Subgroups — #21628

Order 256.

369 conjugacy classes of subgroups.

The subgroup lattice has 253 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3076	C₂	1	1	yes	maximal, characteristic
2	#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#131	C₂	1	1	yes	maximal, characteristic
4	#1986	C₂	1	1	yes	maximal, characteristic, complemented
5	#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	#1922	C₂	1	2	yes	maximal, complemented
7	#1866	C₂	1	2	yes	maximal
8	#1806	C₂	1	1	yes	maximal, characteristic
9	#1773	C₂	1	2	yes	maximal, complemented
10	#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	#1765	C₂	1	2	yes	maximal
12	C₂×#131	C₂²	1	1	yes	characteristic
13	C₂×#131	C₂²	1	1	yes	characteristic
14	#533	C₂²	1	1	yes	characteristic
15	C₄×#41	C₂²	1	1	yes	characteristic
16	#479	C₄	1	2	yes	complemented
17	#478	C₄	1	2	yes	complemented
18	#477	C₄	1	1	yes	characteristic, complemented
19	#476	C₄	1	1	yes	characteristic, complemented
20	#475	C₄	1	1	yes	characteristic, complemented
21	#474	C₄	1	1	yes	characteristic, complemented
22	#446	—	2	1	no	—
23	#446	—	2	1	no	—
24	C₈×Q₈	—	2	1	no	—
25	C₈×Q₈	—	2	1	no	—
26	C₂×#108	C₂²	1	2	yes	—
27	#427	C₂²	1	2	yes	—
28	#424	C₂²	1	2	yes	—
29	#423	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×#104	C₂²	1	2	yes	—
32	#416	C₂²	1	2	yes	—
33	#405	—	2	1	no	—
34	#405	—	2	1	no	—
35	#405	C₂²	1	1	yes	characteristic
36	#405	C₂²	1	1	yes	characteristic
37	#405	C₂²	1	2	yes	—
38	C₄×M₁₆	—	2	1	no	—
39	C₄×M₁₆	C₂²	1	1	yes	characteristic
40	C₂×#97	C₂²	1	2	yes	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
44	#341	C₂²	1	2	yes	—
45	#339	C₂²	1	2	yes	—
46	#339	C₂²	1	2	yes	—
47	#339	C₂²	1	2	yes	—
48	#336	C₂²	1	2	yes	—
49	C₂×#41	C₂³	1	1	yes	characteristic
50	#131	—	2	2	no	—
51	#131	—	2	2	no	—
52	#131	—	2	2	no	—
53	#131	Q₈	1	4	yes	complemented
54	C₂×M₁₆	—	2	1	no	—
55	C₂×M₁₆	—	2	1	no	—
56	C₂×M₁₆	C₂³	1	2	yes	—
57	C₂×M₁₆	C₂³	1	1	yes	characteristic
58	C₂×M₁₆	C₂³	1	1	yes	characteristic
59	C₂²×C₈	—	2	1	no	—
60	C₂²×C₈	—	2	1	no	—
61	C₂²×C₈	C₂³	1	2	yes	—
62	C₂²×C₈	C₂³	1	1	yes	characteristic
63	C₂²×C₈	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
65	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
66	#125	C₂×C₄	1	2	yes	—
67	C₂²⋊Q₈	—	2	1	no	—
68	C₂²⋊Q₈	—	2	1	no	—
69	C₄×Q₈	—	2	1	no	—
70	C₄×Q₈	—	2	1	no	—
71	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄	—	2	1	no	—
74	C₄×D₄	—	2	1	no	—
75	C₄×D₄	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄	C₂×C₄	1	1	yes	characteristic
77	#117	—	2	1	no	—
78	#117	C₂³	1	1	yes	characteristic
79	#117	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄²	—	2	1	no	—
82	C₂×C₄²	C₂³	1	2	yes	—
83	C₂×C₄²	C₂³	1	1	yes	characteristic
84	#108	—	2	4	no	—
85	#107	C₂×C₄	1	2	yes	—
86	#107	C₂×C₄	1	2	yes	—
87	#106	C₂×C₄	1	2	yes	—
88	#106	C₂×C₄	1	2	yes	—
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
94	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
95	C₄⋊C₈	C₂×C₄	1	2	yes	—
96	#104	—	2	4	no	—
97	#103	C₂×C₄	1	2	yes	—
98	#103	C₂×C₄	1	2	yes	—
99	#102	C₂×C₄	1	2	yes	—
100	#102	C₂×C₄	1	2	yes	—
101	#97	—	2	1	no	—
102	#97	—	2	1	no	—
