Subgroups — #72321462

Order 1024.

565 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#483942	C₂	1	1	yes	maximal, characteristic
2	#483575	C₂	1	2	yes	maximal
3	#122969	C₂	1	4	yes	maximal, complemented
4	#122965	C₂	1	2	yes	maximal, complemented
5	#122962	C₂	1	2	yes	maximal, complemented
6	C₂×#7099	C₂	1	1	yes	maximal, characteristic
7	#122469	C₂	1	2	yes	maximal
8	#122219	C₂	1	1	yes	maximal, characteristic
9	#33734	C₄	1	1	yes	characteristic
10	#33476	—	2	2	no	—
11	#17958	C₂²	1	2	yes	—
12	#12085	C₄	1	2	yes	—
13	#12085	C₄	1	2	yes	—
14	#12061	C₂²	1	4	yes	—
15	#12060	—	2	4	no	—
16	#12059	C₂²	1	4	yes	—
17	#12044	C₂²	1	2	yes	—
18	#12044	C₂²	1	2	yes	—
19	#12023	C₄	1	1	yes	characteristic
20	C₂×#1367	C₂²	1	2	yes	—
21	#11672	C₄	1	2	yes	—
22	#11606	—	2	2	no	—
23	#11411	C₂²	1	1	yes	characteristic
24	#11402	C₂²	1	1	yes	characteristic
25	C₂×#1311	C₂²	1	1	yes	characteristic
26	#7099	C₂²	1	1	yes	characteristic
27	#7099	C₂²	1	1	yes	characteristic
28	#7099	C₂²	1	2	yes	—
29	#7071	C₂²	1	4	yes	—
30	#7071	C₂²	1	4	yes	—
31	#7053	C₂²	1	2	yes	—
32	#7053	C₂²	1	2	yes	—
33	#3170	C₂×C₄	1	2	yes	—
34	#2960	—	2	2	no	—
35	#2252	C₈	1	1	yes	characteristic
36	#2247	C₈	1	1	yes	characteristic
37	#2241	C₈	1	2	yes	—
38	#2216	C₈	1	2	yes	—
39	#2216	C₈	1	2	yes	—
40	#2183	—	2	4	no	—
41	#2183	—	4	4	no	—
42	Q₈×C₁₆	—	2	4	no	—
43	C₄×Q₃₂	C₂×C₄	1	2	yes	—
44	C₄×Q₃₂	C₂×C₄	1	2	yes	—
45	#2162	—	2	1	no	—
46	#2151	—	4	2	no	—
47	C₂×C₄⋊C₁₆	—	2	2	no	—
48	#2108	—	4	2	no	—
49	C₄×M₃₂	D₄	1	2	yes	—
50	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
52	#1591	—	2	2	no	—
53	#1591	C₂×C₄	1	4	yes	—
54	C₈×Q₁₆	—	2	2	no	—
55	C₈×Q₁₆	C₂³	1	4	yes	—
56	#1567	C₂×C₄	1	2	yes	—
57	#1493	—	2	2	no	—
58	C₂×C₈²	C₂³	1	1	yes	characteristic
59	#1372	C₂×C₄	1	2	yes	—
60	#1372	C₂×C₄	1	2	yes	—
61	#1367	—	2	2	no	—
62	#1367	C₂³	1	2	yes	—
63	#1367	C₂³	1	2	yes	—
64	Q₈⋊C₁₆	—	2	4	no	—
65	Q₈⋊C₁₆	—	2	4	no	—
66	#1332	C₂×C₄	1	4	yes	—
67	#1332	C₂×C₄	1	4	yes	—
68	#1312	D₄	1	2	yes	—
69	#1312	C₂³	1	2	yes	—
70	#1312	C₂³	1	2	yes	—
71	#1311	—	2	1	no	—
72	#1311	C₂×C₄	1	1	yes	characteristic
73	#1311	C₂×C₄	1	1	yes	characteristic
74	#1311	C₂³	1	1	yes	characteristic
75	#1311	C₂³	1	1	yes	characteristic
76	#533	—	2	2	no	—
77	C₂×Q₃₂	C₂×C₈	1	2	yes	—
78	C₂×Q₃₂	C₂×C₈	1	2	yes	—
79	C₂×M₃₂	—	2	2	no	—
80	C₂²×C₁₆	—	2	1	no	—
81	C₂²×C₁₆	—	4	2	no	—
82	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
84	#500	C₂×C₈	1	2	yes	—
85	#494	C₂×C₈	1	2	yes	—
86	#487	—	2	2	no	—
87	#467	—	2	2	no	—
88	#467	C₂×C₈	1	4	yes	—
89	#446	—	2	4	no	—
90	#446	—	4	2	no	—
91	C₈×Q₈	—	2	4	no	—
