Finite Group Databasebeta

Subgroups — #33660

Order 256.

327 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 111 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, complemented
2	C₂×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#2232	C₂	1	2	yes	maximal, complemented
4	#2224	C₂	1	2	yes	maximal, complemented
5	#2206	C₂	1	2	yes	maximal, complemented
6	#2184	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₆	C₂	1	4	yes	maximal, complemented
8	#2162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈	C₂²	1	1	yes	characteristic
10	#532	—	2	1	no	—
11	C₂×C₄×D₄	—	2	1	no	—
12	C₂×D₁₆	C₂²	1	4	yes	—
13	#500	C₂²	1	1	yes	characteristic, complemented
14	#493	C₂²	1	1	yes	characteristic, complemented
15	#486	—	2	1	no	—
16	#466	—	2	1	no	—
17	#466	C₂²	1	2	yes	—
18	C₄×D₈	—	2	1	no	—
19	C₄×D₈	—	2	1	no	—
20	C₄×D₈	—	2	1	no	—
21	C₄×D₈	C₂²	1	2	yes	—
22	C₄×D₈	C₂²	1	2	yes	—
23	C₄×D₈	C₂²	1	2	yes	complemented
24	C₂×#107	C₂²	1	1	yes	characteristic
25	C₂×#102	—	2	1	no	—
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	#366	C₂²	1	4	yes	complemented
28	C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	#357	C₂²	1	4	yes	—
30	#357	C₂²	1	4	yes	complemented
31	C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₄×C₁₆	C₂²	1	2	yes	complemented
33	C₂×#41	—	4	1	no	—
34	C₂²×D₄	—	2	1	no	—
35	C₂³×C₄	—	4	1	no	—
36	C₂×D₈	—	2	4	no	—
37	C₂×D₈	—	2	1	no	—
38	C₂×D₈	—	2	1	no	—
39	C₂×D₈	C₂³	1	2	yes	—
40	C₂×D₈	C₂³	1	2	yes	—
41	C₂²×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈	—	2	1	no	—
44	#127	—	2	1	no	—
45	#124	—	4	1	no	—
46	#121	—	2	2	no	—
47	#121	—	4	1	no	—
48	C₄×D₄	—	2	2	no	—
49	C₄×D₄	—	2	2	no	—
50	C₄×D₄	—	2	2	no	—
51	C₄×D₄	—	4	1	no	—
52	C₄×D₄	—	4	1	no	—
53	C₄×D₄	—	4	1	no	—
54	C₂×C₄⋊C₄	—	2	1	no	—
55	C₂×C₂²⋊C₄	—	4	1	no	—
56	C₂×C₄²	D₄	1	1	yes	characteristic
57	D₁₆	—	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	#102	—	2	2	no	—
64	#102	—	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	C₂⁴	—	4	1	no	—
72	#41	—	8	4	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×D₄	—	2	4	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	4	2	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	1	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂×D₄	—	4	1	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	—	4	2	no	—
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	—	4	2	no	—
86	C₂²×C₄	—	8	4	no	—
87	C₂²×C₄	D₈	1	1	yes	characteristic
88	C₂²×C₄	D₈	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	D₈	—	2	4	no	—
91	D₈	—	2	2	no	—
92	D₈	—	2	4	no	—
93	D₈	—	4	2	no	—
94	D₈	#41	1	4	yes	complemented
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₂²⋊C₄	—	4	2	no	—
106	C₂²⋊C₄	—	4	2	no	—
107	C₂²⋊C₄	—	4	2	no	—
108	C₂²⋊C₄	—	4	2	no	—
109	C₄²	D₈	1	1	yes	characteristic
110	C₄²	D₈	1	1	yes	characteristic
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₄²	C₂×D₄	1	1	yes	characteristic
113	C₁₆	—	2	4	no	—
114	C₂³	—	4	4	no	—
115	C₂³	—	4	2	no	—
116	C₂³	—	4	2	no	—
117	C₂³	—	8	2	no	—
118	C₂³	—	8	2	no	—
119	C₂³	C₂×D₈	1	1	yes	characteristic
120	Q₈	—	8	2	no	—
121	D₄	—	2	4	no	—
122	D₄	—	4	4	no	—
123	D₄	—	4	2	no	—
124	D₄	—	4	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	8	2	no	—
127	D₄	—	8	4	no	—
128	D₄	—	8	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	—	8	4	no	—
137	C₂×C₄	—	8	4	no	—
138	C₂×C₄	D₁₆	1	4	yes	complemented
139	C₂×C₄	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	C₂×#41	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	4	4	no	—
152	C₂²	—	8	8	no	—
153	C₂²	—	8	2	no	—
154	C₂²	—	8	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	8	2	no	—
158	C₂²	C₂×D₁₆	1	2	yes	—
159	C₂²	C₂²×D₈	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	8	2	no	—
164	C₄	—	8	2	no	—
165	C₄	C₂×D₁₆	1	2	yes	—
166	C₄	C₂×D₁₆	1	2	yes	—
167	C₄	#525	1	1	yes	characteristic
168	C₄	#575	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	8	4	no	—
171	C₂	—	16	2	no	—
172	C₂	#3100	1	1	yes	characteristic
173	C₂	C₂²×D₁₆	1	1	yes	characteristic
174	C₂	#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂²	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. #3169	C₂	semidirect
2. C₂×C₄×D₈	C₂	semidirect
3. #2232	C₂	semidirect
4. #2224	C₂	semidirect
5. #2206	C₂	semidirect
6. #2184	C₂	semidirect
7. C₄×D₁₆	C₂	semidirect
8. #2162	C₂	semidirect
13. #500	C₂²	semidirect
14. #493	C₂²	semidirect
23. C₄×D₈	C₂²	semidirect
27. #366	C₂²	semidirect
28. C₄⋊C₁₆	C₂²	semidirect
30. #357	C₂²	semidirect
32. C₄×C₁₆	C₂²	semidirect
94. D₈	#41	semidirect
138. C₂×C₄	D₁₆	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×D₈	2
#2232	2
#2224	2
#2206	2
#2184	2
C₄×D₁₆	2
#2162	2