Subgroups — #134366

Order 512.

398 conjugacy classes of subgroups.

The subgroup lattice has 293 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	C₄×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	#11592	C₂	1	2	yes	maximal
4	#8807	C₂	1	1	yes	maximal, characteristic, complemented
5	#7176	C₂	1	2	yes	maximal, complemented
6	C₂×#443	C₄	1	1	yes	characteristic
7	C₂×#443	C₄	1	1	yes	characteristic
8	C₄×#135	C₂²	1	1	yes	characteristic
9	C₄×#131	—	2	1	no	—
10	C₂×#364	—	2	1	no	—
11	C₄×M₃₂	C₂²	1	2	yes	—
12	#1831	—	2	1	no	—
13	C₄×#108	C₂²	1	1	yes	characteristic
14	#1772	C₄	1	1	yes	characteristic
15	#1772	C₄	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	C₂×#335	C₂²	1	2	yes	—
21	C₂×#334	—	2	1	no	—
22	#1376	—	2	1	no	—
23	#1332	—	2	2	no	—
24	#1331	—	2	2	no	—
25	C₂×#135	C₂×C₄	1	1	yes	characteristic
26	C₂×#131	—	2	1	no	—
27	C₂×#131	—	2	1	no	—
28	C₄×#41	—	2	1	no	—
29	C₂×M₃₂	—	2	2	no	—
30	#443	—	2	2	no	—
31	#443	—	2	2	no	—
32	#443	—	2	2	no	—
33	#443	C₈	1	4	yes	complemented
34	C₄×Q₁₆	D₄	1	1	yes	characteristic
35	C₄×Q₁₆	Q₈	1	1	yes	characteristic
36	C₄×SD₁₆	—	2	1	no	—
37	C₄×SD₁₆	—	2	1	no	—
38	C₄×D₈	D₄	1	1	yes	characteristic
39	C₄×D₈	Q₈	1	1	yes	characteristic
40	C₂×#108	C₂×C₄	1	1	yes	characteristic
41	C₂×#108	C₂×C₄	1	1	yes	characteristic
42	#427	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₈	—	2	2	no	—
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	4	no	—
62	#346	—	2	2	no	—
63	C₄×C₁₆	—	2	2	no	—
64	#341	—	2	1	no	—
65	#339	—	2	1	no	—
66	#339	—	2	1	no	—
67	#336	—	4	2	no	—
68	#335	—	2	2	no	—
69	#335	—	2	2	no	—
70	#334	—	4	1	no	—
71	#334	—	4	1	no	—
72	C₂×#41	—	2	1	no	—
73	#135	—	2	1	no	—
74	#135	—	2	1	no	—
75	#135	C₂×C₈	1	2	yes	—
76	#135	M₁₆	1	2	yes	—
77	C₂×Q₁₆	C₄⋊C₄	1	1	yes	characteristic
78	C₂×SD₁₆	—	2	1	no	—
79	C₂×D₈	C₄⋊C₄	1	1	yes	characteristic
80	#131	—	2	4	no	—
81	#131	—	2	4	no	—
82	#131	—	4	2	no	—
83	#131	—	4	2	no	—
84	C₂×M₁₆	—	2	1	no	—
85	C₂×M₁₆	—	2	1	no	—
86	C₂×M₁₆	—	4	1	no	—
87	C₂×M₁₆	—	4	1	no	—
88	C₂²×C₈	—	4	1	no	—
89	C₂²×C₈	—	4	1	no	—
90	C₂²×C₈	—	4	2	no	—
91	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	C₄×Q₈	—	2	1	no	—
96	C₄×Q₈	—	2	1	no	—
97	C₄×D₄	—	2	1	no	—
98	C₄×D₄	—	2	1	no	—
99	C₄×D₄	—	4	1	no	—
100	C₄×D₄	—	4	1	no	—
101	#117	—	2	1	no	—
102	#117	—	4	1	no	—
103	C₂×C₄²	—	2	1	no	—
104	C₂×C₄²	—	2	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	—	2	4	no	—
110	M₃₂	—	4	2	no	—
111	C₂×C₁₆	—	4	2	no	—
112	#108	—	2	1	no	—
113	#108	—	2	1	no	—
114	#108	C₂×C₈	1	2	yes	—
115	#108	M₁₆	1	2	yes	—
116	#107	C₄⋊C₄	1	1	yes	characteristic
117	#107	C₄⋊C₄	1	1	yes	characteristic
118	#106	—	2	1	no	—
119	C₄⋊C₈	—	4	2	no	—
120	C₄⋊C₈	—	4	2	no	—
121	#104	—	2	4	no	—
122	#104	—	2	4	no	—
123	#103	—	2	1	no	—
124	#103	—	2	1	no	—
125	#102	—	2	1	no	—
126	#102	—	2	1	no	—
127	#97	—	4	1	no	—
128	#97	—	4	1	no	—
129	#97	—	4	1	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	4	1	no	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	—	4	1	no	—
136	C₄×C₈	—	4	1	no	—
137	C₄×C₈	C₂×C₈	1	2	yes	—
138	C₄×C₈	M₁₆	1	2	yes	—
139	C₄×C₈	D₈	1	1	yes	characteristic
140	C₄×C₈	D₈	1	1	yes	characteristic
141	C₄×C₈	SD₁₆	1	1	yes	characteristic
142	C₄×C₈	SD₁₆	1	1	yes	characteristic
143	C₄×C₈	C₂×D₄	1	1	yes	characteristic
144	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
145	C₄×C₈	#41	1	1	yes	characteristic
146	C₄×C₈	#41	1	1	yes	characteristic
