Finite Group Databasebeta

Subgroups — #30313

Order 256.

295 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 127 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	#2660	C₂	1	1	yes	maximal, characteristic
2	#2647	C₂	1	1	yes	maximal, characteristic
3	#2085	C₂	1	2	yes	maximal
4	#2083	C₂	1	2	yes	maximal
5	#2059	C₂	1	2	yes	maximal
6	#2057	C₂	1	2	yes	maximal
7	#1847	C₂	1	1	yes	maximal, characteristic
8	#1845	C₂	1	1	yes	maximal, characteristic
9	#1836	C₂	1	1	yes	maximal, characteristic
10	#1835	C₂	1	1	yes	maximal, characteristic
11	#1746	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×#125	C₂²	1	1	yes	characteristic
14	C₂×#107	C₂²	1	2	yes	—
15	C₂×#107	C₂²	1	2	yes	—
16	C₂×#107	C₂²	1	2	yes	—
17	C₂×#106	C₂²	1	2	yes	—
18	C₂×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	#400	—	2	2	no	—
21	#398	—	2	2	no	—
22	#398	—	2	2	no	—
23	#395	—	2	2	no	—
24	#389	C₂²	1	2	yes	—
25	#389	C₂²	1	2	yes	—
26	#384	C₂²	1	1	yes	characteristic
27	#384	C₂²	1	1	yes	characteristic
28	#384	C₂²	1	1	yes	characteristic
29	#382	C₂²	1	1	yes	characteristic
30	#382	C₂²	1	1	yes	characteristic
31	#382	C₂²	1	1	yes	characteristic
32	#376	C₂²	1	1	yes	characteristic
33	#340	C₂²	1	2	yes	—
34	#340	C₂²	1	2	yes	—
35	#340	C₂²	1	2	yes	—
36	#340	C₂²	1	2	yes	—
37	#336	C₂²	1	2	yes	—
38	#336	C₂²	1	2	yes	—
39	C₂²×Q₈	—	2	1	no	—
40	C₂²×C₈	C₂³	1	2	yes	—
41	C₂²×C₈	C₂³	1	2	yes	—
42	C₄⋊Q₈	—	4	2	no	—
43	#125	—	4	2	no	—
44	C₂×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄⋊C₄	—	2	2	no	—
51	C₂×C₄⋊C₄	—	2	2	no	—
52	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
54	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄	C₂³	1	2	yes	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	#107	—	4	2	no	—
62	#107	—	4	2	no	—
63	#107	—	4	2	no	—
64	#106	—	4	2	no	—
65	#97	Q₈	1	4	yes	complemented
66	C₄×C₈	Q₈	1	4	yes	complemented
67	#95	—	2	2	no	—
68	#95	—	2	2	no	—
69	#95	—	2	2	no	—
70	#95	—	2	2	no	—
71	#95	—	2	2	no	—
72	#95	D₄	1	2	yes	—
73	#95	D₄	1	2	yes	—
74	C₂×Q₈	—	4	2	no	—
75	C₂×Q₈	—	8	2	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₄	1	2	yes	—
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	2	yes	—
86	C₂²×C₄	C₂⁴	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	C₂×Q₈	1	4	yes	—
92	C₂×C₈	C₂×Q₈	1	4	yes	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	8	2	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₄⋊C₄	—	8	2	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄⋊C₄	—	8	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	C₂×Q₈	1	2	yes	—
111	C₄²	C₂×Q₈	1	2	yes	—
112	C₂³	C₂²×D₄	1	1	yes	characteristic
113	Q₈	—	8	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₄⋊Q₈	1	4	yes	—
138	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
141	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
142	C₈	—	2	4	no	—
143	C₈	—	2	4	no	—
144	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²	#535	1	1	yes	characteristic
146	C₂²	#536	1	1	yes	characteristic
147	C₂²	#576	1	2	yes	—
148	C₂²	#578	1	2	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	8	1	no	—
154	C₄	—	8	2	no	—
155	C₄	—	8	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	8	1	no	—
158	C₄	—	8	2	no	—
159	C₄	#544	1	2	yes	—
160	C₄	#544	1	2	yes	—
161	C₂	#2660	1	1	yes	characteristic
162	C₂	#3164	1	2	yes	—
163	C₂	#3220	1	1	yes	characteristic
164	C₂	#3223	1	1	yes	characteristic
165	C₂	#3243	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
65. #97	Q₈	semidirect
66. C₄×C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2660	2
#2647	2
#2085	2
#2083	2
#2059	2
#2057	2
#1847	2
#1845	2
#1836	2
#1835	2
#1746	2