Finite Group Databasebeta

← #13136403

Subgroups — #13136403

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×#123C211yesmaximal, characteristic, complemented
2#11720188C211yesmaximal, characteristic
3D4×Dic27C211yesmaximal, characteristic, complemented
4#11720178C211yesmaximal, characteristic, complemented
5C2×C4⋊Dic27C211yesmaximal, characteristic, complemented
6#11720147C211yesmaximal, characteristic, complemented
7#11720146C211yesmaximal, characteristic, complemented
8#11720139C211yesmaximal, characteristic, complemented
9#11720138C211yesmaximal, characteristic, complemented
10#11720135C211yesmaximal, characteristic, complemented
11#11720132C211yesmaximal, characteristic, complemented
12#6171⋊C4C211yesmaximal, characteristic, complemented
13#11720128C211yesmaximal, characteristic, complemented
14#11720127C211yesmaximal, characteristic, complemented
15#11720127C211yesmaximal, characteristic, complemented
16#10589767—31nomaximal
17C2×D4×C27C2211yescharacteristic
18C22×C4×C27C2211yescharacteristic
19C2×#6171C2211yescharacteristic
20C2×#6171C2211yescharacteristic
21C22×Dic27C2211yescharacteristic
22C22×Dic27C2211yescharacteristic
23C22×Dic27C2211yescharacteristic
24C2×C4×D27C2211yescharacteristic
25C2×Dic54C2211yescharacteristic
26C4⋊C4×C27C2211yescharacteristic
27C4⋊C4×C27C2211yescharacteristic
28C22⋊C4×C27C2211yescharacteristic
29C22⋊C4×C27C2211yescharacteristic
30C22⋊C4×C27C2211yescharacteristic
31#88372C2211yescharacteristic
32#88372C2211yescharacteristic
33#88372C2211yescharacteristic
34(C2×D27)⋊C4C2211yescharacteristic
35(C2×D27)⋊C4C2211yescharacteristic
36(C2×D27)⋊C4C2211yescharacteristic
37(C2×D27)⋊C4C2211yescharacteristic
38C4⋊Dic27C2211yescharacteristic
39C4⋊Dic27C2211yescharacteristic
40C4⋊Dic27C2211yescharacteristic
41C4⋊Dic27C2211yescharacteristic
42C4⋊Dic27C2211yescharacteristic
43C4⋊Dic27C2211yescharacteristic
44Dic27⋊C4C2211yescharacteristic
45Dic27⋊C4C2211yescharacteristic
46Dic27⋊C4C2211yescharacteristic
47Dic27⋊C4C2211yescharacteristic
48Dic27⋊C4C2211yescharacteristic
49Dic27⋊C4C2211yescharacteristic
50C4×Dic27C2211yescharacteristic
51C4×Dic27C2211yescharacteristic
52C9×#123D311yescharacteristic
53#63501—31no—
54D4×Dic9—31no—
55#63491—31no—
56C2×C4⋊Dic9—31no—
57#63460—31no—
58#63459—31no—
59#63452—31no—
60#63451—31no—
61#63448—31no—
62#63445—31no—
63#611⋊C4—31no—
64#63441—31no—
65#63440—31no—
66#63440—31no—
67C23×C27C2311yescharacteristic
68C23×C27C2311yescharacteristic
69C22×D27C2311yescharacteristic
70D4×C27—22no—
71C2×C4×C27—22no—
72C2×C4×C27C2311yescharacteristic
73C2×C4×C27C2311yescharacteristic
74C2×C4×C27C2311yescharacteristic
75C2×C4×C27C2311yescharacteristic
76C2×C4×C27C2311yescharacteristic
77#6171—22no—
78#6171—41no—
79C2×Dic27—41no—
80C2×Dic27—41no—
81C2×Dic27—41no—
82C2×Dic27C2311yescharacteristic
83C2×Dic27C2311yescharacteristic
84C2×Dic27C2311yescharacteristic
85C2×Dic27C2311yescharacteristic
86C2×Dic27C2311yescharacteristic
87C2×Dic27C2311yescharacteristic
88C2×Dic27C2311yescharacteristic
89C4×D27—41no—
90Dic54—41no—
91#5665—91no—
92C2×D4×C9C2×D311yescharacteristic
93C22×C4×C9C2×D311yescharacteristic
94C2×#611—31no—
95C2×#611—31no—
96C22×Dic9—31no—
97C22×Dic9—31no—
98C22×Dic9—31no—
99C2×C4×D9—31no—
