Finite Group Databasebeta

← #13136396

Subgroups — #13136396

Order 1728.

668 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×D27C211yesmaximal, characteristic, complemented
2C2×#88399C211yesmaximal, characteristic, complemented
3C27×#123C211yesmaximal, characteristic, complemented
4#11720188C211yesmaximal, characteristic, complemented
5C2×(C2×D27)⋊C4C211yesmaximal, characteristic, complemented
6#11720145C211yesmaximal, characteristic, complemented
7#11720144C211yesmaximal, characteristic, complemented
8#11720143C211yesmaximal, characteristic, complemented
9#11720142C211yesmaximal, characteristic, complemented
10#11720134C211yesmaximal, characteristic, complemented
11#11720133C211yesmaximal, characteristic, complemented
12#11720131C211yesmaximal, characteristic, complemented
13#11720131C211yesmaximal, characteristic, complemented
14#6171⋊C4C211yesmaximal, characteristic, complemented
15C22⋊C4×D27C211yesmaximal, characteristic, complemented
16#10589756—31nomaximal
17C23×D27C2211yescharacteristic
18C23×D27C2211yescharacteristic
19C2×D4×C27C2211yescharacteristic
20C22×C4×C27C2211yescharacteristic, complemented
21C2×#6171C2211yescharacteristic
22C2×#6171C2211yescharacteristic
23C2×#6171C2211yescharacteristic, complemented
24C2×#6171C2211yescharacteristic, complemented
25C22×Dic27C2211yescharacteristic, complemented
26D4×D27—24no—
27#88399—24no—
28C2×D108C2211yescharacteristic
29C2×D108C2211yescharacteristic
30C2×D108C2211yescharacteristic, complemented
31C2×D108C2211yescharacteristic, complemented
32C2×C4×D27C2211yescharacteristic
33C2×C4×D27C2211yescharacteristic, complemented
34C2×C4×D27C2211yescharacteristic, complemented
35C2×C4×D27C2211yescharacteristic, complemented
36C2×Dic54C2211yescharacteristic, complemented
37C4⋊C4×C27C2211yescharacteristic, complemented
38C4⋊C4×C27C2211yescharacteristic, complemented
39C22⋊C4×C27C2211yescharacteristic, complemented
40C22⋊C4×C27C2211yescharacteristic, complemented
41C22⋊C4×C27C2211yescharacteristic, complemented
42#88372C2211yescharacteristic, complemented
43(C2×D27)⋊C4C2211yescharacteristic
44(C2×D27)⋊C4C2211yescharacteristic
45(C2×D27)⋊C4C2211yescharacteristic
46(C2×D27)⋊C4C2211yescharacteristic, complemented
47(C2×D27)⋊C4C2211yescharacteristic, complemented
48(C2×D27)⋊C4C2211yescharacteristic, complemented
49(C2×D27)⋊C4C2211yescharacteristic, complemented
50(C2×D27)⋊C4C2211yescharacteristic, complemented
51Dic27⋊C4C2211yescharacteristic, complemented
52Dic27⋊C4C2211yescharacteristic, complemented
53C4×Dic27C2211yescharacteristic, complemented
54C2×D4×D9—31no—
55C2×#3697—31no—
56C9×#123D311yescharacteristic
57#63501—31no—
58C2×(C2×D9)⋊C4—31no—
59#63458—31no—
60#63457—31no—
61#63456—31no—
62#63455—31no—
63#63447—31no—
64#63446—31no—
65#63444—31no—
66#63444—31no—
67#611⋊C4—31no—
68C22⋊C4×D9—31no—
69C23×C27C2311yescharacteristic
70C23×C27C2311yescharacteristic
71C22×D27—22no—
72C22×D27—22no—
73C22×D27—22no—
74C22×D27—22no—
75C22×D27—41no—
76C22×D27—41no—
77C22×D27C2311yescharacteristic
78C22×D27C2311yescharacteristic
79C22×D27C2311yescharacteristic
80C22×D27C2311yescharacteristic
81D4×C27—22no—
82C2×C4×C27—22no—
83C2×C4×C27C2311yescharacteristic
84C2×C4×C27C2311yescharacteristic
85C2×C4×C27C2311yescharacteristic
86C2×C4×C27C2311yescharacteristic
