Finite Group Databasebeta

← #13136398

Subgroups — #13136398

Order 1728.

500 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C27×#123C211yesmaximal, characteristic, complemented
2#11720187C211yesmaximal, characteristic, complemented
3#11720186C211yesmaximal, characteristic, complemented
4#11720179C211yesmaximal, characteristic, complemented
5C4×#6171C211yesmaximal, characteristic, complemented
6#11720147C211yesmaximal, characteristic, complemented
7#11720146C211yesmaximal, characteristic, complemented
8#11720143C211yesmaximal, characteristic, complemented
9#11720142C211yesmaximal, characteristic, complemented
10#11720134C211yesmaximal, characteristic, complemented
11#11720134C211yesmaximal, characteristic, complemented
12#11720133C211yesmaximal, characteristic, complemented
13#11720131C211yesmaximal, characteristic, complemented
14C22⋊C4×D27C211yesmaximal, characteristic, complemented
15#11720128C211yesmaximal, characteristic, complemented
16#10589759—31nomaximal
17C23×D27C2211yescharacteristic
18C2×D4×C27C2211yescharacteristic
19C22×C4×C27C2211yescharacteristic, complemented
20C2×#6171C2211yescharacteristic
21C2×#6171C2211yescharacteristic
22C2×#6171C2211yescharacteristic
23C2×#6171C2211yescharacteristic, complemented
24C2×D108C2211yescharacteristic
25C2×D108C2211yescharacteristic
26C2×C4×D27C2211yescharacteristic
27C2×C4×D27C2211yescharacteristic
28C2×C4×D27C2211yescharacteristic, complemented
29C2×Dic54C2211yescharacteristic, complemented
30C4⋊C4×C27C2211yescharacteristic, complemented
31C4⋊C4×C27C2211yescharacteristic, complemented
32C22⋊C4×C27C2211yescharacteristic
33C22⋊C4×C27C2211yescharacteristic
34C22⋊C4×C27C2211yescharacteristic, complemented
35#88372C2211yescharacteristic
36#88372C2211yescharacteristic
37#88372C2211yescharacteristic, complemented
38(C2×D27)⋊C4C2211yescharacteristic
39(C2×D27)⋊C4C2211yescharacteristic
40(C2×D27)⋊C4C2211yescharacteristic
41(C2×D27)⋊C4C2211yescharacteristic
42(C2×D27)⋊C4C2211yescharacteristic
43(C2×D27)⋊C4C2211yescharacteristic
44(C2×D27)⋊C4C2211yescharacteristic, complemented
45(C2×D27)⋊C4C2211yescharacteristic, complemented
46C4⋊Dic27C2211yescharacteristic, complemented
47C4⋊Dic27C2211yescharacteristic, complemented
48Dic27⋊C4C2211yescharacteristic, complemented
49Dic27⋊C4C2211yescharacteristic, complemented
50C4×Dic27C2211yescharacteristic, complemented
51C4×Dic27C2211yescharacteristic, complemented
52C9×#123D311yescharacteristic
53#63500—31no—
54#63499—31no—
55#63492—31no—
56C4×#611—31no—
57#63460—31no—
58#63459—31no—
59#63456—31no—
60#63455—31no—
61#63447—31no—
62#63447—31no—
63#63446—31no—
64#63444—31no—
65C22⋊C4×D9—31no—
66#63441—31no—
67C23×C27C2311yescharacteristic
68C23×C27C2311yescharacteristic
69C22×D27—22no—
70C22×D27—41no—
71C22×D27—41no—
72C22×D27C2311yescharacteristic
73C22×D27C2311yescharacteristic
74C22×D27C2311yescharacteristic
75D4×C27—41no—
76C2×C4×C27—41no—
77C2×C4×C27C2311yescharacteristic
