Subgroups — #82049

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	#5207	C₂	1	2	yes	maximal
3	#4926	C₂	1	4	yes	maximal, complemented
4	#4925	C₂	1	4	yes	maximal, complemented
5	#4924	C₂	1	2	yes	maximal, complemented
6	C₄×#208	C₂	1	2	yes	maximal, complemented
7	#2920	—	3	1	no	maximal, Sylow
8	C₂³×C₃×C₄	C₄	1	2	yes	—
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²	C₄	1	2	yes	—
11	C₂×C₃×C₄²	C₄	1	2	yes	—
12	C₂×C₃×C₄²	C₄	1	2	yes	—
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄²	C₂²	1	2	yes	—
16	C₂×C₃×C₄²	C₂²	1	2	yes	—
17	C₂×#208	C₂²	1	4	yes	—
18	#830	C₂²	1	8	yes	—
19	C₄⋊(C₃⋊C₈)	C₂²	1	8	yes	complemented
20	#803	C₂²	1	4	yes	complemented
21	C₄×C₃⋊C₈	C₂²	1	4	yes	complemented
22	C₂²×C₄²	D₃	1	1	yes	characteristic
23	#432	—	3	4	no	—
24	#431	—	3	4	no	—
25	#423	—	3	2	no	—
26	#407	—	3	2	no	—
27	C₄×M₁₆	—	3	2	no	—
28	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	—	2	8	no	—
30	C₂²×C₃×C₄	—	2	4	no	—
31	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
35	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
36	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
37	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	2	yes	—
40	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₃×C₄²	C₂×C₄	1	2	yes	—
42	C₃×C₄²	C₂×C₄	1	2	yes	—
43	C₃×C₄²	C₂×C₄	1	2	yes	—
44	C₃×C₄²	C₂×C₄	1	2	yes	—
45	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄²	C₂×C₄	1	2	yes	—
47	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄²	C₂³	1	1	yes	characteristic
49	C₃×C₄²	C₂³	1	2	yes	—
50	C₃×C₄²	C₂³	1	1	yes	characteristic
51	#208	—	2	8	no	—
52	C₂×C₃⋊C₈	C₂³	1	8	yes	—
53	C₂³×C₄	Dic₃	1	2	yes	—
54	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
55	C₂×M₁₆	—	3	4	no	—
56	C₂×C₄²	Dic₃	1	2	yes	—
57	C₂×C₄²	Dic₃	1	2	yes	—
58	C₂×C₄²	Dic₃	1	2	yes	—
59	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄²	C₂×D₃	1	2	yes	—
61	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₂×C₄²	C₂×D₃	1	2	yes	—
63	C₄⋊C₈	—	3	8	no	—
64	C₂²⋊C₈	—	3	8	no	—
65	#97	—	3	4	no	—
66	C₄×C₈	—	3	4	no	—
67	C₂³×C₃	—	2	2	no	—
68	C₂³×C₃	—	2	4	no	—
69	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
70	C₂³×C₃	C₂²×C₄	1	2	yes	—
71	C₂×C₃×C₄	—	2	2	no	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	—	2	4	no	—
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	—	2	4	no	—
77	C₂×C₃×C₄	—	2	4	no	—
78	C₂×C₃×C₄	—	2	4	no	—
79	C₂×C₃×C₄	M₁₆	1	8	yes	complemented
80	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄	#41	1	4	yes	—
87	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
88	C₃⋊C₈	—	2	8	no	—
89	C₂⁴	C₂×Dic₃	1	1	yes	characteristic
90	C₂²×C₄	—	2	8	no	—
