Finite Group Databasebeta

← #13136256

Subgroups — #13136256

Order 1728.

428 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C27×#117C211yesmaximal, characteristic, complemented
2#11720178C211yesmaximal, characteristic
3C2×Dic27⋊C4C211yesmaximal, characteristic, complemented
4#11720147C211yesmaximal, characteristic, complemented
5#11720145C211yesmaximal, characteristic, complemented
6#11720138C211yesmaximal, characteristic, complemented
7#11720138C211yesmaximal, characteristic, complemented
8#11720135C211yesmaximal, characteristic
9#6171⋊C4C211yesmaximal, characteristic
10#11720127C211yesmaximal, characteristic, complemented
11#11720126C211yesmaximal, characteristic, complemented
12#11720125C211yesmaximal, characteristic, complemented
13#11720121C211yesmaximal, characteristic, complemented
14#11720119C211yesmaximal, characteristic, complemented
15C4×Dic54C211yesmaximal, characteristic, complemented
16#10589633—31nomaximal
17C22×C4×C27C2211yescharacteristic
18C2×#6171C2211yescharacteristic
19C22×Dic27C2211yescharacteristic
20C22×Dic27C2211yescharacteristic
21C2×C4×D27C2211yescharacteristic
22C2×Dic54C2211yescharacteristic
23C4⋊C4×C27C2211yescharacteristic
24C4⋊C4×C27C2211yescharacteristic
25C22⋊C4×C27C2211yescharacteristic
26C22⋊C4×C27C2211yescharacteristic
27C42×C27C2211yescharacteristic
28C42×C27C2211yescharacteristic
29#88372C2211yescharacteristic
30(C2×D27)⋊C4C2211yescharacteristic
31(C2×D27)⋊C4C2211yescharacteristic
32(C2×D27)⋊C4C2211yescharacteristic
33(C2×D27)⋊C4C2211yescharacteristic
34(C2×D27)⋊C4C2211yescharacteristic
35C4⋊Dic27C2211yescharacteristic
36C4⋊Dic27C2211yescharacteristic
37C4⋊Dic27C2211yescharacteristic
38Dic27⋊C4C2211yescharacteristic
39Dic27⋊C4C2211yescharacteristic
40Dic27⋊C4C2211yescharacteristic
41Dic27⋊C4C2211yescharacteristic
42Dic27⋊C4C2211yescharacteristic
43Dic27⋊C4C2211yescharacteristic
44Dic27⋊C4C2211yescharacteristic
45Dic27⋊C4C2211yescharacteristic
46Dic27⋊C4C2211yescharacteristic
47Dic27⋊C4C2211yescharacteristic
48Dic27⋊C4C2211yescharacteristic
49C4×Dic27C2211yescharacteristic
50C4×Dic27C2211yescharacteristic
51C4×Dic27C2211yescharacteristic
52C9×#117D311yescharacteristic
53#63491—31no—
54C2×Dic9⋊C4—31no—
55#63460—31no—
56#63458—31no—
57#63451—31no—
58#63451—31no—
59#63448—31no—
60#611⋊C4—31no—
61#63440—31no—
62#63439—31no—
63#63438—31no—
64#63434—31no—
65#63432—31no—
66C4×Dic18—31no—
67C23×C27C2311yescharacteristic
68C22×D27C2311yescharacteristic
69C2×C4×C27—22no—
70C2×C4×C27C2311yescharacteristic
71C2×C4×C27C2311yescharacteristic
72C2×C4×C27C2311yescharacteristic
73C2×C4×C27C2311yescharacteristic
74C2×C4×C27C2311yescharacteristic
75C2×C4×C27C2311yescharacteristic
76#6171—22no—
77C2×Dic27—22no—
78C2×Dic27—22no—
79C2×Dic27C2311yescharacteristic
80C2×Dic27C2311yescharacteristic
81C2×Dic27C2311yescharacteristic
82C2×Dic27C2311yescharacteristic
83C2×Dic27C2311yescharacteristic
84C2×Dic27C2311yescharacteristic
85C2×Dic27C2311yescharacteristic
86C4×D27—22no—
87Dic54—22no—
88#5531—91no—
89C22×C4×C9C2×D311yescharacteristic
90C2×#611—31no—
91C22×Dic9—31no—
92C22×Dic9—31no—
93C2×C4×D9—31no—
94C2×Dic18—31no—
95C9×C4⋊C4C2×D311yescharacteristic
96C9×C4⋊C4C2×D311yescharacteristic
97C9×C22⋊C4C2×D311yescharacteristic
98C9×C22⋊C4C2×D311yescharacteristic
