Subgroups — #60209

Order 256.

939 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3455	C₂	1	1	yes	maximal, characteristic
2	#2969	C₂	1	8	yes	maximal, complemented
3	#2968	C₂	1	8	yes	maximal, complemented
4	#2961	C₂	1	1	yes	maximal, characteristic, complemented
5	#2960	C₂	1	1	yes	maximal, characteristic, complemented
6	#2932	C₂	1	1	yes	maximal, characteristic, complemented
7	#2931	C₂	1	1	yes	maximal, characteristic, complemented
8	#2930	C₂	1	2	yes	maximal, complemented
9	#2929	C₂	1	2	yes	maximal, complemented
10	#2922	C₂	1	2	yes	maximal, complemented
11	#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	#2875	C₂	1	2	yes	maximal, complemented
13	C₂×#405	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#131	C₂²	1	2	yes	—
15	C₂×#131	C₂²	1	2	yes	—
16	C₂²×M₁₆	C₂²	1	1	yes	characteristic
17	C₂³×C₈	C₂²	1	1	yes	characteristic
18	#542	C₄	1	4	yes	—
19	#541	C₄	1	2	yes	—
20	#540	C₄	1	2	yes	—
21	#539	C₄	1	4	yes	—
22	#538	C₄	1	4	yes	—
23	#533	C₂²	1	1	yes	characteristic
24	#532	C₂²	1	1	yes	characteristic
25	#529	C₂²	1	8	yes	—
26	#525	C₂²	1	2	yes	—
27	C₄×#41	C₂²	1	2	yes	—
28	C₂×#117	C₂²	1	1	yes	characteristic
29	#446	C₂²	1	4	yes	complemented
30	#446	C₂²	1	8	yes	complemented
31	C₈×Q₈	C₂²	1	4	yes	complemented
32	#436	C₂²	1	8	yes	—
33	#436	C₂²	1	4	yes	complemented
34	#435	C₂²	1	8	yes	—
35	#435	C₂²	1	8	yes	—
36	#435	C₂²	1	8	yes	complemented
37	C₈×D₄	C₂²	1	8	yes	—
38	C₈×D₄	C₂²	1	4	yes	complemented
39	#433	C₂²	1	8	yes	—
40	#433	C₂²	1	8	yes	complemented
41	#431	C₂²	1	8	yes	complemented
42	#424	C₂²	1	2	yes	—
43	#424	C₂²	1	2	yes	—
44	#423	C₂²	1	1	yes	characteristic
45	#423	C₂²	1	1	yes	characteristic
46	#408	C₂²	1	2	yes	—
47	#408	C₂²	1	2	yes	—
48	#407	C₂²	1	2	yes	—
49	#405	C₂²	1	4	yes	—
50	#405	C₂²	1	4	yes	—
51	#405	C₂²	1	2	yes	—
52	#405	C₂²	1	2	yes	—
53	#405	C₂²	1	4	yes	complemented
54	#405	C₂²	1	4	yes	complemented
55	C₄×M₁₆	C₂²	1	2	yes	—
56	C₄×M₁₆	C₂²	1	4	yes	complemented
57	C₂×#97	C₂²	1	1	yes	characteristic
58	C₂×#97	C₂²	1	1	yes	characteristic
59	C₂×#97	C₂²	1	4	yes	complemented
60	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
62	C₂×#41	C₂³	1	2	yes	—
63	C₂³×C₄	C₂³	1	1	yes	characteristic
64	#131	—	2	8	no	—
65	#131	—	2	8	no	—
66	C₂×M₁₆	—	2	4	no	—
67	C₂×M₁₆	C₂³	1	2	yes	—
68	C₂×M₁₆	C₂³	1	2	yes	—
69	C₂×M₁₆	C₂³	1	2	yes	—
70	C₂×M₁₆	C₂³	1	2	yes	—
71	C₂×M₁₆	C₂³	1	4	yes	—
72	C₂×M₁₆	C₂³	1	4	yes	—
73	C₂²×C₈	—	2	4	no	—
74	C₂²×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₈	C₂³	1	4	yes	—
76	C₂²×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₈	C₂³	1	4	yes	—
78	C₂²×C₈	C₂³	1	4	yes	—
