Subgroups — #93386

Order 512.

436 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1376	C₂	1	1	yes	maximal, characteristic
2	#10859	C₂	1	1	yes	maximal, characteristic
3	C₂×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂²×M₃₂	—	2	1	no	—
5	C₂²×#108	C₄	1	1	yes	characteristic
6	C₂×#427	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	—	2	1	no	—
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#339	—	2	1	no	—
10	#1729	—	2	1	no	—
11	#1376	—	2	1	no	—
12	#1376	C₄	1	2	yes	complemented
13	#1273	—	2	1	no	—
14	#1273	C₄	1	2	yes	complemented
15	C₂²×M₁₆	—	2	1	no	—
16	C₂³×C₈	—	2	2	no	—
17	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₈	D₄	1	1	yes	characteristic
19	C₂×#117	—	2	1	no	—
20	C₂²×C₄²	D₄	1	1	yes	characteristic
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	C₂×M₃₂	—	2	2	no	—
24	C₂×M₃₂	—	4	1	no	—
25	C₂×M₃₂	—	4	1	no	—
26	C₂×M₃₂	—	4	1	no	—
27	C₂²×C₁₆	—	4	1	no	—
28	C₂×#108	—	2	1	no	—
29	C₂×#108	—	4	1	no	—
30	C₂×#108	—	4	1	no	—
31	C₂×#108	C₈	1	2	yes	complemented
32	#427	—	4	1	no	—
33	#427	—	4	1	no	—
34	C₂×#107	C₈	1	2	yes	complemented
35	C₂×#106	—	2	1	no	—
36	C₂×C₄×C₈	—	2	2	no	—
37	C₂×C₄×C₈	—	2	2	no	—
38	C₂×C₄×C₈	—	4	1	no	—
39	C₂×C₄×C₈	—	4	1	no	—
40	C₂×C₄×C₈	—	4	1	no	—
41	C₂×C₄×C₈	—	4	1	no	—
42	C₂×C₄×C₈	—	4	1	no	—
43	C₂×C₄×C₈	—	4	1	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
48	#339	—	4	1	no	—
49	#339	—	4	1	no	—
50	#336	—	4	1	no	—
51	#336	—	4	1	no	—
52	C₂³×C₄	—	2	1	no	—
53	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×M₁₆	—	2	2	no	—
55	C₂×M₁₆	—	4	1	no	—
56	C₂×M₁₆	—	4	1	no	—
57	C₂×M₁₆	—	4	1	no	—
58	C₂×M₁₆	—	4	1	no	—
59	C₂×M₁₆	—	4	1	no	—
60	C₂²×C₈	—	2	1	no	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	—	2	8	no	—
69	C₂²×C₈	—	2	4	no	—
70	C₂²×C₈	—	4	4	no	—
71	C₂²×C₈	—	4	2	no	—
72	C₂²×C₈	—	4	2	no	—
73	C₂²×C₈	—	4	1	no	—
74	C₂²×C₈	C₄²	1	1	yes	characteristic
75	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₈	M₁₆	1	1	yes	characteristic
81	C₂²×C₈	M₁₆	1	1	yes	characteristic
82	C₂²×C₈	D₈	1	1	yes	characteristic
83	C₂²×C₈	SD₁₆	1	1	yes	characteristic
84	C₂²×C₈	SD₁₆	1	1	yes	characteristic
85	C₂²×C₈	Q₁₆	1	1	yes	characteristic
86	#117	—	4	1	no	—
87	#117	—	4	1	no	—
88	#117	—	4	1	no	—
89	#117	—	4	1	no	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₂²⋊C₄	—	4	1	no	—
92	C₂×C₄²	—	2	1	no	—
93	C₂×C₄²	—	2	1	no	—
94	C₂×C₄²	—	2	1	no	—
95	C₂×C₄²	—	4	1	no	—
96	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄²	D₈	1	1	yes	characteristic
99	C₂×C₄²	SD₁₆	1	1	yes	characteristic
100	C₂×C₄²	SD₁₆	1	1	yes	characteristic
101	C₂×C₄²	Q₁₆	1	1	yes	characteristic
102	M₃₂	—	2	4	no	—
103	M₃₂	—	4	2	no	—
104	M₃₂	—	4	2	no	—
105	M₃₂	—	4	2	no	—
106	C₂×C₁₆	—	4	2	no	—
107	C₂×C₁₆	—	4	2	no	—
108	C₂×C₁₆	—	4	2	no	—
109	#108	—	4	2	no	—
110	#108	—	4	2	no	—
111	#107	—	4	2	no	—
112	#106	—	4	2	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	—	4	1	no	—
115	C₄×C₈	—	4	1	no	—
116	C₄×C₈	—	4	1	no	—
117	C₄×C₈	—	4	2	no	—
118	C₄×C₈	—	4	2	no	—
119	C₄×C₈	—	4	2	no	—
120	C₄×C₈	—	4	2	no	—
121	C₂⁴	#104	1	1	yes	characteristic
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	#95	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₄	#99	1	1	yes	characteristic
135	C₂²×C₄	#99	1	1	yes	characteristic
136	C₂²×C₄	#100	1	1	yes	characteristic
137	C₂²×C₄	#101	1	1	yes	characteristic
138	C₂²×C₄	#102	1	1	yes	characteristic
139	C₂²×C₄	#102	1	1	yes	characteristic
140	C₂²×C₄	#103	1	1	yes	characteristic
141	C₂²×C₄	#103	1	1	yes	characteristic
142	C₂²×C₄	#104	1	1	yes	characteristic
143	C₂²×C₄	#104	1	1	yes	characteristic
144	C₂²×C₄	#104	1	1	yes	characteristic
145	M₁₆	—	4	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	8	1	no	—
148	M₁₆	—	8	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	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	8	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	4	8	no	—
