Finite Group Databasebeta

Subgroups — #27694

Order 256.

313 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 117 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	#2932	C₂	1	1	yes	maximal, characteristic
2	#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	#2102	C₂	1	2	yes	maximal, complemented
4	#2098	C₂	1	2	yes	maximal, complemented
5	#1946	C₂	1	2	yes	maximal, complemented
6	#1884	C₂	1	2	yes	maximal
7	#1868	C₂	1	2	yes	maximal, complemented
8	#1814	C₂	1	2	yes	maximal, complemented
9	#1812	C₂	1	1	yes	maximal, characteristic, complemented
10	#568	C₂²	1	2	yes	—
11	C₂×#131	C₂²	1	1	yes	characteristic
12	C₂×#131	C₂²	1	1	yes	characteristic
13	#532	C₂²	1	1	yes	characteristic
14	#435	—	2	2	no	—
15	#435	—	2	2	no	—
16	#430	—	2	2	no	—
17	#430	C₂²	1	2	yes	—
18	C₂×#108	—	2	2	no	—
19	C₂×#108	C₂²	1	2	yes	—
20	#427	C₂²	1	2	yes	—
21	#424	C₂²	1	4	yes	—
22	#423	C₂²	1	1	yes	characteristic
23	#421	C₂²	1	4	yes	—
24	#410	C₂²	1	2	yes	—
25	#408	C₂²	1	1	yes	characteristic
26	#408	C₂²	1	1	yes	characteristic
27	C₂×#97	C₂²	1	1	yes	characteristic
28	#344	C₂²	1	4	yes	—
29	#339	C₂²	1	4	yes	—
30	H(3,ℤ/4)	C₂²	1	2	yes	—
31	C₂×#41	C₂³	1	1	yes	characteristic
32	C₂×#41	C₂³	1	1	yes	characteristic
33	#131	—	2	2	no	—
34	#131	—	2	2	no	—
35	#131	—	2	2	no	—
36	#131	—	2	2	no	—
37	#131	—	4	4	no	—
38	C₂×M₁₆	—	2	1	no	—
39	C₂×M₁₆	—	2	4	no	—
40	C₂×M₁₆	D₄	1	2	yes	—
41	C₂×M₁₆	C₂³	1	1	yes	characteristic
42	C₂×M₁₆	C₂³	1	4	yes	—
43	C₂×M₁₆	C₂³	1	1	yes	characteristic
44	C₂²×C₈	—	2	1	no	—
45	C₂²×C₈	D₄	1	2	yes	—
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈	Q₈	1	1	yes	characteristic
49	#127	Q₈	1	1	yes	characteristic
50	#124	Q₈	1	1	yes	characteristic
51	#124	Q₈	1	1	yes	characteristic
52	#121	—	2	2	no	—
53	C₄×D₄	—	2	2	no	—
54	#117	C₂³	1	4	yes	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	#108	—	2	2	no	—
57	#108	—	2	4	no	—
58	#108	—	2	2	no	—
59	#107	—	2	2	no	—
60	#106	—	2	2	no	—
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	—	4	4	no	—
63	#104	—	4	4	no	—
64	#99	—	2	4	no	—
65	C₂²⋊C₈	—	2	2	no	—
66	C₂²⋊C₈	—	2	2	no	—
67	#97	—	2	1	no	—
68	#97	—	2	1	no	—
69	#41	—	2	2	no	—
70	#41	—	2	2	no	—
71	#41	—	2	2	no	—
72	#41	—	2	2	no	—
73	C₂×Q₈	C₂×Q₈	1	1	yes	characteristic
74	C₂×Q₈	C₂×Q₈	1	1	yes	characteristic
75	C₂×D₄	—	2	1	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	C₂×Q₈	1	1	yes	characteristic
78	C₂×D₄	C₂×Q₈	1	1	yes	characteristic
79	C₂×D₄	#41	1	2	yes	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	C₂×D₄	1	2	yes	—
83	C₂²×C₄	C₂⁴	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	2	4	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	4	no	—
89	M₁₆	—	4	4	no	—
90	M₁₆	—	4	1	no	—
91	M₁₆	—	4	1	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	4	4	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	C₂×D₄	1	2	yes	—
101	C₂×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈	#41	1	2	yes	—
104	C₂×C₈	#41	1	2	yes	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	2	4	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	4	no	—
112	C₄²	C₂×Q₈	1	1	yes	characteristic
113	C₄²	C₂×Q₈	1	1	yes	characteristic
114	C₂³	—	2	1	no	—
115	C₂³	C₂²⋊Q₈	1	2	yes	—
116	C₂³	C₂×#41	1	1	yes	characteristic
117	Q₈	—	2	2	no	—
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	D₄	—	2	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	8	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄	C₂×#41	1	2	yes	—
137	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	4	1	no	—
140	C₈	—	4	1	no	—
141	C₈	—	4	4	no	—
142	C₈	#123	1	1	yes	characteristic
143	C₈	#123	1	1	yes	characteristic
144	C₈	#123	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₂²	#553	1	2	yes	—
152	C₄	—	2	2	no	—
153	C₄	—	4	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	8	4	no	—
157	C₄	C₂×#123	1	1	yes	characteristic
158	C₄	#535	1	1	yes	characteristic
159	C₄	#554	1	2	yes	—
160	C₂	—	2	1	no	—
161	C₂	—	2	2	no	—
162	C₂	—	4	2	no	—
163	C₂	—	8	1	no	—
164	C₂	#2487	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2487
prime radical	#532	C₂²
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
2. #2916	C₂	semidirect
3. #2102	C₂	semidirect
4. #2098	C₂	semidirect
5. #1946	C₂	semidirect
7. #1868	C₂	semidirect
8. #1814	C₂	semidirect
9. #1812	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2932	2
#2916	2
#2102	2
#2098	2
#1946	2
#1884	2
#1868	2
#1814	2
#1812	2