Subgroups — C₂²×C₃×#121 #87316

Order 384.

2528 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	C₂³×C₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×D₄	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#121	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂²×#121	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁵×C₃	C₂²	1	2	yes	—
9	C₂⁵×C₃	C₂²	1	1	yes	characteristic
10	C₂²×C₃×D₄	—	2	16	no	—
11	C₂²×C₃×D₄	—	2	8	no	—
12	C₂²×C₃×D₄	C₂²	1	24	yes	complemented
13	C₂²×C₃×D₄	C₂²	1	12	yes	complemented
14	C₂³×C₃×C₄	—	2	8	no	—
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	2	yes	—
17	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
18	C₂³×C₃×C₄	C₂²	1	12	yes	complemented
19	C₃×#121	C₂²	1	64	yes	direct product factor
20	C₂×C₃×C₄⋊C₄	C₂²	1	12	yes	complemented
21	C₂×C₃×C₂²⋊C₄	C₂²	1	24	yes	complemented
22	C₂³×D₄	C₂×C₃	1	2	yes	complemented
23	C₂³×D₄	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂⁴×C₄	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#121	C₂×C₃	1	24	yes	direct product factor
26	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂²×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
28	C₂⁴×C₃	—	2	4	no	—
29	C₂⁴×C₃	—	2	4	no	—
30	C₂⁴×C₃	—	2	8	no	—
31	C₂⁴×C₃	—	2	8	no	—
32	C₂⁴×C₃	—	2	8	no	—
33	C₂⁴×C₃	D₄	1	8	yes	complemented
34	C₂⁴×C₃	C₂³	1	12	yes	—
35	C₂⁴×C₃	C₂³	1	6	yes	—
36	C₂⁴×C₃	C₂³	1	1	yes	characteristic
37	C₂×C₃×D₄	—	2	96	no	—
38	C₂×C₃×D₄	—	2	48	no	—
39	C₂×C₃×D₄	C₂³	1	32	yes	complemented
40	C₂×C₃×D₄	C₂³	1	16	yes	complemented
41	C₂²×C₃×C₄	—	2	4	no	—
42	C₂²×C₃×C₄	—	2	8	no	—
43	C₂²×C₃×C₄	—	2	48	no	—
44	C₂²×C₃×C₄	D₄	1	8	yes	complemented
45	C₂²×C₃×C₄	C₂³	1	6	yes	—
46	C₂²×C₃×C₄	C₂³	1	12	yes	—
47	C₂²×C₃×C₄	C₂³	1	6	yes	—
48	C₂²×C₃×C₄	C₂³	1	16	yes	complemented
49	C₃×C₄⋊C₄	C₂³	1	16	yes	complemented
50	C₃×C₂²⋊C₄	C₂³	1	32	yes	complemented
51	C₂⁵	C₂²×C₃	1	2	yes	—
52	C₂⁵	C₂²×C₃	1	1	yes	characteristic
53	C₂²×D₄	—	2	16	no	—
54	C₂²×D₄	—	2	8	no	—
55	C₂²×D₄	C₂²×C₃	1	24	yes	complemented
56	C₂²×D₄	C₂²×C₃	1	12	yes	complemented
57	C₂³×C₄	—	2	8	no	—
58	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
59	C₂³×C₄	C₂²×C₃	1	2	yes	—
60	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂³×C₄	C₂²×C₃	1	12	yes	complemented
62	#121	C₂²×C₃	1	64	yes	direct product factor
63	C₂×C₄⋊C₄	C₂²×C₃	1	12	yes	complemented
64	C₂×C₂²⋊C₄	C₂²×C₃	1	24	yes	complemented
65	C₂³×C₃	—	2	32	no	—
66	C₂³×C₃	—	2	12	no	—
67	C₂³×C₃	—	2	12	no	—
68	C₂³×C₃	—	2	24	no	—
69	C₂³×C₃	—	2	24	no	—
70	C₂³×C₃	—	2	24	no	—
