Subgroups — C₃²×#1816 #11768865

Order 1152.

792 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#417	C₂	1	2	yes	maximal
5	C₃²×#343	C₂	1	2	yes	maximal, complemented
6	C₃×#1816	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
8	C₃²×#131	—	2	2	no	—
9	C₃²×#131	C₄	1	4	yes	complemented
10	C₂×C₃²×M₁₆	—	2	1	no	—
11	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
12	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₃²×C₈	—	2	1	no	—
14	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃²×#108	—	2	2	no	—
17	C₃²×#107	—	2	1	no	—
18	C₃²×#107	—	2	1	no	—
19	C₃²×#103	—	2	2	no	—
20	C₃²×#102	—	2	2	no	—
21	C₂×C₃×#131	C₂×C₃	1	4	yes	—
22	C₂×C₃×#108	C₂×C₃	1	4	yes	complemented
23	C₂×C₃×#107	C₂×C₃	1	4	yes	complemented
24	C₃×#417	C₂×C₃	1	8	yes	—
25	C₃×#343	C₂×C₃	1	8	yes	complemented
26	C₃²×#41	—	2	1	no	—
27	C₃²×#41	—	2	1	no	—
28	C₃²×#41	C₂×C₄	1	2	yes	—
29	C₃²×#41	Q₈	1	2	yes	—
30	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₃²×D₄	—	2	1	no	—
32	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
33	C₂²×C₃²×C₄	—	2	1	no	—
34	C₂²×C₃²×C₄	—	2	2	no	—
35	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
36	C₃²×M₁₆	—	2	1	no	—
37	C₃²×M₁₆	—	2	4	no	—
38	C₃²×M₁₆	—	2	2	no	—
39	C₃²×M₁₆	—	2	2	no	—
40	C₃²×M₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₃²×C₈	—	2	1	no	—
42	C₂×C₃²×C₈	—	2	4	no	—
43	C₂×C₃²×C₈	—	2	2	no	—
44	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
47	C₂×C₃²×C₈	D₄	1	2	yes	—
48	C₃²×C₄⋊C₄	—	2	2	no	—
49	C₃²×C₄⋊C₄	—	2	2	no	—
50	#1816	C₃²	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
52	C₃×#131	—	2	8	no	—
53	C₃×#131	C₃×C₄	1	16	yes	complemented
54	C₂×C₃×M₁₆	—	2	4	no	—
55	C₂×C₃×M₁₆	C₂²×C₃	1	8	yes	—
56	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
57	C₂²×C₃×C₈	—	2	4	no	—
58	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
59	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
60	C₃×#108	—	2	8	no	—
61	C₃×#107	—	2	4	no	—
62	C₃×#107	—	2	4	no	—
63	C₃×#103	—	2	8	no	—
64	C₃×#102	—	2	8	no	—
65	C₂³×C₃²	—	2	1	no	—
66	C₂³×C₃²	#41	1	1	yes	characteristic
67	C₃²×Q₈	—	2	1	no	—
68	C₃²×Q₈	C₄⋊C₄	1	2	yes	—
69	C₃²×D₄	—	2	2	no	—
70	C₃²×D₄	—	2	2	no	—
71	C₃²×D₄	—	2	1	no	—
72	C₃²×D₄	C₄⋊C₄	1	2	yes	—
73	C₂×C₃²×C₄	—	2	1	no	—
74	C₂×C₃²×C₄	—	2	2	no	—
75	C₂×C₃²×C₄	—	2	2	no	—
76	C₂×C₃²×C₄	—	4	2	no	—
77	C₂×C₃²×C₄	—	8	2	no	—
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	#41	1	1	yes	characteristic
84	C₃²×C₈	—	2	2	no	—
85	C₃²×C₈	—	4	2	no	—
86	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
87	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×#131	C₂×C₃²	1	1	yes	characteristic
89	C₂×#108	C₂×C₃²	1	1	yes	characteristic, complemented
90	C₂×#107	C₂×C₃²	1	1	yes	characteristic, complemented
91	#417	C₂×C₃²	1	2	yes	—
92	#343	C₂×C₃²	1	2	yes	complemented
93	C₃×#41	—	2	4	no	—
94	C₃×#41	—	2	4	no	—
95	C₃×#41	C₂×C₃×C₄	1	8	yes	—
96	C₃×#41	C₃×Q₈	1	8	yes	—
97	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
98	C₂×C₃×D₄	—	2	4	no	—
99	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
100	C₂²×C₃×C₄	—	2	4	no	—
101	C₂²×C₃×C₄	—	2	8	no	—
102	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
103	C₃×M₁₆	—	2	4	no	—
104	C₃×M₁₆	—	2	16	no	—
105	C₃×M₁₆	—	2	8	no	—
106	C₃×M₁₆	—	2	8	no	—
107	C₃×M₁₆	C₂×C₃×C₄	1	8	yes	—
108	C₂×C₃×C₈	—	2	4	no	—
109	C₂×C₃×C₈	—	2	16	no	—
110	C₂×C₃×C₈	—	2	8	no	—
111	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
112	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
113	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
114	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
115	C₃×C₄⋊C₄	—	2	8	no	—
116	C₃×C₄⋊C₄	—	2	8	no	—
117	C₂²×C₃²	—	2	1	no	—
118	C₂²×C₃²	—	2	1	no	—
119	C₂²×C₃²	—	2	2	no	—
120	C₂²×C₃²	—	4	1	no	—
121	C₂²×C₃²	—	4	1	no	—
122	C₂²×C₃²	#117	1	1	yes	characteristic
123	C₂²×C₃²	#121	1	1	yes	characteristic
124	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
125	C₃²×C₄	—	2	2	no	—
126	C₃²×C₄	—	8	2	no	—
127	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₃²×C₄	#121	1	1	yes	characteristic
130	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×#41	C₂²×C₃²	1	1	yes	characteristic
132	#131	—	2	2	no	—
133	#131	C₃²×C₄	1	4	yes	complemented
134	C₂×M₁₆	—	2	1	no	—
135	C₂×M₁₆	C₂²×C₃²	1	2	yes	—
136	C₂×M₁₆	C₂²×C₃²	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
140	#108	—	2	2	no	—
141	#107	—	2	1	no	—
142	#107	—	2	1	no	—
143	#103	—	2	2	no	—
144	#102	—	2	2	no	—
