Subgroups — #22697

Order 256.

472 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#435	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	#2290	C₂	1	1	yes	maximal, characteristic
4	#1940	C₂	1	1	yes	maximal, characteristic, complemented
5	#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	#1851	C₂	1	1	yes	maximal, characteristic, complemented
7	#1850	C₂	1	1	yes	maximal, characteristic, complemented
8	#1850	C₂	1	1	yes	maximal, characteristic, complemented
9	#1850	C₂	1	1	yes	maximal, characteristic, complemented
10	#1832	C₂	1	1	yes	maximal, characteristic, complemented
11	#1798	C₂	1	1	yes	maximal, characteristic, complemented
12	#1797	C₂	1	1	yes	maximal, characteristic, complemented
13	#1768	C₂	1	1	yes	maximal, characteristic, complemented
14	#1729	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#336	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆	C₂²	1	1	yes	characteristic
17	C₂³×C₈	C₂²	1	1	yes	characteristic
18	C₂³×C₈	C₂²	1	1	yes	characteristic
19	C₂³×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
21	C₂×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄²	C₂²	1	1	yes	characteristic
23	#435	—	2	4	no	—
24	C₈×D₄	—	2	4	no	—
25	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈	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	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×#97	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	#388	C₄	1	1	yes	characteristic
40	#388	C₄	1	1	yes	characteristic
41	#387	C₄	1	1	yes	characteristic
42	#387	C₄	1	1	yes	characteristic
43	#386	C₄	1	1	yes	characteristic
44	#386	C₄	1	1	yes	characteristic
45	#385	C₄	1	1	yes	characteristic
46	#385	C₄	1	1	yes	characteristic
47	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
49	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
50	#376	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	#336	C₂²	1	1	yes	characteristic
60	#336	C₂²	1	1	yes	characteristic
61	C₂²×D₄	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄	C₂³	1	1	yes	characteristic
65	C₂³×C₄	C₂³	1	1	yes	characteristic
66	C₂×M₁₆	—	2	2	no	—
67	C₂×M₁₆	—	2	2	no	—
68	C₂×M₁₆	—	4	1	no	—
69	C₂²×C₈	—	2	2	no	—
70	C₂²×C₈	—	2	2	no	—
71	C₂²×C₈	—	2	2	no	—
72	C₂²×C₈	—	2	2	no	—
73	C₂²×C₈	—	2	2	no	—
74	C₂²×C₈	—	2	2	no	—
75	C₂²×C₈	—	4	1	no	—
76	C₂²×C₈	—	4	1	no	—
77	C₂²×C₈	—	4	1	no	—
78	C₂²×C₈	C₂³	1	1	yes	characteristic
79	C₂²×C₈	C₂³	1	1	yes	characteristic
80	C₂²×C₈	C₂³	1	1	yes	characteristic
81	C₂²×C₈	C₂³	1	1	yes	characteristic
82	C₂²×C₈	C₂³	1	1	yes	characteristic
83	C₂²×C₈	C₂³	1	1	yes	characteristic
84	C₂²×C₈	C₂³	1	1	yes	characteristic
85	C₂²×C₈	C₂³	1	1	yes	characteristic
86	C₄×D₄	D₄	1	4	yes	—
87	C₄×D₄	Q₈	1	4	yes	—
88	#117	—	4	1	no	—
89	#117	—	4	1	no	—
90	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
98	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
100	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
101	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
102	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄²	—	4	1	no	—
104	C₂×C₄²	—	4	1	no	—
105	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²	C₂³	1	1	yes	characteristic
108	C₂×C₄²	C₂³	1	1	yes	characteristic
109	C₂×C₄²	C₂³	1	1	yes	characteristic
110	C₂×C₄²	C₂³	1	1	yes	characteristic
111	C₂×C₄²	C₂³	1	1	yes	characteristic
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	4	1	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₄⋊C₈	—	4	1	no	—
119	C₄⋊C₈	—	4	1	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	2	2	no	—