100C2×Dic18—31no—
101C9×C4⋊C4C2×D311yescharacteristic
102C9×C4⋊C4C2×D311yescharacteristic
103C9×C22⋊C4C2×D311yescharacteristic
104C9×C22⋊C4C2×D311yescharacteristic
105C9×C22⋊C4C2×D311yescharacteristic
106#3676—31no—
107#3676—31no—
108#3676—31no—
109(C2×D9)⋊C4—31no—
110(C2×D9)⋊C4—31no—
111(C2×D9)⋊C4—31no—
112(C2×D9)⋊C4—31no—
113C4⋊Dic9—31no—
114C4⋊Dic9—31no—
115C4⋊Dic9—31no—
116C4⋊Dic9—31no—
117C4⋊Dic9—31no—
118C4⋊Dic9—31no—
119Dic9⋊C4—31no—
120Dic9⋊C4—31no—
121Dic9⋊C4—31no—
122Dic9⋊C4—31no—
123Dic9⋊C4—31no—
124Dic9⋊C4—31no—
125C4×Dic9—31no—
126C4×Dic9—31no—
127C22×C27—21no—
128C22×C27—21no—
129C22×C27—21no—
130C22×C27—21no—
131C22×C27—21no—
132C22×C27#4112yes—
133C22×C27C2411yescharacteristic
134C2×D27—21no—
135C2×D27—21no—
136C2×D27—21no—
137C4×C27—21no—
138C4×C27—21no—
139C4×C27—21no—
140C4×C27—21no—
141C4×C27—21no—
142Dic27—21no—
143Dic27—21no—
144Dic27—21no—
145Dic27—21no—
146Dic27—21no—
147Dic27—21no—
148Dic27—21no—
149C3×#123D911yescharacteristic
150#939—91no—
151D4×Dic3—91no—
152#929—91no—
153C2×C4⋊Dic3—91no—
154#898—91no—
155#897—91no—
156#890—91no—
157#889—91no—
158#886—91no—
159#883—91no—
160#67⋊C4—91no—
161#879—91no—
162#878—91no—
163#878—91no—
164C23×C9C22×D311yescharacteristic
165C23×C9C22×D311yescharacteristic
166C22×D9—31no—
167D4×C9—22no—
168C2×C4×C9—22no—
169C2×C4×C9C22×D311yescharacteristic
170C2×C4×C9C22×D311yescharacteristic
171C2×C4×C9C22×D311yescharacteristic
172C2×C4×C9C22×D311yescharacteristic
173C2×C4×C9C22×D311yescharacteristic
174#611—62no—
175#611—121no—
176C2×Dic9—31no—
177C2×Dic9—31no—
178C2×Dic9—31no—
179C2×Dic9—31no—
180C2×Dic9—31no—
181C2×Dic9—31no—
182C2×Dic9—31no—
183C2×Dic9—121no—
184C2×Dic9—121no—
185C2×Dic9—121no—
186C4×D9—121no—
187Dic18—121no—
188#539—271noSylow
189C2×C27—22no—
190C2×C27—41no—
191C2×C27C2×#4111yescharacteristic
192C2×C27H(5,𝔽2)11yescharacteristic
193C2×C272−1+411yescharacteristic
194D27—41no—
195C2×C3×D4C2×D911yescharacteristic
196C22×C3×C4C2×D911yescharacteristic
197C2×#67—91no—
198C2×#67—91no—
199C22×Dic3—91no—
200C22×Dic3—91no—
201C22×Dic3—91no—
202C2×C4×D3—91no—
203C2×Dic6—91no—
204C3×C4⋊C4C2×D911yescharacteristic
205C3×C4⋊C4C2×D911yescharacteristic
206C3×C22⋊C4C2×D911yescharacteristic
207C3×C22⋊C4C2×D911yescharacteristic
208C3×C22⋊C4C2×D911yescharacteristic
209#217—91no—
210#217—91no—
211#217—91no—
212(C2×D3)⋊C4—91no—
213(C2×D3)⋊C4—91no—
214(C2×D3)⋊C4—91no—
215(C2×D3)⋊C4—91no—
216C4⋊Dic3—91no—
217C4⋊Dic3—91no—
218C4⋊Dic3—91no—
219C4⋊Dic3—91no—
220C4⋊Dic3—91no—
221C4⋊Dic3—91no—
222Dic3⋊C4—91no—
223Dic3⋊C4—91no—
224Dic3⋊C4—91no—
225Dic3⋊C4—91no—
226Dic3⋊C4—91no—
227Dic3⋊C4—91no—
228C4×Dic3—91no—
229C4×Dic3—91no—
230C22×C9—21no—
231C22×C9—21no—
232C22×C9—21no—
233C22×C9—21no—
234C22×C9—21no—
235C22×C9#23712yes—
236C22×C9C23×D311yescharacteristic
237C2×D9—61no—
238C2×D9—61no—
239C2×D9—61no—
240C4×C9—21no—
241C4×C9—21no—
242C4×C9—21no—
243C4×C9—21no—
244C4×C9—21no—
245Dic9—61no—
246Dic9—61no—
247Dic9—61no—
248Dic9—61no—
249Dic9—61no—
250Dic9—61no—
251Dic9—61no—
252#126—271no—
253#126—271no—
254#125—271no—
255#125—271no—
256#123—271no—
257#123—271no—
258#123—271no—
259#123D2711yescharacteristic, complemented
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
