Subgroups — #191313

Order 512.

456 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆²	C₂	1	1	yes	maximal, characteristic
2	#8804	C₂	1	4	yes	maximal, complemented
3	#8803	C₂	1	4	yes	maximal, complemented
4	#7996	C₂	1	1	yes	maximal, characteristic
5	#7992	C₂	1	1	yes	maximal, characteristic
6	#7992	C₂	1	1	yes	maximal, characteristic
7	#7976	C₂	1	1	yes	maximal, characteristic
8	#7974	C₂	1	1	yes	maximal, characteristic
9	#7973	C₂	1	1	yes	maximal, characteristic
10	#2920	C₂²	1	1	yes	characteristic
11	#2920	C₂²	1	1	yes	characteristic
12	#2920	C₂²	1	1	yes	characteristic
13	#2920	C₂²	1	1	yes	characteristic
14	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
16	#2283	C₂²	1	1	yes	characteristic
17	#1455	C₂²	1	4	yes	—
18	#1451	C₄	1	2	yes	complemented
19	#1451	C₄	1	2	yes	complemented
20	C₈×M₁₆	C₄	1	2	yes	complemented
21	C₈×M₁₆	C₄	1	2	yes	complemented
22	#1291	C₂²	1	2	yes	—
23	#1291	C₂²	1	4	yes	—
24	#1291	C₂²	1	2	yes	—
25	#1291	C₂²	1	2	yes	—
26	#1291	C₂²	1	2	yes	—
27	#1288	C₂²	1	4	yes	—
28	#1288	C₂²	1	2	yes	—
29	#1288	C₂²	1	2	yes	—
30	#1288	C₂²	1	4	yes	—
31	C₂²×M₁₆	—	2	1	no	—
32	C₂²×M₁₆	—	2	1	no	—
33	C₂²×C₄²	C₂³	1	1	yes	characteristic
34	#432	—	2	1	no	—
35	#432	—	2	1	no	—
36	#432	—	2	1	no	—
37	#432	—	2	1	no	—
38	#432	—	2	1	no	—
39	#432	C₂³	1	2	yes	—
40	#432	C₂³	1	1	yes	characteristic
41	#432	C₂³	1	1	yes	characteristic
42	#432	C₂³	1	2	yes	—
43	#431	—	2	1	no	—
44	#431	—	2	1	no	—
45	#431	—	2	1	no	—
46	#431	—	2	1	no	—
47	#431	—	2	1	no	—
48	#431	—	2	1	no	—
49	#431	C₂×C₄	1	2	yes	—
50	#431	C₂×C₄	1	2	yes	—
51	#423	—	2	1	no	—
52	#423	—	2	1	no	—
53	#423	—	2	1	no	—
54	#423	—	2	1	no	—
55	#407	—	2	1	no	—
56	#407	—	2	1	no	—
57	#407	—	2	1	no	—
58	#407	—	2	1	no	—
59	C₄×M₁₆	—	2	1	no	—
60	C₄×M₁₆	—	2	1	no	—
61	C₄×M₁₆	—	2	1	no	—
62	C₄×M₁₆	—	2	1	no	—
63	C₄×M₁₆	—	2	1	no	—
64	C₄×M₁₆	—	2	1	no	—
65	C₄×M₁₆	—	2	1	no	—
66	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
67	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
68	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
69	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
70	C₄×M₁₆	C₂³	1	2	yes	—
71	C₄×M₁₆	C₂³	1	2	yes	—
72	C₄×M₁₆	C₂³	1	2	yes	—
73	C₂×#97	—	2	1	no	—
74	C₂×#97	—	2	1	no	—
75	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
76	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
77	#383	C₂³	1	2	yes	—
78	#382	—	2	1	no	—
79	#381	—	2	1	no	—
80	#380	—	2	1	no	—
81	C₄×C₄⋊C₄	—	2	1	no	—
82	C₄×C₂²⋊C₄	—	2	1	no	—
83	#376	—	2	1	no	—
84	#322	C₂×C₄	1	4	yes	—
85	#322	C₂×C₄	1	4	yes	—
