Subgroups — #11341

Order 256.

359 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	C₂×#430	C₂	1	1	yes	maximal, characteristic
2	#2905	C₂	1	1	yes	maximal, characteristic
3	C₂×#423	C₂	1	1	yes	maximal, characteristic
4	C₂×#344	C₂	1	1	yes	maximal, characteristic
5	#1733	C₂	1	1	yes	maximal, characteristic
6	C₂×#339	C₂	1	1	yes	maximal, characteristic
7	C₂×#339	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	C₂²×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×#117	C₂²	1	1	yes	characteristic
13	C₂×#117	C₂²	1	1	yes	characteristic
14	C₂²×C₄²	C₂²	1	1	yes	characteristic
15	#430	—	2	2	no	—
16	#430	—	2	2	no	—
17	C₂×#108	—	2	2	no	—
18	#424	—	2	2	no	—
19	#424	—	2	2	no	—
20	#423	—	2	2	no	—
21	#423	—	2	2	no	—
22	#423	—	2	2	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	#344	—	2	2	no	—
25	#339	—	2	2	no	—
26	#339	—	2	2	no	—
27	#339	C₄	1	2	yes	complemented
28	#339	C₄	1	2	yes	complemented
29	C₂³×C₄	—	2	1	no	—
30	C₂³×C₄	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
43	C₂×M₁₆	—	2	1	no	—
44	C₂×M₁₆	—	2	1	no	—
45	C₂×M₁₆	—	2	1	no	—
46	C₂×M₁₆	—	2	1	no	—
47	C₂×M₁₆	—	2	1	no	—
48	C₂×M₁₆	—	2	1	no	—
49	C₂×M₁₆	—	2	1	no	—
50	C₂×M₁₆	C₂×C₄	1	2	yes	—
51	C₂×M₁₆	D₄	1	2	yes	—
52	C₂×M₁₆	D₄	1	2	yes	—
53	C₂×M₁₆	D₄	1	2	yes	—
54	C₂×M₁₆	Q₈	1	2	yes	—
55	C₂²×C₈	—	2	1	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	D₄	1	2	yes	—
59	#117	—	2	1	no	—
60	#117	—	2	1	no	—
61	#117	—	2	1	no	—
62	#117	—	2	1	no	—
63	#117	—	2	1	no	—
64	#117	C₂×C₄	1	2	yes	—
65	#117	D₄	1	2	yes	—
66	#117	Q₈	1	2	yes	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	Q₈	1	2	yes	—
69	C₂×C₂²⋊C₄	—	2	1	no	—
70	C₂×C₂²⋊C₄	—	2	1	no	—
71	C₂×C₄²	—	2	1	no	—
72	C₂×C₄²	—	2	1	no	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	—	2	1	no	—
75	C₂×C₄²	—	2	1	no	—
76	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²	D₄	1	2	yes	—
79	C₂×C₄²	Q₈	1	2	yes	—
80	#108	—	2	8	no	—
81	C₄⋊C₈	—	2	8	no	—
82	C₄⋊C₈	—	2	8	no	—
83	C₂⁴	#41	1	1	yes	characteristic
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	4	4	no	—
90	C₂²×C₄	—	4	1	no	—
91	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
101	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₄	#41	1	1	yes	characteristic
103	C₂²×C₄	#41	1	1	yes	characteristic
104	C₂²×C₄	#41	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	4	1	no	—
113	M₁₆	—	4	1	no	—
114	M₁₆	—	4	1	no	—
115	M₁₆	—	4	1	no	—
116	M₁₆	—	4	1	no	—
117	M₁₆	—	4	1	no	—
118	M₁₆	—	4	1	no	—
119	M₁₆	—	4	1	no	—
120	M₁₆	C₄⋊C₄	1	4	yes	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	C₄⋊C₄	1	4	yes	—
137	C₂²⋊C₄	—	2	2	no	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	2	2	no	—
140	C₂²⋊C₄	—	4	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	4	1	no	—
149	C₄²	C₄⋊C₄	1	2	yes	—
150	C₄²	C₄⋊C₄	1	2	yes	—
151	C₂³	—	4	2	no	—
152	C₂³	C₄×D₄	1	1	yes	characteristic
153	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂³	#123	1	1	yes	characteristic
157	C₂³	#123	1	1	yes	characteristic
158	C₂³	#125	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
175	C₂×C₄	#120	1	1	yes	characteristic
176	C₂×C₄	#120	1	1	yes	characteristic
177	C₂×C₄	#120	1	1	yes	characteristic
178	C₂×C₄	#121	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	C₂²⋊Q₈	1	1	yes	characteristic
184	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
185	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
186	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
187	C₂×C₄	#123	1	1	yes	characteristic
188	C₂×C₄	#125	1	1	yes	characteristic
189	C₂×C₄	#125	1	1	yes	characteristic
190	C₂×C₄	#127	1	1	yes	characteristic
191	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
192	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
193	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
194	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
195	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
196	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
197	C₈	—	2	4	no	—
198	C₈	—	4	2	no	—
199	C₈	—	4	2	no	—
200	C₈	—	4	2	no	—
201	C₂²	—	2	2	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	4	1	no	—
205	C₂²	—	4	1	no	—
206	C₂²	—	4	1	no	—
207	C₂²	—	4	1	no	—
208	C₂²	#385	1	1	yes	characteristic
209	C₂²	#385	1	1	yes	characteristic
210	C₂²	#389	1	1	yes	characteristic
211	C₂²	#395	1	1	yes	characteristic
212	C₂²	#398	1	1	yes	characteristic
213	C₂²	#398	1	1	yes	characteristic
214	C₂²	#399	1	1	yes	characteristic
215	C₄	—	2	4	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	4	no	—
218	C₄	—	8	2	no	—
219	C₄	#384	1	1	yes	characteristic
220	C₄	#385	1	1	yes	characteristic
221	C₄	#389	1	1	yes	characteristic
222	C₄	#389	1	1	yes	characteristic
223	C₄	#392	1	1	yes	characteristic
224	C₄	#395	1	1	yes	characteristic
225	C₄	#395	1	1	yes	characteristic
226	C₄	#398	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	#1444	1	1	yes	characteristic
234	C₂	#1861	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂³×C₄	C₂³
socle	C₂²	#385
prime radical	C₂⁴	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
27. #339	C₄	semidirect
28. #339	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#430	2
#2905	2
C₂×#423	2
C₂×#344	2
#1733	2
C₂×#339	2