147	#41	—	2	2	no	—
148	#41	—	2	2	no	—
149	#41	—	4	1	no	—
150	#41	—	4	1	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	4	1	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	4	1	no	—
157	C₂²×C₄	—	4	1	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂²×C₄	#137	1	1	yes	characteristic
162	Q₁₆	—	2	1	no	—
163	Q₁₆	C₄⋊C₈	1	2	yes	—
164	SD₁₆	—	2	2	no	—
165	SD₁₆	—	2	2	no	—
166	D₈	—	2	1	no	—
167	D₈	C₄⋊C₈	1	2	yes	—
168	M₁₆	—	2	2	no	—
169	M₁₆	—	2	2	no	—
170	M₁₆	—	4	4	no	—
171	M₁₆	—	4	1	no	—
172	M₁₆	—	4	1	no	—
173	M₁₆	—	4	4	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	4	no	—
182	C₂×C₈	—	8	4	no	—
183	C₂×C₈	—	8	2	no	—
184	C₂×C₈	C₂²⋊C₈	1	2	yes	—
185	C₂×C₈	#102	1	1	yes	characteristic
186	C₂×C₈	#102	1	1	yes	characteristic
187	C₂×C₈	#102	1	1	yes	characteristic
188	C₂×C₈	#102	1	1	yes	characteristic
189	C₂×C₈	#104	1	2	yes	—
190	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₈	#117	1	1	yes	characteristic
192	C₂×C₈	#121	1	1	yes	characteristic
193	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
194	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
195	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
196	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
197	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	4	1	no	—
204	C₂²⋊C₄	—	4	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	C₂²⋊C₈	1	2	yes	—
214	C₄²	#121	1	1	yes	characteristic
215	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
216	C₄²	C₂×D₈	1	1	yes	characteristic
217	C₄²	C₂×SD₁₆	1	1	yes	characteristic
218	C₁₆	—	4	4	no	—
219	C₂³	—	4	1	no	—
220	C₂³	#417	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	Q₈	—	4	1	no	—
223	D₄	—	2	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	#380	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
244	C₂×C₄	#407	1	1	yes	characteristic
245	C₂×C₄	C₂×#102	1	1	yes	characteristic
246	C₂×C₄	C₂×#104	1	1	yes	characteristic
247	C₂×C₄	#421	1	1	yes	characteristic
248	C₂×C₄	#459	1	1	yes	characteristic
249	C₂×C₄	#461	1	1	yes	characteristic
250	C₂×C₄	#474	1	1	yes	characteristic
251	C₂×C₄	#476	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	4	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	16	2	no	—
259	C₈	D₄⋊C₈	1	2	yes	—
260	C₈	D₄⋊C₈	1	2	yes	—
261	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
262	C₈	#423	1	1	yes	characteristic
263	C₈	#431	1	1	yes	characteristic
264	C₈	#432	1	1	yes	characteristic
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	8	1	no	—
269	C₂²	—	8	1	no	—
270	C₂²	#1477	1	1	yes	characteristic
271	C₂²	#1489	1	1	yes	characteristic
272	C₂²	#1802	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	2	no	—
279	C₄	—	8	1	no	—
280	C₄	—	8	1	no	—
281	C₄	C₂×D₄⋊C₈	1	1	yes	characteristic
282	C₄	#1486	1	1	yes	characteristic
283	C₄	#1797	1	1	yes	characteristic
284	C₄	#1806	1	1	yes	characteristic
285	C₄	#2212	1	2	yes	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	8	2	no	—
289	C₂	#8563	1	1	yes	characteristic
290	C₂	#12820	1	1	yes	characteristic
291	C₂	#12822	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1802
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ 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. C₄×#364	C₂	semidirect
4. #8807	C₂	semidirect
5. #7176	C₂	semidirect
33. #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
C₄×#364	2
#11592	2
#8807	2
#7176	2