Finite Group Databasebeta

Subgroups — #33928

Order 256.

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

normal subgroups only

This group has 249 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	#3396	C₂	1	1	yes	maximal, characteristic, complemented
2	#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	#2242	C₂	1	2	yes	maximal, complemented
4	#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	#2228	C₂	1	2	yes	maximal, complemented
6	#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	#2210	C₂	1	2	yes	maximal, complemented
8	#2206	C₂	1	1	yes	maximal, characteristic
9	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆	C₂	1	2	yes	maximal, complemented
12	#559	—	2	1	no	—
13	#554	—	2	1	no	—
14	C₂×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂×D₁₆	C₂²	1	2	yes	—
16	#500	C₂²	1	1	yes	characteristic
17	#499	C₂²	1	2	yes	—
18	#495	C₂²	1	2	yes	—
19	#493	C₂²	1	1	yes	characteristic
20	#478	—	2	2	no	—
21	#476	—	2	1	no	—
22	#463	—	2	2	no	—
23	#460	—	2	1	no	—
24	C₈×Q₈	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆	C₂²	1	1	yes	characteristic
26	C₄×D₈	C₂²	1	2	yes	—
27	C₄×D₈	C₂²	1	1	yes	characteristic
28	C₄×D₈	C₂²	1	2	yes	—
29	#367	C₂²	1	1	yes	characteristic
30	#366	C₂²	1	2	yes	—
31	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆	C₂²	1	2	yes	—
33	#358	C₂²	1	1	yes	characteristic
34	#358	C₂²	1	2	yes	—
35	#357	C₂²	1	1	yes	characteristic
36	#357	C₂²	1	2	yes	—
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	C₄×C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆	C₂³	1	1	yes	characteristic
42	C₂×SD₁₆	—	2	2	no	—
43	C₂×D₈	—	2	1	no	—
44	C₂×D₈	C₂³	1	2	yes	—
45	C₂×D₈	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈	—	2	1	no	—
47	#127	—	2	1	no	—
48	#125	—	2	2	no	—
49	#124	—	2	2	no	—
50	#124	—	4	1	no	—
51	#123	—	4	2	no	—
52	C₂²⋊Q₈	—	4	1	no	—
53	C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈	—	2	1	no	—
55	C₄×Q₈	D₄	1	1	yes	characteristic
56	C₄×D₄	—	2	1	no	—
57	C₄×D₄	—	2	2	no	—
58	C₄×D₄	—	2	2	no	—
59	C₄×D₄	—	4	1	no	—
60	C₂×C₄⋊C₄	—	4	1	no	—
61	SD₃₂	—	2	2	no	—
62	D₁₆	—	2	4	no	—
63	C₂×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₆	C₂³	1	2	yes	—
66	#107	—	2	1	no	—
67	#107	C₂³	1	2	yes	—
68	#107	C₂³	1	1	yes	characteristic
69	#107	C₂³	1	1	yes	characteristic
70	#106	—	2	2	no	—
71	C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₄⋊C₈	D₄	1	2	yes	—
73	#103	—	2	1	no	—
74	#102	—	2	2	no	—
75	#102	—	2	2	no	—
76	#102	—	2	1	no	—
77	C₄×C₈	C₂³	1	2	yes	—
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×Q₈	D₈	1	1	yes	characteristic
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂×D₄	—	4	1	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	4	2	no	—
89	Q₁₆	—	2	1	no	—
90	Q₁₆	—	2	1	no	—
91	SD₁₆	—	8	2	no	—
92	D₈	—	2	2	no	—
93	D₈	—	2	1	no	—
94	D₈	—	2	2	no	—
95	D₈	—	8	1	no	—
96	D₈	#41	1	2	yes	—
97	C₂×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	D₈	1	1	yes	characteristic
108	C₄⋊C₄	D₈	1	2	yes	—
109	C₂²⋊C₄	—	4	1	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂²⋊C₄	—	4	2	no	—
112	C₂²⋊C₄	—	4	2	no	—
113	C₂²⋊C₄	—	4	2	no	—
114	C₄²	—	4	1	no	—
115	C₄²	C₂×D₄	1	2	yes	—
116	C₄²	C₂×D₄	1	1	yes	characteristic
117	C₁₆	—	2	2	no	—
118	C₁₆	—	2	1	no	—
119	C₁₆	—	2	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	4	2	no	—
122	Q₈	—	2	2	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	16	1	no	—
126	D₄	—	2	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	16	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	—	16	2	no	—
138	C₂×C₄	—	16	2	no	—
139	C₂×C₄	C₂×D₈	1	2	yes	—
140	C₂×C₄	C₂×D₈	1	2	yes	—
141	C₂×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	2	no	—
146	C₈	C₂×#41	1	1	yes	characteristic
147	C₈	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²	—	4	2	no	—
149	C₂²	—	8	1	no	—
150	C₂²	—	8	1	no	—
151	C₂²	—	8	2	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	C₂²×D₈	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	2	1	no	—
158	C₄	—	8	1	no	—
159	C₄	—	8	1	no	—
160	C₄	—	8	1	no	—
161	C₄	—	8	2	no	—
162	C₄	#508	1	2	yes	—
163	C₄	#548	1	1	yes	characteristic
164	C₄	#577	1	1	yes	characteristic
165	C₂	—	8	2	no	—
166	C₂	—	16	2	no	—
167	C₂	#3358	1	1	yes	characteristic
168	C₂	C₂×#508	1	1	yes	characteristic
169	C₂	#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#2206	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ 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. #3396	C₂	semidirect
2. #3380	C₂	semidirect
3. #2242	C₂	semidirect
4. #2240	C₂	semidirect
5. #2228	C₂	semidirect
6. #2227	C₂	semidirect
7. #2210	C₂	semidirect
9. Q₈×C₁₆	C₂	semidirect
10. C₄×SD₃₂	C₂	semidirect
11. C₄×D₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3396	2
#3380	2
#2242	2
#2240	2
#2228	2
#2227	2
#2210	2
#2206	2
Q₈×C₁₆	2
C₄×SD₃₂	2
C₄×D₁₆	2