103	#97	—	2	2	no	—
104	#97	—	2	4	no	—
105	#97	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	D₄	1	1	yes	characteristic
111	C₄×C₈	D₄	1	1	yes	characteristic
112	C₄×C₈	D₄	1	1	yes	characteristic
113	C₄×C₈	D₄	1	1	yes	characteristic
114	#95	—	2	1	no	—
115	#41	—	2	1	no	—
116	#41	—	2	1	no	—
117	#41	C₂×Q₈	1	2	yes	—
118	#41	#41	1	2	yes	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	#41	1	1	yes	characteristic
129	C₂²×C₄	#41	1	1	yes	characteristic
130	C₂²×C₄	C₂⁴	1	1	yes	characteristic
131	M₁₆	—	2	4	no	—
132	M₁₆	—	2	1	no	—
133	M₁₆	—	2	1	no	—
134	M₁₆	—	2	4	no	—
135	M₁₆	—	2	2	no	—
136	M₁₆	—	2	2	no	—
137	M₁₆	C₂×Q₈	1	2	yes	—
138	M₁₆	#41	1	2	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₂²×C₄	1	2	yes	—
149	C₂×C₈	C₂²×C₄	1	2	yes	—
150	C₂×C₈	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈	C₂×D₄	1	1	yes	characteristic
153	C₂×C₈	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈	C₂×Q₈	1	2	yes	—
155	C₂×C₈	#41	1	2	yes	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	C₂²×C₄	1	2	yes	—
164	C₄⋊C₄	C₂²×C₄	1	2	yes	—
165	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
166	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
167	C₂²⋊C₄	—	2	1	no	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	1	no	—
170	C₂²⋊C₄	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂²×C₄	1	1	yes	characteristic
178	C₄²	C₂²×C₄	1	1	yes	characteristic
179	C₄²	C₂×D₄	1	1	yes	characteristic
180	C₄²	C₂×D₄	1	1	yes	characteristic
181	C₂³	—	2	1	no	—
182	C₂³	C₂×#41	1	1	yes	characteristic
183	Q₈	—	2	1	no	—
184	Q₈	—	4	2	no	—
185	Q₈	—	4	2	no	—
186	Q₈	C₄×Q₈	1	2	yes	—
187	D₄	—	2	2	no	—
188	D₄	—	2	2	no	—
189	D₄	—	2	1	no	—
190	D₄	C₄×Q₈	1	2	yes	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄	C₄×D₄	1	1	yes	characteristic
208	C₂×C₄	C₄×D₄	1	1	yes	characteristic
209	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
212	C₂×C₄	C₂×#41	1	1	yes	characteristic
213	C₂×C₄	C₂×#41	1	1	yes	characteristic
214	C₂×C₄	C₂×#41	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	C₂²⋊Q₈	1	2	yes	—
223	C₈	C₂²⋊Q₈	1	2	yes	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
230	C₂²	#529	1	1	yes	characteristic
231	C₂²	#533	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	8	2	no	—
241	C₄	#444	1	2	yes	—
242	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
243	C₄	C₄×#41	1	1	yes	characteristic
244	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
245	C₄	#525	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	—	4	2	no	—
249	C₂	C₄×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂	C₂×#444	1	1	yes	characteristic
251	C₂	#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2960	C₂	semidirect
4. #1986	C₂	semidirect
5. #1950	C₂	semidirect
6. #1922	C₂	semidirect
9. #1773	C₂	semidirect
10. #1770	C₂	semidirect
16. #479	C₄	semidirect
17. #478	C₄	semidirect
18. #477	C₄	semidirect
19. #476	C₄	semidirect
20. #475	C₄	semidirect
21. #474	C₄	semidirect
53. #131	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3076	2
#2960	2
C₄×#131	2
#1986	2
#1950	2
#1922	2
#1866	2
#1806	2
#1773	2
#1770	2
#1765	2