Finite Group Databasebeta

← #13136444

Subgroups — #13136444

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×#126C211yesmaximal, characteristic, complemented
2#11720146C211yesmaximal, characteristic, complemented
3#11720145C211yesmaximal, characteristic, complemented
4#11720143C211yesmaximal, characteristic, complemented
5#11720143C211yesmaximal, characteristic, complemented
6#11720142C211yesmaximal, characteristic, complemented
7#11720141C211yesmaximal, characteristic, complemented
8#11720134C211yesmaximal, characteristic, complemented
9#11720132C211yesmaximal, characteristic, complemented
10#11720131C211yesmaximal, characteristic
11C22⋊C4×D27C211yesmaximal, characteristic, complemented
12C22⋊C4×D27C211yesmaximal, characteristic, complemented
13#11720128C211yesmaximal, characteristic, complemented
14C4×D108C211yesmaximal, characteristic, complemented
15#11720121C211yesmaximal, characteristic, complemented
16#10589807—31nomaximal
17C23×D27C2211yescharacteristic
18C2×#6171C2211yescharacteristic
19C2×D108C2211yescharacteristic
20C2×D108C2211yescharacteristic
21C2×C4×D27C2211yescharacteristic
22C2×C4×D27C2211yescharacteristic
23C2×C4×D27C2211yescharacteristic
24C2×C4×D27C2211yescharacteristic
25C2×C4×D27C2211yescharacteristic, complemented
26C2×Dic54C2211yescharacteristic, complemented
27C4⋊C4×C27C2211yescharacteristic, complemented
28C4⋊C4×C27C2211yescharacteristic, complemented
29C4⋊C4×C27C2211yescharacteristic, complemented
30C22⋊C4×C27C2211yescharacteristic
31C22⋊C4×C27C2211yescharacteristic
32C22⋊C4×C27C2211yescharacteristic
33C42×C27C2211yescharacteristic, complemented
34#88372C2211yescharacteristic
35#88372C2211yescharacteristic
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, complemented
43(C2×D27)⋊C4C2211yescharacteristic, complemented
44(C2×D27)⋊C4C2211yescharacteristic, complemented
45C4⋊Dic27C2211yescharacteristic, complemented
46C4⋊Dic27C2211yescharacteristic, complemented
47Dic27⋊C4C2211yescharacteristic, complemented
48Dic27⋊C4C2211yescharacteristic, complemented
49Dic27⋊C4C2211yescharacteristic, complemented
50C4×Dic27C2211yescharacteristic, complemented
51C4×Dic27C2211yescharacteristic, complemented
52C9×#126D311yescharacteristic
53#63459—31no—
54#63458—31no—
55#63456—31no—
56#63456—31no—
57#63455—31no—
58#63454—31no—
59#63447—31no—
60#63445—31no—
61#63444—31no—
62C22⋊C4×D9—31no—
63C22⋊C4×D9—31no—
64#63441—31no—
65C4×D36—31no—
66#63434—31no—
67C23×C27C2311yescharacteristic
68C22×D27—22no—
69C22×D27—22no—
70C22×D27—41no—
71C22×D27C2311yescharacteristic
72C22×D27C2311yescharacteristic
73C22×D27C2311yescharacteristic
74C2×C4×C27C2311yescharacteristic
75C2×C4×C27C2311yescharacteristic
76C2×C4×C27C2311yescharacteristic
77C2×C4×C27C2311yescharacteristic
78C2×C4×C27C2311yescharacteristic
79C2×C4×C27C2311yescharacteristic
80#6171—41no—
81C2×Dic27C2311yescharacteristic
82C2×Dic27C2311yescharacteristic
83C2×Dic27C2311yescharacteristic
84C2×Dic27C2311yescharacteristic
85C2×Dic27C2311yescharacteristic
86D108—22no—
87D108—22no—
88C4×D27—22no—
89C4×D27—22no—
90C4×D27—41no—
91C4×D27—41no—
92C4×D27—41no—
93Dic54—41no—
94#5705—91no—
95C23×D9—31no—
96C2×#611—31no—
97C2×D36—31no—
98C2×D36—31no—
99C2×C4×D9—31no—
100C2×C4×D9—31no—
101C2×C4×D9—31no—
102C2×C4×D9—31no—
103C2×C4×D9—31no—
104C2×Dic18—31no—
105C9×C4⋊C4C2×D311yescharacteristic
106C9×C4⋊C4C2×D311yescharacteristic
107C9×C4⋊C4C2×D311yescharacteristic
108C9×C22⋊C4C2×D311yescharacteristic
109C9×C22⋊C4C2×D311yescharacteristic
