Finite Group Databasebeta

← #13136277

Subgroups — #13136277

Order 1728.

468 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
1C4×D4×C27C211yesmaximal, characteristic, complemented
2#11720188C211yesmaximal, characteristic
3#11720184C211yesmaximal, characteristic, complemented
4C4×#6171C211yesmaximal, characteristic, complemented
5C4×#6171C211yesmaximal, characteristic, complemented
6#11720175C211yesmaximal, characteristic, complemented
7C2×C4×Dic27C211yesmaximal, characteristic, complemented
8#11720147C211yesmaximal, characteristic, complemented
9#11720138C211yesmaximal, characteristic, complemented
10#11720135C211yesmaximal, characteristic, complemented
11#11720134C211yesmaximal, characteristic, complemented
12#11720128C211yesmaximal, characteristic, complemented
13#11720127C211yesmaximal, characteristic, complemented
14#11720121C211yesmaximal, characteristic, complemented
15C4×Dic54C211yesmaximal, characteristic, complemented
16#10589640—31nomaximal
17C2×D4×C27C2211yescharacteristic
18C22×C4×C27C2211yescharacteristic
19C22×C4×C27C2211yescharacteristic
20C2×#6171C2211yescharacteristic
21C2×#6171C2211yescharacteristic
22C22×Dic27C2211yescharacteristic
23C22×Dic27C2211yescharacteristic
24C2×C4×D27C2211yescharacteristic
25C2×Dic54C2211yescharacteristic
26C4⋊C4×C27C2211yescharacteristic
27C22⋊C4×C27C2211yescharacteristic
28C22⋊C4×C27C2211yescharacteristic
29C42×C27C2211yescharacteristic
30#88372C2211yescharacteristic
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
41Dic27⋊C4C2211yescharacteristic
42Dic27⋊C4C2211yescharacteristic
43Dic27⋊C4C2211yescharacteristic
44Dic27⋊C4C2211yescharacteristic
45Dic27⋊C4C2211yescharacteristic
46Dic27⋊C4C2211yescharacteristic
47C4×Dic27C2211yescharacteristic
48C4×Dic27C2211yescharacteristic
49C4×Dic27C2211yescharacteristic
50C4×Dic27C2211yescharacteristic
51C4×Dic27C2211yescharacteristic
52C4×D4×C9D311yescharacteristic
53#63501—31no—
54#63497—31no—
55C4×#611—31no—
56C4×#611—31no—
57#63488—31no—
58C2×C4×Dic9—31no—
59#63460—31no—
60#63451—31no—
61#63448—31no—
62#63447—31no—
63#63441—31no—
64#63440—31no—
65#63434—31no—
66C4×Dic18—31no—
67C23×C27C2311yescharacteristic
68C23×C27C2311yescharacteristic
69C22×D27C2311yescharacteristic
70D4×C27—22no—
71C2×C4×C27—22no—
72C2×C4×C27—22no—
73C2×C4×C27C2311yescharacteristic
74C2×C4×C27C2311yescharacteristic
75C2×C4×C27C2311yescharacteristic
76C2×C4×C27C2311yescharacteristic
77C2×C4×C27C2311yescharacteristic
78#6171—22no—
79#6171—22no—
80C2×Dic27—22no—
81C2×Dic27—22no—
82C2×Dic27C2311yescharacteristic
83C2×Dic27C2311yescharacteristic
84C2×Dic27C2311yescharacteristic
85C2×Dic27C2311yescharacteristic
86C2×Dic27C2311yescharacteristic
87C2×Dic27C2311yescharacteristic
88C2×Dic27C2311yescharacteristic
89C4×D27—22no—
90Dic54—22no—
91#5538—91no—
92C2×D4×C9C2×D311yescharacteristic
93C22×C4×C9C2×D311yescharacteristic
94C22×C4×C9C2×D311yescharacteristic
95C2×#611—31no—
96C2×#611—31no—
97C22×Dic9—31no—
98C22×Dic9—31no—
99C2×C4×D9—31no—
100C2×Dic18—31no—
101C9×C4⋊C4C2×D311yescharacteristic
102C9×C22⋊C4C2×D311yescharacteristic
