Finite Group Databasebeta

← #13136446

Subgroups — #13136446

Order 1728.

436 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C27×#126C211yesmaximal, characteristic, complemented
2#11720147C211yesmaximal, characteristic, complemented
3#11720146C211yesmaximal, characteristic, complemented
4#11720145C211yesmaximal, characteristic, complemented
5#11720143C211yesmaximal, characteristic, complemented
6C4⋊C4×D27C211yesmaximal, characteristic, complemented
7#11720139C211yesmaximal, characteristic, complemented
8#11720135C211yesmaximal, characteristic, complemented
9#11720133C211yesmaximal, characteristic
10#11720132C211yesmaximal, characteristic, complemented
11#11720132C211yesmaximal, characteristic, complemented
12#6171⋊C4C211yesmaximal, characteristic, complemented
13#11720127C211yesmaximal, characteristic, complemented
14C4×D108C211yesmaximal, characteristic, complemented
15#11720119C211yesmaximal, characteristic, complemented
16#10589809—31nomaximal
17C2×#6171C2211yescharacteristic
18C2×#6171C2211yescharacteristic
19C22×Dic27C2211yescharacteristic
20C2×D108C2211yescharacteristic
21C2×C4×D27C2211yescharacteristic
22C2×C4×D27C2211yescharacteristic
23C2×C4×D27C2211yescharacteristic
24C2×C4×D27C2211yescharacteristic
25C2×Dic54C2211yescharacteristic
26C4⋊C4×C27C2211yescharacteristic
27C4⋊C4×C27C2211yescharacteristic
28C4⋊C4×C27C2211yescharacteristic
29C22⋊C4×C27C2211yescharacteristic
30C22⋊C4×C27C2211yescharacteristic
31C22⋊C4×C27C2211yescharacteristic
32C42×C27C2211yescharacteristic
33#88372C2211yescharacteristic
34(C2×D27)⋊C4C2211yescharacteristic
35(C2×D27)⋊C4C2211yescharacteristic
36(C2×D27)⋊C4C2211yescharacteristic
37(C2×D27)⋊C4C2211yescharacteristic
38(C2×D27)⋊C4C2211yescharacteristic
39(C2×D27)⋊C4C2211yescharacteristic
40C4⋊Dic27C2211yescharacteristic
41C4⋊Dic27C2211yescharacteristic
42C4⋊Dic27C2211yescharacteristic
43C4⋊Dic27C2211yescharacteristic
44C4⋊Dic27C2211yescharacteristic
45Dic27⋊C4C2211yescharacteristic
46Dic27⋊C4C2211yescharacteristic
47Dic27⋊C4C2211yescharacteristic
48Dic27⋊C4C2211yescharacteristic
49Dic27⋊C4C2211yescharacteristic
50Dic27⋊C4C2211yescharacteristic
51C4×Dic27C2211yescharacteristic
52C9×#126D311yescharacteristic
53#63460—31no—
54#63459—31no—
55#63458—31no—
56#63456—31no—
57C4⋊C4×D9—31no—
58#63452—31no—
59#63448—31no—
60#63446—31no—
61#63445—31no—
62#63445—31no—
63#611⋊C4—31no—
64#63440—31no—
65C4×D36—31no—
66#63432—31no—
67C23×C27C2311yescharacteristic
68C22×D27C2311yescharacteristic
69C22×D27C2311yescharacteristic
70C2×C4×C27C2311yescharacteristic
71C2×C4×C27C2311yescharacteristic
72C2×C4×C27C2311yescharacteristic
73C2×C4×C27C2311yescharacteristic
74C2×C4×C27C2311yescharacteristic
75C2×C4×C27C2311yescharacteristic
76#6171—22no—
77#6171—41no—
78C2×Dic27—41no—
79C2×Dic27C2311yescharacteristic
80C2×Dic27C2311yescharacteristic
81C2×Dic27C2311yescharacteristic
82C2×Dic27C2311yescharacteristic
83C2×Dic27C2311yescharacteristic
84C2×Dic27C2311yescharacteristic
85D108—22no—
86C4×D27—22no—
87C4×D27—41no—
88C4×D27—41no—
89C4×D27—41no—
90Dic54—41no—
91#5707—91no—
92C2×#611—31no—
93C2×#611—31no—
94C22×Dic9—31no—
95C2×D36—31no—
96C2×C4×D9—31no—
97C2×C4×D9—31no—
98C2×C4×D9—31no—
99C2×C4×D9—31no—
100C2×Dic18—31no—
101C9×C4⋊C4C2×D311yescharacteristic
102C9×C4⋊C4C2×D311yescharacteristic
103C9×C4⋊C4C2×D311yescharacteristic
104C9×C22⋊C4C2×D311yescharacteristic
105C9×C22⋊C4C2×D311yescharacteristic