87C2×C4×C27C2311yescharacteristic
88#6171—22no—
89#6171—22no—
90#6171—41no—
91#6171D414yescomplemented
92C2×Dic27—41no—
93C2×Dic27C2311yescharacteristic
94C2×Dic27C2311yescharacteristic
95C2×Dic27C2311yescharacteristic
96C2×Dic27C2311yescharacteristic
97D108—22no—
98D108—22no—
99D108—22no—
100D108—41no—
101C4×D27—22no—
102C4×D27—22no—
103C4×D27—22no—
104C4×D27—41no—
105Dic54—22no—
106#5654—91no—
107C23×D9—31no—
108C23×D9—31no—
109C2×D4×C9C2×D311yescharacteristic
110C22×C4×C9C2×D311yescharacteristic
111C2×#611—31no—
112C2×#611—31no—
113C2×#611—31no—
114C2×#611—31no—
115C22×Dic9—31no—
116D4×D9—64no—
117#3697—64no—
118C2×D36—31no—
119C2×D36—31no—
120C2×D36—31no—
121C2×D36—31no—
122C2×C4×D9—31no—
123C2×C4×D9—31no—
124C2×C4×D9—31no—
125C2×C4×D9—31no—
126C2×Dic18—31no—
127C9×C4⋊C4C2×D311yescharacteristic
128C9×C4⋊C4C2×D311yescharacteristic
129C9×C22⋊C4C2×D311yescharacteristic
130C9×C22⋊C4C2×D311yescharacteristic
131C9×C22⋊C4C2×D311yescharacteristic
132#3676—31no—
133(C2×D9)⋊C4—31no—
134(C2×D9)⋊C4—31no—
135(C2×D9)⋊C4—31no—
136(C2×D9)⋊C4—31no—
137(C2×D9)⋊C4—31no—
138(C2×D9)⋊C4—31no—
139(C2×D9)⋊C4—31no—
140(C2×D9)⋊C4—31no—
141Dic9⋊C4—31no—
142Dic9⋊C4—31no—
143C4×Dic9—31no—
144C22×C27—21no—
145C22×C27—21no—
146C22×C27—21no—
147C22×C27—21no—
148C22×C27—21no—
149C22×C27C2×D412yes—
150C22×C27C2411yescharacteristic
151C2×D27—21no—
152C2×D27—21no—
153C2×D27—21no—
154C2×D27—21no—
155C2×D27—21no—
156C2×D27—21no—
157C2×D27—21no—
158C2×D27—21no—
159C2×D27—21no—
160C2×D27—21no—
161C2×D27—44no—
162C2×D27—44no—
163C2×D27C2×D412yes—
164C2×D27#4112yescomplemented
165C4×C27—21no—
166C4×C27—21no—
167C4×C27—21no—
168C4×C27—21no—
169C4×C27—21no—
170Dic27—21no—
171Dic27—21no—
172Dic27—21no—
173Dic27C2×D412yes—
174C2×D3×D4—91no—
175C2×#235—91no—
176C3×#123D911yescharacteristic
177#939—91no—
178C2×(C2×D3)⋊C4—91no—
179#896—91no—
180#895—91no—
181#894—91no—
182#893—91no—
183#885—91no—
184#884—91no—
185#882—91no—
186#882—91no—
187#67⋊C4—91no—
188D3×C22⋊C4—91no—
189C23×C9C22×D311yescharacteristic
190C23×C9C22×D311yescharacteristic
191C22×D9—31no—
192C22×D9—31no—
193C22×D9—31no—
194C22×D9—31no—
195C22×D9—62no—
196C22×D9—62no—
197C22×D9—62no—
198C22×D9—62no—
199C22×D9—121no—
200C22×D9—121no—
201D4×C9—22no—
202C2×C4×C9—22no—
203C2×C4×C9C22×D311yescharacteristic
204C2×C4×C9C22×D311yescharacteristic
205C2×C4×C9C22×D311yescharacteristic
206C2×C4×C9C22×D311yescharacteristic
207C2×C4×C9C22×D311yescharacteristic
208#611—34no—
209#611—62no—
210#611—62no—
211#611—121no—
212C2×Dic9—31no—
213C2×Dic9—31no—
214C2×Dic9—31no—
215C2×Dic9—31no—
216C2×Dic9—121no—
217D36—62no—
218D36—62no—
219D36—62no—
220D36—121no—
221C4×D9—62no—
222C4×D9—62no—
223C4×D9—62no—
224C4×D9—121no—
225Dic18—62no—
226#546—271noSylow
227C2×C27—22no—
228C2×C27—41no—
229C2×C27C22×D411yescharacteristic
230C2×C27C2×#4111yescharacteristic
231C2×C27H(5,𝔽2)11yescharacteristic
232D27—22no—
233D27—22no—
234D27—41no—
235D27—41no—
236C23×D3—91no—
237C23×D3—91no—
238C2×C3×D4C2×D911yescharacteristic
239C22×C3×C4C2×D911yescharacteristic
240C2×#67—91no—
241C2×#67—91no—
242C2×#67—91no—
243C2×#67—91no—
244C22×Dic3—91no—
245D3×D4—184no—
246#235—184no—
247C2×D12—91no—
248C2×D12—91no—
249C2×D12—91no—
250C2×D12—91no—
251C2×C4×D3—91no—
252C2×C4×D3—91no—
253C2×C4×D3—91no—
