Finite Group Databasebeta

← C24×#120⋊C3 #72341390

Subgroups — C24×#120⋊C3 #72341390

Order 1536.

96030 conjugacy classes of subgroups; only the 1189 normal subgroups are shown.

normal subgroups only

This group has 96030 conjugacy classes of subgroups — too many to draw. Only its 1189 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
1C24×C22⋊A4C211yesmaximal, characteristic, complemented
2C23×#120⋊C3C2130yesmaximal, direct product factor
3C24×#120C311yesmaximal, characteristic, complemented, Sylow
4C25×A4—41nomaximal
5C23×C22⋊A4C22115yescomplemented
6C22×#120⋊C3C221140yesdirect product factor
7C28C2×C311yescharacteristic, complemented
8C25×D4—31no—
9C23×#120—330no—
10C23×#120C2×C3130yescomplemented
11C24×C22⋊C4—31no—
12C22×C22⋊A4C23135yescomplemented
13C24×A4—41no—
14C24×A4—430no—
15C24×A4—82no—
16C2×#120⋊C3C231120yesdirect product factor
17C27—31no—
18C27—315no—
19C27—62no—
20C27—630no—
21C27A411yescharacteristic, complemented
22C27C22×C3115yescomplemented
23C24×D4—330no—
24C24×D4—330no—
25C24×D4—62no—
26C24×D4—630no—
27C25×C4—31no—
28C22×#120—3280no—
29C22×#120—3420no—
30C22×#120C22×C31140yescomplemented
31C23×C22⋊C4—330no—
32C23×C22⋊C4—330no—
33C25×C3—161no—
34C2×C22⋊A4C24115yescomplemented
35C23×A4—415no—
36C23×A4—4140no—
37C23×A4—830no—
38#120⋊C3C24116yesdirect product factor
39C26—22no—
40C26—330no—
41C26—32no—
42C26—315no—
43C26—315no—
44C26—370no—
45C26—3105no—
46C26—630no—
47C26—630no—
48C26—61no—
49C26—6210no—
50C26—630no—
51C26—62no—
52C26—630no—
53C26—660no—
54C26—6210no—
55C26—6420no—
56C26—615no—
57C26—630no—
58C26—62no—
59C26—6140no—
60C26—660no—
61C26—6420no—
62C26C2×A411yescharacteristic
63C26C2×A4130yescomplemented
64C26C23×C3135yescomplemented
65C23×D4—3420no—
66C23×D4—3140no—
67C23×D4—660no—
68C23×D4—660no—
69C23×D4—6420no—
70C23×D4—6420no—
71C24×C4—330no—
72C24×C4—615no—
73C24×C4—61no—
74C2×#120—31680no—
75C2×#120—3840no—
76C2×#120C23×C31120yescomplemented
77C22×C22⋊C4—3420no—
78C22×C22⋊C4—3140no—
79C24×C3—1630no—
80C24×C3—161no—
81C22⋊A4C2511yescharacteristic, complemented
82C22×A4—435no—
83C22×A4—4120no—
84C22×A4—870no—
85C25—230no—
86C25—315no—
87C25—31no—
88C25—3210no—
89C25—3210no—
90C25—330no—
91C25—330no—
92C25—3105no—
93C25—335no—
94C25—3210no—
95C25—3105no—
96C25—41no—
97C25—670no—
98C25—6210no—
99C25—630no—
100C25—6210no—
101C25—6210no—
102C25—6210no—
103C25—6420no—
104C25—630no—
105C25—630no—
106C25—660no—
107C25—630no—
108C25—6210no—
109C25—6420no—
110C25—61260no—
111C25—6420no—
112C25—61260no—
113C25—6630no—
114C25—6210no—
115C25—630no—
116C25—6210no—
117C25—630no—
118C25—630no—
119C25—6420no—
120C25—6420no—
121C25—6420no—
122C25—6840no—
123C25—61260no—
124C25—6840no—
125C25—65040no—
126C25—6420no—
127C25—660no—
128C25—62no—
129C25—1215no—
130C25—1270no—
131C25C22×A4115yes—
132C25C22×A41140yescomplemented
133C25C24×C3115yescomplemented
134C22×D4—3120no—
135C22×D4—3840no—
136C22×D4—6840no—
137C22×D4—6280no—
138C22×D4—62520no—
139C22×D4—6840no—
140C23×C4—3140no—
141C23×C4—6210no—
142C23×C4—630no—
143#120—31120no—
144#120—3240no—
145#120C24×C3116yescomplemented
146C2×C22⋊C4—3840no—
147C2×C22⋊C4—3120no—
148C23×C3—16140no—
149C23×C3—1615no—
150C2×A4—415no—
151C2×A4—416no—
152C2×A4—830no—
153C24—270no—
154C24—3105no—
155C24—370no—
156C24—315no—
157C24—315no—
158C24—3630no—
159C24—3210no—
160C24—370no—
161C24—3210no—
162C24—3105no—
163C24—315no—
164C24—370no—
165C24—315no—
166C24—430no—
167C24—630no—
168C24—6210no—
169C24—6140no—
170C24—630no—
