Finite Group Databasebeta

← #13136399

Subgroups — #13136399

Order 1728.

496 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C27×#123C211yesmaximal, characteristic, complemented
2#11720186C211yesmaximal, characteristic
3D4×Dic27C211yesmaximal, characteristic, complemented
4C2×(C2×D27)⋊C4C211yesmaximal, characteristic, complemented
5#11720175C211yesmaximal, characteristic, complemented
6#11720147C211yesmaximal, characteristic, complemented
7#11720147C211yesmaximal, characteristic, complemented
8#11720146C211yesmaximal, characteristic, complemented
9#11720141C211yesmaximal, characteristic, complemented
10#11720135C211yesmaximal, characteristic, complemented
11#11720134C211yesmaximal, characteristic, complemented
12#11720132C211yesmaximal, characteristic, complemented
13#6171⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D27C211yesmaximal, characteristic, complemented
15#11720127C211yesmaximal, characteristic, complemented
16#10589760—31nomaximal
17C23×D27C2211yescharacteristic
18C2×D4×C27C2211yescharacteristic
19C22×C4×C27C2211yescharacteristic
20C2×#6171C2211yescharacteristic
21C2×#6171C2211yescharacteristic
22C22×Dic27C2211yescharacteristic
23C22×Dic27C2211yescharacteristic, complemented
24C2×C4×D27C2211yescharacteristic
25C2×C4×D27C2211yescharacteristic
26C2×Dic54C2211yescharacteristic, complemented
27C4⋊C4×C27C2211yescharacteristic, complemented
28C4⋊C4×C27C2211yescharacteristic, complemented
29C22⋊C4×C27C2211yescharacteristic
30C22⋊C4×C27C2211yescharacteristic, complemented
31C22⋊C4×C27C2211yescharacteristic, complemented
32#88372C2211yescharacteristic
33#88372C2211yescharacteristic
34#88372C2211yescharacteristic, complemented
35(C2×D27)⋊C4C2211yescharacteristic
36(C2×D27)⋊C4C2211yescharacteristic
37(C2×D27)⋊C4C2211yescharacteristic
38(C2×D27)⋊C4C2211yescharacteristic
39(C2×D27)⋊C4C2211yescharacteristic
40(C2×D27)⋊C4C2211yescharacteristic
41(C2×D27)⋊C4C2211yescharacteristic
42(C2×D27)⋊C4C2211yescharacteristic
43C4⋊Dic27C2211yescharacteristic, complemented
44C4⋊Dic27C2211yescharacteristic, complemented
45C4⋊Dic27C2211yescharacteristic, complemented
46C4⋊Dic27C2211yescharacteristic, complemented
47Dic27⋊C4C2211yescharacteristic, complemented
48Dic27⋊C4C2211yescharacteristic, complemented
49C4×Dic27C2211yescharacteristic, complemented
50C4×Dic27C2211yescharacteristic, complemented
51C4×Dic27C2211yescharacteristic, complemented
52C9×#123D311yescharacteristic
53#63499—31no—
54D4×Dic9—31no—
55C2×(C2×D9)⋊C4—31no—
56#63488—31no—
57#63460—31no—
58#63460—31no—
59#63459—31no—
60#63454—31no—
61#63448—31no—
62#63447—31no—
63#63445—31no—
64#611⋊C4—31no—
65C22⋊C4×D9—31no—
66#63440—31no—
67C23×C27C2311yescharacteristic
68C23×C27C2311yescharacteristic
69C22×D27—22no—
70C22×D27—22no—
71C22×D27—41no—
72C22×D27C2311yescharacteristic
73C22×D27C2311yescharacteristic
74D4×C27—22no—
75C2×C4×C27—22no—
