Subgroups — #11303

Order 256.

295 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#343	C₂	1	1	yes	maximal, characteristic
2	C₂×#343	C₂	1	1	yes	maximal, characteristic
3	C₂×#341	C₂	1	1	yes	maximal, characteristic
4	#1736	C₂	1	1	yes	maximal, characteristic
5	#1736	C₂	1	1	yes	maximal, characteristic
6	C₂×#340	C₂	1	1	yes	maximal, characteristic
7	#1729	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈	C₂²	1	1	yes	characteristic
12	C₂×#117	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	#343	—	2	2	no	—
16	#343	C₄	1	4	yes	complemented
17	#341	—	2	2	no	—
18	#340	—	2	2	no	—
19	#340	—	2	1	no	—
20	#340	—	2	1	no	—
21	#340	—	2	1	no	—
22	#340	—	2	1	no	—
23	#336	—	2	2	no	—
24	C₂³×C₄	—	2	1	no	—
25	C₂³×C₄	—	2	1	no	—
26	C₂³×C₄	C₂³	1	1	yes	characteristic
27	C₂×M₁₆	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	—	2	1	no	—
33	C₂×M₁₆	—	2	1	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂×M₁₆	—	2	1	no	—
36	C₂×M₁₆	—	2	1	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂×M₁₆	—	2	1	no	—
39	C₂×M₁₆	—	2	1	no	—
40	C₂×M₁₆	—	2	1	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	C₂×C₄	1	2	yes	—
43	C₂×M₁₆	C₂×C₄	1	2	yes	—
44	C₂×M₁₆	D₄	1	2	yes	—
45	C₂²×C₈	—	2	1	no	—
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	—	2	1	no	—
48	C₂²×C₈	—	2	1	no	—
49	C₂²×C₈	—	2	1	no	—
50	C₂²×C₈	—	2	1	no	—
51	C₂²×C₈	—	2	1	no	—
52	C₂²×C₈	—	2	1	no	—
53	C₂²×C₈	—	2	1	no	—
54	C₂²×C₈	D₄	1	2	yes	—
55	#117	—	2	1	no	—
56	#117	—	2	1	no	—
57	#117	—	2	1	no	—
58	#117	—	2	1	no	—
59	C₂×C₄⋊C₄	—	2	1	no	—
60	C₂×C₄⋊C₄	—	2	1	no	—
61	C₂×C₄⋊C₄	—	2	1	no	—
62	C₂×C₄⋊C₄	—	2	1	no	—
63	C₂×C₄⋊C₄	—	2	1	no	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	—	2	1	no	—
66	C₂×C₄⋊C₄	—	2	1	no	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
70	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄	Q₈	1	2	yes	—
73	C₂×C₂²⋊C₄	—	2	1	no	—
74	C₂×C₄²	—	2	1	no	—
75	C₂⁴	#41	1	1	yes	characteristic
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	4	1	no	—
81	C₂²×C₄	—	4	1	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	—	4	1	no	—
84	C₂²×C₄	—	8	1	no	—
85	C₂²×C₄	—	8	1	no	—
86	C₂²×C₄	—	8	1	no	—
87	C₂²×C₄	—	8	1	no	—
88	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄	#41	1	1	yes	characteristic
94	C₂²×C₄	#41	1	1	yes	characteristic
95	C₂²×C₄	#41	1	1	yes	characteristic
96	C₂²×C₄	#41	1	1	yes	characteristic
97	C₂²×C₄	#41	1	1	yes	characteristic
98	C₂²×C₄	#41	1	1	yes	characteristic
99	C₂²×C₄	#41	1	1	yes	characteristic
100	C₂²×C₄	#41	1	1	yes	characteristic
101	C₂²×C₄	#41	1	1	yes	characteristic
102	M₁₆	—	2	2	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	4	1	no	—
109	M₁₆	—	4	1	no	—
110	M₁₆	—	4	1	no	—
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	1	no	—
113	M₁₆	—	4	1	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₂²⋊C₄	—	4	1	no	—
143	C₂²⋊C₄	—	4	1	no	—
144	C₄²	—	4	1	no	—
145	C₄²	—	4	1	no	—
146	C₂³	—	2	1	no	—
147	C₂³	—	2	1	no	—
148	C₂³	—	2	1	no	—
149	C₂³	—	2	1	no	—
150	C₂³	#117	1	1	yes	characteristic
