Subgroups — #134372

Order 512.

392 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#443	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic, complemented
3	#11602	C₂	1	2	yes	maximal
4	#8266	C₂	1	1	yes	maximal, characteristic, complemented
5	#7181	C₂	1	2	yes	maximal, complemented
6	C₂×#443	C₄	1	1	yes	characteristic
7	C₂×#443	C₂²	1	1	yes	characteristic
8	C₄×#135	C₄	1	1	yes	characteristic
9	C₄×#131	—	2	1	no	—
10	C₂×#364	—	2	1	no	—
11	C₂×#364	C₂²	1	2	yes	—
12	#2150	—	2	1	no	—
13	C₄×#108	C₄	1	1	yes	characteristic
14	#1772	C₄	1	1	yes	characteristic
15	#1772	C₂²	1	1	yes	characteristic
16	#1764	—	2	1	no	—
17	#1764	—	2	1	no	—
18	C₄×#104	—	2	1	no	—
19	C₄²×C₈	C₂²	1	1	yes	characteristic
20	#1564	—	2	1	no	—
21	#1450	C₂²	1	2	yes	—
22	#1335	—	2	4	no	—
23	#1276	—	2	1	no	—
24	C₂×#135	C₂×C₄	1	1	yes	characteristic
25	C₂×#131	—	2	1	no	—
26	C₂×#131	—	2	1	no	—
27	C₄×#41	—	2	1	no	—
28	C₂×M₃₂	—	2	2	no	—
29	#443	—	2	1	no	—
30	#443	—	2	1	no	—
31	#443	—	2	1	no	—
32	#443	—	2	1	no	—
33	#443	C₈	1	2	yes	complemented
34	#443	C₈	1	2	yes	complemented
35	#443	D₄	1	2	yes	—
36	#443	Q₈	1	2	yes	—
37	C₄×Q₁₆	—	2	1	no	—
38	C₄×SD₁₆	—	2	2	no	—
39	C₄×D₈	—	2	1	no	—
40	#431	—	2	2	no	—
41	C₂×#108	C₂×C₄	1	1	yes	characteristic
42	C₂×#108	C₂×C₄	1	1	yes	characteristic
43	#427	C₂×C₄	1	1	yes	characteristic
44	C₂×#104	—	2	1	no	—
45	C₂×#104	—	2	1	no	—
46	#416	—	2	1	no	—
47	#405	—	2	1	no	—
48	#405	—	2	1	no	—
49	#405	—	4	1	no	—
50	#405	—	4	1	no	—
51	C₄×M₁₆	—	2	1	no	—
52	C₄×M₁₆	—	4	1	no	—
53	C₂×C₄×C₈	—	4	1	no	—
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	D₄	1	1	yes	characteristic
57	C₂×C₄×C₈	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
60	C₄³	D₄	1	1	yes	characteristic
61	#364	—	2	2	no	—
62	#364	—	2	2	no	—
63	#364	—	4	1	no	—
64	#364	—	4	1	no	—
65	C₄⋊C₁₆	—	4	2	no	—
66	#341	—	2	1	no	—
67	#339	—	2	1	no	—
68	#339	—	2	1	no	—
69	#335	—	2	2	no	—
70	#334	—	2	2	no	—
71	#322	—	2	2	no	—
72	C₈²	—	2	2	no	—
73	C₂×#41	—	2	1	no	—
74	#135	—	2	1	no	—
75	#135	—	2	1	no	—
76	#135	C₄⋊C₄	1	2	yes	—
77	#135	C₂×C₈	1	2	yes	—
78	C₂×Q₁₆	M₁₆	1	1	yes	characteristic
79	C₂×SD₁₆	—	2	1	no	—
80	C₂×D₈	M₁₆	1	1	yes	characteristic
81	#131	—	2	2	no	—
82	#131	—	2	2	no	—
83	#131	—	2	2	no	—
84	#131	—	2	2	no	—
85	#131	—	4	1	no	—
86	#131	—	4	1	no	—
87	#131	—	4	1	no	—
88	#131	—	4	1	no	—
89	C₂×M₁₆	—	2	1	no	—
90	C₂×M₁₆	—	2	1	no	—
91	C₂×M₁₆	—	4	1	no	—
92	C₂×M₁₆	—	4	1	no	—
93	C₂²×C₈	—	4	1	no	—
94	C₂²×C₈	—	4	1	no	—
95	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
99	C₄×Q₈	—	2	2	no	—
100	C₄×D₄	—	2	2	no	—
101	C₄×D₄	—	4	2	no	—
102	#117	—	2	1	no	—
103	#117	—	4	1	no	—
104	C₂×C₄²	—	4	1	no	—
105	C₂×C₄²	—	4	1	no	—
106	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
109	M₃₂	—	4	2	no	—
110	M₃₂	—	4	2	no	—
111	C₂×C₁₆	—	4	2	no	—
112	#108	—	2	1	no	—
113	#108	—	2	1	no	—
114	#108	C₄⋊C₄	1	2	yes	—
115	#108	C₂×C₈	1	2	yes	—
116	#107	—	2	1	no	—
117	#106	M₁₆	1	2	yes	—
118	C₄⋊C₈	—	2	4	no	—
119	#104	—	2	2	no	—
120	#104	—	2	2	no	—
121	#104	—	2	2	no	—
122	#104	—	2	2	no	—
123	#103	—	2	2	no	—
124	#102	—	2	2	no	—
125	C₂²⋊C₈	—	4	2	no	—
126	#97	—	4	1	no	—
127	#97	—	4	1	no	—
128	#97	—	4	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	—	4	1	no	—
136	C₄×C₈	—	4	1	no	—
137	C₄×C₈	—	4	1	no	—
138	C₄×C₈	—	4	2	no	—
139	C₄×C₈	C₂×C₈	1	1	yes	characteristic
140	C₄×C₈	C₂×C₈	1	1	yes	characteristic
141	C₄×C₈	M₁₆	1	1	yes	characteristic
142	C₄×C₈	M₁₆	1	1	yes	characteristic
143	C₄×C₈	D₈	1	1	yes	characteristic
144	C₄×C₈	D₈	1	1	yes	characteristic
145	C₄×C₈	SD₁₆	1	1	yes	characteristic
146	C₄×C₈	SD₁₆	1	1	yes	characteristic
