Subgroups — #33745

Order 256.

248 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	#3175	C₂	1	1	yes	maximal, characteristic
3	#2255	C₂	1	1	yes	maximal, characteristic, complemented
4	#2251	C₂	1	1	yes	maximal, characteristic, complemented
5	#2242	C₂	1	1	yes	maximal, characteristic, complemented
6	#2241	C₂	1	1	yes	maximal, characteristic, complemented
7	#2220	C₂	1	1	yes	maximal, characteristic
8	#2219	C₂	1	1	yes	maximal, characteristic, complemented
9	#2217	C₂	1	1	yes	maximal, characteristic
10	#2216	C₂	1	1	yes	maximal, characteristic, complemented
11	#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	#2175	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
15	#2108	C₂	1	1	yes	maximal, characteristic, complemented
16	#542	—	2	1	no	—
17	#541	—	2	1	no	—
18	C₂×Q₃₂	C₂²	1	1	yes	characteristic
19	C₂×Q₃₂	C₂²	1	1	yes	characteristic
20	C₂×SD₃₂	C₂²	1	1	yes	characteristic
21	C₂×SD₃₂	C₂²	1	1	yes	characteristic
22	C₂×M₃₂	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆	C₂²	1	1	yes	characteristic
24	#500	C₂²	1	1	yes	characteristic
25	#499	C₂²	1	1	yes	characteristic
26	#495	C₂²	1	1	yes	characteristic
27	#494	C₂²	1	1	yes	characteristic
28	#491	—	2	1	no	—
29	#489	—	2	1	no	—
30	#470	—	2	1	no	—
31	#468	—	2	1	no	—
32	#467	C₂²	1	1	yes	characteristic
33	#467	C₂²	1	1	yes	characteristic
34	#467	C₂²	1	1	yes	characteristic
35	#466	C₂²	1	1	yes	characteristic
36	#441	—	2	1	no	—
37	#440	—	2	1	no	—
38	C₄×Q₁₆	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆	C₂²	1	1	yes	characteristic
41	C₄×D₈	C₂²	1	1	yes	characteristic
42	#405	C₂²	1	1	yes	characteristic
43	#367	C₂²	1	1	yes	characteristic
44	#367	C₂²	1	1	yes	characteristic
45	#366	C₂²	1	1	yes	characteristic
46	#366	C₂²	1	1	yes	characteristic
47	#358	C₂²	1	1	yes	characteristic
48	#358	C₂²	1	1	yes	characteristic
49	#358	C₂²	1	1	yes	characteristic
50	#358	C₂²	1	1	yes	characteristic
51	#358	C₂²	1	1	yes	characteristic
52	#358	C₂²	1	1	yes	characteristic
53	#357	C₂²	1	1	yes	characteristic
54	#357	C₂²	1	1	yes	characteristic
55	#346	C₂²	1	1	yes	characteristic
56	#346	C₂²	1	1	yes	characteristic
57	C₄×C₁₆	C₂²	1	1	yes	characteristic
58	C₄×C₁₆	C₂²	1	1	yes	characteristic
59	C₂×Q₁₆	—	2	1	no	—
60	C₂×Q₁₆	C₂³	1	1	yes	characteristic
61	C₂×Q₁₆	C₂³	1	1	yes	characteristic
62	C₂×Q₁₆	C₂³	1	1	yes	characteristic
63	C₂×SD₁₆	—	2	1	no	—
64	C₂×D₈	C₂³	1	1	yes	characteristic
65	C₂×M₁₆	D₄	1	1	yes	characteristic
66	C₂²×C₈	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈	—	2	1	no	—
68	C₄⋊Q₈	—	2	1	no	—
69	#126	—	4	1	no	—
70	#126	—	4	1	no	—
71	#126	—	4	1	no	—
72	#125	—	2	1	no	—
73	#125	—	4	1	no	—
74	#125	—	4	1	no	—
75	#124	—	2	1	no	—
76	#124	—	4	1	no	—
77	#123	—	4	1	no	—
78	C₂²⋊Q₈	—	2	1	no	—
79	C₂²⋊Q₈	—	2	1	no	—
80	C₂²⋊Q₈	—	2	1	no	—
81	C₂²⋊Q₈	—	4	1	no	—
82	#121	—	2	1	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	—	2	1	no	—
86	C₄×D₄	—	2	1	no	—
87	#117	D₄	1	1	yes	characteristic
88	Q₃₂	—	2	2	no	—
89	Q₃₂	—	2	2	no	—
90	SD₃₂	—	2	2	no	—
91	SD₃₂	—	2	2	no	—
92	M₃₂	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	C₂³	1	1	yes	characteristic
95	C₂×C₁₆	C₂³	1	1	yes	characteristic
96	C₂×C₁₆	C₂³	1	1	yes	characteristic
97	C₂×C₁₆	C₂³	1	1	yes	characteristic
98	#107	—	2	1	no	—
99	#107	C₂³	1	1	yes	characteristic
