Finite Group Databasebeta

Subgroups — #128947

Order 512.

283 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 114 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	#21733	C₂	1	1	yes	maximal, characteristic
2	#12660	C₂	1	1	yes	maximal, characteristic
3	#12656	C₂	1	1	yes	maximal, characteristic
4	#11956	C₂	1	1	yes	maximal, characteristic
5	#11956	C₂	1	1	yes	maximal, characteristic
6	#7270	C₂	1	1	yes	maximal, characteristic
7	#7270	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	1	yes	characteristic
9	#2961	C₄	1	2	yes	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	1	yes	characteristic
12	C₂×#364	C₂²	1	1	yes	characteristic
13	C₂×#364	C₂²	1	1	yes	characteristic
14	C₂×#347	—	2	1	no	—
15	C₄×M₃₂	—	2	1	no	—
16	#1995	C₄	1	2	yes	—
17	#1946	—	2	1	no	—
18	#1945	—	2	1	no	—
19	#1848	C₂²	1	1	yes	characteristic
20	#1773	—	2	1	no	—
21	#1772	—	2	1	no	—
22	#1726	C₂²	1	1	yes	characteristic
23	#1377	—	2	1	no	—
24	#1376	—	2	1	no	—
25	#1366	C₄	1	4	yes	complemented
26	#1366	C₄	1	4	yes	complemented
27	#533	C₂×C₄	1	2	yes	—
28	C₂×M₃₂	—	2	1	no	—
29	C₂×M₃₂	—	2	1	no	—
30	C₂×M₃₂	—	2	1	no	—
31	C₂×M₃₂	—	2	1	no	—
32	#500	—	2	1	no	—
33	#499	C₂×C₄	1	2	yes	—
34	#499	C₂×C₄	1	2	yes	—
35	#498	—	2	1	no	—
36	#446	—	4	4	no	—
37	C₂×#108	—	2	1	no	—
38	C₂×#108	—	2	1	no	—
39	#427	—	2	1	no	—
40	#427	—	2	1	no	—
41	#424	—	2	2	no	—
42	#423	C₂×C₄	1	2	yes	—
43	#405	—	2	2	no	—
44	C₂×#97	C₂×C₄	1	1	yes	characteristic
45	C₂×#97	D₄	1	1	yes	characteristic
46	C₂×#97	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	#376	C₂×C₄	1	1	yes	characteristic
51	#364	—	2	1	no	—
52	#364	—	2	1	no	—
53	#364	—	2	1	no	—
54	#364	—	2	1	no	—
55	#364	C₂×C₄	1	2	yes	—
56	#364	C₂×C₄	1	2	yes	—
57	#364	C₂×C₄	1	2	yes	—
58	#364	C₂×C₄	1	2	yes	—
59	#347	—	4	2	no	—
60	#346	—	4	1	no	—
61	C₄×C₁₆	—	4	1	no	—
62	#339	—	2	2	no	—
63	#339	—	2	2	no	—
64	C₂×M₁₆	—	2	2	no	—
65	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₄⋊Q₈	—	2	2	no	—
70	#125	C₄²	1	4	yes	—
71	C₂²⋊Q₈	—	4	2	no	—
72	C₄×Q₈	—	4	2	no	—
73	#117	—	2	2	no	—
74	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
77	M₃₂	—	2	2	no	—
78	M₃₂	—	2	2	no	—
79	M₃₂	—	2	2	no	—
80	M₃₂	—	2	2	no	—
81	M₃₂	—	4	1	no	—
82	M₃₂	—	4	1	no	—
83	M₃₂	—	4	1	no	—
84	M₃₂	—	4	1	no	—
85	C₂×C₁₆	—	4	1	no	—
86	C₂×C₁₆	—	4	1	no	—
87	C₂×C₁₆	—	4	1	no	—
88	C₂×C₁₆	—	4	1	no	—
89	#108	—	4	1	no	—
90	#108	—	4	1	no	—
91	#108	—	4	1	no	—
92	#108	—	4	1	no	—
93	#107	—	2	2	no	—
94	#107	—	2	2	no	—
95	#106	—	2	2	no	—
96	#106	—	2	2	no	—
97	C₄⋊C₈	—	4	2	no	—
98	C₄⋊C₈	—	4	4	no	—
99	#97	—	2	2	no	—
100	#97	—	2	1	no	—
101	#97	—	2	1	no	—
102	#97	—	4	2	no	—
103	#97	SD₁₆	1	2	yes	—
104	#97	Q₁₆	1	2	yes	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	4	2	no	—
110	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₄×C₈	#41	1	1	yes	characteristic
113	C₄×C₈	#41	1	1	yes	characteristic
114	#95	—	4	1	no	—
115	C₂×Q₈	—	4	2	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	M₁₆	—	4	2	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	#100	1	2	yes	—
128	C₂×C₈	#103	1	2	yes	—
129	C₂×C₈	C₄×D₄	1	1	yes	characteristic
130	C₂×C₈	C₄×D₄	1	1	yes	characteristic
131	C₂×C₈	C₄×D₄	1	1	yes	characteristic
132	C₂×C₈	C₄×D₄	1	1	yes	characteristic
133	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×C₈	#135	1	1	yes	characteristic
136	C₂×C₈	#135	1	1	yes	characteristic
137	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₈	#137	1	1	yes	characteristic
139	C₂×C₈	#137	1	1	yes	characteristic
140	C₂×C₈	#137	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	2	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	4	2	no	—
150	C₄²	#103	1	2	yes	—
151	C₄²	C₂×C₄²	1	1	yes	characteristic
152	C₄²	#117	1	1	yes	characteristic
153	C₁₆	—	4	2	no	—
154	C₁₆	—	4	2	no	—
155	C₁₆	—	4	2	no	—
156	C₁₆	—	4	2	no	—
157	C₂³	#413	1	1	yes	characteristic
158	Q₈	—	16	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	1	no	—
166	C₂×C₄	—	8	1	no	—
167	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄	C₂×#100	1	1	yes	characteristic
169	C₂×C₄	C₂×#103	1	1	yes	characteristic
170	C₂×C₄	#416	1	1	yes	characteristic
171	C₂×C₄	#417	1	1	yes	characteristic
172	C₂×C₄	#419	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₈	—	8	2	no	—
176	C₈	C₄×SD₁₆	1	1	yes	characteristic
177	C₈	C₄×SD₁₆	1	1	yes	characteristic
178	C₈	C₄×Q₁₆	1	1	yes	characteristic
179	C₈	C₄×Q₁₆	1	1	yes	characteristic
180	C₈	#440	1	1	yes	characteristic
181	C₈	#440	1	1	yes	characteristic
182	C₈	#441	1	1	yes	characteristic
183	C₈	#441	1	1	yes	characteristic
184	C₂²	—	4	1	no	—
185	C₂²	#1550	1	1	yes	characteristic
186	C₂²	#1554	1	1	yes	characteristic
187	C₂²	C₄×#100	1	1	yes	characteristic
188	C₄	—	4	1	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	1	no	—
191	C₄	—	8	1	no	—
192	C₄	—	8	4	no	—
193	C₄	#1548	1	1	yes	characteristic
194	C₄	#1551	1	1	yes	characteristic
195	C₄	C₄×#103	1	1	yes	characteristic
196	C₄	#1763	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	#8235	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1551
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8235
prime radical	C₂³	#413
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ 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
25. #1366	C₄	semidirect
26. #1366	C₄	semidirect

Maximal subgroups

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

subgroup	index
#21733	2
#12660	2
#12656	2
#11956	2
#7270	2