Finite Group Databasebeta

← C32×D4×C22⋊A4 #72253044

Subgroups — C32×D4×C22⋊A4 #72253044

Order 3456.

12008 conjugacy classes of subgroups; only the 384 normal subgroups are shown.

normal subgroups only

This group has 12008 conjugacy classes of subgroups — too many to draw. Only its 384 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
1C22×C32×C22⋊A4C212yesmaximal, complemented
2C32×C4×C22⋊A4C211yesmaximal, characteristic, complemented
3C3×D4×C22⋊A4C3112yesmaximal, direct product factor
4C24×C32×D4C311yesmaximal, characteristic, complemented
5C2×C32×C22⋊A4—22no—
6C2×C32×C22⋊A4C2211yescharacteristic
7C32×D4×A4—45nomaximal
8C26×C32C2×C312yescomplemented
9C22×C3×C22⋊A4C2×C3124yescomplemented
10C23×C32×D4—35no—
11C23×C32×D4—310no—
12C23×C32×D4—35no—
13C24×C32×C4C2×C311yescharacteristic, complemented
14C3×C4×C22⋊A4C2×C3112yescomplemented
15C32×C22⋊A4D411yescharacteristic, direct product factor
16C22×C32×A4—410no—
17C32×C4×A4—45no—
18C24×C3×D4C3214yescomplemented
19D4×C22⋊A4C3219yesdirect product factor
20C25×C32—22no—
21C25×C32—310no—
22C25×C32—310no—
23C25×C32—610no—
24C25×C32C22×C311yescharacteristic
25C2×C3×C22⋊A4—224no—
26C2×C3×C22⋊A4C22×C3112yes—
27C22×C32×D4—360no—
28C22×C32×D4—330no—
29C22×C32×D4—360no—
30C22×C32×D4—310no—
31C22×C32×D4—310no—
32C22×C32×D4—35no—
33C22×C32×D4—310no—
34C22×C32×D4A415yescomplemented
35C23×C32×C4—35no—
36C23×C32×C4—35no—
37C3×D4×A4—460no—
38C2×C32×A4—45no—
39C2×C32×A4—810no—
40C33×D4—161no—
41C26×C3C2×C3218yescomplemented
42C22×C22⋊A4C2×C32118yescomplemented
43C23×C3×D4—320no—
44C23×C3×D4—340no—
45C23×C3×D4—320no—
46C24×C3×C4C2×C3214yescomplemented
47C4×C22⋊A4C2×C3219yescomplemented
48C24×C32—35no—
49C24×C32—35no—
50C24×C32—360no—
51C24×C32—320no—
52C24×C32—310no—
53C24×C32—610no—
54C24×C32—610no—
55C24×C32—660no—
56C24×C32—660no—
57C24×C32—610no—
58C24×C32—610no—
59C24×C32C3×D411yescharacteristic, complemented
60C24×C32C2×A4110yescomplemented
61C3×C22⋊A4C3×D4112yesdirect product factor
62C22×C3×A4—4120no—
63C2×C32×D4—3120no—
64C2×C32×D4—360no—
65C2×C32×D4—360no—
66C2×C32×D4—330no—
67C2×C32×D4—330no—
68C2×C32×D4—35no—
69C2×C32×D4—35no—
70C2×C32×D4—310no—
71C22×C32×C4—330no—
72C22×C32×C4—310no—
73C22×C32×C4—35no—
74C22×C32×C4C2×A415yescomplemented
75C3×C4×A4—460no—
76C24×D4C3311yescharacteristic, complemented, Sylow
77C22×C33—162no—
78C32×A4—45no—
79C33×C4—161no—
80C25×C3—28no—
81C25×C3—340no—
82C25×C3—340no—
83C25×C3—640no—
84C25×C3C22×C3214yes—
85C2×C22⋊A4—218no—
86C2×C22⋊A4C22×C3219yes—
87C22×C3×D4—3240no—
88C22×C3×D4—3240no—
89C22×C3×D4—3120no—
90C22×C3×D4—340no—
91C22×C3×D4—340no—
92C22×C3×D4—340no—
93C22×C3×D4—320no—
94C22×C3×D4C3×A4120yescomplemented
95C23×C3×C4—320no—
96C23×C3×C4—320no—
97D4×A4—445no—
98C23×C32—210no—
99C23×C32—310no—
100C23×C32—330no—
101C23×C32—35no—
102C23×C32—35no—
103C23×C32—360no—
104C23×C32—310no—
105C23×C32—310no—
106C23×C32—610no—
107C23×C32—630no—
108C23×C32—620no—
109C23×C32—660no—
110C23×C32—660no—
111C23×C32—660no—
112C23×C32—6120no—
113C23×C32—610no—
114C23×C32C22×A415yes—
115C2×C3×A4—460no—
116C2×C3×A4—8120no—
117C32×D4—360no—
118C32×D4—310no—
119C32×D4—310no—
120C32×D4—35no—
121C32×D4—1612no—
122C32×D4C22⋊A411yescharacteristic, direct product factor
123C2×C32×C4—330no—
124C2×C32×C4—35no—
125C2×C32×C4—35no—
126C26C2×C3312yescomplemented
127C23×D4—35no—
128C23×D4—35no—
129C23×D4—310no—
130C24×C4C2×C3311yescharacteristic, complemented
131C2×C33—161no—
132C2×C33—322no—
133C24×C3—320no—
134C24×C3—320no—
135C24×C3—3240no—
136C24×C3—380no—
137C24×C3—340no—
138C24×C3—640no—
139C24×C3—640no—
140C24×C3—640no—
141C24×C3—640no—
142C24×C3—6240no—
143C24×C3—6240no—
144C24×C3C32×D414yescomplemented
145C24×C3C2×C3×A4140yescomplemented
146C22⋊A4C32×D419yesdirect product factor