71	C₂³×C₃	—	4	32	no	—
72	C₂³×C₃	C₂×D₄	1	4	yes	—
73	C₂³×C₃	C₂×D₄	1	4	yes	—
74	C₂³×C₃	C₂×D₄	1	24	yes	complemented
75	C₂³×C₃	#41	1	4	yes	—
76	C₂³×C₃	C₂⁴	1	8	yes	—
77	C₂³×C₃	C₂⁴	1	4	yes	—
78	C₂³×C₃	C₂⁴	1	3	yes	—
79	C₃×D₄	—	2	64	no	—
80	C₃×D₄	—	2	32	no	—
81	C₂×C₃×C₄	—	2	12	no	—
82	C₂×C₃×C₄	—	2	24	no	—
83	C₂×C₃×C₄	—	2	32	no	—
84	C₂×C₃×C₄	C₂×D₄	1	24	yes	complemented
85	C₂×C₃×C₄	C₂⁴	1	4	yes	—
86	C₂×C₃×C₄	C₂⁴	1	8	yes	—
87	C₂×C₃×C₄	C₂⁴	1	4	yes	—
88	C₂⁴	—	2	4	no	—
89	C₂⁴	—	2	4	no	—
90	C₂⁴	—	2	8	no	—
91	C₂⁴	—	2	8	no	—
92	C₂⁴	—	2	8	no	—
93	C₂⁴	C₃×D₄	1	8	yes	complemented
94	C₂⁴	C₂³×C₃	1	12	yes	—
95	C₂⁴	C₂³×C₃	1	6	yes	—
96	C₂⁴	C₂³×C₃	1	1	yes	characteristic
97	C₂×D₄	—	2	96	no	—
98	C₂×D₄	—	2	48	no	—
99	C₂×D₄	C₂³×C₃	1	32	yes	complemented
100	C₂×D₄	C₂³×C₃	1	16	yes	complemented
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	8	no	—
103	C₂²×C₄	—	2	48	no	—
104	C₂²×C₄	C₃×D₄	1	8	yes	complemented
105	C₂²×C₄	C₂³×C₃	1	6	yes	—
106	C₂²×C₄	C₂³×C₃	1	12	yes	—
107	C₂²×C₄	C₂³×C₃	1	6	yes	—
108	C₂²×C₄	C₂³×C₃	1	16	yes	complemented
109	C₄⋊C₄	C₂³×C₃	1	16	yes	complemented
110	C₂²⋊C₄	C₂³×C₃	1	32	yes	complemented
111	C₂²×C₃	—	2	48	no	—
112	C₂²×C₃	—	2	4	no	—
113	C₂²×C₃	—	2	4	no	—
114	C₂²×C₃	—	2	8	no	—
115	C₂²×C₃	—	2	8	no	—
116	C₂²×C₃	—	2	8	no	—
117	C₂²×C₃	—	4	48	no	—
118	C₂²×C₃	#121	1	16	yes	direct product factor
119	C₂²×C₃	C₂²×D₄	1	6	yes	—
120	C₂²×C₃	C₂²×D₄	1	6	yes	—
121	C₂²×C₃	C₂²×D₄	1	8	yes	complemented
122	C₂²×C₃	C₂×#41	1	6	yes	—
123	C₂²×C₃	C₂⁵	1	1	yes	characteristic
124	C₃×C₄	—	2	4	no	—
125	C₃×C₄	—	2	8	no	—
126	C₃×C₄	C₂²×D₄	1	8	yes	complemented
127	C₂³	—	2	32	no	—
128	C₂³	—	2	12	no	—
129	C₂³	—	2	12	no	—
130	C₂³	—	2	24	no	—
131	C₂³	—	2	24	no	—
132	C₂³	—	2	24	no	—
133	C₂³	—	4	32	no	—
134	C₂³	C₂×C₃×D₄	1	4	yes	—
135	C₂³	C₂×C₃×D₄	1	4	yes	—
136	C₂³	C₂×C₃×D₄	1	24	yes	complemented
137	C₂³	C₃×#41	1	4	yes	—
138	C₂³	C₂⁴×C₃	1	8	yes	—
139	C₂³	C₂⁴×C₃	1	4	yes	—
140	C₂³	C₂⁴×C₃	1	3	yes	—
141	D₄	—	2	64	no	—
142	D₄	—	2	32	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	2	24	no	—
145	C₂×C₄	—	2	32	no	—
146	C₂×C₄	C₂×C₃×D₄	1	24	yes	complemented
147	C₂×C₄	C₂⁴×C₃	1	4	yes	—
148	C₂×C₄	C₂⁴×C₃	1	8	yes	—
149	C₂×C₄	C₂⁴×C₃	1	4	yes	—
150	C₂×C₃	—	2	8	no	—
151	C₂×C₃	—	4	8	no	—
152	C₂×C₃	C₂×#121	1	12	yes	direct product factor
153	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
154	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
155	C₂×C₃	C₂²×#41	1	1	yes	characteristic
156	C₂²	—	2	4	no	—
157	C₂²	—	2	4	no	—
158	C₂²	—	2	48	no	—
159	C₂²	—	2	8	no	—
160	C₂²	—	2	8	no	—
