Subgroups — C₂²×C₃×#3675 #13153632

Order 1728.

1050 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₃₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3675	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃×#215	—	3	1	no	maximal
6	C₂²×#3675	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃×D₉	—	2	1	no	—
8	C₂×C₃×Q₈×C₉	—	2	3	no	—
9	C₂×C₃×Q₈×C₉	C₂²	1	6	yes	complemented
10	C₂²×C₃×C₄×C₉	—	2	1	no	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×D₃₆	—	2	3	no	—
13	C₂×C₃×D₃₆	C₂²	1	6	yes	—
14	C₃×#3675	C₂²	1	16	yes	direct product factor
15	C₂×C₃×C₉⋊C₈	C₂²	1	6	yes	complemented
16	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
17	C₂²×C₃×D₁₂	—	3	1	no	—
18	C₂×C₃×#215	—	3	12	no	—
19	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
20	C₂²×Q₈×C₉	—	2	1	no	—
21	C₂²×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²×D₃₆	C₂×C₃	1	1	yes	characteristic
23	C₂×#3675	C₂×C₃	1	12	yes	direct product factor
24	C₂²×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
26	C₂²×C₃×D₉	—	2	6	no	—
27	C₂²×C₃×D₉	—	4	4	no	—
28	C₃×Q₈×C₉	—	2	6	no	—
29	C₃×Q₈×C₉	C₂³	1	4	yes	complemented
30	C₂×C₃×C₄×C₉	—	2	6	no	—
31	C₂×C₃×C₄×C₉	D₄	1	3	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
33	C₃×D₃₆	—	2	6	no	—
34	C₃×D₃₆	C₂³	1	4	yes	—
35	C₃×C₉⋊C₈	C₂³	1	4	yes	complemented
36	C₂²×C₃×SD₁₆	—	9	1	no	—
37	C₂²×#215	—	3	1	no	—
38	C₂³×C₃×D₃	—	6	1	no	—
39	C₂×C₃²×Q₈	—	2	3	no	—
40	C₂×C₃²×Q₈	C₂×D₃	1	6	yes	—
41	C₂²×C₃²×C₄	—	2	1	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃×D₁₂	—	3	6	no	—
44	C₂×C₃×D₁₂	—	6	3	no	—
45	C₂³×D₉	—	2	1	no	—
46	C₃×#215	—	3	16	no	—
47	C₂×C₃×C₃⋊C₈	—	3	6	no	—
48	C₂×Q₈×C₉	—	2	6	no	—
49	C₂×Q₈×C₉	—	2	6	no	—
50	C₂×Q₈×C₉	—	2	3	no	—
51	C₂×Q₈×C₉	C₂²×C₃	1	6	yes	complemented
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	—	2	1	no	—
54	C₂²×C₄×C₉	—	2	1	no	—
55	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
56	C₂×D₃₆	—	2	3	no	—
57	C₂×D₃₆	C₂²×C₃	1	6	yes	—
58	#3675	C₂²×C₃	1	16	yes	direct product factor
59	C₂×C₉⋊C₈	C₂²×C₃	1	6	yes	complemented
60	C₂²×C₃×C₉	SD₁₆	1	4	yes	complemented
61	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
62	C₂×C₃×D₉	—	2	4	no	—
63	C₂×C₃×D₉	—	4	12	no	—
64	C₃×C₄×C₉	—	2	4	no	—
65	C₃×C₄×C₉	C₂×D₄	1	3	yes	—
66	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
67	C₂²×C₃×Q₈	—	2	1	no	—
68	C₂²×C₃×Q₈	D₉	1	1	yes	characteristic, complemented
