Subgroups — #22782

Order 256.

370 conjugacy classes of subgroups.

The subgroup lattice has 281 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#135	C₂	1	1	yes	maximal, characteristic
2	#2925	C₂	1	1	yes	maximal, characteristic, complemented
3	#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	#1938	C₂	1	2	yes	maximal, complemented
5	#1922	C₂	1	2	yes	maximal, complemented
6	#1876	C₂	1	2	yes	maximal
7	#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	#1806	C₂	1	1	yes	maximal, characteristic, complemented
9	#1806	C₂	1	1	yes	maximal, characteristic, complemented
10	#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	#1765	C₂	1	2	yes	maximal, complemented
12	C₂×#135	C₂²	1	1	yes	characteristic
13	C₂×#131	C₂²	1	2	yes	—
14	C₄×#41	C₂²	1	1	yes	characteristic
15	C₄×#41	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆	C₄	1	1	yes	characteristic, complemented
17	C₄×Q₁₆	C₄	1	1	yes	characteristic, complemented
18	C₄×SD₁₆	C₄	1	1	yes	characteristic, complemented
19	C₄×SD₁₆	C₄	1	1	yes	characteristic, complemented
20	C₄×SD₁₆	C₄	1	1	yes	characteristic, complemented
21	C₄×SD₁₆	C₄	1	1	yes	characteristic, complemented
22	C₄×D₈	C₄	1	1	yes	characteristic, complemented
23	C₄×D₈	C₄	1	1	yes	characteristic, complemented
24	#433	—	2	2	no	—
25	#432	—	2	1	no	—
26	#431	—	2	1	no	—
27	#427	C₂²	1	1	yes	characteristic
28	#424	—	2	2	no	—
29	#424	C₂²	1	2	yes	—
30	#423	—	2	1	no	—
31	#423	C₂²	1	1	yes	characteristic
32	#423	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈	—	2	1	no	—
34	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×#104	C₂²	1	2	yes	—
36	C₂×#104	C₂²	1	2	yes	—
37	#416	C₂²	1	1	yes	characteristic
38	#416	C₂²	1	1	yes	characteristic
39	#408	C₂²	1	2	yes	—
40	#405	C₂²	1	2	yes	—
41	C₂×#97	C₂²	1	2	yes	—
42	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
44	#341	C₂²	1	2	yes	—
45	#341	C₂²	1	2	yes	—
46	#339	C₂²	1	2	yes	—
47	#339	C₂²	1	2	yes	—
48	#336	C₂²	1	2	yes	—
49	C₂×#41	C₂³	1	1	yes	characteristic
50	C₂×#41	C₂³	1	1	yes	characteristic
51	#135	—	2	2	no	—
52	#135	—	2	2	no	—
53	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂×D₈	C₂×C₄	1	1	yes	characteristic
57	#131	—	2	4	no	—
58	#131	—	4	2	no	—
59	C₂×M₁₆	—	2	2	no	—
60	C₂×M₁₆	C₂³	1	2	yes	—
61	C₂×M₁₆	C₂³	1	2	yes	—
62	C₂²×C₈	—	2	2	no	—
63	C₂²×C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₈	C₂³	1	2	yes	—
66	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
67	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
69	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄	—	2	1	no	—
71	C₄×D₄	—	2	1	no	—
72	C₄×D₄	—	2	1	no	—
73	C₄×D₄	—	2	1	no	—
74	C₄×D₄	C₂×C₄	1	1	yes	characteristic
75	C₄×D₄	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄	C₂×C₄	1	1	yes	characteristic
77	C₄×D₄	C₂×C₄	1	1	yes	characteristic
78	#117	—	2	1	no	—
79	#117	—	2	1	no	—
80	#117	C₂³	1	1	yes	characteristic
81	#117	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂³	1	2	yes	—
86	C₂×C₄²	C₂³	1	1	yes	characteristic
87	#107	C₂×C₄	1	1	yes	characteristic
88	#107	C₂×C₄	1	1	yes	characteristic
89	#106	C₂×C₄	1	1	yes	characteristic
90	#106	C₂×C₄	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	—	2	1	no	—
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	2	1	no	—
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	D₄	1	1	yes	characteristic
98	C₄⋊C₈	D₄	1	1	yes	characteristic
99	C₄⋊C₈	D₄	1	2	yes	—
100	#104	—	2	4	no	—
101	#104	—	4	2	no	—
102	#103	C₂×C₄	1	1	yes	characteristic
103	#103	C₂×C₄	1	1	yes	characteristic
104	#103	C₂×C₄	1	1	yes	characteristic
105	#103	C₂×C₄	1	1	yes	characteristic
106	#102	C₂×C₄	1	1	yes	characteristic
