Subgroups — #136347

Order 512.

351 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#13177	C₂	1	1	yes	maximal, characteristic
3	#12906	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#8645	C₂	1	1	yes	maximal, characteristic
6	#7274	C₂	1	1	yes	maximal, characteristic
7	#7171	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
9	Q₈×C₄⋊C₄	—	2	1	no	—
10	C₂×#367	—	2	2	no	—
11	C₂×#367	—	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	#1845	C₂²	1	1	yes	characteristic
19	#1803	—	2	1	no	—
20	C₄×#107	—	2	1	no	—
21	C₄×#107	C₂²	1	1	yes	characteristic
22	C₂×#335	C₂²	1	1	yes	characteristic
23	#1380	—	2	1	no	—
24	Q₈⋊C₁₆	C₄	1	4	yes	complemented
25	#1296	—	2	1	no	—
26	C₂×C₄×Q₈	—	4	1	no	—
27	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
28	C₂²×C₁₆	—	2	1	no	—
29	C₂²×C₁₆	—	2	1	no	—
30	C₈×Q₈	—	2	1	no	—
31	C₈×Q₈	—	2	1	no	—
32	C₈×Q₈	C₂×C₄	1	2	yes	—
33	C₈×Q₈	Q₈	1	2	yes	—
34	C₂×#107	—	2	1	no	—
35	C₂×#107	—	2	1	no	—
36	C₂×#107	—	2	1	no	—
37	C₂×#107	—	2	2	no	—
38	C₂×#107	D₄	1	1	yes	characteristic
39	C₂×#107	D₄	1	1	yes	characteristic
40	C₂×#107	D₄	1	2	yes	—
41	C₂×C₄⋊C₈	—	2	1	no	—
42	C₂×C₄⋊C₈	—	2	1	no	—
43	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
44	C₂×#103	—	4	1	no	—
45	C₂×C₄×C₈	—	2	1	no	—
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	#389	—	4	2	no	—
48	#389	—	4	1	no	—
49	#384	—	2	2	no	—
50	#384	—	2	1	no	—
51	#384	—	2	1	no	—
52	C₄×C₄⋊C₄	—	2	2	no	—
53	C₄×C₄⋊C₄	—	2	1	no	—
54	#367	—	4	2	no	—
55	#367	—	4	4	no	—
56	C₄⋊C₁₆	—	2	1	no	—
57	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
58	C₄×C₁₆	—	2	1	no	—
59	C₄×C₁₆	C₂×C₄	1	2	yes	—
60	#340	—	2	2	no	—
61	#340	—	2	1	no	—
62	#340	—	4	1	no	—
63	#335	—	2	1	no	—
64	#335	—	2	1	no	—
65	C₂²×Q₈	D₈	1	1	yes	characteristic
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	—	4	1	no	—
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₄×Q₈	—	2	1	no	—
71	C₄×Q₈	—	2	1	no	—
72	C₄×Q₈	—	8	2	no	—
73	C₄×Q₈	—	8	2	no	—
74	C₄×Q₈	C₄⋊C₄	1	2	yes	—
75	C₄×Q₈	Q₁₆	1	2	yes	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄⋊C₄	—	4	2	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	C₂×C₄⋊C₄	—	4	1	no	—
83	C₂×C₄⋊C₄	—	4	1	no	—
84	C₂×C₄⋊C₄	—	4	2	no	—
85	C₂×C₄⋊C₄	—	4	1	no	—
86	C₂×C₄⋊C₄	—	4	1	no	—
87	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
88	C₂×C₄²	—	2	1	no	—
89	C₂×C₄²	—	4	2	no	—
90	C₂×C₄²	—	4	1	no	—
91	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	4	1	no	—
97	C₂×C₁₆	—	4	1	no	—
98	#107	—	2	1	no	—
99	#107	—	2	1	no	—
100	#107	—	2	1	no	—
101	#107	—	4	1	no	—
102	#107	—	4	1	no	—
103	#107	—	4	2	no	—
104	#107	—	4	2	no	—
105	#107	—	4	1	no	—
106	#107	—	4	1	no	—
107	#107	—	8	2	no	—
108	#107	Q₁₆	1	2	yes	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	—	4	1	no	—
113	C₄⋊C₈	—	4	1	no	—
114	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
115	#103	—	8	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
120	C₄×C₈	#41	1	1	yes	characteristic
121	C₄×C₈	#41	1	1	yes	characteristic
122	#95	—	4	2	no	—
123	#95	—	4	1	no	—
124	C₂×Q₈	—	2	1	no	—
125	C₂×Q₈	—	2	1	no	—
126	C₂×Q₈	—	2	1	no	—
127	C₂×Q₈	#107	1	2	yes	—
128	C₂×Q₈	SD₃₂	1	2	yes	—
129	C₂×Q₈	SD₃₂	1	2	yes	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	4	1	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	8	1	no	—
136	C₂²×C₄	#120	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	8	1	no	—
