Finite Group Databasebeta

← #13154390

Subgroups — #13154390

Order 1728.

1441 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#272C211yesmaximal, characteristic, complemented
2C2×#88748C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#137C211yesmaximal, characteristic, complemented
4#11721870C211yesmaximal, characteristic, complemented
5#11721859C211yesmaximal, characteristic, complemented
6#11721808C212yesmaximal, complemented
7#11721807C212yesmaximal, complemented
8#11721802C212yesmaximal, complemented
9#11721800C212yesmaximal, complemented
10#11721787C211yesmaximal, characteristic, complemented
11M16×#272C211yesmaximal, characteristic, complemented
12C3×#5777—34nomaximal
13#41×H(3,𝔽3)C2211yescharacteristic
14C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
15#88748—21no—
16#88748—21no—
17#88748C2212yes—
18#88748C2212yescomplemented
19#88748C2211yescharacteristic, complemented
20Q8×#272C2211yescharacteristic, complemented
21#88744—21no—
22#88744C2211yescharacteristic, complemented
23D4×#272—21no—
24D4×#272C2211yescharacteristic, complemented
25#88740—21no—
26#88740C2211yescharacteristic
27C2×C4⋊#272—21no—
28C2×C4⋊#272C2211yescharacteristic, complemented
29C2×C4×#272—21no—
30C2×C4×#272—21no—
31C2×C4×#272C2211yescharacteristic
32Q16×H(3,𝔽3)C2212yescomplemented
33SD16×H(3,𝔽3)C2212yescomplemented
34M16×H(3,𝔽3)C2211yescharacteristic, complemented
35#88550C2212yescomplemented
36#88549C2212yescomplemented
37#88549C2212yescomplemented
38#88545C2212yescomplemented
39#88538C2211yescharacteristic, complemented
40#88529C2212yescomplemented
41#88528C2212yescomplemented
42#88527C2212yescomplemented
43C8×#272C2212yescomplemented
44C3×D3×#41—34no—
45C2×C3×#239—34no—
46C32×#137D314yescomplemented
47C3×#949—34no—
48C3×#942—34no—
49C3×#919—38no—
50C3×#918—38no—
51C3×#916—38no—
52C3×#914—38no—
53C3×#908—34no—
54C3×D3×M16—34no—
55C22×#272—21no—
56C22×#272—21no—
57C22×#272D411yescharacteristic
58Q8×H(3,𝔽3)—21no—
59Q8×H(3,𝔽3)C2311yescharacteristic
60Q8×H(3,𝔽3)C2312yes—
61D4×H(3,𝔽3)—21no—
62D4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)—21no—
64C2×C4×H(3,𝔽3)—21no—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66#6235—21no—
67#6235—21no—
68#6235—21no—
69C2×#1084—21no—
70C2×#1084D411yescharacteristic
71C4⋊#272—22no—
72C4⋊#272—22no—
73C4⋊#272—21no—
74C4⋊#272—21no—
75C4⋊#272C2312yes—
76C4⋊#272C2311yescharacteristic
77C4×#272—21no—
78C4×#272—21no—
79C4×#272—21no—
80C4×#272—22no—
81C4×#272—22no—
82C4×#272D411yescharacteristic
83C4×#272D411yescharacteristic
84C4×#272C2311yescharacteristic
85C4×#272C2311yescharacteristic
86#6229—21no—
87#6229C2311yescharacteristic
88C8×H(3,𝔽3)C2312yes—
89#6180C2312yes—
90C3×#575—91no—
91#5777—34no—
92C32×#41C2×D314yes—
93C2×C32×Q8C2×D314yescomplemented
94C3×#239—38no—
95C3×#239—38no—
96C3×#239—34no—
97C3×#239—64no—
98C3×#239—64no—
99C3×D3×Q8—34no—
100C3×#237—34no—
101C3×#237—64no—
102C3×D3×D4—34no—
103C3×D3×D4—64no—
104C3×#235—34no—
105C3×#235—64no—
106C2×C3×D12—34no—
107C2×C3×D12—64no—
108C2×C3×C4×D3—34no—
109C2×C3×C4×D3—64no—
110C2×C3×C4×D3—64no—
111C32×Q16C2×D318yescomplemented
112C32×SD16C2×D318yescomplemented
113C32×M16C2×D314yescomplemented
114C3×#216—38no—
115C3×#215—38no—
116C3×#215—38no—
117C3×#213—38no—
118C3×#208—34no—
119C3×D24—38no—
120C3×#204—38no—
121C3×ΓL(1,𝔽25)—38no—
122C3×D3×C8—38no—
123C22×H(3,𝔽3)—21no—
124C22×H(3,𝔽3)C2×D411yescharacteristic
125C2×#272—22no—
126C2×#272—21no—
127C2×#272—41no—
128C2×#272—42no—
129C2×#272—42no—
