Finite Group Databasebeta

← #13136397

Subgroups — #13136397

Order 1728.

584 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88401C211yesmaximal, characteristic, complemented
2C2×#88399C211yesmaximal, characteristic, complemented
3C27×#123C211yesmaximal, characteristic, complemented
4#11720188C211yesmaximal, characteristic, complemented
5C2×Dic27⋊C4C211yesmaximal, characteristic, complemented
6#11720145C211yesmaximal, characteristic, complemented
7#11720143C211yesmaximal, characteristic, complemented
8C4⋊C4×D27C211yesmaximal, characteristic, complemented
9#11720137C211yesmaximal, characteristic, complemented
10#11720135C211yesmaximal, characteristic, complemented
11#11720133C211yesmaximal, characteristic, complemented
12#11720132C211yesmaximal, characteristic, complemented
13#6171⋊C4C211yesmaximal, characteristic, complemented
14#6171⋊C4C211yesmaximal, characteristic, complemented
15#11720127C211yesmaximal, characteristic, complemented
16#10589757—31nomaximal
17C2×D4×C27C2211yescharacteristic
18C22×C4×C27C2211yescharacteristic
19C2×#6171C2211yescharacteristic
20C2×#6171C2211yescharacteristic
21C2×#6171C2211yescharacteristic
22C2×#6171C2211yescharacteristic
23C22×Dic27C2211yescharacteristic
24C22×Dic27C2211yescharacteristic
25C22×Dic27C2211yescharacteristic
26#88401—24no—
27#88399—24no—
28C2×D108C2211yescharacteristic
29C2×C4×D27C2211yescharacteristic
30C2×C4×D27C2211yescharacteristic
31C2×C4×D27C2211yescharacteristic
32C2×C4×D27C2211yescharacteristic
33C2×Dic54C2211yescharacteristic
34C2×Dic54C2211yescharacteristic
35C4⋊C4×C27C2211yescharacteristic
36C4⋊C4×C27C2211yescharacteristic
37C22⋊C4×C27C2211yescharacteristic
38C22⋊C4×C27C2211yescharacteristic
39C22⋊C4×C27C2211yescharacteristic
40#88372C2211yescharacteristic
41(C2×D27)⋊C4C2211yescharacteristic
42(C2×D27)⋊C4C2211yescharacteristic
43(C2×D27)⋊C4C2211yescharacteristic
44(C2×D27)⋊C4C2211yescharacteristic
45C4⋊Dic27C2211yescharacteristic
46C4⋊Dic27C2211yescharacteristic
47Dic27⋊C4C2211yescharacteristic
48Dic27⋊C4C2211yescharacteristic
49Dic27⋊C4C2211yescharacteristic
50Dic27⋊C4C2211yescharacteristic
51Dic27⋊C4C2211yescharacteristic
52Dic27⋊C4C2211yescharacteristic
53C4×Dic27C2211yescharacteristic
54C2×#3699—31no—
55C2×#3697—31no—
56C9×#123D311yescharacteristic
57#63501—31no—
58C2×Dic9⋊C4—31no—
59#63458—31no—
60#63456—31no—
61C4⋊C4×D9—31no—
62#63450—31no—
63#63448—31no—
64#63446—31no—
65#63445—31no—
66#611⋊C4—31no—
67#611⋊C4—31no—
68#63440—31no—
69C23×C27C2311yescharacteristic
70C23×C27C2311yescharacteristic
71C22×D27C2311yescharacteristic
72C22×D27C2311yescharacteristic
73D4×C27—22no—
74C2×C4×C27—22no—
75C2×C4×C27C2311yescharacteristic
76C2×C4×C27C2311yescharacteristic
77C2×C4×C27C2311yescharacteristic
78C2×C4×C27C2311yescharacteristic
