Finite Group Databasebeta

Subgroups — #134651

Order 512.

280 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 92 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	#21728	C₂	1	1	yes	maximal, characteristic
2	#12882	C₂	1	1	yes	maximal, characteristic
3	#12882	C₂	1	1	yes	maximal, characteristic
4	C₂×#1366	C₂	1	2	yes	maximal
5	#7271	C₂	1	2	yes	maximal
6	C₂×#499	C₂²	1	1	yes	characteristic
7	C₄×#125	C₄	1	2	yes	—
8	C₂×#364	C₂²	1	2	yes	—
9	C₂×#364	C₂²	1	2	yes	—
10	#1991	C₄	1	2	yes	—
11	#1944	—	2	1	no	—
12	#1943	—	2	1	no	—
13	#1846	C₂²	1	1	yes	characteristic
14	#1776	—	2	1	no	—
15	#1776	—	2	1	no	—
16	C₄×#97	C₂²	1	1	yes	characteristic
17	#1377	—	2	1	no	—
18	#1377	—	2	1	no	—
19	#1375	—	2	1	no	—
20	#1375	—	2	1	no	—
21	#1366	—	2	4	no	—
22	C₂×#125	C₂×C₄	1	2	yes	—
23	C₂×M₃₂	—	2	2	no	—
24	C₂×M₃₂	—	2	2	no	—
25	#499	—	2	1	no	—
26	#499	—	2	1	no	—
27	#499	D₄	1	2	yes	—
28	#499	Q₈	1	2	yes	—
29	C₂×#107	—	2	1	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#106	—	2	1	no	—
32	C₂×#106	—	2	1	no	—
33	C₂×#97	D₄	1	1	yes	characteristic
34	C₂×#97	D₄	1	1	yes	characteristic
35	C₂×#97	D₄	1	1	yes	characteristic
36	C₂×#97	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#389	—	2	2	no	—
40	#387	—	4	2	no	—
41	#382	C₂×C₄	1	2	yes	—
42	C₄×C₄⋊C₄	—	2	2	no	—
43	C₄×C₄⋊C₄	—	4	2	no	—
44	C₄³	C₂×C₄	1	1	yes	characteristic
45	#364	—	2	2	no	—
46	#364	—	2	2	no	—
47	#364	—	2	2	no	—
48	#364	—	2	2	no	—
49	#340	—	2	2	no	—
50	#340	—	2	2	no	—
51	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	#125	—	2	2	no	—
56	#125	—	4	2	no	—
57	#125	C₄⋊C₄	1	4	yes	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	C₂×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄⋊C₄	—	4	2	no	—
62	C₂×C₄²	—	2	2	no	—
63	C₂×C₄²	—	4	2	no	—
64	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
67	M₃₂	—	4	2	no	—
68	M₃₂	—	4	2	no	—
69	M₃₂	—	4	2	no	—
70	M₃₂	—	4	2	no	—
71	C₂×C₁₆	—	4	2	no	—
72	C₂×C₁₆	—	4	2	no	—
73	#107	—	2	2	no	—
74	#107	—	2	2	no	—
75	#107	—	4	2	no	—
76	#106	—	2	2	no	—
77	#106	—	2	2	no	—
78	#106	—	4	2	no	—
79	#97	—	2	2	no	—
80	#97	—	2	2	no	—
81	#97	—	2	2	no	—
82	#97	—	2	2	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	C₂×D₄	1	1	yes	characteristic
85	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₄×C₈	#41	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	#95	—	4	2	no	—
89	#95	—	4	2	no	—
90	#95	—	4	2	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	#100	1	2	yes	—
96	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	#121	1	1	yes	characteristic
104	C₂×C₈	#121	1	1	yes	characteristic
105	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₈	#135	1	2	yes	—
108	C₂×C₈	#135	1	2	yes	—
109	C₂×C₈	#137	1	2	yes	—
110	C₂×C₈	#137	1	2	yes	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	4	4	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	8	4	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	8	2	no	—
124	C₄²	—	8	4	no	—
125	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₄²	#117	1	1	yes	characteristic
127	C₁₆	—	8	2	no	—
128	C₁₆	—	8	2	no	—
129	C₂³	C₂×#100	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	#380	1	1	yes	characteristic
144	C₂×C₄	#413	1	1	yes	characteristic
145	C₂×C₄	#416	1	1	yes	characteristic
146	C₂×C₄	#416	1	1	yes	characteristic
147	C₂×C₄	#419	1	1	yes	characteristic
148	C₂×C₄	#419	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	#464	1	2	yes	—
152	C₈	#464	1	2	yes	—
153	C₈	#479	1	2	yes	—
154	C₈	#479	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	#1543	1	1	yes	characteristic
157	C₂²	#1558	1	1	yes	characteristic
158	C₂²	#1800	1	1	yes	characteristic
159	C₄	—	4	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	2	no	—
162	C₄	—	8	1	no	—
163	C₄	—	8	4	no	—
164	C₄	—	16	2	no	—
165	C₄	#1554	1	1	yes	characteristic
166	C₄	#1554	1	1	yes	characteristic
167	C₄	#1805	1	1	yes	characteristic
168	C₄	#1805	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#8594	1	1	yes	characteristic
172	C₂	#11994	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#21728	2
#12882	2
C₂×#1366	2
#7271	2