130C2×#272C2×D411yescharacteristic
131C2×#272C2×D411yescharacteristic
132C4×H(3,𝔽3)—21no—
133C4×H(3,𝔽3)—22no—
134C4×H(3,𝔽3)C2×D411yescharacteristic
135C4×H(3,𝔽3)C2411yescharacteristic
136#1084—21no—
137#1084C2×D411yescharacteristic
138#1084C2×D411yescharacteristic
139C2×C3×#41—91no—
140C2×C3×#41—91no—
141D3×#41—34no—
142C2×#239—34no—
143C3×#137—34no—
144C3×#137—91no—
145C3×#137—92no—
146C3×#137C3⋊D311yescharacteristic
147C3×AGL(1,ℤ/8)—91no—
148C3×AGL(1,ℤ/8)—92no—
149C3×AGL(1,ℤ/8)—91no—
150C3×#135—92no—
151C3×#135—92no—
152C2×C3×M16—91no—
153#949—34no—
154#942—34no—
155#919—38no—
156#918—38no—
157#916—38no—
158#914—38no—
159#908—34no—
160D3×M16—34no—
161C22×C3×D3—34no—
162C22×C3×D3—64no—
163C22×C3×D3—64no—
164C32×Q8—24no—
165C32×Q8C22×D314yes—
166C32×Q8C22×D318yes—
167C32×D4—24no—
168C32×D4C22×D314yes—
169C2×C32×C4—24no—
170C2×C32×C4—24no—
171C2×C32×C4C22×D314yes—
172C3×#67—64no—
173C3×#67—64no—
174C3×#67—64no—
175C2×C3×Dic3—34no—
176C2×C3×Dic3—64no—
177C3×D12—38no—
178C3×D12—34no—
179C3×D12—68no—
180C3×D12—68no—
181C3×D12—64no—
182C3×D12—64no—
183C3×C4×D3—34no—
184C3×C4×D3—34no—
185C3×C4×D3—34no—
186C3×C4×D3—34no—
187C3×C4×D3—64no—
188C3×C4×D3—68no—
189C3×C4×D3—68no—
190C3×C4×D3—64no—
191C3×C4×D3—64no—
192C3×Dic6—34no—
193C3×Dic6—64no—
194C32×C8C22×D318yes—
195C3×C3⋊C8—38no—
196#575—91noSylow
197C2×H(3,𝔽3)—21no—
198C2×H(3,𝔽3)—41no—
199C2×H(3,𝔽3)C22×D411yescharacteristic
200#272—21no—
201#272—21no—
202#272—42no—
203#272—41no—
204C3×#41—34no—
205C3×#41—91no—
206C3×#41—92no—
207C3×#41—92no—
208C3×#41—91no—
209C3×#41—91no—
210C3×#41—181no—
211C3×#41—181no—
212C3×#41—181no—
213C3×#41—181no—
214C3×#41C2×C3⋊D311yescharacteristic
215C2×C3×Q8—34no—
216C2×C3×Q8—91no—
217C2×C3×Q8C2×C3⋊D311yescharacteristic
218C2×C3×D4—91no—
219C2×C3×D4—91no—
220C2×C3×D4—181no—
221C2×C3×D4—181no—
222C22×C3×C4—91no—
223C22×C3×C4—181no—
224C22×C3×C4—181no—
225#239—38no—
226#239—38no—
227#239—34no—
228#239—64no—
229#239—64no—
230D3×Q8—34no—
231#237—34no—
232#237—64no—
233D3×D4—34no—
234D3×D4—64no—
235#235—34no—
236#235—64no—
237C2×D12—34no—
238C2×D12—64no—
239C2×C4×D3—34no—
240C2×C4×D3—64no—
241C2×C4×D3—64no—
242C3×Q16—38no—
243C3×Q16—92no—
244C3×Q16C2×C3⋊D312yes—
245C3×SD16—38no—
246C3×SD16—92no—
247C3×SD16—92no—
248C3×SD16—92no—
249C3×SD16C2×C3⋊D312yes—
250C3×D8—92no—
251C3×D8—92no—
252C3×M16—34no—
253C3×M16—91no—
254C3×M16—92no—
255C3×M16C2×C3⋊D311yescharacteristic
256C2×C3×C8—92no—
257#216—38no—
258#215—38no—
259#215—38no—
260#213—38no—
261#208—34no—
262D24—38no—
263#204—38no—
264ΓL(1,𝔽25)—38no—
265D3×C8—38no—
266C22×C32—24no—
267C22×C32D3×D414yes—
268C2×C3×D3—34no—
269C2×C3×D3—34no—
270C2×C3×D3—68no—
271C2×C3×D3—64no—
272C2×C3×D3—124no—
273C2×C3×D3—128no—
274C2×C3×D3—128no—
275C32×C4—24no—
276C32×C4—28no—
277C32×C4D3×D414yes—
278C32×C4C23×D314yes—
279C3×Dic3—34no—
280C3×Dic3—34no—
281C3×Dic3—64no—
282C2×#41—91no—
283C2×#41—91no—
284#137—91no—
285#137—92no—
286#137#27211yescharacteristic, complemented
287AGL(1,ℤ/8)—91no—
288AGL(1,ℤ/8)—92no—
289AGL(1,ℤ/8)—91no—
290#135—92no—
291#135—92no—
292C2×M16—91no—
293H(3,𝔽3)#57511yescharacteristic, complemented, Sylow
294C23×C3—91no—
295C23×C3—181no—
296C23×C3—181no—
297C22×D3—34no—
298C22×D3—64no—
299C22×D3—64no—
300C3×Q8—21no—
301C3×Q8—34no—
302C3×Q8—38no—
303C3×Q8—64no—
304C3×Q8—91no—
305C3×Q8—181no—
306C3×Q8C22×C3⋊D311yescharacteristic
307C3×Q8C22×C3⋊D312yes—
308C3×D4—21no—
309C3×D4—34no—
310C3×D4—64no—
311C3×D4—92no—
312C3×D4—91no—
313C3×D4—181no—
314C3×D4—182no—
