Subgroups — #483412

Order 512.

401 conjugacy classes of subgroups.

The subgroup lattice has 233 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2166	C₂	1	2	yes	maximal
2	#33324	C₂	1	1	yes	maximal, characteristic
3	#12027	C₂	1	4	yes	maximal, complemented
4	#12026	C₂	1	4	yes	maximal, complemented
5	#12011	C₂	1	2	yes	maximal, complemented
6	#12002	C₂	1	2	yes	maximal, complemented
7	C₂²×M₃₂	C₄	1	1	yes	characteristic
8	C₂²×M₃₂	C₄	1	1	yes	characteristic
9	C₂×#405	C₂²	1	1	yes	characteristic
10	#2166	—	2	2	no	—
11	#2166	—	2	2	no	—
12	#2166	C₂²	1	4	yes	—
13	#2166	C₂²	1	4	yes	—
14	C₂×#364	C₂²	1	4	yes	—
15	C₂×#349	—	2	2	no	—
16	C₂×#347	—	2	2	no	—
17	#2108	C₄	1	2	yes	—
18	#2108	C₄	1	2	yes	—
19	#2108	C₂²	1	2	yes	—
20	#2108	C₂²	1	2	yes	—
21	C₄×M₃₂	C₄	1	1	yes	characteristic
22	C₄×M₃₂	C₄	1	1	yes	characteristic
23	C₄×M₃₂	C₂²	1	1	yes	characteristic
24	C₄×M₃₂	C₂²	1	1	yes	characteristic
25	#1373	C₂²	1	8	yes	complemented
26	#1369	C₂²	1	8	yes	complemented
27	C₂²×M₁₆	D₄	1	1	yes	characteristic
28	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×#117	Q₈	1	1	yes	characteristic
30	C₂×M₃₂	—	2	4	no	—
31	C₂×M₃₂	—	2	4	no	—
32	C₂×M₃₂	—	2	4	no	—
33	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
34	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
35	C₂×M₃₂	C₂×C₄	1	2	yes	—
36	C₂×M₃₂	C₂×C₄	1	2	yes	—
37	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
38	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₆	C₂×C₄	1	2	yes	—
40	C₂²×C₁₆	C₂×C₄	1	2	yes	—
41	#405	C₂×C₄	1	1	yes	characteristic
42	#405	C₂×C₄	1	1	yes	characteristic
43	#405	D₄	1	1	yes	characteristic
44	#405	D₄	1	1	yes	characteristic
45	#405	Q₈	1	1	yes	characteristic
46	#405	Q₈	1	1	yes	characteristic
47	#405	C₂³	1	1	yes	characteristic
48	#405	C₂³	1	1	yes	characteristic
49	C₂×#97	D₄	1	1	yes	characteristic
50	C₂×#97	Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	#364	—	2	4	no	—
54	#364	C₂³	1	8	yes	—
55	#349	—	2	4	no	—
56	#349	—	2	4	no	—
57	#347	—	2	4	no	—
58	#347	—	2	4	no	—
59	#346	C₂×C₄	1	2	yes	—
60	#346	C₂×C₄	1	2	yes	—
61	#346	C₂×C₄	1	2	yes	—
62	#346	C₂³	1	2	yes	—
63	C₄×C₁₆	C₂×C₄	1	2	yes	—
64	C₄×C₁₆	C₂×C₄	1	2	yes	—
65	C₄×C₁₆	C₂×C₄	1	2	yes	—
66	C₄×C₁₆	C₂³	1	2	yes	—
67	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂×M₁₆	—	2	4	no	—
69	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
70	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
71	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
72	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₈	—	2	2	no	—
74	C₂²×C₈	—	2	2	no	—
75	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
83	#117	—	2	1	no	—
84	#117	—	2	1	no	—
85	#117	C₄⋊C₄	1	1	yes	characteristic
86	#117	C₄⋊C₄	1	1	yes	characteristic
87	#117	C₂×Q₈	1	1	yes	characteristic
88	#117	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₂²⋊C₄	—	2	1	no	—
91	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
93	M₃₂	—	2	2	no	—
94	M₃₂	—	2	2	no	—
