Subgroups — #18016

Order 256.

300 conjugacy classes of subgroups.

The subgroup lattice has 215 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3107	C₂	1	1	yes	maximal, characteristic
2	#2990	C₂	1	1	yes	maximal, characteristic
3	#2986	C₂	1	1	yes	maximal, characteristic, complemented
4	#1591	C₂	1	2	yes	maximal, complemented
5	#1587	C₂	1	2	yes	maximal, complemented
6	#1585	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆	C₂	1	2	yes	maximal, complemented
8	#1574	C₂	1	1	yes	maximal, characteristic, complemented
9	#1493	C₂	1	1	yes	maximal, characteristic, complemented
10	#1476	C₂	1	1	yes	maximal, characteristic
11	#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	#533	C₂²	1	1	yes	characteristic
13	C₄×#41	C₂²	1	1	yes	characteristic
14	#485	C₄	1	2	yes	—
15	#475	C₄	1	1	yes	characteristic
16	#474	C₄	1	1	yes	characteristic
17	#462	C₄	1	1	yes	characteristic
18	#461	C₄	1	1	yes	characteristic
19	#452	C₄	1	2	yes	—
20	#446	—	2	1	no	—
21	#446	C₂²	1	2	yes	—
22	#446	C₂²	1	2	yes	—
23	C₈×Q₈	—	2	1	no	—
24	C₈×Q₈	C₂²	1	2	yes	—
25	C₄×Q₁₆	C₂²	1	2	yes	—
26	C₄×SD₁₆	C₂²	1	2	yes	—
27	#435	—	2	2	no	—
28	C₈×D₄	C₂²	1	2	yes	—
29	#433	—	2	2	no	—
30	#432	—	2	1	no	—
31	#431	C₂²	1	1	yes	characteristic
32	#431	C₂²	1	1	yes	characteristic
33	#431	C₂²	1	1	yes	characteristic
34	#423	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	#335	C₂²	1	2	yes	—
38	#334	C₂²	1	2	yes	—
39	Q₈⋊C₈	C₂²	1	2	yes	—
40	Q₈⋊C₈	C₂²	1	2	yes	—
41	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	D₄⋊C₈	C₂²	1	1	yes	characteristic
44	D₄⋊C₈	C₂²	1	1	yes	characteristic
45	#322	C₂²	1	2	yes	—
46	C₈²	C₂²	1	2	yes	—
47	C₂×#41	C₂×C₄	1	1	yes	characteristic
48	C₂×Q₁₆	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆	C₂×C₄	1	2	yes	—
50	C₂×M₁₆	—	2	1	no	—
51	C₂²×C₈	—	2	1	no	—
52	C₂²×C₈	—	2	2	no	—
53	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
55	#125	—	2	1	no	—
56	C₂²⋊Q₈	—	2	1	no	—
57	C₂²⋊Q₈	C₂×C₄	1	2	yes	—
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
60	C₄×Q₈	C₂³	1	1	yes	characteristic
61	C₄×Q₈	C₂³	1	2	yes	—
62	C₄×D₄	—	2	1	no	—
63	C₄×D₄	—	2	1	no	—
64	C₄×D₄	C₂×C₄	1	1	yes	characteristic
65	C₄×D₄	C₂³	1	1	yes	characteristic
66	#117	—	2	1	no	—
67	#117	—	2	1	no	—
68	#117	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²	—	2	1	no	—
70	C₂×C₄²	C₂³	1	1	yes	characteristic
71	#107	C₂×C₄	1	2	yes	—
72	#106	C₂×C₄	1	2	yes	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	2	2	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
78	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
79	C₄⋊C₈	D₄	1	1	yes	characteristic
80	C₄⋊C₈	D₄	1	1	yes	characteristic
81	C₄⋊C₈	C₂³	1	1	yes	characteristic
82	C₄⋊C₈	C₂³	1	1	yes	characteristic
83	C₄⋊C₈	C₂³	1	2	yes	—
84	#103	C₂×C₄	1	2	yes	—
85	#103	C₂×C₄	1	2	yes	—
86	#103	C₂×C₄	1	2	yes	—
87	#102	C₂×C₄	1	2	yes	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	C₂²⋊C₈	C₂×C₄	1	2	yes	—
91	C₂²⋊C₈	D₄	1	2	yes	—
92	#97	—	2	2	no	—
93	#97	—	2	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	C₂³	1	1	yes	characteristic
98	C₄×C₈	C₂³	1	1	yes	characteristic
99	C₄×C₈	C₂³	1	2	yes	—
100	C₄×C₈	C₂³	1	2	yes	—
101	#41	—	2	2	no	—
102	#41	—	4	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×Q₈	C₂²×C₄	1	2	yes	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄	#41	1	1	yes	characteristic
114	Q₁₆	—	4	2	no	—
115	SD₁₆	—	2	4	no	—
116	M₁₆	—	8	1	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	8	2	no	—
124	C₂×C₈	C₂²×C₄	1	2	yes	—
125	C₂×C₈	C₂²×C₄	1	2	yes	—
126	C₂×C₈	C₂×D₄	1	2	yes	—
127	C₂×C₈	C₂×D₄	1	2	yes	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	C₂²×C₄	1	2	yes	—
136	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
137	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	2	1	no	—
140	C₂²⋊C₄	—	2	1	no	—
141	C₂²⋊C₄	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	1	no	—
146	C₄²	C₂²×C₄	1	1	yes	characteristic
147	C₄²	C₂×D₄	1	1	yes	characteristic
148	C₄²	#41	1	1	yes	characteristic
149	C₄²	C₂⁴	1	1	yes	characteristic
150	C₂³	—	2	1	no	—
151	C₂³	C₄×D₄	1	1	yes	characteristic
152	Q₈	—	2	2	no	—
153	Q₈	—	2	1	no	—
154	Q₈	—	2	1	no	—
155	Q₈	—	2	2	no	—
156	Q₈	—	4	2	no	—
157	D₄	—	2	1	no	—
158	D₄	—	4	2	no	—
159	D₄	#131	1	2	yes	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄	C₄×D₄	1	1	yes	characteristic
177	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄	C₂×#41	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	2	2	no	—
183	C₈	—	2	2	no	—
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₈	—	4	1	no	—
187	C₈	—	4	2	no	—
188	C₈	—	4	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	4	1	no	—
194	C₂²	—	4	1	no	—
195	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	4	1	no	—
200	C₄	—	4	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	4	1	no	—
203	C₄	—	4	2	no	—
204	C₄	#443	1	2	yes	—
205	C₄	C₂×#131	1	1	yes	characteristic
206	C₄	#568	1	1	yes	characteristic
207	C₄	#577	1	1	yes	characteristic
208	C₄	#578	1	1	yes	characteristic
209	C₂	—	4	1	no	—
210	C₂	—	4	2	no	—
211	C₂	#2933	1	1	yes	characteristic
212	C₂	#2947	1	1	yes	characteristic
213	C₂	C₂×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2986	C₂	semidirect
4. #1591	C₂	semidirect
5. #1587	C₂	semidirect
6. #1585	C₂	semidirect
7. C₈×SD₁₆	C₂	semidirect
8. #1574	C₂	semidirect
9. #1493	C₂	semidirect
11. #1450	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3107	2
#2990	2
#2986	2
#1591	2
#1587	2
#1585	2
C₈×SD₁₆	2
#1574	2
#1493	2
#1476	2
#1450	2