78C2×C4×C27C2311yescharacteristic
79C2×C4×C27C2311yescharacteristic
80C2×C4×C27C2311yescharacteristic
81C2×C4×C27C2311yescharacteristic
82#6171—22no—
83#6171—41no—
84#6171—41no—
85#6171—41no—
86C2×Dic27C2311yescharacteristic
87C2×Dic27C2311yescharacteristic
88C2×Dic27C2311yescharacteristic
89C2×Dic27C2311yescharacteristic
90C2×Dic27C2311yescharacteristic
91D108—22no—
92D108—41no—
93C4×D27—41no—
94C4×D27—41no—
95C4×D27—41no—
96Dic54—41no—
97#5657—91no—
98C23×D9—31no—
99C2×D4×C9C2×D311yescharacteristic
100C22×C4×C9C2×D311yescharacteristic
101C2×#611—31no—
102C2×#611—31no—
103C2×#611—31no—
104C2×#611—31no—
105C2×D36—31no—
106C2×D36—31no—
107C2×C4×D9—31no—
108C2×C4×D9—31no—
109C2×C4×D9—31no—
110C2×Dic18—31no—
111C9×C4⋊C4C2×D311yescharacteristic
112C9×C4⋊C4C2×D311yescharacteristic
113C9×C22⋊C4C2×D311yescharacteristic
114C9×C22⋊C4C2×D311yescharacteristic
115C9×C22⋊C4C2×D311yescharacteristic
116#3676—31no—
117#3676—31no—
118#3676—31no—
119(C2×D9)⋊C4—31no—
120(C2×D9)⋊C4—31no—
121(C2×D9)⋊C4—31no—
122(C2×D9)⋊C4—31no—
123(C2×D9)⋊C4—31no—
124(C2×D9)⋊C4—31no—
125(C2×D9)⋊C4—31no—
126(C2×D9)⋊C4—31no—
127C4⋊Dic9—31no—
128C4⋊Dic9—31no—
129Dic9⋊C4—31no—
130Dic9⋊C4—31no—
131C4×Dic9—31no—
132C4×Dic9—31no—
133C22×C27—21no—
134C22×C27—21no—
135C22×C27—21no—
136C22×C27—21no—
137C22×C27—21no—
138C22×C27—21no—
139C22×C27C2411yescharacteristic
140C2×D27—21no—
141C2×D27—21no—
142C2×D27—21no—
143C2×D27—21no—
144C2×D27—21no—
145C2×D27—21no—
146C2×D27—21no—
147C2×D27—21no—
148C2×D27—44no—
149C2×D27#4112yes—
150C4×C27—21no—
151C4×C27—21no—
152C4×C27—21no—
153C4×C27—21no—
154C4×C27—21no—
155Dic27—21no—
156Dic27—21no—
157Dic27—21no—
158Dic27—21no—
159Dic27—21no—
160C3×#123D911yescharacteristic
161#938—91no—
162#937—91no—
163#930—91no—
164C4×#67—91no—
165#898—91no—
166#897—91no—
167#894—91no—
168#893—91no—
169#885—91no—
170#885—91no—
171#884—91no—
172#882—91no—
173D3×C22⋊C4—91no—
174#879—91no—
175C23×C9C22×D311yescharacteristic
176C23×C9C22×D311yescharacteristic
177C22×D9—31no—
178C22×D9—31no—
179C22×D9—31no—
180C22×D9—62no—
181C22×D9—121no—
182C22×D9—121no—
183D4×C9—41no—
184C2×C4×C9—41no—
185C2×C4×C9C22×D311yescharacteristic
186C2×C4×C9C22×D311yescharacteristic
187C2×C4×C9C22×D311yescharacteristic
188C2×C4×C9C22×D311yescharacteristic
189C2×C4×C9C22×D311yescharacteristic
190#611—62no—
191#611—121no—
192#611—121no—
193#611—121no—
194C2×Dic9—31no—
195C2×Dic9—31no—
196C2×Dic9—31no—
197C2×Dic9—31no—
198C2×Dic9—31no—
199D36—62no—
200D36—121no—
201C4×D9—121no—
202C4×D9—121no—
203C4×D9—121no—
204Dic18—121no—
205#538—271noSylow
206C2×C27—41no—
207C2×C27—41no—
208C2×C27C2×#4111yescharacteristic
209C2×C27H(5,𝔽2)11yescharacteristic
210C2×C27H(5,𝔽2)11yescharacteristic
211D27—22no—
212D27—41no—
213D27—41no—
214C23×D3—91no—
215C2×C3×D4C2×D911yescharacteristic
216C22×C3×C4C2×D911yescharacteristic
217C2×#67—91no—
218C2×#67—91no—
219C2×#67—91no—
220C2×#67—91no—
221C2×D12—91no—
222C2×D12—91no—
223C2×C4×D3—91no—
224C2×C4×D3—91no—
225C2×C4×D3—91no—