86	#322	C₂×C₄	1	4	yes	—
87	C₈²	C₂×C₄	1	4	yes	—
88	C₂³×C₄	#41	1	1	yes	characteristic
89	C₂³×C₄	#41	1	1	yes	characteristic
90	C₂³×C₄	#41	1	1	yes	characteristic
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	4	4	no	—
99	C₂×M₁₆	—	4	4	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂×M₁₆	—	4	1	no	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂²×C₈	—	2	2	no	—
104	C₂²×C₈	—	2	2	no	—
105	C₂×C₄⋊C₄	—	4	1	no	—
106	C₂×C₂²⋊C₄	—	4	1	no	—
107	C₂×C₄²	—	4	1	no	—
108	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄²	#41	1	1	yes	characteristic
111	C₂×C₄²	#41	1	1	yes	characteristic
112	C₂×C₄²	#41	1	1	yes	characteristic
113	C₂×C₄²	#41	1	1	yes	characteristic
114	C₂×C₄²	#41	1	1	yes	characteristic
115	C₂×C₄²	#41	1	1	yes	characteristic
116	C₂×C₄²	#41	1	1	yes	characteristic
117	C₂×C₄²	#41	1	1	yes	characteristic
118	C₂×C₄²	#41	1	1	yes	characteristic
119	C₂×C₄²	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	1	no	—
123	C₄⋊C₈	—	2	1	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	1	no	—
127	C₄⋊C₈	—	4	1	no	—
128	C₄⋊C₈	—	4	1	no	—
129	C₂²⋊C₈	—	2	2	no	—
130	C₂²⋊C₈	—	2	2	no	—
131	C₂²⋊C₈	—	2	2	no	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	2	1	no	—
134	C₂²⋊C₈	—	2	1	no	—
135	C₂²⋊C₈	—	4	1	no	—
136	C₂²⋊C₈	—	4	1	no	—
137	C₂²⋊C₈	—	4	1	no	—
138	#97	—	2	1	no	—
139	#97	—	2	1	no	—
140	#97	—	2	1	no	—
141	#97	—	2	1	no	—
142	#97	—	2	2	no	—
143	#97	—	2	2	no	—
144	#97	—	2	2	no	—
145	#97	—	2	2	no	—
146	#97	—	2	1	no	—
147	#97	—	2	1	no	—
148	#97	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	4	1	no	—
152	C₄×C₈	—	4	1	no	—
153	C₄×C₈	—	4	1	no	—
154	C₄×C₈	C₂²×C₄	1	2	yes	—
155	C₄×C₈	C₂²×C₄	1	2	yes	—
156	C₄×C₈	C₂²×C₄	1	4	yes	—
157	C₄×C₈	C₂²×C₄	1	2	yes	—
158	C₄×C₈	C₂²×C₄	1	2	yes	—
159	#95	—	2	2	no	—
160	#95	—	2	2	no	—
161	#95	—	2	2	no	—
162	#95	—	4	1	no	—
163	C₂⁴	#126	1	1	yes	characteristic
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	8	1	no	—
168	C₂²×C₄	#117	1	1	yes	characteristic
169	C₂²×C₄	#117	1	1	yes	characteristic
170	C₂²×C₄	#117	1	1	yes	characteristic
171	C₂²×C₄	#117	1	1	yes	characteristic
172	C₂²×C₄	#117	1	1	yes	characteristic
173	C₂²×C₄	#117	1	1	yes	characteristic
174	C₂²×C₄	#126	1	1	yes	characteristic
175	C₂²×C₄	#126	1	1	yes	characteristic
176	C₂²×C₄	#126	1	1	yes	characteristic
177	C₂²×C₄	#126	1	1	yes	characteristic
178	C₂²×C₄	#126	1	1	yes	characteristic
179	C₂²×C₄	#126	1	1	yes	characteristic
180	C₂²×C₄	#126	1	1	yes	characteristic
181	C₂²×C₄	#126	1	1	yes	characteristic
182	C₂²×C₄	#126	1	1	yes	characteristic
183	C₂²×C₄	C₂×#41	1	1	yes	characteristic
