Finite Group Databasebeta

← #13136233

Subgroups — #13136233

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
1C2×C22⋊C4×C27C211yesmaximal, characteristic, complemented
2#11720202C211yesmaximal, characteristic
3C2×#88372C211yesmaximal, characteristic, complemented
4#11720178C211yesmaximal, characteristic, complemented
5C4×#6171C211yesmaximal, characteristic, complemented
6#11720175C211yesmaximal, characteristic, complemented
7#11720173C211yesmaximal, characteristic, complemented
8#11720135C211yesmaximal, characteristic, complemented
9#11720134C211yesmaximal, characteristic, complemented
10#11720132C211yesmaximal, characteristic, complemented
11#6171⋊C4C211yesmaximal, characteristic, complemented
12#11720128C211yesmaximal, characteristic, complemented
13#11720128C211yesmaximal, characteristic, complemented
14#11720127C211yesmaximal, characteristic, complemented
15#11720126C211yesmaximal, characteristic, complemented
16#10589597—31nomaximal
17C24×C27C2211yescharacteristic
18C22×C4×C27C2211yescharacteristic
19C22×C4×C27C2211yescharacteristic
20C2×#6171C2211yescharacteristic
21C2×#6171C2211yescharacteristic
22C2×#6171C2211yescharacteristic
23C22×Dic27C2211yescharacteristic
24C22×Dic27C2211yescharacteristic
25C2×C4×D27C2211yescharacteristic, complemented
26C2×Dic54C2211yescharacteristic, complemented
27C22⋊C4×C27C2211yescharacteristic
28C22⋊C4×C27C2211yescharacteristic
29C22⋊C4×C27C2211yescharacteristic
30C22⋊C4×C27C2211yescharacteristic
31#88372C2211yescharacteristic
32#88372C2211yescharacteristic
33#88372C2211yescharacteristic
34#88372C2211yescharacteristic
35#88372C2211yescharacteristic
36#88372C2211yescharacteristic
37#88372C2211yescharacteristic
38(C2×D27)⋊C4C2211yescharacteristic, complemented
39(C2×D27)⋊C4C2211yescharacteristic, complemented
40(C2×D27)⋊C4C2211yescharacteristic, complemented
41C4⋊Dic27C2211yescharacteristic, complemented
42C4⋊Dic27C2211yescharacteristic, complemented
43C4⋊Dic27C2211yescharacteristic, complemented
44Dic27⋊C4C2211yescharacteristic, complemented
45Dic27⋊C4C2211yescharacteristic, complemented
46Dic27⋊C4C2211yescharacteristic, complemented
47Dic27⋊C4C2211yescharacteristic, complemented
48Dic27⋊C4C2211yescharacteristic, complemented
49C4×Dic27C2211yescharacteristic, complemented
50C4×Dic27C2211yescharacteristic, complemented
51C4×Dic27C2211yescharacteristic, complemented
52C2×C9×C22⋊C4D311yescharacteristic
53#63515—31no—
54C2×#3676—31no—
55#63491—31no—
56C4×#611—31no—
57#63488—31no—
58#63486—31no—
59#63448—31no—
60#63447—31no—
61#63445—31no—
62#611⋊C4—31no—
63#63441—31no—
64#63441—31no—
65#63440—31no—
66#63439—31no—
67C23×C27—22no—
68C23×C27—22no—
69C23×C27—41no—
70C23×C27C2311yescharacteristic
71C23×C27C2311yescharacteristic
72C23×C27C2311yescharacteristic
73C22×D27C2311yescharacteristic
74C2×C4×C27—22no—
75C2×C4×C27—41no—
76C2×C4×C27C2311yescharacteristic
77C2×C4×C27C2311yescharacteristic
78C2×C4×C27C2311yescharacteristic
79C2×C4×C27C2311yescharacteristic
80#6171—22no—
81#6171—22no—
82#6171—41no—
83C2×Dic27—22no—
84C2×Dic27—41no—
85C2×Dic27C2311yescharacteristic
86C2×Dic27C2311yescharacteristic
87C2×Dic27C2311yescharacteristic
88C2×Dic27C2311yescharacteristic
89C2×Dic27C2311yescharacteristic
90C2×Dic27C2311yescharacteristic
91C2×Dic27C2311yescharacteristic
92C4×D27—41no—
93Dic54—41no—
94#5495—91no—
95C24×C9C2×D311yescharacteristic
96C22×C4×C9C2×D311yescharacteristic
97C22×C4×C9C2×D311yescharacteristic
98C2×#611—31no—
99C2×#611—31no—
100C2×#611—31no—
101C22×Dic9—31no—
