Finite Group Databasebeta

← #10594019

Subgroups — #10594019

Order 576.

547 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#876C211yesmaximal, characteristic, complemented
3C3×#875C211yesmaximal, characteristic, complemented
4#63923C211yesmaximal, characteristic, complemented
5#63922C211yesmaximal, characteristic, complemented
6#63921C211yesmaximal, characteristic, complemented
7#63912C211yesmaximal, characteristic, complemented
8#63910C211yesmaximal, characteristic, complemented
9#63907C211yesmaximal, characteristic, complemented
10#63891C211yesmaximal, characteristic, complemented
11#63888C211yesmaximal, characteristic, complemented
12#63886C211yesmaximal, characteristic, complemented
13#63881C211yesmaximal, characteristic, complemented
14#63861C211yesmaximal, characteristic, complemented
15#63853C211yesmaximal, characteristic, complemented
16#5704—31nomaximal
17#5678—31nomaximal
18C22×D32C2211yescharacteristic
19C2×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C4×C3⋊D3C2211yescharacteristic
21C2×C4×C3⋊D3C2211yescharacteristic, complemented
22C2×C3×D12C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic, complemented
24C2×#626C2211yescharacteristic
25C2×#626C2211yescharacteristic
26C2×#626C2211yescharacteristic
27C2×#626C2211yescharacteristic, complemented
28C2×#625C2211yescharacteristic
29C2×#624C2211yescharacteristic
30C2×D3×Dic3C2211yescharacteristic, complemented
31C32×C42C2211yescharacteristic, complemented
32#3752C2211yescharacteristic
33#3752C2211yescharacteristic, complemented
34C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×(C2×D3)⋊C4C2211yescharacteristic
40C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
42C3×Dic3⋊C4C2211yescharacteristic, complemented
43C3×Dic3⋊C4C2211yescharacteristic, complemented
44C3×Dic3⋊C4C2211yescharacteristic, complemented
45#3724C2211yescharacteristic, complemented
46Dic3⋊Dic3C2211yescharacteristic, complemented
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic, complemented
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic, complemented
53C3×#126—31no—
54C3×#124—31no—
55#947—31no—
56#946—31no—
57#939—31no—
58#937—31no—
59#898—31no—
60#896—31no—
61#895—31no—
62#895—31no—
63#894—31no—
64#893—31no—
65#886—31no—
66#885—31no—
67#884—31no—
68#884—31no—
69#884—31no—
70#883—31no—
71#883—31no—
72#882—31no—
73#882—31no—
74#882—31no—
75#67⋊C4—31no—
76D3×C22⋊C4—31no—
77D3×C22⋊C4—31no—
78#879—31no—
79#876D311yescharacteristic, complemented
80#875D311yescharacteristic, complemented
81C4×D12—31no—
82C4×D12—31no—
83C4×D12—61no—
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
96C2×C3⋊Dic3C2311yescharacteristic
97C4⋊(C3⋊D3)—22no—
98C4×C3⋊D3—41no—
99C4×C3⋊D3—41no—
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C3×D12—41no—
106C3×Dic6—41no—
107#626—22no—
108#626—41no—
109#626—41no—
110#626—41no—
111#625—41no—
112#624—41no—
113D3×Dic3—41no—
114#538—91noSylow
115C23×D3—31no—
116C23×D3—31no—
117C2×C3×Q8—31no—
118C2×C3×D4—31no—
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—31no—
123C2×#67—31no—
124C2×#67—31no—
125C22×Dic3—31no—
126C2×D12—31no—
127C2×D12—31no—
128C2×D12—31no—
129C2×D12—31no—
130C2×D12—31no—
131C2×D12—31no—
132C2×D12—61no—
133C2×D12C2×D311yescharacteristic
134C2×C4×D3—31no—
135C2×C4×D3—31no—
136C2×C4×D3—31no—
137C2×C4×D3—31no—
138C2×C4×D3—31no—
139C2×C4×D3—31no—
140C2×C4×D3—31no—
141C2×C4×D3—61no—
142C2×C4×D3—61no—
143C2×Dic6C2×D311yescharacteristic, complemented
144C3×C4⋊C4—31no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C22⋊C4—31no—
