Finite Group Databasebeta

Subgroups — #8070

Order 256.

281 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 115 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	#2920	C₂	1	1	yes	maximal, characteristic
2	#2920	C₂	1	1	yes	maximal, characteristic
3	#2283	C₂	1	1	yes	maximal, characteristic
4	#1300	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²	C₂²	1	1	yes	characteristic
6	#432	—	2	1	no	—
7	#432	C₄	1	2	yes	complemented
8	#432	C₄	1	2	yes	complemented
9	#432	C₂²	1	2	yes	—
10	#431	—	2	1	no	—
11	#431	—	2	1	no	—
12	#431	—	2	1	no	—
13	#431	C₄	1	2	yes	complemented
14	#423	—	2	1	no	—
15	#423	C₄	1	2	yes	complemented
16	#407	—	2	1	no	—
17	#407	—	2	1	no	—
18	C₄×M₁₆	—	2	1	no	—
19	C₄×M₁₆	C₂²	1	2	yes	—
20	#383	C₄	1	2	yes	—
21	#382	—	2	1	no	—
22	#381	—	2	1	no	—
23	#380	C₄	1	2	yes	—
24	C₄×C₄⋊C₄	—	2	1	no	—
25	C₄×C₂²⋊C₄	—	2	1	no	—
26	#376	C₂²	1	2	yes	—
27	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄	D₄	1	1	yes	characteristic
29	C₂³×C₄	D₄	1	1	yes	characteristic
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₂²⋊C₄	—	2	2	no	—
34	C₂×C₄²	—	2	2	no	—
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	D₄	1	1	yes	characteristic
41	C₂×C₄²	D₄	1	1	yes	characteristic
42	C₂×C₄²	D₄	1	1	yes	characteristic
43	C₂×C₄²	D₄	1	1	yes	characteristic
44	C₂×C₄²	Q₈	1	1	yes	characteristic
45	C₂×C₄²	Q₈	1	1	yes	characteristic
46	C₂×C₄²	C₂³	1	1	yes	characteristic
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	2	2	no	—
49	C₄⋊C₈	—	2	2	no	—
50	C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈	—	2	2	no	—
52	C₂²⋊C₈	—	2	2	no	—
53	C₂²⋊C₈	—	2	2	no	—
54	C₂²⋊C₈	—	2	2	no	—
55	#97	—	2	2	no	—
56	#97	C₂×C₄	1	4	yes	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	2	2	no	—
59	#95	—	2	2	no	—
60	#95	—	2	2	no	—
61	#95	—	2	2	no	—
62	#95	C₂×C₄	1	4	yes	—
63	C₂⁴	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄	—	2	4	no	—
65	C₂²×C₄	—	4	2	no	—
66	C₂²×C₄	—	4	2	no	—
67	C₂²×C₄	—	4	2	no	—
68	C₂²×C₄	C₄²	1	1	yes	characteristic
69	C₂²×C₄	C₄²	1	1	yes	characteristic
70	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	M₁₆	—	4	4	no	—
87	M₁₆	—	4	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₂²⋊C₄	—	4	4	no	—
92	C₄²	—	4	4	no	—
93	C₄²	C₄²	1	1	yes	characteristic
94	C₄²	C₄²	1	1	yes	characteristic
95	C₄²	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²	C₄⋊C₄	1	1	yes	characteristic
100	C₄²	C₄⋊C₄	1	1	yes	characteristic
101	C₄²	C₄⋊C₄	1	1	yes	characteristic
102	C₄²	C₄⋊C₄	1	1	yes	characteristic
103	C₄²	C₄⋊C₄	1	1	yes	characteristic
104	C₄²	C₄⋊C₄	1	1	yes	characteristic
105	C₄²	C₂²×C₄	1	1	yes	characteristic
106	C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₄²	C₂×D₄	1	1	yes	characteristic
108	C₄²	C₂×Q₈	1	1	yes	characteristic
109	C₂³	—	2	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	#95	1	1	yes	characteristic
113	C₂³	#95	1	1	yes	characteristic
114	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	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	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	#95	1	1	yes	characteristic
137	C₂×C₄	#95	1	1	yes	characteristic
138	C₂×C₄	#95	1	1	yes	characteristic
139	C₂×C₄	#95	1	1	yes	characteristic
140	C₂×C₄	#95	1	1	yes	characteristic
141	C₂×C₄	#95	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₈	—	4	4	no	—
149	C₈	—	4	4	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	4	4	no	—
159	C₂²	C₂×#95	1	1	yes	characteristic
160	C₂²	#421	1	2	yes	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	8	4	no	—
172	C₄	#344	1	2	yes	—
173	C₄	#421	1	2	yes	—
174	C₂	—	2	2	no	—
175	C₂	—	4	1	no	—
176	C₂	—	4	1	no	—
177	C₂	#1733	1	1	yes	characteristic
178	C₂	#1733	1	1	yes	characteristic
179	C₂	C₂×#344	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₂×#95
prime radical	C₂⁴	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1300	C₂	semidirect
7. #432	C₄	semidirect
8. #432	C₄	semidirect
13. #431	C₄	semidirect
15. #423	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2920	2
#2283	2
#1300	2