151	C₂³	#121	1	1	yes	characteristic
152	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂³	#126	1	1	yes	characteristic
154	C₂³	#126	1	1	yes	characteristic
155	C₂³	#126	1	1	yes	characteristic
156	C₂³	#126	1	1	yes	characteristic
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	1	no	—
168	C₂×C₄	—	8	1	no	—
169	C₂×C₄	—	8	1	no	—
170	C₂×C₄	—	8	1	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	#117	1	1	yes	characteristic
174	C₂×C₄	#117	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	C₄×D₄	1	1	yes	characteristic
178	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
179	C₂×C₄	#121	1	1	yes	characteristic
180	C₂×C₄	#121	1	1	yes	characteristic
181	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
182	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
183	C₂×C₄	#123	1	1	yes	characteristic
184	C₂×C₄	#123	1	1	yes	characteristic
185	C₂×C₄	#123	1	1	yes	characteristic
186	C₂×C₄	#123	1	1	yes	characteristic
187	C₂×C₄	#123	1	1	yes	characteristic
188	C₂×C₄	#123	1	1	yes	characteristic
189	C₂×C₄	#124	1	1	yes	characteristic
190	C₂×C₄	#124	1	1	yes	characteristic
191	C₂×C₄	#124	1	1	yes	characteristic
192	C₂×C₄	#125	1	1	yes	characteristic
193	C₂×C₄	#125	1	1	yes	characteristic
194	C₂×C₄	#125	1	1	yes	characteristic
195	C₂×C₄	#126	1	1	yes	characteristic
196	C₂×C₄	#126	1	1	yes	characteristic
197	C₂×C₄	#126	1	1	yes	characteristic
198	C₂×C₄	#126	1	1	yes	characteristic
199	C₂×C₄	#126	1	1	yes	characteristic
200	C₂×C₄	#126	1	1	yes	characteristic
201	C₈	—	4	2	no	—
202	C₈	—	4	2	no	—
203	C₈	—	4	2	no	—
204	C₈	—	4	2	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	#383	1	1	yes	characteristic
218	C₂²	#387	1	1	yes	characteristic
219	C₂²	#388	1	1	yes	characteristic
220	C₂²	#396	1	1	yes	characteristic
221	C₂²	#397	1	1	yes	characteristic
222	C₂²	#400	1	1	yes	characteristic
223	C₂²	#401	1	1	yes	characteristic
224	C₂²	#471	1	1	yes	characteristic
225	C₂²	#471	1	1	yes	characteristic
226	C₂²	#472	1	1	yes	characteristic
227	C₂²	#473	1	1	yes	characteristic
228	C₄	—	8	2	no	—
229	C₄	—	8	1	no	—
230	C₄	—	8	1	no	—
231	C₄	—	8	2	no	—
232	C₄	#387	1	1	yes	characteristic
233	C₄	#387	1	1	yes	characteristic
234	C₄	#388	1	1	yes	characteristic
235	C₄	#388	1	1	yes	characteristic
236	C₄	#397	1	1	yes	characteristic
237	C₄	#397	1	1	yes	characteristic
238	C₄	#400	1	1	yes	characteristic
239	C₄	#400	1	1	yes	characteristic
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	#1443	1	1	yes	characteristic
245	C₂	#1930	1	1	yes	characteristic
246	C₂	#1931	1	1	yes	characteristic
247	C₂	#2063	1	1	yes	characteristic
248	C₂	#2064	1	1	yes	characteristic
249	C₂	#2078	1	1	yes	characteristic
250	C₂	#2079	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#121
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂³×C₄	C₂³
prime radical	C₂⁴	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #343	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#343	2
C₂×#341	2
#1736	2
C₂×#340	2
#1729	2