76C2×C4×C27C2311yescharacteristic
77C2×C4×C27C2311yescharacteristic
78C2×C4×C27C2311yescharacteristic
79C2×C4×C27C2311yescharacteristic
80C2×C4×C27C2311yescharacteristic
81#6171—22no—
82#6171—41no—
83C2×Dic27—41no—
84C2×Dic27—41no—
85C2×Dic27C2311yescharacteristic
86C2×Dic27C2311yescharacteristic
87C2×Dic27C2311yescharacteristic
88C2×Dic27C2311yescharacteristic
89C2×Dic27C2311yescharacteristic
90C2×Dic27C2311yescharacteristic
91C4×D27—22no—
92C4×D27—41no—
93Dic54—41no—
94#5658—91no—
95C23×D9—31no—
96C2×D4×C9C2×D311yescharacteristic
97C22×C4×C9C2×D311yescharacteristic
98C2×#611—31no—
99C2×#611—31no—
100C22×Dic9—31no—
101C22×Dic9—31no—
102C2×C4×D9—31no—
103C2×C4×D9—31no—
104C2×Dic18—31no—
105C9×C4⋊C4C2×D311yescharacteristic
106C9×C4⋊C4C2×D311yescharacteristic
107C9×C22⋊C4C2×D311yescharacteristic
108C9×C22⋊C4C2×D311yescharacteristic
109C9×C22⋊C4C2×D311yescharacteristic
110#3676—31no—
111#3676—31no—
112#3676—31no—
113(C2×D9)⋊C4—31no—
114(C2×D9)⋊C4—31no—
115(C2×D9)⋊C4—31no—
116(C2×D9)⋊C4—31no—
117(C2×D9)⋊C4—31no—
118(C2×D9)⋊C4—31no—
119(C2×D9)⋊C4—31no—
120(C2×D9)⋊C4—31no—
121C4⋊Dic9—31no—
122C4⋊Dic9—31no—
123C4⋊Dic9—31no—
124C4⋊Dic9—31no—
125Dic9⋊C4—31no—
126Dic9⋊C4—31no—
127C4×Dic9—31no—
128C4×Dic9—31no—
129C4×Dic9—31no—
130C22×C27—21no—
131C22×C27—21no—
132C22×C27—21no—
133C22×C27—21no—
134C22×C27—21no—
135C22×C27#4112yes—
136C22×C27C2411yescharacteristic
137C2×D27—21no—
138C2×D27—21no—
139C2×D27—21no—
140C2×D27—21no—
141C2×D27—21no—
142C2×D27—44no—
143C2×D27#4112yes—
144C4×C27—21no—
145C4×C27—21no—
146C4×C27—21no—
147C4×C27—21no—
148C4×C27—21no—
149Dic27—21no—
150Dic27—21no—
151Dic27—21no—
152Dic27—21no—
153Dic27—21no—
154Dic27—21no—
155C3×#123D911yescharacteristic
156#937—91no—
157D4×Dic3—91no—
158C2×(C2×D3)⋊C4—91no—
159#926—91no—
160#898—91no—
161#898—91no—
162#897—91no—
163#892—91no—
164#886—91no—
165#885—91no—
166#883—91no—
167#67⋊C4—91no—
168D3×C22⋊C4—91no—
169#878—91no—
170C23×C9C22×D311yescharacteristic
171C23×C9C22×D311yescharacteristic
172C22×D9—31no—
173C22×D9—31no—
174C22×D9—62no—
175C22×D9—62no—
176C22×D9—121no—
177D4×C9—22no—
178C2×C4×C9—22no—
179C2×C4×C9C22×D311yescharacteristic
180C2×C4×C9C22×D311yescharacteristic
181C2×C4×C9C22×D311yescharacteristic
182C2×C4×C9C22×D311yescharacteristic
183C2×C4×C9C22×D311yescharacteristic
184#611—62no—
185#611—121no—
186C2×Dic9—31no—
187C2×Dic9—31no—
188C2×Dic9—31no—
189C2×Dic9—31no—
190C2×Dic9—31no—
191C2×Dic9—31no—
192C2×Dic9—121no—
193C2×Dic9—121no—
194C4×D9—62no—
195C4×D9—121no—
196Dic18—121no—
197#551—271noSylow
198C2×C27—22no—
199C2×C27—41no—
200C2×C27C2×#4111yescharacteristic
201C2×C27C2×#4111yescharacteristic
202C2×C27H(5,𝔽2)11yescharacteristic
203D27—22no—
204D27—41no—
205C23×D3—91no—
206C2×C3×D4C2×D911yescharacteristic
207C22×C3×C4C2×D911yescharacteristic
208C2×#67—91no—
209C2×#67—91no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C2×C4×D3—91no—