100	#107	C₂³	1	1	yes	characteristic
101	#107	C₂³	1	1	yes	characteristic
102	#107	C₂³	1	1	yes	characteristic
103	#106	—	2	1	no	—
104	#103	—	2	1	no	—
105	#103	—	2	1	no	—
106	#103	—	2	1	no	—
107	#103	—	2	1	no	—
108	#103	—	2	1	no	—
109	#103	—	2	1	no	—
110	#102	—	2	1	no	—
111	#102	—	2	1	no	—
112	#97	D₄	1	1	yes	characteristic
113	#97	D₄	1	1	yes	characteristic
114	C₄×C₈	C₂³	1	1	yes	characteristic
115	C₄×C₈	C₂³	1	1	yes	characteristic
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	—	4	1	no	—
120	C₂×D₄	—	2	1	no	—
121	C₂×D₄	—	4	1	no	—
122	C₂²×C₄	—	4	1	no	—
123	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
124	Q₁₆	—	2	1	no	—
125	Q₁₆	—	2	1	no	—
126	Q₁₆	—	2	1	no	—
127	Q₁₆	—	2	1	no	—
128	Q₁₆	—	2	1	no	—
129	Q₁₆	—	2	1	no	—
130	Q₁₆	—	4	2	no	—
131	SD₁₆	—	4	2	no	—
132	D₈	—	2	1	no	—
133	D₈	—	2	1	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈	C₂⁴	1	1	yes	characteristic
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	4	1	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄⋊C₄	D₈	1	1	yes	characteristic
157	C₄⋊C₄	D₈	1	1	yes	characteristic
158	C₂²⋊C₄	—	4	1	no	—
159	C₂²⋊C₄	—	4	1	no	—
160	C₂²⋊C₄	—	4	1	no	—
161	C₂²⋊C₄	—	4	1	no	—
162	C₂²⋊C₄	—	4	1	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₂²⋊C₄	—	4	1	no	—
165	C₂²⋊C₄	D₈	1	1	yes	characteristic
166	C₂²⋊C₄	D₈	1	1	yes	characteristic
167	C₄²	—	4	1	no	—
168	C₄²	—	4	1	no	—
169	C₄²	—	4	1	no	—
170	C₄²	C₂×D₄	1	1	yes	characteristic
171	C₄²	C₂×D₄	1	1	yes	characteristic
172	C₁₆	—	2	1	no	—
173	C₁₆	—	2	1	no	—
174	C₁₆	#41	1	2	yes	—
175	C₁₆	#41	1	2	yes	—
176	C₂³	—	4	1	no	—
177	C₂³	C₂×D₈	1	1	yes	characteristic
178	Q₈	—	4	1	no	—
179	Q₈	—	4	1	no	—
180	Q₈	—	4	1	no	—
181	Q₈	—	4	1	no	—
182	Q₈	—	4	1	no	—
183	Q₈	—	4	1	no	—
184	Q₈	—	16	1	no	—
185	D₄	—	4	1	no	—
186	D₄	—	4	1	no	—
187	D₄	—	16	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	16	1	no	—
198	C₂×C₄	C₂×D₈	1	1	yes	characteristic
199	C₂×C₄	C₂×D₈	1	1	yes	characteristic
200	C₂×C₄	C₂×D₈	1	1	yes	characteristic
201	C₂×C₄	C₂×D₈	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈	1	1	yes	characteristic
203	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	#135	1	2	yes	—
207	C₈	C₂×#41	1	1	yes	characteristic
208	C₈	C₂×#41	1	1	yes	characteristic
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	8	1	no	—
213	C₂²	—	8	1	no	—
214	C₂²	—	8	1	no	—
215	C₂²	C₂²×D₈	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	8	1	no	—
222	C₄	—	8	1	no	—
223	C₄	—	8	1	no	—
224	C₄	—	8	1	no	—
225	C₄	—	8	1	no	—
226	C₄	—	8	1	no	—
227	C₄	—	8	1	no	—
228	C₄	#532	1	1	yes	characteristic
229	C₄	C₂×#135	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	16	1	no	—
232	C₂	#3169	1	1	yes	characteristic
233	C₂	#3416	1	1	yes	characteristic
234	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3177	C₂	semidirect
3. #2255	C₂	semidirect
4. #2251	C₂	semidirect
5. #2242	C₂	semidirect
6. #2241	C₂	semidirect
8. #2219	C₂	semidirect
10. #2216	C₂	semidirect
11. #2176	C₂	semidirect
12. #2175	C₂	semidirect
13. C₄×Q₃₂	C₂	semidirect
14. C₄×SD₃₂	C₂	semidirect
15. #2108	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3177	2
#3175	2
#2255	2
#2251	2
#2242	2
#2241	2
#2220	2
#2219	2
#2217	2
#2216	2
#2176	2
#2175	2
C₄×Q₃₂	2
C₄×SD₃₂	2
#2108	2