Subgroups — C₂²×#88806 #13178037

Order 1728.

3452 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#88806	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#3750	—	3	1	no	maximal
6	C₂²×#3724	—	3	4	no	maximal
7	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊Dic₃)	C₄	1	8	yes	complemented
9	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	6	yes	—
10	C₂²×C₃×C₃⋊Dic₃	—	2	4	no	—
11	C₂²×C₃×C₃⋊Dic₃	C₂²	1	6	yes	—
12	C₂²×C₃²×Dic₃	—	2	4	no	—
13	C₂²×C₃²×Dic₃	C₂²	1	6	yes	—
14	#88806	C₂²	1	16	yes	direct product factor
15	C₂³×C₃²×C₄	—	3	1	no	—
16	C₂³×C₃⋊Dic₃	—	3	1	no	—
17	C₂³×C₃⋊Dic₃	—	3	4	no	—
18	C₂³×C₃⋊Dic₃	—	6	4	no	—
19	C₂³×C₃⋊Dic₃	D₃	1	1	yes	characteristic
20	C₂³×C₃×Dic₃	—	3	4	no	—
21	C₂³×C₃×Dic₃	D₃	1	4	yes	—
22	C₂×#3750	—	3	12	no	—
23	C₂×#3724	—	3	48	no	—
24	C₂³×C₃³	C₂×C₄	1	4	yes	—
25	C₂³×C₃³	D₄	1	4	yes	—
26	C₂³×C₃³	Q₈	1	4	yes	—
27	C₂³×C₃³	C₂³	1	3	yes	—
28	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	24	yes	complemented
29	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
30	C₂×C₃×C₃⋊Dic₃	—	2	12	no	—
31	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
32	C₂×C₃²×Dic₃	—	2	12	no	—
33	C₂×C₃²×Dic₃	C₂³	1	4	yes	—
34	C₂²×Dic₃⋊C₄	—	9	4	no	—
35	C₂²×Dic₃⋊C₄	—	9	1	no	—
36	C₂⁴×C₃²	—	2	4	no	—
37	C₂⁴×C₃²	C₂×D₃	1	4	yes	—
38	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃²×C₄	—	3	6	no	—
40	C₂²×C₃²×C₄	—	6	4	no	—
41	C₂²×C₃⋊Dic₃	—	2	4	no	—
42	C₂²×C₃⋊Dic₃	—	3	8	no	—
43	C₂²×C₃⋊Dic₃	—	3	32	no	—
44	C₂²×C₃⋊Dic₃	—	3	6	no	—
45	C₂²×C₃⋊Dic₃	—	3	24	no	—
46	C₂²×C₃⋊Dic₃	—	6	32	no	—
47	C₂²×C₃⋊Dic₃	—	6	24	no	—
48	C₂²×C₃⋊Dic₃	C₂×D₃	1	6	yes	—
49	C₂²×C₃×Dic₃	—	2	16	no	—
50	C₂²×C₃×Dic₃	—	3	24	no	—
51	C₂²×C₃×Dic₃	—	6	16	no	—
52	C₂²×C₃×Dic₃	C₂×D₃	1	24	yes	—
53	#3750	—	3	16	no	—
54	#3724	—	3	64	no	—
55	C₂²×C₃³	C₄⋊C₄	1	16	yes	complemented
56	C₂²×C₃³	C₂²×C₄	1	6	yes	—
57	C₂²×C₃³	C₂×D₄	1	6	yes	—
58	C₂²×C₃³	C₂×Q₈	1	6	yes	—
59	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
60	C₃⋊(C₃⋊Dic₃)	C₂²×C₄	1	8	yes	complemented
61	C₃×C₃⋊Dic₃	—	2	4	no	—
62	C₃²×Dic₃	—	2	4	no	—
63	C₂³×C₃×C₄	—	3	4	no	—
64	C₂³×C₃×C₄	—	9	1	no	—
65	C₂³×Dic₃	—	3	4	no	—
66	C₂³×Dic₃	—	9	4	no	—
67	C₂³×Dic₃	—	9	1	no	—
68	C₂³×Dic₃	—	18	4	no	—
69	C₂³×Dic₃	C₃⋊D₃	1	1	yes	characteristic
