Finite Group Databasebeta

← C3×#64241 #11724296

Subgroups — C3×#64241 #11724296

Order 864.

496 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 58 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×C3×D32C211yesmaximal, characteristic, complemented
2C2×C3×GO+(4,𝔽2)C212yesmaximal, complemented
3C3×#3811C211yesmaximal, characteristic, complemented
4C3×C22⋊#157C211yesmaximal, characteristic, complemented
5C3×D32⋊C4C212yesmaximal, complemented
6#64241C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊D3C2211yescharacteristic
8C22×C32×D3—21no—
9C2×C3×D32—21no—
10C2×C3×D32—24no—
11C2×C3×D32C2212yes—
12C2×C3×D32C2211yescharacteristic, complemented
13C2×C3×#157C2212yescomplemented
14C3×GO+(4,𝔽2)—24no—
15C32×#67—21no—
16C3×#626—21no—
17C3×#624C2211yescharacteristic, complemented
18C23×C3×D3—61no—
19C22×D32C2×C311yescharacteristic, complemented
20C2×GO+(4,𝔽2)C2×C312yescomplemented
21C3×D3×D4—61no—
22#3811C2×C311yescharacteristic, complemented
23C22⋊#157C2×C311yescharacteristic, complemented
24D32⋊C4C2×C312yescomplemented
25C22×C33D411yescharacteristic, complemented
26C2×C3×C3⋊D3—22no—
27C2×C3×C3⋊D3D411yescharacteristic, complemented
28C2×C3×C3⋊D3C2311yescharacteristic
29C2×C32×D3—24no—
30C2×C32×D3—22no—
31C2×C32×D3—21no—
32C3×D32—24no—
33C3×D32—22no—
34C3×D32—22no—
35C3×D32D414yescomplemented
36C3×#157—22no—
37C32×Dic3—21no—
38C3×#120—91nomaximal
39C23×C32—61no—
40C22×C3⋊D3C22×C311yescharacteristic
41C22×C3×D3—21no—
42C22×C3×D3—21no—
43C22×C3×D3—41no—
44C22×C3×D3—62no—
45C22×C3×D3—61no—
46C22×C3×D3—61no—
47C22×C3×D3—61no—
48C22×C3×D3—68no—
49C22×C3×D3—62no—
50C2×D32—21no—
51C2×D32—24no—
52C2×D32C22×C312yes—
53C2×D32C22×C311yescharacteristic, complemented
54C2×#157C22×C312yescomplemented
55GO+(4,𝔽2)—24no—
56C32×D4—61no—
57C3×#67—21no—
58C3×#67—21no—
59C3×#67—41no—
60C3×#67—61no—
61C3×D12—61no—
62C3×C4×D3—61no—
63#626—21no—
64#624C22×C311yescharacteristic, complemented
65C2×C33—21no—
66C2×C33C2×D411yescharacteristic
67C3×C3⋊D3—21no—
68C3×C3⋊D3C2×D412yes—
69C32×D3—24no—
70C32×D3—41no—
71C24×C3—91no—
72C23×D3—61no—
73C2×C3×D4—91no—
74C2×C3×D4—92no—
75D3×D4—61no—
76C3×C22⋊C4—92no—
77C3×C22⋊C4—91no—
78C22×C32—21no—
79C22×C32—21no—
80C22×C32—41no—
81C22×C32—41no—
82C22×C32—64no—
83C22×C32—62no—
84C22×C32—61no—
85C22×C32C3×D411yescharacteristic, complemented
86C2×C3⋊D3—22no—
87C2×C3⋊D3C3×D411yescharacteristic, complemented
88C2×C3⋊D3C23×C311yescharacteristic
89C2×C3×D3—24no—
90C2×C3×D3—24no—
91C2×C3×D3—22no—
92C2×C3×D3—22no—
93C2×C3×D3—21no—
94C2×C3×D3—21no—
95C2×C3×D3—42no—
96C2×C3×D3—44no—
97C2×C3×D3—41no—
98C2×C3×D3—64no—
99C2×C3×D3—61no—
100C2×C3×D3—61no—
101C2×C3×D3—61no—
102C2×C3×D3—61no—
103C2×C3×D3—64no—
104C2×C3×D3—68no—
105C2×C3×D3—64no—
106C2×C3×D3—122no—
107C2×C3×D3—122no—
108D32—24no—
109D32—22no—
110D32—22no—
111D32C3×D414yescomplemented
112#157—22no—
113C32×C4—61no—
114C3×Dic3—21no—
115C3×Dic3—21no—
116C3×Dic3—41no—
117#120—91noSylow
118C33#12011yescharacteristic, complemented, Sylow
119C23×C3—61no—
120C23×C3—61no—
121C23×C3—92no—
122C23×C3—91no—
123C23×C3—91no—
124C23×C3—121no—
125C23×C3—184no—
126C23×C3—181no—
127C22×D3—21no—
128C22×D3—62no—
129C22×D3—61no—
130C22×D3—61no—
131C22×D3—61no—
132C22×D3—68no—
133C22×D3—62no—
134C3×D4—61no—
135C3×D4—61no—
136C3×D4—121no—
137C3×D4—181no—
138C3×D4—184no—
139C2×C3×C4—91no—
