Subgroups — C₃²×#2139 #11769188

Order 1152.

696 conjugacy classes of subgroups.

The subgroup lattice has 222 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	C₂×C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#107	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×#2139	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	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
15	C₃²×M₃₂	—	2	2	no	—
16	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
17	C₃²×#107	—	2	1	no	—
18	C₃²×#107	C₂²	1	2	yes	—
19	C₂×C₃×#135	C₂×C₃	1	4	yes	—
20	C₂×C₃×M₃₂	C₂×C₃	1	4	yes	complemented
21	C₂×C₃×#107	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	—	4	1	no	—
31	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
32	C₃²×Q₁₆	—	2	1	no	—
33	C₃²×Q₁₆	C₂×C₄	1	2	yes	complemented
34	C₃²×SD₁₆	—	2	4	no	—
35	C₃²×D₈	—	2	1	no	—
36	C₃²×D₈	C₂×C₄	1	2	yes	complemented
37	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	D₄	1	2	yes	—
40	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
41	C₃²×C₄⋊C₄	—	2	2	no	—
42	C₃²×C₄⋊C₄	—	4	1	no	—
43	C₃²×C₁₆	—	2	2	no	—
44	#2139	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	C₂×C₃×C₄⋊C₄	—	2	4	no	—
53	C₃×M₃₂	—	2	8	no	—
54	C₂×C₃×C₁₆	C₂²×C₃	1	8	yes	—
55	C₃×#107	—	2	4	no	—
56	C₃×#107	C₂²×C₃	1	8	yes	—
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	—	8	2	no	—
68	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
69	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
70	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
71	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
73	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×#135	C₂×C₃²	1	1	yes	characteristic
76	C₂×M₃₂	C₂×C₃²	1	1	yes	characteristic, complemented
77	C₂×#107	C₂×C₃²	1	1	yes	characteristic, complemented
78	#358	C₂×C₃²	1	2	yes	complemented
79	#357	C₂×C₃²	1	2	yes	complemented
80	C₃×#41	—	2	16	no	—
81	C₃×#41	—	4	8	no	—
82	C₂×C₃×Q₈	—	2	4	no	—
83	C₂×C₃×D₄	—	2	4	no	—
84	C₂×C₃×D₄	—	4	4	no	—
85	C₂²×C₃×C₄	—	4	4	no	—
86	C₂²×C₃×C₄	—	4	4	no	—
87	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
88	C₃×Q₁₆	—	2	4	no	—
89	C₃×Q₁₆	C₂×C₃×C₄	1	8	yes	complemented
90	C₃×SD₁₆	—	2	16	no	—
91	C₃×D₈	—	2	4	no	—
92	C₃×D₈	C₂×C₃×C₄	1	8	yes	complemented
93	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
94	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
95	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
96	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
97	C₃×C₄⋊C₄	—	2	8	no	—
98	C₃×C₄⋊C₄	—	4	4	no	—
99	C₃×C₁₆	—	2	8	no	—
100	C₂²×C₃²	—	2	1	no	—
101	C₂²×C₃²	—	2	1	no	—
102	C₂²×C₃²	—	4	2	no	—
103	C₂²×C₃²	—	8	1	no	—
104	C₂²×C₃²	—	8	1	no	—
105	C₂²×C₃²	#102	1	2	yes	—
106	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
107	C₃²×C₄	—	4	2	no	—
108	C₃²×C₄	—	8	2	no	—
109	C₃²×C₄	#102	1	2	yes	—
110	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×#41	—	2	1	no	—
113	#135	—	2	2	no	—
114	#135	C₃²×C₄	1	4	yes	complemented
115	C₂×Q₁₆	C₂²×C₃²	1	1	yes	characteristic
116	C₂×SD₁₆	—	2	1	no	—
117	C₂×D₈	C₂²×C₃²	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
119	C₂×C₄⋊C₄	—	2	1	no	—
120	M₃₂	—	2	2	no	—
121	C₂×C₁₆	C₂²×C₃²	1	2	yes	—
122	#107	—	2	1	no	—
123	#107	C₂²×C₃²	1	2	yes	—
124	C₂³×C₃	—	4	4	no	—
125	C₂³×C₃	C₃×D₈	1	4	yes	—
126	C₃×Q₈	—	2	8	no	—
127	C₃×Q₈	—	4	4	no	—
128	C₃×D₄	—	2	8	no	—
129	C₃×D₄	—	4	16	no	—
130	C₃×D₄	—	4	4	no	—
131	C₂×C₃×C₄	—	4	4	no	—
132	C₂×C₃×C₄	—	4	16	no	—
133	C₂×C₃×C₄	—	4	8	no	—
134	C₂×C₃×C₄	—	8	8	no	—
135	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	8	yes	—
136	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
