Finite Group Databasebeta

Subgroups — #33454

Order 256.

348 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 159 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×#509	C₂	1	1	yes	maximal, characteristic
2	#2951	C₂	1	1	yes	maximal, characteristic, complemented
3	#2948	C₂	1	1	yes	maximal, characteristic, complemented
4	#2177	C₂	1	2	yes	maximal, complemented
5	#2175	C₂	1	2	yes	maximal, complemented
6	C₄×SD₃₂	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₆	C₂	1	2	yes	maximal, complemented
8	#2159	C₂	1	1	yes	maximal, characteristic, complemented
9	#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#357	C₂	1	1	yes	maximal, characteristic, complemented
11	#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	C₂²	1	1	yes	characteristic
13	C₂²×D₈	C₂²	1	1	yes	characteristic
14	#520	—	2	1	no	—
15	#518	—	2	1	no	—
16	#509	C₄	1	8	yes	complemented
17	C₂×SD₃₂	C₂²	1	2	yes	—
18	C₂×D₁₆	C₂²	1	2	yes	—
19	C₂×M₃₂	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆	C₂²	1	1	yes	characteristic
21	#442	—	2	2	no	—
22	#440	—	2	2	no	—
23	C₄×Q₁₆	C₂²	1	2	yes	complemented
24	C₄×D₈	C₂²	1	2	yes	—
25	C₄×D₈	C₂²	1	2	yes	—
26	C₄×D₈	C₂²	1	2	yes	complemented
27	#427	C₂²	1	1	yes	characteristic
28	C₂×#107	C₂²	1	1	yes	characteristic
29	#418	—	2	1	no	—
30	#417	—	2	1	no	—
31	#405	C₂²	1	1	yes	characteristic
32	#367	C₂²	1	2	yes	complemented
33	#366	C₂²	1	2	yes	complemented
34	#358	C₂²	1	1	yes	characteristic, complemented
35	#358	C₂²	1	1	yes	characteristic, complemented
36	#357	C₂²	1	2	yes	—
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	1	yes	characteristic, complemented
39	#357	C₂²	1	1	yes	characteristic, complemented
40	#346	C₂²	1	2	yes	complemented
41	C₄×C₁₆	C₂²	1	2	yes	complemented
42	C₂×#41	—	2	1	no	—
43	C₂²×D₄	—	2	1	no	—
44	#135	—	2	2	no	—
45	#135	C₂×C₄	1	4	yes	complemented
46	C₂×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆	—	2	1	no	—
48	C₂×D₈	—	2	1	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	C₂×C₄	1	4	yes	complemented
51	C₂×D₈	C₂³	1	2	yes	—
52	C₂×D₈	C₂³	1	1	yes	characteristic
53	C₂×M₁₆	D₄	1	1	yes	characteristic
54	C₂²×C₈	C₂³	1	1	yes	characteristic
55	C₄×Q₈	—	2	2	no	—
56	C₄×D₄	—	2	2	no	—
57	C₄×D₄	—	2	2	no	—
58	C₄×D₄	—	2	2	no	—
59	C₄×D₄	—	4	1	no	—
60	C₄×D₄	—	4	1	no	—
61	C₄×D₄	—	4	2	no	—
62	#117	—	2	1	no	—
63	#117	—	4	1	no	—
64	#117	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄	—	2	1	no	—
66	C₂×C₄⋊C₄	—	4	1	no	—
67	C₂×C₂²⋊C₄	—	4	2	no	—
68	SD₃₂	C₂×C₄	1	8	yes	complemented
69	D₁₆	C₂×C₄	1	8	yes	complemented
70	M₃₂	C₂×C₄	1	4	yes	complemented
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₁₆	C₂³	1	2	yes	—
75	#107	—	2	1	no	—
76	#107	C₂³	1	2	yes	—
77	#107	C₂³	1	1	yes	characteristic
78	#107	C₂³	1	1	yes	characteristic
79	#106	—	2	1	no	—
80	#103	—	2	2	no	—
81	#102	—	2	2	no	—
82	#102	—	2	2	no	—
83	#102	—	2	2	no	—
84	#97	D₄	1	2	yes	—
85	C₄×C₈	C₂³	1	2	yes	—
86	C₂⁴	—	4	1	no	—
87	#41	—	2	4	no	—
88	#41	—	4	2	no	—
89	C₂×Q₈	—	2	1	no	—