79C2×C4×C27C2311yescharacteristic
80#6171—22no—
81#6171—22no—
82#6171—22no—
83#6171D414yescomplemented
84C2×Dic27—22no—
85C2×Dic27—22no—
86C2×Dic27—22no—
87C2×Dic27C2311yescharacteristic
88C2×Dic27C2311yescharacteristic
89C2×Dic27C2311yescharacteristic
90C2×Dic27C2311yescharacteristic
91C2×Dic27C2311yescharacteristic
92C2×Dic27C2311yescharacteristic
93D108—22no—
94C4×D27—22no—
95C4×D27—22no—
96C4×D27—22no—
97C4×D27—22no—
98Dic54—22no—
99Dic54—22no—
100#5655—91no—
101C2×D4×C9C2×D311yescharacteristic
102C22×C4×C9C2×D311yescharacteristic
103C2×#611—31no—
104C2×#611—31no—
105C2×#611—31no—
106C2×#611—31no—
107C22×Dic9—31no—
108C22×Dic9—31no—
109C22×Dic9—31no—
110#3699—64no—
111#3697—64no—
112C2×D36—31no—
113C2×C4×D9—31no—
114C2×C4×D9—31no—
115C2×C4×D9—31no—
116C2×C4×D9—31no—
117C2×Dic18—31no—
118C2×Dic18—31no—
119C9×C4⋊C4C2×D311yescharacteristic
120C9×C4⋊C4C2×D311yescharacteristic
121C9×C22⋊C4C2×D311yescharacteristic
122C9×C22⋊C4C2×D311yescharacteristic
123C9×C22⋊C4C2×D311yescharacteristic
124#3676—31no—
125(C2×D9)⋊C4—31no—
126(C2×D9)⋊C4—31no—
127(C2×D9)⋊C4—31no—
128(C2×D9)⋊C4—31no—
129C4⋊Dic9—31no—
130C4⋊Dic9—31no—
131Dic9⋊C4—31no—
132Dic9⋊C4—31no—
133Dic9⋊C4—31no—
134Dic9⋊C4—31no—
135Dic9⋊C4—31no—
136Dic9⋊C4—31no—
137C4×Dic9—31no—
138C22×C27—21no—
139C22×C27—21no—
140C22×C27—21no—
141C22×C27—21no—
142C22×C27—21no—
143C22×C27C2×D412yes—
144C22×C27C2411yescharacteristic
145C2×D27—21no—
146C2×D27—21no—
147C2×D27—21no—
148C2×D27—21no—
149C2×D27—21no—
150C2×D27C2×D412yes—
151C4×C27—21no—
152C4×C27—21no—
153C4×C27—21no—
154C4×C27—21no—
155C4×C27—21no—
156Dic27—21no—
157Dic27—21no—
158Dic27—21no—
159Dic27—21no—
160Dic27C2×D412yes—
161Dic27#4112yescomplemented
162C2×#237—91no—
163C2×#235—91no—
164C3×#123D911yescharacteristic
165#939—91no—
166C2×Dic3⋊C4—91no—
167#896—91no—
168#894—91no—
169D3×C4⋊C4—91no—
170#888—91no—
171#886—91no—
172#884—91no—
173#883—91no—
174#67⋊C4—91no—
175#67⋊C4—91no—
176#878—91no—
177C23×C9C22×D311yescharacteristic
178C23×C9C22×D311yescharacteristic
179C22×D9—31no—
180C22×D9—31no—
181D4×C9—22no—
182C2×C4×C9—22no—
183C2×C4×C9C22×D311yescharacteristic
184C2×C4×C9C22×D311yescharacteristic
185C2×C4×C9C22×D311yescharacteristic
186C2×C4×C9C22×D311yescharacteristic
187C2×C4×C9C22×D311yescharacteristic
188#611—34no—
189#611—62no—
190#611—62no—
191#611—62no—
192C2×Dic9—31no—
193C2×Dic9—31no—
194C2×Dic9—31no—
195C2×Dic9—31no—
196C2×Dic9—31no—
197C2×Dic9—31no—
198C2×Dic9—62no—
199C2×Dic9—62no—
200C2×Dic9—62no—
201D36—62no—
202C4×D9—62no—
203C4×D9—62no—
204C4×D9—62no—
205C4×D9—62no—
206Dic18—62no—
207Dic18—62no—
208#547—271noSylow
209C2×C27—22no—
210C2×C27—41no—
211C2×C27C22×D411yescharacteristic
212C2×C27C2×#4111yescharacteristic
213C2×C272−1+411yescharacteristic
214D27—22no—
