Subgroups — #213242

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21734	C₂	1	2	yes	maximal
2	#19604	C₂	1	1	yes	maximal, characteristic
3	#17977	C₂	1	2	yes	maximal
4	#17958	C₂	1	2	yes	maximal
5	#9861	C₂	1	2	yes	maximal
6	#9848	C₂	1	2	yes	maximal
7	#9845	C₂	1	2	yes	maximal
8	#8823	C₂	1	1	yes	maximal, characteristic
9	C₈×#97	C₂	1	1	yes	maximal, characteristic
10	#3170	C₂²	1	2	yes	—
11	#3170	C₂²	1	2	yes	—
12	#3160	C₂²	1	2	yes	—
13	#2960	C₂²	1	4	yes	—
14	#1994	C₂²	1	4	yes	—
15	#1946	—	2	4	no	—
16	#1848	C₂²	1	2	yes	—
17	#1773	—	2	4	no	—
18	#1744	C₂²	1	1	yes	characteristic
19	C₄×#97	C₂²	1	1	yes	characteristic
20	C₄²×C₈	C₂²	1	1	yes	characteristic
21	#1713	C₄	1	1	yes	characteristic, complemented
22	#1713	C₄	1	1	yes	characteristic, complemented
23	#1710	C₄	1	2	yes	complemented
24	#1707	C₄	1	2	yes	complemented
25	#1707	C₄	1	2	yes	complemented
26	#1591	—	2	4	no	—
27	#1584	—	2	8	no	—
28	C₈×Q₁₆	—	2	4	no	—
29	#1567	C₂²	1	2	yes	—
30	#1567	C₂²	1	2	yes	—
31	#1493	C₂²	1	4	yes	—
32	#1493	C₂²	1	4	yes	—
33	C₂×#322	C₂²	1	1	yes	characteristic
34	C₂×#322	C₂²	1	1	yes	characteristic
35	C₂×C₈²	C₂²	1	1	yes	characteristic
36	C₂×C₈²	C₂²	1	1	yes	characteristic
37	#533	C₂³	1	4	yes	—
38	#500	C₂×C₄	1	2	yes	—
39	#500	C₂×C₄	1	2	yes	—
40	#500	C₂×C₄	1	2	yes	—
41	#500	C₂×C₄	1	2	yes	—
42	#499	—	2	4	no	—
43	#498	C₂×C₄	1	4	yes	—
44	#494	C₂×C₄	1	2	yes	—
45	#494	C₂×C₄	1	2	yes	—
46	#487	C₂×C₄	1	4	yes	—
47	#487	C₂×C₄	1	4	yes	—
48	#467	—	2	4	no	—
49	#467	—	2	4	no	—
50	#446	—	2	4	no	—
51	#446	—	2	4	no	—
52	C₈×Q₈	—	2	4	no	—
53	C₈×Q₈	—	2	4	no	—
54	C₄×Q₁₆	—	2	4	no	—
55	C₄×Q₁₆	—	2	4	no	—
56	C₂×#108	—	2	4	no	—
57	#427	—	2	4	no	—
58	#424	—	2	4	no	—
59	#423	C₂³	1	4	yes	—
60	#405	—	2	4	no	—
61	C₂×#97	D₄	1	2	yes	—
62	C₂×#97	D₄	1	2	yes	—
63	C₂×C₄×C₈	—	2	2	no	—
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₄³	C₂³	1	1	yes	characteristic
71	#339	—	2	8	no	—
72	#336	—	2	2	no	—
73	#335	—	2	4	no	—
74	#335	—	2	4	no	—
75	Q₈⋊C₈	—	2	8	no	—
76	Q₈⋊C₈	—	2	8	no	—
77	#322	—	2	2	no	—
78	#322	D₄	1	4	yes	—
79	C₈²	C₂×C₄	1	1	yes	characteristic
80	C₈²	C₂×C₄	1	1	yes	characteristic
81	C₈²	C₂×C₄	1	1	yes	characteristic
82	C₈²	C₂×C₄	1	1	yes	characteristic
83	C₈²	D₄	1	4	yes	—
84	C₂×Q₁₆	—	2	4	no	—
85	C₂×Q₁₆	—	2	4	no	—
86	C₂×M₁₆	—	2	4	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₂²×C₈	—	2	1	no	—
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₄	1	2	yes	—
91	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₈	#41	1	2	yes	—
93	C₂²×C₈	#41	1	2	yes	—
94	C₄⋊Q₈	C₂²×C₄	1	4	yes	—
95	C₄⋊Q₈	C₂²×C₄	1	4	yes	—
96	#125	—	2	4	no	—
97	C₂²⋊Q₈	—	2	4	no	—
98	C₂²⋊Q₈	—	2	4	no	—
99	C₄×Q₈	—	2	4	no	—
100	C₄×Q₈	—	2	4	no	—
101	#117	—	2	4	no	—
102	C₂×C₄²	—	2	1	no	—
103	C₂×C₄²	—	2	1	no	—
104	C₂×C₄²	C₂×D₄	1	2	yes	—
105	C₂×C₄²	C₂⁴	1	1	yes	characteristic
106	#108	—	4	8	no	—
107	#107	—	2	4	no	—
108	#107	—	2	4	no	—
109	#107	—	2	4	no	—
110	#107	—	2	4	no	—
111	#106	—	2	8	no	—
112	C₄⋊C₈	—	2	4	no	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	4	4	no	—
