Finite Group Databasebeta

Subgroups — #33649

Order 256.

290 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×#427	C₂	1	1	yes	maximal, characteristic
3	#2237	C₂	1	4	yes	maximal, complemented
4	#2236	C₂	1	2	yes	maximal, complemented
5	#2235	C₂	1	2	yes	maximal, complemented
6	#2159	C₂	1	2	yes	maximal, complemented
7	#2141	C₂	1	2	yes	maximal, complemented
8	#2112	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Q₈	—	2	1	no	—
12	C₂×#117	—	2	1	no	—
13	C₂×M₃₂	C₂²	1	2	yes	—
14	#467	—	2	1	no	—
15	#467	—	2	1	no	—
16	#467	C₂²	1	2	yes	—
17	#467	C₂²	1	2	yes	—
18	#427	—	2	1	no	—
19	#427	—	2	1	no	—
20	#427	C₂²	1	2	yes	—
21	#427	C₂²	1	2	yes	—
22	C₂×#107	C₂²	1	1	yes	characteristic
23	C₂×#107	C₂²	1	1	yes	characteristic
24	C₂×#107	C₂²	1	1	yes	characteristic
25	C₂×#106	—	2	1	no	—
26	C₂×#103	—	2	1	no	—
27	#367	C₂²	1	4	yes	complemented
28	#366	C₂²	1	4	yes	complemented
29	#358	C₂²	1	4	yes	complemented
30	#358	C₂²	1	4	yes	complemented
31	C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₂²×Q₈	—	2	1	no	—
34	C₂³×C₄	D₄	1	1	yes	characteristic
35	C₂×Q₁₆	—	2	1	no	—
36	C₂×Q₁₆	—	2	4	no	—
37	C₂×Q₁₆	C₂³	1	2	yes	—
38	C₂²×C₈	—	2	4	no	—
39	C₂²×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²⋊Q₈	—	2	2	no	—
46	C₂²⋊Q₈	—	2	2	no	—
47	C₂²⋊Q₈	—	4	1	no	—
48	C₂²⋊Q₈	—	4	1	no	—
49	#117	—	2	2	no	—
50	#117	—	2	2	no	—
51	#117	—	4	1	no	—
52	#117	—	4	1	no	—
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄⋊C₄	—	2	1	no	—
55	C₂×C₄⋊C₄	—	2	1	no	—
56	C₂×C₄⋊C₄	—	4	1	no	—
57	C₂×C₂²⋊C₄	—	4	1	no	—
58	C₂×C₂²⋊C₄	—	4	1	no	—
59	C₂×C₄²	—	4	1	no	—
60	M₃₂	—	2	4	no	—
61	C₂×C₁₆	C₂³	1	4	yes	—
62	#107	—	2	1	no	—
63	#107	—	2	1	no	—
64	#107	—	2	1	no	—
65	#107	C₂³	1	2	yes	—
66	#107	C₂³	1	2	yes	—
67	#107	C₂³	1	2	yes	—
68	#106	—	2	2	no	—
69	#106	—	2	2	no	—
70	#103	—	2	2	no	—
71	#103	—	2	2	no	—
72	C₂⁴	D₈	1	1	yes	characteristic
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	4	1	no	—
75	C₂×Q₈	—	4	2	no	—
76	C₂×Q₈	—	4	2	no	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	4	1	no	—
80	C₂²×C₄	—	4	1	no	—
81	C₂²×C₄	—	4	1	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	D₈	1	1	yes	characteristic
84	C₂²×C₄	D₈	1	1	yes	characteristic
85	C₂²×C₄	D₈	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	Q₁₆	—	2	4	no	—
90	Q₁₆	—	4	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	#41	1	4	yes	—
100	C₂×C₈	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂²⋊C₄	—	4	2	no	—
112	C₂²⋊C₄	—	4	2	no	—
113	C₄²	—	4	2	no	—
114	C₄²	—	4	2	no	—
115	C₁₆	—	2	4	no	—
116	C₂³	—	2	2	no	—
117	C₂³	—	2	2	no	—
118	C₂³	—	4	1	no	—
119	C₂³	C₂×D₈	1	1	yes	characteristic
120	C₂³	C₂×D₈	1	1	yes	characteristic
121	C₂³	C₂×D₈	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	8	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	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	#123	1	2	yes	—
141	C₂×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄	#137	1	2	yes	—
145	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	#123	1	2	yes	—
149	C₈	C₂×#41	1	2	yes	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	4	4	no	—
158	C₂²	#480	1	2	yes	—
159	C₂²	#510	1	2	yes	—
160	C₂²	C₂²×D₈	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	8	2	no	—
164	C₄	—	8	2	no	—
165	C₄	—	8	2	no	—
166	C₄	—	8	2	no	—
167	C₄	#480	1	2	yes	—
168	C₄	C₂×#123	1	1	yes	characteristic
169	C₄	C₂×#137	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	—	2	2	no	—
172	C₂	—	4	1	no	—
173	C₂	C₂×#480	1	1	yes	characteristic
174	C₂	C₂×#510	1	1	yes	characteristic
175	C₂	#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂⁴	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2237	C₂	semidirect
4. #2236	C₂	semidirect
5. #2235	C₂	semidirect
6. #2159	C₂	semidirect
7. #2141	C₂	semidirect
27. #367	C₂²	semidirect
28. #366	C₂²	semidirect
29. #358	C₂²	semidirect
30. #358	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#467	2
C₂×#427	2
#2237	2
#2236	2
#2235	2
#2159	2
#2141	2
#2112	2