Finite Group Databasebeta

← #10594652

Subgroups — #10594652

Order 576.

547 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64104C211yesmaximal, characteristic, complemented
2C3×#898C211yesmaximal, characteristic, complemented
3C3×#896C211yesmaximal, characteristic, complemented
4#63923C211yesmaximal, characteristic, complemented
5#63923C211yesmaximal, characteristic, complemented
6D3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7#63910C211yesmaximal, characteristic, complemented
8#63909C211yesmaximal, characteristic, complemented
9#63907C211yesmaximal, characteristic, complemented
10#63897C211yesmaximal, characteristic, complemented
11#63881C211yesmaximal, characteristic, complemented
12#63880C211yesmaximal, characteristic, complemented
13#63864C211yesmaximal, characteristic, complemented
14#63853C211yesmaximal, characteristic, complemented
15#63848C211yesmaximal, characteristic, complemented
16#5704—31nomaximal
17#5644—31nomaximal
18C22×D32C2211yescharacteristic
19C2×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
22C2×C4×C3⋊D3C2211yescharacteristic, complemented
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×Dic6C2211yescharacteristic, complemented
25C2×#626C2211yescharacteristic
26C2×#626C2211yescharacteristic
27C2×#626C2211yescharacteristic
28C2×#626C2211yescharacteristic
29C2×#624C2211yescharacteristic, complemented
30C2×D3×Dic3C2211yescharacteristic
31C32×C4⋊C4C2211yescharacteristic, complemented
32#3752C2211yescharacteristic
33#3752C2211yescharacteristic, complemented
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×C4⋊Dic3C2211yescharacteristic, complemented
40C3×Dic3⋊C4C2211yescharacteristic, complemented
41C3×Dic3⋊C4C2211yescharacteristic, complemented
42C3×Dic3⋊C4C2211yescharacteristic, complemented
43C3×C4×Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic, complemented
49#3722C2211yescharacteristic, complemented
50#3722C2211yescharacteristic, complemented
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic, complemented
53C3×#126—31no—
54C3×C22⋊Q8—31no—
55#947—31no—
56#947—31no—
57#930—31no—
58C2×(C2×D3)⋊C4—31no—
59#898—31no—
60#898—31no—
61#898D311yescharacteristic, complemented
62#896D311yescharacteristic, complemented
63#895—31no—
64#895—31no—
65#895—31no—
66#895—31no—
67#895—61no—
68#894—31no—
69#894—31no—
70#894—31no—
71#893—31no—
72#885—31no—
73#884—31no—
74#883—31no—
75#882—31no—
76#882—31no—
77#882—31no—
78#882—31no—
79#67⋊C4—31no—
80#67⋊C4—31no—
81D3×C22⋊C4—31no—
82#875—31no—
83C4×D12—31no—
84C22×C3⋊D3C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3⋊D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C22×C3×D3C2311yescharacteristic
89C2×D32—22no—
90C2×D32—41no—
91C2×D32—41no—
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C2×C32×C4C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C4⋊(C3⋊D3)—41no—
97C4⋊(C3⋊D3)—41no—
98C4⋊(C3⋊D3)—41no—
99C4×C3⋊D3—41no—
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C3×C4×D3—41no—
107C3×Dic6—41no—
108#626—22no—
109#626—22no—
110#626—41no—
111#626—41no—
112#624—41no—
113D3×Dic3—41no—
114#538—91noSylow
115C23×D3—31no—
116C23×D3—31no—
117C2×C3×Q8—31no—
118C22×C3×C4—31no—
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—31no—
123C22×Dic3—31no—
124C2×D12—31no—
125C2×D12—31no—
126C2×D12—31no—
127C2×D12—31no—
128C2×D12—31no—
129C2×D12—31no—
130C2×D12—31no—
131C2×D12—31no—
132C2×D12—31no—
133C2×D12—31no—
134C2×D12—61no—
135C2×D12—61no—
136C2×D12—61no—
137C2×C4×D3—31no—
138C2×C4×D3—31no—
139C2×C4×D3—31no—
140C2×C4×D3—31no—
141C2×C4×D3—31no—
142C2×C4×D3—61no—
143C2×C4×D3C2×D311yescharacteristic
144C2×Dic6C2×D311yescharacteristic, complemented
145C3×C4⋊C4—21no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4C2×D311yescharacteristic, complemented
