Finite Group Databasebeta

Subgroups — #33966

Order 256.

283 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3396	C₂	1	1	yes	maximal, characteristic, complemented
2	#3391	C₂	1	1	yes	maximal, characteristic, complemented
3	#2250	C₂	1	2	yes	maximal, complemented
4	#2246	C₂	1	1	yes	maximal, characteristic, complemented
5	#2210	C₂	1	2	yes	maximal, complemented
6	#2207	C₂	1	1	yes	maximal, characteristic, complemented
7	#2206	C₂	1	2	yes	maximal, complemented
8	#2206	C₂	1	1	yes	maximal, characteristic, complemented
9	#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	#2177	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#559	—	2	1	no	—
13	#550	—	2	1	no	—
14	C₂×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂×SD₃₂	C₂²	1	2	yes	—
16	C₂×D₁₆	C₂²	1	2	yes	—
17	C₂×D₁₆	C₂²	1	2	yes	—
18	C₂×D₁₆	C₂²	1	2	yes	—
19	#495	C₂²	1	2	yes	—
20	#493	C₂²	1	2	yes	—
21	#493	C₂²	1	1	yes	characteristic
22	#493	C₂²	1	1	yes	characteristic
23	#463	—	2	2	no	—
24	#460	—	2	1	no	—
25	#459	—	2	1	no	—
26	#459	—	2	2	no	—
27	C₈×Q₈	C₂²	1	1	yes	characteristic
28	C₄×Q₁₆	C₂²	1	1	yes	characteristic
29	C₄×D₈	C₂²	1	1	yes	characteristic
30	C₄×D₈	C₂²	1	2	yes	—
31	C₄×D₈	C₂²	1	2	yes	—
32	#367	C₂²	1	1	yes	characteristic
33	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₄⋊C₁₆	C₂²	1	2	yes	—
35	#358	C₂²	1	1	yes	characteristic
36	#357	C₂²	1	1	yes	characteristic
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	#346	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×#41	—	4	1	no	—
42	C₂×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂×SD₁₆	—	2	2	no	—
44	C₂×D₈	—	2	2	no	—
45	C₂×D₈	—	2	1	no	—
46	C₂×D₈	—	2	1	no	—
47	C₂×D₈	C₂³	1	2	yes	—
48	C₂×D₈	C₂³	1	1	yes	characteristic
49	C₂×D₈	C₂³	1	2	yes	—
50	#127	—	2	2	no	—
51	#127	—	2	1	no	—
52	#127	—	2	1	no	—
53	#124	—	2	2	no	—
54	#124	—	4	1	no	—
55	#123	—	4	2	no	—
56	#121	—	4	1	no	—
57	#121	—	4	2	no	—
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	D₄	1	1	yes	characteristic
60	C₄×D₄	—	2	2	no	—
61	C₄×D₄	—	2	1	no	—
62	C₄×D₄	—	2	2	no	—
63	C₄×D₄	—	4	1	no	—
64	SD₃₂	—	2	2	no	—
65	SD₃₂	—	4	2	no	—
66	D₁₆	—	2	4	no	—
67	D₁₆	—	4	2	no	—
68	D₁₆	—	4	2	no	—
69	C₂×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₁₆	C₂³	1	2	yes	—
71	C₂×C₁₆	C₂³	1	1	yes	characteristic
72	#107	C₂³	1	1	yes	characteristic
73	#107	C₂³	1	1	yes	characteristic
74	C₄⋊C₈	D₄	1	1	yes	characteristic
75	C₄⋊C₈	D₄	1	2	yes	—
76	#103	—	2	1	no	—
77	#102	—	2	2	no	—
78	#102	—	2	1	no	—
79	#102	—	2	2	no	—
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	2	yes	—
82	#41	—	8	4	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	D₈	1	1	yes	characteristic
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	—	4	1	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂×D₄	—	4	1	no	—
91	C₂×D₄	—	4	2	no	—
92	C₂×D₄	—	4	1	no	—
93	C₂×D₄	—	4	1	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	—	4	1	no	—
96	C₂²×C₄	—	4	2	no	—
97	Q₁₆	—	2	1	no	—
98	Q₁₆	—	2	1	no	—
99	SD₁₆	—	8	2	no	—
100	D₈	—	2	2	no	—
101	D₈	—	2	2	no	—
102	D₈	—	2	1	no	—
103	D₈	—	2	1	no	—
104	D₈	—	2	2	no	—
105	D₈	—	2	2	no	—
106	D₈	—	8	1	no	—
107	D₈	—	8	1	no	—
108	D₈	—	8	2	no	—
109	C₂×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	D₈	1	1	yes	characteristic
116	C₄⋊C₄	D₈	1	2	yes	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	4	1	no	—
119	C₂²⋊C₄	—	4	2	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₂²⋊C₄	—	4	2	no	—
122	C₄²	—	4	1	no	—
123	C₄²	C₂×D₄	1	2	yes	—
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	2	2	no	—
127	C₁₆	#41	1	2	yes	complemented
128	C₂³	—	4	2	no	—
129	C₂³	—	4	2	no	—
130	C₂³	—	4	1	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	2	no	—
138	D₄	—	4	1	no	—
139	D₄	—	4	2	no	—
140	D₄	—	8	4	no	—
141	D₄	—	8	2	no	—
142	D₄	—	8	2	no	—
143	D₄	—	16	2	no	—
144	D₄	—	16	1	no	—
145	D₄	—	16	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	16	2	no	—
152	C₂×C₄	C₂×D₈	1	2	yes	—
153	C₂×C₄	C₂×D₈	1	2	yes	—
154	C₂×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	2	no	—
159	C₈	C₂×#41	1	1	yes	characteristic
160	C₈	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²	—	8	2	no	—
162	C₂²	—	8	1	no	—
163	C₂²	—	8	2	no	—
164	C₂²	—	8	2	no	—
165	C₂²	—	8	1	no	—
166	C₂²	—	8	2	no	—
167	C₂²	—	8	2	no	—
168	C₂²	—	8	1	no	—
169	C₂²	—	8	2	no	—
170	C₂²	C₂²×D₈	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	2	no	—
173	C₄	—	2	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	#509	1	2	yes	—
178	C₄	#550	1	1	yes	characteristic
179	C₄	#576	1	1	yes	characteristic
180	C₂	—	16	2	no	—
181	C₂	—	16	1	no	—
182	C₂	—	16	2	no	—
183	C₂	#3391	1	1	yes	characteristic
184	C₂	C₂×#509	1	1	yes	characteristic
185	C₂	#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3396	C₂	semidirect
2. #3391	C₂	semidirect
3. #2250	C₂	semidirect
4. #2246	C₂	semidirect
5. #2210	C₂	semidirect
6. #2207	C₂	semidirect
7. #2206	C₂	semidirect
8. #2206	C₂	semidirect
9. #2183	C₂	semidirect
10. #2177	C₂	semidirect
11. C₄×SD₃₂	C₂	semidirect
127. C₁₆	#41	semidirect

Maximal subgroups

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

subgroup	index
#3396	2
#3391	2
#2250	2
#2246	2
#2210	2
#2207	2
#2206	2
#2183	2
#2177	2
C₄×SD₃₂	2