Subgroups — C₂×C₃²×#399 #11769442

Order 1152.

2724 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₂²×C₃²×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₃²×#399	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃²×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#399	C₃	1	4	yes	maximal, direct product factor
6	C₂⁵×C₃²	C₂²	1	1	yes	characteristic
7	C₂³×C₃²×C₄	C₂²	1	3	yes	—
8	C₂³×C₃²×C₄	C₂²	1	3	yes	—
9	C₂×C₃²×C₄⋊C₄	—	2	12	no	—
10	C₂×C₃²×C₄⋊C₄	C₂²	1	12	yes	complemented
11	C₂×C₃²×C₂²⋊C₄	—	2	12	no	—
12	C₂×C₃²×C₂²⋊C₄	C₂²	1	12	yes	—
13	C₃²×#95	C₂²	1	4	yes	complemented
14	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	12	yes	complemented
15	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	12	yes	—
16	C₃×#399	C₂×C₃	1	32	yes	direct product factor
17	C₂×C₃×#95	C₂×C₃	1	4	yes	complemented
18	C₂⁴×C₃²	—	2	6	no	—
19	C₂⁴×C₃²	—	2	6	no	—
20	C₂⁴×C₃²	Q₈	1	4	yes	—
21	C₂⁴×C₃²	C₂³	1	2	yes	—
22	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
23	C₂²×C₃²×C₄	—	2	12	no	—
24	C₂²×C₃²×C₄	—	2	6	no	—
25	C₂²×C₃²×C₄	—	2	12	no	—
26	C₂²×C₃²×C₄	D₄	1	12	yes	—
27	C₂²×C₃²×C₄	C₂³	1	6	yes	—
28	C₂²×C₃²×C₄	C₂³	1	6	yes	—
29	C₃²×C₄⋊C₄	—	2	24	no	—
30	C₃²×C₂²⋊C₄	—	2	24	no	—
31	C₂×#399	C₃²	1	1	yes	characteristic, direct product factor, Sylow
32	C₂⁵×C₃	C₂²×C₃	1	4	yes	—
33	C₂³×C₃×C₄	C₂²×C₃	1	12	yes	—
34	C₂³×C₃×C₄	C₂²×C₃	1	12	yes	—
35	C₂×C₃×C₄⋊C₄	—	2	48	no	—
36	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	48	yes	complemented
37	C₂×C₃×C₂²⋊C₄	—	2	48	no	—
38	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	48	yes	—
39	C₃×#95	C₂²×C₃	1	16	yes	complemented
40	C₂³×C₃²	—	2	24	no	—
41	C₂³×C₃²	—	2	6	no	—
42	C₂³×C₃²	—	2	6	no	—
43	C₂³×C₃²	—	4	12	no	—
44	C₂³×C₃²	—	4	4	no	—
45	C₂³×C₃²	C₂×D₄	1	6	yes	—
46	C₂³×C₃²	C₂×Q₈	1	4	yes	—
47	C₂³×C₃²	C₂×Q₈	1	2	yes	—
48	C₂³×C₃²	#41	1	6	yes	—
49	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
50	C₂×C₃²×C₄	—	2	24	no	—
51	C₂×C₃²×C₄	—	2	12	no	—
52	C₂×C₃²×C₄	—	2	6	no	—
53	C₂×C₃²×C₄	—	2	12	no	—
54	C₂×C₃²×C₄	—	4	12	no	—
55	C₂×C₃²×C₄	C₂×D₄	1	12	yes	—
56	C₂²×C₄⋊C₄	C₂×C₃²	1	3	yes	complemented
57	C₂²×C₂²⋊C₄	C₂×C₃²	1	3	yes	—
58	#399	C₂×C₃²	1	8	yes	direct product factor
59	C₂×#95	C₂×C₃²	1	1	yes	characteristic, complemented
60	C₂⁴×C₃	—	2	24	no	—
61	C₂⁴×C₃	—	2	24	no	—
