Subgroups — #191267

Order 512.

492 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17793	C₂	1	1	yes	maximal, characteristic
2	C₂×#1568	C₂	1	1	yes	maximal, characteristic
3	#17323	C₂	1	1	yes	maximal, characteristic
4	#8799	C₂	1	4	yes	maximal, complemented
5	#8797	C₂	1	4	yes	maximal, complemented
6	#7970	C₂	1	2	yes	maximal
7	#7968	C₂	1	2	yes	maximal
8	#2920	C₂²	1	2	yes	—
9	#2920	C₂²	1	1	yes	characteristic
10	#2920	C₂²	1	1	yes	characteristic
11	C₂×#432	C₂²	1	2	yes	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#1568	—	2	1	no	—
14	#1568	—	2	1	no	—
15	#1568	C₂²	1	2	yes	—
16	#1568	C₂²	1	2	yes	—
17	#1568	C₂²	1	2	yes	—
18	#1568	C₂²	1	2	yes	—
19	#1567	—	2	1	no	—
20	#1566	—	2	1	no	—
21	#1564	—	2	1	no	—
22	#1564	—	2	1	no	—
23	C₂×#335	—	2	1	no	—
24	C₂×#334	—	2	1	no	—
25	#1452	C₂²	1	2	yes	—
26	#1451	C₄	1	2	yes	—
27	#1451	C₄	1	2	yes	—
28	#1450	C₂²	1	2	yes	—
29	C₈×M₁₆	C₄	1	2	yes	—
30	C₈×M₁₆	C₄	1	2	yes	—
31	#1287	C₂²	1	8	yes	—
32	#1287	C₂²	1	8	yes	—
33	C₂³×C₈	Q₈	1	1	yes	characteristic
34	C₂³×C₈	Q₈	1	1	yes	characteristic
35	C₂²×C₄²	C₂³	1	1	yes	characteristic
36	#432	—	2	1	no	—
37	#432	—	2	1	no	—
38	#432	—	2	1	no	—
39	#432	—	2	1	no	—
40	#432	—	2	2	no	—
41	#432	—	2	2	no	—
42	#432	—	2	2	no	—
43	#432	—	2	2	no	—
44	#432	C₂³	1	4	yes	—
45	#432	C₂³	1	4	yes	—
46	#431	—	2	2	no	—
47	#431	—	2	2	no	—
48	#431	C₂×C₄	1	2	yes	—
49	#431	C₂×C₄	1	2	yes	—
50	#431	C₂³	1	2	yes	—
51	#431	C₂³	1	2	yes	—
52	#423	—	2	1	no	—
53	#423	—	2	1	no	—
54	#423	—	2	2	no	—
55	C₂×C₄⋊C₈	—	2	2	no	—
56	#407	—	2	1	no	—
57	#407	—	2	1	no	—
58	#407	—	2	2	no	—
59	C₂×C₂²⋊C₈	—	2	2	no	—
60	C₄×M₁₆	—	2	2	no	—
61	C₄×M₁₆	C₂×C₄	1	2	yes	—
62	C₄×M₁₆	C₂×C₄	1	2	yes	—
63	C₂×#97	—	2	2	no	—
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	D₄	1	1	yes	characteristic
70	C₂×C₄×C₈	D₄	1	1	yes	characteristic
71	C₂×C₄×C₈	D₄	1	1	yes	characteristic
72	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
76	#335	—	2	2	no	—
77	#335	—	2	2	no	—
78	#335	—	2	2	no	—
79	#335	—	2	2	no	—
80	#334	—	2	2	no	—
81	#334	—	2	2	no	—
82	#334	—	2	2	no	—
83	#334	—	2	2	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	C₂×Q₈	1	1	yes	characteristic
89	C₂³×C₄	#41	1	1	yes	characteristic
90	C₂³×C₄	#41	1	1	yes	characteristic
91	C₂×M₁₆	—	2	2	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	4	2	no	—
94	C₂²×C₈	—	4	2	no	—
95	C₂²×C₈	—	4	2	no	—
96	C₂²×C₈	—	4	2	no	—
97	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
106	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄²	C₂×Q₈	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	#41	1	1	yes	characteristic
120	C₂×C₄²	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	4	no	—
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	—	4	2	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	4	no	—
134	C₂²⋊C₈	—	2	4	no	—
135	C₂²⋊C₈	—	4	2	no	—
136	C₂²⋊C₈	—	4	2	no	—
137	#97	—	2	2	no	—
138	#97	—	2	2	no	—
139	#97	—	2	4	no	—
140	#97	—	2	4	no	—
141	#97	—	4	2	no	—
142	C₄×C₈	—	4	2	no	—
143	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₈	C₂²×C₄	1	4	yes	—
152	C₄×C₈	C₂²×C₄	1	4	yes	—
153	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
156	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₄×C₈	C₂×D₄	1	1	yes	characteristic