70	C₂×Dic₃⋊C₄	—	9	48	no	—
71	C₂×Dic₃⋊C₄	—	9	12	no	—
72	C₂³×C₃²	—	2	16	no	—
73	C₂³×C₃²	—	2	16	no	—
74	C₂³×C₃²	—	2	16	no	—
75	C₂³×C₃²	—	2	12	no	—
76	C₂³×C₃²	Dic₆	1	16	yes	—
77	C₂³×C₃²	Dic₆	1	4	yes	—
78	C₂³×C₃²	C₄×D₃	1	16	yes	—
79	C₂³×C₃²	C₄×D₃	1	4	yes	—
80	C₂³×C₃²	#67	1	16	yes	—
81	C₂³×C₃²	#67	1	4	yes	—
82	C₂³×C₃²	C₂²×D₃	1	12	yes	—
83	C₂³×C₃²	C₂²×D₃	1	3	yes	—
84	C₂×C₃²×C₄	—	3	4	no	—
85	C₂×C₃²×C₄	—	6	12	no	—
86	C₂×C₃⋊Dic₃	—	2	12	no	—
87	C₂×C₃⋊Dic₃	—	3	24	no	—
88	C₂×C₃⋊Dic₃	—	3	4	no	—
89	C₂×C₃⋊Dic₃	—	3	96	no	—
90	C₂×C₃⋊Dic₃	—	3	16	no	—
91	C₂×C₃⋊Dic₃	—	6	96	no	—
92	C₂×C₃⋊Dic₃	—	6	16	no	—
93	C₂×C₃⋊Dic₃	C₂²×D₃	1	4	yes	—
94	C₂×C₃×Dic₃	—	2	48	no	—
95	C₂×C₃×Dic₃	—	3	16	no	—
96	C₂×C₃×Dic₃	—	6	48	no	—
97	C₂×C₃×Dic₃	C₂²×D₃	1	16	yes	—
98	C₂²×C₄⋊C₄	—	27	1	no	Sylow
99	C₂×C₃³	C₂×C₄⋊C₄	1	12	yes	complemented
100	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
103	C₂⁴×C₃	—	2	4	no	—
104	C₂⁴×C₃	D₃²	1	4	yes	—
105	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	—	3	24	no	—
107	C₂²×C₃×C₄	—	6	16	no	—
108	C₂²×C₃×C₄	—	9	6	no	—
109	C₂²×C₃×C₄	—	18	4	no	—
110	C₂²×Dic₃	—	2	4	no	—
111	C₂²×Dic₃	—	3	24	no	—
112	C₂²×Dic₃	—	6	16	no	—
113	C₂²×Dic₃	—	9	32	no	—
114	C₂²×Dic₃	—	9	8	no	—
115	C₂²×Dic₃	—	9	24	no	—
116	C₂²×Dic₃	—	9	6	no	—
117	C₂²×Dic₃	—	18	32	no	—
118	C₂²×Dic₃	—	18	24	no	—
119	C₂²×Dic₃	C₂×C₃⋊D₃	1	6	yes	—
120	Dic₃⋊C₄	—	9	64	no	—
121	Dic₃⋊C₄	—	9	16	no	—
122	C₂²×C₃²	—	2	4	no	—
123	C₂²×C₃²	—	2	64	no	—
124	C₂²×C₃²	—	2	24	no	—
125	C₂²×C₃²	—	2	24	no	—
126	C₂²×C₃²	—	2	24	no	—
127	C₂²×C₃²	Dic₃⋊C₄	1	64	yes	complemented
128	C₂²×C₃²	Dic₃⋊C₄	1	16	yes	complemented
129	C₂²×C₃²	C₂×Dic₆	1	24	yes	—
130	C₂²×C₃²	C₂×Dic₆	1	6	yes	—
131	C₂²×C₃²	C₂×C₄×D₃	1	24	yes	—
132	C₂²×C₃²	C₂×C₄×D₃	1	6	yes	—
133	C₂²×C₃²	C₂×#67	1	24	yes	—
134	C₂²×C₃²	C₂×#67	1	6	yes	—
135	C₂²×C₃²	C₂³×D₃	1	4	yes	—
136	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
137	C₃²×C₄	—	6	4	no	—
138	C₃⋊Dic₃	—	2	4	no	—
139	C₃⋊Dic₃	—	3	8	no	—
140	C₃⋊Dic₃	—	3	32	no	—
141	C₃⋊Dic₃	—	6	32	no	—
142	C₃×Dic₃	—	2	16	no	—
143	C₃×Dic₃	—	6	16	no	—
144	C₂³×C₄	—	3	1	no	—
145	C₂³×C₄	—	9	1	no	—
146	C₂³×C₄	—	27	1	no	—
147	C₂×C₄⋊C₄	—	27	12	no	—
148	C₃³	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	—	2	16	no	—
