Finite Group Databasebeta

Subgroups — #144188

Order 512.

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

normal subgroups only

This group has 411 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	#33728	C₂	1	1	yes	maximal, characteristic, complemented
2	#33704	C₂	1	1	yes	maximal, characteristic
3	#19652	C₂	1	1	yes	maximal, characteristic
4	#13288	C₂	1	1	yes	maximal, characteristic, complemented
5	#13206	C₂	1	1	yes	maximal, characteristic
6	#9755	C₂	1	1	yes	maximal, characteristic, complemented
7	#7140	C₂	1	1	yes	maximal, characteristic, complemented
8	#3208	—	2	2	no	—
9	#3160	C₂²	1	1	yes	characteristic
10	#3160	C₂²	1	1	yes	characteristic
11	C₂×#493	C₂²	1	1	yes	characteristic
12	#2255	—	2	2	no	—
13	#2255	—	2	2	no	—
14	#2244	—	2	2	no	—
15	#2240	—	2	2	no	—
16	#2158	—	2	2	no	—
17	#2140	—	2	2	no	—
18	C₂×#357	—	2	2	no	—
19	C₄×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×#346	C₂²	1	1	yes	characteristic
21	#1846	C₂²	1	1	yes	characteristic
22	#1715	—	2	2	no	—
23	#1706	—	2	2	no	—
24	#1518	—	2	2	no	—
25	#1450	C₂²	1	1	yes	characteristic
26	#1386	—	2	2	no	—
27	C₂²×D₈	—	2	1	no	—
28	C₂²×D₈	—	2	1	no	—
29	#533	—	2	1	no	—
30	#533	—	2	1	no	—
31	C₂×#127	—	2	1	no	—
32	C₂×M₃₂	D₄	1	2	yes	—
33	C₂²×C₁₆	D₄	1	2	yes	—
34	#500	D₄	1	2	yes	—
35	#500	D₄	1	2	yes	—
36	#499	—	2	1	no	—
37	#499	—	2	1	no	—
38	#499	—	2	1	no	—
39	#499	—	2	1	no	—
40	#498	—	2	1	no	—
41	#498	—	2	1	no	—
42	#493	—	2	1	no	—
43	#493	—	2	1	no	—
44	#493	—	2	1	no	—
45	#493	D₄	1	2	yes	—
46	#492	—	2	4	no	—
47	#486	—	2	4	no	—
48	#460	—	4	4	no	—
49	#450	—	4	4	no	—
50	#431	D₄	1	2	yes	—
51	#427	—	2	1	no	—
52	#427	—	2	1	no	—
53	#427	—	2	1	no	—
54	#427	—	2	1	no	—
55	C₂×#107	—	2	1	no	—
56	C₂×#106	—	2	1	no	—
57	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
58	#382	—	2	1	no	—
59	#367	—	2	4	no	—
60	#366	—	2	4	no	—
61	#357	—	2	8	no	—
62	#357	—	4	4	no	—
63	#346	—	2	1	no	—
64	#346	—	2	1	no	—
65	#346	—	2	1	no	—
66	C₄×C₁₆	—	2	1	no	—
67	#328	—	4	4	no	—
68	#322	—	2	1	no	—
69	C₈²	—	2	1	no	—
70	C₂²×D₄	—	4	1	no	—
71	C₂²×D₄	—	4	2	no	—
72	C₂×SD₁₆	—	4	8	no	—
73	C₂×D₈	—	2	2	no	—
74	C₂×D₈	—	2	2	no	—
75	C₂×D₈	—	4	1	no	—
76	C₂×D₈	—	4	1	no	—
77	C₂×D₈	—	4	2	no	—
78	C₂×D₈	—	4	2	no	—
79	C₂×D₈	—	4	2	no	—
80	C₂×D₈	—	4	2	no	—
81	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄⋊Q₈	—	2	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	#127	—	2	2	no	—
86	#127	—	4	1	no	—
87	#127	—	4	2	no	—
88	#125	—	4	1	no	—
89	#125	—	4	1	no	—
90	C₂²⋊Q₈	—	4	2	no	—
91	C₂²⋊Q₈	—	4	2	no	—
92	C₄×Q₈	—	4	2	no	—
93	C₄×Q₈	—	4	2	no	—
94	#117	—	4	1	no	—
95	#117	—	4	1	no	—
96	C₂×C₄⋊C₄	—	4	1	no	—
97	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
98	M₃₂	—	2	2	no	—
99	M₃₂	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	SD₁₆	1	4	yes	complemented
104	#107	—	2	2	no	—
105	#107	—	2	2	no	—
106	#107	—	2	2	no	—
107	#107	—	2	2	no	—
108	#107	—	4	2	no	—
109	#106	—	4	1	no	—
110	#106	—	4	1	no	—
111	#106	—	4	1	no	—
112	#106	—	4	1	no	—
