Finite Group Databasebeta

← C23×D32⋊D3 #13181848

Subgroups — C23×D32⋊D3 #13181848

Order 1728.

4488 conjugacy classes of subgroups; only the 575 normal subgroups are shown.

normal subgroups only

This group has 4488 conjugacy classes of subgroups — too many to draw. Only its 575 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
1C23×#1113C211yesmaximal, characteristic, complemented
2C23×C3×D32C211yesmaximal, characteristic, complemented
3C23×C3⋊#157C211yesmaximal, characteristic, complemented
4C22×D32⋊D3C2128yesmaximal, direct product factor
5C23×GO+(4,𝔽2)—31nomaximal
6C23×C3×C3⋊D3—21no—
7C23×C3×C3⋊D3C2211yescharacteristic
8C23×C32×D3—21no—
9C22×#1113—27no—
10C22×#1113C22114yescomplemented
11C22×C3×D32—27no—
12C22×C3×D32C22114yescomplemented
13C22×C3⋊#157C22114yescomplemented
14C2×D32⋊D3C221112yesdirect product factor
15C24×C3×D3—61no—
16C23×D32—31no—
17C23×D32—61no—
18C23×D32D311yescharacteristic, complemented
19C23×#157—31no—
20C22×GO+(4,𝔽2)—328no—
21C23×C33D411yescharacteristic, complemented
22C22×C3×C3⋊D3—214no—
23C22×C3×C3⋊D3D417yescomplemented
24C22×C3×C3⋊D3C2317yes—
25C22×C32×D3—214no—
26C2×#1113—242no—
27C2×#1113C23128yescomplemented
28C2×C3×D32—242no—
29C2×C3×D32C23128yescomplemented
30C2×C3⋊#157C23128yescomplemented
31D32⋊D3C23164yesdirect product factor
32C23×#67—91nomaximal
33C24×C32—61no—
34C23×C3⋊D3—21no—
35C23×C3⋊D3C2×D311yescharacteristic
36C23×C3×D3—21no—
37C23×C3×D3—21no—
38C23×C3×D3—41no—
39C23×C3×D3—614no—
40C23×C3×D3—614no—
41C23×C3×D3—61no—
42C23×C3×D3—61no—
43C23×C3×D3—61no—
44C23×C3×D3—61no—
45C23×C3×D3—121no—
46C22×D32—27no—
47C22×D32—314no—
48C22×D32—614no—
49C22×D32—614no—
50C22×D32—67no—
51C22×D32C2×D3114yescomplemented
52C22×#157—314no—
53C2×GO+(4,𝔽2)—3112no—
54C22×C33C2×D417yescomplemented
55C2×C3×C3⋊D3—228no—
56C2×C3×C3⋊D3C2×D4121yescomplemented
57C2×C3×C3⋊D3C2417yes—
58C2×C32×D3—228no—
59#1113—228no—
60#1113C2418yescomplemented
61C3×D32—228no—
62C3×D32C2418yescomplemented
63C3⋊#157C2418yescomplemented
64C25×C3—91no—
65C24×D3—61no—
66C24×D3—91no—
67C24×D3—181no—
68C22×#67—928no—
69C23×Dic3—91no—
70C23×C32—21no—
71C23×C32—21no—
72C23×C32—41no—
73C23×C32—41no—
74C23×C32—614no—
75C23×C32#6711yescharacteristic, complemented
76C22×C3⋊D3—214no—
77C22×C3⋊D3#6717yescomplemented
78C22×C3⋊D3C22×D317yes—
79C22×C3×D3—214no—
80C22×C3×D3—214no—
81C22×C3×D3—414no—
82C22×C3×D3—628no—
83C22×C3×D3—67no—
84C22×C3×D3—67no—
85C22×C3×D3—67no—
86C22×C3×D3—67no—
87C22×C3×D3—684no—
88C22×C3×D3—614no—
89C22×C3×D3—614no—
90C22×C3×D3—1214no—
91C2×D32—242no—
92C2×D32—328no—
