Subgroups — C₂×C₉×#396 #11767110

Order 1152.

1362 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₂²×C₉×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₉×#396	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₉×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#396	C₃	1	1	yes	maximal, characteristic
6	C₂⁵×C₉	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₂²	1	3	yes	—
8	C₂³×C₄×C₉	C₂²	1	3	yes	—
9	C₂×C₉×C₄⋊C₄	—	2	12	no	—
10	C₂×C₉×C₄⋊C₄	C₂²	1	12	yes	complemented
11	C₂×C₉×C₂²⋊C₄	—	2	12	no	—
12	C₂×C₉×C₂²⋊C₄	C₂²	1	12	yes	—
13	C₉×#95	C₂²	1	4	yes	complemented
14	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	3	yes	—
15	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	3	yes	—
16	C₃×#396	C₂×C₃	1	8	yes	—
17	C₂×C₃×#95	C₂×C₃	1	1	yes	characteristic
18	C₂⁴×C₉	—	2	6	no	—
19	C₂⁴×C₉	—	2	6	no	—
20	C₂⁴×C₉	Q₈	1	4	yes	—
21	C₂⁴×C₉	C₂³	1	2	yes	—
22	C₂⁴×C₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₉	—	2	6	no	—
24	C₂²×C₄×C₉	—	2	6	no	—
25	C₂²×C₄×C₉	—	2	6	no	—
26	C₂²×C₄×C₉	—	2	6	no	—
27	C₂²×C₄×C₉	—	2	6	no	—
28	C₂²×C₄×C₉	D₄	1	12	yes	—
29	C₂²×C₄×C₉	C₂³	1	6	yes	—
30	C₂²×C₄×C₉	C₂³	1	6	yes	—
31	C₉×C₄⋊C₄	—	2	24	no	—
32	C₉×C₂²⋊C₄	—	2	24	no	—
33	C₂×#396	C₉	1	1	yes	characteristic, direct product factor, Sylow
34	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
35	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
36	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
37	C₂×C₃×C₄⋊C₄	—	2	12	no	—
38	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	12	yes	—
39	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
40	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
41	C₃×#95	C₂²×C₃	1	4	yes	—
42	C₂³×C₉	—	2	24	no	—
43	C₂³×C₉	—	2	6	no	—
44	C₂³×C₉	—	2	6	no	—
45	C₂³×C₉	—	4	12	no	—
46	C₂³×C₉	—	4	4	no	—
47	C₂³×C₉	C₂×D₄	1	6	yes	—
48	C₂³×C₉	C₂×Q₈	1	4	yes	—
49	C₂³×C₉	C₂×Q₈	1	2	yes	—
50	C₂³×C₉	#41	1	6	yes	—
51	C₂³×C₉	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₉	—	2	24	no	—
53	C₂×C₄×C₉	—	2	6	no	—
54	C₂×C₄×C₉	—	2	6	no	—
55	C₂×C₄×C₉	—	2	6	no	—
56	C₂×C₄×C₉	—	2	6	no	—
57	C₂×C₄×C₉	—	2	6	no	—
58	C₂×C₄×C₉	—	4	12	no	—
59	C₂×C₄×C₉	C₂×D₄	1	12	yes	—
60	C₂²×C₄⋊C₄	C₂×C₉	1	3	yes	complemented
61	C₂²×C₂²⋊C₄	C₂×C₉	1	3	yes	—
62	#396	C₂×C₉	1	8	yes	direct product factor
63	C₂×#95	C₂×C₉	1	1	yes	characteristic, complemented