254C2×C4×D3—91no—
255C2×Dic6—91no—
256C3×C4⋊C4C2×D911yescharacteristic
257C3×C4⋊C4C2×D911yescharacteristic
258C3×C22⋊C4C2×D911yescharacteristic
259C3×C22⋊C4C2×D911yescharacteristic
260C3×C22⋊C4C2×D911yescharacteristic
261#217—91no—
262(C2×D3)⋊C4—91no—
263(C2×D3)⋊C4—91no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—91no—
266(C2×D3)⋊C4—91no—
267(C2×D3)⋊C4—91no—
268(C2×D3)⋊C4—91no—
269(C2×D3)⋊C4—91no—
270Dic3⋊C4—91no—
271Dic3⋊C4—91no—
272C4×Dic3—91no—
273C22×C9—21no—
274C22×C9—21no—
275C22×C9—21no—
276C22×C9—21no—
277C22×C9—21no—
278C22×C9D3×D412yes—
279C22×C9C23×D311yescharacteristic
280C2×D9—32no—
281C2×D9—32no—
282C2×D9—61no—
283C2×D9—61no—
284C2×D9—61no—
285C2×D9—61no—
286C2×D9—61no—
287C2×D9—61no—
288C2×D9—61no—
289C2×D9—61no—
290C2×D9—61no—
291C2×D9—61no—
292C2×D9—124no—
293C2×D9—124no—
294C4×C9—21no—
295C4×C9—21no—
296C4×C9—21no—
297C4×C9—21no—
298C4×C9—21no—
299Dic9—32no—
300Dic9—61no—
301Dic9—61no—
302Dic9—61no—
303C2×#41—271no—
304C22×D4—271no—
305#124—271no—
306#123—271no—
307#123D2711yescharacteristic, complemented
308C22⋊Q8—271no—
309#121—271no—
310#121—271no—
311#121—271no—
312#120—271no—
313#120—271no—
314C4×D4—271no—
315C4×D4—271no—
316C2×C22⋊C4—271no—
317C2×C22⋊C4—271no—
318C27#54611yescharacteristic, complemented, Sylow
319C23×C3C22×D911yescharacteristic
320C23×C3C22×D911yescharacteristic
321C22×D3—91no—
322C22×D3—91no—
323C22×D3—91no—
324C22×D3—91no—
325C22×D3—182no—
326C22×D3—182no—
327C22×D3—182no—
328C22×D3—182no—
329C22×D3—361no—
330C22×D3—361no—
331C3×D4—22no—
332C2×C3×C4—22no—
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C22×D911yescharacteristic
338#67—94no—
339#67—182no—
340#67—182no—
341#67—361no—
342C2×Dic3—91no—
343C2×Dic3—91no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—361no—
347D12—182no—
348D12—182no—
349D12—182no—
350D12—361no—
351C4×D3—182no—
352C4×D3—182no—
353C4×D3—182no—
354C4×D3—361no—
355Dic6—182no—
356C2×C9—22no—
357C2×C9—41no—
358C2×C9C2×D3×D411yescharacteristic
359C2×C9D3×#4111yescharacteristic
360C2×C9#100911yescharacteristic
361D9—62no—
362D9—62no—
363D9—121no—
364D9—121no—
365C24—271no—
366C24—271no—
367#41—544no—
368C2×Q8—271no—
369C2×D4—271no—
370C2×D4—271no—
371C2×D4—271no—
372C2×D4—271no—
373C2×D4—271no—
374C2×D4—271no—
375C2×D4—271no—
376C2×D4—271no—
377C2×D4—544no—
378C2×D4C2×D2711yescharacteristic
379C22×C4—271no—
380C22×C4—271no—
381C22×C4—271no—
382C22×C4—271no—
383C22×C4—271no—
384C22×C4C2×D2711yescharacteristic, complemented
385C4⋊C4—271no—
386C4⋊C4—271no—
387C4⋊C4C2×D2711yescharacteristic, complemented
388C4⋊C4C2×D2711yescharacteristic, complemented
389C22⋊C4—271no—
390C22⋊C4—271no—
391C22⋊C4—271no—
392C22⋊C4—271no—
393C22⋊C4—271no—
394C22⋊C4—271no—
395C22⋊C4—271no—
396C22⋊C4—271no—
397C22⋊C4—271no—
398C22⋊C4C2×D2711yescharacteristic, complemented
399C22⋊C4C2×D2711yescharacteristic, complemented
400C22⋊C4C2×D2711yescharacteristic, complemented
401C42—271no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3D4×D912yes—
408C22×C3C23×D911yescharacteristic
409C2×D3—92no—
410C2×D3—92no—
411C2×D3—181no—
412C2×D3—181no—
413C2×D3—181no—
414C2×D3—181no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—181no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—181no—
421C2×D3—364no—