90	C₂×D₄	—	2	4	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	4	1	no	—
94	C₂×D₄	—	4	1	no	—
95	C₂×D₄	—	4	2	no	—
96	C₂²×C₄	—	4	1	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
102	Q₁₆	—	2	1	no	—
103	Q₁₆	C₂²×C₄	1	2	yes	—
104	SD₁₆	—	2	4	no	—
105	D₈	—	2	4	no	—
106	D₈	—	2	2	no	—
107	D₈	—	2	1	no	—
108	D₈	C₂²×C₄	1	2	yes	—
109	D₈	C₂²×C₄	1	4	yes	—
110	M₁₆	—	2	2	no	—
111	C₂×C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈	#41	1	2	yes	—
115	C₂×C₈	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	D₈	1	1	yes	characteristic
122	C₄⋊C₄	D₈	1	1	yes	characteristic
123	C₂²⋊C₄	—	4	1	no	—
124	C₂²⋊C₄	—	4	2	no	—
125	C₂²⋊C₄	—	4	2	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₂²⋊C₄	D₈	1	2	yes	—
128	C₄²	—	4	1	no	—
129	C₄²	—	4	2	no	—
130	C₄²	C₂×D₄	1	2	yes	—
131	C₁₆	—	2	1	no	—
132	C₁₆	—	2	1	no	—
133	C₁₆	C₂²×C₄	1	4	yes	—
134	C₂³	—	4	1	no	—
135	C₂³	—	4	2	no	—
136	C₂³	—	4	4	no	—
137	C₂³	—	16	1	no	—
138	C₂³	—	16	1	no	—
139	C₂³	C₂×D₈	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	Q₈	—	4	1	no	—
142	D₄	—	2	4	no	—
143	D₄	—	2	2	no	—
144	D₄	—	4	4	no	—
145	D₄	—	4	2	no	—
146	D₄	—	4	1	no	—
147	D₄	—	4	4	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	—	16	2	no	—
155	C₂×C₄	—	16	2	no	—
156	C₂×C₄	—	16	1	no	—
157	C₂×C₄	C₄×D₄	1	2	yes	—
158	C₂×C₄	C₂×D₈	1	2	yes	—
159	C₂×C₄	C₂×D₈	1	2	yes	—
160	C₂×C₄	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄	#135	1	2	yes	—
162	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	C₄×D₄	1	2	yes	—
165	C₈	C₂³×C₄	1	1	yes	characteristic
166	C₈	C₂×#41	1	1	yes	characteristic
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	4	2	no	—
170	C₂²	—	4	4	no	—
171	C₂²	—	8	1	no	—
172	C₂²	—	8	2	no	—
173	C₂²	—	8	2	no	—
174	C₂²	—	8	1	no	—
175	C₂²	—	16	4	no	—
176	C₂²	C₄×D₈	1	2	yes	—
177	C₂²	C₂²×D₈	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	8	1	no	—
182	C₄	—	8	2	no	—
183	C₄	—	8	1	no	—
184	C₄	C₄×D₈	1	2	yes	—
185	C₄	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄	C₂×#135	1	1	yes	characteristic
187	C₂	—	2	2	no	—
188	C₂	—	8	2	no	—
189	C₂	—	8	4	no	—
190	C₂	C₂×C₄×D₈	1	1	yes	characteristic
191	C₂	#3415	1	1	yes	characteristic
192	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#509	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2951	C₂	semidirect
3. #2948	C₂	semidirect
4. #2177	C₂	semidirect
5. #2175	C₂	semidirect
6. C₄×SD₃₂	C₂	semidirect
7. C₄×D₁₆	C₂	semidirect
8. #2159	C₂	semidirect
9. #2138	C₂	semidirect
10. C₂×#357	C₂	semidirect
11. #2108	C₂	semidirect
16. #509	C₄	semidirect
23. C₄×Q₁₆	C₂²	semidirect
26. C₄×D₈	C₂²	semidirect
32. #367	C₂²	semidirect
33. #366	C₂²	semidirect
34. #358	C₂²	semidirect
35. #358	C₂²	semidirect
38. #357	C₂²	semidirect
39. #357	C₂²	semidirect
40. #346	C₂²	semidirect
41. C₄×C₁₆	C₂²	semidirect
45. #135	C₂×C₄	semidirect
50. C₂×D₈	C₂×C₄	semidirect
68. SD₃₂	C₂×C₄	semidirect
69. D₁₆	C₂×C₄	semidirect
70. M₃₂	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509	2
#2951	2
#2948	2
#2177	2
#2175	2
C₄×SD₃₂	2
C₄×D₁₆	2
#2159	2
#2138	2
C₂×#357	2
#2108	2