151C3×C4⋊C4C2×D311yescharacteristic, complemented
152C3×C22⋊C4—31no—
153C3×C22⋊C4—31no—
154C3×C22⋊C4—31no—
155C3×C22⋊C4—31no—
156C3×C22⋊C4—31no—
157C3×C42—31no—
158#217—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)⋊C4—31no—
167(C2×D3)⋊C4—31no—
168(C2×D3)⋊C4—31no—
169(C2×D3)⋊C4—31no—
170(C2×D3)⋊C4—31no—
171(C2×D3)⋊C4—31no—
172(C2×D3)⋊C4—31no—
173(C2×D3)⋊C4—31no—
174(C2×D3)⋊C4—31no—
175(C2×D3)⋊C4—31no—
176(C2×D3)⋊C4—61no—
177(C2×D3)⋊C4—61no—
178(C2×D3)⋊C4C2×D311yescharacteristic
179(C2×D3)⋊C4C2×D311yescharacteristic
180(C2×D3)⋊C4C2×D311yescharacteristic
181(C2×D3)⋊C4C2×D311yescharacteristic
182(C2×D3)⋊C4C2×D311yescharacteristic
183C4⋊Dic3—31no—
184C4⋊Dic3C2×D311yescharacteristic, complemented
185Dic3⋊C4—31no—
186Dic3⋊C4C2×D311yescharacteristic, complemented
187Dic3⋊C4C2×D311yescharacteristic, complemented
188Dic3⋊C4C2×D311yescharacteristic, complemented
189C4×Dic3—31no—
190C4×Dic3—31no—
191C4×Dic3C2×D311yescharacteristic, complemented
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—21no—
202C2×C3×D3—21no—
203C2×C3×D3—21no—
204C2×C3×D3—21no—
205C2×C3×D3—21no—
206C2×C3×D3—21no—
207C2×C3×D3#4112yes—
208D32—44no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C3⋊Dic3—21no—
213C3×Dic3—21no—
214C3×Dic3—21no—
215C3×Dic3—21no—
216C3×Dic3—21no—
217C3×Dic3—21no—
218C3×Dic3—21no—
219#126—31no—
220#126—91no—
221#124—91no—
222#124—91no—
223#123—91no—
224#123—91no—
225C22⋊Q8—31no—
226#121—91no—
227#121—91no—
228#121—91no—
229#120—91no—
230#120—91no—
231C4×D4—91no—
232C4×D4—91no—
233C2×C22⋊C4—91no—
234C23×C3—31no—
235C23×C3—31no—
236C22×D3—31no—
237C22×D3—31no—
238C22×D3—31no—
239C22×D3—31no—
240C22×D3—31no—
241C22×D3—31no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—61no—
245C22×D3—62no—
246C22×D3—62no—
247C22×D3—121no—
248C22×D3—121no—
249C22×D3—121no—
250C22×D3—121no—
251C22×D3C22×D311yescharacteristic
252C22×D3C22×D311yescharacteristic
253C3×Q8—121no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—121no—
264C2×C3×C4C22×D311yescharacteristic
265C2×C3×C4C22×D311yescharacteristic
266C2×C3×C4C22×D311yescharacteristic
267C2×C3×C4C22×D311yescharacteristic
268C2×C3×C4C22×D311yescharacteristic
269C2×C3×C4C22×D311yescharacteristic
270#67—62no—
271#67—62no—
272#67—121no—
273#67—121no—
274C2×Dic3—31no—
275C2×Dic3—31no—
276C2×Dic3—61no—
277C2×Dic3—121no—
278C2×Dic3C22×D311yescharacteristic
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282C2×Dic3C22×D311yescharacteristic
283C2×Dic3C22×D311yescharacteristic
284D12—62no—
285D12—62no—
286D12—121no—
287D12—121no—
288D12—121no—
289D12—121no—
290D12—121no—
291D12—121no—
292D12—122no—
293D12—122no—
294D12—122no—
295D12—121no—
296D12—121no—
297C4×D3—41no—
298C4×D3—121no—
299C4×D3—121no—
300C4×D3—121no—
301C4×D3—121no—
302C4×D3—121no—
303C4×D3—122no—
304Dic6—41no—
305C2×C32C2×#4111yescharacteristic
306C2×C32H(5,𝔽2)11yescharacteristic
307C2×C32H(5,𝔽2)11yescharacteristic
308C3⋊D3—41no—
309C3⋊D3—41no—
310C3⋊D3—41no—
311C3×D3—22no—
312C3×D3—41no—
313C24—91no—
314C2×Q8—31no—
315C2×D4—91no—
316C2×D4—91no—
317C2×D4—91no—
318C2×D4—91no—
319C2×D4—91no—
320C2×D4—91no—
321C2×D4—91no—
322C22×C4—31no—
323C22×C4—91no—
324C22×C4—91no—
325C22×C4—91no—
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4—31no—
329C4⋊C4—31no—
330C4⋊C4—91no—
331C4⋊C4D3211yescharacteristic, complemented
332C22⋊C4—31no—
333C22⋊C4—31no—
334C22⋊C4—31no—
335C22⋊C4—31no—
336C22⋊C4—31no—
337C22⋊C4—91no—
338C22⋊C4—91no—
339C22⋊C4—91no—
340C22⋊C4—91no—
341C22⋊C4—91no—
342C22⋊C4—91no—
343C22⋊C4—91no—
344C22⋊C4—91no—
345C22⋊C4—91no—
346C42—31no—
347C42—91no—
348C22×C3—21no—
349C22×C3—32no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3C23×D311yescharacteristic
356C22×C3C23×D311yescharacteristic
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—21no—