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
93	C₂²×C₄	C₂×Dic₃	1	2	yes	—
94	C₂²×C₄	C₂×Dic₃	1	2	yes	—
95	C₂²×C₄	C₂×Dic₃	1	2	yes	—
96	C₂²×C₄	C₂×Dic₃	1	2	yes	—
97	C₂²×C₄	C₂×Dic₃	1	2	yes	—
98	C₂²×C₄	C₂×Dic₃	1	2	yes	—
99	C₂²×C₄	C₂×Dic₃	1	2	yes	—
100	C₂²×C₄	C₂²×D₃	1	2	yes	—
101	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
102	M₁₆	—	6	8	no	—
103	C₂×C₈	—	3	8	no	—
104	C₄²	C₂×Dic₃	1	2	yes	—
105	C₄²	C₂×Dic₃	1	2	yes	—
106	C₄²	C₂×Dic₃	1	2	yes	—
107	C₄²	C₂×Dic₃	1	2	yes	—
108	C₄²	C₂×Dic₃	1	1	yes	characteristic
109	C₄²	C₂×Dic₃	1	2	yes	—
110	C₄²	C₂×Dic₃	1	1	yes	characteristic
111	C₄²	C₂²×D₃	1	1	yes	characteristic
112	C₄²	C₂²×D₃	1	2	yes	—
113	C₄²	C₂²×D₃	1	1	yes	characteristic
114	C₂²×C₃	—	2	8	no	—
115	C₂²×C₃	—	2	2	no	—
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	#117	1	2	yes	—
119	C₂²×C₃	C₂×M₁₆	1	4	yes	—
120	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
121	C₃×C₄	—	2	2	no	—
122	C₃×C₄	—	2	2	no	—
123	C₃×C₄	—	2	2	no	—
124	C₃×C₄	#117	1	2	yes	—
125	C₃×C₄	C₂×M₁₆	1	4	yes	—
126	C₃×C₄	C₂×M₁₆	1	4	yes	—
127	C₃×C₄	C₂×#41	1	2	yes	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	2	no	—
130	C₂³	C₂²×Dic₃	1	2	yes	—
131	C₂³	C₂²×Dic₃	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	#208	1	8	yes	complemented
141	C₂×C₄	#235	1	4	yes	—
142	C₂×C₄	C₂²×Dic₃	1	2	yes	—
143	C₂×C₄	C₂²×Dic₃	1	2	yes	—
144	C₂×C₄	C₂²×Dic₃	1	2	yes	—
145	C₂×C₄	C₂²×Dic₃	1	2	yes	—
146	C₂×C₄	C₂²×Dic₃	1	2	yes	—
147	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
148	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
149	C₈	—	6	8	no	—
150	C₂×C₃	—	2	2	no	—
151	C₂×C₃	—	2	4	no	—
152	C₂×C₃	C₂×#117	1	1	yes	characteristic
153	C₂×C₃	C₂²×M₁₆	1	2	yes	—
154	C₂²	—	2	8	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	C₂×#208	1	4	yes	—
159	C₂²	#926	1	2	yes	—
160	C₂²	C₂³×Dic₃	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	C₂×#208	1	4	yes	—
165	C₄	C₂×#208	1	4	yes	—
166	C₄	#926	1	2	yes	—
167	C₄	C₂×#235	1	2	yes	—
168	C₃	#2920	1	1	yes	characteristic, complemented, Sylow
169	C₂	—	2	4	no	—
170	C₂	—	2	2	no	—
171	C₂	C₂²×#208	1	2	yes	—
172	C₂	C₂×#926	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂²×C₃×C₄²	C₂
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	#2920
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. #4926	C₂	semidirect
4. #4925	C₂	semidirect
5. #4924	C₂	semidirect
6. C₄×#208	C₂	semidirect
19. C₄⋊(C₃⋊C₈)	C₂²	semidirect
20. #803	C₂²	semidirect
21. C₄×C₃⋊C₈	C₂²	semidirect
79. C₂×C₃×C₄	M₁₆	semidirect
140. C₂×C₄	#208	semidirect
168. C₃	#2920	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄²	2
#5207	2
#4926	2
#4925	2
#4924	2
C₄×#208	2
#2920	3