150	C₂³×C₃	—	2	16	no	—
151	C₂³×C₃	—	2	16	no	—
152	C₂³×C₃	—	2	12	no	—
153	C₂³×C₃	#624	1	16	yes	—
154	C₂³×C₃	#625	1	16	yes	—
155	C₂³×C₃	#627	1	16	yes	—
156	C₂³×C₃	#634	1	4	yes	—
157	C₂³×C₃	C₄×C₃⋊D₃	1	4	yes	—
158	C₂³×C₃	#638	1	4	yes	—
159	C₂³×C₃	C₂×D₃²	1	12	yes	—
160	C₂³×C₃	C₂²×C₃⋊D₃	1	3	yes	—
161	C₂×C₃×C₄	—	3	16	no	—
162	C₂×C₃×C₄	—	6	48	no	—
163	C₂×C₃×C₄	—	9	4	no	—
164	C₂×C₃×C₄	—	18	12	no	—
165	C₂×Dic₃	—	2	12	no	—
166	C₂×Dic₃	—	3	16	no	—
167	C₂×Dic₃	—	6	48	no	—
168	C₂×Dic₃	—	9	16	no	—
169	C₂×Dic₃	—	9	4	no	—
170	C₂×Dic₃	—	9	96	no	—
171	C₂×Dic₃	—	9	24	no	—
172	C₂×Dic₃	—	18	16	no	—
173	C₂×Dic₃	—	18	96	no	—
174	C₂×Dic₃	C₂²×C₃⋊D₃	1	4	yes	—
175	C₂×C₃²	—	2	4	no	—
176	C₂×C₃²	—	2	4	no	—
177	C₂×C₃²	—	2	4	no	—
178	C₂×C₃²	—	2	48	no	—
179	C₂×C₃²	C₂×Dic₃⋊C₄	1	48	yes	complemented
180	C₂×C₃²	C₂×Dic₃⋊C₄	1	12	yes	complemented
181	C₂×C₃²	C₂²×Dic₆	1	4	yes	—
182	C₂×C₃²	C₂²×Dic₆	1	1	yes	characteristic
183	C₂×C₃²	C₂²×C₄×D₃	1	4	yes	—
184	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
185	C₂×C₃²	C₂²×#67	1	4	yes	—
186	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
187	C₂⁴	D₃×C₃⋊D₃	1	1	yes	characteristic
188	C₂²×C₄	—	3	6	no	—
189	C₂²×C₄	—	6	4	no	—
190	C₂²×C₄	—	9	6	no	—
191	C₂²×C₄	—	18	4	no	—
192	C₂²×C₄	—	27	8	no	—
193	C₂²×C₄	—	27	6	no	—
194	C₄⋊C₄	—	27	16	no	—
195	C₂²×C₃	—	2	4	no	—
196	C₂²×C₃	—	2	24	no	—
197	C₂²×C₃	—	2	24	no	—
198	C₂²×C₃	—	2	24	no	—
199	C₂²×C₃	—	2	64	no	—
200	C₂²×C₃	#3724	1	64	yes	complemented
201	C₂²×C₃	#3750	1	16	yes	complemented
202	C₂²×C₃	C₂×#624	1	24	yes	—
203	C₂²×C₃	C₂×#625	1	24	yes	—
204	C₂²×C₃	C₂×#627	1	24	yes	—
205	C₂²×C₃	C₂×#634	1	6	yes	—
206	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	6	yes	—
207	C₂²×C₃	C₂×#638	1	6	yes	—
208	C₂²×C₃	C₂²×D₃²	1	4	yes	—
209	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
210	C₃×C₄	—	6	16	no	—
211	C₃×C₄	—	18	4	no	—
212	Dic₃	—	2	4	no	—
213	Dic₃	—	6	16	no	—
214	Dic₃	—	9	32	no	—
215	Dic₃	—	9	8	no	—
216	Dic₃	—	18	32	no	—
217	C₃²	—	2	4	no	—
218	C₃²	C₂²×Dic₃⋊C₄	1	4	yes	complemented
219	C₃²	C₂²×Dic₃⋊C₄	1	1	yes	characteristic, complemented
220	C₂³	#6289	1	4	yes	—
221	C₂³	#6290	1	4	yes	—
222	C₂³	#6293	1	4	yes	—
223	C₂³	C₂×D₃×C₃⋊D₃	1	3	yes	—
224	C₂×C₄	—	3	4	no	—
225	C₂×C₄	—	6	12	no	—