64	C₂⁴×C₃	—	2	6	no	—
65	C₂⁴×C₃	—	2	6	no	—
66	C₂⁴×C₃	C₃×Q₈	1	4	yes	—
67	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
68	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
69	C₂²×C₃×C₄	—	2	6	no	—
70	C₂²×C₃×C₄	—	2	6	no	—
71	C₂²×C₃×C₄	—	2	6	no	—
72	C₂²×C₃×C₄	—	2	6	no	—
73	C₂²×C₃×C₄	—	2	6	no	—
74	C₂²×C₃×C₄	C₃×D₄	1	12	yes	—
75	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
76	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
77	C₃×C₄⋊C₄	—	2	24	no	—
78	C₃×C₂²⋊C₄	—	2	24	no	—
79	C₂²×C₉	—	2	12	no	—
80	C₂²×C₉	—	4	6	no	—
81	C₂²×C₉	—	4	2	no	—
82	C₂²×C₉	—	4	16	no	—
83	C₂²×C₉	#121	1	12	yes	—
84	C₂²×C₉	C₂²⋊Q₈	1	12	yes	—
85	C₂²×C₉	#126	1	4	yes	—
86	C₂²×C₉	C₂²×D₄	1	3	yes	—
87	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
88	C₂²×C₉	C₂×#41	1	3	yes	—
89	C₄×C₉	—	2	12	no	—
90	C₄×C₉	—	4	6	no	—
91	C₂⁵	C₂²×C₉	1	1	yes	characteristic
92	C₂³×C₄	C₂²×C₉	1	3	yes	—
93	C₂³×C₄	C₂²×C₉	1	3	yes	—
94	C₂×C₄⋊C₄	—	2	12	no	—
95	C₂×C₄⋊C₄	C₂²×C₉	1	12	yes	complemented
96	C₂×C₂²⋊C₄	—	2	12	no	—
97	C₂×C₂²⋊C₄	C₂²×C₉	1	12	yes	—
98	#95	C₂²×C₉	1	4	yes	complemented
99	C₂³×C₃	—	2	24	no	—
100	C₂³×C₃	—	2	6	no	—
101	C₂³×C₃	—	2	6	no	—
102	C₂³×C₃	—	4	12	no	—
103	C₂³×C₃	—	4	4	no	—
104	C₂³×C₃	C₂×C₃×D₄	1	6	yes	—
105	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
106	C₂³×C₃	C₂×C₃×Q₈	1	2	yes	—
107	C₂³×C₃	C₃×#41	1	6	yes	—
108	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
109	C₂×C₃×C₄	—	2	24	no	—
110	C₂×C₃×C₄	—	2	6	no	—
111	C₂×C₃×C₄	—	2	6	no	—
112	C₂×C₃×C₄	—	2	6	no	—
113	C₂×C₃×C₄	—	2	6	no	—
114	C₂×C₃×C₄	—	2	6	no	—
115	C₂×C₃×C₄	—	4	12	no	—
116	C₂×C₃×C₄	C₂×C₃×D₄	1	12	yes	—
117	C₂×C₉	—	4	4	no	—
118	C₂×C₉	#396	1	8	yes	direct product factor
119	C₂×C₉	C₂×#121	1	3	yes	—
120	C₂×C₉	C₂×C₂²⋊Q₈	1	3	yes	—
121	C₂×C₉	C₂×#126	1	1	yes	characteristic
122	C₂⁴	—	2	6	no	—
123	C₂⁴	—	2	6	no	—
124	C₂⁴	Q₈×C₉	1	4	yes	—
125	C₂⁴	C₂³×C₉	1	2	yes	—
126	C₂⁴	C₂³×C₉	1	1	yes	characteristic
127	C₂²×C₄	—	2	6	no	—
128	C₂²×C₄	—	2	6	no	—
129	C₂²×C₄	—	2	6	no	—
130	C₂²×C₄	—	2	6	no	—
131	C₂²×C₄	—	2	6	no	—
132	C₂²×C₄	D₄×C₉	1	12	yes	—
133	C₂²×C₄	C₂³×C₉	1	6	yes	—
134	C₂²×C₄	C₂³×C₉	1	6	yes	—
135	C₄⋊C₄	—	2	24	no	—
136	C₂²⋊C₄	—	2	24	no	—
137	C₂²×C₃	—	2	12	no	—
138	C₂²×C₃	—	4	16	no	—
139	C₂²×C₃	—	4	6	no	—
140	C₂²×C₃	—	4	2	no	—
141	C₂²×C₃	C₃×#121	1	12	yes	—