215D27—41no—
216C2×C3×D4C2×D911yescharacteristic
217C22×C3×C4C2×D911yescharacteristic
218C2×#67—91no—
219C2×#67—91no—
220C2×#67—91no—
221C2×#67—91no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C22×Dic3—91no—
225#237—184no—
226#235—184no—
227C2×D12—91no—
228C2×C4×D3—91no—
229C2×C4×D3—91no—
230C2×C4×D3—91no—
231C2×C4×D3—91no—
232C2×Dic6—91no—
233C2×Dic6—91no—
234C3×C4⋊C4C2×D911yescharacteristic
235C3×C4⋊C4C2×D911yescharacteristic
236C3×C22⋊C4C2×D911yescharacteristic
237C3×C22⋊C4C2×D911yescharacteristic
238C3×C22⋊C4C2×D911yescharacteristic
239#217—91no—
240(C2×D3)⋊C4—91no—
241(C2×D3)⋊C4—91no—
242(C2×D3)⋊C4—91no—
243(C2×D3)⋊C4—91no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246Dic3⋊C4—91no—
247Dic3⋊C4—91no—
248Dic3⋊C4—91no—
249Dic3⋊C4—91no—
250Dic3⋊C4—91no—
251Dic3⋊C4—91no—
252C4×Dic3—91no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9D3×D412yes—
259C22×C9C23×D311yescharacteristic
260C2×D9—32no—
261C2×D9—61no—
262C2×D9—61no—
263C2×D9—61no—
264C2×D9—61no—
265C2×D9—61no—
266C4×C9—21no—
267C4×C9—21no—
268C4×C9—21no—
269C4×C9—21no—
270C4×C9—21no—
271Dic9—32no—
272Dic9—32no—
273Dic9—61no—
274Dic9—61no—
275Dic9—61no—
276Dic9—61no—
277C2×#41—271no—
278C2×#41—271no—
279C4⋊Q8—271no—
280#123—271no—
281#123—271no—
282#123—271no—
283#123D2711yescharacteristic, complemented
284C22⋊Q8—271no—
285C22⋊Q8—271no—
286#121—271no—
287#121—271no—
288C4×D4—271no—
289C4×D4—271no—
290C2×C4⋊C4—271no—
291C2×C4⋊C4—271no—
292C27#54711yescharacteristic, complemented, Sylow
293C23×C3C22×D911yescharacteristic
294C23×C3C22×D911yescharacteristic
295C22×D3—91no—
296C22×D3—91no—
297C3×D4—22no—
298C2×C3×C4—22no—
299C2×C3×C4C22×D911yescharacteristic
300C2×C3×C4C22×D911yescharacteristic
301C2×C3×C4C22×D911yescharacteristic
302C2×C3×C4C22×D911yescharacteristic
303C2×C3×C4C22×D911yescharacteristic
304#67—94no—
305#67—182no—
306#67—182no—
307#67—182no—
308C2×Dic3—91no—
309C2×Dic3—91no—
310C2×Dic3—91no—
311C2×Dic3—91no—
312C2×Dic3—91no—
313C2×Dic3—91no—
314C2×Dic3—182no—
315C2×Dic3—182no—
316C2×Dic3—182no—
317D12—182no—
318C4×D3—182no—
319C4×D3—182no—
320C4×D3—182no—
321C4×D3—182no—
322Dic6—182no—
323Dic6—182no—
324C2×C9—22no—
325C2×C9—41no—
326C2×C9C2×D3×D411yescharacteristic
327C2×C9D3×#4111yescharacteristic
328C2×C9#101011yescharacteristic
329D9—62no—
330D9—121no—
331#41—544no—
332#41—544no—
333C2×Q8—271no—
334C2×Q8—271no—
335C2×D4—271no—
336C2×D4—271no—
337C2×D4—271no—
338C2×D4—271no—
339C2×D4—271no—
340C2×D4C2×D2711yescharacteristic
341C22×C4—271no—
342C22×C4—271no—
343C22×C4—271no—
344C22×C4—271no—
345C22×C4—271no—
346C22×C4—271no—
347C22×C4—271no—
348C22×C4C2×D2711yescharacteristic
349C4⋊C4—271no—
350C4⋊C4—271no—
351C4⋊C4—271no—
352C4⋊C4—271no—
353C4⋊C4—271no—
354C4⋊C4—271no—
355C4⋊C4—271no—
356C4⋊C4—271no—
