Subgroups — #22982

Order 256.

360 conjugacy classes of subgroups.

The subgroup lattice has 245 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	#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	#2906	C₂	1	1	yes	maximal, characteristic
4	#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	#1945	C₂	1	2	yes	maximal, complemented
7	#1922	C₂	1	2	yes	maximal, complemented
8	#1866	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#1764	C₂	1	2	yes	maximal
11	C₄×#104	C₂	1	1	yes	maximal, characteristic
12	C₂×#131	C₂²	1	2	yes	—
13	#533	C₂²	1	1	yes	characteristic
14	C₄×#41	C₂²	1	1	yes	characteristic
15	#479	C₄	1	2	yes	complemented
16	#478	C₄	1	2	yes	complemented
17	#477	C₄	1	1	yes	characteristic, complemented
18	#476	C₄	1	1	yes	characteristic, complemented
19	#475	C₄	1	1	yes	characteristic, complemented
20	#474	C₄	1	1	yes	characteristic, complemented
21	#446	—	2	2	no	—
22	#446	—	2	1	no	—
23	C₈×Q₈	—	2	1	no	—
24	C₂×#108	C₂²	1	2	yes	—
25	#427	C₂²	1	2	yes	—
26	#424	—	2	2	no	—
27	#424	C₂²	1	2	yes	—
28	#424	C₂²	1	2	yes	—
29	#423	—	2	1	no	—
30	#423	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈	—	2	1	no	—
32	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×#104	C₂²	1	1	yes	characteristic
35	C₂×#104	C₂²	1	1	yes	characteristic
36	#416	C₂²	1	2	yes	—
37	#405	C₂²	1	2	yes	—
38	C₄×M₁₆	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂²	1	2	yes	—
40	C₄³	C₂²	1	1	yes	characteristic
41	#341	C₂²	1	2	yes	—
42	#339	C₂²	1	2	yes	—
43	#339	C₂²	1	1	yes	characteristic
44	#339	C₂²	1	1	yes	characteristic
45	#339	C₂²	1	2	yes	—
46	#336	C₂²	1	2	yes	—
47	C₂×#41	C₂³	1	1	yes	characteristic
48	#131	—	2	4	no	—
49	#131	—	4	2	no	—
50	C₂×M₁₆	—	2	2	no	—
51	C₂×M₁₆	C₂³	1	2	yes	—
52	C₂×M₁₆	C₂³	1	1	yes	characteristic
53	C₂×M₁₆	C₂³	1	1	yes	characteristic
54	C₂²×C₈	—	2	2	no	—
55	C₂²×C₈	C₂³	1	2	yes	—
56	C₂²×C₈	C₂³	1	2	yes	—
57	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
59	#125	C₂×C₄	1	2	yes	—
60	C₂²⋊Q₈	—	2	1	no	—
61	C₂²⋊Q₈	—	2	1	no	—
62	C₄×Q₈	—	2	1	no	—
63	C₄×Q₈	—	2	1	no	—
64	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
66	C₄×D₄	—	2	1	no	—
67	C₄×D₄	—	2	1	no	—
68	C₄×D₄	C₂×C₄	1	1	yes	characteristic
69	C₄×D₄	C₂×C₄	1	1	yes	characteristic
70	#117	—	2	1	no	—
71	#117	C₂³	1	1	yes	characteristic
72	#117	C₂³	1	2	yes	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	—	2	1	no	—
75	C₂×C₄²	—	2	1	no	—
76	C₂×C₄²	C₂³	1	1	yes	characteristic
77	C₂×C₄²	C₂³	1	1	yes	characteristic
78	C₂×C₄²	C₂³	1	1	yes	characteristic
79	#108	—	4	2	no	—
80	#107	C₂×C₄	1	2	yes	—
81	#107	C₂×C₄	1	2	yes	—
82	#106	C₂×C₄	1	2	yes	—
83	#106	C₂×C₄	1	2	yes	—
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	2	2	no	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
91	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
92	C₄⋊C₈	C₂×C₄	1	2	yes	—
93	C₄⋊C₈	D₄	1	1	yes	characteristic
94	C₄⋊C₈	D₄	1	1	yes	characteristic
95	C₄⋊C₈	D₄	1	2	yes	—