103C9×C22⋊C4C2×D311yescharacteristic
104C42×C9C2×D311yescharacteristic
105#3676—31no—
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—
116Dic9⋊C4—31no—
117Dic9⋊C4—31no—
118Dic9⋊C4—31no—
119Dic9⋊C4—31no—
120Dic9⋊C4—31no—
121Dic9⋊C4—31no—
122C4×Dic9—31no—
123C4×Dic9—31no—
124C4×Dic9—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#4112yes—
142Dic27—21no—
143Dic27—21no—
144Dic27—21no—
145Dic27—21no—
146Dic27—21no—
147Dic27—21no—
148Dic27#4112yes—
149C3×C4×D4D911yescharacteristic
150#939—91no—
151#935—91no—
152C4×#67—91no—
153C4×#67—91no—
154#926—91no—
155C2×C4×Dic3—91no—
156#898—91no—
157#889—91no—
158#886—91no—
159#885—91no—
160#879—91no—
161#878—91no—
162#872—91no—
163C4×Dic6—91no—
164C23×C9C22×D311yescharacteristic
165C23×C9C22×D311yescharacteristic
166C22×D9—31no—
167D4×C9—22no—
168C2×C4×C9—22no—
169C2×C4×C9—22no—
170C2×C4×C9C22×D311yescharacteristic
171C2×C4×C9C22×D311yescharacteristic
172C2×C4×C9C22×D311yescharacteristic
173C2×C4×C9C22×D311yescharacteristic
174C2×C4×C9C22×D311yescharacteristic
175#611—62no—
176#611—62no—
177C2×Dic9—31no—
178C2×Dic9—31no—
179C2×Dic9—31no—
180C2×Dic9—31no—
181C2×Dic9—31no—
182C2×Dic9—31no—
183C2×Dic9—31no—
184C2×Dic9—62no—
185C2×Dic9—62no—
186C4×D9—62no—
187Dic18—62no—
188#529—271noSylow
189C2×C27—22no—
190C2×C27—41no—
191C2×C27C2×#4111yescharacteristic
192C2×C27C2×#4111yescharacteristic
193C2×C27C2×#4111yescharacteristic
194D27—41no—
195C2×C3×D4C2×D911yescharacteristic
196C22×C3×C4C2×D911yescharacteristic
197C22×C3×C4C2×D911yescharacteristic
198C2×#67—91no—
199C2×#67—91no—
200C22×Dic3—91no—
201C22×Dic3—91no—
202C2×C4×D3—91no—
203C2×Dic6—91no—
204C3×C4⋊C4C2×D911yescharacteristic
205C3×C22⋊C4C2×D911yescharacteristic
206C3×C22⋊C4C2×D911yescharacteristic
207C3×C42C2×D911yescharacteristic
208#217—91no—
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—
219Dic3⋊C4—91no—
220Dic3⋊C4—91no—
221Dic3⋊C4—91no—
222Dic3⋊C4—91no—
223Dic3⋊C4—91no—
224Dic3⋊C4—91no—
225C4×Dic3—91no—
226C4×Dic3—91no—
227C4×Dic3—91no—
228C4×Dic3—91no—
229C4×Dic3—91no—
230C22×C9—21no—
231C22×C9—21no—
232C22×C9—21no—
233C22×C9—21no—
234C22×C9—21no—
235C22×C9#23512yes—
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#23712yes—
245Dic9—32no—
246Dic9—61no—
247Dic9—61no—
248Dic9—61no—
249Dic9—61no—
250Dic9—61no—
251Dic9—61no—
252#126—271no—
253#126—271no—
254#125—271no—
255#124—271no—
256#123—271no—
257#123—271no—
258C22⋊Q8—271no—
259#121—271no—
260C4×Q8—271no—
261C4×D4—271no—
262C4×D4—271no—
263C4×D4D2711yescharacteristic, complemented
264#117—271no—
265#117—271no—
266C2×C42—271no—
267C27#52911yescharacteristic, complemented, Sylow
268C23×C3C22×D911yescharacteristic
269C23×C3C22×D911yescharacteristic
270C22×D3—91no—
271C3×D4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4C22×D911yescharacteristic
275C2×C3×C4C22×D911yescharacteristic
276C2×C3×C4C22×D911yescharacteristic
277C2×C3×C4C22×D911yescharacteristic
278C2×C3×C4C22×D911yescharacteristic
279#67—182no—
280#67—182no—
281C2×Dic3—91no—
282C2×Dic3—91no—
283C2×Dic3—91no—
284C2×Dic3—91no—
285C2×Dic3—91no—
286C2×Dic3—91no—