269C23×C3C22×D911yescharacteristic
270C22×D3—91no—
271C3×D4—22no—
272C2×C3×C4—22no—
273C2×C3×C4C22×D911yescharacteristic
274C2×C3×C4C22×D911yescharacteristic
275C2×C3×C4C22×D911yescharacteristic
276C2×C3×C4C22×D911yescharacteristic
277C2×C3×C4C22×D911yescharacteristic
278#67—182no—
279#67—361no—
280C2×Dic3—91no—
281C2×Dic3—91no—
282C2×Dic3—91no—
283C2×Dic3—91no—
284C2×Dic3—91no—
285C2×Dic3—91no—
286C2×Dic3—91no—
287C2×Dic3—361no—
288C2×Dic3—361no—
289C2×Dic3—361no—
290C4×D3—361no—
291Dic6—361no—
292C2×C9—22no—
293C2×C9—41no—
294C2×C9C2×#23711yescharacteristic
295C2×C9#100911yescharacteristic
296C2×C9#101011yescharacteristic
297D9—121no—
298C2×Q8—271no—
299C2×D4—271no—
300C2×D4—271no—
301C2×D4C2×D2711yescharacteristic
302C22×C4—271no—
303C22×C4—271no—
304C22×C4—271no—
305C22×C4—271no—
306C22×C4C2×D2711yescharacteristic
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⋊C4—271no—
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—
332C42—271no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3#369912yes—
339C22×C3C23×D911yescharacteristic
340C2×D3—181no—
341C2×D3—181no—
342C2×D3—181no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348Dic3—181no—
349Dic3—181no—
350Dic3—181no—
351Dic3—181no—
352Dic3—181no—
353Dic3—181no—
354Dic3—181no—
355C9#566511yescharacteristic
356C23—271no—
357C23C22×D2711yescharacteristic
358C23C22×D2711yescharacteristic
359Q8—1081no—
360D4—22no—
361D4—542no—
362D4—1081no—
363C2×C4—22no—
364C2×C4—271no—
365C2×C4—271no—
366C2×C4—271no—
367C2×C4—271no—
368C2×C4—271no—
369C2×C4—271no—
370C2×C4—271no—
371C2×C4—1081no—
372C2×C4—1081no—
373C2×C4—1081no—
374C2×C4—1081no—
375C2×C4C22×D2711yescharacteristic
376C2×C4C22×D2711yescharacteristic
377C2×C4C22×D2711yescharacteristic
378C2×C4C22×D2711yescharacteristic
379C2×C4C22×D2711yescharacteristic
380C2×C3—22no—
381C2×C3—41no—
382C2×C3C2×#369911yescharacteristic
383C2×C3#6371511yescharacteristic
384C2×C3#6371611yescharacteristic
385D3—361no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—541no—
392C22—541no—
393C22—541no—
394C22#8840112yes—
395C22C23×D2711yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—541no—
402C4—541no—
403C4—541no—
404C4—541no—
405C4—541no—
406C4—541no—
407C4—541no—
408C3#1058976711yescharacteristic
409C2—22no—
410C2—41no—
411C2—1081no—
412C2C2×#8840111yescharacteristic
413C2#1172067511yescharacteristic
414C2#1172067611yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#123C2
socleC22×C3C23×D9
prime radical#11720188C2
hypercenter#123D27
hypocenterC27#539
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C27×#123C2semidirect
3. D4×Dic27C2semidirect
4. #11720178C2semidirect
5. C2×C4⋊Dic27C2semidirect
6. #11720147C2semidirect
7. #11720146C2semidirect
8. #11720139C2semidirect
9. #11720138C2semidirect
10. #11720135C2semidirect
11. #11720132C2semidirect
12. #6171⋊C4C2semidirect
13. #11720128C2semidirect
14. #11720127C2semidirect
15. #11720127C2semidirect
259. #123D27semidirect
267. C27#539semidirect

Maximal subgroups

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

subgroupindex
C27×#1232
#117201882
D4×Dic272
#117201782
C2×C4⋊Dic272
#117201472
#117201462
#117201392
#117201382
#117201352
#117201322
#6171⋊C42
#117201282
#117201272
#105897673