Finite Group Databasebeta

Subgroups — #33672

Order 256.

293 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 105 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	#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#2232	C₂	1	2	yes	maximal, complemented
4	#2227	C₂	1	2	yes	maximal, complemented
5	#2207	C₂	1	2	yes	maximal, complemented
6	#2185	C₂	1	2	yes	maximal, complemented
7	#2175	C₂	1	4	yes	maximal, complemented
8	#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
10	#532	—	2	1	no	—
11	C₂×C₄×Q₈	—	2	1	no	—
12	C₂×SD₃₂	C₂²	1	4	yes	—
13	#500	C₂²	1	1	yes	characteristic
14	#493	C₂²	1	1	yes	characteristic
15	#486	—	2	1	no	—
16	#466	—	2	1	no	—
17	#466	C₂²	1	2	yes	—
18	C₄×Q₁₆	—	2	1	no	—
19	C₄×Q₁₆	—	2	1	no	—
20	C₄×Q₁₆	C₂²	1	2	yes	—
21	C₄×Q₁₆	C₂²	1	2	yes	—
22	C₄×D₈	—	2	1	no	—
23	C₄×D₈	C₂²	1	2	yes	—
24	C₂×#107	C₂²	1	1	yes	characteristic
25	C₂×#103	—	2	1	no	—
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	#366	C₂²	1	4	yes	—
28	C₄⋊C₁₆	C₂²	1	2	yes	—
29	#358	C₂²	1	4	yes	—
30	#357	C₂²	1	4	yes	—
31	C₂²⋊C₁₆	C₂²	1	2	yes	—
32	#346	C₂²	1	2	yes	—
33	C₂×#41	—	4	1	no	—
34	C₂²×Q₈	—	2	1	no	—
35	C₂×Q₁₆	—	2	4	no	—
36	C₂×Q₁₆	—	2	1	no	—
37	C₂×Q₁₆	C₂³	1	2	yes	—
38	C₂×D₈	—	2	1	no	—
39	C₂×D₈	C₂³	1	2	yes	—
40	C₂²×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈	—	2	1	no	—
43	#127	—	2	1	no	—
44	#124	—	4	1	no	—
45	#121	—	2	2	no	—
46	#121	—	4	1	no	—
47	C₄×Q₈	—	2	2	no	—
48	C₄×Q₈	—	2	2	no	—
49	C₄×Q₈	—	4	1	no	—
50	C₄×Q₈	—	4	1	no	—
51	C₄×D₄	—	2	2	no	—
52	C₄×D₄	—	4	1	no	—
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄⋊C₄	—	4	1	no	—
55	C₂×C₄²	—	4	1	no	—
56	C₂×C₄²	D₄	1	1	yes	characteristic
57	SD₃₂	—	2	8	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	4	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	4	2	no	—
75	C₂×Q₈	—	4	1	no	—
76	C₂×Q₈	—	4	1	no	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	1	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂²×C₄	—	4	1	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	—	4	2	no	—
84	C₂²×C₄	D₈	1	1	yes	characteristic
85	C₂²×C₄	D₈	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	Q₁₆	—	2	4	no	—
88	Q₁₆	—	2	2	no	—
89	Q₁₆	#41	1	4	yes	complemented
90	D₈	—	2	4	no	—
91	D₈	—	4	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	2	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	C₄²	—	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	—	2	4	no	—
116	Q₈	—	4	4	no	—
117	Q₈	—	4	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	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	8	4	no	—
133	C₂×C₄	—	8	2	no	—
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	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	C₂×#41	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	8	2	no	—
147	C₂²	—	8	2	no	—
148	C₂²	—	8	2	no	—
149	C₂²	#509	1	2	yes	—
150	C₂²	C₂²×D₈	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	4	no	—
156	C₄	—	8	2	no	—
157	C₄	—	8	2	no	—
158	C₄	#510	1	2	yes	—
159	C₄	#525	1	1	yes	characteristic
160	C₄	#575	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	—	16	2	no	—
163	C₂	#3100	1	1	yes	characteristic
164	C₂	C₂×#509	1	1	yes	characteristic
165	C₂	C₂×#510	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
2. C₂×C₄×Q₁₆	C₂	semidirect
3. #2232	C₂	semidirect
4. #2227	C₂	semidirect
5. #2207	C₂	semidirect
6. #2185	C₂	semidirect
7. #2175	C₂	semidirect
8. #2163	C₂	semidirect
89. Q₁₆	#41	semidirect

Maximal subgroups

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

subgroup	index
#3169	2
C₂×C₄×Q₁₆	2
#2232	2
#2227	2
#2207	2
#2185	2
#2175	2
#2163	2