158	C₄×C₈	C₂×D₄	1	1	yes	characteristic
159	C₄×C₈	C₂×D₄	1	1	yes	characteristic
160	C₄×C₈	C₂×D₄	1	1	yes	characteristic
161	C₂⁴	#125	1	1	yes	characteristic
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#124	1	1	yes	characteristic
173	C₂²×C₄	#124	1	1	yes	characteristic
174	C₂²×C₄	#124	1	1	yes	characteristic
175	C₂²×C₄	#124	1	1	yes	characteristic
176	C₂²×C₄	#125	1	1	yes	characteristic
177	C₂²×C₄	#125	1	1	yes	characteristic
178	C₂²×C₄	#125	1	1	yes	characteristic
179	C₂²×C₄	#125	1	1	yes	characteristic
180	C₂²×C₄	#125	1	1	yes	characteristic
181	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×#41	1	1	yes	characteristic
183	C₂²×C₄	C₂×#41	1	1	yes	characteristic
184	M₁₆	—	4	4	no	—
185	M₁₆	—	4	4	no	—
186	M₁₆	—	8	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	8	4	no	—
204	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
214	C₄²	#117	1	1	yes	characteristic
215	C₄²	#117	1	1	yes	characteristic
216	C₄²	#117	1	1	yes	characteristic
217	C₄²	#117	1	1	yes	characteristic
218	C₄²	#124	1	1	yes	characteristic
219	C₄²	#124	1	1	yes	characteristic
220	C₄²	#124	1	1	yes	characteristic
221	C₄²	#124	1	1	yes	characteristic
222	C₄²	#125	1	1	yes	characteristic
223	C₄²	#125	1	1	yes	characteristic
224	C₄²	C₂³×C₄	1	1	yes	characteristic
225	C₄²	C₂²×D₄	1	1	yes	characteristic
226	C₄²	C₂×#41	1	1	yes	characteristic
227	C₄²	C₂×#41	1	1	yes	characteristic
228	C₂³	—	2	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	#382	1	1	yes	characteristic
232	C₂³	#382	1	1	yes	characteristic
233	C₂³	C₂×#125	1	1	yes	characteristic
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	—	2	2	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	—	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	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	#382	1	1	yes	characteristic
256	C₂×C₄	#382	1	1	yes	characteristic
257	C₂×C₄	#382	1	1	yes	characteristic
258	C₂×C₄	#382	1	1	yes	characteristic
259	C₂×C₄	#382	1	1	yes	characteristic
260	C₂×C₄	#382	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×#117	1	1	yes	characteristic
263	C₂×C₄	C₂×#117	1	1	yes	characteristic
264	C₂×C₄	C₂×#124	1	1	yes	characteristic
265	C₂×C₄	C₂×#124	1	1	yes	characteristic
266	C₂×C₄	C₂×#125	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	4	no	—
274	C₈	—	4	4	no	—
275	C₈	—	8	4	no	—
276	C₈	—	8	4	no	—
277	C₈	#430	1	2	yes	—
278	C₈	#430	1	2	yes	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	4	4	no	—
288	C₂²	#1833	1	2	yes	—
289	C₂²	C₂×#382	1	1	yes	characteristic
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	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	#1833	1	2	yes	—
301	C₄	C₂×#430	1	1	yes	characteristic
302	C₄	C₂×#430	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	4	1	no	—
305	C₂	—	4	1	no	—
306	C₂	C₂×#1833	1	1	yes	characteristic
307	C₂	C₂×#1833	1	1	yes	characteristic
308	C₂	#20769	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#125
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
4. #8799	C₂	semidirect
5. #8797	C₂	semidirect

Maximal subgroups

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

subgroup	index
#17793	2
C₂×#1568	2
#17323	2
#8799	2
#8797	2
#7970	2
#7968	2