Finite Group Databasebeta

Subgroups — #9056

Order 256.

216 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 71 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	#3170	C₂	1	1	yes	maximal, characteristic
2	#2962	C₂	1	1	yes	maximal, characteristic
3	#1986	C₂	1	1	yes	maximal, characteristic
4	#1503	C₂	1	1	yes	maximal, characteristic
5	#1481	C₂	1	1	yes	maximal, characteristic
6	#1279	C₂	1	1	yes	maximal, characteristic
7	#1278	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	#533	C₂²	1	1	yes	characteristic
10	#500	C₄	1	1	yes	characteristic, complemented
11	#494	C₄	1	1	yes	characteristic, complemented
12	#487	C₄	1	1	yes	characteristic, complemented
13	#487	C₄	1	1	yes	characteristic, complemented
14	#467	—	2	2	no	—
15	#446	—	2	2	no	—
16	#446	—	2	1	no	—
17	C₈×Q₈	—	2	1	no	—
18	C₄×Q₁₆	—	2	2	no	—
19	#431	C₂²	1	1	yes	characteristic
20	#424	—	2	1	no	—
21	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	#405	—	2	1	no	—
23	C₄×M₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	#339	—	2	1	no	—
27	#339	—	2	1	no	—
28	#328	—	2	2	no	—
29	Q₈⋊C₈	—	2	1	no	—
30	Q₈⋊C₈	—	2	1	no	—
31	C₂×Q₁₆	—	2	2	no	—
32	C₂×M₁₆	D₄	1	1	yes	characteristic
33	C₂×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	—	2	1	no	—
36	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
40	#125	—	2	1	no	—
41	#125	—	2	1	no	—
42	C₂²⋊Q₈	—	2	1	no	—
43	C₂²⋊Q₈	—	2	1	no	—
44	C₂²⋊Q₈	—	2	1	no	—
45	C₂²⋊Q₈	D₄	1	2	yes	—
46	C₄×Q₈	—	2	1	no	—
47	C₄×Q₈	—	2	1	no	—
48	C₄×Q₈	—	2	1	no	—
49	C₄×Q₈	D₄	1	2	yes	—
50	#117	—	2	1	no	—
51	#117	—	2	1	no	—
52	C₂×C₄⋊C₄	—	2	1	no	—
53	C₂×C₄²	—	2	1	no	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#107	—	2	2	no	—
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	D₄	1	1	yes	characteristic
61	C₄⋊C₈	D₄	1	1	yes	characteristic
62	#103	—	2	2	no	—
63	#103	—	2	2	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	#97	—	2	2	no	—
66	#97	—	2	1	no	—
67	C₄×C₈	—	2	2	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	#95	—	2	1	no	—
74	C₂×Q₈	—	2	1	no	—
75	C₂×Q₈	—	2	1	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄	—	4	1	no	—
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄	#41	1	1	yes	characteristic
82	Q₁₆	—	4	4	no	—
83	M₁₆	—	2	1	no	—
84	M₁₆	—	2	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	Q₁₆	1	2	yes	complemented
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	4	1	no	—
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	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	4	1	no	—
115	C₄²	—	4	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	C₂²×C₄	1	1	yes	characteristic
118	C₄²	C₂×D₄	1	1	yes	characteristic
119	C₄²	C₂×D₄	1	1	yes	characteristic
120	C₄²	#41	1	1	yes	characteristic
121	C₂³	#121	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	4	1	no	—
126	Q₈	—	4	1	no	—
127	Q₈	—	4	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	8	1	no	—
137	C₂×C₄	—	8	1	no	—
138	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄	#120	1	1	yes	characteristic
142	C₂×C₄	#121	1	1	yes	characteristic
143	C₂×C₄	#123	1	1	yes	characteristic
144	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₄	#135	1	1	yes	characteristic
146	C₂×C₄	#137	1	1	yes	characteristic
147	C₂×C₄	#137	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	2	no	—
150	C₈	—	4	1	no	—
151	C₈	—	4	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	4	1	no	—
154	C₈	—	8	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#386	1	1	yes	characteristic
158	C₂²	#462	1	1	yes	characteristic
159	C₂²	#464	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	4	2	no	—
164	C₄	—	4	2	no	—
165	C₄	—	8	2	no	—
166	C₄	—	8	1	no	—
167	C₄	—	8	1	no	—
168	C₄	C₄×Q₁₆	1	1	yes	characteristic
169	C₄	#441	1	1	yes	characteristic
170	C₄	#443	1	1	yes	characteristic
171	C₄	#444	1	1	yes	characteristic
172	C₄	#451	1	1	yes	characteristic
173	C₄	#452	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#1876	1	1	yes	characteristic
177	C₂	#1878	1	1	yes	characteristic
178	C₂	#1884	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#386
prime radical	C₂³	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
10. #500	C₄	semidirect
11. #494	C₄	semidirect
12. #487	C₄	semidirect
13. #487	C₄	semidirect
86. M₁₆	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3170	2
#2962	2
#1986	2
#1503	2
#1481	2
#1279	2
#1278	2