69	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
70	C₂²×C₃×D₄	—	9	1	no	—
71	C₂²×D₁₂	—	3	1	no	—
72	C₂×C₃×SD₁₆	—	9	12	no	—
73	C₂²×C₃×C₈	—	9	1	no	—
74	C₂×#215	—	3	12	no	—
75	C₂²×C₃⋊C₈	—	3	1	no	—
76	C₂³×C₃²	#67	1	1	yes	characteristic
77	C₂²×C₃×D₃	—	6	6	no	—
78	C₂²×C₃×D₃	—	12	4	no	—
79	C₃²×Q₈	—	2	6	no	—
80	C₃²×Q₈	C₂²×D₃	1	4	yes	—
81	C₂×C₃²×C₄	—	2	6	no	—
82	C₂×C₃²×C₄	#67	1	3	yes	—
83	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
84	C₃×D₁₂	—	3	4	no	—
85	C₃×D₁₂	—	6	6	no	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	C₃×D₄	1	1	yes	characteristic
88	C₂²×D₉	—	2	6	no	—
89	C₂²×D₉	—	4	4	no	—
90	C₃×C₃⋊C₈	—	3	4	no	—
91	Q₈×C₉	—	2	12	no	—
92	Q₈×C₉	—	2	6	no	—
93	Q₈×C₉	—	2	4	no	—
94	Q₈×C₉	C₂³×C₃	1	4	yes	complemented
95	C₂×C₄×C₉	—	2	12	no	—
96	C₂×C₄×C₉	—	2	6	no	—
97	C₂×C₄×C₉	—	2	3	no	—
98	C₂×C₄×C₉	—	2	3	no	—
99	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
100	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
101	D₃₆	—	2	6	no	—
102	D₃₆	C₂³×C₃	1	4	yes	—
103	C₉⋊C₈	C₂³×C₃	1	4	yes	complemented
104	C₂²×SD₁₆	—	9	1	no	Sylow
105	C₂×C₃×C₉	C₂×SD₁₆	1	6	yes	complemented
106	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
107	C₃×D₉	—	4	4	no	—
108	C₂⁴×C₃	—	18	1	no	—
109	C₂³×D₃	—	6	1	no	—
110	C₂×C₃×Q₈	—	2	6	no	—
111	C₂×C₃×Q₈	—	2	3	no	—
112	C₂×C₃×Q₈	—	2	3	no	—
113	C₂×C₃×Q₈	—	2	6	no	—
114	C₂×C₃×Q₈	C₂×D₉	1	6	yes	complemented
115	C₂×C₃×Q₈	C₂×C₃×D₃	1	6	yes	—
116	C₂×C₃×D₄	—	9	6	no	—
117	C₂×C₃×D₄	—	18	3	no	—
118	C₂²×C₃×C₄	—	2	2	no	—
119	C₂²×C₃×C₄	—	2	1	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	2	1	no	—
122	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
124	C₂×D₁₂	—	3	6	no	—
125	C₂×D₁₂	—	6	3	no	—
126	C₃×SD₁₆	—	9	16	no	—
127	C₂×C₃×C₈	—	9	6	no	—
128	#215	—	3	16	no	—
129	C₂×C₃⋊C₈	—	3	6	no	—
130	C₂²×C₃²	#215	1	4	yes	—
131	C₂²×C₃²	C₂×#67	1	3	yes	—
132	C₂×C₃×D₃	—	6	4	no	—
133	C₂×C₃×D₃	—	12	12	no	—
134	C₃²×C₄	—	2	4	no	—
135	C₃²×C₄	C₂×#67	1	3	yes	—
136	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
137	C₂²×C₉	—	2	4	no	—
138	C₂²×C₉	—	2	3	no	—
139	C₂²×C₉	C₃×SD₁₆	1	4	yes	complemented
140	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
141	C₂×D₉	—	2	4	no	—
142	C₂×D₉	—	4	12	no	—
143	C₄×C₉	—	2	8	no	—
144	C₄×C₉	—	2	4	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	2	3	no	—
147	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
148	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
