Finite Group Databasebeta

Subgroups — #12815

Order 256.

211 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₄×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#359	C₂	1	2	yes	maximal
4	#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	#1382	C₂	1	2	yes	maximal, complemented
6	C₂×#135	C₂²	1	1	yes	characteristic
7	C₄×#41	—	2	1	no	—
8	C₂²×C₁₆	C₂²	1	2	yes	—
9	C₄×Q₁₆	C₄	1	1	yes	characteristic
10	C₄×Q₁₆	C₄	1	1	yes	characteristic
11	C₄×SD₁₆	—	2	1	no	—
12	C₄×SD₁₆	—	2	1	no	—
13	C₄×D₈	C₄	1	1	yes	characteristic
14	C₄×D₈	C₄	1	1	yes	characteristic
15	C₂×#108	—	2	1	no	—
16	C₂×#108	C₂²	1	2	yes	—
17	#427	C₂²	1	1	yes	characteristic
18	#423	—	2	1	no	—
19	#416	—	2	1	no	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆	—	2	1	no	—
22	C₄⋊C₁₆	—	2	1	no	—
23	#359	—	2	4	no	—
24	C₂×#41	—	2	1	no	—
25	#135	—	2	1	no	—
26	#135	—	2	1	no	—
27	#135	D₄	1	2	yes	—
28	#135	Q₈	1	2	yes	—
29	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×SD₁₆	—	2	1	no	—
31	C₂×D₈	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆	—	2	2	no	—
33	C₂²×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	C₄×Q₈	—	2	1	no	—
36	C₄×Q₈	—	2	1	no	—
37	C₄×D₄	—	2	1	no	—
38	C₄×D₄	—	2	1	no	—
39	C₄×D₄	—	4	1	no	—
40	C₄×D₄	—	4	1	no	—
41	#117	—	2	1	no	—
42	#117	—	4	1	no	—
43	C₂×C₄²	—	4	1	no	—
44	C₂×C₄²	D₄	1	1	yes	characteristic
45	C₂×C₁₆	—	2	2	no	—
46	C₂×C₁₆	—	2	2	no	—
47	C₂×C₁₆	—	2	2	no	—
48	#108	—	2	2	no	—
49	#108	—	2	2	no	—
50	#108	—	4	2	no	—
51	#107	C₂×C₄	1	1	yes	characteristic
52	#107	C₂×C₄	1	1	yes	characteristic
53	#106	—	2	1	no	—
54	C₄⋊C₈	—	4	1	no	—
55	C₄⋊C₈	—	4	1	no	—
56	#103	—	2	1	no	—
57	#103	—	2	1	no	—
58	#102	—	2	1	no	—
59	#102	—	2	1	no	—
60	C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈	D₄	1	1	yes	characteristic
63	C₄×C₈	D₄	1	1	yes	characteristic
64	#41	—	2	2	no	—
65	#41	—	2	2	no	—
66	#41	—	4	1	no	—
67	#41	—	4	1	no	—
68	C₂×Q₈	—	2	1	no	—
69	C₂×D₄	—	2	1	no	—
70	C₂×D₄	—	4	1	no	—
71	C₂²×C₄	—	4	1	no	—
72	C₂²×C₄	—	4	1	no	—
73	C₂²×C₄	—	4	1	no	—
74	C₂²×C₄	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄	SD₁₆	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	Q₁₆	—	2	1	no	—
78	Q₁₆	C₄⋊C₄	1	2	yes	—
79	SD₁₆	—	2	2	no	—
80	SD₁₆	—	2	2	no	—
81	D₈	—	2	1	no	—
82	D₈	C₄⋊C₄	1	2	yes	—
83	M₁₆	—	4	2	no	—
84	M₁₆	—	4	2	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈	#41	1	1	yes	characteristic
95	C₂×C₈	#41	1	1	yes	characteristic
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₂²⋊C₄	—	4	1	no	—
101	C₂²⋊C₄	—	4	1	no	—
102	C₂²⋊C₄	—	4	1	no	—
103	C₄²	—	4	1	no	—
104	C₄²	—	4	1	no	—
105	C₄²	—	4	1	no	—
106	C₄²	C₂²⋊C₄	1	1	yes	characteristic
107	C₄²	C₂²⋊C₄	1	1	yes	characteristic
108	C₄²	D₈	1	1	yes	characteristic
109	C₄²	D₈	1	1	yes	characteristic
110	C₁₆	—	4	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	C₂×SD₁₆	1	1	yes	characteristic
113	Q₈	—	2	2	no	—
114	Q₈	—	4	1	no	—
115	D₄	—	2	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	4	1	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	#102	1	1	yes	characteristic
129	C₂×C₄	#102	1	1	yes	characteristic
130	C₂×C₄	#102	1	1	yes	characteristic
131	C₂×C₄	#102	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄	#121	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄	#137	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	8	2	no	—
141	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₈	#117	1	1	yes	characteristic
143	C₈	#121	1	1	yes	characteristic
144	C₈	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	8	1	no	—
150	C₂²	C₂×#102	1	1	yes	characteristic
151	C₂²	#461	1	1	yes	characteristic
152	C₂²	#476	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	8	1	no	—
158	C₄	—	8	1	no	—
159	C₄	#359	1	2	yes	complemented
160	C₄	#380	1	1	yes	characteristic
161	C₄	#417	1	1	yes	characteristic
162	C₄	#459	1	1	yes	characteristic
163	C₄	#474	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	—	8	2	no	—
167	C₂	#1802	1	1	yes	characteristic
168	C₂	C₂×#359	1	1	yes	characteristic
169	C₂	#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#102
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆	C₂	semidirect
4. #1848	C₂	semidirect
5. #1382	C₂	semidirect
159. C₄	#359	semidirect

Maximal subgroups

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

subgroup	index
C₄×#135	2
C₂×C₄⋊C₁₆	2
C₂×#359	2
#1848	2
#1382	2