92	C₈×Q₈	—	4	2	no	—
93	C₄×Q₁₆	—	2	2	no	—
94	C₄×Q₁₆	C₂²×C₄	1	4	yes	—
95	#424	—	4	2	no	—
96	#423	—	2	2	no	—
97	#405	—	4	2	no	—
98	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₈	#41	1	1	yes	characteristic
101	#366	C₂×C₈	1	2	yes	—
102	#366	C₂×C₈	1	2	yes	—
103	C₄⋊C₁₆	—	2	2	no	—
104	C₄⋊C₁₆	—	2	2	no	—
105	C₄⋊C₁₆	—	4	1	no	—
106	C₄⋊C₁₆	—	4	4	no	—
107	C₄⋊C₁₆	—	4	4	no	—
108	C₄⋊C₁₆	—	4	2	no	—
109	#358	C₂×C₈	1	4	yes	—
110	#358	C₂×C₈	1	4	yes	—
111	C₂²⋊C₁₆	—	4	1	no	—
112	#346	—	2	2	no	—
113	#346	—	4	4	no	—
114	C₄×C₁₆	—	2	1	no	—
115	C₄×C₁₆	—	2	2	no	—
116	C₄×C₁₆	—	4	4	no	—
117	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
118	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
119	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
120	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
121	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
122	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
123	C₄×C₁₆	C₂×D₄	1	4	yes	—
124	#335	—	2	2	no	—
125	#335	C₂²×C₄	1	4	yes	—
126	Q₈⋊C₈	—	2	4	no	—
127	Q₈⋊C₈	—	2	4	no	—
128	C₈²	C₂²×C₄	1	1	yes	characteristic
129	C₈²	C₂×D₄	1	1	yes	characteristic
130	C₈²	#41	1	1	yes	characteristic
131	C₈²	C₂⁴	1	1	yes	characteristic
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	C₂²×C₈	1	4	yes	—
134	C₂×M₁₆	—	4	2	no	—
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	C₂²×C₈	1	1	yes	characteristic
137	C₂²×C₈	#131	1	1	yes	characteristic
138	C₂²×C₈	AGL(1,ℤ/8)	1	2	yes	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	#125	—	4	2	no	—
142	C₂²⋊Q₈	—	2	4	no	—
143	C₂²⋊Q₈	—	4	2	no	—
144	C₄×Q₈	—	2	4	no	—
145	C₄×Q₈	—	4	2	no	—
146	#117	—	4	2	no	—
147	C₂×C₄²	C₄×D₄	1	1	yes	characteristic
148	Q₃₂	—	2	4	no	—
149	Q₃₂	—	4	2	no	—
150	M₃₂	—	4	4	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	1	no	—
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	—	4	1	no	—
156	C₂×C₁₆	—	4	2	no	—
157	C₂×C₁₆	—	4	2	no	—
158	C₂×C₁₆	—	4	2	no	—
159	C₂×C₁₆	—	4	2	no	—
160	C₂×C₁₆	—	8	4	no	—
161	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
165	#107	—	2	2	no	—
166	#107	C₂²×C₈	1	4	yes	—
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	4	4	no	—
169	C₄⋊C₈	—	4	4	no	—
170	C₄⋊C₈	—	4	2	no	—
171	#103	—	2	4	no	—
172	#103	—	2	4	no	—
173	#97	—	4	4	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	4	4	no	—
176	C₄×C₈	C₄×D₄	1	1	yes	characteristic
177	C₄×C₈	C₄×D₄	1	1	yes	characteristic
178	C₄×C₈	C₄×D₄	1	1	yes	characteristic
179	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
180	C₄×C₈	#131	1	1	yes	characteristic
181	C₄×C₈	AGL(1,ℤ/8)	1	2	yes	—