110C9×C22⋊C4C2×D311yescharacteristic
111C42×C9C2×D311yescharacteristic
112#3676—31no—
113#3676—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—
121(C2×D9)⋊C4—31no—
122(C2×D9)⋊C4—31no—
123C4⋊Dic9—31no—
124C4⋊Dic9—31no—
125Dic9⋊C4—31no—
126Dic9⋊C4—31no—
127Dic9⋊C4—31no—
128C4×Dic9—31no—
129C4×Dic9—31no—
130C22×C27—21no—
131C22×C27—21no—
132C22×C27—21no—
133C22×C27C2411yescharacteristic
134C2×D27—21no—
135C2×D27—21no—
136C2×D27—21no—
137C2×D27—21no—
138C2×D27—21no—
139C2×D27—21no—
140C2×D27—21no—
141C2×D27—44no—
142C2×D27#4112yes—
143C2×D27#4112yes—
144C4×C27—21no—
145C4×C27—21no—
146C4×C27—21no—
147C4×C27—21no—
148C4×C27—21no—
149C4×C27—21no—
150Dic27—21no—
151Dic27—21no—
152Dic27—21no—
153Dic27—21no—
154Dic27—21no—
155C3×#126D911yescharacteristic
156#897—91no—
157#896—91no—
158#894—91no—
159#894—91no—
160#893—91no—
161#892—91no—
162#885—91no—
163#883—91no—
164#882—91no—
165D3×C22⋊C4—91no—
166D3×C22⋊C4—91no—
167#879—91no—
168C4×D12—91no—
169#872—91no—
170C23×C9C22×D311yescharacteristic
171C22×D9—31no—
172C22×D9—31no—
173C22×D9—31no—
174C22×D9—62no—
175C22×D9—62no—
176C22×D9—121no—
177C2×C4×C9C22×D311yescharacteristic
178C2×C4×C9C22×D311yescharacteristic
179C2×C4×C9C22×D311yescharacteristic
180C2×C4×C9C22×D311yescharacteristic
181C2×C4×C9C22×D311yescharacteristic
182C2×C4×C9C22×D311yescharacteristic
183#611—121no—
184C2×Dic9—31no—
185C2×Dic9—31no—
186C2×Dic9—31no—
187C2×Dic9—31no—
188C2×Dic9—31no—
189D36—62no—
190D36—62no—
191C4×D9—62no—
192C4×D9—62no—
193C4×D9—121no—
194C4×D9—121no—
195C4×D9—121no—
196Dic18—121no—
197#551—271noSylow
198C2×C27—41no—
199C2×C27C2×#4111yescharacteristic
200C2×C27C2×#4111yescharacteristic
201C2×C27H(5,𝔽2)11yescharacteristic
202D27—22no—
203D27—22no—
204D27—41no—
205C23×D3—91no—
206C2×#67—91no—
207C2×D12—91no—
208C2×D12—91no—
209C2×C4×D3—91no—
210C2×C4×D3—91no—
211C2×C4×D3—91no—
212C2×C4×D3—91no—
213C2×C4×D3—91no—
214C2×Dic6—91no—
215C3×C4⋊C4C2×D911yescharacteristic
216C3×C4⋊C4C2×D911yescharacteristic
217C3×C4⋊C4C2×D911yescharacteristic
218C3×C22⋊C4C2×D911yescharacteristic
219C3×C22⋊C4C2×D911yescharacteristic
220C3×C22⋊C4C2×D911yescharacteristic
221C3×C42C2×D911yescharacteristic
222#217—91no—
223#217—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—
231(C2×D3)⋊C4—91no—
232(C2×D3)⋊C4—91no—
233C4⋊Dic3—91no—
234C4⋊Dic3—91no—
235Dic3⋊C4—91no—
236Dic3⋊C4—91no—
237Dic3⋊C4—91no—
238C4×Dic3—91no—
239C4×Dic3—91no—
240C22×C9—21no—
241C22×C9—21no—
242C22×C9—21no—
243C22×C9C23×D311yescharacteristic
244C2×D9—32no—
245C2×D9—32no—
246C2×D9—61no—
247C2×D9—61no—
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—
259C4×C9—21no—
260Dic9—61no—
261Dic9—61no—
262Dic9—61no—
263Dic9—61no—
264Dic9—61no—
265#126—271no—
266#126D2711yescharacteristic, complemented
267#124—271no—
268#123—271no—
269#123—271no—
270#123—271no—
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
282C22×D3—91no—
283C22×D3—91no—
284C22×D3—91no—
285C22×D3—182no—
286C22×D3—182no—
287C22×D3—361no—
288C2×C3×C4C22×D911yescharacteristic
289C2×C3×C4C22×D911yescharacteristic
290C2×C3×C4C22×D911yescharacteristic
291C2×C3×C4C22×D911yescharacteristic
292C2×C3×C4C22×D911yescharacteristic
293C2×C3×C4C22×D911yescharacteristic
294#67—361no—
295C2×Dic3—91no—
296C2×Dic3—91no—
297C2×Dic3—91no—
298C2×Dic3—91no—
299C2×Dic3—91no—
300D12—182no—
301D12—182no—
302C4×D3—182no—
303C4×D3—182no—
304C4×D3—361no—
305C4×D3—361no—
306C4×D3—361no—
307Dic6—361no—
308C2×C9—41no—
309C2×C9D3×#4111yescharacteristic
