Finite Group Databasebeta

Subgroups — #33629

Order 256.

287 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 123 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×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#135	C₂	1	1	yes	maximal, characteristic
3	#2227	C₂	1	2	yes	maximal, complemented
4	#2226	C₂	1	2	yes	maximal, complemented
5	#2225	C₂	1	2	yes	maximal, complemented
6	#2224	C₂	1	2	yes	maximal, complemented
7	#2159	C₂	1	2	yes	maximal, complemented
8	#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	#2139	C₂	1	2	yes	maximal
10	C₂×#135	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈	—	2	1	no	—
12	C₄×#41	—	2	1	no	—
13	C₂×M₃₂	C₂²	1	2	yes	—
14	#500	—	2	1	no	—
15	#500	—	2	1	no	—
16	#500	C₂²	1	2	yes	—
17	#500	C₂²	1	2	yes	—
18	C₄×Q₁₆	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×SD₁₆	—	2	1	no	—
21	C₄×SD₁₆	—	2	1	no	—
22	C₄×D₈	C₂²	1	1	yes	characteristic
23	C₄×D₈	C₂²	1	1	yes	characteristic
24	#427	C₂²	1	1	yes	characteristic
25	C₂×#107	—	2	1	no	—
26	C₂×#107	C₂²	1	2	yes	—
27	#416	—	2	1	no	—
28	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
29	#367	C₂²	1	4	yes	—
30	#366	C₂²	1	4	yes	—
31	C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₄⋊C₁₆	C₂²	1	2	yes	—
33	#358	C₂²	1	4	yes	—
34	#357	C₂²	1	4	yes	—
35	C₂×#41	—	2	1	no	—
36	C₂²×Q₈	—	4	1	no	—
37	#135	—	2	2	no	—
38	#135	Q₈	1	4	yes	complemented
39	C₂×Q₁₆	C₂³	1	1	yes	characteristic
40	C₂×SD₁₆	—	2	1	no	—
41	C₂×D₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈	—	2	2	no	—
45	C₄⋊Q₈	—	2	2	no	—
46	C₄⋊Q₈	—	4	1	no	—
47	C₄⋊Q₈	—	4	1	no	—
48	C₄×Q₈	—	2	1	no	—
49	C₄×Q₈	—	2	1	no	—
50	C₄×D₄	—	2	1	no	—
51	C₄×D₄	—	2	1	no	—
52	C₄×D₄	—	4	1	no	—
53	C₄×D₄	—	4	1	no	—
54	#117	—	2	1	no	—
55	#117	—	4	1	no	—
56	C₂×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄⋊C₄	—	4	1	no	—
58	C₂×C₄²	—	4	1	no	—
59	C₂×C₄²	D₄	1	1	yes	characteristic
60	M₃₂	—	2	4	no	—
61	C₂×C₁₆	C₂³	1	4	yes	—
62	#107	—	2	2	no	—
63	#107	—	2	2	no	—
64	#107	—	2	2	no	—
65	#107	C₂³	1	4	yes	—
66	#107	C₂³	1	1	yes	characteristic
67	#107	C₂³	1	1	yes	characteristic
68	#106	—	2	1	no	—
69	#103	—	2	1	no	—
70	#103	—	2	1	no	—
71	#102	—	2	1	no	—
72	#102	—	2	1	no	—
73	C₄×C₈	D₄	1	1	yes	characteristic
74	C₄×C₈	D₄	1	1	yes	characteristic
75	C₄×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₈	C₂³	1	1	yes	characteristic
77	#41	—	2	4	no	—
78	#41	—	4	2	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	8	4	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	D₈	1	1	yes	characteristic
90	C₂²×C₄	D₈	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	Q₁₆	—	2	1	no	—
93	Q₁₆	C₂×Q₈	1	2	yes	complemented
94	SD₁₆	—	2	4	no	—
95	D₈	—	2	1	no	—
96	D₈	C₂×Q₈	1	2	yes	complemented
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈	C₂×Q₈	1	2	yes	—
102	C₂×C₈	#41	1	2	yes	—
103	C₂×C₈	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	4	1	no	—
114	C₂²⋊C₄	—	4	1	no	—
115	C₄²	—	4	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₄²	D₈	1	1	yes	characteristic
119	C₄²	D₈	1	1	yes	characteristic
120	C₄²	C₂×D₄	1	1	yes	characteristic
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₁₆	—	2	4	no	—
123	C₂³	—	4	1	no	—
124	C₂³	C₂×D₈	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	4	1	no	—
127	Q₈	—	8	4	no	—
128	Q₈	—	8	4	no	—
129	D₄	—	2	2	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	4	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄	#137	1	2	yes	—
149	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	C₂²⋊Q₈	1	2	yes	—
153	C₈	C₂²×Q₈	1	1	yes	characteristic
154	C₈	C₂×#41	1	1	yes	characteristic
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	8	1	no	—
159	C₂²	—	8	1	no	—
160	C₂²	#474	1	2	yes	—
161	C₂²	C₂²×D₈	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	8	1	no	—
166	C₄	—	8	4	no	—
167	C₄	—	8	1	no	—
168	C₄	#474	1	2	yes	—
169	C₄	#509	1	2	yes	—
170	C₄	#510	1	2	yes	—
171	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₄	C₂×#137	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	8	2	no	—
175	C₂	C₂×#474	1	1	yes	characteristic
176	C₂	C₂×#509	1	1	yes	characteristic
177	C₂	C₂×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×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
1. C₂×#500	C₂	semidirect
3. #2227	C₂	semidirect
4. #2226	C₂	semidirect
5. #2225	C₂	semidirect
6. #2224	C₂	semidirect
7. #2159	C₂	semidirect
8. #2150	C₂	semidirect
38. #135	Q₈	semidirect
93. Q₁₆	C₂×Q₈	semidirect
96. D₈	C₂×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500	2
C₄×#135	2
#2227	2
#2226	2
#2225	2
#2224	2
#2159	2
#2150	2
#2139	2