Finite Group Databasebeta

Subgroups — #19211

Order 256.

285 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 121 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	#3249	C₂	1	2	yes	maximal
2	#3160	C₂	1	1	yes	maximal, characteristic
3	#2960	C₂	1	1	yes	maximal, characteristic
4	#1692	C₂	1	4	yes	maximal, complemented
5	#1676	C₂	1	2	yes	maximal, complemented
6	#1674	C₂	1	2	yes	maximal, complemented
7	C₂×#335	C₂	1	1	yes	maximal, characteristic
8	#1550	C₂	1	2	yes	maximal
9	#533	C₂²	1	2	yes	—
10	#533	C₂²	1	1	yes	characteristic
11	#500	C₂²	1	1	yes	characteristic
12	#500	C₂²	1	1	yes	characteristic
13	#499	—	2	1	no	—
14	#499	—	2	1	no	—
15	#498	C₂²	1	2	yes	—
16	#477	—	2	2	no	—
17	#477	C₂²	1	4	yes	—
18	#467	—	2	4	no	—
19	#462	—	2	2	no	—
20	#462	C₂²	1	4	yes	—
21	#446	—	2	1	no	—
22	#446	C₂²	1	2	yes	—
23	C₈×Q₈	—	2	1	no	—
24	C₈×Q₈	C₂²	1	2	yes	—
25	#427	—	2	1	no	—
26	#427	—	2	1	no	—
27	#424	—	2	1	no	—
28	#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈	C₂²	1	2	yes	—
30	#405	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#335	C₂²	1	2	yes	—
33	#335	C₂²	1	2	yes	—
34	#333	C₂²	1	4	yes	—
35	#332	C₂²	1	4	yes	—
36	C₂×Q₁₆	—	2	4	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₈	D₄	1	1	yes	characteristic
41	C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₄⋊Q₈	C₂³	1	2	yes	—
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	#125	—	2	2	no	—
46	#125	—	2	1	no	—
47	C₂²⋊Q₈	—	2	1	no	—
48	C₂²⋊Q₈	—	2	2	no	—
49	C₂²⋊Q₈	—	2	2	no	—
50	C₂²⋊Q₈	D₄	1	2	yes	—
51	C₄×Q₈	—	2	1	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₄×Q₈	C₂³	1	2	yes	—
55	#117	—	2	1	no	—
56	#117	—	2	2	no	—
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	#107	—	2	1	no	—
59	#107	—	2	1	no	—
60	#107	—	2	1	no	—
61	#107	—	2	1	no	—
62	#107	—	2	4	no	—
63	#106	—	2	2	no	—
64	#106	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	C₂³	1	2	yes	—
70	C₄⋊C₈	C₂³	1	4	yes	—
71	#103	—	2	4	no	—
72	#103	—	2	4	no	—
73	#97	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	D₄	1	1	yes	characteristic
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	C₂×D₄	1	2	yes	—
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	#41	1	1	yes	characteristic
86	Q₁₆	—	8	4	no	—
87	M₁₆	—	4	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	8	2	no	—
92	C₂×C₈	Q₁₆	1	4	yes	complemented
93	C₂×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	4	2	no	—
113	C₄²	C₂×D₄	1	1	yes	characteristic
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₄²	#41	1	1	yes	characteristic
116	C₄²	C₂⁴	1	1	yes	characteristic
117	C₂³	#121	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	Q₈	—	2	2	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	8	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	#121	1	1	yes	characteristic
132	C₂×C₄	#121	1	1	yes	characteristic
133	C₂×C₄	#121	1	1	yes	characteristic
134	C₂×C₄	C₂×Q₁₆	1	2	yes	—
135	C₂×C₄	#137	1	2	yes	—
136	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄	C₂×#41	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	4	2	no	—
142	C₈	—	4	4	no	—
143	C₈	C₂×Q₁₆	1	2	yes	—
144	C₈	C₂×Q₁₆	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#462	1	2	yes	—
148	C₂²	C₂×#121	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	8	2	no	—
155	C₄	—	8	2	no	—
156	C₄	#462	1	2	yes	—
157	C₄	C₂²×Q₁₆	1	1	yes	characteristic
158	C₄	C₂×#137	1	1	yes	characteristic
159	C₄	#576	1	1	yes	characteristic
160	C₄	#578	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₂×#462	1	1	yes	characteristic
163	C₂	#3061	1	1	yes	characteristic
164	C₂	#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1692	C₂	semidirect
5. #1676	C₂	semidirect
6. #1674	C₂	semidirect
92. C₂×C₈	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3249	2
#3160	2
#2960	2
#1692	2
#1676	2
#1674	2
C₂×#335	2
#1550	2