137	C₂×C₃×C₄	C₃×SD₁₆	1	8	yes	—
138	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
139	C₃×C₈	C₃×C₂²⋊C₄	1	8	yes	—
140	C₃×C₈	C₂²×C₃×C₄	1	4	yes	—
141	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
142	C₂×C₃²	—	2	2	no	—
143	C₂×C₃²	—	8	2	no	—
144	C₂×C₃²	C₂×#102	1	1	yes	characteristic
145	C₂×C₃²	#509	1	1	yes	characteristic
146	C₂×C₃²	#510	1	1	yes	characteristic
147	#41	—	2	4	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×D₄	—	2	1	no	—
151	C₂×D₄	—	4	1	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
155	Q₁₆	—	2	1	no	—
156	Q₁₆	C₂×C₃²×C₄	1	2	yes	complemented
157	SD₁₆	—	2	4	no	—
158	D₈	—	2	1	no	—
159	D₈	C₂×C₃²×C₄	1	2	yes	complemented
160	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
161	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
162	C₂×C₈	C₃²×D₄	1	2	yes	—
163	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	4	1	no	—
166	C₁₆	—	2	2	no	—
167	C₂²×C₃	—	2	4	no	—
168	C₂²×C₃	—	2	4	no	—
169	C₂²×C₃	—	4	8	no	—
170	C₂²×C₃	—	8	4	no	—
171	C₂²×C₃	—	8	4	no	—
172	C₂²×C₃	C₃×#102	1	8	yes	—
173	C₂²×C₃	C₂×C₃×D₈	1	4	yes	—
174	C₃×C₄	—	4	8	no	—
175	C₃×C₄	—	8	8	no	—
176	C₃×C₄	C₃×#102	1	8	yes	—
177	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
178	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
179	C₃²	#2139	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³	—	4	1	no	—
181	C₂³	C₃²×D₈	1	1	yes	characteristic
182	Q₈	—	2	2	no	—
183	Q₈	—	4	1	no	—
184	D₄	—	2	2	no	—
185	D₄	—	4	4	no	—
186	D₄	—	4	1	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	C₃²×C₂²⋊C₄	1	2	yes	—
192	C₂×C₄	C₃²×D₈	1	1	yes	characteristic
193	C₂×C₄	C₃²×SD₁₆	1	2	yes	—
194	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
195	C₈	C₃²×C₂²⋊C₄	1	2	yes	—
196	C₈	C₂²×C₃²×C₄	1	1	yes	characteristic
197	C₈	C₂×C₃²×D₄	1	1	yes	characteristic
198	C₂×C₃	—	2	8	no	—
199	C₂×C₃	—	8	8	no	—
200	C₂×C₃	C₂×C₃×#102	1	4	yes	—
201	C₂×C₃	C₃×#509	1	4	yes	—
202	C₂×C₃	C₃×#510	1	4	yes	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	8	1	no	—
207	C₂²	—	8	1	no	—
208	C₂²	C₃²×#102	1	2	yes	—
209	C₂²	C₂×C₃²×D₈	1	1	yes	characteristic
210	C₄	—	4	2	no	—
211	C₄	—	8	2	no	—
212	C₄	C₃²×#102	1	2	yes	—
213	C₄	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
214	C₄	C₂×C₃²×SD₁₆	1	1	yes	characteristic
215	C₃	C₃×#2139	1	4	yes	direct product factor
216	C₂	—	2	2	no	—
217	C₂	—	8	2	no	—
218	C₂	C₂×C₃²×#102	1	1	yes	characteristic
219	C₂	C₃²×#509	1	1	yes	characteristic
220	C₂	C₃²×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₈
commutator	C₈	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₂×C₃²×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ 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. C₂×C₃²×#107	C₂	semidirect
4. C₃²×#358	C₂	semidirect
5. C₃²×#357	C₂	semidirect
6. C₃×#2139	C₃	direct (N × H)
9. C₃²×#135	C₄	semidirect
20. C₂×C₃×M₃₂	C₂×C₃	semidirect
21. C₂×C₃×#107	C₂×C₃	semidirect
22. C₃×#358	C₂×C₃	semidirect
23. C₃×#357	C₂×C₃	semidirect
33. C₃²×Q₁₆	C₂×C₄	semidirect
36. C₃²×D₈	C₂×C₄	semidirect
44. #2139	C₃²	direct (N × H)
47. C₃×#135	C₃×C₄	semidirect
76. C₂×M₃₂	C₂×C₃²	semidirect
77. C₂×#107	C₂×C₃²	semidirect
78. #358	C₂×C₃²	semidirect
79. #357	C₂×C₃²	semidirect
89. C₃×Q₁₆	C₂×C₃×C₄	semidirect
92. C₃×D₈	C₂×C₃×C₄	semidirect
114. #135	C₃²×C₄	semidirect
156. Q₁₆	C₂×C₃²×C₄	semidirect
159. D₈	C₂×C₃²×C₄	semidirect
179. C₃²	#2139	direct (N × H)
215. C₃	C₃×#2139	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
C₂×C₃²×M₃₂	2
C₂×C₃²×#107	2
C₃²×#358	2
C₃²×#357	2
C₃×#2139	3

Subgroups — C32×#2139 #11769188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#2139 #11769188