102C22×Dic9—31no—
103C2×C4×D9—31no—
104C2×Dic18—31no—
105C9×C22⋊C4C2×D311yescharacteristic
106C9×C22⋊C4C2×D311yescharacteristic
107C9×C22⋊C4C2×D311yescharacteristic
108C9×C22⋊C4C2×D311yescharacteristic
109#3676—31no—
110#3676—31no—
111#3676—31no—
112#3676—31no—
113#3676—31no—
114#3676—31no—
115#3676—31no—
116(C2×D9)⋊C4—31no—
117(C2×D9)⋊C4—31no—
118(C2×D9)⋊C4—31no—
119C4⋊Dic9—31no—
120C4⋊Dic9—31no—
121C4⋊Dic9—31no—
122Dic9⋊C4—31no—
123Dic9⋊C4—31no—
124Dic9⋊C4—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—21no—
136C22×C27—21no—
137C22×C27—44no—
138C22×C27#4112yes—
139C22×C27#4112yes—
140C22×C27C2411yescharacteristic
141C2×D27—21no—
142C2×D27—21no—
143C2×D27—21no—
144C4×C27—21no—
145C4×C27—21no—
146C4×C27—21no—
147C4×C27—21no—
148Dic27—21no—
149Dic27—21no—
150Dic27—21no—
151Dic27—21no—
152Dic27—21no—
153Dic27—21no—
154Dic27—21no—
155C2×C3×C22⋊C4D911yescharacteristic
156#953—91no—
157C2×#217—91no—
158#929—91no—
159C4×#67—91no—
160#926—91no—
161#924—91no—
162#886—91no—
163#885—91no—
164#883—91no—
165#67⋊C4—91no—
166#879—91no—
167#879—91no—
168#878—91no—
169#877—91no—
170C23×C9—22no—
171C23×C9—22no—
172C23×C9—41no—
173C23×C9C22×D311yescharacteristic
174C23×C9C22×D311yescharacteristic
175C23×C9C22×D311yescharacteristic
176C22×D9—31no—
177C2×C4×C9—22no—
178C2×C4×C9—41no—
179C2×C4×C9C22×D311yescharacteristic
180C2×C4×C9C22×D311yescharacteristic
181C2×C4×C9C22×D311yescharacteristic
182C2×C4×C9C22×D311yescharacteristic
183#611—62no—
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—31no—
193C2×Dic9—62no—
194C2×Dic9—121no—
195C4×D9—121no—
196Dic18—121no—
197#551—271noSylow
198C2×C27—22no—
199C2×C27—22no—
200C2×C27—41no—
201C2×C27C2×#4111yescharacteristic
202C2×C27C2×#4111yescharacteristic
203C2×C27H(5,𝔽2)11yescharacteristic
204D27—41no—
205C24×C3C2×D911yescharacteristic
206C22×C3×C4C2×D911yescharacteristic
207C22×C3×C4C2×D911yescharacteristic
208C2×#67—91no—
209C2×#67—91no—
210C2×#67—91no—
211C22×Dic3—91no—
212C22×Dic3—91no—
213C2×C4×D3—91no—
214C2×Dic6—91no—
215C3×C22⋊C4C2×D911yescharacteristic
216C3×C22⋊C4C2×D911yescharacteristic
217C3×C22⋊C4C2×D911yescharacteristic
218C3×C22⋊C4C2×D911yescharacteristic
219#217—91no—
220#217—91no—
221#217—91no—
222#217—91no—
223#217—91no—
224#217—91no—
225#217—91no—
226(C2×D3)⋊C4—91no—
227(C2×D3)⋊C4—91no—
228(C2×D3)⋊C4—91no—
229C4⋊Dic3—91no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232Dic3⋊C4—91no—
233Dic3⋊C4—91no—
234Dic3⋊C4—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—21no—
246C22×C9—21no—
247C22×C9—44no—
248C22×C9#23512yes—
249C22×C9#23712yes—
250C22×C9C23×D311yescharacteristic
251C2×D9—61no—
252C2×D9—61no—
253C2×D9—61no—
254C4×C9—21no—
255C4×C9—21no—
256C4×C9—21no—
257C4×C9—21no—
258Dic9—61no—
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#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⋊C4D2711yescharacteristic, complemented
280C27#55111yescharacteristic, complemented, Sylow
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—41no—
284C23×C3C22×D911yescharacteristic
285C23×C3C22×D911yescharacteristic
286C23×C3C22×D911yescharacteristic
287C22×D3—91no—
288C2×C3×C4—22no—
289C2×C3×C4—41no—
290C2×C3×C4C22×D911yescharacteristic
291C2×C3×C4C22×D911yescharacteristic
292C2×C3×C4C22×D911yescharacteristic
293C2×C3×C4C22×D911yescharacteristic
294#67—182no—
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—91no—
304C2×Dic3—182no—
305C2×Dic3—361no—
306C4×D3—361no—
307Dic6—361no—
308C2×C9—22no—
309C2×C9—22no—
310C2×C9—41no—
311C2×C9C2×#23511yescharacteristic