226C2×Dic6—91no—
227C3×C4⋊C4C2×D911yescharacteristic
228C3×C4⋊C4C2×D911yescharacteristic
229C3×C22⋊C4C2×D911yescharacteristic
230C3×C22⋊C4C2×D911yescharacteristic
231C3×C22⋊C4C2×D911yescharacteristic
232#217—91no—
233#217—91no—
234#217—91no—
235(C2×D3)⋊C4—91no—
236(C2×D3)⋊C4—91no—
237(C2×D3)⋊C4—91no—
238(C2×D3)⋊C4—91no—
239(C2×D3)⋊C4—91no—
240(C2×D3)⋊C4—91no—
241(C2×D3)⋊C4—91no—
242(C2×D3)⋊C4—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—91no—
245Dic3⋊C4—91no—
246Dic3⋊C4—91no—
247C4×Dic3—91no—
248C4×Dic3—91no—
249C22×C9—21no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9—21no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9C23×D311yescharacteristic
256C2×D9—32no—
257C2×D9—61no—
258C2×D9—61no—
259C2×D9—61no—
260C2×D9—61no—
261C2×D9—61no—
262C2×D9—61no—
263C2×D9—61no—
264C2×D9—61no—
265C2×D9—124no—
266C4×C9—21no—
267C4×C9—21no—
268C4×C9—21no—
269C4×C9—21no—
270C4×C9—21no—
271Dic9—61no—
272Dic9—61no—
273Dic9—61no—
274Dic9—61no—
275Dic9—61no—
276#126—271no—
277#126—271no—
278#124—271no—
279#124—271no—
280#123—271no—
281#123D2711yescharacteristic, complemented
282C22⋊Q8—271no—
283#121—271no—
284#121—271no—
285#121—271no—
286#120—271no—
287#120—271no—
288C4×D4—271no—
289C4×D4—271no—
290C2×C22⋊C4—271no—
291C27#53811yescharacteristic, complemented, Sylow
292C23×C3C22×D911yescharacteristic
293C23×C3C22×D911yescharacteristic
294C22×D3—91no—
295C22×D3—91no—
296C22×D3—91no—
297C22×D3—182no—
298C22×D3—361no—
299C22×D3—361no—
300C3×D4—41no—
301C2×C3×C4—41no—
302C2×C3×C4C22×D911yescharacteristic
303C2×C3×C4C22×D911yescharacteristic
304C2×C3×C4C22×D911yescharacteristic
305C2×C3×C4C22×D911yescharacteristic
306C2×C3×C4C22×D911yescharacteristic
307#67—182no—
308#67—361no—
309#67—361no—
310#67—361no—
311C2×Dic3—91no—
312C2×Dic3—91no—
313C2×Dic3—91no—
314C2×Dic3—91no—
315C2×Dic3—91no—
316D12—182no—
317D12—361no—
318C4×D3—361no—
319C4×D3—361no—
320C4×D3—361no—
321Dic6—361no—
322C2×C9—41no—
323C2×C9—41no—
324C2×C9#100411yescharacteristic
325C2×C9D3×#4111yescharacteristic
326C2×C9#100911yescharacteristic
327D9—62no—
328D9—121no—
329D9—121no—
330C24—271no—
331C2×Q8—271no—
332C2×D4—271no—
333C2×D4—271no—
334C2×D4—271no—
335C2×D4—271no—
336C2×D4—271no—
337C2×D4—271no—
338C2×D4C2×D2711yescharacteristic
339C22×C4—271no—
340C22×C4—271no—
341C22×C4—271no—
342C22×C4C2×D2711yescharacteristic, complemented
343C4⋊C4—271no—
344C4⋊C4—271no—
345C4⋊C4—271no—
346C4⋊C4—271no—
347C4⋊C4C2×D2711yescharacteristic, complemented
348C4⋊C4C2×D2711yescharacteristic, complemented
349C22⋊C4—271no—
350C22⋊C4—271no—
351C22⋊C4—271no—
352C22⋊C4—271no—
353C22⋊C4—271no—
354C22⋊C4—271no—
355C22⋊C4—271no—
356C22⋊C4—271no—
357C22⋊C4—271no—
358C22⋊C4—271no—
359C22⋊C4—271no—
360C22⋊C4C2×D2711yescharacteristic
361C22⋊C4C2×D2711yescharacteristic
362C22⋊C4C2×D2711yescharacteristic, complemented
363C42—271no—
364C42—271no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3C23×D911yescharacteristic
372C2×D3—92no—