62	C₂⁴×C₃	C₃×Q₈	1	16	yes	—
63	C₂⁴×C₃	C₂³×C₃	1	8	yes	—
64	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
65	C₂²×C₃×C₄	—	2	48	no	—
66	C₂²×C₃×C₄	—	2	24	no	—
67	C₂²×C₃×C₄	—	2	48	no	—
68	C₂²×C₃×C₄	C₃×D₄	1	48	yes	—
69	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
70	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
71	C₃×C₄⋊C₄	—	2	96	no	—
72	C₃×C₂²⋊C₄	—	2	96	no	—
73	C₂²×C₃²	—	2	12	no	—
74	C₂²×C₃²	—	4	6	no	—
75	C₂²×C₃²	—	4	2	no	—
76	C₂²×C₃²	—	4	16	no	—
77	C₂²×C₃²	C₂²⋊Q₈	1	12	yes	—
78	C₂²×C₃²	#123	1	12	yes	—
79	C₂²×C₃²	#127	1	4	yes	—
80	C₂²×C₃²	C₂²×D₄	1	3	yes	—
81	C₂²×C₃²	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₃²	C₂×#41	1	3	yes	—
83	C₃²×C₄	—	2	12	no	—
84	C₃²×C₄	—	4	6	no	—
85	C₂⁵	C₂²×C₃²	1	1	yes	characteristic
86	C₂³×C₄	C₂²×C₃²	1	3	yes	—
87	C₂³×C₄	C₂²×C₃²	1	3	yes	—
88	C₂×C₄⋊C₄	—	2	12	no	—
89	C₂×C₄⋊C₄	C₂²×C₃²	1	12	yes	complemented
90	C₂×C₂²⋊C₄	—	2	12	no	—
91	C₂×C₂²⋊C₄	C₂²×C₃²	1	12	yes	—
92	#95	C₂²×C₃²	1	4	yes	complemented
93	C₂³×C₃	—	2	96	no	—
94	C₂³×C₃	—	2	24	no	—
95	C₂³×C₃	—	2	24	no	—
96	C₂³×C₃	—	4	48	no	—
97	C₂³×C₃	—	4	16	no	—
98	C₂³×C₃	C₂×C₃×D₄	1	24	yes	—
99	C₂³×C₃	C₂×C₃×Q₈	1	16	yes	—
100	C₂³×C₃	C₂×C₃×Q₈	1	8	yes	—
101	C₂³×C₃	C₃×#41	1	24	yes	—
102	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
103	C₂×C₃×C₄	—	2	96	no	—
104	C₂×C₃×C₄	—	2	48	no	—
105	C₂×C₃×C₄	—	2	24	no	—
106	C₂×C₃×C₄	—	2	48	no	—
107	C₂×C₃×C₄	—	4	48	no	—
108	C₂×C₃×C₄	C₂×C₃×D₄	1	48	yes	—
109	C₂×C₃²	—	4	4	no	—
110	C₂×C₃²	#399	1	8	yes	direct product factor
111	C₂×C₃²	C₂×C₂²⋊Q₈	1	3	yes	—
112	C₂×C₃²	C₂×#123	1	3	yes	—
113	C₂×C₃²	C₂×#127	1	1	yes	characteristic
114	C₂⁴	—	2	6	no	—
115	C₂⁴	—	2	6	no	—
116	C₂⁴	C₃²×Q₈	1	4	yes	—
117	C₂⁴	C₂³×C₃²	1	2	yes	—
118	C₂⁴	C₂³×C₃²	1	1	yes	characteristic
119	C₂²×C₄	—	2	12	no	—
120	C₂²×C₄	—	2	6	no	—
121	C₂²×C₄	—	2	12	no	—
122	C₂²×C₄	C₃²×D₄	1	12	yes	—
123	C₂²×C₄	C₂³×C₃²	1	6	yes	—
124	C₂²×C₄	C₂³×C₃²	1	6	yes	—
125	C₄⋊C₄	—	2	24	no	—
126	C₂²⋊C₄	—	2	24	no	—
127	C₂²×C₃	—	2	48	no	—
128	C₂²×C₃	—	4	64	no	—
129	C₂²×C₃	—	4	24	no	—
130	C₂²×C₃	—	4	8	no	—
131	C₂²×C₃	C₃×C₂²⋊Q₈	1	48	yes	—
132	C₂²×C₃	C₃×#123	1	48	yes	—
133	C₂²×C₃	C₃×#127	1	16	yes	—
134	C₂²×C₃	C₂²×C₃×D₄	1	12	yes	—
135	C₂²×C₃	C₂²×C₃×Q₈	1	4	yes	—
136	C₂²×C₃	C₂×C₃×#41	1	12	yes	—
137	C₃×C₄	—	2	48	no	—
138	C₃×C₄	—	4	24	no	—
