Finite Group Databasebeta

← #127×C22⋊A4 #72228364

Subgroups — #127×C22⋊A4 #72228364

Order 1536.

43916 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 43916 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C22⋊A4C216yesmaximal, complemented
2C42×C22⋊A4C211yesmaximal, characteristic, complemented
3C24×#127C311yesmaximal, characteristic, complemented, Sylow
4C23×C22⋊A4C2214yes—
5D4×C22⋊A4—212no—
6C2×C4×C22⋊A4C2213yes—
7A4×#127—45nomaximal
8C25×D4C2×C316yescomplemented
9C23×#127—330no—
10C23×#127—35no—
11C23×#127—35no—
12C24×C42C2×C311yescharacteristic, complemented
13C22×C22⋊A4—212no—
14C22×C22⋊A4C2311yescharacteristic
15C4×C22⋊A4D416yescomplemented
16C2×D4×A4—430no—
17C42×A4—45no—
18C27C22×C314yes—
19C24×D4—212no—
20C24×D4—330no—
21C24×D4—330no—
22C24×D4—360no—
23C24×D4—660no—
24C25×C4C22×C313yes—
25C22×#127—3240no—
26C22×#127—3180no—
27C22×#127—330no—
28C22×#127—340no—
29C22×#127—330no—
30C22×#127—3180no—
31C22×#127—310no—
32C22×#127—35no—
33C22×#127—330no—
34C22×#127A415yescomplemented
35C23×C42—315no—
36C23×C42—35no—
37C2×C22⋊A4—44no—
38C2×C22⋊A4C2×D413yes—
39C23×A4—420no—
40D4×A4—860no—
41C2×C4×A4—415no—
42C3×#127—161no—
43C26—212no—
44C26—320no—
45C26—320no—
46C26—660no—
47C26C23×C311yescharacteristic
48C23×D4—3360no—
49C23×D4—3360no—
50C23×D4—3180no—
51C23×D4—360no—
52C23×D4—360no—
53C23×D4—360no—
54C23×D4—330no—
55C23×D4—660no—
56C23×D4—660no—
57C23×D4—6120no—
58C23×D4—6720no—
59C23×D4—6360no—
60C23×D4—6360no—
61C23×D4—660no—
62C23×D4—6120no—
63C23×D4—660no—
64C23×D4C2×A4130yescomplemented
65C24×C4—315no—
66C24×C4—315no—
67C24×C4—330no—
68C24×C4C3×D416yescomplemented
69C2×#127—3480no—
70C2×#127—3480no—
71C2×#127—3360no—
72C2×#127—3360no—
73C2×#127—3360no—
74C2×#127—3180no—
75C2×#127—3180no—
76C2×#127—330no—
77C2×#127—390no—
78C2×#127—35no—
79C2×#127—35no—
80C2×#127—330no—
81C22×C42—360no—
82C22×C42—310no—
83C22×C42—390no—
84C22×C42—310no—
85C22×C42—315no—
86C22×C42C2×A415yescomplemented
87C22⋊A4#12711yescharacteristic, direct product factor
88C22×A4—45no—
89C22×A4—860no—
90C2×C3×D4—166no—
91C4×A4—430no—
92C3×C42—161no—
93C25—315no—
94C25—35no—
95C25—3120no—
96C25—340no—
97C25—320no—
98C25—44no—
99C25—660no—
100C25—660no—
101C25—6360no—
102C25—6360no—
103C25—660no—
104C25—660no—
105C25—1260no—
106C25—1240no—
107C25—12240no—
108C25C2×C3×D413yes—
109C25C22×A4120yes—
110C22×D4—260no—
111C22×D4—3720no—
112C22×D4—3360no—
113C22×D4—3360no—
114C22×D4—3180no—
115C22×D4—3180no—
116C22×D4—330no—
117C22×D4—330no—
118C22×D4—360no—
119C22×D4—6120no—
120C22×D4—6120no—
121C22×D4—6360no—
122C22×D4—6360no—
123C22×D4—61440no—
124C22×D4—61440no—
125C22×D4—6720no—
126C22×D4—6720no—
127C22×D4—6360no—
128C22×D4—6720no—
129C22×D4—6360no—
130C22×D4—6720no—
131C22×D4—6360no—
132C22×D4—6720no—
133C22×D4—6120no—
134C22×D4—6180no—
135C22×D4—660no—
136C22×D4—660no—
137C23×C4—390no—
138C23×C4—330no—
139C23×C4—3180no—
140C23×C4—3180no—
141C23×C4—330no—
142C23×C4—315no—
143C23×C4—330no—
144C23×C4—330no—
145C23×C4—330no—
146C23×C4C22×A4115yes—
147#127—3720no—
148#127—3120no—
149#127—3240no—
150#127—340no—
151#127—3180no—
152#127—330no—
153#127—330no—
154#127—35no—
155#127C22⋊A411yescharacteristic, direct product factor
156C2×C42—3120no—
157C2×C42—390no—
158C2×C42—390no—
159C2×C42—35no—
