Finite Group Databasebeta

Subgroups — #122891

Order 512.

250 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 82 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	#19604	C₂	1	1	yes	maximal, characteristic
2	#12013	C₂	1	1	yes	maximal, characteristic
3	#12004	C₂	1	1	yes	maximal, characteristic
4	#7098	C₂	1	2	yes	maximal, complemented
5	#7097	C₂	1	2	yes	maximal, complemented
6	#3170	—	2	1	no	—
7	#3170	C₄	1	2	yes	—
8	#3160	—	2	1	no	—
9	#2163	—	2	1	no	—
10	#2150	—	2	1	no	—
11	#1713	C₂²	1	1	yes	characteristic
12	#1713	C₂²	1	1	yes	characteristic
13	#1710	—	2	1	no	—
14	#1707	—	2	1	no	—
15	#1707	C₄	1	2	yes	—
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	#1370	—	2	1	no	—
18	#1370	C₂²	1	2	yes	—
19	#1370	C₂²	1	2	yes	—
20	#1367	—	2	1	no	—
21	#533	—	2	2	no	—
22	C₂×M₃₂	—	4	1	no	—
23	#500	—	2	1	no	—
24	#500	—	2	1	no	—
25	#500	—	2	1	no	—
26	#500	C₂×C₄	1	2	yes	—
27	#499	—	4	1	no	—
28	#499	—	4	1	no	—
29	#498	—	2	2	no	—
30	#494	—	2	1	no	—
31	#494	C₂×C₄	1	2	yes	—
32	#487	—	2	2	no	—
33	#487	—	2	2	no	—
34	#467	—	4	1	no	—
35	#467	—	4	1	no	—
36	#467	—	4	2	no	—
37	C₄×Q₁₆	—	4	2	no	—
38	C₄×Q₁₆	—	4	1	no	—
39	C₄×Q₁₆	—	4	1	no	—
40	#427	—	4	1	no	—
41	#427	—	4	1	no	—
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₄⋊C₁₆	—	2	2	no	—
46	C₄⋊C₁₆	—	2	2	no	—
47	C₄⋊C₁₆	—	4	1	no	—
48	C₂²⋊C₁₆	—	4	1	no	—
49	#346	—	4	1	no	—
50	C₈²	C₂×C₄	1	1	yes	characteristic
51	C₈²	D₄	1	1	yes	characteristic
52	C₈²	D₄	1	1	yes	characteristic
53	C₈²	C₂³	1	1	yes	characteristic
54	C₂×Q₁₆	—	4	2	no	—
55	C₂×Q₁₆	—	4	1	no	—
56	C₂×Q₁₆	—	4	1	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	D₈	1	1	yes	characteristic
59	C₂²×C₈	D₈	1	1	yes	characteristic
60	C₂²×C₈	SD₁₆	1	1	yes	characteristic
61	C₂²×C₈	SD₁₆	1	1	yes	characteristic
62	C₄⋊Q₈	—	2	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	#125	—	4	2	no	—
65	C₂²⋊Q₈	—	4	2	no	—
66	C₂²⋊Q₈	—	4	2	no	—
67	C₄×Q₈	—	4	2	no	—
68	C₄×Q₈	—	4	2	no	—
69	#117	—	4	2	no	—
70	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
71	M₃₂	—	8	2	no	—
72	C₂×C₁₆	—	4	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	#107	—	4	1	no	—
75	#107	—	4	1	no	—
76	#107	—	4	1	no	—
77	#107	—	4	1	no	—
78	#107	—	4	1	no	—
79	#107	—	4	1	no	—
80	#107	—	4	2	no	—
81	#106	—	4	2	no	—
82	#106	—	4	2	no	—
83	#103	—	4	4	no	—
84	#103	—	4	2	no	—
85	#103	—	4	2	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×C₈	D₈	1	1	yes	characteristic
91	C₄×C₈	D₈	1	1	yes	characteristic
92	C₄×C₈	SD₁₆	1	1	yes	characteristic
93	C₄×C₈	SD₁₆	1	1	yes	characteristic
94	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₄×C₈	C₂×D₄	1	1	yes	characteristic
96	C₄×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×Q₈	—	4	2	no	—
98	C₂×Q₈	—	4	2	no	—
99	C₂²×C₄	#100	1	1	yes	characteristic
100	C₂²×C₄	#102	1	1	yes	characteristic
101	C₂²×C₄	#102	1	1	yes	characteristic
102	Q₁₆	—	8	4	no	—
103	Q₁₆	—	8	2	no	—
104	Q₁₆	—	8	1	no	—
105	Q₁₆	—	8	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	#102	1	1	yes	characteristic
115	C₂×C₈	#102	1	1	yes	characteristic
116	C₂×C₈	#102	1	1	yes	characteristic
117	C₂×C₈	#102	1	1	yes	characteristic
118	C₂×C₈	C₂×D₈	1	1	yes	characteristic
119	C₂×C₈	C₂×D₈	1	1	yes	characteristic
120	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	8	2	no	—
127	C₄⋊C₄	—	8	2	no	—
128	C₂²⋊C₄	—	8	2	no	—
129	C₄²	—	8	2	no	—
130	C₄²	#100	1	1	yes	characteristic
131	C₄²	#102	1	1	yes	characteristic
132	C₄²	#102	1	1	yes	characteristic
133	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₁₆	—	8	2	no	—
135	C₁₆	—	8	2	no	—
136	C₂³	#331	1	1	yes	characteristic
137	Q₈	—	8	4	no	—
138	Q₈	—	8	2	no	—
139	Q₈	—	8	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	#331	1	1	yes	characteristic
148	C₂×C₄	#331	1	1	yes	characteristic
149	C₂×C₄	#331	1	1	yes	characteristic
150	C₂×C₄	C₂×#100	1	1	yes	characteristic
151	C₂×C₄	C₂×#102	1	1	yes	characteristic
152	C₂×C₄	C₂×#102	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	4	1	no	—
160	C₈	#362	1	2	yes	—
161	C₈	#509	1	1	yes	characteristic
162	C₈	#510	1	1	yes	characteristic
163	C₈	#510	1	2	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	C₂×#331	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	16	2	no	—
173	C₄	—	16	2	no	—
174	C₄	#2139	1	1	yes	characteristic
175	C₄	#2141	1	1	yes	characteristic
176	C₄	C₂×#362	1	1	yes	characteristic
177	C₄	#2148	1	1	yes	characteristic
178	C₂	—	4	1	no	—
179	C₂	#11920	1	1	yes	characteristic
180	C₂	#11930	1	1	yes	characteristic
181	C₂	#11932	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
prime radical	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7098	C₂	semidirect
5. #7097	C₂	semidirect

Maximal subgroups

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

subgroup	index
#19604	2
#12013	2
#12004	2
#7098	2
#7097	2