139	C₃²	C₂×#399	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³	—	2	6	no	—
141	C₂³	—	2	6	no	—
142	C₂³	—	2	24	no	—
143	C₂³	—	4	12	no	—
144	C₂³	—	4	4	no	—
145	C₂³	C₂×C₃²×D₄	1	6	yes	—
146	C₂³	C₂×C₃²×Q₈	1	4	yes	—
147	C₂³	C₂×C₃²×Q₈	1	2	yes	—
148	C₂³	C₃²×#41	1	6	yes	—
149	C₂³	C₂⁴×C₃²	1	1	yes	characteristic
150	C₂×C₄	—	2	12	no	—
151	C₂×C₄	—	2	24	no	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	12	no	—
154	C₂×C₄	—	4	12	no	—
155	C₂×C₄	C₂×C₃²×D₄	1	12	yes	—
156	C₂×C₃	—	4	16	no	—
157	C₂×C₃	C₃×#399	1	32	yes	direct product factor
158	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	12	yes	—
159	C₂×C₃	C₂×C₃×#123	1	12	yes	—
160	C₂×C₃	C₂×C₃×#127	1	4	yes	—
161	C₂²	—	2	12	no	—
162	C₂²	—	4	16	no	—
163	C₂²	—	4	6	no	—
164	C₂²	—	4	2	no	—
165	C₂²	C₃²×C₂²⋊Q₈	1	12	yes	—
166	C₂²	C₃²×#123	1	12	yes	—
167	C₂²	C₃²×#127	1	4	yes	—
168	C₂²	C₂²×C₃²×D₄	1	3	yes	—
169	C₂²	C₂²×C₃²×Q₈	1	1	yes	characteristic
170	C₂²	C₂×C₃²×#41	1	3	yes	—
171	C₄	—	2	12	no	—
172	C₄	—	4	6	no	—
173	C₃	C₂×C₃×#399	1	4	yes	direct product factor
174	C₂	—	4	4	no	—
175	C₂	C₃²×#399	1	8	yes	direct product factor
176	C₂	C₂×C₃²×C₂²⋊Q₈	1	3	yes	—
177	C₂	C₂×C₃²×#123	1	3	yes	—
178	C₂	C₂×C₃²×#127	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃²	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃²
prime radical	C₂⁵×C₃²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₄⋊C₄	C₂	semidirect
3. C₃²×#399	C₂	direct (N × H)
4. C₂×C₃²×#95	C₂	semidirect
5. C₂×C₃×#399	C₃	direct (N × H)
10. C₂×C₃²×C₄⋊C₄	C₂²	semidirect
13. C₃²×#95	C₂²	semidirect
14. C₂²×C₃×C₄⋊C₄	C₂×C₃	semidirect
16. C₃×#399	C₂×C₃	direct (N × H)
17. C₂×C₃×#95	C₂×C₃	semidirect
31. C₂×#399	C₃²	direct (N × H)
36. C₂×C₃×C₄⋊C₄	C₂²×C₃	semidirect
39. C₃×#95	C₂²×C₃	semidirect
56. C₂²×C₄⋊C₄	C₂×C₃²	semidirect
58. #399	C₂×C₃²	direct (N × H)
59. C₂×#95	C₂×C₃²	semidirect
89. C₂×C₄⋊C₄	C₂²×C₃²	semidirect
92. #95	C₂²×C₃²	semidirect
110. C₂×C₃²	#399	direct (N × H)
139. C₃²	C₂×#399	direct (N × H)
157. C₂×C₃	C₃×#399	direct (N × H)
173. C₃	C₂×C₃×#399	direct (N × H)
175. C₂	C₃²×#399	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×C₃²×C₄⋊C₄	2
C₂²×C₃²×C₂²⋊C₄	2
C₃²×#399	2
C₂×C₃²×#95	2
C₂×C₃×#399	3

Subgroups — C2×C32×#399 #11769442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#399 #11769442