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—62no—
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—61no—
366C2×D3—61no—
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—61no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—61no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—61no—
380C2×D3—61no—
381C2×D3—121no—
382C2×D3—121no—
383C2×D3—121no—
384C2×D3—121no—
385C2×D3—121no—
386C2×D3—121no—
387C2×D3—121no—
388C2×D3—121no—
389C2×D3—124no—
390C2×D3—124no—
391C2×D3#23912yes—
392C3×C4—22no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—41no—
400C3×C4—41no—
401C3×C4—61no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407Dic3—21no—
408Dic3—21no—
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413Dic3—61no—
414Dic3—61no—
415Dic3—121no—
416C32#53811yescharacteristic, complemented, Sylow
417C23—31no—
418C23—31no—
419C23—91no—
420C23—91no—
421C23—91no—
422C23—182no—
423C23—361no—
424C23—361no—
425Q8—121no—
426D4—182no—
427D4—182no—
428D4—361no—
429D4—361no—
430D4—361no—
431D4—361no—
432D4—361no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—91no—
440C2×C4—121no—
441C2×C4—361no—
442C2×C4—361no—
443C2×C4—361no—
444C2×C4C2×D3211yescharacteristic
445C2×C4C2×D3211yescharacteristic
446C2×C4C2×D3211yescharacteristic
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—62no—
451C2×C3—121no—
452C2×C3#100411yescharacteristic
453C2×C3C2×#23911yescharacteristic
454C2×C3D3×#4111yescharacteristic
455C2×C3#100911yescharacteristic
456C2×C3#100911yescharacteristic
457C2×C3#100911yescharacteristic
458D3—22no—
459D3—41no—
460D3—122no—
461D3—121no—
462D3—121no—
463D3—121no—
464D3—121no—
465D3—121no—
466D3—121no—
467D3—241no—
468D3—241no—
469C22—32no—
470C22—61no—
471C22—61no—
472C22—61no—
473C22—61no—
474C22—61no—
475C22—181no—
476C22—181no—
477C22—181no—
478C22—181no—
479C22—181no—
480C22—181no—
481C22—181no—
482C22—181no—
483C22—181no—
484C22—364no—
485C22C22×D3211yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—61no—
490C4—61no—
491C4—61no—
492C4—61no—
493C4—61no—
494C4—61no—
495C4—181no—
496C3—21no—
497C3#564411yescharacteristic, complemented
498C3#570411yescharacteristic, complemented
499C2—62no—
500C2—121no—
501C2—361no—
502C2—361no—
503C2—361no—
504C2#6430811yescharacteristic
505C2#6431411yescharacteristic
506C2D3×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64104C2semidirect
2. C3×#898C2semidirect
3. C3×#896C2semidirect
4. #63923C2semidirect
5. #63923C2semidirect
6. D3×(C2×D3)⋊C4C2semidirect
7. #63910C2semidirect
8. #63909C2semidirect
9. #63907C2semidirect
10. #63897C2semidirect
11. #63881C2semidirect
12. #63880C2semidirect
13. #63864C2semidirect
14. #63853C2semidirect
15. #63848C2semidirect
21. C2×C4⋊(C3⋊D3)C22semidirect
22. C2×C4×C3⋊D3C22semidirect
24. C2×C3×Dic6C22semidirect
29. C2×#624C22semidirect
31. C32×C4⋊C4C22semidirect
33. #3752C22semidirect
39. C3×C4⋊Dic3C22semidirect
40. C3×Dic3⋊C4C22semidirect
41. C3×Dic3⋊C4C22semidirect
42. C3×Dic3⋊C4C22semidirect
43. C3×C4×Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
48. #3722C22semidirect
49. #3722C22semidirect
50. #3722C22semidirect
52. Dic32C22semidirect
61. #898D3semidirect
62. #896D3semidirect
144. C2×Dic6C2×D3semidirect
150. C3×C4⋊C4C2×D3semidirect
151. C3×C4⋊C4C2×D3semidirect
184. C4⋊Dic3C2×D3semidirect
186. Dic3⋊C4C2×D3semidirect
187. Dic3⋊C4C2×D3semidirect
188. Dic3⋊C4C2×D3semidirect
191. C4×Dic3C2×D3semidirect
331. C4⋊C4D32semidirect
416. C32#538semidirect
497. C3#5644semidirect
498. C3#5704semidirect

Maximal subgroups

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

subgroupindex
#641042
C3×#8982
C3×#8962
#639232
D3×(C2×D3)⋊C42
#639102
#639092
#639072
#638972
#638812
#638802
#638642
#638532
#638482
#57043
#56443