Finite Group Databasebeta

Subgroups — #21710

Order 256.

507 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 187 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	#3150	C₂	1	1	yes	maximal, characteristic
2	C₂×#444	C₂	1	2	yes	maximal, complemented
3	C₂×#443	C₂	1	2	yes	maximal, complemented
4	#2932	C₂	1	2	yes	maximal, complemented
5	#1975	C₂	1	2	yes	maximal, complemented
6	#1974	C₂	1	2	yes	maximal, complemented
7	#1962	C₂	1	2	yes	maximal, complemented
8	#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135	C₂²	1	2	yes	—
11	C₂×#135	C₂²	1	2	yes	—
12	C₂×#131	C₂²	1	4	yes	—
13	#532	C₂²	1	2	yes	—
14	#497	C₄	1	4	yes	complemented
15	#496	C₄	1	4	yes	complemented
16	#444	—	2	8	no	—
17	#443	—	2	8	no	—
18	#435	—	2	8	no	—
19	C₂×#108	C₂²	1	2	yes	—
20	C₂×#108	C₂²	1	2	yes	—
21	#423	C₂²	1	2	yes	—
22	C₂×#104	C₂²	1	4	yes	—
23	C₂×#104	C₂²	1	4	yes	—
24	#408	C₂²	1	4	yes	—
25	C₂×#97	C₂²	1	1	yes	characteristic
26	C₂×#97	C₂²	1	1	yes	characteristic
27	C₂×#97	C₂²	1	1	yes	characteristic
28	C₂×#97	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₄³	C₂²	1	1	yes	characteristic
32	C₂×#41	C₂³	1	4	yes	—
33	#135	—	2	4	no	—
34	#135	—	2	4	no	—
35	#135	—	2	4	no	—
36	#135	—	2	4	no	—
37	C₂×Q₁₆	C₂×C₄	1	2	yes	—
38	C₂×Q₁₆	C₂×C₄	1	2	yes	—
39	C₂×SD₁₆	C₂×C₄	1	4	yes	—
40	C₂×SD₁₆	C₂×C₄	1	4	yes	—
41	C₂×D₈	C₂×C₄	1	2	yes	—
42	C₂×D₈	C₂×C₄	1	2	yes	—
43	#131	—	2	8	no	—
44	#131	—	2	8	no	—
45	C₂×M₁₆	—	2	4	no	—
46	C₂×M₁₆	C₂³	1	4	yes	—
47	C₂²×C₈	—	2	4	no	—
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₈	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈	C₂×C₄	1	2	yes	—
53	#127	C₂×C₄	1	2	yes	—
54	#124	C₂×C₄	1	4	yes	—
55	#121	—	2	4	no	—
56	C₄×D₄	—	2	4	no	—
57	C₂×C₄²	—	2	2	no	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	#108	—	2	4	no	—
62	#108	—	2	4	no	—
63	C₄⋊C₈	—	2	4	no	—
64	#104	—	2	8	no	—
65	#104	—	2	8	no	—
66	C₂²⋊C₈	—	2	8	no	—
67	#97	—	2	2	no	—
68	#97	C₂×C₄	1	4	yes	—
69	#97	D₄	1	4	yes	—
70	#97	D₄	1	4	yes	complemented
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	D₄	1	4	yes	complemented
73	#41	—	2	4	no	—
74	#41	—	2	4	no	—
75	#41	—	2	4	no	—
76	#41	—	2	4	no	—
77	C₂×Q₈	C₂²×C₄	1	4	yes	—
78	C₂×D₄	—	2	4	no	—
79	C₂×D₄	—	2	4	no	—
80	C₂×D₄	C₂²×C₄	1	4	yes	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	4	no	—
84	C₂²×C₄	#41	1	2	yes	—
85	C₂²×C₄	C₂⁴	1	1	yes	characteristic
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	2	4	no	—
88	SD₁₆	—	2	8	no	—
89	SD₁₆	—	2	8	no	—
90	D₈	—	2	4	no	—
91	D₈	—	2	4	no	—
92	M₁₆	—	2	4	no	—
93	M₁₆	—	2	4	no	—
94	M₁₆	—	4	8	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	4	8	no	—
99	C₂×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₈	C₂×D₄	1	2	yes	—
103	C₂×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₈	C₂×D₄	1	2	yes	—
106	C₂×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₈	#41	1	2	yes	—
108	C₂×C₈	#41	1	2	yes	—
109	C₄⋊C₄	—	2	4	no	—
110	C₂²⋊C₄	—	2	8	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	4	4	no	—
114	C₄²	C₂²×C₄	1	2	yes	—
115	C₄²	C₂×D₄	1	2	yes	—
116	C₄²	C₂×D₄	1	2	yes	—
117	C₄²	C₂×D₄	1	2	yes	—
118	C₂³	—	2	4	no	—
119	C₂³	C₂×#41	1	1	yes	characteristic
120	Q₈	—	2	4	no	—
121	Q₈	—	2	4	no	—
122	D₄	—	2	4	no	—
123	D₄	—	2	4	no	—
124	D₄	—	4	4	no	—
125	D₄	—	4	4	no	—
126	D₄	—	8	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	C₄×D₄	1	4	yes	—
136	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄	C₂×#41	1	1	yes	characteristic
141	C₂×C₄	C₂×#41	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	4	4	no	—
144	C₈	C₄×D₄	1	4	yes	—
145	C₈	C₄×D₄	1	4	yes	—
146	C₈	#127	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	8	no	—
150	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂²	#532	1	1	yes	characteristic
152	C₂²	#532	1	1	yes	characteristic
153	C₄	—	2	4	no	—
154	C₄	—	2	4	no	—
155	C₄	—	4	2	no	—
156	C₄	—	4	4	no	—
157	C₄	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄	C₂×#127	1	1	yes	characteristic
160	C₄	#532	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	8	4	no	—
164	C₂	C₄×#127	1	1	yes	characteristic
165	C₂	#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#444	C₂	semidirect
3. C₂×#443	C₂	semidirect
4. #2932	C₂	semidirect
5. #1975	C₂	semidirect
6. #1974	C₂	semidirect
7. #1962	C₂	semidirect
8. #1849	C₂	semidirect
9. C₄×#97	C₂	semidirect
14. #497	C₄	semidirect
15. #496	C₄	semidirect
70. #97	D₄	semidirect
72. 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
#3150	2
C₂×#444	2
C₂×#443	2
#2932	2
#1975	2
#1974	2
#1962	2
#1849	2
C₄×#97	2