Subgroups — C₃²×#2138 #11769187

Order 1152.

684 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#358	C₂	1	2	yes	maximal, complemented
5	C₃²×#357	C₂	1	2	yes	maximal, complemented
6	C₃×#2138	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×#41	—	2	1	no	—
8	C₃²×#135	—	2	2	no	—
9	C₃²×#135	C₄	1	4	yes	complemented
10	C₂×C₃²×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃²×SD₁₆	—	2	1	no	—
12	C₂×C₃²×D₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃²×#117	—	2	1	no	—
15	C₂×C₃²×C₁₆	—	2	2	no	—
16	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
17	C₃²×#107	C₂²	1	2	yes	—
18	C₃²×#106	—	2	1	no	—
19	C₂×C₃×#135	C₂×C₃	1	4	yes	—
20	C₂²×C₃×C₁₆	C₂×C₃	1	4	yes	complemented
21	C₃×#427	C₂×C₃	1	4	yes	complemented
22	C₃×#358	C₂×C₃	1	8	yes	complemented
23	C₃×#357	C₂×C₃	1	8	yes	complemented
24	C₃²×#41	—	2	4	no	—
25	C₃²×#41	—	4	2	no	—
26	C₂×C₃²×Q₈	—	2	1	no	—
27	C₂×C₃²×D₄	—	2	1	no	—
28	C₂×C₃²×D₄	—	4	1	no	—
29	C₂²×C₃²×C₄	—	4	1	no	—
30	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
31	C₃²×Q₁₆	—	2	1	no	—
32	C₃²×Q₁₆	C₂×C₄	1	2	yes	—
33	C₃²×SD₁₆	—	2	4	no	—
34	C₃²×D₈	—	2	1	no	—
35	C₃²×D₈	C₂×C₄	1	2	yes	—
36	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃²×C₈	D₄	1	2	yes	—
39	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
40	C₃²×C₄⋊C₄	—	2	2	no	—
41	C₃²×C₂²⋊C₄	—	4	1	no	—
42	C₃²×C₄²	—	4	1	no	—
43	C₃²×C₁₆	—	2	2	no	—
44	#2138	C₃²	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₃×#41	—	2	4	no	—
46	C₃×#135	—	2	8	no	—
47	C₃×#135	C₃×C₄	1	16	yes	complemented
48	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
49	C₂×C₃×SD₁₆	—	2	4	no	—
50	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
51	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
52	C₃×#117	—	2	4	no	—
53	C₂×C₃×C₁₆	—	2	8	no	—
54	C₂×C₃×C₁₆	C₂²×C₃	1	8	yes	—
55	C₃×#107	C₂²×C₃	1	8	yes	—
56	C₃×#106	—	2	4	no	—
57	C₂³×C₃²	—	4	1	no	—
58	C₂³×C₃²	D₈	1	1	yes	characteristic
59	C₃²×Q₈	—	2	2	no	—
60	C₃²×Q₈	—	4	1	no	—
61	C₃²×D₄	—	2	2	no	—
62	C₃²×D₄	—	4	4	no	—
63	C₃²×D₄	—	4	1	no	—
64	C₂×C₃²×C₄	—	4	1	no	—
65	C₂×C₃²×C₄	—	4	4	no	—
66	C₂×C₃²×C₄	—	4	2	no	—
67	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
68	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
70	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
72	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×#135	C₂×C₃²	1	1	yes	characteristic
75	C₂²×C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
76	#427	C₂×C₃²	1	1	yes	characteristic, complemented
77	#358	C₂×C₃²	1	2	yes	complemented
78	#357	C₂×C₃²	1	2	yes	complemented
79	C₃×#41	—	2	16	no	—
80	C₃×#41	—	4	8	no	—
81	C₂×C₃×Q₈	—	2	4	no	—
82	C₂×C₃×D₄	—	2	4	no	—
83	C₂×C₃×D₄	—	4	4	no	—
84	C₂²×C₃×C₄	—	4	4	no	—
85	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
86	C₃×Q₁₆	—	2	4	no	—
87	C₃×Q₁₆	C₂×C₃×C₄	1	8	yes	—
88	C₃×SD₁₆	—	2	16	no	—
89	C₃×D₈	—	2	4	no	—
90	C₃×D₈	C₂×C₃×C₄	1	8	yes	—
91	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
92	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
93	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
94	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
95	C₃×C₄⋊C₄	—	2	8	no	—
96	C₃×C₂²⋊C₄	—	4	4	no	—
97	C₃×C₄²	—	4	4	no	—
98	C₃×C₁₆	—	2	8	no	—
99	C₂²×C₃²	—	2	2	no	—
100	C₂²×C₃²	—	4	2	no	—
101	C₂²×C₃²	—	8	2	no	—
102	C₂²×C₃²	#102	1	2	yes	—
103	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
104	C₃²×C₄	—	4	2	no	—
105	C₃²×C₄	—	8	2	no	—
106	C₃²×C₄	#102	1	2	yes	—
107	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×#41	—	2	1	no	—
110	#135	—	2	2	no	—
111	#135	C₃²×C₄	1	4	yes	complemented
112	C₂×Q₁₆	C₂²×C₃²	1	1	yes	characteristic
113	C₂×SD₁₆	—	2	1	no	—
114	C₂×D₈	C₂²×C₃²	1	1	yes	characteristic
