Subgroups — #105206

Order 512.

299 conjugacy classes of subgroups.

The subgroup lattice has 269 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#9896	C₂	1	1	yes	maximal, characteristic
2	#9896	C₂	1	1	yes	maximal, characteristic
3	#9896	C₂	1	1	yes	maximal, characteristic
4	#9602	C₂	1	1	yes	maximal, characteristic
5	#9601	C₂	1	1	yes	maximal, characteristic
6	#9601	C₂	1	1	yes	maximal, characteristic
7	#7050	C₂	1	1	yes	maximal, characteristic
8	#2868	C₂²	1	1	yes	characteristic
9	#1842	C₂²	1	1	yes	characteristic
10	#1842	C₂²	1	1	yes	characteristic
11	#1842	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×#335	—	2	1	no	—
16	C₂×#335	—	2	1	no	—
17	C₂×#334	—	2	1	no	—
18	#1296	—	2	1	no	—
19	#1296	—	2	1	no	—
20	#1296	—	2	1	no	—
21	#1296	—	2	1	no	—
22	#1296	—	2	1	no	—
23	#1296	—	2	1	no	—
24	#1277	—	2	1	no	—
25	#1277	—	2	1	no	—
26	#1277	—	2	1	no	—
27	C₂×C₄⋊Q₈	—	2	1	no	—
28	C₂×C₄⋊Q₈	—	2	1	no	—
29	C₂×C₄⋊Q₈	—	2	1	no	—
30	C₂×C₄⋊C₈	—	2	1	no	—
31	C₂×C₄⋊C₈	—	2	1	no	—
32	C₂×C₄⋊C₈	—	2	1	no	—
33	C₂×C₄⋊C₈	—	2	1	no	—
34	C₂×C₄⋊C₈	—	2	1	no	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	—	2	1	no	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	—	2	1	no	—
46	C₂×C₄×C₈	—	2	1	no	—
47	C₂×C₄×C₈	—	2	1	no	—
48	#384	—	2	1	no	—
49	#384	—	2	1	no	—
50	#384	—	2	1	no	—
51	#384	Q₈	1	2	yes	—
52	C₄³	C₂³	1	1	yes	characteristic
53	#336	—	2	1	no	—
54	#336	—	2	1	no	—
55	#336	—	2	1	no	—
56	#335	—	4	2	no	—
57	#335	—	4	2	no	—
58	#334	—	4	2	no	—
59	C₂²×Q₈	—	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	—	4	1	no	—
67	C₂²×C₈	—	4	1	no	—
68	C₂²×C₈	—	4	1	no	—
69	C₄⋊Q₈	—	8	1	no	—
70	C₄⋊Q₈	—	8	1	no	—
71	C₄⋊Q₈	—	8	1	no	—
72	C₄⋊Q₈	—	8	1	no	—
73	C₄⋊Q₈	—	8	1	no	—
74	C₄⋊Q₈	—	8	1	no	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	4	1	no	—
79	C₂×C₄⋊C₄	—	4	1	no	—
80	C₂×C₄⋊C₄	—	4	1	no	—
81	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄²	#41	1	1	yes	characteristic
86	C₂×C₄²	#41	1	1	yes	characteristic
87	C₂×C₄²	#41	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	—	2	1	no	—
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	2	1	no	—
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	—	4	1	no	—
98	C₄⋊C₈	—	4	1	no	—
99	C₄⋊C₈	—	4	1	no	—
100	C₄⋊C₈	—	4	1	no	—
101	C₄⋊C₈	—	4	1	no	—
102	C₄⋊C₈	—	4	1	no	—
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₄⋊C₈	—	4	1	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄⋊C₈	—	4	1	no	—
108	C₄⋊C₈	—	4	1	no	—
109	C₄⋊C₈	—	8	1	no	—
110	C₄⋊C₈	—	8	1	no	—
111	C₄⋊C₈	—	8	1	no	—
112	C₄⋊C₈	SD₁₆	1	2	yes	—
113	C₄⋊C₈	Q₁₆	1	2	yes	—
114	C₄⋊C₈	Q₁₆	1	2	yes	—
115	C₄×C₈	—	4	1	no	—
116	C₄×C₈	—	4	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	—	4	1	no	—
119	C₄×C₈	—	4	1	no	—
120	C₄×C₈	—	4	1	no	—
121	C₂×Q₈	—	8	2	no	—
122	C₂×Q₈	—	8	2	no	—
123	C₂×Q₈	—	8	2	no	—
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	#121	1	1	yes	characteristic
126	C₂²×C₄	#121	1	1	yes	characteristic
127	C₂²×C₄	#121	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂²×C₄	#126	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	8	1	no	—
151	C₂×C₈	—	8	1	no	—
152	C₂×C₈	—	8	1	no	—
153	C₂×C₈	—	8	1	no	—
154	C₂×C₈	—	8	1	no	—
155	C₂×C₈	—	8	1	no	—
156	C₂×C₈	—	8	1	no	—