422C2×D3—364no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428Dic3—92no—
429Dic3—181no—
430Dic3—181no—
431Dic3—181no—
432C9#565411yescharacteristic
433C23—271no—
434C23—271no—
435C23—271no—
436C23—271no—
437C23—542no—
438C23—542no—
439C23—542no—
440C23—542no—
441C23—1081no—
442C23—1081no—
443C23C22×D2711yescharacteristic
444C23C22×D2711yescharacteristic
445Q8—542no—
446D4—22no—
447D4—274no—
448D4—542no—
449D4—542no—
450D4—542no—
451D4—542no—
452D4—542no—
453D4—1081no—
454D4—1081no—
455C2×C4—22no—
456C2×C4—271no—
457C2×C4—271no—
458C2×C4—271no—
459C2×C4—271no—
460C2×C4—542no—
461C2×C4—542no—
462C2×C4—542no—
463C2×C4—1081no—
464C2×C4—1081no—
465C2×C4C22×D2711yescharacteristic
466C2×C4C22×D2711yescharacteristic
467C2×C4C22×D2711yescharacteristic
468C2×C4C22×D2711yescharacteristic
469C2×C4C22×D2711yescharacteristic
470C2×C3—22no—
471C2×C3—41no—
472C2×C3C2×D4×D911yescharacteristic
473C2×C3#41×D911yescharacteristic
474C2×C3#6371511yescharacteristic
475D3—182no—
476D3—182no—
477D3—361no—
478D3—361no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—272no—
485C22—272no—
486C22—541no—
487C22—541no—
488C22—541no—
489C22—541no—
490C22—541no—
491C22—541no—
492C22—541no—
493C22—541no—
494C22—541no—
495C22—541no—
496C22—1084no—
497C22—1084no—
498C22D4×D2712yes—
499C22C23×D2711yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—272no—
506C4—541no—
507C4—541no—
508C4—541no—
509C3#1058975611yescharacteristic
510C2—22no—
511C2—41no—
512C2—542no—
513C2—542no—
514C2—1081no—
515C2—1081no—
516C2C2×D4×D2711yescharacteristic
517C2#41×D2711yescharacteristic
518C2#1172067511yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#123C2
socleC22×C3C23×D9
hypercenter#123D27
hypocenterC27#546
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×D4×D27C2semidirect
2. C2×#88399C2semidirect
3. C27×#123C2semidirect
4. #11720188C2semidirect
5. C2×(C2×D27)⋊C4C2semidirect
6. #11720145C2semidirect
7. #11720144C2semidirect
8. #11720143C2semidirect
9. #11720142C2semidirect
10. #11720134C2semidirect
11. #11720133C2semidirect
12. #11720131C2semidirect
13. #11720131C2semidirect
14. #6171⋊C4C2semidirect
15. C22⋊C4×D27C2semidirect
20. C22×C4×C27C22semidirect
23. C2×#6171C22semidirect
24. C2×#6171C22semidirect
25. C22×Dic27C22semidirect
30. C2×D108C22semidirect
31. C2×D108C22semidirect
33. C2×C4×D27C22semidirect
34. C2×C4×D27C22semidirect
35. C2×C4×D27C22semidirect
36. C2×Dic54C22semidirect
37. C4⋊C4×C27C22semidirect
38. C4⋊C4×C27C22semidirect
39. C22⋊C4×C27C22semidirect
40. C22⋊C4×C27C22semidirect
41. C22⋊C4×C27C22semidirect
42. #88372C22semidirect
46. (C2×D27)⋊C4C22semidirect
47. (C2×D27)⋊C4C22semidirect
48. (C2×D27)⋊C4C22semidirect
49. (C2×D27)⋊C4C22semidirect
50. (C2×D27)⋊C4C22semidirect
51. Dic27⋊C4C22semidirect
52. Dic27⋊C4C22semidirect
53. C4×Dic27C22semidirect
91. #6171D4semidirect
164. C2×D27#41semidirect
307. #123D27semidirect
318. C27#546semidirect
384. C22×C4C2×D27semidirect
387. C4⋊C4C2×D27semidirect
388. C4⋊C4C2×D27semidirect
398. C22⋊C4C2×D27semidirect
399. C22⋊C4C2×D27semidirect
400. C22⋊C4C2×D27semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D272
C2×#883992
C27×#1232
#117201882
C2×(C2×D27)⋊C42
#117201452
#117201442
#117201432
#117201422
#117201342
#117201332
#117201312
#6171⋊C42
C22⋊C4×D272
#105897563