Finite Group Databasebeta

Subgroups — #12996

Order 256.

256 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 75 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×#472	C₂	1	1	yes	maximal, characteristic
2	#2913	C₂	1	1	yes	maximal, characteristic, complemented
3	#2140	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#350	C₂	1	1	yes	maximal, characteristic, complemented
6	#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)	—	2	1	no	—
9	C₂²×D₈	C₂²	1	1	yes	characteristic
10	C₂×#131	C₂²	1	1	yes	characteristic
11	#520	—	2	1	no	—
12	C₂×M₃₂	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆	C₂²	1	1	yes	characteristic
14	#472	—	2	2	no	—
15	#472	C₄	1	4	yes	complemented
16	C₂×#108	C₂²	1	1	yes	characteristic
17	#428	—	2	1	no	—
18	#428	—	2	2	no	—
19	#427	—	2	1	no	—
20	#427	C₂²	1	1	yes	characteristic
21	C₂×#107	—	2	1	no	—
22	C₂×#107	C₂²	1	1	yes	characteristic
23	C₂×#100	—	2	1	no	—
24	#357	—	2	2	no	—
25	#357	—	2	1	no	—
26	#357	—	2	1	no	—
27	#350	—	2	2	no	—
28	C₂×#41	D₄	1	1	yes	characteristic
29	C₂²×D₄	—	2	1	no	—
30	AGL(1,ℤ/8)	—	2	4	no	—
31	AGL(1,ℤ/8)	—	4	2	no	—
32	C₂×SD₁₆	—	4	1	no	—
33	C₂×D₈	—	2	1	no	—
34	C₂×D₈	—	2	1	no	—
35	C₂×D₈	—	2	1	no	—
36	C₂×D₈	—	2	1	no	—
37	C₂×D₈	—	2	1	no	—
38	C₂×D₈	—	4	1	no	—
39	C₂×D₈	C₂×C₄	1	2	yes	complemented
40	#131	—	2	1	no	—
41	#131	—	2	1	no	—
42	#131	C₂×C₄	1	2	yes	—
43	#131	D₄	1	2	yes	—
44	C₂×M₁₆	—	2	1	no	—
45	C₂×M₁₆	—	2	1	no	—
46	C₂×M₁₆	D₄	1	1	yes	characteristic
47	C₂²×C₈	—	2	1	no	—
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₄×Q₈	—	4	1	no	—
50	C₄×D₄	—	4	2	no	—
51	C₄×D₄	—	4	1	no	—
52	#117	—	2	2	no	—
53	#117	—	2	1	no	—
54	C₂×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄⋊C₄	—	2	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	2	1	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	2	1	no	—
61	C₂×C₁₆	—	2	1	no	—
62	#108	—	2	1	no	—
63	#108	C₂×C₄	1	2	yes	complemented
64	#107	—	2	2	no	—
65	#107	—	2	1	no	—
66	#107	D₄	1	1	yes	characteristic
67	#107	D₄	1	2	yes	—
68	#107	D₄	1	1	yes	characteristic
69	#106	—	2	2	no	—
70	#106	—	2	1	no	—
71	#100	—	2	4	no	—
72	C₂⁴	—	4	1	no	—
73	#41	—	2	1	no	—
74	#41	—	2	1	no	—
75	#41	C₂²⋊C₄	1	2	yes	—
76	#41	SD₁₆	1	2	yes	—
77	C₂×Q₈	D₈	1	1	yes	characteristic
78	C₂×D₄	—	2	1	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂×D₄	—	4	1	no	—
82	C₂×D₄	—	4	1	no	—
83	C₂×D₄	—	4	1	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂×D₄	D₈	1	1	yes	characteristic
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	SD₁₆	—	4	4	no	—
91	D₈	—	2	2	no	—
92	D₈	—	2	2	no	—
93	D₈	—	4	4	no	—
94	D₈	—	4	1	no	—
95	D₈	—	4	1	no	—
96	M₁₆	—	2	1	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	2	2	no	—
99	M₁₆	—	2	2	no	—
100	M₁₆	—	4	1	no	—
101	M₁₆	C₂²⋊C₄	1	2	yes	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈	#41	1	1	yes	characteristic
111	C₂×C₈	#41	1	1	yes	characteristic
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	4	1	no	—
119	C₄²	—	4	2	no	—
120	C₄²	—	4	1	no	—
121	C₁₆	—	4	1	no	—
122	C₁₆	—	4	1	no	—
123	C₂³	—	2	1	no	—
124	C₂³	—	4	2	no	—
125	C₂³	—	8	1	no	—
126	C₂³	—	8	1	no	—
127	C₂³	—	16	1	no	—
128	C₂³	—	16	1	no	—
129	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
130	Q₈	—	2	1	no	—
131	Q₈	#102	1	2	yes	—
132	D₄	—	2	2	no	—
133	D₄	—	2	2	no	—
134	D₄	—	2	1	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	8	1	no	—
138	D₄	—	8	1	no	—
139	D₄	#102	1	2	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	16	1	no	—
147	C₂×C₄	—	16	2	no	—
148	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#120	1	1	yes	characteristic
150	C₂×C₄	#123	1	1	yes	characteristic
151	C₂×C₄	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₄	#137	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	4	2	no	—
156	C₈	C₄×D₄	1	1	yes	characteristic
157	C₈	C₄×D₄	1	1	yes	characteristic
158	C₈	#121	1	1	yes	characteristic
159	C₈	#121	1	1	yes	characteristic
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	8	2	no	—
166	C₂²	—	16	2	no	—
167	C₂²	—	16	2	no	—
168	C₂²	—	16	1	no	—
169	C₂²	—	16	1	no	—
170	C₂²	#417	1	1	yes	characteristic
171	C₂²	#447	1	1	yes	characteristic
172	C₂²	#481	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	8	2	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	#386	1	1	yes	characteristic
178	C₄	C₂×#102	1	1	yes	characteristic
179	C₄	#448	1	1	yes	characteristic
180	C₄	#480	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	—	4	2	no	—
184	C₂	—	16	2	no	—
185	C₂	#1874	1	1	yes	characteristic
186	C₂	#2192	1	1	yes	characteristic
187	C₂	#2193	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#447
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ 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. #2913	C₂	semidirect
3. #2140	C₂	semidirect
4. C₂×#357	C₂	semidirect
5. C₂×#350	C₂	semidirect
6. #1387	C₂	semidirect
7. #1382	C₂	semidirect
15. #472	C₄	semidirect
39. C₂×D₈	C₂×C₄	semidirect
63. #108	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#472	2
#2913	2
#2140	2
C₂×#357	2
C₂×#350	2
#1387	2
#1382	2