115	C₄⋊C₈	—	4	4	no	—
116	#103	—	2	8	no	—
117	#103	—	2	8	no	—
118	#97	—	2	4	no	—
119	#97	—	4	4	no	—
120	#97	Q₁₆	1	8	yes	complemented
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	4	4	no	—
124	C₄×C₈	—	4	4	no	—
125	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
127	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
128	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
129	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
130	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₄×C₈	C₂×D₄	1	2	yes	—
132	C₄×C₈	C₂×D₄	1	2	yes	—
133	C₄×C₈	C₂×D₄	1	2	yes	—
134	C₄×C₈	C₂×D₄	1	2	yes	—
135	C₄×C₈	C₂×D₄	1	2	yes	—
136	C₄×C₈	C₂×D₄	1	2	yes	—
137	C₄×C₈	#41	1	2	yes	—
138	C₂×Q₈	—	2	4	no	—
139	C₂×Q₈	—	2	4	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×#41	1	1	yes	characteristic
143	C₂²×C₄	C₂×#41	1	1	yes	characteristic
144	Q₁₆	—	4	8	no	—
145	Q₁₆	—	8	4	no	—
146	M₁₆	—	4	8	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	C₄×D₄	1	2	yes	—
156	C₂×C₈	C₄×D₄	1	2	yes	—
157	C₂×C₈	C₄×D₄	1	2	yes	—
158	C₂×C₈	C₄×D₄	1	2	yes	—
159	C₂×C₈	C₄×D₄	1	2	yes	—
160	C₂×C₈	C₄×D₄	1	2	yes	—
161	C₂×C₈	#127	1	2	yes	—
162	C₂×C₈	#127	1	2	yes	—
163	C₂×C₈	C₂×Q₁₆	1	4	yes	—
164	C₂×C₈	C₂×Q₁₆	1	4	yes	—
165	C₂×C₈	#135	1	4	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₂²⋊C₄	—	4	4	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	4	2	no	—
175	C₄²	—	4	2	no	—
176	C₄²	—	4	4	no	—
177	C₄²	C₂×Q₁₆	1	4	yes	—
178	C₄²	C₂³×C₄	1	1	yes	characteristic
179	C₄²	C₂²×D₄	1	1	yes	characteristic
180	C₄²	C₂²×D₄	1	1	yes	characteristic
181	C₄²	C₂×#41	1	1	yes	characteristic
182	C₂³	#532	1	1	yes	characteristic
183	Q₈	—	4	8	no	—
184	Q₈	—	4	8	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
195	C₂×C₄	C₂×#127	1	1	yes	characteristic
196	C₂×C₄	#532	1	1	yes	characteristic
197	C₂×C₄	#532	1	1	yes	characteristic
198	C₂×C₄	C₂²×Q₁₆	1	2	yes	—
199	C₂×C₄	C₂×#135	1	2	yes	—
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	4	2	no	—
206	C₈	—	8	4	no	—
207	C₈	C₄×Q₁₆	1	4	yes	—
208	C₈	C₄×Q₁₆	1	4	yes	—
209	C₈	#444	1	4	yes	—
210	C₈	#494	1	4	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₄×#127	1	1	yes	characteristic
214	C₂²	#3170	1	2	yes	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	4	no	—
218	C₄	—	8	4	no	—
219	C₄	—	8	4	no	—
220	C₄	C₂×C₄×Q₁₆	1	2	yes	—
221	C₄	C₂×#444	1	2	yes	—
222	C₄	#2957	1	2	yes	—
223	C₄	C₂×#494	1	1	yes	characteristic
224	C₄	#3147	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	C₄×#494	1	1	yes	characteristic
227	C₂	#21706	1	1	yes	characteristic
228	C₂	#21707	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	C₂³	#532
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
21. #1713	C₄	semidirect
22. #1713	C₄	semidirect
23. #1710	C₄	semidirect
24. #1707	C₄	semidirect
25. #1707	C₄	semidirect
120. #97	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#21734	2
#19604	2
#17977	2
#17958	2
#9861	2
#9848	2
#9845	2
#8823	2
C₈×#97	2