79	C₂²×C₈	C₂³	1	1	yes	characteristic
80	C₂²×C₈	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×C₄	1	2	yes	—
83	#127	C₂×C₄	1	1	yes	characteristic
84	#126	C₂×C₄	1	8	yes	—
85	#126	C₂×C₄	1	8	yes	—
86	#125	C₂×C₄	1	8	yes	—
87	#125	C₂×C₄	1	2	yes	—
88	#124	C₂×C₄	1	8	yes	—
89	#124	C₂×C₄	1	2	yes	—
90	#123	C₂×C₄	1	4	yes	—
91	#123	C₂×C₄	1	8	yes	—
92	#123	C₂×C₄	1	8	yes	—
93	C₂²⋊Q₈	C₂×C₄	1	4	yes	—
94	C₂²⋊Q₈	C₂×C₄	1	8	yes	—
95	C₂²⋊Q₈	C₂×C₄	1	4	yes	—
96	#121	C₂×C₄	1	4	yes	—
97	#121	C₂×C₄	1	8	yes	—
98	#121	C₂×C₄	1	4	yes	—
99	#120	C₂×C₄	1	4	yes	—
100	C₄×Q₈	C₂×C₄	1	4	yes	—
101	C₄×Q₈	C₂³	1	4	yes	—
102	C₄×D₄	C₂×C₄	1	8	yes	—
103	C₄×D₄	C₂×C₄	1	4	yes	—
104	C₄×D₄	C₂³	1	8	yes	—
105	C₄×D₄	C₂³	1	4	yes	—
106	#117	C₂×C₄	1	2	yes	—
107	#117	C₂×C₄	1	2	yes	—
108	#117	C₂³	1	4	yes	—
109	#117	C₂³	1	2	yes	—
110	#117	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
113	C₂×C₄²	C₂³	1	1	yes	characteristic
114	C₂×C₄²	C₂³	1	1	yes	characteristic
115	C₂×C₄²	C₂³	1	4	yes	—
116	C₄⋊C₈	C₂³	1	4	yes	—
117	C₄⋊C₈	C₂³	1	4	yes	—
118	C₄⋊C₈	C₂³	1	8	yes	—
119	C₄⋊C₈	C₂³	1	8	yes	—
120	C₂²⋊C₈	C₂³	1	8	yes	—
121	C₂²⋊C₈	C₂³	1	8	yes	—
122	C₂²⋊C₈	C₂³	1	8	yes	—
123	#97	C₂³	1	2	yes	—
124	#97	C₂³	1	2	yes	—
125	#97	C₂³	1	4	yes	—
126	#97	C₂³	1	4	yes	—
127	#97	C₂³	1	4	yes	—
128	#97	C₂³	1	4	yes	—
129	C₄×C₈	C₂³	1	4	yes	—
130	C₄×C₈	C₂³	1	2	yes	—
131	C₄×C₈	C₂³	1	2	yes	—
132	C₄×C₈	C₂³	1	4	yes	—
133	C₄×C₈	C₂³	1	4	yes	—
134	C₄×C₈	C₂³	1	4	yes	—
135	C₂⁴	C₂²×C₄	1	1	yes	characteristic
136	#41	—	2	4	no	—
137	#41	—	2	4	no	—
138	C₂×Q₈	C₂²×C₄	1	2	yes	—
139	C₂×Q₈	C₂²×C₄	1	4	yes	—
140	C₂×D₄	C₂²×C₄	1	4	yes	—
141	C₂×D₄	C₂²×C₄	1	2	yes	—
142	C₂×D₄	C₂²×C₄	1	8	yes	—
143	C₂×D₄	C₂²×C₄	1	4	yes	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄	1	2	yes	—
148	C₂²×C₄	C₂²×C₄	1	2	yes	—
149	C₂²×C₄	C₂²×C₄	1	2	yes	—
150	C₂²×C₄	C₂²×C₄	1	8	yes	—
151	C₂²×C₄	C₂⁴	1	4	yes	—
152	C₂²×C₄	C₂⁴	1	2	yes	—
153	C₂²×C₄	C₂⁴	1	1	yes	characteristic
154	M₁₆	—	2	4	no	—
155	M₁₆	—	2	4	no	—
156	M₁₆	—	2	4	no	—
157	M₁₆	—	2	4	no	—
158	M₁₆	—	2	8	no	—
159	M₁₆	—	2	8	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	8	no	—
163	C₂×C₈	—	2	8	no	—
164	C₂×C₈	—	2	8	no	—
165	C₂×C₈	#41	1	4	yes	—
166	C₂×C₈	#41	1	4	yes	—
167	C₂×C₈	C₂⁴	1	2	yes	—
168	C₂×C₈	C₂⁴	1	2	yes	—
169	C₂×C₈	C₂⁴	1	4	yes	—
170	C₂×C₈	C₂⁴	1	4	yes	—
171	C₂×C₈	C₂⁴	1	1	yes	characteristic
172	C₂×C₈	C₂⁴	1	1	yes	characteristic
173	C₂×C₈	C₂⁴	1	2	yes	—
174	C₄⋊C₄	C₂²×C₄	1	4	yes	—
175	C₄⋊C₄	C₂²×C₄	1	4	yes	—
176	C₄⋊C₄	C₂²×C₄	1	4	yes	—
177	C₄⋊C₄	C₂²×C₄	1	8	yes	—
178	C₄⋊C₄	C₂²×C₄	1	4	yes	—
179	C₄⋊C₄	C₂²×C₄	1	8	yes	—
180	C₄⋊C₄	C₂²×C₄	1	8	yes	—
181	C₄⋊C₄	C₂²×C₄	1	4	yes	—
