Finite Group Databasebeta

Subgroups — #12943

Order 256.

211 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 69 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	#2943	C₂	1	1	yes	maximal, characteristic
2	#2159	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#360	C₂	1	1	yes	maximal, characteristic
4	#2139	C₂	1	1	yes	maximal, characteristic
5	#1944	C₂	1	1	yes	maximal, characteristic, complemented
6	#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	#1377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	C₂²	1	1	yes	characteristic
9	C₄×#41	—	2	1	no	—
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	#442	—	2	1	no	—
13	#441	—	2	1	no	—
14	#440	—	2	1	no	—
15	#440	—	2	1	no	—
16	#427	C₂²	1	1	yes	characteristic
17	C₂×#107	C₂²	1	1	yes	characteristic
18	C₂×#106	C₂²	1	1	yes	characteristic
19	#416	—	2	1	no	—
20	C₂×#97	C₂²	1	1	yes	characteristic
21	#389	—	2	1	no	—
22	#367	—	2	1	no	—
23	#366	—	2	1	no	—
24	#360	C₄	1	4	yes	complemented
25	#358	—	2	1	no	—
26	#357	—	2	1	no	—
27	#340	—	2	1	no	—
28	C₂×#41	—	2	1	no	—
29	#135	—	2	1	no	—
30	#135	—	2	1	no	—
31	#135	C₂×C₄	1	2	yes	—
32	#135	Q₈	1	2	yes	—
33	C₂×Q₁₆	D₄	1	1	yes	characteristic
34	C₂×SD₁₆	—	2	1	no	—
35	C₂×D₈	D₄	1	1	yes	characteristic
36	C₂²×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₄×Q₈	—	2	1	no	—
39	C₄×Q₈	—	2	1	no	—
40	C₄×D₄	—	2	1	no	—
41	C₄×D₄	—	2	1	no	—
42	C₄×D₄	—	4	1	no	—
43	C₄×D₄	—	4	1	no	—
44	#117	—	2	1	no	—
45	#117	—	4	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄⋊C₄	—	2	1	no	—
48	C₂×C₄⋊C₄	—	4	1	no	—
49	C₂×C₄²	—	4	1	no	—
50	C₂×C₄²	D₄	1	1	yes	characteristic
51	M₃₂	—	2	1	no	—
52	M₃₂	—	2	1	no	—
53	M₃₂	—	2	1	no	—
54	M₃₂	C₂×C₄	1	2	yes	complemented
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	2	1	no	—
57	#107	—	2	1	no	—
58	#107	—	2	1	no	—
59	#107	D₄	1	1	yes	characteristic
60	#107	D₄	1	1	yes	characteristic
61	#106	—	2	1	no	—
62	#106	—	2	1	no	—
63	#106	C₂×C₄	1	2	yes	complemented
64	#103	—	2	1	no	—
65	#103	—	2	1	no	—
66	#102	—	2	1	no	—
67	#102	—	2	1	no	—
68	#97	—	2	1	no	—
69	#97	—	2	1	no	—
70	#95	—	4	1	no	—
71	#41	—	2	2	no	—
72	#41	—	2	2	no	—
73	#41	—	4	1	no	—
74	#41	—	4	1	no	—
75	C₂×Q₈	—	2	1	no	—
76	C₂×D₄	—	2	1	no	—
77	C₂×D₄	—	4	1	no	—
78	C₂²×C₄	—	4	1	no	—
79	C₂²×C₄	—	4	1	no	—
80	C₂²×C₄	—	4	1	no	—
81	C₂²×C₄	—	4	1	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	D₈	1	1	yes	characteristic
84	C₂²×C₄	D₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	Q₁₆	—	2	1	no	—
87	Q₁₆	C₄⋊C₄	1	2	yes	complemented
88	SD₁₆	—	2	2	no	—
89	SD₁₆	—	2	2	no	—
90	D₈	—	2	1	no	—
91	D₈	C₄⋊C₄	1	2	yes	complemented
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₈	#41	1	1	yes	characteristic
100	C₂×C₈	#41	1	1	yes	characteristic
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₂²⋊C₄	—	4	1	no	—
111	C₂²⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	4	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₁₆	—	2	2	no	—
119	C₁₆	—	4	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	C₂×D₈	1	1	yes	characteristic
122	Q₈	—	2	2	no	—
123	Q₈	—	4	1	no	—
124	D₄	—	2	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	4	2	no	—
127	D₄	—	4	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄	#120	1	1	yes	characteristic
147	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₄	#135	1	1	yes	characteristic
149	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄	#137	1	1	yes	characteristic
151	C₈	—	4	1	no	—
152	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈	C₄×D₄	1	1	yes	characteristic
154	C₈	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈	#123	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	8	1	no	—
160	C₂²	—	8	1	no	—
161	C₂²	C₄×D₈	1	1	yes	characteristic
162	C₂²	#447	1	1	yes	characteristic
163	C₂²	#474	1	1	yes	characteristic
164	C₄	—	4	2	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	8	2	no	—
168	C₄	—	8	1	no	—
169	C₄	—	8	1	no	—
170	C₄	—	16	1	no	—
171	C₄	#385	1	1	yes	characteristic
172	C₄	#440	1	1	yes	characteristic
173	C₄	#452	1	1	yes	characteristic
174	C₄	#478	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	2	no	—
178	C₂	#1862	1	1	yes	characteristic
179	C₂	#2190	1	1	yes	characteristic
180	C₂	#2191	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#447
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ 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. #2159	C₂	semidirect
5. #1944	C₂	semidirect
6. #1386	C₂	semidirect
7. #1377	C₂	semidirect
24. #360	C₄	semidirect
54. M₃₂	C₂×C₄	semidirect
63. #106	C₂×C₄	semidirect
87. Q₁₆	C₄⋊C₄	semidirect
91. D₈	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#2943	2
#2159	2
C₂×#360	2
#2139	2
#1944	2
#1386	2
#1377	2