Finite Group Databasebeta

Subgroups — #33933

Order 256.

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

normal subgroups only

This group has 266 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	#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	#2244	C₂	1	2	yes	maximal, complemented
4	#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	#2228	C₂	1	2	yes	maximal, complemented
6	#2226	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×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#554	—	2	1	no	—
13	#550	—	2	1	no	—
14	C₂×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×D₁₆	C₂²	1	2	yes	—
16	#500	C₂²	1	1	yes	characteristic
17	#499	C₂²	1	2	yes	—
18	#493	C₂²	1	1	yes	characteristic
19	#493	C₂²	1	2	yes	—
20	#478	—	2	2	no	—
21	#476	—	2	1	no	—
22	#459	—	2	1	no	—
23	#459	—	2	2	no	—
24	C₈×Q₈	C₂²	1	1	yes	characteristic
25	C₄×D₈	C₂²	1	1	yes	characteristic
26	C₄×D₈	C₂²	1	2	yes	—
27	C₄×D₈	C₂²	1	1	yes	characteristic
28	C₄×D₈	C₂²	1	2	yes	—
29	#367	C₂²	1	2	yes	—
30	#366	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆	C₂²	1	2	yes	—
33	#357	C₂²	1	1	yes	characteristic
34	#357	C₂²	1	1	yes	characteristic
35	#357	C₂²	1	2	yes	—
36	#357	C₂²	1	2	yes	—
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×D₈	—	2	2	no	—
43	C₂×D₈	—	2	1	no	—
44	C₂×D₈	C₂³	1	1	yes	characteristic
45	C₂×D₈	C₂³	1	2	yes	—
46	C₂×D₈	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈	—	2	1	no	—
48	#127	—	2	2	no	—
49	#127	—	2	1	no	—
50	#125	—	2	2	no	—
51	C₂²⋊Q₈	—	4	1	no	—
52	C₂²⋊Q₈	—	4	2	no	—
53	#121	—	4	1	no	—
54	#121	—	4	2	no	—
55	C₄×Q₈	D₄	1	1	yes	characteristic
56	C₄×D₄	—	2	2	no	—
57	C₄×D₄	—	2	1	no	—
58	C₄×D₄	—	2	1	no	—
59	C₄×D₄	—	2	2	no	—
60	C₂×C₄⋊C₄	—	4	1	no	—
61	D₁₆	—	2	2	no	—
62	D₁₆	—	2	4	no	—
63	C₂×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₆	C₂³	1	2	yes	—
66	#107	—	2	1	no	—
67	#107	C₂³	1	1	yes	characteristic
68	#107	C₂³	1	2	yes	—
69	#107	C₂³	1	1	yes	characteristic
70	#106	—	2	2	no	—
71	C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₄⋊C₈	D₄	1	2	yes	—
73	#102	—	2	2	no	—
74	#102	—	2	1	no	—
75	#102	—	2	2	no	—
76	#102	—	2	1	no	—
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	2	yes	—
79	#41	—	8	4	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×Q₈	D₈	1	1	yes	characteristic
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	—	4	2	no	—
86	C₂×D₄	—	4	1	no	—
87	C₂×D₄	—	4	1	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	—	4	1	no	—
92	C₂²×C₄	—	4	2	no	—
93	D₈	—	2	1	no	—
94	D₈	—	2	2	no	—
95	D₈	—	2	1	no	—
96	D₈	—	2	1	no	—
97	D₈	—	2	2	no	—
98	D₈	—	8	2	no	—
99	D₈	—	8	1	no	—
100	D₈	#41	1	2	yes	complemented
101	C₂×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	D₈	1	1	yes	characteristic
111	C₄⋊C₄	D₈	1	2	yes	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	4	2	no	—
114	C₂²⋊C₄	—	4	1	no	—
115	C₂²⋊C₄	—	4	2	no	—
116	C₄²	C₂×D₄	1	2	yes	—
117	C₄²	C₂×D₄	1	1	yes	characteristic
118	C₁₆	—	2	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	2	1	no	—
121	C₂³	—	4	1	no	—
122	C₂³	—	4	2	no	—
123	C₂³	—	4	1	no	—
124	Q₈	—	2	2	no	—
125	Q₈	—	16	1	no	—
126	D₄	—	2	2	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	8	4	no	—
133	D₄	—	8	2	no	—
134	D₄	—	16	2	no	—
135	D₄	—	16	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	16	2	no	—
143	C₂×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄	C₂×D₈	1	2	yes	—
145	C₂×C₄	C₂×D₈	1	2	yes	—
146	C₂×C₄	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	1	no	—
150	C₈	C₂×#41	1	1	yes	characteristic
151	C₈	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂²	—	4	2	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	8	2	no	—
158	C₂²	—	8	1	no	—
159	C₂²	—	8	1	no	—
160	C₂²	—	8	2	no	—
161	C₂²	C₂²×D₈	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	1	no	—
165	C₄	—	8	1	no	—
166	C₄	—	8	1	no	—
167	C₄	—	8	2	no	—
168	C₄	#509	1	2	yes	—
169	C₄	#548	1	1	yes	characteristic
170	C₄	#577	1	1	yes	characteristic
171	C₂	—	8	2	no	—
172	C₂	—	16	2	no	—
173	C₂	—	16	1	no	—
174	C₂	#3358	1	1	yes	characteristic
175	C₂	C₂×#509	1	1	yes	characteristic
176	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. #3391	C₂	semidirect
2. #3380	C₂	semidirect
3. #2244	C₂	semidirect
4. #2240	C₂	semidirect
5. #2228	C₂	semidirect
6. #2226	C₂	semidirect
7. #2206	C₂	semidirect
8. #2206	C₂	semidirect
9. #2183	C₂	semidirect
10. #2177	C₂	semidirect
11. C₄×D₁₆	C₂	semidirect
100. D₈	#41	semidirect

Maximal subgroups

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

subgroup	index
#3391	2
#3380	2
#2244	2
#2240	2
#2228	2
#2226	2
#2206	2
#2183	2
#2177	2
C₄×D₁₆	2