Subgroups — #136995

Order 512.

401 conjugacy classes of subgroups.

The subgroup lattice has 289 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×#1674	C₂	1	1	yes	maximal, characteristic
3	#13282	C₂	1	1	yes	maximal, characteristic
4	#13275	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#7274	C₂	1	1	yes	maximal, characteristic
7	#7171	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	C₂²	1	1	yes	characteristic
9	C₂×#462	—	2	1	no	—
10	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
11	Q₈×C₄⋊C₄	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1947	—	2	2	no	—
15	#1947	—	2	1	no	—
16	#1845	—	2	1	no	—
17	#1845	C₂²	1	1	yes	characteristic
18	C₄×#107	—	2	1	no	—
19	C₄×#107	C₂²	1	1	yes	characteristic
20	#1674	—	2	2	no	—
21	#1674	C₄	1	4	yes	complemented
22	C₂×#335	C₂²	1	1	yes	characteristic
23	C₂×#332	—	2	1	no	—
24	#1380	—	2	1	no	—
25	#1380	—	2	1	no	—
26	#1380	—	2	2	no	—
27	Q₈⋊C₁₆	—	2	2	no	—
28	C₂²×Q₁₆	—	4	1	no	—
29	C₂×C₄⋊Q₈	—	2	1	no	—
30	C₂×C₄×Q₈	—	4	1	no	—
31	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	—	2	1	no	—
33	C₂²×C₁₆	—	2	1	no	—
34	#500	—	2	1	no	—
35	#500	—	2	1	no	—
36	#500	—	2	1	no	—
37	#500	C₂×C₄	1	2	yes	—
38	#462	—	2	4	no	—
39	#462	—	4	2	no	—
40	C₈×Q₈	—	2	1	no	—
41	C₈×Q₈	—	2	1	no	—
42	C₈×Q₈	C₂×C₄	1	2	yes	—
43	C₈×Q₈	D₄	1	2	yes	—
44	C₂×#107	—	2	1	no	—
45	C₂×#107	—	2	1	no	—
46	C₂×#107	—	2	1	no	—
47	C₂×#107	—	2	2	no	—
48	C₂×#107	D₄	1	1	yes	characteristic
49	C₂×#107	D₄	1	1	yes	characteristic
50	C₂×#107	D₄	1	2	yes	—
51	C₂×C₄⋊C₈	—	2	1	no	—
52	C₂×C₄⋊C₈	—	2	1	no	—
53	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₂×#103	—	4	1	no	—
55	C₂×C₄×C₈	—	2	1	no	—
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	#389	—	4	2	no	—
58	#389	—	4	1	no	—
59	#384	—	2	2	no	—
60	#384	—	2	1	no	—
61	C₄×C₄⋊C₄	—	2	2	no	—
62	C₄×C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₁₆	—	2	1	no	—
64	C₄⋊C₁₆	—	2	1	no	—
65	C₄×C₁₆	—	2	1	no	—
66	C₄×C₁₆	—	2	1	no	—
67	#340	—	2	2	no	—
68	#340	—	2	1	no	—
69	#335	—	2	1	no	—
70	#335	C₂×C₄	1	2	yes	—
71	#332	—	2	4	no	—
72	C₂²×Q₈	—	4	1	no	—
73	C₂²×Q₈	D₈	1	1	yes	characteristic
74	C₂×Q₁₆	—	4	4	no	—
75	C₂×Q₁₆	—	8	2	no	—
76	C₂×Q₁₆	—	8	2	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	—	4	1	no	—
79	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
81	C₄⋊Q₈	—	2	2	no	—
82	C₄⋊Q₈	—	4	1	no	—
83	C₄⋊Q₈	—	4	1	no	—
84	C₄⋊Q₈	—	4	1	no	—
85	C₄×Q₈	—	2	1	no	—
86	C₄×Q₈	—	2	1	no	—
87	C₄×Q₈	—	8	2	no	—
88	C₄×Q₈	—	8	2	no	—
89	C₄×Q₈	C₂²⋊C₄	1	2	yes	—
90	C₄×Q₈	SD₁₆	1	2	yes	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	—	2	1	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	—	4	1	no	—
96	C₂×C₄⋊C₄	—	4	1	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄⋊C₄	—	4	1	no	—
99	C₂×C₄⋊C₄	—	4	2	no	—
100	C₂×C₄⋊C₄	—	4	1	no	—
101	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
102	C₂×C₄²	—	2	1	no	—
103	C₂×C₄²	—	4	2	no	—
104	C₂×C₄²	—	4	1	no	—
105	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	4	1	no	—
108	C₂×C₁₆	—	4	1	no	—
109	C₂×C₁₆	—	4	1	no	—
110	C₂×C₁₆	—	4	1	no	—
111	C₂×C₁₆	—	4	1	no	—
112	#107	—	2	1	no	—
113	#107	—	2	2	no	—
114	#107	—	2	2	no	—
115	#107	—	2	1	no	—
116	#107	—	2	1	no	—
117	#107	—	4	1	no	—
118	#107	—	4	1	no	—
119	#107	—	4	1	no	—
120	#107	—	4	1	no	—
121	#107	—	4	2	no	—
122	#107	—	8	2	no	—
123	#107	Q₁₆	1	2	yes	complemented
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	—	4	1	no	—
129	C₄⋊C₈	C₂²⋊C₄	1	2	yes	—
130	#103	—	4	4	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₄×C₈	C₂×D₄	1	1	yes	characteristic
135	C₄×C₈	#41	1	1	yes	characteristic
136	C₄×C₈	#41	1	1	yes	characteristic
137	#95	—	4	2	no	—
138	#95	—	4	1	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	8	1	no	—
144	C₂×Q₈	—	8	1	no	—
