Finite Group Databasebeta

Subgroups — #12806

Order 256.

210 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 63 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	#2930	C₂	1	1	yes	maximal, characteristic
2	#2120	C₂	1	2	yes	maximal
3	#1813	C₂	1	1	yes	maximal, characteristic, complemented
4	#1812	C₂	1	1	yes	maximal, characteristic, complemented
5	#1395	C₂	1	2	yes	maximal, complemented
6	C₂×#131	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆	C₄	1	1	yes	characteristic, complemented
8	#525	C₂²	1	1	yes	characteristic
9	#436	—	2	1	no	—
10	#436	—	2	1	no	—
11	#435	—	2	1	no	—
12	#435	—	2	1	no	—
13	C₂×#108	—	2	1	no	—
14	#427	—	2	1	no	—
15	#424	C₄	1	1	yes	characteristic, complemented
16	#413	C₂²	1	2	yes	—
17	#410	C₂²	1	2	yes	—
18	#408	C₂²	1	1	yes	characteristic
19	#407	C₄	1	1	yes	characteristic, complemented
20	#405	C₄	1	1	yes	characteristic, complemented
21	#352	—	2	2	no	—
22	AΣL(1,𝔽₁₆)	—	2	2	no	—
23	#338	—	2	1	no	—
24	H(3,ℤ/4)	—	2	1	no	—
25	C₂×#41	C₂³	1	1	yes	characteristic
26	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
27	#131	—	4	1	no	—
28	#131	—	4	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	4	1	no	—
33	C₂×M₁₆	—	4	1	no	—
34	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×M₁₆	D₄	1	1	yes	characteristic
37	C₂×M₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₈	D₄	1	1	yes	characteristic
42	#123	D₄	1	1	yes	characteristic
43	#123	Q₈	1	1	yes	characteristic
44	C₂²⋊Q₈	—	2	1	no	—
45	#121	—	2	1	no	—
46	#120	D₄	1	1	yes	characteristic
47	#120	Q₈	1	1	yes	characteristic
48	C₄×D₄	—	2	1	no	—
49	C₄×D₄	—	2	1	no	—
50	#117	—	2	2	no	—
51	#117	C₂×C₄	1	1	yes	characteristic
52	#108	—	4	1	no	—
53	#108	—	4	1	no	—
54	#107	—	4	1	no	—
55	#106	—	4	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	2	1	no	—
58	#101	—	2	2	no	—
59	#100	—	2	2	no	—
60	#99	—	2	2	no	—
61	#99	—	2	2	no	—
62	C₂²⋊C₈	—	2	1	no	—
63	C₂²⋊C₈	—	2	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	2	1	no	—
66	#97	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
69	#41	—	4	1	no	—
70	#41	—	4	1	no	—
71	C₂×Q₈	#41	1	1	yes	characteristic
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	C₂×D₄	1	1	yes	characteristic
75	C₂×D₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×D₄	#41	1	1	yes	characteristic
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	4	1	no	—
80	C₂²×C₄	—	4	1	no	—
81	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	M₁₆	—	4	1	no	—
86	M₁₆	—	4	1	no	—
87	M₁₆	—	4	1	no	—
88	M₁₆	—	4	2	no	—
89	M₁₆	—	4	1	no	—
90	M₁₆	—	4	1	no	—
91	M₁₆	—	4	1	no	—
92	M₁₆	—	4	2	no	—
93	M₁₆	—	4	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₂²⋊C₄	—	2	1	no	—
113	C₂²⋊C₄	—	2	1	no	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	2	1	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	C₄⋊C₄	1	1	yes	characteristic
118	C₂²⋊C₄	C₄⋊C₄	1	1	yes	characteristic
119	C₄²	—	2	1	no	—
120	C₄²	—	4	2	no	—
121	C₂³	—	2	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	8	1	no	—
124	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂³	#121	1	1	yes	characteristic
126	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
127	Q₈	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	D₄	—	8	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#117	1	1	yes	characteristic
149	C₂×C₄	#121	1	1	yes	characteristic
150	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	4	1	no	—
155	C₈	—	4	1	no	—
156	C₈	—	8	2	no	—
157	C₈	#99	1	1	yes	characteristic
158	C₈	#99	1	1	yes	characteristic
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	8	2	no	—
169	C₂²	#380	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	16	2	no	—
177	C₄	C₂×#99	1	1	yes	characteristic
178	C₄	#410	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	4	1	no	—
181	C₂	—	4	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	8	1	no	—
184	C₂	#1799	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#410
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×#41	C₂³
socle	C₂	#1799
prime radical	#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1813	C₂	semidirect
4. #1812	C₂	semidirect
5. #1395	C₂	semidirect
7. C₂²×M₁₆	C₄	semidirect
15. #424	C₄	semidirect
19. #407	C₄	semidirect
20. #405	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2930	2
#2120	2
#1813	2
#1812	2
#1395	2