182	C₂²⋊C₄	C₂²×C₄	1	4	yes	—
183	C₂²⋊C₄	C₂²×C₄	1	4	yes	—
184	C₂²⋊C₄	C₂²×C₄	1	4	yes	—
185	C₂²⋊C₄	C₂²×C₄	1	8	yes	—
186	C₂²⋊C₄	C₂²×C₄	1	8	yes	—
187	C₂²⋊C₄	C₂²×C₄	1	8	yes	—
188	C₂²⋊C₄	C₂²×C₄	1	8	yes	—
189	C₄²	C₂²×C₄	1	2	yes	—
190	C₄²	C₂²×C₄	1	1	yes	characteristic
191	C₄²	C₂²×C₄	1	1	yes	characteristic
192	C₄²	C₂²×C₄	1	8	yes	—
193	C₄²	C₂⁴	1	2	yes	—
194	C₄²	C₂⁴	1	2	yes	—
195	C₄²	C₂⁴	1	4	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	—	4	2	no	—
198	C₂³	C₂³×C₄	1	2	yes	—
199	C₂³	C₂³×C₄	1	4	yes	—
200	C₂³	C₂³×C₄	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	4	no	—
203	D₄	—	2	8	no	—
204	D₄	—	2	4	no	—
205	D₄	—	4	4	no	—
206	D₄	—	4	8	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	8	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂³×C₄	1	4	yes	—
217	C₂×C₄	C₂³×C₄	1	4	yes	—
218	C₂×C₄	C₂³×C₄	1	2	yes	—
219	C₂×C₄	C₂³×C₄	1	8	yes	—
220	C₂×C₄	C₂³×C₄	1	2	yes	—
221	C₂×C₄	C₂³×C₄	1	2	yes	—
222	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
223	C₂×C₄	C₂×#41	1	2	yes	—
224	C₂×C₄	C₂×#41	1	2	yes	—
225	C₂×C₄	C₂⁵	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	2	4	no	—
229	C₈	—	2	4	no	—
230	C₈	C₂×#41	1	4	yes	—
231	C₈	C₂×#41	1	2	yes	—
232	C₈	C₂×#41	1	2	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	8	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	4	4	no	—
239	C₂²	C₄×#41	1	2	yes	—
240	C₂²	C₂⁴×C₄	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	4	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	8	no	—
246	C₄	—	2	2	no	—
247	C₄	C₄×#41	1	2	yes	—
248	C₄	C₂²×#41	1	1	yes	characteristic
249	C₄	C₂²×#41	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	4	4	no	—
252	C₂	—	4	2	no	—
253	C₂	C₂×C₄×#41	1	1	yes	characteristic
254	C₂	#3573	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁵
commutator, socle	C₂²	C₂⁴×C₄
prime radical	#3455	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2969	C₂	semidirect
3. #2968	C₂	semidirect
4. #2961	C₂	semidirect
5. #2960	C₂	semidirect
6. #2932	C₂	semidirect
7. #2931	C₂	semidirect
8. #2930	C₂	semidirect
9. #2929	C₂	semidirect
10. #2922	C₂	semidirect
11. #2921	C₂	semidirect
12. #2875	C₂	semidirect
13. C₂×#405	C₂	semidirect
29. #446	C₂²	semidirect
30. #446	C₂²	semidirect
31. C₈×Q₈	C₂²	semidirect
33. #436	C₂²	semidirect
36. #435	C₂²	semidirect
38. C₈×D₄	C₂²	semidirect
40. #433	C₂²	semidirect
41. #431	C₂²	semidirect
53. #405	C₂²	semidirect
54. #405	C₂²	semidirect
56. C₄×M₁₆	C₂²	semidirect
59. C₂×#97	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3455	2
#2969	2
#2968	2
#2961	2
#2960	2
#2932	2
#2931	2
#2930	2
#2929	2
#2922	2
#2921	2
#2875	2
C₂×#405	2