Subgroups — #191270

Order 512.

502 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17797	C₂	1	1	yes	maximal, characteristic
2	#17797	C₂	1	1	yes	maximal, characteristic
3	#17314	C₂	1	1	yes	maximal, characteristic
4	#8797	C₂	1	4	yes	maximal, complemented
5	#8796	C₂	1	4	yes	maximal, complemented
6	#7966	C₂	1	1	yes	maximal, characteristic
7	#7965	C₂	1	1	yes	maximal, characteristic
8	#7964	C₂	1	1	yes	maximal, characteristic
9	C₂×#1287	C₂	1	1	yes	maximal, characteristic
10	#2920	C₂²	1	1	yes	characteristic
11	#2920	C₂²	1	1	yes	characteristic
12	C₂×#432	C₂²	1	1	yes	characteristic
13	C₂×#432	C₂²	1	1	yes	characteristic
14	C₂×#431	C₂²	1	1	yes	characteristic
15	C₂×#431	C₂²	1	1	yes	characteristic
16	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
17	#1569	—	2	2	no	—
18	#1569	—	2	2	no	—
19	#1568	C₂²	1	4	yes	—
20	#1568	C₂²	1	4	yes	—
21	#1565	—	2	2	no	—
22	#1565	—	2	2	no	—
23	#1450	C₂²	1	2	yes	—
24	#1450	C₂²	1	2	yes	—
25	C₈×M₁₆	C₄	1	2	yes	—
26	C₈×M₁₆	C₄	1	4	yes	—
27	C₈×M₁₆	C₄	1	2	yes	—
28	#1287	C₂²	1	2	yes	—
29	#1287	C₂²	1	2	yes	—
30	#1287	C₂²	1	4	yes	—
31	#1287	C₂²	1	2	yes	—
32	#1287	C₂²	1	2	yes	—
33	#1287	C₂²	1	4	yes	—
34	C₂²×M₁₆	Q₈	1	2	yes	—
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	1	no	—
41	#432	—	2	1	no	—
42	#432	—	2	1	no	—
43	#432	—	2	1	no	—
44	#432	C₂³	1	2	yes	—
45	#432	C₂³	1	2	yes	—
46	#432	C₂³	1	2	yes	—
47	#432	C₂³	1	2	yes	—
48	#431	—	2	1	no	—
49	#431	—	2	1	no	—
50	#431	—	2	1	no	—
51	#431	—	2	1	no	—
52	#431	—	2	1	no	—
53	#431	—	2	1	no	—
54	#431	—	2	1	no	—
55	#431	—	2	1	no	—
56	#431	C₂×C₄	1	2	yes	—
57	#431	C₂×C₄	1	2	yes	—
58	#431	C₂³	1	1	yes	characteristic
59	#431	C₂³	1	1	yes	characteristic
60	#431	C₂³	1	2	yes	—
61	#423	—	2	1	no	—
62	#423	—	2	1	no	—
63	C₂×C₄⋊C₈	—	2	1	no	—
64	C₂×C₄⋊C₈	—	2	1	no	—
65	C₂×C₄⋊C₈	—	2	1	no	—
66	C₂×C₄⋊C₈	—	2	1	no	—
67	#407	—	2	1	no	—
68	#407	—	2	1	no	—
69	C₂×C₂²⋊C₈	—	2	1	no	—
70	C₂×C₂²⋊C₈	—	2	1	no	—
71	C₂×C₂²⋊C₈	—	2	1	no	—
72	C₂×C₂²⋊C₈	—	2	1	no	—
73	C₄×M₁₆	—	2	1	no	—
74	C₄×M₁₆	—	2	1	no	—
75	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
76	C₄×M₁₆	C₂×C₄	1	2	yes	—
77	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
78	C₄×M₁₆	D₄	1	2	yes	—
79	C₄×M₁₆	D₄	1	2	yes	—
80	C₂×#97	—	2	1	no	—
81	C₂×#97	—	2	1	no	—
82	C₂×#97	Q₈	1	2	yes	—
83	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
84	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
85	C₂×C₄×C₈	C₂³	1	2	yes	—
86	#335	—	2	4	no	—
87	#335	—	2	4	no	—
88	#334	—	2	4	no	—
89	#334	—	2	4	no	—
90	#322	C₂×C₄	1	4	yes	—
91	#322	C₂×C₄	1	4	yes	—
92	C₈²	C₂×C₄	1	4	yes	—
93	C₈²	C₂×C₄	1	4	yes	—
94	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
95	C₂³×C₄	#41	1	1	yes	characteristic
96	C₂³×C₄	#41	1	1	yes	characteristic
97	C₂×M₁₆	—	2	8	no	—
98	C₂×M₁₆	—	4	1	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
101	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
102	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
103	C₂²×C₈	—	2	2	no	—
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	4	1	no	—
106	C₂²×C₈	—	4	1	no	—
107	C₂²×C₈	C₂×Q₈	1	4	yes	—
108	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄²	C₂×Q₈	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	2	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	1	no	—
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	4	1	no	—
130	C₄⋊C₈	—	4	1	no	—
131	C₄⋊C₈	—	4	1	no	—
132	C₄⋊C₈	—	4	1	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	2	2	no	—
135	C₂²⋊C₈	—	2	1	no	—
136	C₂²⋊C₈	—	2	1	no	—
137	C₂²⋊C₈	—	2	2	no	—
138	C₂²⋊C₈	—	2	2	no	—
139	C₂²⋊C₈	—	2	2	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₂²⋊C₈	—	4	1	no	—
143	C₂²⋊C₈	—	4	1	no	—
144	C₂²⋊C₈	—	4	1	no	—
145	C₂²⋊C₈	—	4	1	no	—
146	#97	—	2	2	no	—
147	#97	—	2	2	no	—