106C9×C22⋊C4C2×D311yescharacteristic
107C42×C9C2×D311yescharacteristic
108#3676—31no—
109(C2×D9)⋊C4—31no—
110(C2×D9)⋊C4—31no—
111(C2×D9)⋊C4—31no—
112(C2×D9)⋊C4—31no—
113(C2×D9)⋊C4—31no—
114(C2×D9)⋊C4—31no—
115C4⋊Dic9—31no—
116C4⋊Dic9—31no—
117C4⋊Dic9—31no—
118C4⋊Dic9—31no—
119C4⋊Dic9—31no—
120Dic9⋊C4—31no—
121Dic9⋊C4—31no—
122Dic9⋊C4—31no—
123Dic9⋊C4—31no—
124Dic9⋊C4—31no—
125Dic9⋊C4—31no—
126C4×Dic9—31no—
127C22×C27—21no—
128C22×C27—21no—
129C22×C27—21no—
130C22×C27C2411yescharacteristic
131C2×D27—21no—
132C2×D27—21no—
133C2×D27—21no—
134C2×D27—21no—
135C2×D27—21no—
136C2×D27#4112yes—
137C4×C27—21no—
138C4×C27—21no—
139C4×C27—21no—
140C4×C27—21no—
141C4×C27—21no—
142C4×C27—21no—
143Dic27—21no—
144Dic27—21no—
145Dic27—21no—
146Dic27—21no—
147Dic27—21no—
148Dic27—21no—
149C3×#126D911yescharacteristic
150#898—91no—
151#897—91no—
152#896—91no—
153#894—91no—
154D3×C4⋊C4—91no—
155#890—91no—
156#886—91no—
157#884—91no—
158#883—91no—
159#883—91no—
160#67⋊C4—91no—
161#878—91no—
162C4×D12—91no—
163#870—91no—
164C23×C9C22×D311yescharacteristic
165C22×D9—31no—
166C22×D9—31no—
167C2×C4×C9C22×D311yescharacteristic
168C2×C4×C9C22×D311yescharacteristic
169C2×C4×C9C22×D311yescharacteristic
170C2×C4×C9C22×D311yescharacteristic
171C2×C4×C9C22×D311yescharacteristic
172C2×C4×C9C22×D311yescharacteristic
173#611—62no—
174#611—121no—
175C2×Dic9—31no—
176C2×Dic9—31no—
177C2×Dic9—31no—
178C2×Dic9—31no—
179C2×Dic9—31no—
180C2×Dic9—31no—
181C2×Dic9—121no—
182D36—62no—
183C4×D9—62no—
184C4×D9—121no—
185C4×D9—121no—
186C4×D9—121no—
187Dic18—121no—
188#539—271noSylow
189C2×C27—41no—
190C2×C27C2×#4111yescharacteristic
191C2×C27H(5,𝔽2)11yescharacteristic
192C2×C272−1+411yescharacteristic
193D27—22no—
194D27—41no—
195C2×#67—91no—
196C2×#67—91no—
197C22×Dic3—91no—
198C2×D12—91no—
199C2×C4×D3—91no—
200C2×C4×D3—91no—
201C2×C4×D3—91no—
202C2×C4×D3—91no—
203C2×Dic6—91no—
204C3×C4⋊C4C2×D911yescharacteristic
205C3×C4⋊C4C2×D911yescharacteristic
206C3×C4⋊C4C2×D911yescharacteristic
207C3×C22⋊C4C2×D911yescharacteristic
208C3×C22⋊C4C2×D911yescharacteristic
209C3×C22⋊C4C2×D911yescharacteristic
210C3×C42C2×D911yescharacteristic
211#217—91no—
212(C2×D3)⋊C4—91no—
213(C2×D3)⋊C4—91no—
214(C2×D3)⋊C4—91no—
215(C2×D3)⋊C4—91no—
216(C2×D3)⋊C4—91no—
217(C2×D3)⋊C4—91no—
218C4⋊Dic3—91no—
219C4⋊Dic3—91no—
220C4⋊Dic3—91no—
221C4⋊Dic3—91no—
222C4⋊Dic3—91no—
223Dic3⋊C4—91no—
224Dic3⋊C4—91no—
225Dic3⋊C4—91no—
226Dic3⋊C4—91no—
227Dic3⋊C4—91no—
228Dic3⋊C4—91no—
229C4×Dic3—91no—
230C22×C9—21no—
231C22×C9—21no—
232C22×C9—21no—
233C22×C9C23×D311yescharacteristic
234C2×D9—32no—
235C2×D9—61no—
236C2×D9—61no—
237C2×D9—61no—
238C2×D9—61no—
239C2×D9—61no—
240C4×C9—21no—
241C4×C9—21no—
242C4×C9—21no—
243C4×C9—21no—
244C4×C9—21no—
245C4×C9—21no—
246Dic9—61no—
247Dic9—61no—
248Dic9—61no—
249Dic9—61no—
250Dic9—61no—
251Dic9—61no—
252#126—271no—
253#126D2711yescharacteristic, complemented
254#125—271no—
255#125—271no—
256#123—271no—
257#123—271no—
258#123—271no—
259#123—271no—
260C22⋊Q8—271no—
261C22⋊Q8—271no—
262C22⋊Q8—271no—
263#121—271no—
264C4×D4—271no—
265C4×D4—271no—
266C2×C4⋊C4—271no—
267C27#53911yescharacteristic, complemented, Sylow
268C23×C3C22×D911yescharacteristic
269C22×D3—91no—
270C22×D3—91no—
271C2×C3×C4C22×D911yescharacteristic
272C2×C3×C4C22×D911yescharacteristic
273C2×C3×C4C22×D911yescharacteristic
274C2×C3×C4C22×D911yescharacteristic
275C2×C3×C4C22×D911yescharacteristic