315C3×D4—182no—
316C3×D4—181no—
317C3×D4—181no—
318C3×D4—181no—
319C3×D4—181no—
320C3×D4C22×C3⋊D311yescharacteristic
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—34no—
324C2×C3×C4—64no—
325C2×C3×C4—64no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—181no—
332C2×C3×C4—182no—
333C2×C3×C4—182no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4C22×C3⋊D311yescharacteristic
338#67—64no—
339#67—64no—
340#67—64no—
341C2×Dic3—34no—
342C2×Dic3—64no—
343D12—38no—
344D12—34no—
345D12—64no—
346D12—68no—
347D12—68no—
348D12—64no—
349C4×D3—34no—
350C4×D3—34no—
351C4×D3—34no—
352C4×D3—34no—
353C4×D3—64no—
354C4×D3—68no—
355C4×D3—68no—
356C4×D3—64no—
357C4×D3—64no—
358Dic6—34no—
359Dic6—64no—
360C3×C8—38no—
361C3×C8—92no—
362C3×C8C22×C3⋊D312yes—
363C3⋊C8—38no—
364C2×C32—24no—
365C2×C32—44no—
366C2×C32C2×D3×D414yes—
367C3×D3—64no—
368C3×D3—64no—
369C3×D3—128no—
370C3×D3—124no—
371#41—91no—
372#41—92no—
373#41—92no—
374#41—91no—
375#41—91no—
376#41—181no—
377#41—181no—
378#41—181no—
379#41—181no—
380#41C2×#27211yescharacteristic
381C2×Q8—91no—
382C2×Q8C2×#27211yescharacteristic, complemented
383C2×D4—91no—
384C2×D4—91no—
385C2×D4—181no—
386C2×D4—181no—
387C22×C4—91no—
388C22×C4—181no—
389C22×C4—181no—
390Q16—92no—
391Q16C2×#27212yescomplemented
392SD16—92no—
393SD16—92no—
394SD16—92no—
395SD16C2×#27212yescomplemented
396D8—92no—
397D8—92no—
398M16—91no—
399M16—92no—
400M16C2×#27211yescharacteristic, complemented
401C2×C8—92no—
402C22×C3—21no—
403C22×C3—34no—
404C22×C3—64no—
405C22×C3—91no—
406C22×C3—91no—
407C22×C3—182no—
408C22×C3—181no—
409C22×C3—361no—
410C22×C3—362no—
411C22×C3—362no—
412C22×C3D4×C3⋊D311yescharacteristic
413C2×D3—34no—
414C2×D3—34no—
415C2×D3—68no—
416C2×D3—64no—
417C2×D3—124no—
418C2×D3—128no—
419C2×D3—128no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—34no—
423C3×C4—34no—
424C3×C4—64no—
425C3×C4—68no—
426C3×C4—91no—
427C3×C4—91no—
428C3×C4—181no—
429C3×C4D4×C3⋊D311yescharacteristic
430C3×C4C23×C3⋊D311yescharacteristic
431Dic3—34no—
432Dic3—34no—
433Dic3—64no—
434C32#577714yescomplemented
435C23—91no—
436C23—181no—
437C23—181no—
438Q8—21no—
439Q8—91no—
440Q8—181no—
441Q8C22×#27211yescharacteristic
442Q8C22×#27212yes—
443D4—21no—
444D4—92no—
445D4—91no—
446D4—181no—
447D4—182no—
448D4—182no—
449D4—181no—
450D4—181no—
451D4—181no—
452D4—181no—
453D4C22×#27211yescharacteristic
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—181no—
462C2×C4—182no—
463C2×C4—182no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4C22×#27211yescharacteristic
468C8—92no—
469C8C22×#27212yes—
470C2×C3—21no—
471C2×C3—34no—
472C2×C3—41no—
473C2×C3—64no—
474C2×C3—124no—
475C2×C3—181no—
476C2×C3—181no—
477C2×C3—362no—
478C2×C3—361no—
479C2×C3C2×D4×C3⋊D311yescharacteristic
480D3—64no—
481D3—64no—
482D3—128no—
483D3—124no—
484C22—21no—
485C22—91no—
486C22—91no—
487C22—182no—
488C22—181no—
489C22—361no—
490C22—362no—
491C22—362no—
492C22D4×#27211yescharacteristic
493C4—21no—
494C4—22no—
495C4—91no—
496C4—91no—
497C4—181no—
498C4D4×#27211yescharacteristic
499C4C23×#27211yescharacteristic
500C3—34no—
501C3#1059624811yescharacteristic
502C2—21no—
503C2—41no—
504C2—181no—
505C2—181no—
506C2—362no—
507C2—361no—
508C2C2×D4×#27211yescharacteristic
subgroupG/H
center, socleC2×C3C2×D4×C3⋊D3
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C23×C3⋊D3
FittingH(3,𝔽3)×#137C2
hypercenterC3×#137C3⋊D3
hypocenterH(3,𝔽3)#575
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#137