160C2×C42—315no—
161C23×C3—164no—
162C2×A4—415no—
163C2×A4—1620no—
164C3×D4—3212no—
165C2×C3×C4—163no—
166C24—260no—
167C24—315no—
168C24—310no—
169C24—390no—
170C24—360no—
171C24—310no—
172C24—315no—
173C24—315no—
174C24—3120no—
175C24—320no—
176C24—320no—
177C24—6180no—
178C24—6120no—
179C24—6360no—
180C24—6360no—
181C24—6360no—
182C24—6720no—
183C24—660no—
184C24—660no—
185C24—1220no—
186C24—1260no—
187C24—1220no—
188C24—1260no—
189C24—12360no—
190C24—12360no—
191C24—12480no—
192C24—12480no—
193C24—12360no—
194C24—12360no—
195C24C3×#12711yescharacteristic, complemented
196C24C23×A415yes—
197C2×D4—3360no—
198C2×D4—360no—
199C2×D4—360no—
200C2×D4—330no—
201C2×D4—660no—
202C2×D4—660no—
203C2×D4—6720no—
204C2×D4—6720no—
205C2×D4—6120no—
206C2×D4—61440no—
207C2×D4—6360no—
208C2×D4—61440no—
209C2×D4—6720no—
210C2×D4—6720no—
211C2×D4—6360no—
212C2×D4—6360no—
213C2×D4—6360no—
214C2×D4—660no—
215C2×D4—6120no—
216C2×D4—660no—
217C2×D4C2×C22⋊A416yescomplemented
218C22×C4—390no—
219C22×C4—390no—
220C22×C4—3360no—
221C22×C4—3180no—
222C22×C4—3180no—
223C22×C4—360no—
224C22×C4—3180no—
225C22×C4—330no—
226C22×C4—315no—
227C22×C4—315no—
228C22×C4—330no—
229C22×C4D4×A4130yescomplemented
230C42—360no—
231C42—310no—
232C42—315no—
233C42C2×C22⋊A411yescharacteristic, complemented
234C22×C3—161no—
235C22×C3—3212no—
236A4—45no—
237C3×C4—166no—
238C23—315no—
239C23—390no—
240C23—330no—
241C23—390no—
242C23—390no—
243C23—3120no—
244C23—35no—
245C23—315no—
246C23—35no—
247C23—320no—
248C23—420no—
249C23—660no—
250C23—660no—
251C23—6360no—
252C23—6360no—
253C23—660no—
254C23—660no—
255C23—1260no—
256C23—12180no—
257C23—1240no—
258C23—12360no—
259C23—1240no—
260C23—12240no—
261C23—12120no—
262C23—12360no—
263C23—12720no—
264C23—12720no—
265C23—12120no—
266C23—1220no—
267C23C2×D4×A4115yes—
268C23C22×C22⋊A414yes—
269D4—212no—
270D4—6120no—
271D4—6720no—
272D4—6120no—
273D4—660no—
274C2×C4—3180no—
275C2×C4—330no—
276C2×C4—3180no—
277C2×C4—315no—
278C2×C4—330no—
279C2×C4—330no—
280C2×C4—330no—
281C2×C4C22×C22⋊A413yes—
282C2×C3—163no—
283C2×C3—644no—
284C22—212no—
285C22—315no—
286C22—310no—
287C22—360no—
288C22—390no—
289C22—310no—
290C22—315no—
291C22—315no—
292C22—660no—
293C22—1220no—
294C22—1260no—
295C22—1220no—
296C22—1260no—
297C22—12360no—
298C22—12120no—
299C22A4×#12715yescomplemented
300C22C23×C22⋊A411yescharacteristic
301C4—330no—
302C4D4×C22⋊A416yescomplemented
303C3—161noSylow
304C2—315no—
305C2—35no—
306C2—44no—
307C2—1220no—
308C2C2×D4×C22⋊A413yes—
subgroupG/H
center, FrattiniC22C23×C22⋊A4
commutator, socle, cosocleC26C23×C3
FittingC24×#127C3
hypercenter#127C22⋊A4
hypocenterC24C3×#127
LCSG ▷ C26 ▷ C24
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×D4×C22⋊A4C2semidirect
2. C42×C22⋊A4C2semidirect
3. C24×#127C3semidirect
8. C25×D4C2×C3semidirect
12. C24×C42C2×C3semidirect
15. C4×C22⋊A4D4semidirect
34. C22×#127A4semidirect
64. C23×D4C2×A4semidirect
68. C24×C4C3×D4semidirect
86. C22×C42C2×A4semidirect
87. C22⋊A4#127direct (N × H)
155. #127C22⋊A4direct (N × H)
195. C24C3×#127semidirect
217. C2×D4C2×C22⋊A4semidirect
229. C22×C4D4×A4semidirect
233. C42C2×C22⋊A4semidirect
299. C22A4×#127semidirect
302. C4D4×C22⋊A4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C22⋊A42
C42×C22⋊A42
C24×#1273
A4×#1274