142	C₂²×C₃	C₃×C₂²⋊Q₈	1	12	yes	—
143	C₂²×C₃	C₃×#126	1	4	yes	—
144	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
145	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
146	C₂²×C₃	C₂×C₃×#41	1	3	yes	—
147	C₃×C₄	—	2	12	no	—
148	C₃×C₄	—	4	6	no	—
149	C₉	C₂×#396	1	1	yes	characteristic, direct product factor, Sylow
150	C₂³	—	2	24	no	—
151	C₂³	—	2	6	no	—
152	C₂³	—	2	6	no	—
153	C₂³	—	4	12	no	—
154	C₂³	—	4	4	no	—
155	C₂³	C₂×D₄×C₉	1	6	yes	—
156	C₂³	C₂×Q₈×C₉	1	4	yes	—
157	C₂³	C₂×Q₈×C₉	1	2	yes	—
158	C₂³	C₉×#41	1	6	yes	—
159	C₂³	C₂⁴×C₉	1	1	yes	characteristic
160	C₂×C₄	—	2	24	no	—
161	C₂×C₄	—	2	6	no	—
162	C₂×C₄	—	2	6	no	—
163	C₂×C₄	—	2	6	no	—
164	C₂×C₄	—	2	6	no	—
165	C₂×C₄	—	2	6	no	—
166	C₂×C₄	—	4	12	no	—
167	C₂×C₄	C₂×D₄×C₉	1	12	yes	—
168	C₂×C₃	—	4	4	no	—
169	C₂×C₃	C₃×#396	1	8	yes	—
170	C₂×C₃	C₂×C₃×#121	1	3	yes	—
171	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	3	yes	—
172	C₂×C₃	C₂×C₃×#126	1	1	yes	characteristic
173	C₂²	—	2	12	no	—
174	C₂²	—	4	6	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	16	no	—
177	C₂²	C₉×#121	1	12	yes	—
178	C₂²	C₉×C₂²⋊Q₈	1	12	yes	—
179	C₂²	C₉×#126	1	4	yes	—
180	C₂²	C₂²×D₄×C₉	1	3	yes	—
181	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
182	C₂²	C₂×C₉×#41	1	3	yes	—
183	C₄	—	2	12	no	—
184	C₄	—	4	6	no	—
185	C₃	C₂×C₃×#396	1	1	yes	characteristic
186	C₂	—	4	4	no	—
187	C₂	C₉×#396	1	8	yes	direct product factor
188	C₂	C₂×C₉×#121	1	3	yes	—
189	C₂	C₂×C₉×C₂²⋊Q₈	1	3	yes	—
190	C₂	C₂×C₉×#126	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₉	C₂³
commutator	C₂³	C₂⁴×C₉
Frattini, cosocle	C₂³×C₃	C₂⁴×C₃
socle	C₂⁴×C₃	C₂³×C₃
prime radical	C₂⁵×C₃	C₂²×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×C₄⋊C₄	C₂	semidirect
3. C₉×#396	C₂	direct (N × H)
4. C₂×C₉×#95	C₂	semidirect
10. C₂×C₉×C₄⋊C₄	C₂²	semidirect
13. C₉×#95	C₂²	semidirect
33. C₂×#396	C₉	direct (N × H)
60. C₂²×C₄⋊C₄	C₂×C₉	semidirect
62. #396	C₂×C₉	direct (N × H)
63. C₂×#95	C₂×C₉	semidirect
95. C₂×C₄⋊C₄	C₂²×C₉	semidirect
98. #95	C₂²×C₉	semidirect
118. C₂×C₉	#396	direct (N × H)
149. C₉	C₂×#396	direct (N × H)
187. C₂	C₉×#396	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×C₉×C₄⋊C₄	2
C₂²×C₉×C₂²⋊C₄	2
C₉×#396	2
C₂×C₉×#95	2
C₂×C₃×#396	3

Subgroups — C2×C9×#396 #11767110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#396 #11767110