148C3×C22⋊C4—31no—
149C3×C22⋊C4—31no—
150C3×C22⋊C4—31no—
151C3×C22⋊C4—31no—
152C3×C22⋊C4—31no—
153C3×C22⋊C4—31no—
154C3×C42—21no—
155C3×C42C2×D311yescharacteristic, complemented
156C3×C42C2×D311yescharacteristic, complemented
157#217—31no—
158#217—31no—
159#217—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—61no—
172(C2×D3)⋊C4—61no—
173(C2×D3)⋊C4C2×D311yescharacteristic
174(C2×D3)⋊C4C2×D311yescharacteristic
175(C2×D3)⋊C4C2×D311yescharacteristic
176(C2×D3)⋊C4C2×D311yescharacteristic
177(C2×D3)⋊C4C2×D311yescharacteristic
178(C2×D3)⋊C4C2×D311yescharacteristic, complemented
179(C2×D3)⋊C4C2×D311yescharacteristic, complemented
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—61no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4—31no—
187Dic3⋊C4C2×D311yescharacteristic, complemented
188Dic3⋊C4C2×D311yescharacteristic, complemented
189Dic3⋊C4C2×D311yescharacteristic, complemented
190C4×Dic3—31no—
191C4×Dic3—31no—
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#4112yes—
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—21no—
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—31no—
222#124—91no—
223#123—91no—
224#123—91no—
225C22⋊Q8—91no—
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—
236C23×C3—31no—
237C22×D3—31no—
238C22×D3—31no—
239C22×D3—31no—
240C22×D3—31no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—62no—
244C22×D3—62no—
245C22×D3—121no—
246C22×D3—121no—
247C22×D3—121no—
248C22×D3—121no—
249C22×D3C22×D311yescharacteristic
250C22×D3C22×D311yescharacteristic
251C22×D3C22×D311yescharacteristic
252C3×Q8—121no—
253C3×D4—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×C4C22×D311yescharacteristic
263C2×C3×C4C22×D311yescharacteristic
264C2×C3×C4C22×D311yescharacteristic
265C2×C3×C4C22×D311yescharacteristic
266C2×C3×C4C22×D311yescharacteristic
267C2×C3×C4C22×D311yescharacteristic
268#67—62no—
269#67—121no—
270#67—121no—
271#67—121no—
272#67—121no—
273#67—121no—
274C2×Dic3—31no—
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—31no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—121no—
281C2×Dic3C22×D311yescharacteristic
282C2×Dic3C22×D311yescharacteristic
283C2×Dic3C22×D311yescharacteristic
284C2×Dic3C22×D311yescharacteristic
285C2×Dic3C22×D311yescharacteristic
286D12—41no—
287D12—62no—
288D12—62no—
289D12—64no—
290D12—62no—
291D12—121no—
292D12—121no—
293D12—121no—
294C4×D3—121no—
295C4×D3—121no—
296C4×D3—121no—
297C4×D3—121no—
298C4×D3—121no—
299C4×D3—121no—
300C4×D3—121no—
301C4×D3—122no—
302C4×D3—122no—
303Dic6—41no—
304C2×C32C2×#4111yescharacteristic
305C2×C32H(5,𝔽2)11yescharacteristic
306C2×C32H(5,𝔽2)11yescharacteristic
307C3⋊D3—22no—
308C3⋊D3—41no—
309C3×D3—41no—
310C3×D3—41no—
311C3×D3—41no—
312C24—91no—
313C2×Q8—31no—
314C2×D4—31no—
315C2×D4—91no—
316C2×D4—91no—
317C2×D4—91no—
318C2×D4—91no—
319C2×D4—91no—
320C2×D4—91no—
321C22×C4—91no—
322C22×C4—91no—
323C22×C4—91no—
324C22×C4—91no—
325C4⋊C4—31no—
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—91no—
331C22⋊C4—31no—
332C22⋊C4—31no—
333C22⋊C4—31no—
334C22⋊C4—31no—
335C22⋊C4—31no—
336C22⋊C4—31no—
337C22⋊C4—31no—
338C22⋊C4—91no—
339C22⋊C4—91no—
340C22⋊C4—91no—
341C22⋊C4—91no—
342C22⋊C4—91no—
343C22⋊C4—91no—
344C22⋊C4—91no—
345C42—91no—
346C42D3211yescharacteristic, complemented
347C22×C3—21no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3C23×D311yescharacteristic
358C22×C3C23×D311yescharacteristic
359C2×D3—21no—
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—21no—