145	C₂×Q₈	—	8	1	no	—
146	C₂×Q₈	—	8	1	no	—
147	C₂×Q₈	—	8	1	no	—
148	C₂×Q₈	#102	1	2	yes	—
149	C₂×Q₈	D₁₆	1	2	yes	—
150	C₂×Q₈	SD₃₂	1	2	yes	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	1	no	—
156	C₂²×C₄	—	8	1	no	—
157	C₂²×C₄	#120	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
159	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
160	Q₁₆	—	8	8	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	8	1	no	—
168	C₂×C₈	—	8	1	no	—
169	C₂×C₈	C₄×D₄	1	1	yes	characteristic
170	C₂×C₈	C₄×D₄	1	1	yes	characteristic
171	C₂×C₈	#121	1	1	yes	characteristic
172	C₂×C₈	#121	1	1	yes	characteristic
173	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
174	C₂×C₈	#135	1	1	yes	characteristic
175	C₂×C₈	#137	1	1	yes	characteristic
176	C₂×C₈	#137	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	8	4	no	—
189	C₄⋊C₄	—	8	1	no	—
190	C₄⋊C₄	—	8	1	no	—
191	C₄⋊C₄	—	8	1	no	—
192	C₄⋊C₄	—	8	1	no	—
193	C₄⋊C₄	—	16	1	no	—
194	C₄⋊C₄	—	16	2	no	—
195	C₄⋊C₄	#102	1	2	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	8	4	no	—
199	C₄²	—	8	1	no	—
200	C₄²	—	8	1	no	—
201	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₄²	#123	1	1	yes	characteristic
203	C₄²	C₂×SD₁₆	1	1	yes	characteristic
204	C₄²	#137	1	1	yes	characteristic
205	C₁₆	—	4	2	no	—
206	C₁₆	—	8	1	no	—
207	C₂³	#447	1	1	yes	characteristic
208	Q₈	—	2	2	no	—
209	Q₈	—	2	2	no	—
210	Q₈	—	2	2	no	—
211	Q₈	—	8	2	no	—
212	Q₈	—	8	2	no	—
213	Q₈	—	16	1	no	—
214	Q₈	—	16	1	no	—
215	Q₈	#357	1	4	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	1	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	1	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	16	1	no	—
235	C₂×C₄	—	16	1	no	—
236	C₂×C₄	#386	1	1	yes	characteristic
237	C₂×C₄	C₂×#102	1	1	yes	characteristic
238	C₂×C₄	#417	1	1	yes	characteristic
239	C₂×C₄	#448	1	1	yes	characteristic
240	C₂×C₄	#451	1	1	yes	characteristic
241	C₂×C₄	#452	1	1	yes	characteristic
242	C₂×C₄	#455	1	1	yes	characteristic
243	C₂×C₄	#456	1	1	yes	characteristic
244	C₂×C₄	#480	1	1	yes	characteristic
245	C₂×C₄	#481	1	1	yes	characteristic
246	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
247	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
248	C₂×C₄	#509	1	1	yes	characteristic
249	C₂×C₄	#510	1	1	yes	characteristic
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	4	2	no	—
253	C₈	—	8	2	no	—
254	C₈	C₄×Q₁₆	1	1	yes	characteristic
255	C₈	#441	1	1	yes	characteristic
256	C₈	#462	1	1	yes	characteristic
257	C₈	#464	1	1	yes	characteristic
258	C₂²	#1641	1	1	yes	characteristic
259	C₂²	#1649	1	1	yes	characteristic
260	C₂²	#1874	1	1	yes	characteristic
261	C₂²	#2184	1	1	yes	characteristic
262	C₂²	#2185	1	1	yes	characteristic
263	C₂²	#2194	1	1	yes	characteristic
264	C₂²	#2195	1	1	yes	characteristic
265	C₄	—	2	4	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	2	no	—
270	C₄	—	16	1	no	—
271	C₄	—	16	2	no	—
272	C₄	—	16	2	no	—
273	C₄	#1645	1	1	yes	characteristic
274	C₄	#1653	1	1	yes	characteristic
275	C₄	#1878	1	1	yes	characteristic
276	C₄	#1882	1	1	yes	characteristic
277	C₄	C₂×#357	1	1	yes	characteristic
278	C₄	#2139	1	1	yes	characteristic
279	C₄	#2232	1	1	yes	characteristic
280	C₄	#2233	1	1	yes	characteristic
281	C₂	#8987	1	1	yes	characteristic
282	C₂	#12380	1	1	yes	characteristic
283	C₂	#12382	1	1	yes	characteristic
284	C₂	#12404	1	1	yes	characteristic
285	C₂	#12406	1	1	yes	characteristic
286	C₂	#12991	1	1	yes	characteristic
287	C₂	#12997	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ 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
21. #1674	C₄	semidirect
123. #107	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#107	2
C₂×#1674	2
#13282	2
#13275	2
C₂×Q₈⋊C₁₆	2
#7274	2
#7171	2