312C2×C9C2×#23711yescharacteristic
313C2×C9#100411yescharacteristic
314D9—121no—
315C24C2×D2711yescharacteristic
316C2×Q8—271no—
317C2×D4—271no—
318C2×D4—271no—
319C2×D4—271no—
320C22×C4—271no—
321C22×C4—271no—
322C22×C4—271no—
323C22×C4C2×D2711yescharacteristic
324C22×C4C2×D2711yescharacteristic
325C4⋊C4—271no—
326C4⋊C4—271no—
327C4⋊C4—271no—
328C4⋊C4—271no—
329C4⋊C4—271no—
330C4⋊C4—271no—
331C4⋊C4—271no—
332C4⋊C4—271no—
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⋊C4C2×D2711yescharacteristic
344C22⋊C4C2×D2711yescharacteristic
345C22⋊C4C2×D2711yescharacteristic
346C22⋊C4C2×D2711yescharacteristic
347C42—271no—
348C42—271no—
349C42—271no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—44no—
358C22×C3#369712yes—
359C22×C3#369912yes—
360C22×C3C23×D911yescharacteristic
361C2×D3—181no—
362C2×D3—181no—
363C2×D3—181no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368Dic3—181no—
369Dic3—181no—
370Dic3—181no—
371Dic3—181no—
372Dic3—181no—
373Dic3—181no—
374Dic3—181no—
375C9#549511yescharacteristic
376C23—22no—
377C23—22no—
378C23—41no—
379C23—271no—
380C23C22×D2711yescharacteristic
381C23C22×D2711yescharacteristic
382C23C22×D2711yescharacteristic
383Q8—1081no—
384D4—542no—
385D4—542no—
386D4—1081no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—271no—
390C2×C4—271no—
391C2×C4—271no—
392C2×C4—271no—
393C2×C4—271no—
394C2×C4—271no—
395C2×C4—271no—
396C2×C4—542no—
397C2×C4—1081no—
398C2×C4—1081no—
399C2×C4C22×D2711yescharacteristic
400C2×C4C22×D2711yescharacteristic
401C2×C4C22×D2711yescharacteristic
402C2×C4C22×D2711yescharacteristic
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—41no—
406C2×C3C2×#369711yescharacteristic
407C2×C3C2×#369911yescharacteristic
408C2×C3#6371011yescharacteristic
409D3—361no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—44no—
418C22—541no—
419C22—541no—
420C22—541no—
421C22#8839912yes—
422C22#8840112yes—
423C22C23×D2711yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—541no—
429C4—541no—
430C4—541no—
431C4—541no—
432C4—541no—
433C4—541no—
434C4—541no—
435C3#1058959711yescharacteristic
436C2—22no—
437C2—22no—
438C2—41no—
439C2—1081no—
440C2C2×#8839911yescharacteristic
441C2C2×#8840111yescharacteristic
442C2#1172067011yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC2×C22⋊C4×C27C2
socleC22×C3C23×D9
prime radical#11720202C2
hypercenterC2×C22⋊C4D27
hypocenterC27#551
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊C4×C27C2semidirect
3. C2×#88372C2semidirect
4. #11720178C2semidirect
5. C4×#6171C2semidirect
6. #11720175C2semidirect
7. #11720173C2semidirect
8. #11720135C2semidirect
9. #11720134C2semidirect
10. #11720132C2semidirect
11. #6171⋊C4C2semidirect
12. #11720128C2semidirect
13. #11720128C2semidirect
14. #11720127C2semidirect
15. #11720126C2semidirect
25. C2×C4×D27C22semidirect
26. C2×Dic54C22semidirect
38. (C2×D27)⋊C4C22semidirect
39. (C2×D27)⋊C4C22semidirect
40. (C2×D27)⋊C4C22semidirect
41. C4⋊Dic27C22semidirect
42. C4⋊Dic27C22semidirect
43. C4⋊Dic27C22semidirect
44. Dic27⋊C4C22semidirect
45. Dic27⋊C4C22semidirect
46. Dic27⋊C4C22semidirect
47. Dic27⋊C4C22semidirect
48. Dic27⋊C4C22semidirect
49. C4×Dic27C22semidirect
50. C4×Dic27C22semidirect
51. C4×Dic27C22semidirect
279. C2×C22⋊C4D27semidirect
280. C27#551semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×C272
#117202022
C2×#883722
#117201782
C4×#61712
#117201752
#117201732
#117201352
#117201342
#117201322
#6171⋊C42
#117201282
#117201272
#117201262
#105895973