95	M₃₂	—	2	4	no	—
96	M₃₂	—	2	4	no	—
97	M₃₂	—	2	8	no	—
98	M₃₂	—	4	4	no	—
99	M₃₂	M₁₆	1	4	yes	—
100	M₃₂	M₁₆	1	4	yes	—
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	4	4	no	—
104	C₂×C₁₆	C₂²×C₄	1	2	yes	—
105	C₂×C₁₆	C₂²×C₄	1	2	yes	—
106	C₂×C₁₆	C₂²×C₄	1	2	yes	—
107	C₂×C₁₆	C₂²×C₄	1	2	yes	—
108	#97	C₄⋊C₄	1	1	yes	characteristic
109	#97	C₄⋊C₄	1	1	yes	characteristic
110	#97	C₄⋊C₄	1	1	yes	characteristic
111	#97	C₄⋊C₄	1	1	yes	characteristic
112	#97	C₂×D₄	1	1	yes	characteristic
113	#97	C₂×D₄	1	1	yes	characteristic
114	#97	C₂×Q₈	1	1	yes	characteristic
115	#97	C₂×Q₈	1	1	yes	characteristic
116	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₄×C₈	C₂×D₄	1	1	yes	characteristic
122	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
123	C₄×C₈	C₂⁴	1	1	yes	characteristic
124	C₂⁴	#108	1	1	yes	characteristic
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	#108	1	1	yes	characteristic
130	C₂²×C₄	#108	1	1	yes	characteristic
131	C₂²×C₄	#108	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
135	M₁₆	—	2	2	no	—
136	M₁₆	—	2	2	no	—
137	M₁₆	—	2	2	no	—
138	M₁₆	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₈	C₂×M₁₆	1	2	yes	—
154	C₂×C₈	C₂×M₁₆	1	2	yes	—
155	C₂×C₈	#135	1	4	yes	—
156	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₂²⋊C₄	—	2	1	no	—
163	C₂²⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₂²⋊C₄	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₄²	C₂²×Q₈	1	1	yes	characteristic
172	C₁₆	—	2	2	no	—
173	C₁₆	—	2	2	no	—
174	C₁₆	—	4	8	no	—
175	C₁₆	C₂×M₁₆	1	4	yes	—
176	C₁₆	C₂×M₁₆	1	4	yes	—
177	C₂³	—	2	2	no	—
178	C₂³	—	8	1	no	—
179	C₂³	C₂×#108	1	1	yes	characteristic
180	C₂³	C₂×#108	1	1	yes	characteristic
181	C₂³	C₂×#108	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	#423	1	2	yes	—
196	C₂×C₄	#427	1	2	yes	—
197	C₂×C₄	C₂×#108	1	1	yes	characteristic
198	C₂×C₄	C₂×#108	1	1	yes	characteristic
199	C₂×C₄	C₂×#108	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	#423	1	2	yes	—
206	C₈	#427	1	2	yes	—
207	C₈	C₂²×M₁₆	1	1	yes	characteristic
208	C₈	C₂²×M₁₆	1	1	yes	characteristic
209	C₈	C₂×#135	1	2	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	8	2	no	—
216	C₂²	#1568	1	2	yes	—
217	C₂²	C₂²×#108	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	1	no	—
224	C₄	#1568	1	2	yes	—
225	C₄	C₂×#423	1	1	yes	characteristic
226	C₄	C₂×#427	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	2	no	—
229	C₂	—	4	1	no	—
230	C₂	—	4	1	no	—
231	C₂	C₂×#1568	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#427
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×#1568
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ C₂×#405 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12027	C₂	semidirect
4. #12026	C₂	semidirect
5. #12011	C₂	semidirect
6. #12002	C₂	semidirect
25. #1373	C₂²	semidirect
26. #1369	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2166	2
#33324	2
#12027	2
#12026	2
#12011	2
#12002	2