Subgroups — #128954

Order 512.

310 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21732	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#11939	C₂	1	1	yes	maximal, characteristic
4	#11934	C₂	1	1	yes	maximal, characteristic
5	#8267	C₂	1	1	yes	maximal, characteristic
6	#7277	C₂	1	1	yes	maximal, characteristic
7	#7276	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	1	yes	characteristic
9	#2960	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	—	2	1	no	—
13	#1994	—	2	1	no	—
14	#1945	—	2	1	no	—
15	#1945	C₄	1	2	yes	—
16	#1848	C₂²	1	1	yes	characteristic
17	#1772	—	2	1	no	—
18	#1772	C₄	1	2	yes	—
19	C₄²×C₈	C₂²	1	1	yes	characteristic
20	#1564	C₂²	1	1	yes	characteristic
21	#1564	C₂²	1	1	yes	characteristic
22	#1452	—	2	1	no	—
23	#1374	—	2	1	no	—
24	#1365	C₄	1	2	yes	complemented
25	#1364	C₄	1	2	yes	complemented
26	#1363	C₄	1	4	yes	complemented
27	#1276	—	2	1	no	—
28	#533	—	2	1	no	—
29	C₂²×C₁₆	—	2	1	no	—
30	C₂²×C₁₆	—	2	1	no	—
31	#500	C₂×C₄	1	1	yes	characteristic
32	#500	C₂×C₄	1	1	yes	characteristic
33	#499	—	2	1	no	—
34	#499	C₂×C₄	1	2	yes	—
35	#498	—	2	1	no	—
36	#446	—	4	1	no	—
37	#446	—	4	1	no	—
38	C₈×Q₈	—	4	1	no	—
39	C₈×Q₈	—	4	1	no	—
40	#431	—	2	1	no	—
41	#431	—	2	1	no	—
42	C₂×#108	—	2	1	no	—
43	C₂×#108	C₂×C₄	1	2	yes	—
44	#427	—	2	1	no	—
45	#427	C₂×C₄	1	2	yes	—
46	#424	—	2	2	no	—
47	#423	—	2	1	no	—
48	#405	—	2	2	no	—
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₄³	D₄	1	1	yes	characteristic
56	C₄⋊C₁₆	—	2	1	no	—
57	C₄⋊C₁₆	—	2	1	no	—
58	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
59	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
60	C₄×C₁₆	—	4	2	no	—
61	#339	—	2	2	no	—
62	#339	—	2	2	no	—
63	#335	—	2	1	no	—
64	#335	C₂×C₄	1	2	yes	—
65	#334	—	2	1	no	—
66	#334	C₂×C₄	1	2	yes	—
67	#322	—	4	1	no	—
68	#322	—	4	1	no	—
69	C₂×M₁₆	—	2	2	no	—
70	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₄⋊Q₈	—	2	1	no	—
75	C₄⋊Q₈	—	2	1	no	—
76	#125	—	2	2	no	—
77	C₂²⋊Q₈	—	4	1	no	—
78	C₂²⋊Q₈	—	4	1	no	—
79	C₄×Q₈	—	4	1	no	—
80	C₄×Q₈	—	4	1	no	—
81	#117	—	2	2	no	—
82	C₂×C₄²	—	2	1	no	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	4	1	no	—
90	C₂×C₁₆	—	4	1	no	—
91	C₂×C₁₆	—	4	1	no	—
92	C₂×C₁₆	—	4	1	no	—
93	#108	—	4	2	no	—
94	#108	—	4	1	no	—
95	#108	—	4	1	no	—
96	#107	—	2	1	no	—
97	#107	—	2	1	no	—
98	#107	C₄²	1	2	yes	—
99	#107	C₄²	1	2	yes	—
100	#106	—	2	2	no	—
101	#106	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	4	2	no	—
105	C₄⋊C₈	—	4	2	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄⋊C₈	—	4	1	no	—
108	C₂²⋊C₈	—	4	1	no	—
109	C₂²⋊C₈	—	4	1	no	—
110	#97	—	4	2	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	4	1	no	—
117	C₄×C₈	—	4	2	no	—
118	C₄×C₈	—	4	1	no	—
119	C₄×C₈	—	4	1	no	—
120	C₄×C₈	D₈	1	1	yes	characteristic
121	C₄×C₈	D₈	1	1	yes	characteristic
122	C₄×C₈	SD₁₆	1	1	yes	characteristic
123	C₄×C₈	SD₁₆	1	1	yes	characteristic
124	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₄×C₈	#41	1	1	yes	characteristic
127	C₄×C₈	#41	1	1	yes	characteristic
128	C₂×Q₈	—	4	1	no	—
129	C₂×Q₈	—	4	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄	#135	1	1	yes	characteristic
134	C₂²×C₄	#135	1	1	yes	characteristic
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	#102	1	1	yes	characteristic
147	C₂×C₈	#102	1	1	yes	characteristic
148	C₂×C₈	#102	1	1	yes	characteristic
149	C₂×C₈	#102	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
151	C₂×C₈	#117	1	1	yes	characteristic
152	C₂×C₈	C₄×D₄	1	1	yes	characteristic
153	C₂×C₈	C₄×D₄	1	1	yes	characteristic
154	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₈	#137	1	1	yes	characteristic
156	C₂×C₈	#137	1	1	yes	characteristic
157	C₂×C₈	#137	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₂²⋊C₄	—	4	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	#101	1	2	yes	—
172	C₄²	C₄×D₄	1	1	yes	characteristic
173	C₄²	C₄×D₄	1	1	yes	characteristic
174	C₄²	C₂×D₈	1	1	yes	characteristic
175	C₄²	C₂×SD₁₆	1	1	yes	characteristic
176	C₁₆	—	4	2	no	—
177	C₁₆	—	4	2	no	—
178	C₂³	#416	1	1	yes	characteristic
179	Q₈	—	8	2	no	—
180	Q₈	—	8	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₄	C₂×#101	1	1	yes	characteristic
193	C₂×C₄	#413	1	1	yes	characteristic
194	C₂×C₄	C₂×#102	1	1	yes	characteristic
195	C₂×C₄	#417	1	1	yes	characteristic
196	C₂×C₄	#419	1	1	yes	characteristic
197	C₂×C₄	C₄×D₈	1	1	yes	characteristic
198	C₂×C₄	C₄×D₈	1	1	yes	characteristic
199	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
200	C₂×C₄	C₄×SD₁₆	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	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	8	2	no	—
208	C₈	—	8	2	no	—
209	C₈	—	8	2	no	—
210	C₈	#440	1	1	yes	characteristic
211	C₈	#440	1	1	yes	characteristic
212	C₈	#441	1	1	yes	characteristic
213	C₈	#441	1	1	yes	characteristic
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	#1552	1	1	yes	characteristic
217	C₂²	#1554	1	1	yes	characteristic
218	C₂²	C₄×#102	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	2	no	—
226	C₄	#1546	1	1	yes	characteristic
227	C₄	#1547	1	1	yes	characteristic
228	C₄	C₄×#101	1	1	yes	characteristic
229	C₄	#1763	1	1	yes	characteristic
230	C₄	#2178	1	2	yes	—
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	#8236	1	1	yes	characteristic
234	C₂	#12641	1	1	yes	characteristic
235	C₂	#12646	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	C₂³	#416
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ 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
24. #1365	C₄	semidirect
25. #1364	C₄	semidirect
26. #1363	C₄	semidirect

Maximal subgroups

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

subgroup	index
#21732	2
C₄×C₄⋊C₁₆	2
#11939	2
#11934	2
#8267	2
#7277	2
#7276	2