Finite Group Databasebeta

Subgroups — #9176

Order 256.

245 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 75 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	#3078	C₂	1	1	yes	maximal, characteristic
2	#2351	C₂	1	1	yes	maximal, characteristic
3	#1518	C₂	1	1	yes	maximal, characteristic
4	#1503	C₂	1	1	yes	maximal, characteristic
5	#1479	C₂	1	1	yes	maximal, characteristic
6	#1303	C₂	1	1	yes	maximal, characteristic
7	#1286	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈	—	2	1	no	—
10	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
11	#479	—	2	2	no	—
12	#477	C₄	1	2	yes	complemented
13	#475	C₄	1	2	yes	complemented
14	#431	C₂²	1	1	yes	characteristic
15	#431	C₂²	1	1	yes	characteristic
16	#428	—	2	1	no	—
17	#423	C₂²	1	1	yes	characteristic
18	#419	—	2	1	no	—
19	#389	—	2	1	no	—
20	#389	—	2	2	no	—
21	#382	—	2	1	no	—
22	#382	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄	—	2	1	no	—
24	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	#328	—	2	1	no	—
26	#328	—	2	1	no	—
27	#328	—	2	2	no	—
28	Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈	D₄	1	1	yes	characteristic
30	C₂×M₁₆	—	2	1	no	—
31	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	#125	—	2	1	no	—
34	C₂²⋊Q₈	—	2	1	no	—
35	C₂²⋊Q₈	—	2	1	no	—
36	C₄×Q₈	—	2	1	no	—
37	C₄×Q₈	—	2	1	no	—
38	C₄×Q₈	—	2	1	no	—
39	C₄×Q₈	—	2	1	no	—
40	C₄×Q₈	—	4	2	no	—
41	C₄×Q₈	—	4	2	no	—
42	C₄×Q₈	C₂×C₄	1	2	yes	—
43	C₄×Q₈	D₄	1	2	yes	—
44	#117	—	2	1	no	—
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄⋊C₄	—	2	1	no	—
49	C₂×C₄⋊C₄	—	2	1	no	—
50	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₂×C₄²	—	2	1	no	—
52	C₂×C₄²	—	2	2	no	—
53	C₂×C₄²	—	2	1	no	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#107	—	2	2	no	—
56	#106	—	2	2	no	—
57	C₄⋊C₈	—	2	1	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	C₂×C₄	1	2	yes	—
61	#103	—	2	2	no	—
62	#103	—	2	2	no	—
63	C₂²⋊C₈	—	2	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	#95	—	2	2	no	—
68	#95	—	2	1	no	—
69	#95	D₄	1	1	yes	characteristic
70	#95	D₄	1	1	yes	characteristic
71	#95	D₄	1	2	yes	—
72	C₂×Q₈	—	2	1	no	—
73	C₂×Q₈	—	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	C₂×Q₈	SD₁₆	1	2	yes	—
79	C₂×Q₈	Q₁₆	1	2	yes	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	M₁₆	—	4	1	no	—
89	M₁₆	—	4	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	—	4	1	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
108	C₂²⋊C₄	—	4	1	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	4	4	no	—
112	C₄²	—	4	1	no	—
113	C₄²	—	4	1	no	—
114	C₄²	—	4	1	no	—
115	C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₄²	C₂×D₄	1	1	yes	characteristic
117	C₄²	#41	1	1	yes	characteristic
118	C₄²	#41	1	1	yes	characteristic
119	C₂³	#120	1	1	yes	characteristic
120	Q₈	—	2	2	no	—
121	Q₈	—	2	2	no	—
122	Q₈	—	2	2	no	—
123	Q₈	—	8	1	no	—
124	Q₈	—	8	1	no	—
125	Q₈	#103	1	4	yes	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	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	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄	#121	1	1	yes	characteristic
148	C₂×C₄	#121	1	1	yes	characteristic
149	C₂×C₄	#123	1	1	yes	characteristic
150	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
152	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄	#137	1	1	yes	characteristic
154	C₈	—	4	2	no	—
155	C₈	—	8	1	no	—
156	C₈	—	8	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#386	1	1	yes	characteristic
160	C₂²	#450	1	1	yes	characteristic
161	C₂²	#451	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	2	no	—
169	C₄	—	8	2	no	—
170	C₄	—	8	1	no	—
171	C₄	—	8	1	no	—
172	C₄	—	8	1	no	—
173	C₄	C₂×#103	1	1	yes	characteristic
174	C₄	#417	1	1	yes	characteristic
175	C₄	#483	1	1	yes	characteristic
176	C₄	#484	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#1872	1	1	yes	characteristic
180	C₂	#1905	1	1	yes	characteristic
181	C₂	#1906	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
12. #477	C₄	semidirect
13. #475	C₄	semidirect

Maximal subgroups

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

subgroup	index
#3078	2
#2351	2
#1518	2
#1503	2
#1479	2
#1303	2
#1286	2