Finite Group Databasebeta

Subgroups — #19330

Order 256.

308 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 107 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	#3235	C₂	1	2	yes	maximal
2	#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	#2931	C₂	1	1	yes	maximal, characteristic
4	#1703	C₂	1	4	yes	maximal, complemented
5	#1675	C₂	1	2	yes	maximal, complemented
6	#1669	C₂	1	2	yes	maximal, complemented
7	#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	#1559	C₂	1	2	yes	maximal, complemented
9	C₂×#131	—	2	1	no	—
10	#533	C₂²	1	2	yes	—
11	#532	C₂²	1	1	yes	characteristic
12	#501	—	2	2	no	—
13	#500	C₂²	1	1	yes	characteristic
14	#494	C₂²	1	1	yes	characteristic
15	#490	—	2	2	no	—
16	#487	C₂²	1	2	yes	—
17	#482	—	2	4	no	—
18	#476	C₂²	1	4	yes	—
19	#474	C₂²	1	4	yes	—
20	#467	—	2	1	no	—
21	#467	—	2	1	no	—
22	C₄×Q₁₆	—	2	1	no	—
23	C₄×Q₁₆	—	2	1	no	—
24	#436	—	2	1	no	—
25	#436	C₂²	1	2	yes	—
26	C₈×D₄	—	2	1	no	—
27	C₈×D₄	C₂²	1	2	yes	—
28	#432	C₂²	1	2	yes	—
29	#423	C₂²	1	1	yes	characteristic
30	#408	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#335	C₂²	1	2	yes	—
33	#334	C₂²	1	2	yes	—
34	#333	C₂²	1	4	yes	—
35	#332	C₂²	1	4	yes	—
36	C₂×#41	—	2	1	no	—
37	C₂×Q₁₆	—	2	1	no	—
38	C₂×Q₁₆	—	2	1	no	—
39	#131	—	4	4	no	—
40	C₂×M₁₆	—	2	1	no	—
41	C₂×M₁₆	—	2	2	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₈	D₄	1	1	yes	characteristic
45	C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₄⋊Q₈	C₂³	1	2	yes	—
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	#127	D₄	1	1	yes	characteristic
49	#125	—	2	2	no	—
50	#124	—	2	1	no	—
51	C₂²⋊Q₈	—	2	2	no	—
52	C₂²⋊Q₈	—	2	2	no	—
53	#121	—	2	1	no	—
54	#121	D₄	1	2	yes	—
55	C₄×Q₈	—	2	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	C₄×D₄	—	2	1	no	—
58	C₄×D₄	C₂³	1	2	yes	—
59	#117	—	2	2	no	—
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	#107	—	2	1	no	—
62	#107	—	2	1	no	—
63	#107	—	2	4	no	—
64	#106	—	2	4	no	—
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	C₂³	1	2	yes	—
67	C₄⋊C₈	C₂³	1	4	yes	—
68	#103	—	2	2	no	—
69	#103	—	2	2	no	—
70	#102	—	2	4	no	—
71	#102	—	2	4	no	—
72	C₂²⋊C₈	—	2	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	2	2	no	—
75	#97	—	2	2	no	—
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	D₄	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	#41	—	4	2	no	—
81	#41	—	4	2	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	1	no	—
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	C₂×D₄	1	2	yes	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	#41	1	1	yes	characteristic
93	Q₁₆	—	4	2	no	—
94	Q₁₆	—	4	2	no	—
95	M₁₆	—	4	4	no	—
96	M₁₆	—	4	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	C₂×D₄	1	2	yes	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₄²	—	4	2	no	—
118	C₄²	C₂×D₄	1	1	yes	characteristic
119	C₄²	C₂×D₄	1	1	yes	characteristic
120	C₄²	#41	1	1	yes	characteristic
121	C₄²	C₂⁴	1	1	yes	characteristic
122	C₂³	—	2	2	no	—
123	C₂³	#121	1	1	yes	characteristic
124	Q₈	—	4	2	no	—
125	Q₈	—	4	2	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	8	2	no	—
128	D₄	—	2	4	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	#121	1	1	yes	characteristic
142	C₂×C₄	#121	1	1	yes	characteristic
143	C₂×C₄	#121	1	1	yes	characteristic
144	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄	C₂×#41	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	2	no	—
151	C₈	—	4	4	no	—
152	C₈	AGL(1,ℤ/8)	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	C₂×#121	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	8	2	no	—
166	C₄	—	8	2	no	—
167	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₄	#575	1	1	yes	characteristic
169	C₄	#578	1	2	yes	—
170	C₂	—	4	1	no	—
171	C₂	—	8	2	no	—
172	C₂	#3036	1	1	yes	characteristic
173	C₂	#3059	1	1	yes	characteristic
174	C₂	#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#532	C₂²
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
2. #3170	C₂	semidirect
4. #1703	C₂	semidirect
5. #1675	C₂	semidirect
6. #1669	C₂	semidirect
7. #1568	C₂	semidirect
8. #1559	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3235	2
#3170	2
#2931	2
#1703	2
#1675	2
#1669	2
#1568	2
#1559	2