesmaximal, characteristic, complemented
9#63881C211yesmaximal, characteristic, complemented
10#63853C211yesmaximal, characteristic, complemented
11#63852C211yesmaximal, characteristic, complemented
12#63848C211yesmaximal, characteristic, complemented
13Dic3×Dic6C211yesmaximal, characteristic, complemented
14#63841C211yesmaximal, characteristic, complemented
15#63840C211yesmaximal, characteristic, complemented
16#5710—31nomaximal
17#5643—31nomaximal
18C2×C4×C3⋊D3C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×C3×C4×D3C2211yescharacteristic
21C2×C3×Dic6C2211yescharacteristic
22C2×#627C2211yescharacteristic
23C2×#626C2211yescharacteristic
24C2×#626C2211yescharacteristic
25C2×#625C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C32×C4⋊C4C2211yescharacteristic
28#3752C2211yescharacteristic
29#3752C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2211yescharacteristic
31C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×(C2×D3)⋊C4C2211yescharacteristic
33C3×(C2×D3)⋊C4C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×C4⋊Dic3C2211yescharacteristic
38C3×Dic3⋊C4C2211yescharacteristic
39C3×Dic3⋊C4C2211yescharacteristic
40C3×Dic3⋊C4C2211yescharacteristic
41C3×C4×Dic3C2211yescharacteristic
42Dic3⋊Dic3C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50Dic32C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#126—31no—
54C3×C22⋊Q8—31no—
55#947—31no—
56Q8×Dic3—31no—
57#930—31no—
58#926—31no—
59#898—31no—
60#898—31no—
61#898D311yescharacteristic, complemented
62#897—31no—
63#897—31no—
64#897—31no—
65#897—61no—
66#896D311yescharacteristic, complemented
67#894—31no—
68#893—31no—
69#892—31no—
70#888—31no—
71#888—31no—
72#886—31no—
73#885—31no—
74#885—31no—
75#885—31no—
76#885—31no—
77#884—31no—
78#883—31no—
79#883—31no—
80#67⋊C4—31no—
81#878—31no—
82#875—31no—
83C4×Dic6—31no—
84C22×C3⋊D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C32×C4C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C4×C3⋊D3—41no—
94#634—41no—
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C3×C4×D3—22no—
102C3×Dic6—22no—
103#627—41no—
104#626—22no—
105#626—41no—
106#625—41no—
107D3×Dic3—41no—
108#542—91noSylow
109C2×C3×Q8—31no—
110C22×C3×C4—31no—
111C2×#67—31no—
112C2×#67—31no—
113C2×#67—31no—
114C2×#67—31no—
115C22×Dic3—31no—
116C2×D12—31no—
117C2×D12—31no—
118C2×C4×D3—31no—
119C2×C4×D3—31no—
120C2×C4×D3—31no—
121C2×C4×D3—61no—
122C2×C4×D3C2×D311yescharacteristic
123C2×Dic6—31no—
124C2×Dic6—31no—
125C2×Dic6—31no—
126C2×Dic6—31no—
127C2×Dic6—61no—
128C2×Dic6C2×D311yescharacteristic
129C3×C4⋊C4—21no—
130C3×C4⋊C4—31no—
131C3×C4⋊C4—31no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4C2×D311yescharacteristic
135C3×C4⋊C4C2×D311yescharacteristic
136C3×C22⋊C4—31no—
137C3×C22⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4—31no—
140C3×C22⋊C4—31no—
141C3×C42—31no—
142#217—31no—
143#217—31no—
144#217—31no—
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)⋊C4—61no—
158(C2×D3)⋊C4C2×D311yescharacteristic
159(C2×D3)⋊C4C2×D311yescharacteristic
160(C2×D3)⋊C4C2×D311yescharacteristic
161(C2×D3)⋊C4C2×D311yescharacteristic
162(C2×D3)⋊C4C2×D311yescharacteristic
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—61no—
172C4⋊Dic3C2×D311yescharacteristic
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4C2×D311yescharacteristic
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×C32C2411yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3—21no—
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—
198C32×C4—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—31no—
210#126—91no—
211#124—91no—
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—31no—
224C23×C3—31no—
225C22×D3—31no—
226C22×D3—31no—
227C22×D3—61no—
228C22×D3C22×D311yescharacteristic
229C22×D3C22×D311yescharacteristic
230C3×Q8—62no—
231C2×C3×C4—21no—
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—62no—
241C2×C3×C4C22×D311yescharacteristic
242C2×C3×C4C22×D311yescharacteristic
243C2×C3×C4C22×D311yescharacteristic
244C2×C3×C4C22×D311yescharacteristic
245C2×C3×C4C22×D311yescharacteristic
246C2×C3×C4C22×D311yescharacteristic