148	#97	—	2	2	no	—
149	#97	—	2	2	no	—
150	#97	—	4	1	no	—
151	#97	—	4	1	no	—
152	#97	C₄⋊C₄	1	2	yes	—
153	#97	C₄⋊C₄	1	4	yes	—
154	#97	C₄⋊C₄	1	2	yes	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	4	1	no	—
158	C₄×C₈	—	4	1	no	—
159	C₄×C₈	C₂²×C₄	1	2	yes	—
160	C₄×C₈	C₂²×C₄	1	4	yes	—
161	C₄×C₈	C₂²×C₄	1	2	yes	—
162	C₄×C₈	C₂²×C₄	1	2	yes	—
163	C₄×C₈	C₂²×C₄	1	2	yes	—
164	C₄×C₈	C₂×D₄	1	4	yes	—
165	C₂⁴	#125	1	1	yes	characteristic
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	8	1	no	—
168	C₂²×C₄	—	8	1	no	—
169	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄⋊C₄	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	#117	1	1	yes	characteristic
175	C₂²×C₄	#124	1	1	yes	characteristic
176	C₂²×C₄	#124	1	1	yes	characteristic
177	C₂²×C₄	#124	1	1	yes	characteristic
178	C₂²×C₄	#124	1	1	yes	characteristic
179	C₂²×C₄	#125	1	1	yes	characteristic
180	C₂²×C₄	#125	1	1	yes	characteristic
181	C₂²×C₄	#125	1	1	yes	characteristic
182	C₂²×C₄	#125	1	1	yes	characteristic
183	C₂²×C₄	#125	1	1	yes	characteristic
184	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×#41	1	1	yes	characteristic
186	C₂²×C₄	C₂×#41	1	1	yes	characteristic
187	M₁₆	—	2	4	no	—
188	M₁₆	—	2	8	no	—
189	M₁₆	—	8	2	no	—
190	M₁₆	—	8	2	no	—
191	M₁₆	#108	1	8	yes	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	8	2	no	—
201	C₂×C₈	—	8	2	no	—
202	C₂×C₈	—	8	2	no	—
203	C₂×C₈	—	8	2	no	—
204	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
205	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
206	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₄²	#117	1	1	yes	characteristic
209	C₄²	#117	1	1	yes	characteristic
210	C₄²	#117	1	1	yes	characteristic
211	C₄²	#117	1	1	yes	characteristic
212	C₄²	#124	1	1	yes	characteristic
213	C₄²	#124	1	1	yes	characteristic
214	C₄²	#124	1	1	yes	characteristic
215	C₄²	#124	1	1	yes	characteristic
216	C₄²	#125	1	1	yes	characteristic
217	C₄²	#125	1	1	yes	characteristic
218	C₄²	C₂³×C₄	1	1	yes	characteristic
219	C₄²	C₂²×D₄	1	1	yes	characteristic
220	C₄²	C₂×#41	1	1	yes	characteristic
221	C₄²	C₂×#41	1	1	yes	characteristic
222	C₂³	—	2	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	#382	1	1	yes	characteristic
226	C₂³	#382	1	1	yes	characteristic
227	C₂³	C₂×#125	1	1	yes	characteristic
228	C₂×C₄	—	2	2	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	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	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	#382	1	1	yes	characteristic
246	C₂×C₄	#382	1	1	yes	characteristic
247	C₂×C₄	#382	1	1	yes	characteristic
248	C₂×C₄	#382	1	1	yes	characteristic
249	C₂×C₄	#382	1	1	yes	characteristic
250	C₂×C₄	#382	1	1	yes	characteristic
251	C₂×C₄	C₂×#108	1	4	yes	—
252	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×#117	1	1	yes	characteristic
254	C₂×C₄	C₂×#117	1	1	yes	characteristic
255	C₂×C₄	C₂×#124	1	1	yes	characteristic
256	C₂×C₄	C₂×#124	1	1	yes	characteristic
257	C₂×C₄	C₂×#125	1	1	yes	characteristic
258	C₈	—	2	4	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	4	4	no	—
262	C₈	—	8	2	no	—
263	C₈	—	8	2	no	—
264	C₈	—	8	2	no	—
265	C₈	—	8	2	no	—
266	C₈	C₂×#108	1	8	yes	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	#1837	1	2	yes	—
277	C₂²	C₂×#382	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	#1837	1	2	yes	—
289	C₄	C₂²×#108	1	2	yes	—
290	C₂	—	2	2	no	—
291	C₂	—	4	1	no	—
292	C₂	—	4	1	no	—
293	C₂	#20757	1	1	yes	characteristic
294	C₂	#20757	1	1	yes	characteristic
295	C₂	C₂×#1837	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. #8797	C₂	semidirect
5. #8796	C₂	semidirect

Maximal subgroups

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

subgroup	index
#17797	2
#17314	2
#8797	2
#8796	2
#7966	2
#7965	2
#7964	2
C₂×#1287	2