357C4⋊C4C2×D2711yescharacteristic
358C4⋊C4C2×D2711yescharacteristic
359C22⋊C4—271no—
360C22⋊C4—271no—
361C22⋊C4—271no—
362C22⋊C4—271no—
363C22⋊C4—271no—
364C22⋊C4C2×D2711yescharacteristic
365C22⋊C4C2×D2711yescharacteristic
366C22⋊C4C2×D2711yescharacteristic
367C42—271no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3D4×D912yes—
374C22×C3C23×D911yescharacteristic
375C2×D3—92no—
376C2×D3—181no—
377C2×D3—181no—
378C2×D3—181no—
379C2×D3—181no—
380C2×D3—181no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386Dic3—92no—
387Dic3—92no—
388Dic3—181no—
389Dic3—181no—
390Dic3—181no—
391Dic3—181no—
392C9#565511yescharacteristic
393C23—271no—
394C23—271no—
395C23C22×D2711yescharacteristic
396C23C22×D2711yescharacteristic
397Q8—542no—
398Q8—542no—
399D4—22no—
400D4—274no—
401D4—542no—
402D4—542no—
403D4—542no—
404D4—542no—
405C2×C4—22no—
406C2×C4—271no—
407C2×C4—271no—
408C2×C4—271no—
409C2×C4—271no—
410C2×C4—271no—
411C2×C4—271no—
412C2×C4—542no—
413C2×C4—542no—
414C2×C4—542no—
415C2×C4—542no—
416C2×C4—542no—
417C2×C4—542no—
418C2×C4—542no—
419C2×C4C22×D2711yescharacteristic
420C2×C4C22×D2711yescharacteristic
421C2×C4C22×D2711yescharacteristic
422C2×C4C22×D2711yescharacteristic
423C2×C4C22×D2711yescharacteristic
424C2×C3—22no—
425C2×C3—41no—
426C2×C3C2×D4×D911yescharacteristic
427C2×C3#41×D911yescharacteristic
428C2×C3#6371611yescharacteristic
429D3—182no—
430D3—361no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—272no—
437C22—541no—
438C22—541no—
439C22—541no—
440C22—541no—
441C22—541no—
442C22D4×D2712yes—
443C22C23×D2711yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—272no—
450C4—272no—
451C4—541no—
452C4—541no—
453C4—541no—
454C4—541no—
455C3#1058975711yescharacteristic
456C2—22no—
457C2—41no—
458C2—542no—
459C2—1081no—
460C2C2×D4×D2711yescharacteristic
461C2#41×D2711yescharacteristic
462C2#1172067611yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#123C2
socleC22×C3C23×D9
hypercenter#123D27
hypocenterC27#547
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#88401C2semidirect
2. C2×#88399C2semidirect
3. C27×#123C2semidirect
4. #11720188C2semidirect
5. C2×Dic27⋊C4C2semidirect
6. #11720145C2semidirect
7. #11720143C2semidirect
8. C4⋊C4×D27C2semidirect
9. #11720137C2semidirect
10. #11720135C2semidirect
11. #11720133C2semidirect
12. #11720132C2semidirect
13. #6171⋊C4C2semidirect
14. #6171⋊C4C2semidirect
15. #11720127C2semidirect
83. #6171D4semidirect
161. Dic27#41semidirect
283. #123D27semidirect
292. C27#547semidirect

Maximal subgroups

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

subgroupindex
C2×#884012
C2×#883992
C27×#1232
#117201882
C2×Dic27⋊C42
#117201452
#117201432
C4⋊C4×D272
#117201372
#117201352
#117201332
#117201322
#6171⋊C42
#117201272
#105897573