93C2×D32—628no—
94C2×D32—684no—
95C2×D32—642no—
96C2×D32C22×D3128yescomplemented
97C2×#157—328no—
98GO+(4,𝔽2)—364no—
99C23×D4—271noSylow
100C2×C33C22×D417yescomplemented
101C3×C3⋊D3—28no—
102C3×C3⋊D3C22×D417yescomplemented
103C3×C3⋊D3C2511yescharacteristic
104C32×D3—28no—
105C24×C3—61no—
106C24×C3—61no—
107C24×C3—914no—
108C24×C3—91no—
109C24×C3—121no—
110C24×C3—181no—
111C24×C3—187no—
112C23×D3—21no—
113C23×D3—614no—
114C23×D3—61no—
115C23×D3—61no—
116C23×D3—614no—
117C23×D3—61no—
118C23×D3—61no—
119C23×D3—914no—
120C23×D3—121no—
121C23×D3—1814no—
122C23×D3—187no—
123C23×D3—1814no—
124C2×#67—9112no—
125C22×Dic3—914no—
126C22×C32—27no—
127C22×C32—27no—
128C22×C32—47no—
129C22×C32—47no—
130C22×C32—628no—
131C22×C32C2×#6717yescomplemented
132C2×C3⋊D3—228no—
133C2×C3⋊D3C2×#67121yescomplemented
134C2×C3⋊D3C23×D317yes—
135C2×C3×D3—228no—
136C2×C3×D3—228no—
137C2×C3×D3—428no—
138C2×C3×D3—68no—
139C2×C3×D3—621no—
140C2×C3×D3—621no—
141C2×C3×D3—67no—
142C2×C3×D3—67no—
143C2×C3×D3—656no—
144C2×C3×D3—628no—
145C2×C3×D3—628no—
146C2×C3×D3—1228no—
147D32—228no—
148D32—38no—
149D32—68no—
150D32—656no—
151D32—628no—
152D32C23×D318yescomplemented
153#157—38no—
154C25—91no—
155C25—271no—
156C22×D4—2728no—
157C23×C4—271no—
158C33C23×D411yescharacteristic, complemented, Sylow
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—614no—
164C23×C3—614no—
165C23×C3—928no—
166C23×C3—97no—
167C23×C3—97no—
168C23×C3—1214no—
169C23×C3—1814no—
170C23×C3—1842no—
171C23×C3GO+(4,𝔽2)11yescharacteristic, complemented
172C22×D3—214no—
173C22×D3—628no—
174C22×D3—67no—
175C22×D3—67no—
176C22×D3—67no—
177C22×D3—67no—
178C22×D3—684no—
179C22×D3—614no—
180C22×D3—614no—
181C22×D3—928no—
182C22×D3—1214no—
183C22×D3—1828no—
184C22×D3—1842no—
185C22×D3—1884no—
186#67—964no—
187C2×Dic3—928no—
188C2×C32—27no—
189C2×C32—27no—
190C2×C32—47no—
191C2×C32—47no—
192C2×C32—68no—
193C2×C32C22×#6717yescomplemented
194C3⋊D3—28no—
195C3⋊D3C22×#6717yescomplemented
196C3⋊D3C24×D311yescharacteristic
197C3×D3—28no—
198C3×D3—28no—
199C3×D3—48no—
200C3×D3—67no—
201C3×D3—67no—
202C3×D3—61no—
203C3×D3—61no—
204C3×D3—68no—
205C3×D3—68no—
206C3×D3—128no—
207C24—61no—
208C24—914no—
209C24—91no—
210C24—187no—
211C24—181no—
212C24—2714no—
213C24—547no—
214C2×D4—27112no—
215C22×C4—2714no—
216C22×C3—27no—
217C22×C3—27no—
218C22×C3—47no—
219C22×C3—47no—
220C22×C3—628no—
221C22×C3—628no—
222C22×C3—98no—
223C22×C3—97no—
224C22×C3—921no—
225C22×C3—1228no—
226C22×C3—1828no—
227C22×C3—1828no—
228C22×C3C2×GO+(4,𝔽2)17yescomplemented
229C2×D3—228no—
230C2×D3—68no—
231C2×D3—621no—
