Finite Group Databasebeta

Subgroups — #29385

Order 256.

331 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 101 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	#2956	C₂	1	1	yes	maximal, characteristic, complemented
2	#2921	C₂	1	1	yes	maximal, characteristic, complemented
3	#2102	C₂	1	1	yes	maximal, characteristic, complemented
4	#2091	C₂	1	1	yes	maximal, characteristic, complemented
5	#2050	C₂	1	1	yes	maximal, characteristic, complemented
6	#2010	C₂	1	1	yes	maximal, characteristic
7	#1932	C₂	1	2	yes	maximal, complemented
8	#1897	C₂	1	2	yes	maximal, complemented
9	#1891	C₂	1	2	yes	maximal, complemented
10	#1861	C₂	1	2	yes	maximal, complemented
11	#1841	C₂	1	1	yes	maximal, characteristic, complemented
12	#575	C₂²	1	1	yes	characteristic
13	#568	C₂²	1	2	yes	—
14	#525	C₂²	1	2	yes	—
15	C₂×#117	C₂²	1	1	yes	characteristic
16	#444	—	2	2	no	—
17	#443	—	2	2	no	—
18	#433	—	2	2	no	—
19	#433	—	2	2	no	—
20	#430	C₂²	1	1	yes	characteristic
21	#430	C₂²	1	2	yes	—
22	#423	C₂²	1	1	yes	characteristic
23	#423	C₂²	1	1	yes	characteristic
24	#421	C₂²	1	2	yes	—
25	C₂×#104	C₂²	1	2	yes	—
26	C₂×#104	C₂²	1	2	yes	—
27	#413	C₂²	1	2	yes	—
28	#407	C₂²	1	1	yes	characteristic
29	#407	C₂²	1	1	yes	characteristic
30	#407	C₂²	1	2	yes	—
31	C₄×M₁₆	C₂²	1	1	yes	characteristic
32	C₄×M₁₆	C₂²	1	1	yes	characteristic
33	#344	C₂²	1	2	yes	—
34	#344	C₂²	1	2	yes	—
35	#339	C₂²	1	2	yes	—
36	#339	C₂²	1	2	yes	—
37	#338	C₂²	1	1	yes	characteristic
38	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
39	C₂×#41	C₂³	1	2	yes	—
40	C₂³×C₄	C₂³	1	1	yes	characteristic
41	#137	—	2	2	no	—
42	AGL(1,ℤ/8)	—	2	2	no	—
43	#135	—	2	2	no	—
44	#131	—	2	4	no	—
45	#131	—	4	2	no	—
46	C₂×M₁₆	—	4	2	no	—
47	C₂×M₁₆	C₂³	1	2	yes	—
48	C₂×M₁₆	C₂³	1	2	yes	—
49	C₂×M₁₆	C₂³	1	1	yes	characteristic
50	C₂×M₁₆	C₂³	1	1	yes	characteristic
51	C₂×M₁₆	C₂³	1	1	yes	characteristic
52	C₂×M₁₆	C₂³	1	1	yes	characteristic
53	#123	—	2	2	no	—
54	C₂²⋊Q₈	—	2	2	no	—
55	#121	—	2	2	no	—
56	#120	—	2	2	no	—
57	C₄×D₄	—	4	2	no	—
58	#117	—	2	1	no	—
59	#117	—	2	1	no	—
60	#117	—	2	2	no	—
61	#117	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄	—	2	1	no	—
63	C₂×C₂²⋊C₄	—	2	1	no	—
64	C₂×C₄²	C₂³	1	1	yes	characteristic
65	C₂×C₄²	C₂³	1	1	yes	characteristic
66	#108	—	2	2	no	—
67	#108	—	4	2	no	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	2	2	no	—
70	#104	—	2	2	no	—
71	#104	—	2	2	no	—
72	#104	—	2	4	no	—
73	#104	—	4	2	no	—
74	#101	—	2	2	no	—
75	#100	—	2	2	no	—
76	C₂²⋊C₈	—	2	2	no	—
77	C₂²⋊C₈	—	2	2	no	—
78	C₂²⋊C₈	—	4	2	no	—
79	#97	—	2	1	no	—
80	#97	D₄	1	2	yes	complemented
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	D₄	1	2	yes	complemented
83	C₂⁴	#41	1	1	yes	characteristic
84	#41	—	2	2	no	—
85	#41	—	2	2	no	—
86	#41	—	2	2	no	—
87	#41	—	2	2	no	—
88	C₂×Q₈	#41	1	2	yes	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	4	2	no	—
91	C₂×D₄	#41	1	2	yes	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	#41	1	1	yes	characteristic
100	C₂²×C₄	C₂⁴	1	1	yes	characteristic
101	Q₁₆	—	2	2	no	—
102	SD₁₆	—	2	4	no	—
103	D₈	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	4	4	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	4	no	—
118	C₂×C₈	C₂×D₄	1	2	yes	—
119	C₂×C₈	C₂×D₄	1	2	yes	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	4	2	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	2	no	—
132	C₄²	C₂×D₄	1	1	yes	characteristic
133	C₄²	C₂×D₄	1	1	yes	characteristic
134	C₄²	#41	1	2	yes	—
135	C₂³	—	2	1	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	8	1	no	—
139	C₂³	C₂×#41	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	Q₈	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	2	2	no	—
144	D₄	—	4	2	no	—
145	D₄	—	4	2	no	—
146	D₄	—	8	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄	C₂×#41	1	2	yes	—
163	C₂×C₄	C₂×#41	1	1	yes	characteristic
164	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	4	1	no	—
168	C₈	—	4	2	no	—
169	C₈	—	4	1	no	—
170	C₈	—	4	2	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	8	2	no	—
178	C₂²	#525	1	1	yes	characteristic
179	C₂²	#551	1	2	yes	—
180	C₄	—	4	1	no	—
181	C₄	—	4	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	8	2	no	—
186	C₄	#525	1	1	yes	characteristic
187	C₄	#535	1	1	yes	characteristic
188	C₄	#553	1	2	yes	—
189	C₂	—	2	1	no	—
190	C₂	—	2	2	no	—
191	C₂	—	8	1	no	—
192	C₂	—	8	2	no	—
193	C₂	#2598	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2598
prime radical	#2010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2956	C₂	semidirect
2. #2921	C₂	semidirect
3. #2102	C₂	semidirect
4. #2091	C₂	semidirect
5. #2050	C₂	semidirect
7. #1932	C₂	semidirect
8. #1897	C₂	semidirect
9. #1891	C₂	semidirect
10. #1861	C₂	semidirect
11. #1841	C₂	semidirect
80. #97	D₄	semidirect
82. C₄×C₈	D₄	semidirect

Maximal subgroups

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

subgroup	index
#2956	2
#2921	2
#2102	2
#2091	2
#2050	2
#2010	2
#1932	2
#1897	2
#1891	2
#1861	2
#1841	2