184	C₂²×C₄	C₂×#41	1	1	yes	characteristic
185	C₂²×C₄	C₂×#41	1	1	yes	characteristic
186	M₁₆	—	2	4	no	—
187	M₁₆	—	2	4	no	—
188	M₁₆	—	4	4	no	—
189	M₁₆	—	4	4	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	2	no	—
192	M₁₆	—	8	2	no	—
193	M₁₆	—	8	2	no	—
194	M₁₆	—	8	2	no	—
195	M₁₆	—	8	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₄⋊C₄	—	8	2	no	—
208	C₂²⋊C₄	—	8	2	no	—
209	C₄²	—	8	2	no	—
210	C₄²	#117	1	1	yes	characteristic
211	C₄²	#117	1	1	yes	characteristic
212	C₄²	#117	1	1	yes	characteristic
213	C₄²	#117	1	1	yes	characteristic
214	C₄²	#117	1	1	yes	characteristic
215	C₄²	#117	1	1	yes	characteristic
216	C₄²	#126	1	1	yes	characteristic
217	C₄²	#126	1	1	yes	characteristic
218	C₄²	#126	1	1	yes	characteristic
219	C₄²	#126	1	1	yes	characteristic
220	C₄²	#126	1	1	yes	characteristic
221	C₄²	#126	1	1	yes	characteristic
222	C₄²	C₂³×C₄	1	1	yes	characteristic
223	C₄²	C₂×#41	1	1	yes	characteristic
224	C₄²	C₂×#41	1	1	yes	characteristic
225	C₄²	C₂×#41	1	1	yes	characteristic
226	C₂³	—	4	1	no	—
227	C₂³	—	4	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	#383	1	1	yes	characteristic
230	C₂³	#383	1	1	yes	characteristic
231	C₂³	C₂×#126	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	#383	1	1	yes	characteristic
254	C₂×C₄	#383	1	1	yes	characteristic
255	C₂×C₄	#383	1	1	yes	characteristic
256	C₂×C₄	#383	1	1	yes	characteristic
257	C₂×C₄	#383	1	1	yes	characteristic
258	C₂×C₄	#383	1	1	yes	characteristic
259	C₂×C₄	C₂×#117	1	1	yes	characteristic
260	C₂×C₄	C₂×#117	1	1	yes	characteristic
261	C₂×C₄	C₂×#117	1	1	yes	characteristic
262	C₂×C₄	C₂×#126	1	1	yes	characteristic
263	C₂×C₄	C₂×#126	1	1	yes	characteristic
264	C₂×C₄	C₂×#126	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	4	2	no	—
269	C₈	—	4	4	no	—
270	C₈	—	8	2	no	—
271	C₈	—	8	2	no	—
272	C₈	—	8	1	no	—
273	C₈	—	8	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	C₂×#383	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	16	2	no	—
296	C₄	#1841	1	2	yes	—
297	C₄	#1841	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	—	4	1	no	—
300	C₂	—	4	1	no	—
301	C₂	C₂×#1841	1	1	yes	characteristic
302	C₂	C₂×#1841	1	1	yes	characteristic
303	C₂	#20776	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#383
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #8804	C₂	semidirect
3. #8803	C₂	semidirect
18. #1451	C₄	semidirect
19. #1451	C₄	semidirect
20. C₈×M₁₆	C₄	semidirect
21. C₈×M₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
M₁₆²	2
#8804	2
#8803	2
#7996	2
#7992	2
#7976	2
#7974	2
#7973	2