367C2×D3—21no—
368C2×D3—32no—
369C2×D3—32no—
370C2×D3—62no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—62no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—61no—
380C2×D3—61no—
381C2×D3—61no—
382C2×D3—121no—
383C2×D3—121no—
384C2×D3—121no—
385C2×D3—121no—
386C2×D3—124no—
387C2×D3—124no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—41no—
397C3×C4—61no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—61no—
402Dic3—21no—
403Dic3—21no—
404Dic3—21no—
405Dic3—21no—
406Dic3—21no—
407Dic3—61no—
408Dic3—61no—
409Dic3—61no—
410Dic3—61no—
411Dic3—121no—
412Dic3—121no—
413C32#53811yescharacteristic, complemented, Sylow
414C23—31no—
415C23—31no—
416C23—31no—
417C23—91no—
418C23—91no—
419C23—182no—
420C23—361no—
421C23—361no—
422Q8—121no—
423D4—121no—
424D4—182no—
425D4—182no—
426D4—361no—
427D4—361no—
428D4—361no—
429D4—361no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—91no—
436C2×C4—91no—
437C2×C4—361no—
438C2×C4—361no—
439C2×C4—361no—
440C2×C4—361no—
441C2×C4C2×D3211yescharacteristic
442C2×C4C2×D3211yescharacteristic
443C2×C4C2×D3211yescharacteristic
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—121no—
448C2×C3—121no—
449C2×C3—121no—
450C2×C3#100411yescharacteristic
451C2×C3D3×#4111yescharacteristic
452C2×C3D3×#4111yescharacteristic
453C2×C3#100911yescharacteristic
454C2×C3#100911yescharacteristic
455C2×C3#100911yescharacteristic
456D3—41no—
457D3—41no—
458D3—41no—
459D3—62no—
460D3—62no—
461D3—122no—
462D3—122no—
463D3—121no—
464D3—121no—
465C22—61no—
466C22—61no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22—61no—
471C22—61no—
472C22—61no—
473C22—61no—
474C22—92no—
475C22—181no—
476C22—181no—
477C22—181no—
478C22—181no—
479C22—181no—
480C22—364no—
481C22C22×D3211yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—61no—
486C4—61no—
487C4—61no—
488C4—61no—
489C4—61no—
490C4—181no—
491C4—181no—
492C3—21no—
493C3#567811yescharacteristic, complemented
494C3#570411yescharacteristic, complemented
495C2—121no—
496C2—121no—
497C2—121no—
498C2—182no—
499C2—361no—
500C2#6430611yescharacteristic
501C2#6430711yescharacteristic
502C2#6430811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊D3)C2semidirect
2. C3×#876C2semidirect
3. C3×#875C2semidirect
4. #63923C2semidirect
5. #63922C2semidirect
6. #63921C2semidirect
7. #63912C2semidirect
8. #63910C2semidirect
9. #63907C2semidirect
10. #63891C2semidirect
11. #63888C2semidirect
12. #63886C2semidirect
13. #63881C2semidirect
14. #63861C2semidirect
15. #63853C2semidirect
21. C2×C4×C3⋊D3C22semidirect
23. C2×C3×Dic6C22semidirect
27. C2×#626C22semidirect
30. C2×D3×Dic3C22semidirect
31. C32×C42C22semidirect
33. #3752C22semidirect
34. C4⋊(C3⋊Dic3)C22semidirect
40. C3×(C2×D3)⋊C4C22semidirect
41. C3×(C2×D3)⋊C4C22semidirect
42. C3×Dic3⋊C4C22semidirect
43. C3×Dic3⋊C4C22semidirect
44. C3×Dic3⋊C4C22semidirect
45. #3724C22semidirect
46. Dic3⋊Dic3C22semidirect
48. #3722C22semidirect
52. Dic32C22semidirect
79. #876D3semidirect
80. #875D3semidirect
143. C2×Dic6C2×D3semidirect
155. C3×C42C2×D3semidirect
156. C3×C42C2×D3semidirect
178. (C2×D3)⋊C4C2×D3semidirect
179. (C2×D3)⋊C4C2×D3semidirect
187. Dic3⋊C4C2×D3semidirect
188. Dic3⋊C4C2×D3semidirect
189. Dic3⋊C4C2×D3semidirect
346. C42D32semidirect
413. C32#538semidirect
493. C3#5678semidirect
494. C3#5704semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D3)2
C3×#8762
C3×#8752
#639232
#639222
#639212
#639122
#639102
#639072
#638912
#638882
#638862
#638812
#638612
#638532
#57043
#56783