Subgroups — #33436

Order 256.

346 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#2929	C₂	1	1	yes	maximal, characteristic
4	#2169	C₂	1	2	yes	maximal, complemented
5	#2168	C₂	1	2	yes	maximal, complemented
6	#2168	C₂	1	2	yes	maximal, complemented
7	D₄×C₁₆	C₂	1	2	yes	maximal, complemented
8	#2151	C₂	1	1	yes	maximal, characteristic, complemented
9	#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131	C₂²	1	1	yes	characteristic
13	C₂³×C₈	C₂²	1	1	yes	characteristic
14	#525	C₄	1	1	yes	characteristic
15	#504	—	2	4	no	—
16	C₂×M₃₂	—	2	4	no	—
17	C₂×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×M₃₂	C₂²	1	2	yes	—
19	C₂×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×M₃₂	C₂²	1	2	yes	complemented
21	C₂²×C₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆	C₂²	1	2	yes	—
24	C₂²×C₁₆	C₂²	1	2	yes	complemented
25	#435	C₄	1	2	yes	—
26	#435	C₄	1	2	yes	—
27	C₈×D₄	C₂²	1	2	yes	—
28	C₈×D₄	C₂²	1	2	yes	—
29	#424	C₄	1	1	yes	characteristic
30	#408	C₄	1	1	yes	characteristic
31	#407	C₄	1	1	yes	characteristic
32	#405	C₂²	1	1	yes	characteristic
33	C₄⋊C₁₆	C₂²	1	2	yes	complemented
34	C₄⋊C₁₆	C₂²	1	2	yes	complemented
35	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆	C₂²	1	2	yes	complemented
40	C₂²⋊C₁₆	C₂²	1	2	yes	complemented
41	#346	C₂²	1	2	yes	complemented
42	C₄×C₁₆	C₂²	1	2	yes	complemented
43	C₂×#41	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
45	#131	—	2	2	no	—
46	#131	—	2	2	no	—
47	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆	C₂×C₄	1	2	yes	—
49	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈	—	2	2	no	—
51	C₂²×C₈	—	2	2	no	—
52	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂³	1	1	yes	characteristic
56	C₂²×C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₈	C₂³	1	2	yes	—
58	C₂²×C₈	C₂³	1	1	yes	characteristic
59	#123	C₈	1	2	yes	—
60	C₂²⋊Q₈	C₈	1	1	yes	characteristic
61	C₂²⋊Q₈	C₈	1	1	yes	characteristic
62	#121	C₈	1	1	yes	characteristic
63	#121	C₈	1	1	yes	characteristic
64	#120	C₈	1	2	yes	—
65	C₄×D₄	C₂×C₄	1	2	yes	—
66	C₄×D₄	C₂×C₄	1	2	yes	—
67	#117	C₂×C₄	1	1	yes	characteristic
68	M₃₂	—	2	2	no	—
69	M₃₂	—	2	4	no	—
70	M₃₂	—	2	4	no	—
71	M₃₂	D₄	1	4	yes	complemented
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	4	no	—
75	C₂×C₁₆	—	2	4	no	—
76	C₂×C₁₆	C₂³	1	1	yes	characteristic
77	C₂×C₁₆	C₂³	1	1	yes	characteristic
78	C₂×C₁₆	C₂³	1	1	yes	characteristic
79	C₂×C₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₁₆	C₂³	1	2	yes	—
81	C₂×C₁₆	C₂³	1	2	yes	—
82	C₄⋊C₈	C₂×C₄	1	2	yes	—
83	C₄⋊C₈	C₂×C₄	1	2	yes	—
84	C₂²⋊C₈	C₂×C₄	1	2	yes	—
85	C₂²⋊C₈	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈	C₂×C₄	1	2	yes	—
87	C₂²⋊C₈	C₂×C₄	1	2	yes	—
88	#97	C₂×C₄	1	2	yes	—
89	C₄×C₈	C₂³	1	2	yes	—
90	C₂⁴	C₂×C₈	1	1	yes	characteristic
