Finite Group Databasebeta

← C2×C3×#63735 #13166755

Subgroups — C2×C3×#63735 #13166755

Order 1728.

408 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 66 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×C3×#3718C211yesmaximal, characteristic, complemented
2C2×C3×#3713C211yesmaximal, characteristic
3C2×C3×#3708C211yesmaximal, characteristic, complemented
4C3×#63735C214yesmaximal, direct product factor
5C2×#63735C311yesmaximal, characteristic, direct product factor
6C2×C32×D12—21no—
7C2×C32×Dic6—21no—
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9C3×#3718—21no—
10C3×#3718C2212yescomplemented
11C3×#3713—21no—
12C3×#3713C2212yes—
13C3×#3708C2212yescomplemented
14C2×C3×#216—61no—
15C2×C3×#213—61no—
16C2×#3718C2×C311yescharacteristic, complemented
17C2×#3713C2×C311yescharacteristic
18C2×#3708C2×C311yescharacteristic, complemented
19#63735C2×C314yesdirect product factor
20C22×C32×D3—41no—
21C2×C33×C4D411yescharacteristic
22C2×C32×Dic3—41no—
23C32×D12—22no—
24C32×D12—41no—
25C32×Dic6—22no—
26C32×Dic6—41no—
27C3×C3⋊(C3⋊C8)D411yescharacteristic
28C3×C3⋊(C3⋊C8)C2311yescharacteristic
29C2×C3×SD32—91nomaximal
30C2×C32×Q8—61no—
31C2×C32×D4—61no—
32C2×C3×D12—21no—
33C2×C3×D12—21no—
34C2×C3×D12—41no—
35C2×C3×Dic6—21no—
36C2×C3×Dic6—21no—
37C2×C3×Dic6—41no—
38C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
39C3×#216—62no—
40C3×#216—62no—
41C3×#213—62no—
42C3×#213—62no—
43C2×C3×C3⋊C8—61no—
44C2×C3×C3⋊C8—61no—
45#3718—21no—
46#3718C22×C312yescomplemented
47#3713—21no—
48#3713C22×C312yes—
49#3708C22×C312yescomplemented
50C22×C33D811yescharacteristic
51C2×C32×D3—42no—
52C2×C32×D3—82no—
53C33×C4D811yescharacteristic
54C33×C4C2×D411yescharacteristic
55C32×Dic3—42no—
56C2×C3×Q16—91no—
57C2×C3×D8—91no—
58C2×#216—61no—
59C2×#213—61no—
60C3×SD32—94no—
61C2×C3×C16—91no—
62C23×C32—121no—
63C22×C3×D3—41no—
64C22×C3×D3—42no—
65C22×C3×D3—41no—
66C32×Q8—62no—
67C32×Q8—121no—
68C32×D4—62no—
69C32×D4—121no—
70C2×C32×C4—21no—
71C2×C32×C4—21no—
72C2×C32×C4—41no—
73C2×C32×C4—41no—
74C2×C32×C4—121no—
75C2×C32×C4C3×D411yescharacteristic
76C2×C3×Dic3—41no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—41no—
79C3×D12—22no—
80C3×D12—22no—
81C3×D12—42no—
82C3×D12—41no—
83C3×D12—42no—
84C3×D12—41no—
85C3×Dic6—22no—
86C3×Dic6—22no—
87C3×Dic6—42no—
88C3×Dic6—41no—
89C3×Dic6—42no—
90C3×Dic6—41no—
91C3⋊(C3⋊C8)C3×D411yescharacteristic
92C3⋊(C3⋊C8)C23×C311yescharacteristic
93C3×C3⋊C8—61no—
94C3×C3⋊C8—61no—
95C3×C3⋊C8—61no—
96C3×C3⋊C8—61no—
97C2×SD32—91noSylow
98C2×C33SD3212yescomplemented
99C2×C33C2×D811yescharacteristic
100C32×D3—82no—
101C2×C3×Q8—61no—
102C2×C3×Q8—61no—
103C2×C3×Q8—121no—
104C2×C3×D4—61no—
105C2×C3×D4—61no—
106C2×C3×D4—121no—
107C2×D12—21no—
108C2×Dic6—21no—
109C3×Q16—92no—
110C3×Q16—181no—
111C3×D8—92no—
112C3×D8—181no—
113C2×C3×C8—91no—
114#216—62no—
115#216—62no—
116#213—62no—
117#213—62no—
118C2×C3⋊C8—61no—
119C2×C3⋊C8—61no—
120C3×C16—92no—
121C22×C32—21no—
122C22×C32—21no—
123C22×C32—41no—
124C22×C32—41no—
125C22×C32—122no—
126C22×C32—242no—
127C22×C32C3×D811yescharacteristic
128C2×C3×D3—42no—
129C2×C3×D3—44no—
130C2×C3×D3—42no—
131C2×C3×D3—82no—
132C2×C3×D3—84no—
133C2×C3×D3—82no—
134C32×C4—21no—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4—41no—
139C32×C4—41no—
140C32×C4—41no—
141C32×C4—41no—