149	C₂²×Q₈	C₃×D₉	1	1	yes	characteristic, complemented
150	C₂²×D₄	—	9	1	no	—
151	C₂×SD₁₆	—	9	12	no	—
152	C₂²×C₈	—	9	1	no	—
153	C₃×C₉	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₃	—	2	1	no	—
155	C₂³×C₃	—	18	6	no	—
156	C₂³×C₃	—	36	4	no	—
157	C₂³×C₃	#611	1	1	yes	characteristic
158	C₂³×C₃	C₃×#67	1	1	yes	characteristic
159	C₂²×D₃	—	6	6	no	—
160	C₂²×D₃	—	12	4	no	—
161	C₃×Q₈	—	2	12	no	—
162	C₃×Q₈	—	2	6	no	—
163	C₃×Q₈	—	2	6	no	—
164	C₃×Q₈	—	2	4	no	—
165	C₃×Q₈	C₂²×D₉	1	4	yes	complemented
166	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
167	C₃×D₄	—	9	4	no	—
168	C₃×D₄	—	18	6	no	—
169	C₂×C₃×C₄	—	2	12	no	—
170	C₂×C₃×C₄	—	2	6	no	—
171	C₂×C₃×C₄	—	2	6	no	—
172	C₂×C₃×C₄	—	2	3	no	—
173	C₂×C₃×C₄	—	2	3	no	—
174	C₂×C₃×C₄	#611	1	3	yes	—
175	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
176	C₂×C₃×C₄	C₃×#67	1	3	yes	—
177	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
178	D₁₂	—	3	4	no	—
179	D₁₂	—	6	6	no	—
180	C₃×C₈	—	9	4	no	—
181	C₃⋊C₈	—	3	4	no	—
182	C₂×C₃²	C₂×#215	1	6	yes	—
183	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
184	C₃×D₃	—	12	4	no	—
185	C₂×C₉	—	2	6	no	—
186	C₂×C₉	—	2	1	no	—
187	C₂×C₉	C₂×C₃×SD₁₆	1	6	yes	complemented
188	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
189	D₉	—	4	4	no	—
190	C₂⁴	—	18	1	no	—
191	C₂×Q₈	—	2	3	no	—
192	C₂×Q₈	C₂×C₃×D₉	1	6	yes	complemented
193	C₂×D₄	—	9	6	no	—
194	C₂×D₄	—	18	3	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
197	SD₁₆	—	9	16	no	—
198	C₂×C₈	—	9	6	no	—
199	C₂²×C₃	—	2	3	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	18	4	no	—
202	C₂²×C₃	—	36	12	no	—
203	C₂²×C₃	#3675	1	4	yes	direct product factor
204	C₂²×C₃	C₂×#611	1	3	yes	—
205	C₂²×C₃	C₃×#215	1	4	yes	—
206	C₂²×C₃	C₂×C₃×#67	1	3	yes	—
207	C₂×D₃	—	6	4	no	—
208	C₂×D₃	—	12	12	no	—
209	C₃×C₄	—	2	8	no	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	3	no	—
214	C₃×C₄	C₂×#611	1	3	yes	—
215	C₃×C₄	C₂³×D₉	1	1	yes	characteristic
216	C₃×C₄	C₂×C₃×#67	1	3	yes	—
217	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
218	C₃²	C₂²×#215	1	1	yes	characteristic
219	C₉	—	2	1	no	—
220	C₉	C₂²×C₃×SD₁₆	1	1	yes	characteristic, complemented
221	C₂³	—	18	6	no	—
222	C₂³	—	36	4	no	—
223	C₂³	C₃×#611	1	1	yes	characteristic
224	Q₈	—	2	6	no	—
225	Q₈	C₂²×C₃×D₉	1	4	yes	complemented
226	D₄	—	9	4	no	—
227	D₄	—	18	6	no	—
228	C₂×C₄	—	2	6	no	—