99C42×C9C2×D311yescharacteristic
100C42×C9C2×D311yescharacteristic
101#3676—31no—
102(C2×D9)⋊C4—31no—
103(C2×D9)⋊C4—31no—
104(C2×D9)⋊C4—31no—
105(C2×D9)⋊C4—31no—
106(C2×D9)⋊C4—31no—
107C4⋊Dic9—31no—
108C4⋊Dic9—31no—
109C4⋊Dic9—31no—
110Dic9⋊C4—31no—
111Dic9⋊C4—31no—
112Dic9⋊C4—31no—
113Dic9⋊C4—31no—
114Dic9⋊C4—31no—
115Dic9⋊C4—31no—
116Dic9⋊C4—31no—
117Dic9⋊C4—31no—
118Dic9⋊C4—31no—
119Dic9⋊C4—31no—
120Dic9⋊C4—31no—
121C4×Dic9—31no—
122C4×Dic9—31no—
123C4×Dic9—31no—
124C22×C27—21no—
125C22×C27—21no—
126C22×C27#4112yes—
127C22×C27C2411yescharacteristic
128C2×D27—21no—
129C2×D27—21no—
130C2×D27—21no—
131C4×C27—21no—
132C4×C27—21no—
133C4×C27—21no—
134C4×C27—21no—
135C4×C27—21no—
136C4×C27—21no—
137Dic27—21no—
138Dic27—21no—
139Dic27—21no—
140Dic27—21no—
141Dic27—21no—
142Dic27—21no—
143Dic27#4112yes—
144C3×#117D911yescharacteristic
145#929—91no—
146C2×Dic3⋊C4—91no—
147#898—91no—
148#896—91no—
149#889—91no—
150#889—91no—
151#886—91no—
152#67⋊C4—91no—
153#878—91no—
154#877—91no—
155#876—91no—
156#872—91no—
157#870—91no—
158C4×Dic6—91no—
159C23×C9C22×D311yescharacteristic
160C22×D9—31no—
161C2×C4×C9—22no—
162C2×C4×C9C22×D311yescharacteristic
163C2×C4×C9C22×D311yescharacteristic
164C2×C4×C9C22×D311yescharacteristic
165C2×C4×C9C22×D311yescharacteristic
166C2×C4×C9C22×D311yescharacteristic
167C2×C4×C9C22×D311yescharacteristic
168#611—62no—
169C2×Dic9—31no—
170C2×Dic9—31no—
171C2×Dic9—31no—
172C2×Dic9—31no—
173C2×Dic9—31no—
174C2×Dic9—31no—
175C2×Dic9—31no—
176C2×Dic9—62no—
177C2×Dic9—62no—
178C4×D9—62no—
179Dic18—62no—
180#552—271noSylow
181C2×C27—22no—
182C2×C27C2×#4111yescharacteristic
183C2×C27C2×#4111yescharacteristic
184C2×C272−1+411yescharacteristic
185D27—41no—
186C22×C3×C4C2×D911yescharacteristic
187C2×#67—91no—
188C22×Dic3—91no—
189C22×Dic3—91no—
190C2×C4×D3—91no—
191C2×Dic6—91no—
192C3×C4⋊C4C2×D911yescharacteristic
193C3×C4⋊C4C2×D911yescharacteristic
194C3×C22⋊C4C2×D911yescharacteristic
195C3×C22⋊C4C2×D911yescharacteristic
196C3×C42C2×D911yescharacteristic
197C3×C42C2×D911yescharacteristic
198#217—91no—
199(C2×D3)⋊C4—91no—
200(C2×D3)⋊C4—91no—
201(C2×D3)⋊C4—91no—
202(C2×D3)⋊C4—91no—
203(C2×D3)⋊C4—91no—
204C4⋊Dic3—91no—
205C4⋊Dic3—91no—
206C4⋊Dic3—91no—
207Dic3⋊C4—91no—
208Dic3⋊C4—91no—
209Dic3⋊C4—91no—
210Dic3⋊C4—91no—
211Dic3⋊C4—91no—
212Dic3⋊C4—91no—
213Dic3⋊C4—91no—
214Dic3⋊C4—91no—
215Dic3⋊C4—91no—
216Dic3⋊C4—91no—
217Dic3⋊C4—91no—
218C4×Dic3—91no—
219C4×Dic3—91no—
220C4×Dic3—91no—
221C22×C9—21no—
222C22×C9—21no—
223C22×C9#23512yes—
224C22×C9C23×D311yescharacteristic
225C2×D9—61no—
226C2×D9—61no—
227C2×D9—61no—
228C4×C9—21no—
229C4×C9—21no—
230C4×C9—21no—
231C4×C9—21no—
232C4×C9—21no—
233C4×C9—21no—
234Dic9—32no—
235Dic9—61no—
236Dic9—61no—
237Dic9—61no—
238Dic9—61no—
239Dic9—61no—
240Dic9—61no—
241#126—271no—
242#126—271no—
243#125—271no—
244#125—271no—
245#125—271no—
246#123—271no—
247#123—271no—
248C22⋊Q8—271no—
249C22⋊Q8—271no—
250C4×Q8—271no—
251C4×D4—271no—
252#117—271no—
253#117—271no—
254#117D2711yescharacteristic, complemented
255C2×C4⋊C4—271no—
256C27#55211yescharacteristic, complemented, Sylow
257C23×C3C22×D911yescharacteristic
258C22×D3—91no—
259C2×C3×C4—22no—
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4C22×D911yescharacteristic