226	C₂×C₄	—	9	4	no	—
227	C₂×C₄	—	18	12	no	—
228	C₂×C₄	—	27	4	no	—
229	C₂×C₄	—	27	24	no	—
230	C₂×C₃	—	2	4	no	—
231	C₂×C₃	—	2	4	no	—
232	C₂×C₃	—	2	4	no	—
233	C₂×C₃	—	2	48	no	—
234	C₂×C₃	C₂×#3724	1	48	yes	complemented
235	C₂×C₃	C₂×#3750	1	12	yes	complemented
236	C₂×C₃	C₂²×#624	1	4	yes	—
237	C₂×C₃	C₂²×#625	1	4	yes	—
238	C₂×C₃	C₂²×#627	1	4	yes	—
239	C₂×C₃	C₂²×#634	1	1	yes	characteristic
240	C₂×C₃	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
241	C₂×C₃	C₂²×#638	1	1	yes	characteristic
242	C₂²	#88806	1	16	yes	direct product factor
243	C₂²	C₂×#6289	1	6	yes	—
244	C₂²	C₂×#6290	1	6	yes	—
245	C₂²	C₂×#6293	1	6	yes	—
246	C₂²	C₂²×D₃×C₃⋊D₃	1	1	yes	characteristic
247	C₄	—	6	4	no	—
248	C₄	—	18	4	no	—
249	C₄	—	27	8	no	—
250	C₃	—	2	4	no	—
251	C₃	C₂²×#3724	1	4	yes	complemented
252	C₃	C₂²×#3750	1	1	yes	characteristic, complemented
253	C₂	C₂×#88806	1	12	yes	direct product factor
254	C₂	C₂²×#6289	1	1	yes	characteristic
255	C₂	C₂²×#6290	1	1	yes	characteristic
256	C₂	C₂²×#6293	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	D₃×C₃⋊D₃
commutator	C₂×C₃³	C₂³×C₄
Frattini	C₂²	C₂²×D₃×C₃⋊D₃
Fitting, socle, prime radical	C₂⁴×C₃³	C₂²
cosocle	C₂²×C₃³	C₂⁴
hypocenter	C₃³	C₂²×C₄⋊C₄
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
4. C₂×#88806	C₂	direct (N × H)
8. C₂²×C₃⋊(C₃⋊Dic₃)	C₄	semidirect
14. #88806	C₂²	direct (N × H)
28. C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	semidirect
55. C₂²×C₃³	C₄⋊C₄	semidirect
60. C₃⋊(C₃⋊Dic₃)	C₂²×C₄	semidirect
99. C₂×C₃³	C₂×C₄⋊C₄	semidirect
127. C₂²×C₃²	Dic₃⋊C₄	semidirect
128. C₂²×C₃²	Dic₃⋊C₄	semidirect
148. C₃³	C₂²×C₄⋊C₄	semidirect
179. C₂×C₃²	C₂×Dic₃⋊C₄	semidirect
180. C₂×C₃²	C₂×Dic₃⋊C₄	semidirect
200. C₂²×C₃	#3724	semidirect
201. C₂²×C₃	#3750	semidirect
218. C₃²	C₂²×Dic₃⋊C₄	semidirect
219. C₃²	C₂²×Dic₃⋊C₄	semidirect
234. C₂×C₃	C₂×#3724	semidirect
235. C₂×C₃	C₂×#3750	semidirect
242. C₂²	#88806	direct (N × H)
251. C₃	C₂²×#3724	semidirect
252. C₃	C₂²×#3750	semidirect
253. C₂	C₂×#88806	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃⋊(C₃⋊Dic₃)	2
C₂³×C₃×C₃⋊Dic₃	2
C₂³×C₃²×Dic₃	2
C₂×#88806	2
C₂²×#3750	3
C₂²×#3724	3

Subgroups — C22×#88806 #13178037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#88806 #13178037