Subgroups — C₃²×#2115 #11769164

Order 1152.

660 conjugacy classes of subgroups.

The subgroup lattice has 183 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×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₃²×#349	C₂	1	4	yes	maximal, complemented
4	C₃×#2115	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×#41	C₄	1	1	yes	characteristic
6	C₃²×#131	—	2	2	no	—
7	C₃²×#131	—	2	2	no	—
8	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
9	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃²×C₈	C₄	1	2	yes	—
11	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
12	C₃²×M₃₂	—	2	2	no	—
13	C₃²×M₃₂	C₂²	1	4	yes	—
14	C₂×C₃²×C₁₆	—	2	2	no	—
15	C₂×C₃×#131	C₂×C₃	1	4	yes	—
16	C₂×C₃×M₃₂	C₂×C₃	1	8	yes	complemented
17	C₃×#349	C₂×C₃	1	16	yes	complemented
18	C₃²×#41	—	2	2	no	—
19	C₃²×#41	—	2	2	no	—
20	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
21	C₂×C₃²×D₄	C₈	1	2	yes	—
22	C₂×C₃²×D₄	C₈	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₃²×M₁₆	—	2	2	no	—
26	C₃²×M₁₆	—	2	4	no	—
27	C₂×C₃²×C₈	—	2	2	no	—
28	C₂×C₃²×C₈	—	2	2	no	—
29	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	D₄	1	2	yes	—
32	C₂×C₃²×C₈	D₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
34	C₃²×C₁₆	—	2	4	no	—
35	#2115	C₃²	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃×#41	C₃×C₄	1	4	yes	—
37	C₃×#131	—	2	8	no	—
38	C₃×#131	—	2	8	no	—
39	C₂×C₃×M₁₆	C₃×C₄	1	4	yes	—
40	C₂×C₃×M₁₆	C₂²×C₃	1	8	yes	—
41	C₂²×C₃×C₈	C₃×C₄	1	8	yes	—
42	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
43	C₃×M₃₂	—	2	8	no	—
44	C₃×M₃₂	C₂²×C₃	1	16	yes	—
45	C₂×C₃×C₁₆	—	2	8	no	—
46	C₂³×C₃²	C₂×C₈	1	2	yes	—
47	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
48	C₃²×Q₈	—	2	2	no	—
49	C₃²×D₄	—	2	4	no	—
50	C₃²×D₄	—	2	2	no	—
51	C₂×C₃²×C₄	—	2	2	no	—
52	C₂×C₃²×C₄	—	2	2	no	—
53	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
54	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
55	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃²×C₄	M₁₆	1	2	yes	—
57	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₃²×C₈	—	2	2	no	—
59	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₃²×C₈	C₂×D₄	1	2	yes	—
62	C₂×#131	C₂×C₃²	1	1	yes	characteristic
63	C₂×M₃₂	C₂×C₃²	1	2	yes	complemented
64	#349	C₂×C₃²	1	4	yes	complemented
65	C₃×#41	—	2	8	no	—
66	C₃×#41	—	2	8	no	—
67	C₂×C₃×Q₈	C₃×C₈	1	4	yes	—
68	C₂×C₃×D₄	C₃×C₈	1	8	yes	—
69	C₂×C₃×D₄	C₃×C₈	1	4	yes	—
70	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
71	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
72	C₃×M₁₆	—	2	8	no	—
73	C₃×M₁₆	—	2	16	no	—
74	C₂×C₃×C₈	—	2	8	no	—
75	C₂×C₃×C₈	—	2	8	no	—
76	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
77	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
78	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
79	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
80	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
81	C₃×C₁₆	—	2	16	no	—
82	C₂²×C₃²	—	2	2	no	—
83	C₂²×C₃²	—	4	1	no	—
84	C₂²×C₃²	—	4	2	no	—
85	C₂²×C₃²	C₂²⋊C₈	1	2	yes	—
86	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
87	C₃²×C₄	—	2	2	no	—
88	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
89	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
91	C₂×#41	C₃²×C₄	1	1	yes	characteristic
92	#131	—	2	2	no	—
93	#131	—	2	2	no	—
94	C₂×M₁₆	C₃²×C₄	1	1	yes	characteristic
95	C₂×M₁₆	C₂²×C₃²	1	2	yes	—
96	C₂²×C₈	C₃²×C₄	1	2	yes	—
97	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
98	M₃₂	—	2	2	no	—
