Finite Group Databasebeta

Subgroups — #9562

Order 256.

199 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 81 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	#1988	C₂	1	1	yes	maximal, characteristic
2	#1956	C₂	1	1	yes	maximal, characteristic
3	#1948	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂×#332	C₂	1	1	yes	maximal, characteristic
6	#1296	C₂	1	1	yes	maximal, characteristic
7	#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#106	—	2	1	no	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×#103	—	2	1	no	—
14	C₂×#103	—	2	1	no	—
15	C₂×C₄×C₈	—	2	1	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#384	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	#340	—	2	1	no	—
20	#340	—	2	1	no	—
21	#340	—	2	1	no	—
22	#336	—	2	1	no	—
23	#332	C₄	1	4	yes	complemented
24	C₂²×Q₈	—	2	1	no	—
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	—	2	1	no	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₈	D₄	1	1	yes	characteristic
30	C₄⋊Q₈	—	2	1	no	—
31	C₄⋊Q₈	—	2	1	no	—
32	C₄⋊Q₈	—	2	1	no	—
33	C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄	—	2	1	no	—
35	C₂×C₄⋊C₄	—	2	1	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄⋊C₄	—	2	1	no	—
39	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₂×C₄²	—	2	1	no	—
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	#106	—	4	2	no	—
44	C₄⋊C₈	—	2	1	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₄⋊C₈	C₂×C₄	1	2	yes	—
50	#103	—	4	2	no	—
51	#103	—	4	2	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	—	4	2	no	—
55	#95	—	2	1	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	4	1	no	—
58	C₂×Q₈	—	4	1	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	4	1	no	—
65	C₂²×C₄	—	4	1	no	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄	#41	1	1	yes	characteristic
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	—	4	1	no	—
78	C₂×C₈	—	4	1	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	SD₁₆	1	2	yes	—
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	4	1	no	—
90	C₄⋊C₄	—	4	1	no	—
91	C₄⋊C₄	—	4	1	no	—
92	C₄⋊C₄	—	4	1	no	—
93	C₄⋊C₄	—	4	1	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	SD₁₆	1	2	yes	—
96	C₄⋊C₄	Q₁₆	1	2	yes	—
97	C₄²	—	4	1	no	—
98	C₄²	—	4	1	no	—
99	C₄²	C₂²×C₄	1	1	yes	characteristic
100	C₄²	#41	1	1	yes	characteristic
101	C₄²	#41	1	1	yes	characteristic
102	C₄²	#41	1	1	yes	characteristic
103	C₂³	#121	1	1	yes	characteristic
104	Q₈	—	4	2	no	—
105	Q₈	—	4	2	no	—
106	Q₈	—	8	1	no	—
107	Q₈	—	8	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	8	1	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	#117	1	1	yes	characteristic
123	C₂×C₄	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄	#123	1	1	yes	characteristic
126	C₂×C₄	#124	1	1	yes	characteristic
127	C₂×C₄	#126	1	1	yes	characteristic
128	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₄	#135	1	1	yes	characteristic
132	C₂×C₄	#135	1	1	yes	characteristic
133	C₂×C₄	#135	1	1	yes	characteristic
134	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	4	2	no	—
138	C₈	—	4	2	no	—
139	C₈	—	4	1	no	—
140	C₈	—	8	1	no	—
141	C₂²	#388	1	1	yes	characteristic
142	C₂²	#460	1	1	yes	characteristic
143	C₂²	#463	1	1	yes	characteristic
144	C₂²	#465	1	1	yes	characteristic
145	C₂²	#467	1	1	yes	characteristic
146	C₂²	#471	1	1	yes	characteristic
147	C₂²	#473	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	16	1	no	—
154	C₄	C₄×SD₁₆	1	1	yes	characteristic
155	C₄	C₄×SD₁₆	1	1	yes	characteristic
156	C₄	C₄×Q₁₆	1	1	yes	characteristic
157	C₄	#442	1	1	yes	characteristic
158	C₄	#481	1	1	yes	characteristic
159	C₄	#482	1	1	yes	characteristic
160	C₄	#487	1	1	yes	characteristic
161	C₄	#488	1	1	yes	characteristic
162	C₂	#1666	1	1	yes	characteristic
163	C₂	#1670	1	1	yes	characteristic
164	C₂	#1679	1	1	yes	characteristic
165	C₂	#1683	1	1	yes	characteristic
166	C₂	#1926	1	1	yes	characteristic
167	C₂	#1930	1	1	yes	characteristic
168	C₂	#1933	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
23. #332	C₄	semidirect

Maximal subgroups

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

subgroup	index
#1988	2
#1956	2
#1948	2
C₈×C₄⋊C₄	2
C₂×#332	2
#1296	2
#1277	2