Finite Group Databasebeta

Subgroups — #33812

Order 256.

267 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 107 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	#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	#3169	C₂	1	1	yes	maximal, characteristic
3	#2226	C₂	1	2	yes	maximal, complemented
4	#2225	C₂	1	2	yes	maximal, complemented
5	#2217	C₂	1	4	yes	maximal, complemented
6	#2208	C₂	1	2	yes	maximal, complemented
7	#2207	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#533	—	2	1	no	—
10	#532	—	2	1	no	—
11	C₂×SD₃₂	C₂²	1	4	yes	—
12	C₂²×C₁₆	C₂²	1	2	yes	—
13	#500	C₂²	1	1	yes	characteristic
14	#500	C₂²	1	1	yes	characteristic
15	#494	C₂²	1	1	yes	characteristic
16	#493	C₂²	1	1	yes	characteristic
17	#487	—	2	1	no	—
18	#486	—	2	1	no	—
19	#467	—	2	1	no	—
20	#467	C₂²	1	2	yes	—
21	#466	—	2	1	no	—
22	#466	C₂²	1	2	yes	—
23	C₄×Q₁₆	—	2	1	no	—
24	C₄×Q₁₆	C₂²	1	2	yes	—
25	C₄×D₈	—	2	1	no	—
26	C₄×D₈	C₂²	1	2	yes	—
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	#367	C₂²	1	4	yes	—
29	C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₄⋊C₁₆	C₂²	1	2	yes	—
31	#358	C₂²	1	4	yes	—
32	#357	C₂²	1	4	yes	—
33	C₂×#41	—	4	1	no	—
34	C₂×Q₁₆	—	2	1	no	—
35	C₂×Q₁₆	C₂³	1	2	yes	—
36	C₂×D₈	—	2	1	no	—
37	C₂×D₈	C₂³	1	2	yes	—
38	C₂²×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈	—	2	1	no	—
41	C₄⋊Q₈	—	2	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	#127	—	2	1	no	—
44	#125	—	4	1	no	—
45	#124	—	4	1	no	—
46	C₂²⋊Q₈	—	2	2	no	—
47	C₂²⋊Q₈	—	4	1	no	—
48	#121	—	2	2	no	—
49	#121	—	4	1	no	—
50	C₄×Q₈	—	2	2	no	—
51	C₄×Q₈	—	4	1	no	—
52	C₄×D₄	—	2	2	no	—
53	C₄×D₄	—	4	1	no	—
54	#117	—	4	1	no	—
55	C₂×C₄²	D₄	1	1	yes	characteristic
56	SD₃₂	—	4	4	no	—
57	C₂×C₁₆	—	4	2	no	—
58	C₂×C₁₆	C₂³	1	4	yes	—
59	#107	—	2	1	no	—
60	#107	—	2	1	no	—
61	#107	C₂³	1	2	yes	—
62	#107	C₂³	1	2	yes	—
63	#103	—	2	2	no	—
64	#103	—	2	2	no	—
65	#102	—	2	2	no	—
66	#102	—	2	2	no	—
67	C₄×C₈	D₄	1	1	yes	characteristic
68	C₄×C₈	D₄	1	1	yes	characteristic
69	C₄×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₈	C₂³	1	1	yes	characteristic
71	#41	—	8	4	no	—
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×Q₈	—	4	1	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	4	2	no	—
77	C₂×D₄	—	4	1	no	—
78	C₂×D₄	—	4	1	no	—
79	C₂²×C₄	—	4	2	no	—
80	C₂²×C₄	D₈	1	1	yes	characteristic
81	C₂²×C₄	D₈	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
83	Q₁₆	—	2	2	no	—
84	Q₁₆	—	2	2	no	—
85	Q₁₆	—	4	2	no	—
86	D₈	—	2	2	no	—
87	D₈	—	2	2	no	—
88	D₈	—	4	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₂²⋊C₄	—	4	2	no	—
105	C₂²⋊C₄	—	4	2	no	—
106	C₂²⋊C₄	—	4	2	no	—
107	C₄²	—	4	2	no	—
108	C₄²	D₈	1	1	yes	characteristic
109	C₄²	D₈	1	1	yes	characteristic
110	C₄²	C₂×D₄	1	1	yes	characteristic
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₁₆	—	2	4	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×D₈	1	1	yes	characteristic
115	Q₈	—	4	2	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	8	2	no	—
118	Q₈	—	8	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	8	2	no	—
122	D₄	—	8	4	no	—
123	D₄	—	8	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	SD₃₂	1	4	yes	complemented
132	C₂×C₄	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	H(5,𝔽₂)	1	1	yes	characteristic
142	C₈	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	8	2	no	—
146	C₂²	—	8	2	no	—
147	C₂²	—	8	2	no	—
148	C₂²	C₂×SD₃₂	1	2	yes	—
149	C₂²	C₂²×D₈	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	8	2	no	—
154	C₄	—	8	2	no	—
155	C₄	—	8	2	no	—
156	C₄	C₂×SD₃₂	1	2	yes	—
157	C₄	C₂×SD₃₂	1	2	yes	—
158	C₄	#537	1	1	yes	characteristic
159	C₄	#575	1	1	yes	characteristic
160	C₂	—	2	2	no	—
161	C₂	—	16	2	no	—
162	C₂	#3245	1	1	yes	characteristic
163	C₂	C₂²×SD₃₂	1	1	yes	characteristic
164	C₂	#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×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. #3170	C₂	semidirect
3. #2226	C₂	semidirect
4. #2225	C₂	semidirect
5. #2217	C₂	semidirect
6. #2208	C₂	semidirect
7. #2207	C₂	semidirect
8. C₂×C₄⋊C₁₆	C₂	semidirect
131. C₂×C₄	SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
#3170	2
#3169	2
#2226	2
#2225	2
#2217	2
#2208	2
#2207	2
C₂×C₄⋊C₁₆	2