373C2×D3—181no—
374C2×D3—181no—
375C2×D3—181no—
376C2×D3—181no—
377C2×D3—181no—
378C2×D3—181no—
379C2×D3—181no—
380C2×D3—181no—
381C2×D3—364no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387Dic3—181no—
388Dic3—181no—
389Dic3—181no—
390Dic3—181no—
391Dic3—181no—
392C9#565711yescharacteristic
393C23—271no—
394C23—271no—
395C23—271no—
396C23—542no—
397C23—1081no—
398C23—1081no—
399C23C22×D2711yescharacteristic
400C23C22×D2711yescharacteristic
401Q8—1081no—
402D4—41no—
403D4—542no—
404D4—542no—
405D4—1081no—
406D4—1081no—
407D4—1081no—
408D4—1081no—
409C2×C4—41no—
410C2×C4—271no—
411C2×C4—271no—
412C2×C4—271no—
413C2×C4—271no—
414C2×C4—271no—
415C2×C4—1081no—
416C2×C4—1081no—
417C2×C4—1081no—
418C2×C4C22×D2711yescharacteristic
419C2×C4C22×D2711yescharacteristic
420C2×C4C22×D2711yescharacteristic
421C2×C4C22×D2711yescharacteristic
422C2×C4C22×D2711yescharacteristic
423C2×C3—41no—
424C2×C3—41no—
425C2×C3#6371011yescharacteristic
426C2×C3#41×D911yescharacteristic
427C2×C3#6371511yescharacteristic
428D3—182no—
429D3—361no—
430D3—361no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—272no—
438C22—541no—
439C22—541no—
440C22—541no—
441C22—541no—
442C22—541no—
443C22—541no—
444C22—541no—
445C22—541no—
446C22—1084no—
447C22C23×D2711yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—541no—
454C4—541no—
455C4—541no—
456C4—541no—
457C4—541no—
458C3#1058975911yescharacteristic
459C2—41no—
460C2—41no—
461C2—542no—
462C2—1081no—
463C2—1081no—
464C2#1172067011yescharacteristic
465C2#41×D2711yescharacteristic
466C2#1172067511yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#123C2
socleC22×C3C23×D9
hypercenter#123D27
hypocenterC27#538
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C27×#123C2semidirect
2. #11720187C2semidirect
3. #11720186C2semidirect
4. #11720179C2semidirect
5. C4×#6171C2semidirect
6. #11720147C2semidirect
7. #11720146C2semidirect
8. #11720143C2semidirect
9. #11720142C2semidirect
10. #11720134C2semidirect
11. #11720134C2semidirect
12. #11720133C2semidirect
13. #11720131C2semidirect
14. C22⋊C4×D27C2semidirect
15. #11720128C2semidirect
19. C22×C4×C27C22semidirect
23. C2×#6171C22semidirect
28. C2×C4×D27C22semidirect
29. C2×Dic54C22semidirect
30. C4⋊C4×C27C22semidirect
31. C4⋊C4×C27C22semidirect
34. C22⋊C4×C27C22semidirect
37. #88372C22semidirect
44. (C2×D27)⋊C4C22semidirect
45. (C2×D27)⋊C4C22semidirect
46. C4⋊Dic27C22semidirect
47. C4⋊Dic27C22semidirect
48. Dic27⋊C4C22semidirect
49. Dic27⋊C4C22semidirect
50. C4×Dic27C22semidirect
51. C4×Dic27C22semidirect
281. #123D27semidirect
291. C27#538semidirect
342. C22×C4C2×D27semidirect
347. C4⋊C4C2×D27semidirect
348. C4⋊C4C2×D27semidirect
362. C22⋊C4C2×D27semidirect

Maximal subgroups

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

subgroupindex
C27×#1232
#117201872
#117201862
#117201792
C4×#61712
#117201472
#117201462
#117201432
#117201422
#117201342
#117201332
#117201312
C22⋊C4×D272
#117201282
#105897593