276C2×C3×C4C22×D911yescharacteristic
277#67—182no—
278#67—361no—
279C2×Dic3—91no—
280C2×Dic3—91no—
281C2×Dic3—91no—
282C2×Dic3—91no—
283C2×Dic3—91no—
284C2×Dic3—91no—
285C2×Dic3—361no—
286D12—182no—
287C4×D3—182no—
288C4×D3—361no—
289C4×D3—361no—
290C4×D3—361no—
291Dic6—361no—
292C2×C9—41no—
293C2×C9D3×#4111yescharacteristic
294C2×C9#100911yescharacteristic
295C2×C9#101011yescharacteristic
296D9—62no—
297D9—121no—
298C2×Q8—271no—
299C2×D4—271no—
300C2×D4—271no—
301C2×D4—271no—
302C22×C4—271no—
303C22×C4—271no—
304C22×C4—271no—
305C22×C4—271no—
306C22×C4—271no—
307C4⋊C4—271no—
308C4⋊C4—271no—
309C4⋊C4—271no—
310C4⋊C4—271no—
311C4⋊C4—271no—
312C4⋊C4—271no—
313C4⋊C4—271no—
314C4⋊C4—271no—
315C4⋊C4—271no—
316C4⋊C4—271no—
317C4⋊C4—271no—
318C4⋊C4C2×D2711yescharacteristic
319C4⋊C4C2×D2711yescharacteristic
320C4⋊C4C2×D2711yescharacteristic
321C22⋊C4—271no—
322C22⋊C4—271no—
323C22⋊C4—271no—
324C22⋊C4—271no—
325C22⋊C4—271no—
326C22⋊C4—271no—
327C22⋊C4—271no—
328C22⋊C4C2×D2711yescharacteristic
329C22⋊C4C2×D2711yescharacteristic
330C22⋊C4C2×D2711yescharacteristic
331C42—271no—
332C42C2×D2711yescharacteristic
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3C23×D911yescharacteristic
337C2×D3—92no—
338C2×D3—181no—
339C2×D3—181no—
340C2×D3—181no—
341C2×D3—181no—
342C2×D3—181no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349Dic3—181no—
350Dic3—181no—
351Dic3—181no—
352Dic3—181no—
353Dic3—181no—
354Dic3—181no—
355C9#570711yescharacteristic
356C23—271no—
357C23—271no—
358C23C22×D2711yescharacteristic
359Q8—1081no—
360D4—542no—
361D4—542no—
362D4—1081no—
363C2×C4—271no—
364C2×C4—271no—
365C2×C4—271no—
366C2×C4—271no—
367C2×C4—271no—
368C2×C4—271no—
369C2×C4—542no—
370C2×C4—1081no—
371C2×C4—1081no—
372C2×C4—1081no—
373C2×C4—1081no—
374C2×C4C22×D2711yescharacteristic
375C2×C4C22×D2711yescharacteristic
376C2×C4C22×D2711yescharacteristic
377C2×C4C22×D2711yescharacteristic
378C2×C4C22×D2711yescharacteristic
379C2×C4C22×D2711yescharacteristic
380C2×C3—41no—
381C2×C3#41×D911yescharacteristic
382C2×C3#6371511yescharacteristic
383C2×C3#6371611yescharacteristic
384D3—182no—
385D3—361no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—272no—
390C22—541no—
391C22—541no—
392C22—541no—
393C22—541no—
394C22—541no—
395C22C23×D2711yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—541no—
403C4—541no—
404C4—541no—
405C4—541no—
406C4—541no—
407C4—541no—
408C3#1058980911yescharacteristic
409C2—41no—
410C2—542no—
411C2—1081no—
412C2#41×D2711yescharacteristic
413C2#1172067511yescharacteristic
414C2#1172067611yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#126C2
socleC22×C3C23×D9
prime radical#11720133C2
hypercenter#126D27
hypocenterC27#539
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. #11720147C2semidirect
3. #11720146C2semidirect
4. #11720145C2semidirect
5. #11720143C2semidirect
6. C4⋊C4×D27C2semidirect
7. #11720139C2semidirect
8. #11720135C2semidirect
10. #11720132C2semidirect
11. #11720132C2semidirect
12. #6171⋊C4C2semidirect
13. #11720127C2semidirect
14. C4×D108C2semidirect
15. #11720119C2semidirect
253. #126D27semidirect
267. C27#539semidirect

Maximal subgroups

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

subgroupindex
C27×#1262
#117201472
#117201462
#117201452
#117201432
C4⋊C4×D272
#117201392
#117201352
#117201332
#117201322
#6171⋊C42
#117201272
C4×D1082
#117201192
#105898093