161	C₂²	—	2	8	no	—
162	C₂²	—	4	48	no	—
163	C₂²	C₃×#121	1	16	yes	direct product factor
164	C₂²	C₂²×C₃×D₄	1	6	yes	—
165	C₂²	C₂²×C₃×D₄	1	6	yes	—
166	C₂²	C₂²×C₃×D₄	1	8	yes	complemented
167	C₂²	C₂×C₃×#41	1	6	yes	—
168	C₂²	C₂⁵×C₃	1	1	yes	characteristic
169	C₄	—	2	4	no	—
170	C₄	—	2	8	no	—
171	C₄	C₂²×C₃×D₄	1	8	yes	complemented
172	C₃	C₂²×#121	1	1	yes	characteristic, direct product factor, Sylow
173	C₂	—	2	8	no	—
174	C₂	—	4	8	no	—
175	C₂	C₂×C₃×#121	1	12	yes	direct product factor
176	C₂	C₂³×C₃×D₄	1	1	yes	characteristic
177	C₂	C₂³×C₃×D₄	1	1	yes	characteristic
178	C₂	C₂²×C₃×#41	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×D₄	C₂	semidirect
2. C₂³×C₃×D₄	C₂	semidirect
3. C₂⁴×C₃×C₄	C₂	semidirect
4. C₂×C₃×#121	C₂	direct (N × H)
5. C₂²×C₃×C₄⋊C₄	C₂	semidirect
6. C₂²×C₃×C₂²⋊C₄	C₂	semidirect
7. C₂²×#121	C₃	direct (N × H)
12. C₂²×C₃×D₄	C₂²	semidirect
13. C₂²×C₃×D₄	C₂²	semidirect
18. C₂³×C₃×C₄	C₂²	semidirect
19. C₃×#121	C₂²	direct (N × H)
20. C₂×C₃×C₄⋊C₄	C₂²	semidirect
21. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
22. C₂³×D₄	C₂×C₃	semidirect
23. C₂³×D₄	C₂×C₃	semidirect
24. C₂⁴×C₄	C₂×C₃	semidirect
25. C₂×#121	C₂×C₃	direct (N × H)
26. C₂²×C₄⋊C₄	C₂×C₃	semidirect
27. C₂²×C₂²⋊C₄	C₂×C₃	semidirect
33. C₂⁴×C₃	D₄	semidirect
39. C₂×C₃×D₄	C₂³	semidirect
40. C₂×C₃×D₄	C₂³	semidirect
44. C₂²×C₃×C₄	D₄	semidirect
48. C₂²×C₃×C₄	C₂³	semidirect
49. C₃×C₄⋊C₄	C₂³	semidirect
50. C₃×C₂²⋊C₄	C₂³	semidirect
55. C₂²×D₄	C₂²×C₃	semidirect
56. C₂²×D₄	C₂²×C₃	semidirect
61. C₂³×C₄	C₂²×C₃	semidirect
62. #121	C₂²×C₃	direct (N × H)
63. C₂×C₄⋊C₄	C₂²×C₃	semidirect
64. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
74. C₂³×C₃	C₂×D₄	semidirect
84. C₂×C₃×C₄	C₂×D₄	semidirect
93. C₂⁴	C₃×D₄	semidirect
99. C₂×D₄	C₂³×C₃	semidirect
100. C₂×D₄	C₂³×C₃	semidirect
104. C₂²×C₄	C₃×D₄	semidirect
108. C₂²×C₄	C₂³×C₃	semidirect
109. C₄⋊C₄	C₂³×C₃	semidirect
110. C₂²⋊C₄	C₂³×C₃	semidirect
118. C₂²×C₃	#121	direct (N × H)
121. C₂²×C₃	C₂²×D₄	semidirect
126. C₃×C₄	C₂²×D₄	semidirect
136. C₂³	C₂×C₃×D₄	semidirect
146. C₂×C₄	C₂×C₃×D₄	semidirect
152. C₂×C₃	C₂×#121	direct (N × H)
163. C₂²	C₃×#121	direct (N × H)
166. C₂²	C₂²×C₃×D₄	semidirect
171. C₄	C₂²×C₃×D₄	semidirect
172. C₃	C₂²×#121	direct (N × H)
175. C₂	C₂×C₃×#121	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×D₄	2
C₂⁴×C₃×C₄	2
C₂×C₃×#121	2
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₂²⋊C₄	2
C₂²×#121	3

Subgroups — C22×C3×#121 #87316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#121 #87316