Finite Group Databasebeta

Subgroups — #10848

Order 256.

401 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₂×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#2285	C₂	1	1	yes	maximal, characteristic
4	C₂×#344	C₂	1	4	yes	maximal
5	C₂²×M₁₆	C₂²	1	2	yes	—
6	C₂²×M₁₆	C₂²	1	2	yes	—
7	C₂×#117	C₂²	1	2	yes	—
8	C₂²×C₄²	C₂²	1	1	yes	characteristic
9	#423	—	2	1	no	—
10	#423	—	2	1	no	—
11	#423	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈	—	2	2	no	—
13	C₄×M₁₆	—	2	2	no	—
14	C₄×M₁₆	—	2	2	no	—
15	C₂×#97	C₄	1	2	yes	complemented
16	C₂×C₄×C₈	C₄	1	2	yes	complemented
17	#384	—	2	1	no	—
18	#382	—	2	1	no	—
19	#380	C₄	1	4	yes	—
20	C₄×C₄⋊C₄	—	2	2	no	—
21	#344	—	2	8	no	—
22	C₂³×C₄	C₂×C₄	1	2	yes	—
23	C₂³×C₄	C₂³	1	1	yes	characteristic
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	—	2	2	no	—
26	C₂×M₁₆	—	2	2	no	—
27	C₂×M₁₆	—	2	2	no	—
28	C₂×M₁₆	—	2	2	no	—
29	C₂×M₁₆	—	2	4	no	—
30	C₂×M₁₆	—	2	4	no	—
31	C₂×M₁₆	C₂×C₄	1	4	yes	—
32	C₂×M₁₆	D₄	1	4	yes	—
33	C₂×M₁₆	Q₈	1	4	yes	—
34	C₂²×C₈	—	2	2	no	—
35	C₂²×C₈	C₂×C₄	1	4	yes	—
36	#117	—	2	4	no	—
37	#117	—	2	4	no	—
38	C₂×C₄⋊C₄	—	2	4	no	—
39	C₂×C₄⋊C₄	—	2	2	no	—
40	C₂×C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₄²	—	2	2	no	—
42	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²	C₂×C₄	1	2	yes	—
45	C₂×C₄²	D₄	1	2	yes	—
46	C₂×C₄²	D₄	1	2	yes	—
47	C₂×C₄²	D₄	1	1	yes	characteristic
48	C₂×C₄²	D₄	1	1	yes	characteristic
49	C₂×C₄²	Q₈	1	1	yes	characteristic
50	C₂×C₄²	Q₈	1	1	yes	characteristic
51	C₄⋊C₈	—	2	4	no	—
52	C₄⋊C₈	—	2	4	no	—
53	#97	—	2	4	no	—
54	C₄×C₈	—	2	4	no	—
55	#95	—	2	4	no	—
56	C₂⁴	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄	—	2	4	no	—
58	C₂²×C₄	—	4	4	no	—
59	C₂²×C₄	C₄²	1	2	yes	—
60	C₂²×C₄	C₄²	1	2	yes	—
61	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄	C₂×D₄	1	2	yes	—
71	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×Q₈	1	2	yes	—
73	C₂²×C₄	#41	1	2	yes	—
74	C₂²×C₄	#41	1	2	yes	—
75	M₁₆	—	2	4	no	—
76	M₁₆	—	2	4	no	—
77	M₁₆	—	2	4	no	—
78	M₁₆	—	2	4	no	—
79	M₁₆	—	2	8	no	—
80	M₁₆	—	4	2	no	—
81	M₁₆	—	4	2	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	C₄⋊C₄	1	8	yes	complemented
87	C₄⋊C₄	—	4	8	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₂²⋊C₄	—	2	8	no	—
91	C₄²	—	4	4	no	—
92	C₄²	C₂²⋊C₄	1	2	yes	—
93	C₄²	C₂²⋊C₄	1	2	yes	—
94	C₄²	C₂²⋊C₄	1	2	yes	—
95	C₄²	C₂²⋊C₄	1	2	yes	—
96	C₄²	C₄⋊C₄	1	1	yes	characteristic
97	C₄²	C₄⋊C₄	1	1	yes	characteristic
98	C₄²	C₄⋊C₄	1	1	yes	characteristic
99	C₄²	C₄⋊C₄	1	2	yes	—
100	C₄²	C₄⋊C₄	1	2	yes	—
101	C₄²	C₄⋊C₄	1	1	yes	characteristic
102	C₂³	—	4	2	no	—
103	C₂³	C₂×C₄²	1	1	yes	characteristic
104	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³	C₄×D₄	1	2	yes	—
107	C₂³	C₄×Q₈	1	2	yes	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	8	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	#95	1	2	yes	—
117	C₂×C₄	#95	1	2	yes	—
118	C₂×C₄	#95	1	2	yes	—
119	C₂×C₄	#95	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄	#117	1	2	yes	—
126	C₂×C₄	C₄×D₄	1	2	yes	—
127	C₂×C₄	C₄×Q₈	1	2	yes	—
128	C₂×C₄	#121	1	2	yes	—
129	C₂×C₄	#121	1	2	yes	—
130	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄	#124	1	1	yes	characteristic
133	C₂×C₄	#124	1	1	yes	characteristic
134	C₂×C₄	#125	1	2	yes	—
135	C₂×C₄	#127	1	1	yes	characteristic
136	C₂×C₄	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
138	C₈	—	2	8	no	—
139	C₈	—	4	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×#95	1	1	yes	characteristic
146	C₂²	C₄×C₄⋊C₄	1	2	yes	—
147	C₂²	#389	1	2	yes	—
148	C₂²	#390	1	1	yes	characteristic
149	C₂²	#391	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	8	4	no	—
154	C₄	#344	1	4	yes	complemented
155	C₄	#380	1	2	yes	—
156	C₄	#382	1	1	yes	characteristic
157	C₄	#384	1	1	yes	characteristic
158	C₄	#389	1	2	yes	—
159	C₄	#390	1	1	yes	characteristic
160	C₄	#391	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	2	no	—
163	C₂	—	2	2	no	—
164	C₂	—	2	1	no	—
165	C₂	C₄⋊#95	1	1	yes	characteristic
166	C₂	C₂×#344	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²
Frattini, cosocle	C₂³×C₄	C₂³
socle	C₂²	C₂×#95
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
11. #423	C₄	semidirect
15. C₂×#97	C₄	semidirect
16. C₂×C₄×C₈	C₄	semidirect
86. C₂×C₈	C₄⋊C₄	semidirect
154. C₄	#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423	2
C₂×C₄×M₁₆	2
#2285	2
C₂×#344	2