Finite Group Databasebeta

Subgroups — #33967

Order 256.

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

normal subgroups only

This group has 255 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
2	#3395	C₂	1	1	yes	maximal, characteristic, complemented
3	#2251	C₂	1	2	yes	maximal, complemented
4	#2248	C₂	1	1	yes	maximal, characteristic
5	#2211	C₂	1	2	yes	maximal, complemented
6	#2209	C₂	1	1	yes	maximal, characteristic, complemented
7	#2208	C₂	1	2	yes	maximal, complemented
8	#2207	C₂	1	1	yes	maximal, characteristic
9	#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	#2175	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#559	—	2	1	no	—
13	#557	—	2	1	no	—
14	C₂×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂	C₂²	1	2	yes	—
16	C₂×SD₃₂	C₂²	1	2	yes	—
17	C₂×SD₃₂	C₂²	1	2	yes	—
18	C₂×SD₃₂	C₂²	1	2	yes	—
19	#495	C₂²	1	2	yes	—
20	#494	C₂²	1	2	yes	—
21	#494	C₂²	1	1	yes	characteristic
22	#493	C₂²	1	1	yes	characteristic
23	#463	—	2	2	no	—
24	#462	—	2	1	no	—
25	#462	—	2	2	no	—
26	#460	—	2	1	no	—
27	C₈×Q₈	C₂²	1	1	yes	characteristic
28	C₄×Q₁₆	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆	C₂²	1	2	yes	—
30	C₄×Q₁₆	C₂²	1	1	yes	characteristic
31	C₄×D₈	C₂²	1	2	yes	—
32	#366	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	#358	C₂²	1	1	yes	characteristic
37	#358	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×Q₁₆	—	2	2	no	—
42	C₂×Q₁₆	—	2	1	no	—
43	C₂×Q₁₆	C₂³	1	2	yes	—
44	C₂×Q₁₆	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆	—	2	2	no	—
47	C₂×D₈	—	2	1	no	—
48	C₂×D₈	C₂³	1	2	yes	—
49	C₄⋊Q₈	—	2	2	no	—
50	C₄⋊Q₈	—	2	1	no	—
51	#127	—	2	1	no	—
52	#125	—	4	1	no	—
53	#125	—	4	2	no	—
54	#124	—	2	2	no	—
55	#124	—	4	1	no	—
56	#123	—	4	2	no	—
57	C₄×Q₈	—	2	2	no	—
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	—	2	1	no	—
60	C₄×Q₈	—	4	1	no	—
61	C₄×Q₈	D₄	1	1	yes	characteristic
62	C₄×D₄	—	2	2	no	—
63	C₄×D₄	—	4	1	no	—
64	Q₃₂	—	2	2	no	—
65	Q₃₂	—	4	2	no	—
66	SD₃₂	—	2	4	no	—
67	SD₃₂	—	4	2	no	—
68	SD₃₂	—	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	2	no	—
77	#103	—	2	1	no	—
78	#103	—	2	1	no	—
79	#102	—	2	2	no	—
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	2	yes	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	D₈	1	1	yes	characteristic
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	4	1	no	—
88	C₂×D₄	—	4	1	no	—
89	C₂²×C₄	—	4	2	no	—
90	Q₁₆	—	2	2	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	—	2	1	no	—
93	Q₁₆	—	2	1	no	—
94	Q₁₆	—	2	2	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	8	1	no	—
97	Q₁₆	—	8	2	no	—
98	SD₁₆	—	8	2	no	—
99	D₈	—	2	2	no	—
100	D₈	—	2	2	no	—
101	D₈	—	8	1	no	—
102	C₂×C₈	C₂×D₄	1	2	yes	—
103	C₂×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	D₈	1	1	yes	characteristic
115	C₄⋊C₄	D₈	1	2	yes	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	4	2	no	—
119	C₄²	—	4	1	no	—
120	C₄²	—	4	2	no	—
121	C₄²	—	4	1	no	—
122	C₄²	C₂×D₄	1	2	yes	—
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₁₆	—	2	1	no	—
125	C₁₆	—	2	2	no	—
126	C₁₆	#41	1	2	yes	—
127	C₂³	—	4	2	no	—
128	Q₈	—	2	2	no	—
129	Q₈	—	4	2	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	4	2	no	—
134	Q₈	—	4	1	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	8	2	no	—
138	D₄	—	16	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	16	2	no	—
145	C₂×C₄	C₂×D₈	1	2	yes	—
146	C₂×C₄	C₂×D₈	1	2	yes	—
147	C₂×C₄	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	2	no	—
152	C₈	C₂×#41	1	1	yes	characteristic
153	C₈	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂²	—	8	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	C₂²×D₈	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	8	1	no	—
162	C₄	—	8	2	no	—
163	C₄	—	8	1	no	—
164	C₄	—	8	1	no	—
165	C₄	—	8	1	no	—
166	C₄	#510	1	2	yes	—
167	C₄	#550	1	1	yes	characteristic
168	C₄	#576	1	1	yes	characteristic
169	C₂	—	16	2	no	—
170	C₂	#3391	1	1	yes	characteristic
171	C₂	C₂×#510	1	1	yes	characteristic
172	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#493	C₂²
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
2. #3395	C₂	semidirect
3. #2251	C₂	semidirect
5. #2211	C₂	semidirect
6. #2209	C₂	semidirect
7. #2208	C₂	semidirect
9. #2183	C₂	semidirect
10. #2175	C₂	semidirect
11. C₄×Q₃₂	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3396	2
#3395	2
#2251	2
#2248	2
#2211	2
#2209	2
#2208	2
#2207	2
#2183	2
#2175	2
C₄×Q₃₂	2