287C2×Dic3—91no—
288C2×Dic3—182no—
289C2×Dic3—182no—
290C4×D3—182no—
291Dic6—182no—
292C2×C9—22no—
293C2×C9—41no—
294C2×C9C2×#23511yescharacteristic
295C2×C9C2×#23711yescharacteristic
296C2×C9D3×#4111yescharacteristic
297D9—121no—
298C2×Q8—271no—
299C2×D4—271no—
300C2×D4—271no—
301C2×D4C2×D2711yescharacteristic
302C22×C4—271no—
303C22×C4—271no—
304C22×C4—271no—
305C22×C4C2×D2711yescharacteristic
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⋊C4C2×D2711yescharacteristic
317C22⋊C4—271no—
318C22⋊C4—271no—
319C22⋊C4—271no—
320C22⋊C4—271no—
321C22⋊C4—271no—
322C22⋊C4—271no—
323C22⋊C4—271no—
324C22⋊C4—271no—
325C22⋊C4C2×D2711yescharacteristic
326C22⋊C4C2×D2711yescharacteristic
327C42—271no—
328C42—271no—
329C42—271no—
330C42—271no—
331C42—271no—
332C42C2×D2711yescharacteristic
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3#369712yes—
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#369912yes—
348Dic3—92no—
349Dic3—181no—
350Dic3—181no—
351Dic3—181no—
352Dic3—181no—
353Dic3—181no—
354Dic3—181no—
355C9#553811yescharacteristic
356C23—271no—
357C23C22×D2711yescharacteristic
358C23C22×D2711yescharacteristic
359Q8—542no—
360D4—22no—
361D4—542no—
362D4—542no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—271no—
366C2×C4—271no—
367C2×C4—271no—
368C2×C4—271no—
369C2×C4—271no—
370C2×C4—271no—
371C2×C4—271no—
372C2×C4—542no—
373C2×C4—542no—
374C2×C4—542no—
375C2×C4C22×D2711yescharacteristic
376C2×C4C22×D2711yescharacteristic
377C2×C4C22×D2711yescharacteristic
378C2×C4C22×D2711yescharacteristic
379C2×C4C22×D2711yescharacteristic
380C2×C3—22no—
381C2×C3—41no—
382C2×C3C2×#369711yescharacteristic
383C2×C3C2×#369911yescharacteristic
384C2×C3#41×D911yescharacteristic
385D3—361no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—541no—
392C22—541no—
393C22—541no—
394C22#8839912yes—
395C22C23×D2711yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—272no—
401C4—541no—
402C4—541no—
403C4—541no—
404C4—541no—
405C4—541no—
406C4—541no—
407C4#8840112yes—
408C3#1058964011yescharacteristic
409C2—22no—
410C2—41no—
411C2—1081no—
412C2C2×#8839911yescharacteristic
413C2C2×#8840111yescharacteristic
414C2#41×D2711yescharacteristic
subgroupG/H
centerC2×C4C22×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC4×D4×C27C2
socleC22×C3C23×D9
prime radical#11720188C2
hypercenterC4×D4D27
hypocenterC27#529
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×D4×C27C2semidirect
3. #11720184C2semidirect
4. C4×#6171C2semidirect
5. C4×#6171C2semidirect
6. #11720175C2semidirect
7. C2×C4×Dic27C2semidirect
8. #11720147C2semidirect
9. #11720138C2semidirect
10. #11720135C2semidirect
11. #11720134C2semidirect
12. #11720128C2semidirect
13. #11720127C2semidirect
14. #11720121C2semidirect
15. C4×Dic54C2semidirect
263. C4×D4D27semidirect
267. C27#529semidirect

Maximal subgroups

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

subgroupindex
C4×D4×C272
#117201882
#117201842
C4×#61712
#117201752
C2×C4×Dic272
#117201472
#117201382
#117201352
#117201342
#117201282
#117201272
#117201212
C4×Dic542
#105896403