122	C₂²⋊C₈	—	2	2	no	—
123	C₂²⋊C₈	—	2	2	no	—
124	#97	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	#95	C₂×C₄	1	1	yes	characteristic
127	#95	C₂×C₄	1	1	yes	characteristic
128	#95	C₂×C₄	1	1	yes	characteristic
129	#95	C₂×C₄	1	1	yes	characteristic
130	#95	C₂×C₄	1	1	yes	characteristic
131	#95	C₂×C₄	1	1	yes	characteristic
132	#95	C₂×C₄	1	1	yes	characteristic
133	#95	C₂×C₄	1	1	yes	characteristic
134	#95	C₂×C₄	1	1	yes	characteristic
135	#95	C₂×C₄	1	1	yes	characteristic
136	C₂⁴	C₂²×C₄	1	1	yes	characteristic
137	C₂⁴	C₂²×C₄	1	1	yes	characteristic
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	C₄⋊C₄	1	4	yes	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄	1	2	yes	—
168	C₂²×C₄	C₂×D₄	1	2	yes	—
169	C₂²×C₄	C₂×Q₈	1	2	yes	—
170	C₂²×C₄	C₂×Q₈	1	2	yes	—
171	C₂²×C₄	C₂⁴	1	1	yes	characteristic
172	M₁₆	—	4	4	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	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	#41	1	2	yes	—
199	C₂×C₈	#41	1	2	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₄⋊C₄	—	4	1	no	—
204	C₄⋊C₄	C₄⋊C₄	1	4	yes	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	1	no	—
211	C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
212	C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	C₂×D₄	1	2	yes	—
224	C₄²	C₂×Q₈	1	2	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	C₂×C₄⋊C₄	1	2	yes	—
236	C₂³	C₂×C₄⋊C₄	1	2	yes	—
237	C₂³	#131	1	2	yes	—
238	C₂³	#131	1	2	yes	—
239	C₂³	C₂³×C₄	1	1	yes	characteristic
240	D₄	—	2	8	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	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
283	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
284	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
286	C₂×C₄	#131	1	2	yes	—
287	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
289	C₂×C₄	C₂×#41	1	1	yes	characteristic
290	C₂×C₄	C₂×#41	1	1	yes	characteristic
291	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
292	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	—	4	1	no	—
296	C₈	—	4	1	no	—
297	C₈	—	4	1	no	—
298	C₈	—	4	1	no	—
299	C₈	—	4	1	no	—
300	C₈	—	4	1	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	#424	1	4	yes	—
315	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
316	C₂²	C₄×#41	1	1	yes	characteristic
317	C₂²	#518	1	1	yes	characteristic
318	C₂²	C₂×#131	1	1	yes	characteristic
319	C₂²	C₂×#131	1	1	yes	characteristic
320	C₂²	C₂×#131	1	1	yes	characteristic
321	C₂²	#568	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	#546	1	1	yes	characteristic
335	C₄	#546	1	1	yes	characteristic
336	C₄	#554	1	1	yes	characteristic
337	C₄	#554	1	1	yes	characteristic
338	C₂	—	2	4	no	—
339	C₂	—	4	2	no	—
340	C₂	#2349	1	1	yes	characteristic
341	C₂	C₂×#424	1	1	yes	characteristic
342	C₂	#2906	1	1	yes	characteristic
343	C₂	#2966	1	1	yes	characteristic
344	C₂	#2968	1	1	yes	characteristic
345	C₂	#2976	1	1	yes	characteristic
346	C₂	#2977	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator, socle	C₂³	C₂³×C₄
prime radical	C₂²×D₄	C₂×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
4. #1940	C₂	semidirect
5. #1939	C₂	semidirect
6. #1851	C₂	semidirect
7. #1850	C₂	semidirect
8. #1850	C₂	semidirect
9. #1850	C₂	semidirect
10. #1832	C₂	semidirect
11. #1798	C₂	semidirect
12. #1797	C₂	semidirect
13. #1768	C₂	semidirect
14. #1729	C₂	semidirect
15. C₂×#336	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#435	2
C₂×C₈×D₄	2
#2290	2
#1940	2
#1939	2
#1851	2
#1850	2
#1832	2
#1798	2
#1797	2
#1768	2
#1729	2
C₂×#336	2