147C22×A4—490no—
148C2×C3×D4—3480no—
149C2×C3×D4—3240no—
150C2×C3×D4—3240no—
151C2×C3×D4—3120no—
152C2×C3×D4—3120no—
153C2×C3×D4—320no—
154C2×C3×D4—320no—
155C2×C3×D4—340no—
156C22×C3×C4—3120no—
157C22×C3×C4—340no—
158C22×C3×C4—320no—
159C22×C3×C4C2×C3×A4120yescomplemented
160C4×A4—445no—
161C22×C32—35no—
162C22×C32—330no—
163C22×C32—310no—
164C22×C32—35no—
165C22×C32—310no—
166C22×C32—610no—
167C22×C32—610no—
168C22×C32—610no—
169C22×C32—610no—
170C22×C32—660no—
171C22×C32—660no—
172C22×C32—1624no—
173C22×C32D4×A415yescomplemented
174C22×C32C2×C22⋊A412yescomplemented
175C3×A4—460no—
176C32×C4—35no—
177C32×C4—1612no—
178C32×C4C2×C22⋊A411yescharacteristic, complemented
179C25—22no—
180C25—310no—
181C25—310no—
182C25—610no—
183C25C22×C3311yescharacteristic
184C22×D4—360no—
185C22×D4—360no—
186C22×D4—330no—
187C22×D4—310no—
188C22×D4—310no—
189C22×D4—310no—
190C22×D4—35no—
191C22×D4C32×A415yescomplemented
192C23×C4—35no—
193C23×C4—35no—
194C33—161noSylow
195C23×C3—240no—
196C23×C3—340no—
197C23×C3—3120no—
198C23×C3—320no—
199C23×C3—320no—
200C23×C3—3240no—
201C23×C3—340no—
202C23×C3—340no—
203C23×C3—640no—
204C23×C3—6120no—
205C23×C3—680no—
206C23×C3—6240no—
207C23×C3—6240no—
208C23×C3—6240no—
209C23×C3—6480no—
210C23×C3—640no—
211C23×C3C22×C3×A4120yes—
212C2×A4—445no—
213C2×A4—890no—
214C3×D4—3240no—
215C3×D4—340no—
216C3×D4—340no—
217C3×D4—320no—
218C3×D4—169no—
219C3×D4C3×C22⋊A414yesdirect product factor
220C2×C3×C4—3120no—
221C2×C3×C4—320no—
222C2×C3×C4—320no—
223C2×C32—22no—
224C2×C32—35no—
225C2×C32—35no—
226C2×C32—610no—
227C2×C32—1612no—
228C2×C32—3224no—
229C2×C32C22×C22⋊A411yescharacteristic
230C24—35no—
231C24—35no—
232C24—360no—
233C24—320no—
234C24—310no—
235C24—610no—
236C24—610no—
237C24—610no—
238C24—610no—
239C24—660no—
240C24—660no—
241C24C33×D411yescharacteristic, complemented
242C24C2×C32×A4110yescomplemented
243C2×D4—3120no—
244C2×D4—360no—
245C2×D4—360no—
246C2×D4—330no—
247C2×D4—330no—
248C2×D4—35no—
249C2×D4—35no—
250C2×D4—310no—
251C22×C4—330no—
252C22×C4—310no—
253C22×C4—35no—
254C22×C4C2×C32×A415yescomplemented
255C22×C3—320no—
256C22×C3—340no—
257C22×C3—3120no—
258C22×C3—320no—
259C22×C3—340no—
260C22×C3—640no—
261C22×C3—640no—
262C22×C3—640no—
263C22×C3—640no—
264C22×C3—6240no—
265C22×C3—6240no—
266C22×C3—1618no—
267C22×C3C3×D4×A4120yescomplemented
268C22×C3C2×C3×C22⋊A418yescomplemented
269A4—445no—
270C3×C4—320no—
271C3×C4—169no—
272C3×C4C2×C3×C22⋊A414yescomplemented
273C32—1612no—
274C32D4×C22⋊A411yescharacteristic, direct product factor
275C23—210no—
276C23—310no—
277C23—330no—
278C23—35no—
279C23—35no—
280C23—360no—
281C23—310no—
282C23—310no—
283C23—610no—
284C23—630no—
285C23—620no—
286C23—660no—
287C23—660no—
288C23—660no—
289C23—6120no—
290C23—610no—
291C23C22×C32×A415yes—
292D4—360no—
293D4—310no—
294D4—310no—
295D4—35no—
296D4C32×C22⋊A411yescharacteristic, direct product factor
297C2×C4—330no—
298C2×C4—35no—
299C2×C4—35no—
300C2×C3—28no—
301C2×C3—320no—
302C2×C3—320no—
303C2×C3—640no—
304C2×C3—169no—
305C2×C3—3218no—
306C2×C3C22×C3×C22⋊A414yes—
307C22—35no—
308C22—310no—
309C22—330no—
310C22—35no—
311C22—310no—
312C22—610no—
313C22—610no—
314C22—610no—
315C22—610no—
316C22—660no—
317C22—660no—
318C22C32×D4×A415yescomplemented
319C22C2×C32×C22⋊A412yescomplemented
320C4—35no—
321C4C2×C32×C22⋊A411yescharacteristic, complemented
322C3—169no—
323C3C3×D4×C22⋊A414yesdirect product factor
324C2—22no—
325C2—35no—
326C2—35no—
327C2—610no—
328C2C22×C32×C22⋊A411yescharacteristic
subgroupG/H
centerC2×C32C22×C22⋊A4
commutator, cosocleC25C22×C33
FrattiniC2C22×C32×C22⋊A4
FittingC24×C32×D4C3
socleC25×C32C22×C3
hypercenterC32×D4C22⋊A4
hypocenterC24C33×D4
LCSG ▷ C25 ▷ C24
UCSC1 ◁ C2×C32 ◁ C32×D4