229	C₂×C₄	C₃×#611	1	3	yes	—
230	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
231	C₈	—	9	4	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	6	no	—
234	C₂×C₃	—	36	4	no	—
235	C₂×C₃	C₂×#3675	1	6	yes	direct product factor
236	C₂×C₃	C₂²×#611	1	1	yes	characteristic
237	C₂×C₃	C₂×C₃×#215	1	6	yes	—
238	C₂×C₃	C₂²×C₃×#67	1	1	yes	characteristic
239	D₃	—	12	4	no	—
240	C₂²	—	18	4	no	—
241	C₂²	—	36	12	no	—
242	C₂²	C₃×#3675	1	4	yes	direct product factor
243	C₂²	C₂×C₃×#611	1	3	yes	—
244	C₄	—	2	4	no	—
245	C₄	C₂×C₃×#611	1	3	yes	—
246	C₄	C₂³×C₃×D₉	1	1	yes	characteristic
247	C₃	—	2	1	no	—
248	C₃	C₂²×#3675	1	1	yes	characteristic, direct product factor
249	C₃	C₂²×C₃×#215	1	1	yes	characteristic
250	C₂	—	36	4	no	—
251	C₂	C₂×C₃×#3675	1	6	yes	direct product factor
252	C₂	C₂²×C₃×#611	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	#611
commutator, cosocle	C₄×C₉	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle	C₂³×C₃²	#67
prime radical	C₂²×C₃×D₃₆	C₂
hypercenter	C₂²×C₃×Q₈	D₉
hypocenter	C₉	C₂²×C₃×SD₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈×C₉	C₂	semidirect
3. C₂×C₃×#3675	C₂	direct (N × H)
4. C₂²×C₃×C₉⋊C₈	C₂	semidirect
6. C₂²×#3675	C₃	direct (N × H)
9. C₂×C₃×Q₈×C₉	C₂²	semidirect
14. C₃×#3675	C₂²	direct (N × H)
15. C₂×C₃×C₉⋊C₈	C₂²	semidirect
21. C₂²×Q₈×C₉	C₂×C₃	semidirect
23. C₂×#3675	C₂×C₃	direct (N × H)
24. C₂²×C₉⋊C₈	C₂×C₃	semidirect
29. C₃×Q₈×C₉	C₂³	semidirect
35. C₃×C₉⋊C₈	C₂³	semidirect
51. C₂×Q₈×C₉	C₂²×C₃	semidirect
58. #3675	C₂²×C₃	direct (N × H)
59. C₂×C₉⋊C₈	C₂²×C₃	semidirect
60. C₂²×C₃×C₉	SD₁₆	semidirect
68. C₂²×C₃×Q₈	D₉	semidirect
94. Q₈×C₉	C₂³×C₃	semidirect
103. C₉⋊C₈	C₂³×C₃	semidirect
105. C₂×C₃×C₉	C₂×SD₁₆	semidirect
114. C₂×C₃×Q₈	C₂×D₉	semidirect
139. C₂²×C₉	C₃×SD₁₆	semidirect
149. C₂²×Q₈	C₃×D₉	semidirect
153. C₃×C₉	C₂²×SD₁₆	semidirect
165. C₃×Q₈	C₂²×D₉	semidirect
187. C₂×C₉	C₂×C₃×SD₁₆	semidirect
192. C₂×Q₈	C₂×C₃×D₉	semidirect
203. C₂²×C₃	#3675	direct (N × H)
220. C₉	C₂²×C₃×SD₁₆	semidirect
225. Q₈	C₂²×C₃×D₉	semidirect
235. C₂×C₃	C₂×#3675	direct (N × H)
242. C₂²	C₃×#3675	direct (N × H)
248. C₃	C₂²×#3675	direct (N × H)
251. C₂	C₂×C₃×#3675	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈×C₉	2
C₂²×C₃×D₃₆	2
C₂×C₃×#3675	2
C₂²×C₃×C₉⋊C₈	2
C₂²×C₃×#215	3
C₂²×#3675	3

Subgroups — C22×C3×#3675 #13153632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#3675 #13153632