140C2×C3×C4—92no—
141#67—21no—
142#67—61no—
143D12—61no—
144C4×D3—61no—
145C2×C32—22no—
146C2×C32—21no—
147C2×C32—21no—
148C2×C32—21no—
149C2×C32—41no—
150C2×C32—41no—
151C2×C32—41no—
152C2×C32—42no—
153C2×C32—64no—
154C2×C32—81no—
155C2×C32—121no—
156C2×C32C2×C3×D411yescharacteristic
157C3⋊D3—21no—
158C3⋊D3C2×C3×D412yes—
159C3×D3—24no—
160C3×D3—24no—
161C3×D3—44no—
162C3×D3—41no—
163C3×D3—41no—
164C3×D3—62no—
165C3×D3—62no—
166C3×D3—62no—
167C3×D3—81no—
168C3×D3—121no—
169C24—91no—
170C2×D4—91no—
171C2×D4—92no—
172C22⋊C4—92no—
173C22⋊C4—91no—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3—41no—
177C22×C3—41no—
178C22×C3—64no—
179C22×C3—62no—
180C22×C3—64no—
181C22×C3—61no—
182C22×C3—62no—
183C22×C3—61no—
184C22×C3—94no—
185C22×C3—91no—
186C22×C3—91no—
187C22×C3—124no—
188C22×C3—122no—
189C22×C3—121no—
190C22×C3—182no—
191C22×C3—182no—
192C22×C3—184no—
193C22×C3—182no—
194C22×C3GO+(4,𝔽2)11yescharacteristic, complemented
195C2×D3—24no—
196C2×D3—22no—
197C2×D3—21no—
198C2×D3—64no—
199C2×D3—61no—
200C2×D3—61no—
201C2×D3—61no—
202C2×D3—61no—
203C2×D3—64no—
204C2×D3—68no—
205C2×D3—64no—
206C2×D3—122no—
207C2×D3—122no—
208C3×C4—61no—
209C3×C4—61no—
210C3×C4—121no—
211C3×C4—182no—
212Dic3—21no—
213C32—21no—
214C32—21no—
215C32—41no—
216C32—41no—
217C32C3×#12011yescharacteristic, complemented
218C23—61no—
219C23—92no—
220C23—91no—
221C23—91no—
222C23—184no—
223C23—181no—
224D4—61no—
225D4—181no—
226D4—184no—
227C2×C4—91no—
228C2×C4—92no—
229C2×C3—22no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—41no—
234C2×C3—42no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—64no—
238C2×C3—64no—
239C2×C3—81no—
240C2×C3—92no—
241C2×C3—124no—
242C2×C3—121no—
243C2×C3—121no—
244C2×C3—181no—
245C2×C3—241no—
246C2×C3C2×GO+(4,𝔽2)11yescharacteristic
247D3—24no—
248D3—41no—
249D3—62no—
250D3—62no—
251D3—62no—
252D3—121no—
253C22—64no—
254C22—62no—
255C22—61no—
256C22—94no—
257C22—91no—
258C22—91no—
259C22—182no—
260C22—182no—
261C22—184no—
262C22—182no—
263C22C3×GO+(4,𝔽2)11yescharacteristic, complemented
264C4—61no—
265C4—182no—
266C3—21no—
267C3—21no—
268C3—41no—
269C3—41no—
270C3#6424111yescharacteristic, direct product factor
271C2—21no—
272C2—64no—
273C2—92no—
274C2—121no—
275C2—181no—
276C2C2×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C23×C3
FrattiniC2C2×C3×GO+(4,𝔽2)
FittingC22×C33D4
socleC2×C33C2×D4
hypercenterC22×C3GO+(4,𝔽2)
hypocenterC32C3×#120
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×D32C2semidirect
2. C2×C3×GO+(4,𝔽2)C2semidirect
3. C3×#3811C2semidirect
4. C3×C22⋊#157C2semidirect
5. C3×D32⋊C4C2semidirect
6. #64241C3direct (N × H)
12. C2×C3×D32C22semidirect
13. C2×C3×#157C22semidirect
17. C3×#624C22semidirect
19. C22×D32C2×C3semidirect
20. C2×GO+(4,𝔽2)C2×C3semidirect
22. #3811C2×C3semidirect
23. C22⋊#157C2×C3semidirect
24. D32⋊C4C2×C3semidirect
25. C22×C33D4semidirect
27. C2×C3×C3⋊D3D4semidirect
35. C3×D32D4semidirect
53. C2×D32C22×C3semidirect
54. C2×#157C22×C3semidirect
64. #624C22×C3semidirect
85. C22×C32C3×D4semidirect
87. C2×C3⋊D3C3×D4semidirect
111. D32C3×D4semidirect
118. C33#120semidirect
194. C22×C3GO+(4,𝔽2)semidirect
217. C32C3×#120semidirect
263. C22C3×GO+(4,𝔽2)semidirect
270. C3#64241direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D322
C2×C3×GO+(4,𝔽2)2
C3×#38112
C3×C22⋊#1572
C3×D32⋊C42
#642413
C3×#1209