Finite Group Databasebeta

Subgroups — #93279

Order 512.

320 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 118 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	#12599	C₂	1	1	yes	maximal, characteristic
2	C₂×#1374	C₂	1	1	yes	maximal, characteristic
3	#8144	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₁₆	C₄	1	2	yes	—
5	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
7	C₂×#346	—	2	1	no	—
8	C₄×C₂²⋊C₈	C₄	1	2	yes	—
9	C₂×#322	—	2	1	no	—
10	#1374	—	2	1	no	—
11	#1374	—	2	1	no	—
12	#1374	—	2	1	no	—
13	#1273	—	2	1	no	—
14	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₆	—	2	2	no	—
18	C₂²×C₁₆	—	2	2	no	—
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	—	4	2	no	—
21	C₂²×C₁₆	—	4	2	no	—
22	C₂²×C₁₆	C₈	1	4	yes	complemented
23	C₂×C₂²⋊C₈	—	2	2	no	—
24	C₂×C₄×C₈	—	2	2	no	—
25	C₂×C₄×C₈	—	4	2	no	—
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
30	C₂²⋊C₁₆	—	2	2	no	—
31	C₂²⋊C₁₆	C₈	1	4	yes	complemented
32	#346	—	4	2	no	—
33	#336	C₈	1	4	yes	—
34	#322	—	4	2	no	—
35	C₂³×C₄	C₄²	1	1	yes	characteristic
36	C₂³×C₄	M₁₆	1	2	yes	—
37	C₂²×C₈	—	2	2	no	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	—	2	4	no	—
41	C₂²×C₈	—	4	1	no	—
42	C₂²×C₈	—	4	1	no	—
43	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂×C₈	1	2	yes	—
48	C₂²×C₈	C₂×C₈	1	2	yes	—
49	C₂²×C₈	M₁₆	1	2	yes	—
50	C₂²×C₈	M₁₆	1	2	yes	—
51	C₂×C₄²	—	4	1	no	—
52	C₂×C₄²	—	4	1	no	—
53	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄²	C₂×C₈	1	2	yes	—
56	C₂×C₁₆	—	2	4	no	—
57	C₂×C₁₆	—	2	4	no	—
58	C₂×C₁₆	—	4	2	no	—
59	C₂×C₁₆	—	4	2	no	—
60	C₂×C₁₆	—	4	4	no	—
61	C₂×C₁₆	—	4	2	no	—
62	C₂×C₁₆	—	4	2	no	—
63	C₂×C₁₆	—	4	4	no	—
64	C₂²⋊C₈	—	4	4	no	—
65	C₄×C₈	—	2	2	no	—
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	—	4	1	no	—
68	C₄×C₈	—	4	1	no	—
69	C₄×C₈	—	4	2	no	—
70	C₄×C₈	—	4	2	no	—
71	C₂⁴	#97	1	1	yes	characteristic
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	2	1	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	2	4	no	—
78	C₂²×C₄	—	4	2	no	—
79	C₂²×C₄	#95	1	1	yes	characteristic
80	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
81	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
82	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
83	C₂²×C₄	#97	1	1	yes	characteristic
84	C₂²×C₄	#97	1	1	yes	characteristic
85	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
86	C₂²×C₄	C₄⋊C₈	1	2	yes	—
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	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	4	no	—
100	C₂×C₈	—	4	4	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	C₂²⋊C₈	1	2	yes	—
103	C₂×C₈	C₂²⋊C₈	1	2	yes	—
104	C₂×C₈	#99	1	2	yes	—
105	C₂×C₈	#100	1	1	yes	characteristic
106	C₂×C₈	#101	1	1	yes	characteristic
107	C₂×C₈	C₄⋊C₈	1	2	yes	—
108	C₂×C₈	C₄⋊C₈	1	2	yes	—
109	C₄²	—	2	1	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	4	2	no	—
114	C₁₆	—	4	4	no	—
115	C₁₆	—	4	4	no	—
116	C₂³	—	2	1	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	2	no	—
119	C₂³	#322	1	1	yes	characteristic
120	C₂³	#322	1	1	yes	characteristic
121	C₂³	#336	1	1	yes	characteristic
122	C₂³	#347	1	4	yes	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	C₈²	1	1	yes	characteristic
140	C₂×C₄	#322	1	1	yes	characteristic
141	C₂×C₄	#336	1	1	yes	characteristic
142	C₂×C₄	#336	1	1	yes	characteristic
143	C₂×C₄	#342	1	1	yes	characteristic
144	C₂×C₄	#343	1	1	yes	characteristic
145	C₂×C₄	#344	1	1	yes	characteristic
146	C₂×C₄	#344	1	1	yes	characteristic
147	C₂×C₄	#350	1	4	yes	—
148	C₈	—	2	2	no	—
149	C₈	—	2	2	no	—
150	C₈	—	2	2	no	—
151	C₈	—	8	4	no	—
152	C₈	#323	1	2	yes	—
153	C₈	#324	1	2	yes	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#1273	1	1	yes	characteristic
161	C₂²	#1282	1	1	yes	characteristic
162	C₂²	#1283	1	1	yes	characteristic
163	C₂²	#1313	1	2	yes	—
164	C₂²	#1313	1	2	yes	—
165	C₂²	#1375	1	2	yes	—
166	C₂²	#1379	1	2	yes	—
167	C₄	—	2	4	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	2	no	—
171	C₄	#1280	1	1	yes	characteristic
172	C₄	#1281	1	1	yes	characteristic
173	C₄	#1282	1	1	yes	characteristic
174	C₄	#1283	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	#7016	1	1	yes	characteristic
178	C₂	#7137	1	1	yes	characteristic
179	C₂	#7141	1	1	yes	characteristic
180	C₂	#7200	1	2	yes	—
181	C₂	#7210	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂×C₄	C₈²
Frattini, cosocle	C₂²×C₄×C₈	C₂²
socle	C₂³	#336
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
22. C₂²×C₁₆	C₈	semidirect
31. C₂²⋊C₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
#12599	2
C₂×#1374	2
#8144	2