99	M₃₂	C₂²×C₃²	1	4	yes	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂³×C₃	C₂×C₃×C₈	1	8	yes	—
102	C₂³×C₃	C₂×C₃×C₈	1	4	yes	—
103	C₃×Q₈	—	2	8	no	—
104	C₃×D₄	—	2	16	no	—
105	C₃×D₄	—	2	8	no	—
106	C₂×C₃×C₄	—	2	8	no	—
107	C₂×C₃×C₄	—	2	8	no	—
108	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	8	yes	—
109	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
110	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
111	C₂×C₃×C₄	C₃×M₁₆	1	8	yes	—
112	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
113	C₃×C₈	—	2	8	no	—
114	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
115	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
116	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
117	C₂×C₃²	—	2	1	no	—
118	C₂×C₃²	—	2	2	no	—
119	C₂×C₃²	—	4	2	no	—
120	C₂×C₃²	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	#41	—	2	2	no	—
122	#41	—	2	2	no	—
123	C₂×Q₈	C₃²×C₈	1	1	yes	characteristic
124	C₂×D₄	C₃²×C₈	1	2	yes	—
125	C₂×D₄	C₃²×C₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₃²×C₄	1	2	yes	—
127	C₂²×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	4	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
133	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
134	C₂×C₈	C₃²×D₄	1	2	yes	—
135	C₂×C₈	C₃²×D₄	1	2	yes	—
136	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
137	C₁₆	—	2	4	no	—
138	C₂²×C₃	—	2	8	no	—
139	C₂²×C₃	—	4	4	no	—
140	C₂²×C₃	—	4	8	no	—
141	C₂²×C₃	C₃×C₂²⋊C₈	1	8	yes	—
142	C₂²×C₃	C₂²×C₃×C₈	1	4	yes	—
143	C₃×C₄	—	2	8	no	—
144	C₃×C₄	C₃×C₂²⋊C₈	1	8	yes	—
145	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
146	C₃×C₄	C₂×C₃×M₁₆	1	4	yes	—
147	C₃²	#2115	1	1	yes	characteristic, direct product factor, Sylow
148	C₂³	C₂×C₃²×C₈	1	2	yes	—
149	C₂³	C₂×C₃²×C₈	1	1	yes	characteristic
150	Q₈	—	2	2	no	—
151	D₄	—	2	4	no	—
152	D₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	C₃²×C₂²⋊C₄	1	2	yes	—
156	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
157	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
158	C₂×C₄	C₃²×M₁₆	1	2	yes	—
159	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	C₃²×C₂²⋊C₄	1	1	yes	characteristic
162	C₈	C₃²×C₂²⋊C₄	1	1	yes	characteristic
163	C₈	C₂×C₃²×D₄	1	2	yes	—
164	C₂×C₃	—	2	4	no	—
165	C₂×C₃	—	2	8	no	—
166	C₂×C₃	—	4	8	no	—
167	C₂×C₃	C₂×C₃×C₂²⋊C₈	1	4	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	2	no	—
171	C₂²	C₃²×C₂²⋊C₈	1	2	yes	—
172	C₂²	C₂²×C₃²×C₈	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	C₃²×C₂²⋊C₈	1	2	yes	—
175	C₄	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
176	C₄	C₂×C₃²×M₁₆	1	1	yes	characteristic
177	C₃	C₃×#2115	1	4	yes	direct product factor
178	C₂	—	2	1	no	—
179	C₂	—	2	2	no	—
180	C₂	—	4	2	no	—
181	C₂	C₂×C₃²×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₈	C₂²⋊C₄
commutator	C₂²	C₂²×C₃²×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂×C₃²	C₂×C₂²⋊C₈
prime radical	C₂×C₃²×#41	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₈ ◁ C₂²×C₃²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×M₃₂	C₂	semidirect
3. C₃²×#349	C₂	semidirect
4. C₃×#2115	C₃	direct (N × H)
16. C₂×C₃×M₃₂	C₂×C₃	semidirect
17. C₃×#349	C₂×C₃	semidirect
35. #2115	C₃²	direct (N × H)
63. C₂×M₃₂	C₂×C₃²	semidirect
64. #349	C₂×C₃²	semidirect
147. C₃²	#2115	direct (N × H)
177. C₃	C₃×#2115	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×M₃₂	2
C₃²×#349	2
C₃×#2115	3

Subgroups — C32×#2115 #11769164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#2115 #11769164