157	C₂×C₈	—	8	1	no	—
158	C₂×C₈	—	8	1	no	—
159	C₄⋊C₄	—	8	2	no	—
160	C₄⋊C₄	—	8	2	no	—
161	C₄⋊C₄	—	8	2	no	—
162	C₄⋊C₄	—	8	1	no	—
163	C₄⋊C₄	—	8	1	no	—
164	C₄⋊C₄	—	8	1	no	—
165	C₄⋊C₄	—	8	1	no	—
166	C₄⋊C₄	—	8	1	no	—
167	C₄⋊C₄	—	8	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	4	1	no	—
175	C₄²	C₂×SD₁₆	1	1	yes	characteristic
176	C₄²	C₂×Q₁₆	1	1	yes	characteristic
177	C₄²	C₂×Q₁₆	1	1	yes	characteristic
178	C₄²	#135	1	1	yes	characteristic
179	C₄²	#135	1	1	yes	characteristic
180	C₄²	#135	1	1	yes	characteristic
181	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₄²	#137	1	1	yes	characteristic
184	C₄²	#137	1	1	yes	characteristic
185	C₄²	#137	1	1	yes	characteristic
186	C₄²	#137	1	1	yes	characteristic
187	C₂³	#396	1	1	yes	characteristic
188	Q₈	—	16	4	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	#461	1	1	yes	characteristic
202	C₂×C₄	#462	1	1	yes	characteristic
203	C₂×C₄	#462	1	1	yes	characteristic
204	C₂×C₄	#463	1	1	yes	characteristic
205	C₂×C₄	#463	1	1	yes	characteristic
206	C₂×C₄	#464	1	1	yes	characteristic
207	C₂×C₄	#465	1	1	yes	characteristic
208	C₂×C₄	#467	1	1	yes	characteristic
209	C₂×C₄	#467	1	1	yes	characteristic
210	C₂×C₄	#468	1	1	yes	characteristic
211	C₂×C₄	#468	1	1	yes	characteristic
212	C₂×C₄	#470	1	1	yes	characteristic
213	C₂×C₄	#471	1	1	yes	characteristic
214	C₂×C₄	#471	1	1	yes	characteristic
215	C₂×C₄	#471	1	1	yes	characteristic
216	C₂×C₄	#472	1	1	yes	characteristic
217	C₂×C₄	#472	1	1	yes	characteristic
218	C₂×C₄	#472	1	1	yes	characteristic
219	C₂×C₄	#475	1	1	yes	characteristic
220	C₂×C₄	#477	1	1	yes	characteristic
221	C₂×C₄	#477	1	1	yes	characteristic
222	C₂×C₄	#479	1	1	yes	characteristic
223	C₂×C₄	#479	1	1	yes	characteristic
224	C₂×C₄	#479	1	1	yes	characteristic
225	C₈	—	4	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	8	1	no	—
229	C₈	—	8	1	no	—
230	C₈	—	8	1	no	—
231	C₈	—	16	1	no	—
232	C₈	—	16	1	no	—
233	C₈	—	16	1	no	—
234	C₂²	#2063	1	1	yes	characteristic
235	C₂²	#2063	1	1	yes	characteristic
236	C₂²	#2063	1	1	yes	characteristic
237	C₂²	#2066	1	1	yes	characteristic
238	C₂²	#2068	1	1	yes	characteristic
239	C₂²	#2068	1	1	yes	characteristic
240	C₂²	#2070	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	4	1	no	—
248	C₄	—	16	2	no	—
249	C₄	#1672	1	1	yes	characteristic
250	C₄	#1675	1	1	yes	characteristic
251	C₄	#1676	1	1	yes	characteristic
252	C₄	#1677	1	1	yes	characteristic
253	C₄	#1683	1	1	yes	characteristic
254	C₄	#1684	1	1	yes	characteristic
255	C₄	#1691	1	1	yes	characteristic
256	C₄	#1692	1	1	yes	characteristic
257	C₄	#1696	1	1	yes	characteristic
258	C₄	#1697	1	1	yes	characteristic
259	C₄	#1698	1	1	yes	characteristic
260	C₄	#1703	1	1	yes	characteristic
261	C₂	#10589	1	1	yes	characteristic
262	C₂	#10593	1	1	yes	characteristic
263	C₂	#10601	1	1	yes	characteristic
264	C₂	#10606	1	1	yes	characteristic
265	C₂	#10606	1	1	yes	characteristic
266	C₂	#10610	1	1	yes	characteristic
267	C₂	#10612	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#396
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Maximal subgroups

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

subgroup	index
#9896	2
#9602	2
#9601	2
#7050	2