142C32×C4—122no—
143C32×C4C3×D811yescharacteristic
144C32×C4C2×C3×D411yescharacteristic
145C3×Dic3—42no—
146C3×Dic3—44no—
147C3×Dic3—42no—
148C2×Q16—91no—
149C2×D8—91no—
150SD32—94no—
151C2×C16—91no—
152C33C2×SD3211yescharacteristic, complemented, Sylow
153C23×C3—121no—
154C23×C3—122no—
155C23×C3—121no—
156C22×D3—41no—
157C3×Q8—62no—
158C3×Q8—62no—
159C3×Q8—122no—
160C3×Q8—121no—
161C3×Q8—122no—
162C3×Q8—121no—
163C3×D4—62no—
164C3×D4—62no—
165C3×D4—122no—
166C3×D4—121no—
167C3×D4—122no—
168C3×D4—121no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—121no—
174C2×C3×C4—122no—
175C2×C3×C4—121no—
176C2×C3×C4GO+(4,𝔽2)11yescharacteristic
177C2×Dic3—41no—
178D12—22no—
179D12—41no—
180Dic6—22no—
181Dic6—41no—
182C3×C8—91no—
183C3×C8—91no—
184C3⋊C8—61no—
185C3⋊C8—61no—
186C3⋊C8—61no—
187C3⋊C8—61no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—42no—
193C2×C32—42no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—242no—
197C2×C32C3×SD3212yescomplemented
198C2×C32C2×C3×D811yescharacteristic
199C3×D3—82no—
200C3×D3—84no—
201C3×D3—82no—
202C2×Q8—61no—
203C2×D4—61no—
204Q16—92no—
205Q16—181no—
206D8—92no—
207D8—181no—
208C2×C8—91no—
209C16—92no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—122no—
215C22×C3—124no—
216C22×C3—122no—
217C22×C3—242no—
218C22×C3—244no—
219C22×C3—242no—
220C22×C3#377411yescharacteristic
221C2×D3—42no—
222C2×D3—82no—
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—41no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4—122no—
232C3×C4—124no—
233C3×C4—122no—
234C3×C4#377411yescharacteristic
235C3×C4C2×GO+(4,𝔽2)11yescharacteristic
236Dic3—42no—
237C32—21no—
238C32—21no—
239C32—41no—
240C32—41no—
241C32C2×C3×SD3211yescharacteristic, complemented
242C23—121no—
243Q8—62no—
244Q8—121no—
245D4—62no—
246D4—121no—
247C2×C4—121no—
248C2×C4C3×GO+(4,𝔽2)11yescharacteristic
249C8—91no—
250C8—91no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—42no—
258C2×C3—42no—
259C2×C3—242no—
260C2×C3—244no—
261C2×C3—242no—
262C2×C3#6373512yesdirect product factor
263C2×C3C2×#377411yescharacteristic
264D3—82no—
265C22—122no—
266C22—242no—
267C22C3×#377411yescharacteristic
268C4—122no—
269C4C3×#377411yescharacteristic
270C4C2×C3×GO+(4,𝔽2)11yescharacteristic
271C3—21no—
272C3—21no—
273C3—41no—
274C3—41no—
275C3C2×#6373511yescharacteristic, direct product factor
276C2—242no—
277C2C3×#6373512yesdirect product factor
278C2C2×C3×#377411yescharacteristic
subgroupG/H
centerC22×C3#3774
commutator, cosocleC3⋊(C3⋊C8)C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33D8
prime radicalC2×C3×#3713C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×SD32
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3718C2semidirect
3. C2×C3×#3708C2semidirect
4. C3×#63735C2direct (N × H)
5. C2×#63735C3direct (N × H)
10. C3×#3718C22semidirect
13. C3×#3708C22semidirect
16. C2×#3718C2×C3semidirect
18. C2×#3708C2×C3semidirect
19. #63735C2×C3direct (N × H)
46. #3718C22×C3semidirect
49. #3708C22×C3semidirect
98. C2×C33SD32semidirect
152. C33C2×SD32semidirect
197. C2×C32C3×SD32semidirect
241. C32C2×C3×SD32semidirect
262. C2×C3#63735direct (N × H)
275. C3C2×#63735direct (N × H)
277. C2C3×#63735direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37182
C2×C3×#37132
C2×C3×#37082
C3×#637352
C2×#637353
C2×C3×SD329