Complements

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

NH ≅ G/Ntype
1. C22×C32×C22⋊A4C2semidirect
2. C32×C4×C22⋊A4C2semidirect
3. C3×D4×C22⋊A4C3direct (N × H)
4. C24×C32×D4C3semidirect
8. C26×C32C2×C3semidirect
9. C22×C3×C22⋊A4C2×C3semidirect
13. C24×C32×C4C2×C3semidirect
14. C3×C4×C22⋊A4C2×C3semidirect
15. C32×C22⋊A4D4direct (N × H)
18. C24×C3×D4C32semidirect
19. D4×C22⋊A4C32direct (N × H)
34. C22×C32×D4A4semidirect
41. C26×C3C2×C32semidirect
42. C22×C22⋊A4C2×C32semidirect
46. C24×C3×C4C2×C32semidirect
47. C4×C22⋊A4C2×C32semidirect
59. C24×C32C3×D4semidirect
60. C24×C32C2×A4semidirect
61. C3×C22⋊A4C3×D4direct (N × H)
74. C22×C32×C4C2×A4semidirect
76. C24×D4C33semidirect
94. C22×C3×D4C3×A4semidirect
122. C32×D4C22⋊A4direct (N × H)
126. C26C2×C33semidirect
130. C24×C4C2×C33semidirect
144. C24×C3C32×D4semidirect
145. C24×C3C2×C3×A4semidirect
146. C22⋊A4C32×D4direct (N × H)
159. C22×C3×C4C2×C3×A4semidirect
173. C22×C32D4×A4semidirect
174. C22×C32C2×C22⋊A4semidirect
178. C32×C4C2×C22⋊A4semidirect
191. C22×D4C32×A4semidirect
219. C3×D4C3×C22⋊A4direct (N × H)
241. C24C33×D4semidirect
242. C24C2×C32×A4semidirect
254. C22×C4C2×C32×A4semidirect
267. C22×C3C3×D4×A4semidirect
268. C22×C3C2×C3×C22⋊A4semidirect
272. C3×C4C2×C3×C22⋊A4semidirect
274. C32D4×C22⋊A4direct (N × H)
296. D4C32×C22⋊A4direct (N × H)
318. C22C32×D4×A4semidirect
319. C22C2×C32×C22⋊A4semidirect
321. C4C2×C32×C22⋊A4semidirect
323. C3C3×D4×C22⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C22⋊A42
C32×C4×C22⋊A42
C3×D4×C22⋊A43
C24×C32×D43
C32×D4×A44