Finite Group Databasebeta

Subgroups — #27833

Order 256.

221 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 97 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	#2103	C₂	1	2	yes	maximal, complemented
2	#2080	C₂	1	2	yes	maximal
3	#1953	C₂	1	1	yes	maximal, characteristic, complemented
4	#1945	C₂	1	1	yes	maximal, characteristic, complemented
5	#1932	C₂	1	2	yes	maximal
6	#1841	C₂	1	2	yes	maximal, complemented
7	#1833	C₂	1	1	yes	maximal, characteristic, complemented
8	#1814	C₂	1	1	yes	maximal, characteristic
9	#1813	C₂	1	1	yes	maximal, characteristic
10	C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
11	#1764	C₂	1	1	yes	maximal, characteristic
12	C₂×#131	C₂²	1	1	yes	characteristic
13	#430	C₂²	1	2	yes	—
14	C₂×#108	C₂²	1	1	yes	characteristic
15	C₂×#108	C₂²	1	1	yes	characteristic
16	C₂×#108	C₂²	1	1	yes	characteristic
17	#427	C₂²	1	1	yes	characteristic
18	#424	C₂²	1	2	yes	—
19	#421	C₂²	1	2	yes	—
20	C₂×#104	C₂²	1	1	yes	characteristic
21	#416	C₂²	1	1	yes	characteristic
22	#413	C₂²	1	2	yes	—
23	#405	C₂²	1	1	yes	characteristic
24	C₄×M₁₆	C₂²	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	#344	C₂²	1	2	yes	—
27	#344	C₂²	1	2	yes	—
28	#344	C₂²	1	2	yes	—
29	#341	C₂²	1	2	yes	—
30	#341	C₂²	1	1	yes	characteristic
31	#339	C₂²	1	1	yes	characteristic
32	#339	C₂²	1	2	yes	—
33	#338	C₂²	1	2	yes	—
34	H(3,ℤ/4)	C₂²	1	2	yes	—
35	C₂×#41	C₂³	1	1	yes	characteristic
36	#131	—	2	1	no	—
37	#131	—	2	1	no	—
38	#131	—	2	1	no	—
39	#131	—	2	1	no	—
40	C₂×M₁₆	—	2	1	no	—
41	C₂×M₁₆	C₂³	1	1	yes	characteristic
42	C₂×M₁₆	C₂³	1	2	yes	—
43	C₂×M₁₆	C₂³	1	2	yes	—
44	C₂×M₁₆	C₂³	1	2	yes	—
45	C₂×M₁₆	C₂³	1	1	yes	characteristic
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	#117	C₂³	1	1	yes	characteristic
50	#117	C₂³	1	2	yes	—
51	C₂×C₄²	C₂³	1	1	yes	characteristic
52	#108	—	2	1	no	—
53	#108	—	2	1	no	—
54	#108	—	2	1	no	—
55	#108	—	2	1	no	—
56	#108	—	2	1	no	—
57	#108	—	2	1	no	—
58	#108	—	4	2	no	—
59	#107	—	2	1	no	—
60	#106	—	2	1	no	—
61	C₄⋊C₈	—	4	2	no	—
62	#104	—	2	1	no	—
63	#104	—	2	1	no	—
64	#104	—	4	2	no	—
65	#103	—	2	1	no	—
66	#102	—	2	1	no	—
67	#101	—	2	2	no	—
68	#100	—	2	2	no	—
69	#97	—	2	1	no	—
70	#97	—	2	2	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	—	2	2	no	—
75	#41	—	2	1	no	—
76	#41	—	2	1	no	—
77	#41	—	2	1	no	—
78	#41	—	2	1	no	—
79	C₂×Q₈	#41	1	1	yes	characteristic
80	C₂×D₄	—	2	1	no	—
81	C₂×D₄	#41	1	1	yes	characteristic
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	C₂⁴	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	1	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	1	no	—
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	4	1	no	—
96	M₁₆	—	4	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	#41	1	1	yes	characteristic
108	C₂×C₈	#41	1	1	yes	characteristic
109	C₂×C₈	#41	1	1	yes	characteristic
110	C₂×C₈	#41	1	1	yes	characteristic
111	C₂×C₈	#41	1	1	yes	characteristic
112	C₂×C₈	#41	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	2	no	—
120	C₄²	#41	1	1	yes	characteristic
121	C₄²	#41	1	1	yes	characteristic
122	C₂³	—	2	1	no	—
123	C₂³	2₋¹⁺⁴	1	1	yes	characteristic
124	Q₈	—	2	1	no	—
125	Q₈	—	2	1	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	C₂×#41	1	1	yes	characteristic
139	C₂×C₄	C₂×#41	1	1	yes	characteristic
140	C₂×C₄	C₂×#41	1	1	yes	characteristic
141	C₂×C₄	C₂×#41	1	1	yes	characteristic
142	C₂×C₄	C₂×#41	1	1	yes	characteristic
143	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈	—	4	1	no	—
145	C₈	—	4	2	no	—
146	C₈	—	4	1	no	—
147	C₈	—	4	2	no	—
148	C₈	—	4	1	no	—
149	C₈	—	4	2	no	—
150	C₈	#126	1	1	yes	characteristic
151	C₈	#126	1	1	yes	characteristic
152	C₈	#126	1	1	yes	characteristic
153	C₈	#126	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	#542	1	1	yes	characteristic
158	C₂²	#552	1	1	yes	characteristic
159	C₂²	#552	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	1	no	—
163	C₄	—	8	2	no	—
164	C₄	—	8	1	no	—
165	C₄	C₂×#126	1	1	yes	characteristic
166	C₄	#529	1	1	yes	characteristic
167	C₄	#553	1	1	yes	characteristic
168	C₄	#555	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	—	8	1	no	—
173	C₂	#2495	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2495
prime radical	C₂×#41	C₂³
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. #2103	C₂	semidirect
3. #1953	C₂	semidirect
4. #1945	C₂	semidirect
6. #1841	C₂	semidirect
7. #1833	C₂	semidirect
10. C₄×#108	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2103	2
#2080	2
#1953	2
#1945	2
#1932	2
#1841	2
#1833	2
#1814	2
#1813	2
C₄×#108	2
#1764	2