Subgroups — #23239

Order 256.

392 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#1985	C₂	1	1	yes	maximal, characteristic, complemented
3	#1984	C₂	1	1	yes	maximal, characteristic, complemented
4	#1917	C₂	1	1	yes	maximal, characteristic, complemented
5	#1916	C₂	1	1	yes	maximal, characteristic, complemented
6	#1916	C₂	1	1	yes	maximal, characteristic, complemented
7	#1916	C₂	1	1	yes	maximal, characteristic, complemented
8	#1851	C₂	1	1	yes	maximal, characteristic
9	#1851	C₂	1	1	yes	maximal, characteristic
10	#1771	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#1753	C₂	1	1	yes	maximal, characteristic
13	#1753	C₂	1	1	yes	maximal, characteristic
14	C₄×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
15	#1728	C₂	1	1	yes	maximal, characteristic
16	C₂²×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈	C₄	1	1	yes	characteristic
19	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
20	C₂²×C₄²	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×#97	C₂²	1	1	yes	characteristic
28	C₂×#97	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	#391	C₄	1	1	yes	characteristic
34	#389	C₄	1	1	yes	characteristic
35	#387	C₄	1	1	yes	characteristic
36	#387	C₄	1	1	yes	characteristic
37	#385	C₄	1	1	yes	characteristic
38	#385	C₄	1	1	yes	characteristic
39	#380	C₄	1	1	yes	characteristic
40	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
45	#336	C₂²	1	1	yes	characteristic
46	#336	C₂²	1	1	yes	characteristic
47	#336	C₂²	1	1	yes	characteristic
48	#336	C₂²	1	1	yes	characteristic
49	#336	C₂²	1	1	yes	characteristic
50	#336	C₂²	1	1	yes	characteristic
51	#336	C₂²	1	1	yes	characteristic
52	#336	C₂²	1	1	yes	characteristic
53	#336	C₂²	1	1	yes	characteristic
54	#336	C₂²	1	1	yes	characteristic
55	#336	C₂²	1	1	yes	characteristic
56	#336	C₂²	1	1	yes	characteristic
57	#336	C₂²	1	1	yes	characteristic
58	#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄	C₂³	1	1	yes	characteristic
63	C₂×M₁₆	—	2	2	no	—
64	C₂×M₁₆	—	2	2	no	—
65	C₂×M₁₆	—	2	2	no	—
66	C₂×M₁₆	—	2	2	no	—
67	C₂×M₁₆	—	4	1	no	—
68	C₂×M₁₆	—	4	1	no	—
69	C₂²×C₈	C₂³	1	1	yes	characteristic
70	C₂²×C₈	C₂³	1	1	yes	characteristic
71	C₂²×C₈	C₂³	1	1	yes	characteristic
72	C₂²×C₈	C₂³	1	1	yes	characteristic
73	C₂²×C₈	C₂³	1	1	yes	characteristic
74	C₂²×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₈	C₂³	1	1	yes	characteristic
77	C₂²⋊Q₈	—	2	4	no	—
78	C₄×Q₈	—	4	1	no	—
79	C₄×Q₈	—	4	1	no	—
80	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²	—	4	1	no	—
94	C₂×C₄²	—	4	1	no	—
95	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂³	1	1	yes	characteristic
98	C₂×C₄²	C₂³	1	1	yes	characteristic
99	C₂×C₄²	C₂³	1	1	yes	characteristic
100	C₂×C₄²	C₂³	1	1	yes	characteristic
101	C₂×C₄²	C₂³	1	1	yes	characteristic
102	C₂×C₄²	C₂³	1	1	yes	characteristic
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	4	1	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₂²⋊C₈	Q₈	1	4	yes	complemented
109	#97	—	2	2	no	—
110	#97	—	4	1	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	4	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	—	4	1	no	—
115	#95	C₂×C₄	1	1	yes	characteristic
116	#95	C₂×C₄	1	1	yes	characteristic
117	#95	C₂×C₄	1	1	yes	characteristic
118	#95	C₂×C₄	1	1	yes	characteristic
119	#95	C₂×C₄	1	1	yes	characteristic
120	#95	C₂×C₄	1	1	yes	characteristic
121	#95	C₂×C₄	1	1	yes	characteristic
122	#95	C₂×C₄	1	1	yes	characteristic
123	#95	C₂×C₄	1	1	yes	characteristic
124	#95	C₂×C₄	1	1	yes	characteristic
125	C₂⁴	C₂²×C₄	1	1	yes	characteristic
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	4	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	4	1	no	—
139	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂×Q₈	1	2	yes	—
153	C₂²×C₄	#41	1	2	yes	—
154	C₂²×C₄	C₂⁴	1	1	yes	characteristic
155	M₁₆	—	4	4	no	—
156	M₁₆	—	4	4	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	C₂×Q₈	1	2	yes	—
178	C₂×C₈	C₂×Q₈	1	2	yes	—
179	C₂×C₈	#41	1	2	yes	—
180	C₂×C₈	#41	1	2	yes	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	M₁₆	1	4	yes	complemented
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	4	1	no	—
193	C₂²⋊C₄	—	4	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	4	1	no	—
207	C₄²	—	4	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	C₄×Q₈	1	2	yes	—
215	C₂³	C₂³×C₄	1	1	yes	characteristic
216	Q₈	—	4	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₄×Q₈	1	2	yes	—
257	C₂×C₄	C₂×M₁₆	1	2	yes	—
258	C₂×C₄	C₂×M₁₆	1	2	yes	—
259	C₂×C₄	C₂×M₁₆	1	2	yes	—
260	C₂×C₄	#131	1	2	yes	—
261	C₂×C₄	#131	1	2	yes	—
262	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
263	C₂×C₄	C₂×#41	1	1	yes	characteristic
264	C₂×C₄	C₂×#41	1	1	yes	characteristic
265	C₂×C₄	C₂×#41	1	1	yes	characteristic
266	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
267	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
284	C₂²	C₄×#41	1	1	yes	characteristic
285	C₂²	#518	1	1	yes	characteristic
286	C₂²	C₂²×M₁₆	1	1	yes	characteristic
287	C₂²	C₂×#131	1	1	yes	characteristic
288	C₂²	C₂×#131	1	1	yes	characteristic
289	C₂²	#568	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	#551	1	1	yes	characteristic
304	C₄	#551	1	1	yes	characteristic
305	C₄	#554	1	1	yes	characteristic
306	C₄	#554	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	#2368	1	1	yes	characteristic
309	C₂	#2960	1	1	yes	characteristic
310	C₂	Q₈×M₁₆	1	1	yes	characteristic
311	C₂	#2965	1	1	yes	characteristic
312	C₂	#2969	1	1	yes	characteristic
313	C₂	#2974	1	1	yes	characteristic
314	C₂	#2976	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. #1985	C₂	semidirect
3. #1984	C₂	semidirect
4. #1917	C₂	semidirect
5. #1916	C₂	semidirect
6. #1916	C₂	semidirect
7. #1916	C₂	semidirect
10. #1771	C₂	semidirect
11. C₈×C₄⋊C₄	C₂	semidirect
108. C₂²⋊C₈	Q₈	semidirect
189. C₄⋊C₄	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊Q₈	2
#1985	2
#1984	2
#1917	2
#1916	2
#1851	2
#1771	2
C₈×C₄⋊C₄	2
#1753	2
C₄×C₂²⋊C₈	2
#1728	2