Finite Group Databasebeta

Subgroups — #27987

Order 256.

319 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 109 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	#3057	C₂	1	1	yes	maximal, characteristic
2	#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	#2082	C₂	1	2	yes	maximal, complemented
4	#2081	C₂	1	2	yes	maximal, complemented
5	#1977	C₂	1	2	yes	maximal, complemented
6	#1915	C₂	1	1	yes	maximal, characteristic, complemented
7	#1914	C₂	1	1	yes	maximal, characteristic
8	#1910	C₂	1	1	yes	maximal, characteristic
9	#1909	C₂	1	1	yes	maximal, characteristic, complemented
10	#1833	C₂	1	2	yes	maximal, complemented
11	#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆	C₂²	1	1	yes	characteristic
13	#568	C₂²	1	1	yes	characteristic
14	#568	C₂²	1	1	yes	characteristic
15	C₂×#131	C₂²	1	1	yes	characteristic
16	#536	C₂²	1	1	yes	characteristic
17	#535	C₂²	1	1	yes	characteristic
18	#468	—	2	2	no	—
19	#465	—	2	2	no	—
20	#430	—	2	2	no	—
21	#430	C₂²	1	2	yes	—
22	C₂×#108	—	2	2	no	—
23	C₂×#108	C₂²	1	2	yes	—
24	#427	C₂²	1	1	yes	characteristic
25	#419	C₂²	1	1	yes	characteristic
26	#419	C₂²	1	2	yes	—
27	#418	C₂²	1	1	yes	characteristic
28	#418	C₂²	1	2	yes	—
29	#413	C₂²	1	1	yes	characteristic
30	C₂×#101	C₂²	1	2	yes	—
31	C₂×#100	C₂²	1	2	yes	—
32	#410	C₂²	1	1	yes	characteristic
33	#405	C₂²	1	2	yes	—
34	#344	C₂²	1	2	yes	—
35	#344	C₂²	1	2	yes	—
36	#344	C₂²	1	2	yes	—
37	H(3,ℤ/4)	C₂²	1	2	yes	—
38	H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#41	C₂³	1	1	yes	characteristic
40	C₂²×Q₈	C₂³	1	1	yes	characteristic
41	C₂²×D₄	C₂³	1	1	yes	characteristic
42	C₂×SD₁₆	—	2	1	no	—
43	C₂×SD₁₆	—	2	1	no	—
44	C₂×SD₁₆	—	2	2	no	—
45	C₂×SD₁₆	—	2	2	no	—
46	#131	—	2	2	no	—
47	#131	—	2	2	no	—
48	#131	—	4	2	no	—
49	#131	—	4	2	no	—
50	C₂×M₁₆	—	2	2	no	—
51	C₂×M₁₆	—	2	2	no	—
52	C₂×M₁₆	D₄	1	2	yes	—
53	C₂×M₁₆	C₂³	1	2	yes	—
54	C₂×M₁₆	C₂³	1	2	yes	—
55	C₂×M₁₆	C₂³	1	2	yes	—
56	C₂×M₁₆	C₂³	1	1	yes	characteristic
57	C₂²×C₈	D₄	1	2	yes	—
58	C₂²×C₈	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈	—	2	1	no	—
60	#127	—	2	1	no	—
61	#124	—	2	1	no	—
62	#124	—	2	1	no	—
63	#123	—	2	2	no	—
64	C₂²⋊Q₈	—	2	2	no	—
65	#121	—	2	2	no	—
66	#120	—	2	2	no	—
67	#117	C₂³	1	1	yes	characteristic
68	#117	C₂³	1	1	yes	characteristic
69	#117	C₂³	1	2	yes	—
70	#108	—	2	2	no	—
71	#108	—	2	4	no	—
72	#108	—	2	2	no	—
73	#107	—	2	1	no	—
74	#106	—	2	1	no	—
75	#103	—	2	2	no	—
76	#103	—	4	2	no	—
77	#102	—	2	2	no	—
78	#102	—	4	2	no	—
79	#101	—	2	1	no	—
80	#101	—	4	2	no	—
81	#100	—	2	1	no	—
82	#100	—	4	2	no	—
83	#99	—	2	2	no	—
84	#97	—	2	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₂⁴	—	2	1	no	—
87	#41	—	2	2	no	—
88	#41	—	2	2	no	—
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	#41	1	1	yes	characteristic
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	2	no	—
97	C₂×D₄	—	2	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	#41	1	2	yes	—
100	C₂×D₄	#41	1	1	yes	characteristic
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	C₂×D₄	1	2	yes	—
103	C₂²×C₄	C₂⁴	1	1	yes	characteristic
104	SD₁₆	—	4	1	no	—
105	SD₁₆	—	4	1	no	—
106	SD₁₆	—	4	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	4	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	4	2	no	—
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	C₂×D₄	1	2	yes	—
125	C₂×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈	#41	1	2	yes	—
128	C₂×C₈	#41	1	2	yes	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	2	1	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₂²⋊C₄	—	4	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₂³	—	4	1	no	—
142	C₂³	—	4	2	no	—
143	C₂³	—	16	1	no	—
144	C₂³	#124	1	2	yes	—
145	C₂³	C₂×#41	1	1	yes	characteristic
146	Q₈	—	2	2	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	4	2	no	—
150	D₄	—	2	4	no	—
151	D₄	—	2	2	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	D₄	—	4	2	no	—
155	D₄	—	16	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	#124	1	2	yes	—
164	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄	C₂×#41	1	2	yes	—
166	C₂×C₄	C₂×#41	1	1	yes	characteristic
167	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	—	4	2	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	2	no	—
172	C₈	#123	1	1	yes	characteristic
173	C₈	#123	1	1	yes	characteristic
174	C₈	#123	1	2	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	8	2	no	—
179	C₂²	—	8	1	no	—
180	C₂²	—	16	2	no	—
181	C₂²	C₂×#124	1	1	yes	characteristic
182	C₂²	#553	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	8	1	no	—
185	C₄	—	8	2	no	—
186	C₄	—	8	1	no	—
187	C₄	—	8	1	no	—
188	C₄	C₂×#123	1	1	yes	characteristic
189	C₄	#535	1	1	yes	characteristic
190	C₄	#553	1	2	yes	—
191	C₂	—	2	1	no	—
192	C₂	—	2	2	no	—
193	C₂	—	4	2	no	—
194	C₂	—	16	1	no	—
195	C₂	#2504	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2504
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ 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. #2082	C₂	semidirect
4. #2081	C₂	semidirect
5. #1977	C₂	semidirect
6. #1915	C₂	semidirect
9. #1909	C₂	semidirect
10. #1833	C₂	semidirect
11. #1757	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3057	2
#2916	2
#2082	2
#2081	2
#1977	2
#1915	2
#1914	2
#1910	2
#1909	2
#1833	2
#1757	2