115	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
116	#117	—	2	1	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	C₂²×C₃²	1	2	yes	—
119	#107	C₂²×C₃²	1	2	yes	—
120	#106	—	2	1	no	—
121	C₂³×C₃	—	4	4	no	—
122	C₂³×C₃	C₃×D₈	1	4	yes	—
123	C₃×Q₈	—	2	8	no	—
124	C₃×Q₈	—	4	4	no	—
125	C₃×D₄	—	2	8	no	—
126	C₃×D₄	—	4	16	no	—
127	C₃×D₄	—	4	4	no	—
128	C₂×C₃×C₄	—	4	4	no	—
129	C₂×C₃×C₄	—	4	16	no	—
130	C₂×C₃×C₄	—	4	8	no	—
131	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	8	yes	—
132	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
133	C₂×C₃×C₄	C₃×SD₁₆	1	8	yes	—
134	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
135	C₃×C₈	C₃×C₂²⋊C₄	1	8	yes	—
136	C₃×C₈	C₂²×C₃×C₄	1	4	yes	—
137	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
138	C₂×C₃²	—	2	2	no	—
139	C₂×C₃²	—	8	2	no	—
140	C₂×C₃²	C₂×#102	1	1	yes	characteristic
141	C₂×C₃²	#508	1	2	yes	—
142	#41	—	2	4	no	—
143	#41	—	4	2	no	—
144	C₂×Q₈	—	2	1	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
149	Q₁₆	—	2	1	no	—
150	Q₁₆	C₂×C₃²×C₄	1	2	yes	—
151	SD₁₆	—	2	4	no	—
152	D₈	—	2	1	no	—
153	D₈	C₂×C₃²×C₄	1	2	yes	—
154	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
155	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
156	C₂×C₈	C₃²×D₄	1	2	yes	—
157	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₁₆	—	2	2	no	—
162	C₂²×C₃	—	2	8	no	—
163	C₂²×C₃	—	4	8	no	—
164	C₂²×C₃	—	8	8	no	—
165	C₂²×C₃	C₃×#102	1	8	yes	—
166	C₂²×C₃	C₂×C₃×D₈	1	4	yes	—
167	C₃×C₄	—	4	8	no	—
168	C₃×C₄	—	8	8	no	—
169	C₃×C₄	C₃×#102	1	8	yes	—
170	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
171	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
172	C₃²	#2138	1	1	yes	characteristic, direct product factor, Sylow
173	C₂³	—	4	1	no	—
174	C₂³	C₃²×D₈	1	1	yes	characteristic
175	Q₈	—	2	2	no	—
176	Q₈	—	4	1	no	—
177	D₄	—	2	2	no	—
178	D₄	—	4	4	no	—
179	D₄	—	4	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	C₃²×C₂²⋊C₄	1	2	yes	—
184	C₂×C₄	C₃²×D₈	1	1	yes	characteristic
185	C₂×C₄	C₃²×SD₁₆	1	2	yes	—
186	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
187	C₈	C₃²×C₂²⋊C₄	1	2	yes	—
188	C₈	C₂²×C₃²×C₄	1	1	yes	characteristic
189	C₈	C₂×C₃²×D₄	1	1	yes	characteristic
190	C₂×C₃	—	2	8	no	—
191	C₂×C₃	—	8	8	no	—
192	C₂×C₃	C₂×C₃×#102	1	4	yes	—
193	C₂×C₃	C₃×#508	1	8	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	8	2	no	—
197	C₂²	C₃²×#102	1	2	yes	—
198	C₂²	C₂×C₃²×D₈	1	1	yes	characteristic
199	C₄	—	4	2	no	—
200	C₄	—	8	2	no	—
201	C₄	C₃²×#102	1	2	yes	—
202	C₄	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
203	C₄	C₂×C₃²×SD₁₆	1	1	yes	characteristic
204	C₃	C₃×#2138	1	4	yes	direct product factor
205	C₂	—	2	2	no	—
206	C₂	—	8	2	no	—
207	C₂	C₂×C₃²×#102	1	1	yes	characteristic
208	C₂	C₃²×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₃²×C₄	D₈
commutator	C₈	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂²×C₃²	C₂×D₈
prime radical	C₂×C₃²×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₁₆	C₂	semidirect
3. C₃²×#427	C₂	semidirect
4. C₃²×#358	C₂	semidirect
5. C₃²×#357	C₂	semidirect
6. C₃×#2138	C₃	direct (N × H)
9. C₃²×#135	C₄	semidirect
20. C₂²×C₃×C₁₆	C₂×C₃	semidirect
21. C₃×#427	C₂×C₃	semidirect
22. C₃×#358	C₂×C₃	semidirect
23. C₃×#357	C₂×C₃	semidirect
44. #2138	C₃²	direct (N × H)
47. C₃×#135	C₃×C₄	semidirect
75. C₂²×C₁₆	C₂×C₃²	semidirect
76. #427	C₂×C₃²	semidirect
77. #358	C₂×C₃²	semidirect
78. #357	C₂×C₃²	semidirect
111. #135	C₃²×C₄	semidirect
172. C₃²	#2138	direct (N × H)
204. C₃	C₃×#2138	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×#135	2
C₂²×C₃²×C₁₆	2
C₃²×#427	2
C₃²×#358	2
C₃²×#357	2
C₃×#2138	3

Subgroups — C32×#2138 #11769187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#2138 #11769187