145	C₂×C₈	—	8	1	no	—
146	C₂×C₈	—	8	1	no	—
147	C₂×C₈	C₄×D₄	1	1	yes	characteristic
148	C₂×C₈	C₄×D₄	1	1	yes	characteristic
149	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
150	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
152	C₂×C₈	#135	1	1	yes	characteristic
153	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₈	#137	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	4	1	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	8	4	no	—
165	C₄⋊C₄	—	8	1	no	—
166	C₄⋊C₄	—	8	1	no	—
167	C₄⋊C₄	—	8	1	no	—
168	C₄⋊C₄	—	8	1	no	—
169	C₄⋊C₄	—	8	1	no	—
170	C₄⋊C₄	—	8	1	no	—
171	C₄⋊C₄	—	16	1	no	—
172	C₄⋊C₄	—	16	1	no	—
173	C₄⋊C₄	—	16	2	no	—
174	C₄⋊C₄	#107	1	2	yes	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	8	4	no	—
178	C₄²	—	8	1	no	—
179	C₄²	—	8	1	no	—
180	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₄²	#123	1	1	yes	characteristic
182	C₄²	C₂×Q₁₆	1	1	yes	characteristic
183	C₄²	#137	1	1	yes	characteristic
184	C₁₆	—	2	4	no	—
185	C₁₆	—	4	2	no	—
186	C₂³	#447	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	2	2	no	—
190	Q₈	#367	1	4	yes	complemented
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	8	1	no	—
201	C₂×C₄	—	8	1	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	—	8	1	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	—	8	1	no	—
208	C₂×C₄	—	16	1	no	—
209	C₂×C₄	—	16	1	no	—
210	C₂×C₄	—	16	1	no	—
211	C₂×C₄	#385	1	1	yes	characteristic
212	C₂×C₄	C₂×#107	1	1	yes	characteristic
213	C₂×C₄	#428	1	1	yes	characteristic
214	C₂×C₄	#449	1	1	yes	characteristic
215	C₂×C₄	#451	1	1	yes	characteristic
216	C₂×C₄	#451	1	1	yes	characteristic
217	C₂×C₄	#453	1	1	yes	characteristic
218	C₂×C₄	#456	1	1	yes	characteristic
219	C₂×C₄	#480	1	1	yes	characteristic
220	C₂×C₄	#484	1	1	yes	characteristic
221	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
222	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
223	C₂×C₄	#509	1	1	yes	characteristic
224	C₂×C₄	#510	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	4	2	no	—
228	C₈	—	16	1	no	—
229	C₈	C₄×Q₁₆	1	1	yes	characteristic
230	C₈	#440	1	1	yes	characteristic
231	C₈	#477	1	1	yes	characteristic
232	C₈	#479	1	1	yes	characteristic
233	C₂²	#1621	1	1	yes	characteristic
234	C₂²	#1649	1	1	yes	characteristic
235	C₂²	#1852	1	1	yes	characteristic
236	C₂²	#2185	1	1	yes	characteristic
237	C₂²	#2187	1	1	yes	characteristic
238	C₂²	#2193	1	1	yes	characteristic
239	C₂²	#2194	1	1	yes	characteristic
240	C₄	—	2	4	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	8	2	no	—
245	C₄	—	16	1	no	—
246	C₄	—	16	2	no	—
247	C₄	—	32	1	no	—
248	C₄	#1633	1	1	yes	characteristic
249	C₄	#1648	1	1	yes	characteristic
250	C₄	#1859	1	1	yes	characteristic
251	C₄	#1863	1	1	yes	characteristic
252	C₄	C₂×#367	1	1	yes	characteristic
253	C₄	#2159	1	1	yes	characteristic
254	C₄	#2233	1	1	yes	characteristic
255	C₄	#2235	1	1	yes	characteristic
256	C₂	#8884	1	1	yes	characteristic
257	C₂	#12330	1	1	yes	characteristic
258	C₂	#12332	1	1	yes	characteristic
259	C₂	#12406	1	1	yes	characteristic
260	C₂	#12408	1	1	yes	characteristic
261	C₂	#12920	1	1	yes	characteristic
262	C₂	#12926	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
24. Q₈⋊C₁₆	C₄	semidirect
190. Q₈	#367	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
#13177	2
#12906	2
C₂×Q₈⋊C₁₆	2
#8645	2
#7274	2
#7171	2