Complements

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

NH ≅ G/Ntype
1. #41×#272C2semidirect
2. C2×#88748C2semidirect
3. H(3,𝔽3)×#137C2semidirect
4. #11721870C2semidirect
5. #11721859C2semidirect
6. #11721808C2semidirect
7. #11721807C2semidirect
8. #11721802C2semidirect
9. #11721800C2semidirect
10. #11721787C2semidirect
11. M16×#272C2semidirect
14. C2×Q8×H(3,𝔽3)C22semidirect
18. #88748C22semidirect
19. #88748C22semidirect
20. Q8×#272C22semidirect
22. #88744C22semidirect
24. D4×#272C22semidirect
28. C2×C4⋊#272C22semidirect
32. Q16×H(3,𝔽3)C22semidirect
33. SD16×H(3,𝔽3)C22semidirect
34. M16×H(3,𝔽3)C22semidirect
35. #88550C22semidirect
36. #88549C22semidirect
37. #88549C22semidirect
38. #88545C22semidirect
39. #88538C22semidirect
40. #88529C22semidirect
41. #88528C22semidirect
42. #88527C22semidirect
43. C8×#272C22semidirect
46. C32×#137D3semidirect
93. C2×C32×Q8C2×D3semidirect
111. C32×Q16C2×D3semidirect
112. C32×SD16C2×D3semidirect
113. C32×M16C2×D3semidirect
286. #137#272semidirect
293. H(3,𝔽3)#575semidirect
382. C2×Q8C2×#272semidirect
391. Q16C2×#272semidirect
395. SD16C2×#272semidirect
400. M16C2×#272semidirect
434. C32#5777semidirect

Maximal subgroups

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

subgroupindex
#41×#2722
C2×#887482
H(3,𝔽3)×#1372
#117218702
#117218592
#117218082
#117218072
#117218022
#117218002
#117217872
M16×#2722
C3×#57773