107	#102	C₂×C₄	1	1	yes	characteristic
108	#102	C₂×C₄	1	1	yes	characteristic
109	#102	C₂×C₄	1	1	yes	characteristic
110	C₂²⋊C₈	—	2	2	no	—
111	C₂²⋊C₈	—	2	2	no	—
112	#97	—	2	2	no	—
113	#97	—	4	2	no	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	C₂×C₄	1	1	yes	characteristic
116	C₄×C₈	C₂×C₄	1	1	yes	characteristic
117	C₄×C₈	C₂×C₄	1	1	yes	characteristic
118	C₄×C₈	C₂×C₄	1	1	yes	characteristic
119	#95	—	2	1	no	—
120	#41	—	2	1	no	—
121	#41	—	2	1	no	—
122	#41	—	2	1	no	—
123	#41	—	2	1	no	—
124	#41	—	2	1	no	—
125	#41	—	2	1	no	—
126	#41	#41	1	2	yes	—
127	#41	#41	1	2	yes	—
128	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
133	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	#41	1	1	yes	characteristic
142	C₂²×C₄	#41	1	1	yes	characteristic
143	C₂²×C₄	C₂⁴	1	1	yes	characteristic
144	Q₁₆	—	2	2	no	—
145	SD₁₆	—	2	2	no	—
146	SD₁₆	—	2	2	no	—
147	D₈	—	2	2	no	—
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	4	4	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	4	4	no	—
162	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₈	C₂×D₄	1	2	yes	—
167	C₂×C₈	C₂×D₄	1	2	yes	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
174	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
175	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
176	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₂²⋊C₄	—	2	1	no	—
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	2	1	no	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	C₂²×C₄	1	1	yes	characteristic
192	C₄²	C₂²×C₄	1	1	yes	characteristic
193	C₄²	C₂×D₄	1	1	yes	characteristic
194	C₄²	C₂×D₄	1	1	yes	characteristic
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	C₂×#41	1	1	yes	characteristic
198	Q₈	—	2	1	no	—
199	Q₈	—	2	1	no	—
200	Q₈	—	2	1	no	—
201	Q₈	#117	1	2	yes	—
202	D₄	—	2	2	no	—
203	D₄	—	2	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	2	1	no	—
207	D₄	—	4	1	no	—
208	D₄	—	4	1	no	—
209	D₄	#117	1	2	yes	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	1	no	—
228	C₂×C₄	—	8	1	no	—
229	C₂×C₄	C₄×D₄	1	1	yes	characteristic
230	C₂×C₄	C₄×D₄	1	1	yes	characteristic
231	C₂×C₄	C₄×D₄	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄	1	1	yes	characteristic
233	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂×#41	1	1	yes	characteristic
236	C₂×C₄	C₂×#41	1	1	yes	characteristic
237	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
238	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	—	4	2	no	—
246	C₈	—	4	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂²	#552	1	1	yes	characteristic
257	C₂²	#553	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	8	2	no	—
268	C₄	#444	1	2	yes	—
269	C₄	C₂×#117	1	1	yes	characteristic
270	C₄	#520	1	1	yes	characteristic
271	C₄	#546	1	1	yes	characteristic
272	C₄	#548	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	4	2	no	—
276	C₂	—	8	1	no	—
277	C₂	#2352	1	1	yes	characteristic
278	C₂	C₂×#444	1	1	yes	characteristic
279	C₂	#2957	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×#135	C₂²
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. #2925	C₂	semidirect
3. #2906	C₂	semidirect
4. #1938	C₂	semidirect
5. #1922	C₂	semidirect
7. #1837	C₂	semidirect
8. #1806	C₂	semidirect
9. #1806	C₂	semidirect
10. #1770	C₂	semidirect
11. #1765	C₂	semidirect
16. C₄×Q₁₆	C₄	semidirect
17. C₄×Q₁₆	C₄	semidirect
18. C₄×SD₁₆	C₄	semidirect
19. C₄×SD₁₆	C₄	semidirect
20. C₄×SD₁₆	C₄	semidirect
21. C₄×SD₁₆	C₄	semidirect
22. C₄×D₈	C₄	semidirect
23. C₄×D₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₄×#135	2
#2925	2
#2906	2
#1938	2
#1922	2
#1876	2
#1837	2
#1806	2
#1770	2
#1765	2