232C2×D3—67no—
233C2×D3—621no—
234C2×D3—67no—
235C2×D3—656no—
236C2×D3—628no—
237C2×D3—628no—
238C2×D3—98no—
239C2×D3—1228no—
240C2×D3—188no—
241C2×D3—1828no—
242C2×D3—1856no—
243Dic3—98no—
244C32—21no—
245C32—21no—
246C32—41no—
247C32—41no—
248C32C23×#6711yescharacteristic, complemented
249C23—614no—
250C23—928no—
251C23—97no—
252C23—97no—
253C23—1842no—
254C23—1814no—
255C23—2728no—
256C23—5442no—
257C23D32⋊D311yescharacteristic, direct product factor
258D4—2764no—
259C2×C4—2728no—
260C2×C3—27no—
261C2×C3—27no—
262C2×C3—47no—
263C2×C3—47no—
264C2×C3—68no—
265C2×C3—68no—
266C2×C3—91no—
267C2×C3—97no—
268C2×C3—128no—
269C2×C3—188no—
270C2×C3C22×GO+(4,𝔽2)17yescomplemented
271D3—28no—
272D3—67no—
273D3—61no—
274D3—67no—
275D3—61no—
276D3—68no—
277D3—68no—
278D3—128no—
279C22—628no—
280C22—98no—
281C22—97no—
282C22—921no—
283C22—1828no—
284C22—1828no—
285C22—278no—
286C22—5428no—
287C22C2×D32⋊D317yesdirect product factor
288C4—278no—
289C3—21no—
290C3—21no—
291C3—41no—
292C3—41no—
293C3C23×GO+(4,𝔽2)11yescharacteristic, complemented
294C2—68no—
295C2—91no—
296C2—97no—
297C2—188no—
298C2C22×D32⋊D317yesdirect product factor
subgroupG/H
center, hypercenterC23D32⋊D3
commutator, cosocleC3×C3⋊D3C25
FrattiniC1G
Fitting, socleC23×C33D4
hypocenterC33C23×D4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×#1113C2semidirect
2. C23×C3×D32C2semidirect
3. C23×C3⋊#157C2semidirect
4. C22×D32⋊D3C2direct (N × H)
10. C22×#1113C22semidirect
12. C22×C3×D32C22semidirect
13. C22×C3⋊#157C22semidirect
14. C2×D32⋊D3C22direct (N × H)
18. C23×D32D3semidirect
21. C23×C33D4semidirect
23. C22×C3×C3⋊D3D4semidirect
27. C2×#1113C23semidirect
29. C2×C3×D32C23semidirect
30. C2×C3⋊#157C23semidirect
31. D32⋊D3C23direct (N × H)
51. C22×D32C2×D3semidirect
54. C22×C33C2×D4semidirect
56. C2×C3×C3⋊D3C2×D4semidirect
60. #1113C24semidirect
62. C3×D32C24semidirect
63. C3⋊#157C24semidirect
75. C23×C32#67semidirect
77. C22×C3⋊D3#67semidirect
96. C2×D32C22×D3semidirect
100. C2×C33C22×D4semidirect
102. C3×C3⋊D3C22×D4semidirect
131. C22×C32C2×#67semidirect
133. C2×C3⋊D3C2×#67semidirect
152. D32C23×D3semidirect
158. C33C23×D4semidirect
171. C23×C3GO+(4,𝔽2)semidirect
193. C2×C32C22×#67semidirect
195. C3⋊D3C22×#67semidirect
228. C22×C3C2×GO+(4,𝔽2)semidirect
248. C32C23×#67semidirect
257. C23D32⋊D3direct (N × H)
270. C2×C3C22×GO+(4,𝔽2)semidirect
287. C22C2×D32⋊D3direct (N × H)
293. C3C23×GO+(4,𝔽2)semidirect
298. C2C22×D32⋊D3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#11132
C23×C3×D322
C23×C3⋊#1572
C22×D32⋊D32
C23×GO+(4,𝔽2)3
C23×#679