262C2×C3×C4C22×D911yescharacteristic
263C2×C3×C4C22×D911yescharacteristic
264C2×C3×C4C22×D911yescharacteristic
265C2×C3×C4C22×D911yescharacteristic
266#67—182no—
267C2×Dic3—91no—
268C2×Dic3—91no—
269C2×Dic3—91no—
270C2×Dic3—91no—
271C2×Dic3—91no—
272C2×Dic3—91no—
273C2×Dic3—91no—
274C2×Dic3—182no—
275C2×Dic3—182no—
276C4×D3—182no—
277Dic6—182no—
278C2×C9—22no—
279C2×C9C2×#23511yescharacteristic
280C2×C9D3×#4111yescharacteristic
281C2×C9#101011yescharacteristic
282D9—121no—
283C2×Q8—271no—
284C2×D4—271no—
285C22×C4—271no—
286C22×C4—271no—
287C22×C4—271no—
288C22×C4C2×D2711yescharacteristic
289C4⋊C4—271no—
290C4⋊C4—271no—
291C4⋊C4—271no—
292C4⋊C4—271no—
293C4⋊C4—271no—
294C4⋊C4—271no—
295C4⋊C4—271no—
296C4⋊C4—271no—
297C4⋊C4—271no—
298C4⋊C4—271no—
299C4⋊C4—271no—
300C4⋊C4—271no—
301C4⋊C4—271no—
302C4⋊C4—271no—
303C4⋊C4C2×D2711yescharacteristic
304C4⋊C4C2×D2711yescharacteristic
305C22⋊C4—271no—
306C22⋊C4—271no—
307C22⋊C4—271no—
308C22⋊C4—271no—
309C22⋊C4—271no—
310C22⋊C4—271no—
311C22⋊C4C2×D2711yescharacteristic
312C22⋊C4C2×D2711yescharacteristic
313C42—271no—
314C42—271no—
315C42—271no—
316C42C2×D2711yescharacteristic
317C42C2×D2711yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3#369712yes—
321C22×C3C23×D911yescharacteristic
322C2×D3—181no—
323C2×D3—181no—
324C2×D3—181no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331Dic3—92no—
332Dic3—181no—
333Dic3—181no—
334Dic3—181no—
335Dic3—181no—
336Dic3—181no—
337Dic3—181no—
338C9#553111yescharacteristic
339C23—271no—
340C23C22×D2711yescharacteristic
341Q8—542no—
342D4—542no—
343C2×C4—22no—
344C2×C4—271no—
345C2×C4—271no—
346C2×C4—271no—
347C2×C4—271no—
348C2×C4—271no—
349C2×C4—271no—
350C2×C4—271no—
351C2×C4—542no—
352C2×C4—542no—
353C2×C4—542no—
354C2×C4C22×D2711yescharacteristic
355C2×C4C22×D2711yescharacteristic
356C2×C4C22×D2711yescharacteristic
357C2×C4C22×D2711yescharacteristic
358C2×C4C22×D2711yescharacteristic
359C2×C4C22×D2711yescharacteristic
360C2×C3—22no—
361C2×C3C2×#369711yescharacteristic
362C2×C3#41×D911yescharacteristic
363C2×C3#6371611yescharacteristic
364D3—361no—
365C22—21no—
366C22—21no—
367C22—541no—
368C22—541no—
369C22—541no—
370C22#8839912yes—
371C22C23×D2711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—272no—
379C4—541no—
380C4—541no—
381C4—541no—
382C4—541no—
383C4—541no—
384C4—541no—
385C3#1058963311yescharacteristic
386C2—22no—
387C2—1081no—
388C2C2×#8839911yescharacteristic
389C2#41×D2711yescharacteristic
390C2#1172067611yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#117C2
socleC22×C3C23×D9
prime radicalC2×#6171C22
hypercenter#117D27
hypocenterC27#552
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C27×#117C2semidirect
3. C2×Dic27⋊C4C2semidirect
4. #11720147C2semidirect
5. #11720145C2semidirect
6. #11720138C2semidirect
7. #11720138C2semidirect
10. #11720127C2semidirect
11. #11720126C2semidirect
12. #11720125C2semidirect
13. #11720121C2semidirect
14. #11720119C2semidirect
15. C4×Dic54C2semidirect
254. #117D27semidirect
256. C27#552semidirect

Maximal subgroups

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

subgroupindex
C27×#1172
#117201782
C2×Dic27⋊C42
#117201472
#117201452
#117201382
#117201352
#6171⋊C42
#117201272
#117201262
#117201252
#117201212
#117201192
C4×Dic542
#105896333