182	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
183	C₄×C₈	C₂²×D₄	1	1	yes	characteristic
184	C₄×C₈	C₂×#41	1	1	yes	characteristic
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂²×C₄	C₈×D₄	1	1	yes	characteristic
188	C₂²×C₄	#442	1	1	yes	characteristic
189	C₂²×C₄	#444	1	1	yes	characteristic
190	Q₁₆	—	2	4	no	—
191	Q₁₆	—	2	4	no	—
192	Q₁₆	—	8	2	no	—
193	M₁₆	—	8	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	8	1	no	—
202	C₂×C₈	C₈×D₄	1	1	yes	characteristic
203	C₂×C₈	C₈×D₄	1	1	yes	characteristic
204	C₂×C₈	#442	1	2	yes	—
205	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
206	C₂×C₈	C₂×#131	1	1	yes	characteristic
207	C₂×C₈	C₂×AGL(1,ℤ/8)	1	2	yes	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	—	4	4	no	—
216	C₄²	C₈×D₄	1	1	yes	characteristic
217	C₄²	#442	1	1	yes	characteristic
218	C₄²	#444	1	1	yes	characteristic
219	C₄²	C₂×C₄×D₄	1	1	yes	characteristic
220	C₁₆	—	2	2	no	—
221	C₁₆	—	2	1	no	—
222	C₁₆	—	2	1	no	—
223	C₁₆	—	2	1	no	—
224	C₁₆	—	4	1	no	—
225	C₁₆	—	4	4	no	—
226	C₁₆	—	8	2	no	—
227	C₁₆	—	8	2	no	—
228	C₁₆	—	8	2	no	—
229	C₁₆	#504	1	2	yes	—
230	C₂³	#1580	1	1	yes	characteristic
231	Q₈	—	4	4	no	—
232	Q₈	—	4	4	no	—
233	Q₈	—	16	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	#1580	1	1	yes	characteristic
240	C₂×C₄	#1580	1	1	yes	characteristic
241	C₂×C₄	#1580	1	1	yes	characteristic
242	C₂×C₄	#2180	1	4	yes	—
243	C₂×C₄	C₂×C₈×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂×#442	1	1	yes	characteristic
245	C₂×C₄	C₂×#444	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	4	1	no	—
252	C₈	—	4	1	no	—
253	C₈	—	8	4	no	—
254	C₈	C₂×#504	1	1	yes	characteristic
255	C₈	#3407	1	1	yes	characteristic
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	#12066	1	2	yes	—
259	C₂²	C₂×#1580	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	4	no	—
264	C₄	—	8	4	no	—
265	C₄	#12066	1	2	yes	—
266	C₄	#33438	1	1	yes	characteristic
267	C₄	#33443	1	1	yes	characteristic
268	C₄	C₂×#2180	1	2	yes	—
269	C₂	—	2	2	no	—
270	C₂	#483585	1	1	yes	characteristic
271	C₂	C₂×#12066	1	1	yes	characteristic
272	C₂	#483601	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂²	C₂×#1580
prime radical	C₂³	#1580
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #11402 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #122969	C₂	semidirect
4. #122965	C₂	semidirect
5. #122962	C₂	semidirect

Maximal subgroups

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

subgroup	index
#483942	2
#483575	2
#122969	2
#122965	2
#122962	2
C₂×#7099	2
#122469	2
#122219	2