176	C₂×C₈	—	4	8	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	8	1	no	—
180	C₂×C₈	—	8	1	no	—
181	C₂×C₈	C₄×C₈	1	1	yes	characteristic
182	C₂×C₈	#97	1	1	yes	characteristic
183	C₂×C₈	#102	1	1	yes	characteristic
184	C₂×C₈	#103	1	1	yes	characteristic
185	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
186	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
187	C₂×C₈	#106	1	1	yes	characteristic
188	C₂×C₈	#107	1	1	yes	characteristic
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	—	8	1	no	—
192	C₂²⋊C₄	—	8	1	no	—
193	C₄²	—	2	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	—	8	1	no	—
200	C₄²	—	8	1	no	—
201	C₄²	#102	1	1	yes	characteristic
202	C₄²	#103	1	1	yes	characteristic
203	C₄²	#106	1	1	yes	characteristic
204	C₄²	#107	1	1	yes	characteristic
205	C₁₆	—	4	4	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	D₄⋊C₈	1	1	yes	characteristic
209	C₂³	Q₈⋊C₈	1	1	yes	characteristic
210	C₂³	#327	1	1	yes	characteristic
211	C₂³	#328	1	1	yes	characteristic
212	C₂³	#329	1	1	yes	characteristic
213	C₂³	#330	1	1	yes	characteristic
214	C₂³	#339	1	1	yes	characteristic
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	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	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	#323	1	1	yes	characteristic
231	C₂×C₄	#323	1	1	yes	characteristic
232	C₂×C₄	#324	1	1	yes	characteristic
233	C₂×C₄	#324	1	1	yes	characteristic
234	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
235	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
236	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
237	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
238	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
239	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
240	C₂×C₄	#327	1	1	yes	characteristic
241	C₂×C₄	#327	1	1	yes	characteristic
242	C₂×C₄	#327	1	1	yes	characteristic
243	C₂×C₄	#328	1	1	yes	characteristic
244	C₂×C₄	#328	1	1	yes	characteristic
245	C₂×C₄	#328	1	1	yes	characteristic
246	C₂×C₄	#329	1	1	yes	characteristic
247	C₂×C₄	#330	1	1	yes	characteristic
248	C₂×C₄	#331	1	1	yes	characteristic
249	C₂×C₄	#332	1	1	yes	characteristic
250	C₂×C₄	#332	1	1	yes	characteristic
251	C₂×C₄	#333	1	1	yes	characteristic
252	C₂×C₄	#336	1	1	yes	characteristic
253	C₂×C₄	#339	1	1	yes	characteristic
254	C₂×C₄	#340	1	1	yes	characteristic
255	C₂×C₄	#340	1	1	yes	characteristic
256	C₂×C₄	#342	1	1	yes	characteristic
257	C₂×C₄	#343	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	4	8	no	—
267	C₈	—	8	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#1270	1	1	yes	characteristic
277	C₂²	#1271	1	1	yes	characteristic
278	C₂²	#1271	1	1	yes	characteristic
279	C₂²	#1272	1	1	yes	characteristic
280	C₂²	#1279	1	1	yes	characteristic
281	C₂²	#1294	1	1	yes	characteristic
282	C₂²	#1295	1	1	yes	characteristic
283	C₂²	#1335	1	2	yes	—
284	C₂²	#1336	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	16	2	no	—
290	C₄	#1270	1	1	yes	characteristic
291	C₄	#1271	1	1	yes	characteristic
292	C₄	#1271	1	1	yes	characteristic
293	C₄	#1272	1	1	yes	characteristic
294	C₄	#1279	1	1	yes	characteristic
295	C₄	#1280	1	1	yes	characteristic
296	C₄	#1294	1	1	yes	characteristic
297	C₄	#1296	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#7025	1	1	yes	characteristic
303	C₂	#7114	1	2	yes	—
304	C₂	#7119	1	2	yes	—
305	C₂	#7179	1	1	yes	characteristic
306	C₂	#7182	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂²
socle	C₂³	D₄⋊C₈
prime radical	C₂⁴	#104
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #1376	C₄	semidirect
14. #1273	C₄	semidirect
31. C₂×#108	C₈	semidirect
34. C₂×#107	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1376	2
#10859	2
C₂×#1273	2