310C2×C9D3×#4111yescharacteristic
311C2×C9#100911yescharacteristic
312D9—62no—
313D9—62no—
314D9—121no—
315C24—271no—
316C2×Q8—271no—
317C2×D4—271no—
318C2×D4—271no—
319C2×D4—271no—
320C22×C4—271no—
321C22×C4—271no—
322C22×C4—271no—
323C22×C4—271no—
324C22×C4—271no—
325C4⋊C4—271no—
326C4⋊C4—271no—
327C4⋊C4—271no—
328C4⋊C4—271no—
329C4⋊C4—271no—
330C4⋊C4C2×D2711yescharacteristic, complemented
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
346C22⋊C4C2×D2711yescharacteristic
347C42—271no—
348C42—271no—
349C42C2×D2711yescharacteristic, complemented
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3C23×D911yescharacteristic
354C2×D3—92no—
355C2×D3—92no—
356C2×D3—181no—
357C2×D3—181no—
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—
369C3×C4—21no—
370Dic3—181no—
371Dic3—181no—
372Dic3—181no—
373Dic3—181no—
374Dic3—181no—
375C9#570511yescharacteristic
376C23—271no—
377C23—271no—
378C23—271no—
379C23—542no—
380C23—542no—
381C23—1081no—
382C23C22×D2711yescharacteristic
383Q8—1081no—
384D4—542no—
385D4—542no—
386D4—1081no—
387C2×C4—271no—
388C2×C4—271no—
389C2×C4—271no—
390C2×C4—271no—
391C2×C4—271no—
392C2×C4—542no—
393C2×C4—542no—
394C2×C4—1081no—
395C2×C4—1081no—
396C2×C4—1081no—
397C2×C4C22×D2711yescharacteristic
398C2×C4C22×D2711yescharacteristic
399C2×C4C22×D2711yescharacteristic
400C2×C4C22×D2711yescharacteristic
401C2×C4C22×D2711yescharacteristic
402C2×C4C22×D2711yescharacteristic
403C2×C3—41no—
404C2×C3#41×D911yescharacteristic
405C2×C3#41×D911yescharacteristic
406C2×C3#6371511yescharacteristic
407D3—182no—
408D3—182no—
409D3—361no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—272no—
414C22—272no—
415C22—541no—
416C22—541no—
417C22—541no—
418C22—541no—
419C22—541no—
420C22—541no—
421C22—541no—
422C22—1084no—
423C22C23×D2711yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—541no—
431C4—541no—
432C4—541no—
433C4—541no—
434C4—541no—
435C3#1058980711yescharacteristic
436C2—41no—
437C2—542no—
438C2—542no—
439C2—1081no—
440C2#41×D2711yescharacteristic
441C2#41×D2711yescharacteristic
442C2#1172067511yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#126C2
socleC22×C3C23×D9
prime radical#11720131C2
hypercenter#126D27
hypocenterC27#551
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C27×#126C2semidirect
2. #11720146C2semidirect
3. #11720145C2semidirect
4. #11720143C2semidirect
5. #11720143C2semidirect
6. #11720142C2semidirect
7. #11720141C2semidirect
8. #11720134C2semidirect
9. #11720132C2semidirect
11. C22⋊C4×D27C2semidirect
12. C22⋊C4×D27C2semidirect
13. #11720128C2semidirect
14. C4×D108C2semidirect
15. #11720121C2semidirect
25. C2×C4×D27C22semidirect
26. C2×Dic54C22semidirect
27. C4⋊C4×C27C22semidirect
28. C4⋊C4×C27C22semidirect
29. C4⋊C4×C27C22semidirect
33. C42×C27C22semidirect
42. (C2×D27)⋊C4C22semidirect
43. (C2×D27)⋊C4C22semidirect
44. (C2×D27)⋊C4C22semidirect
45. C4⋊Dic27C22semidirect
46. C4⋊Dic27C22semidirect
47. Dic27⋊C4C22semidirect
48. Dic27⋊C4C22semidirect
49. Dic27⋊C4C22semidirect
50. C4×Dic27C22semidirect
51. C4×Dic27C22semidirect
266. #126D27semidirect
280. C27#551semidirect
330. C4⋊C4C2×D27semidirect
331. C4⋊C4C2×D27semidirect
332. C4⋊C4C2×D27semidirect
349. C42C2×D27semidirect

Maximal subgroups

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

subgroupindex
C27×#1262
#117201462
#117201452
#117201432
#117201422
#117201412
#117201342
#117201322
#117201312
C22⋊C4×D272
#117201282
C4×D1082
#117201212
#105898073