213C2×C4×D3—91no—
214C2×Dic6—91no—
215C3×C4⋊C4C2×D911yescharacteristic
216C3×C4⋊C4C2×D911yescharacteristic
217C3×C22⋊C4C2×D911yescharacteristic
218C3×C22⋊C4C2×D911yescharacteristic
219C3×C22⋊C4C2×D911yescharacteristic
220#217—91no—
221#217—91no—
222#217—91no—
223(C2×D3)⋊C4—91no—
224(C2×D3)⋊C4—91no—
225(C2×D3)⋊C4—91no—
226(C2×D3)⋊C4—91no—
227(C2×D3)⋊C4—91no—
228(C2×D3)⋊C4—91no—
229(C2×D3)⋊C4—91no—
230(C2×D3)⋊C4—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—91no—
233C4⋊Dic3—91no—
234C4⋊Dic3—91no—
235Dic3⋊C4—91no—
236Dic3⋊C4—91no—
237C4×Dic3—91no—
238C4×Dic3—91no—
239C4×Dic3—91no—
240C22×C9—21no—
241C22×C9—21no—
242C22×C9—21no—
243C22×C9—21no—
244C22×C9—21no—
245C22×C9#23712yes—
246C22×C9C23×D311yescharacteristic
247C2×D9—32no—
248C2×D9—61no—
249C2×D9—61no—
250C2×D9—61no—
251C2×D9—61no—
252C2×D9—61no—
253C2×D9—124no—
254C4×C9—21no—
255C4×C9—21no—
256C4×C9—21no—
257C4×C9—21no—
258C4×C9—21no—
259Dic9—61no—
260Dic9—61no—
261Dic9—61no—
262Dic9—61no—
263Dic9—61no—
264Dic9—61no—
265#126—271no—
266#126—271no—
267#124—271no—
268#123—271no—
269#123—271no—
270#123D2711yescharacteristic, complemented
271C22⋊Q8—271no—
272C22⋊Q8—271no—
273#120—271no—
274C4×D4—271no—
275C4×D4—271no—
276#117—271no—
277#117—271no—
278C2×C22⋊C4—271no—
279C2×C22⋊C4—271no—
280C27#55111yescharacteristic, complemented, Sylow
281C23×C3C22×D911yescharacteristic
282C23×C3C22×D911yescharacteristic
283C22×D3—91no—
284C22×D3—91no—
285C22×D3—182no—
286C22×D3—182no—
287C22×D3—361no—
288C3×D4—22no—
289C2×C3×C4—22no—
290C2×C3×C4C22×D911yescharacteristic
291C2×C3×C4C22×D911yescharacteristic
292C2×C3×C4C22×D911yescharacteristic
293C2×C3×C4C22×D911yescharacteristic
294C2×C3×C4C22×D911yescharacteristic
295#67—182no—
296#67—361no—
297C2×Dic3—91no—
298C2×Dic3—91no—
299C2×Dic3—91no—
300C2×Dic3—91no—
301C2×Dic3—91no—
302C2×Dic3—91no—
303C2×Dic3—361no—
304C2×Dic3—361no—
305C4×D3—182no—
306C4×D3—361no—
307Dic6—361no—
308C2×C9—22no—
309C2×C9—41no—
310C2×C9C2×#23711yescharacteristic
311C2×C9D3×#4111yescharacteristic
312C2×C9#100911yescharacteristic
313D9—62no—
314D9—121no—
315C24—271no—
316C2×Q8—271no—
317C2×D4—271no—
318C2×D4—271no—
319C2×D4C2×D2711yescharacteristic
320C22×C4—271no—
321C22×C4—271no—
322C22×C4—271no—
323C22×C4—271no—
324C22×C4C2×D2711yescharacteristic
325C4⋊C4—271no—
326C4⋊C4—271no—
327C4⋊C4—271no—
328C4⋊C4—271no—
329C4⋊C4—271no—
330C4⋊C4—271no—
331C4⋊C4C2×D2711yescharacteristic, complemented
332C4⋊C4C2×D2711yescharacteristic, complemented
333C22⋊C4—271no—
334C22⋊C4—271no—
335C22⋊C4—271no—
336C22⋊C4—271no—
337C22⋊C4—271no—
338C22⋊C4—271no—
339C22⋊C4—271no—
340C22⋊C4—271no—
341C22⋊C4—271no—
342C22⋊C4—271no—
343C22⋊C4—271no—
344C22⋊C4C2×D2711yescharacteristic
345C22⋊C4C2×D2711yescharacteristic, complemented
346C22⋊C4C2×D2711yescharacteristic, complemented
347C42—271no—
348C42—271no—