91	#41	—	2	2	no	—
92	#41	—	2	2	no	—
93	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
94	C₂×D₄	C₂×C₈	1	2	yes	—
95	C₂×D₄	C₂×C₈	1	2	yes	—
96	C₂×D₄	C₂×C₈	1	1	yes	characteristic
97	C₂×D₄	C₂×C₈	1	2	yes	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄	C₂²×C₄	1	2	yes	—
106	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₂²×C₄	1	2	yes	—
120	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄	1	2	yes	—
125	C₂×C₈	#41	1	2	yes	—
126	C₂×C₈	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
128	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
129	C₄⋊C₄	C₂×C₈	1	2	yes	—
130	C₄⋊C₄	C₂×C₈	1	2	yes	—
131	C₂²⋊C₄	C₂×C₈	1	2	yes	—
132	C₂²⋊C₄	C₂×C₈	1	2	yes	—
133	C₂²⋊C₄	C₂×C₈	1	2	yes	—
134	C₂²⋊C₄	C₂×C₈	1	2	yes	—
135	C₂²⋊C₄	C₂×C₈	1	2	yes	—
136	C₄²	C₂²×C₄	1	2	yes	—
137	C₁₆	—	2	1	no	—
138	C₁₆	—	2	1	no	—
139	C₁₆	—	2	2	no	—
140	C₁₆	—	2	1	no	—
141	C₁₆	—	2	1	no	—
142	C₁₆	C₂×D₄	1	4	yes	—
143	C₂³	—	2	2	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	4	1	no	—
146	C₂³	C₂²×C₈	1	1	yes	characteristic
147	C₂³	C₂²×C₈	1	2	yes	—
148	C₂³	C₂²×C₈	1	1	yes	characteristic
149	C₂³	C₂²×C₈	1	1	yes	characteristic
150	Q₈	—	2	2	no	—
151	D₄	—	2	4	no	—
152	D₄	—	2	4	no	—
153	D₄	—	2	2	no	—
154	D₄	—	4	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	C₄×D₄	1	2	yes	—
165	C₂×C₄	C₂²×C₈	1	2	yes	—
166	C₂×C₄	C₂²×C₈	1	2	yes	—
167	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₈	1	2	yes	—
170	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₄	#131	1	2	yes	—
172	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	C₄×D₄	1	2	yes	—
178	C₈	C₂²×D₄	1	1	yes	characteristic
179	C₈	C₂×#41	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	4	no	—
191	C₂²	C₈×D₄	1	2	yes	—
192	C₂²	#504	1	2	yes	complemented
193	C₂²	C₂³×C₈	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	C₈×D₄	1	2	yes	—
200	C₄	C₂×C₄×D₄	1	1	yes	characteristic
201	C₄	C₂×#131	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	2	2	no	—
204	C₂	—	4	2	no	—
205	C₂	—	4	1	no	—
206	C₂	C₂×C₈×D₄	1	1	yes	characteristic
207	C₂	C₂×#504	1	1	yes	characteristic
208	C₂	#3407	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#504	C₂	semidirect
2. C₂²×M₃₂	C₂	semidirect
4. #2169	C₂	semidirect
5. #2168	C₂	semidirect
6. #2168	C₂	semidirect
7. D₄×C₁₆	C₂	semidirect
8. #2151	C₂	semidirect
9. #2113	C₂	semidirect
10. C₂×C₂²⋊C₁₆	C₂	semidirect
11. #2108	C₂	semidirect
20. C₂×M₃₂	C₂²	semidirect
24. C₂²×C₁₆	C₂²	semidirect
33. C₄⋊C₁₆	C₂²	semidirect
34. C₄⋊C₁₆	C₂²	semidirect
39. C₂²⋊C₁₆	C₂²	semidirect
40. C₂²⋊C₁₆	C₂²	semidirect
41. #346	C₂²	semidirect
42. C₄×C₁₆	C₂²	semidirect
71. M₃₂	D₄	semidirect
192. C₂²	#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504	2
C₂²×M₃₂	2
#2929	2
#2169	2
#2168	2
D₄×C₁₆	2
#2151	2
#2113	2
C₂×C₂²⋊C₁₆	2
#2108	2