Finite Group Databasebeta

Subgroups — #150389

Order 512.

218 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 70 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	#33735	C₂	1	1	yes	maximal, characteristic
2	#13640	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	#7421	C₂	1	2	yes	maximal, complemented
5	#7420	C₂	1	2	yes	maximal
6	#3168	—	2	1	no	—
7	#2253	C₄	1	1	yes	characteristic
8	#2249	C₄	1	1	yes	characteristic
9	#2242	—	2	1	no	—
10	#2217	—	2	1	no	—
11	#2216	C₂²	1	1	yes	characteristic
12	#2215	C₂²	1	1	yes	characteristic
13	C₄×Q₃₂	C₄	1	1	yes	characteristic
14	C₄×SD₃₂	—	2	1	no	—
15	C₄×D₁₆	C₄	1	1	yes	characteristic
16	C₂×#368	—	2	1	no	—
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₄×#108	—	2	1	no	—
19	C₄⋊C₃₂	—	2	1	no	—
20	C₂²⋊C₃₂	C₂²	1	2	yes	—
21	#1397	—	2	1	no	—
22	#1381	C₂²	1	2	yes	—
23	#529	—	4	1	no	—
24	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
25	C₂×SD₃₂	—	2	1	no	—
26	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	D₄	1	1	yes	characteristic
28	C₂²×C₁₆	C₂³	1	1	yes	characteristic
29	#499	—	2	1	no	—
30	#495	—	2	1	no	—
31	#488	—	4	1	no	—
32	#467	—	2	1	no	—
33	#466	—	2	1	no	—
34	#465	—	4	1	no	—
35	C₄×Q₁₆	—	2	1	no	—
36	C₄×SD₁₆	—	4	1	no	—
37	C₄×D₈	—	2	1	no	—
38	C₂×#108	—	2	2	no	—
39	C₄×M₁₆	—	4	1	no	—
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃₂	—	2	2	no	—
42	#368	—	2	4	no	—
43	#367	—	2	1	no	—
44	#366	C₂×C₄	1	1	yes	characteristic
45	#366	C₂×C₄	1	1	yes	characteristic
46	#358	—	2	1	no	—
47	#358	—	2	1	no	—
48	#357	—	2	1	no	—
49	#357	—	2	1	no	—
50	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
51	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
52	C₄×C₁₆	D₄	1	1	yes	characteristic
53	C₄×C₁₆	D₄	1	1	yes	characteristic
54	#341	—	4	1	no	—
55	C₂×Q₁₆	—	2	1	no	—
56	C₂×SD₁₆	—	4	1	no	—
57	C₂×D₈	—	2	1	no	—
58	C₂×M₁₆	—	4	2	no	—
59	C₂²×C₈	D₈	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	#126	—	8	1	no	—
62	#125	—	4	1	no	—
63	#124	—	4	1	no	—
64	#123	—	8	1	no	—
65	C₂²⋊Q₈	—	4	1	no	—
66	#121	—	4	1	no	—
67	C₄×Q₈	—	4	1	no	—
68	C₄×D₄	—	4	1	no	—
69	C₄×D₄	—	8	1	no	—
70	#117	—	8	1	no	—
71	C₂×C₄²	D₈	1	1	yes	characteristic
72	Q₃₂	—	4	1	no	—
73	SD₃₂	—	4	2	no	—
74	D₁₆	—	4	1	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	—	4	1	no	—
77	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
81	#108	—	2	4	no	—
82	#108	—	4	2	no	—
83	#107	—	2	1	no	—
84	#107	—	2	1	no	—
85	#106	—	4	1	no	—
86	#103	—	4	1	no	—
87	#103	—	4	1	no	—
88	#102	—	4	1	no	—
89	#102	—	4	1	no	—
90	#97	—	8	1	no	—
91	C₄×C₈	—	8	1	no	—
92	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₈	SD₁₆	1	1	yes	characteristic
95	C₄×C₈	SD₁₆	1	1	yes	characteristic
96	C₃₂	—	4	2	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂×D₄	—	4	1	no	—
99	C₂×D₄	—	8	1	no	—
100	C₂²×C₄	—	8	1	no	—
101	C₂²×C₄	SD₃₂	1	1	yes	characteristic
102	C₂²×C₄	SD₃₂	1	1	yes	characteristic
103	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
104	Q₁₆	—	4	1	no	—
105	Q₁₆	—	4	1	no	—
106	SD₁₆	—	8	2	no	—
107	D₈	—	4	1	no	—
108	D₈	—	4	1	no	—
109	M₁₆	—	4	4	no	—
110	M₁₆	—	8	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	8	2	no	—
115	C₂×C₈	#102	1	1	yes	characteristic
116	C₂×C₈	#102	1	1	yes	characteristic
117	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₄⋊C₄	—	8	1	no	—
123	C₄⋊C₄	—	8	1	no	—
124	C₄⋊C₄	—	8	1	no	—
125	C₂²⋊C₄	—	8	1	no	—
126	C₂²⋊C₄	—	8	1	no	—
127	C₂²⋊C₄	—	8	1	no	—
128	C₂²⋊C₄	—	8	1	no	—
129	C₂²⋊C₄	—	8	1	no	—
130	C₄²	—	8	1	no	—
131	C₄²	#102	1	1	yes	characteristic
132	C₄²	#102	1	1	yes	characteristic
133	C₄²	D₁₆	1	1	yes	characteristic
134	C₄²	D₁₆	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	2	1	no	—
137	C₁₆	—	2	1	no	—
138	C₁₆	#104	1	2	yes	—
139	C₂³	—	8	1	no	—
140	C₂³	C₂×SD₃₂	1	1	yes	characteristic
141	Q₈	—	8	1	no	—
142	Q₈	—	8	1	no	—
143	D₄	—	8	1	no	—
144	D₄	—	8	1	no	—
145	D₄	—	16	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	#357	1	1	yes	characteristic
155	C₂×C₄	#357	1	1	yes	characteristic
156	C₂×C₄	#357	1	1	yes	characteristic
157	C₂×C₄	#357	1	1	yes	characteristic
158	C₂×C₄	C₂×#102	1	1	yes	characteristic
159	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	4	no	—
164	C₈	#410	1	1	yes	characteristic
165	C₈	C₂×#104	1	1	yes	characteristic
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	16	1	no	—
170	C₂²	—	16	1	no	—
171	C₂²	—	16	1	no	—
172	C₂²	C₂×#357	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	16	1	no	—
179	C₄	—	16	1	no	—
180	C₄	—	16	1	no	—
181	C₄	#1502	1	1	yes	characteristic
182	C₄	#2145	1	1	yes	characteristic
183	C₂	—	4	1	no	—
184	C₂	—	32	1	no	—
185	C₂	#11742	1	1	yes	characteristic
186	C₂	#13620	1	1	yes	characteristic
187	C₂	#13625	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#2215	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13640	C₂	semidirect
3. C₄×#368	C₂	semidirect
4. #7421	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33735	2
#13640	2
C₄×#368	2
#7421	2
#7420	2