96	#104	—	2	2	no	—
97	#104	Q₈	1	4	yes	complemented
98	#103	C₂×C₄	1	2	yes	—
99	#103	C₂×C₄	1	2	yes	—
100	#102	C₂×C₄	1	2	yes	—
101	#102	C₂×C₄	1	2	yes	—
102	#97	—	2	2	no	—
103	#97	—	2	1	no	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	4	2	no	—
107	#41	—	2	1	no	—
108	#41	—	2	1	no	—
109	#41	C₂×Q₈	1	2	yes	—
110	#41	#41	1	2	yes	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄	—	2	1	no	—
115	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	#41	1	1	yes	characteristic
122	C₂²×C₄	#41	1	1	yes	characteristic
123	C₂²×C₄	C₂⁴	1	1	yes	characteristic
124	M₁₆	—	2	1	no	—
125	M₁₆	—	2	1	no	—
126	M₁₆	—	2	2	no	—
127	M₁₆	—	2	1	no	—
128	M₁₆	—	2	2	no	—
129	M₁₆	—	4	4	no	—
130	M₁₆	C₂×Q₈	1	2	yes	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	C₂²×C₄	1	2	yes	—
140	C₂×C₈	C₂²×C₄	1	2	yes	—
141	C₂×C₈	C₂×D₄	1	2	yes	—
142	C₂×C₈	C₂×D₄	1	2	yes	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	C₂²×C₄	1	2	yes	—
150	C₄⋊C₄	C₂²×C₄	1	2	yes	—
151	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
153	C₂²⋊C₄	—	2	1	no	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	2	1	no	—
156	C₂²⋊C₄	—	2	1	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	4	2	no	—
166	C₄²	C₂²×C₄	1	1	yes	characteristic
167	C₄²	C₂²×C₄	1	1	yes	characteristic
168	C₄²	C₂×D₄	1	1	yes	characteristic
169	C₄²	C₂×D₄	1	1	yes	characteristic
170	C₄²	C₂×Q₈	1	2	yes	—
171	C₂³	—	2	1	no	—
172	C₂³	C₂×#41	1	1	yes	characteristic
173	Q₈	—	2	1	no	—
174	Q₈	—	4	2	no	—
175	Q₈	—	8	1	no	—
176	Q₈	C₄×Q₈	1	2	yes	—
177	D₄	—	2	2	no	—
178	D₄	—	2	2	no	—
179	D₄	—	2	1	no	—
180	D₄	C₄×Q₈	1	2	yes	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	8	1	no	—
198	C₂×C₄	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
205	C₂×C₄	C₂×#41	1	1	yes	characteristic
206	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₈	—	4	2	no	—
214	C₈	—	4	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	1	no	—
220	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂²	#552	1	1	yes	characteristic
222	C₂²	#554	1	1	yes	characteristic
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	2	no	—
233	C₄	#443	1	2	yes	—
234	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
235	C₄	#520	1	1	yes	characteristic
236	C₄	#546	1	1	yes	characteristic
237	C₄	D₄×Q₈	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	4	2	no	—
241	C₂	#2360	1	1	yes	characteristic
242	C₂	C₂×#443	1	1	yes	characteristic
243	C₂	#2956	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. #2962	C₂	semidirect
4. #1994	C₂	semidirect
5. #1950	C₂	semidirect
6. #1945	C₂	semidirect
7. #1922	C₂	semidirect
9. C₄×C₄⋊C₈	C₂	semidirect
15. #479	C₄	semidirect
16. #478	C₄	semidirect
17. #477	C₄	semidirect
18. #476	C₄	semidirect
19. #475	C₄	semidirect
20. #474	C₄	semidirect
97. #104	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
#2962	2
#2906	2
#1994	2
#1950	2
#1945	2
#1922	2
#1866	2
C₄×C₄⋊C₈	2
#1764	2
C₄×#104	2