145	C₂³×C₃	—	2	4	no	—
146	C₂³×C₃	C₃×#41	1	4	yes	—
147	C₃×Q₈	—	2	4	no	—
148	C₃×Q₈	C₃×C₄⋊C₄	1	8	yes	—
149	C₃×D₄	—	2	8	no	—
150	C₃×D₄	—	2	8	no	—
151	C₃×D₄	—	2	4	no	—
152	C₃×D₄	C₃×C₄⋊C₄	1	8	yes	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	8	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	4	8	no	—
157	C₂×C₃×C₄	—	8	8	no	—
158	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
159	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
160	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
161	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
162	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
163	C₂×C₃×C₄	C₃×#41	1	4	yes	—
164	C₃×C₈	—	2	8	no	—
165	C₃×C₈	—	4	8	no	—
166	C₃×C₈	C₃×C₂²⋊C₄	1	8	yes	—
167	C₃×C₈	C₃×C₂²⋊C₄	1	8	yes	—
168	C₂×C₃²	—	2	1	no	—
169	C₂×C₃²	—	2	1	no	—
170	C₂×C₃²	—	4	2	no	—
171	C₂×C₃²	#380	1	1	yes	characteristic
172	C₂×C₃²	#472	1	1	yes	characteristic
173	C₂×C₃²	#473	1	1	yes	characteristic
174	#41	—	2	1	no	—
175	#41	—	2	1	no	—
176	#41	C₂×C₃²×C₄	1	2	yes	—
177	#41	C₃²×Q₈	1	2	yes	—
178	C₂×Q₈	C₃²×D₄	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₃²×D₄	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
184	M₁₆	—	2	1	no	—
185	M₁₆	—	2	4	no	—
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	2	no	—
188	M₁₆	C₂×C₃²×C₄	1	2	yes	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
193	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
194	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
195	C₂×C₈	C₃²×D₄	1	2	yes	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₂²×C₃	—	2	4	no	—
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	8	no	—
201	C₂²×C₃	—	4	4	no	—
202	C₂²×C₃	—	4	4	no	—
203	C₂²×C₃	C₃×#117	1	4	yes	—
204	C₂²×C₃	C₃×#121	1	4	yes	—
205	C₂²×C₃	C₃×C₂²⋊Q₈	1	4	yes	—
206	C₃×C₄	—	2	8	no	—
207	C₃×C₄	—	8	8	no	—
208	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
209	C₃×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
210	C₃×C₄	C₃×#121	1	4	yes	—
211	C₃×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
212	C₃²	#1816	1	1	yes	characteristic, direct product factor, Sylow
213	C₂³	—	2	1	no	—
214	C₂³	C₃²×#41	1	1	yes	characteristic
215	Q₈	—	2	1	no	—
216	Q₈	C₃²×C₄⋊C₄	1	2	yes	—
217	D₄	—	2	2	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	C₃²×C₄⋊C₄	1	2	yes	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
227	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
228	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
229	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
230	C₂×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
231	C₂×C₄	C₃²×#41	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	4	2	no	—
234	C₈	C₃²×C₂²⋊C₄	1	2	yes	—
235	C₈	C₃²×C₂²⋊C₄	1	2	yes	—
236	C₂×C₃	—	2	4	no	—
237	C₂×C₃	—	2	4	no	—
238	C₂×C₃	—	4	8	no	—
239	C₂×C₃	C₃×#380	1	4	yes	—
240	C₂×C₃	C₃×#472	1	4	yes	—
241	C₂×C₃	C₃×#473	1	4	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	C₃²×#117	1	1	yes	characteristic
248	C₂²	C₃²×#121	1	1	yes	characteristic
249	C₂²	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	8	2	no	—
252	C₄	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
253	C₄	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
254	C₄	C₃²×#121	1	1	yes	characteristic
255	C₄	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
256	C₃	C₃×#1816	1	4	yes	direct product factor
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	4	2	no	—
260	C₂	C₃²×#380	1	1	yes	characteristic
261	C₂	C₃²×#472	1	1	yes	characteristic
262	C₂	C₃²×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	#121
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
prime radical	C₂×C₃²×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#108	C₂	semidirect
3. C₂×C₃²×#107	C₂	semidirect
5. C₃²×#343	C₂	semidirect
6. C₃×#1816	C₃	direct (N × H)
9. C₃²×#131	C₄	semidirect
22. C₂×C₃×#108	C₂×C₃	semidirect
23. C₂×C₃×#107	C₂×C₃	semidirect
25. C₃×#343	C₂×C₃	semidirect
50. #1816	C₃²	direct (N × H)
53. C₃×#131	C₃×C₄	semidirect
89. C₂×#108	C₂×C₃²	semidirect
90. C₂×#107	C₂×C₃²	semidirect
92. #343	C₂×C₃²	semidirect
133. #131	C₃²×C₄	semidirect
212. C₃²	#1816	direct (N × H)
256. C₃	C₃×#1816	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
C₂×C₃²×#108	2
C₂×C₃²×#107	2
C₃²×#417	2
C₃²×#343	2
C₃×#1816	3

Subgroups — C32×#1816 #11768865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1816 #11768865