147	C₄×C₈	C₂×D₄	1	1	yes	characteristic
148	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
149	C₄×C₈	#41	1	1	yes	characteristic
150	C₄×C₈	#41	1	1	yes	characteristic
151	#41	—	2	2	no	—
152	#41	—	2	2	no	—
153	#41	—	4	1	no	—
154	#41	—	4	1	no	—
155	C₂×Q₈	—	2	1	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	4	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	Q₁₆	—	2	1	no	—
165	Q₁₆	C₄⋊C₈	1	2	yes	—
166	SD₁₆	—	2	2	no	—
167	SD₁₆	—	2	2	no	—
168	D₈	—	2	1	no	—
169	D₈	C₄⋊C₈	1	2	yes	—
170	M₁₆	—	2	2	no	—
171	M₁₆	—	2	2	no	—
172	M₁₆	—	4	1	no	—
173	M₁₆	—	4	1	no	—
174	M₁₆	—	4	2	no	—
175	M₁₆	—	4	2	no	—
176	M₁₆	—	4	2	no	—
177	M₁₆	—	4	2	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
190	C₂×C₈	#102	1	1	yes	characteristic
191	C₂×C₈	#102	1	1	yes	characteristic
192	C₂×C₈	#104	1	1	yes	characteristic
193	C₂×C₈	#104	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₈	#117	1	1	yes	characteristic
196	C₂×C₈	#121	1	1	yes	characteristic
197	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
198	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
199	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
200	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
201	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
202	C₂×C₈	C₂×D₈	1	1	yes	characteristic
203	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
204	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂×C₈	#137	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	4	2	no	—
214	C₄²	—	4	4	no	—
215	C₄²	—	4	1	no	—
216	C₄²	C₂²⋊C₈	1	1	yes	characteristic
217	C₄²	C₂²⋊C₈	1	1	yes	characteristic
218	C₄²	#102	1	1	yes	characteristic
219	C₄²	#102	1	1	yes	characteristic
220	C₄²	#121	1	1	yes	characteristic
221	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
222	C₁₆	—	8	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	#421	1	1	yes	characteristic
225	Q₈	—	2	2	no	—
226	Q₈	—	4	1	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	#380	1	1	yes	characteristic
246	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
247	C₂×C₄	#407	1	1	yes	characteristic
248	C₂×C₄	C₂×#102	1	1	yes	characteristic
249	C₂×C₄	#417	1	1	yes	characteristic
250	C₂×C₄	C₂×#104	1	1	yes	characteristic
251	C₈	—	4	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	4	1	no	—
254	C₈	—	8	4	no	—
255	C₈	D₄⋊C₈	1	1	yes	characteristic
256	C₈	D₄⋊C₈	1	1	yes	characteristic
257	C₈	D₄⋊C₈	1	1	yes	characteristic
258	C₈	D₄⋊C₈	1	1	yes	characteristic
259	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
260	C₈	#423	1	1	yes	characteristic
261	C₈	#431	1	1	yes	characteristic
262	C₈	#432	1	1	yes	characteristic
263	C₈	#459	1	1	yes	characteristic
264	C₈	#461	1	1	yes	characteristic
265	C₈	#474	1	1	yes	characteristic
266	C₈	#476	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	#1477	1	1	yes	characteristic
271	C₂²	#1486	1	1	yes	characteristic
272	C₂²	#1806	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	2	no	—
280	C₄	—	8	2	no	—
281	C₄	C₂×D₄⋊C₈	1	1	yes	characteristic
282	C₄	#1489	1	1	yes	characteristic
283	C₄	#1797	1	1	yes	characteristic
284	C₄	#1802	1	1	yes	characteristic
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	8	2	no	—
289	C₂	#8563	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#459
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8563
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #12900	C₂	semidirect
4. #8266	C₂	semidirect
5. #7181	C₂	semidirect
33. #443	C₈	semidirect
34. #443	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#443	2
#12900	2
#11602	2
#8266	2
#7181	2