247#67—62no—
248#67—121no—
249#67—121no—
250#67—121no—
251C2×Dic3—31no—
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—31no—
255C2×Dic3—31no—
256C2×Dic3—31no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—121no—
261C2×Dic3C22×D311yescharacteristic
262C2×Dic3C22×D311yescharacteristic
263C2×Dic3C22×D311yescharacteristic
264C2×Dic3C22×D311yescharacteristic
265C2×Dic3C22×D311yescharacteristic
266C2×Dic3C22×D311yescharacteristic
267D12—62no—
268D12—121no—
269C4×D3—22no—
270C4×D3—121no—
271C4×D3—121no—
272C4×D3—121no—
273C4×D3—122no—
274Dic6—22no—
275Dic6—121no—
276Dic6—121no—
277Dic6—121no—
278Dic6—121no—
279Dic6—122no—
280C2×C32C2×#4111yescharacteristic
281C2×C32H(5,𝔽2)11yescharacteristic
282C2×C322−1+411yescharacteristic
283C3⋊D3—41no—
284C3×D3—41no—
285C3×D3—41no—
286C2×Q8—31no—
287C2×Q8—91no—
288C2×Q8—91no—
289C2×D4—91no—
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—31no—
299C4⋊C4—91no—
300C4⋊C4—91no—
301C4⋊C4—91no—
302C4⋊C4—91no—
303C4⋊C4—91no—
304C4⋊C4D3211yescharacteristic
305C22⋊C4—31no—
306C22⋊C4—31no—
307C22⋊C4—31no—
308C22⋊C4—31no—
309C22⋊C4—31no—
310C22⋊C4—91no—
311C22⋊C4—91no—
312C22⋊C4—91no—
313C22⋊C4—91no—
314C22⋊C4—91no—
315C22⋊C4—91no—
316C22⋊C4—91no—
317C42—31no—
318C42—91no—
319C42—91no—
320C42—91no—
321C22×C3—21no—
322C22×C3—61no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—61no—
326C22×C3—61no—
327C22×C3—61no—
328C22×C3C23×D311yescharacteristic
329C22×C3C23×D311yescharacteristic
330C2×D3—21no—
331C2×D3—21no—
332C2×D3—21no—
333C2×D3—21no—
334C2×D3—21no—
335C2×D3—21no—
336C2×D3—62no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—61no—
340C2×D3—61no—
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—121no—
344C2×D3—121no—
345C3×C4—22no—
346C3×C4—21no—
347C3×C4—21no—
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—
360Dic3—21no—
361Dic3—21no—
362Dic3—21no—
363Dic3—21no—
364Dic3—21no—
365Dic3—61no—
366Dic3—61no—
367Dic3—61no—
368Dic3—61no—
369Dic3—61no—
370Dic3—61no—
371Dic3—121no—
372Dic3—121no—
373Dic3—121no—
374Dic3#23712yes—
375C32#54211yescharacteristic, complemented, Sylow
376C23—31no—
377C23—31no—
378C23—91no—
379Q8—62no—
380Q8—361no—
381Q8—361no—
382D4—182no—
383D4—361no—
384D4—361no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—62no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—361no—
396C2×C4—361no—
397C2×C4C2×D3211yescharacteristic
398C2×C4C2×D3211yescharacteristic
399C2×C4C2×D3211yescharacteristic
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—121no—
404C2×C3—121no—
405C2×C3C2×#23711yescharacteristic
406C2×C3#100711yescharacteristic
407C2×C3D3×#4111yescharacteristic
408C2×C3#100911yescharacteristic
409C2×C3#100911yescharacteristic
410C2×C3#101011yescharacteristic
411D3—41no—
412D3—41no—
413D3—122no—
414D3—121no—
415D3—121no—
416C22—61no—
417C22—61no—
418C22—61no—
419C22—61no—
420C22—61no—
421C22—61no—
422C22—181no—
423C22—181no—
424C22—181no—
425C22C22×D3211yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—32no—
430C4—61no—
431C4—61no—
432C4—61no—
433C4—61no—
434C4—61no—
435C4—181no—
436C4—181no—
437C4—181no—
438C3—21no—
439C3#564311yescharacteristic, complemented
440C3#571011yescharacteristic, complemented
441C2—121no—
442C2—121no—
443C2—361no—
444C2#6430811yescharacteristic
445C2D3×#23711yescharacteristic
446C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radical#63906C2
hypercenterC4⋊C4D32
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64106C2semidirect
2. C3×#898C2semidirect
3. C3×#896C2semidirect
4. #63913C2semidirect
5. #63907C2semidirect
7. #63893C2semidirect
8. #63891C2semidirect
9. #63881C2semidirect
10. #63853C2semidirect
11. #63852C2semidirect
12. #63848C2semidirect
13. Dic3×Dic6C2semidirect
14. #63841C2semidirect
15. #63840C2semidirect
61. #898D3semidirect
66. #896D3semidirect
375. C32#542semidirect
439. C3#5643semidirect
440. C3#5710semidirect

Maximal subgroups

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

subgroupindex
#641062
C3×#8982
C3×#8962
#639132
#639072
#639062
#638932
#638912
#638812
#638532
#638522
#638482
Dic3×Dic62
#638412
#638402
#57103
#56433