Finite Group Databasebeta

Subgroups — #33906

Order 256.

247 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3339	C₂	1	2	yes	maximal, complemented
2	#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	#2234	C₂	1	2	yes	maximal, complemented
4	#2233	C₂	1	2	yes	maximal, complemented
5	#2228	C₂	1	2	yes	maximal, complemented
6	#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	#2215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#2158	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#366	C₂	1	1	yes	maximal, characteristic, complemented
12	#553	—	2	2	no	—
13	C₂×D₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆	C₂²	1	1	yes	characteristic
16	#499	C₂²	1	2	yes	—
17	#482	—	2	2	no	—
18	#481	—	2	2	no	—
19	#478	—	2	2	no	—
20	#466	C₂²	1	2	yes	—
21	#466	C₂²	1	2	yes	—
22	C₄×D₈	C₂²	1	2	yes	—
23	C₈×D₄	C₂²	1	1	yes	characteristic
24	#427	C₂²	1	2	yes	—
25	C₂×#107	C₂²	1	2	yes	—
26	#367	C₂²	1	2	yes	—
27	#367	C₂²	1	2	yes	—
28	#366	C₂²	1	1	yes	characteristic
29	#366	C₂²	1	1	yes	characteristic
30	#366	C₂²	1	2	yes	—
31	#366	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	#357	C₂²	1	2	yes	—
34	#357	C₂²	1	2	yes	—
35	#357	C₂²	1	2	yes	—
36	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₆	C₂²	1	1	yes	characteristic
39	C₂×D₈	C₂³	1	2	yes	—
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	#126	—	4	2	no	—
43	#125	—	2	2	no	—
44	#123	—	4	2	no	—
45	#121	—	2	2	no	—
46	#121	—	2	2	no	—
47	#121	—	4	2	no	—
48	C₄×D₄	—	2	2	no	—
49	C₄×D₄	—	4	2	no	—
50	C₄×D₄	D₄	1	1	yes	characteristic
51	#117	—	2	2	no	—
52	C₂×C₄⋊C₄	—	2	2	no	—
53	D₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	2	2	no	—
56	C₂×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₆	C₂³	1	1	yes	characteristic
60	#107	—	2	2	no	—
61	#107	C₂³	1	2	yes	—
62	#107	C₂³	1	2	yes	—
63	#107	C₂³	1	2	yes	—
64	#106	—	2	2	no	—
65	#106	—	2	2	no	—
66	C₄⋊C₈	D₄	1	1	yes	characteristic
67	#102	—	2	2	no	—
68	#102	—	2	2	no	—
69	#102	—	2	2	no	—
70	C₂²⋊C₈	D₄	1	1	yes	characteristic
71	C₂²⋊C₈	D₄	1	1	yes	characteristic
72	C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	4	2	no	—
75	C₂×D₄	—	4	2	no	—
76	C₂×D₄	D₈	1	1	yes	characteristic
77	C₂²×C₄	—	4	2	no	—
78	C₂²×C₄	—	4	2	no	—
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
81	D₈	—	2	2	no	—
82	D₈	—	2	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	D₈	1	1	yes	characteristic
98	C₂²⋊C₄	—	4	2	no	—
99	C₂²⋊C₄	—	4	2	no	—
100	C₂²⋊C₄	—	4	2	no	—
101	C₂²⋊C₄	—	4	2	no	—
102	C₂²⋊C₄	D₈	1	1	yes	characteristic
103	C₂²⋊C₄	D₈	1	1	yes	characteristic
104	C₄²	—	4	2	no	—
105	C₄²	C₂×D₄	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	2	1	no	—
109	C₁₆	#41	1	2	yes	—
110	C₂³	—	4	2	no	—
111	C₂³	C₂×D₈	1	1	yes	characteristic
112	C₂³	C₂×D₈	1	1	yes	characteristic
113	D₄	—	2	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	D₄	—	8	4	no	—
117	D₄	—	16	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	16	2	no	—
126	C₂×C₄	—	16	2	no	—
127	C₂×C₄	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	C₂×#41	1	1	yes	characteristic
136	C₈	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	8	2	no	—
143	C₂²	—	8	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	#508	1	2	yes	—
146	C₂²	C₂²×D₈	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	8	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	8	2	no	—
154	C₄	#547	1	1	yes	characteristic
155	C₄	#578	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	—	4	1	no	—
158	C₂	—	16	2	no	—
159	C₂	#3334	1	1	yes	characteristic
160	C₂	C₂×#508	1	1	yes	characteristic
161	C₂	#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ 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. #3339	C₂	semidirect
2. #2253	C₂	semidirect
3. #2234	C₂	semidirect
4. #2233	C₂	semidirect
5. #2228	C₂	semidirect
6. #2215	C₂	semidirect
7. #2215	C₂	semidirect
8. C₄×D₁₆	C₂	semidirect
9. D₄×C₁₆	C₂	semidirect
10. #2158	C₂	semidirect
11. C₂×#366	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3339	2
#2253	2
#2234	2
#2233	2
#2228	2
#2215	2
C₄×D₁₆	2
D₄×C₁₆	2
#2158	2
C₂×#366	2