113	#106	—	4	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	#102	—	4	8	no	—
116	C₂²⋊C₈	D₈	1	4	yes	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	C₂×D₄	1	1	yes	characteristic
119	C₄×C₈	C₂×D₄	1	1	yes	characteristic
120	C₄×C₈	C₂×D₄	1	1	yes	characteristic
121	C₄×C₈	#41	1	1	yes	characteristic
122	#95	—	4	2	no	—
123	C₂⁴	—	8	1	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×D₄	—	4	2	no	—
127	C₂×D₄	—	4	4	no	—
128	C₂×D₄	—	8	2	no	—
129	C₂×D₄	—	8	1	no	—
130	C₂×D₄	—	8	1	no	—
131	C₂×D₄	—	8	1	no	—
132	C₂×D₄	—	8	1	no	—
133	C₂×D₄	—	8	1	no	—
134	C₂×D₄	—	16	4	no	—
135	C₂²×C₄	—	8	1	no	—
136	C₂²×C₄	#127	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₈	1	2	yes	—
138	SD₁₆	—	16	8	no	—
139	D₈	—	4	4	no	—
140	D₈	—	4	4	no	—
141	D₈	—	8	2	no	—
142	D₈	—	8	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	#120	1	1	yes	characteristic
145	C₂×C₈	#120	1	1	yes	characteristic
146	C₂×C₈	#121	1	1	yes	characteristic
147	C₂×C₈	#121	1	1	yes	characteristic
148	C₂×C₈	C₂×SD₁₆	1	2	yes	—
149	C₂×C₈	#135	1	2	yes	—
150	C₂×C₈	#137	1	2	yes	—
151	C₂×C₈	#137	1	2	yes	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	8	2	no	—
157	C₄⋊C₄	—	8	2	no	—
158	C₄⋊C₄	—	8	1	no	—
159	C₄⋊C₄	—	8	1	no	—
160	C₂²⋊C₄	—	8	1	no	—
161	C₂²⋊C₄	—	8	1	no	—
162	C₄²	—	8	1	no	—
163	C₄²	—	8	1	no	—
164	C₄²	#120	1	1	yes	characteristic
165	C₄²	#121	1	1	yes	characteristic
166	C₄²	AGL(1,ℤ/8)	1	2	yes	—
167	C₁₆	—	2	4	no	—
168	C₁₆	—	4	2	no	—
169	C₂³	—	8	2	no	—
170	C₂³	—	16	4	no	—
171	C₂³	—	16	1	no	—
172	C₂³	—	16	1	no	—
173	C₂³	#493	1	1	yes	characteristic
174	Q₈	—	8	4	no	—
175	Q₈	—	16	2	no	—
176	D₄	—	8	8	no	—
177	D₄	—	8	2	no	—
178	D₄	—	8	2	no	—
179	D₄	—	16	4	no	—
180	D₄	—	16	1	no	—
181	D₄	—	16	1	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	16	2	no	—
186	C₂×C₄	—	16	1	no	—
187	C₂×C₄	#392	1	1	yes	characteristic
188	C₂×C₄	#447	1	2	yes	—
189	C₂×C₄	#459	1	2	yes	—
190	C₂×C₄	#492	1	1	yes	characteristic
191	C₂×C₄	#495	1	1	yes	characteristic
192	C₂×C₄	#496	1	1	yes	characteristic
193	C₂×C₄	#497	1	1	yes	characteristic
194	C₂×C₄	#497	1	1	yes	characteristic
195	C₈	—	4	4	no	—
196	C₈	#448	1	2	yes	—
197	C₈	#452	1	2	yes	—
198	C₈	#461	1	2	yes	—
199	C₈	#464	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	—	16	4	no	—
202	C₂²	—	16	2	no	—
203	C₂²	—	32	2	no	—
204	C₂²	—	32	2	no	—
205	C₂²	#1706	1	1	yes	characteristic
206	C₂²	#1722	1	1	yes	characteristic
207	C₂²	#2011	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	16	2	no	—
211	C₄	—	16	2	no	—
212	C₄	—	32	1	no	—
213	C₄	#1714	1	1	yes	characteristic
214	C₄	#1719	1	1	yes	characteristic
215	C₄	#2013	1	1	yes	characteristic
216	C₄	#2015	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	—	32	2	no	—
220	C₂	#10308	1	1	yes	characteristic
221	C₂	#13334	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33728	C₂	semidirect
4. #13288	C₂	semidirect
6. #9755	C₂	semidirect
7. #7140	C₂	semidirect
103. C₂×C₁₆	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#33728	2
#33704	2
#19652	2
#13288	2
#13206	2
#9755	2
#7140	2