349C42—271no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3#369912yes—
356C22×C3C23×D911yescharacteristic
357C2×D3—92no—
358C2×D3—181no—
359C2×D3—181no—
360C2×D3—181no—
361C2×D3—181no—
362C2×D3—181no—
363C2×D3—364no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369Dic3—181no—
370Dic3—181no—
371Dic3—181no—
372Dic3—181no—
373Dic3—181no—
374Dic3—181no—
375C9#565811yescharacteristic
376C23—271no—
377C23—271no—
378C23—542no—
379C23—542no—
380C23—1081no—
381C23C22×D2711yescharacteristic
382C23C22×D2711yescharacteristic
383Q8—1081no—
384D4—22no—
385D4—542no—
386D4—1081no—
387C2×C4—22no—
388C2×C4—271no—
389C2×C4—271no—
390C2×C4—271no—
391C2×C4—271no—
392C2×C4—271no—
393C2×C4—271no—
394C2×C4—542no—
395C2×C4—1081no—
396C2×C4—1081no—
397C2×C4—1081no—
398C2×C4C22×D2711yescharacteristic
399C2×C4C22×D2711yescharacteristic
400C2×C4C22×D2711yescharacteristic
401C2×C4C22×D2711yescharacteristic
402C2×C4C22×D2711yescharacteristic
403C2×C3—22no—
404C2×C3—41no—
405C2×C3C2×#369911yescharacteristic
406C2×C3#41×D911yescharacteristic
407C2×C3#6371511yescharacteristic
408D3—182no—
409D3—361no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—272no—
416C22—541no—
417C22—541no—
418C22—541no—
419C22—541no—
420C22—541no—
421C22—1084no—
422C22#8840112yes—
423C22C23×D2711yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—541no—
430C4—541no—
431C4—541no—
432C4—541no—
433C4—541no—
434C4—541no—
435C3#1058976011yescharacteristic
436C2—22no—
437C2—41no—
438C2—542no—
439C2—1081no—
440C2C2×#8840111yescharacteristic
441C2#41×D2711yescharacteristic
442C2#1172067511yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#123C2
socleC22×C3C23×D9
prime radical#11720186C2
hypercenter#123D27
hypocenterC27#551
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C27×#123C2semidirect
3. D4×Dic27C2semidirect
4. C2×(C2×D27)⋊C4C2semidirect
5. #11720175C2semidirect
6. #11720147C2semidirect
7. #11720147C2semidirect
8. #11720146C2semidirect
9. #11720141C2semidirect
10. #11720135C2semidirect
11. #11720134C2semidirect
12. #11720132C2semidirect
13. #6171⋊C4C2semidirect
14. C22⋊C4×D27C2semidirect
15. #11720127C2semidirect
23. C22×Dic27C22semidirect
26. C2×Dic54C22semidirect
27. C4⋊C4×C27C22semidirect
28. C4⋊C4×C27C22semidirect
30. C22⋊C4×C27C22semidirect
31. C22⋊C4×C27C22semidirect
34. #88372C22semidirect
43. C4⋊Dic27C22semidirect
44. C4⋊Dic27C22semidirect
45. C4⋊Dic27C22semidirect
46. C4⋊Dic27C22semidirect
47. Dic27⋊C4C22semidirect
48. Dic27⋊C4C22semidirect
49. C4×Dic27C22semidirect
50. C4×Dic27C22semidirect
51. C4×Dic27C22semidirect
270. #123D27semidirect
280. C27#551semidirect
331. C4⋊C4C2×D27semidirect
332. C4⋊C4C2×D27semidirect
345. C22⋊C4C2×D27semidirect
346. 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
#117201862
D4×Dic272
C2×(C2×D27)⋊C42
#117201752
#117201472
#117201462
#117201412
#117201352
#117201342
#117201322
#6171⋊C42
C22⋊C4×D272
#117201272
#105897603