171C24—6630no—
172C24—6210no—
173C24—61260no—
174C24—61260no—
175C24—670no—
176C24—6210no—
177C24—6420no—
178C24—6210no—
179C24—6420no—
180C24—6210no—
181C24—6210no—
182C24—6420no—
183C24—61260no—
184C24—61260no—
185C24—6420no—
186C24—6210no—
187C24—6420no—
188C24—6210no—
189C24—6210no—
190C24—670no—
191C24—6210no—
192C24—6140no—
193C24—6420no—
194C24—61260no—
195C24—62520no—
196C24—6420no—
197C24—62520no—
198C24—65040no—
199C24—63360no—
200C24—61260no—
201C24—65040no—
202C24—6420no—
203C24—6140no—
204C24—6420no—
205C24—630no—
206C24—630no—
207C24—12210no—
208C24—12420no—
209C24#120⋊C311yescharacteristic, direct product factor
210C24C23×A4135yes—
211C24C23×A41120yescomplemented
212C24C25×C311yescharacteristic, complemented
213C2×D4—316no—
214C2×D4—3240no—
215C2×D4—6240no—
216C2×D4—61680no—
217C2×D4—6240no—
218C2×D4—61680no—
219C22×C4—3120no—
220C22×C4—6420no—
221C22×C4—6140no—
222C22⋊C4—3240no—
223C22⋊C4—316no—
224C22×C3—16120no—
225C22×C3—1635no—
226A4—41no—
227A4—82no—
228C23—230no—
229C23—3105no—
230C23—3210no—
231C23—335no—
232C23—3105no—
233C23—3210no—
234C23—330no—
235C23—330no—
236C23—3210no—
237C23—315no—
238C23—31no—
239C23—4140no—
240C23—62no—
241C23—630no—
242C23—6120no—
243C23—6210no—
244C23—630no—
245C23—6420no—
246C23—61260no—
247C23—630no—
248C23—6210no—
249C23—6420no—
250C23—6210no—
251C23—61260no—
252C23—6630no—
253C23—630no—
254C23—660no—
255C23—6420no—
256C23—6120no—
257C23—6210no—
258C23—630no—
259C23—630no—
260C23—6210no—
261C23—660no—
262C23—6420no—
263C23—6840no—
264C23—6840no—
265C23—6420no—
266C23—65040no—
267C23—6420no—
268C23—6420no—
269C23—61260no—
270C23—670no—
271C23—6210no—
272C23—12420no—
273C23—12280no—
274C23C2×#120⋊C3115yesdirect product factor
275C23C24×A4115yes—
276C23C24×A4116yescomplemented
277D4—632no—
278D4—6480no—
279C2×C4—316no—
280C2×C4—6120no—
281C2×C4—6120no—
282C2×C3—1616no—
283C2×C3—1615no—
284C22—22no—
285C22—315no—
286C22—370no—
287C22—315no—
288C22—3105no—
289C22—32no—
290C22—330no—
291C22—4120no—
292C22—616no—
293C22—62no—
294C22—630no—
295C22—660no—
296C22—630no—
297C22—6420no—
298C22—6210no—
299C22—630no—
300C22—62no—
301C22—630no—
302C22—660no—
303C22—6140no—
304C22—6420no—
305C22—630no—
306C22—6210no—
307C22—12120no—
308C22C22×#120⋊C3135yesdirect product factor
309C22C25×A411yescharacteristic
310C4—616no—
311C3—161noSylow
312C2—31no—
313C2—315no—
314C2—416no—
315C2—62no—
316C2—630no—
317C2C23×#120⋊C3115yesdirect product factor
subgroupG/H
center, hypercenterC24#120⋊C3
commutator, cosocle, hypocenterC24C25×C3
FrattiniC22C25×A4
FittingC24×#120C3
socleC26C2×A4
LCSG ▷ C24
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×C22⋊A4C2semidirect
2. C23×#120⋊C3C2direct (N × H)
3. C24×#120C3semidirect
5. C23×C22⋊A4C22semidirect
6. C22×#120⋊C3C22direct (N × H)
7. C28C2×C3semidirect
10. C23×#120C2×C3semidirect
12. C22×C22⋊A4C23semidirect
16. C2×#120⋊C3C23direct (N × H)
21. C27A4semidirect
22. C27C22×C3semidirect
30. C22×#120C22×C3semidirect
34. C2×C22⋊A4C24semidirect
38. #120⋊C3C24direct (N × H)
63. C26C2×A4semidirect
64. C26C23×C3semidirect
76. C2×#120C23×C3semidirect
81. C22⋊A4C25semidirect
132. C25C22×A4semidirect
133. C25C24×C3semidirect
145. #120C24×C3semidirect
209. C24#120⋊C3direct (N × H)
211. C24C23×A4semidirect
212. C24C25×C3semidirect
274. C23C2×#120⋊C3direct (N × H)
276. C23C24×A4semidirect
308. C22C22×#120⋊C3direct (N × H)
317. C2C23×#120⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C22⋊A42
C23×#120⋊C32
C24×#1203
C25×A44