Subgroups — #146997

Order 512.

694 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×#1845	C₂	1	1	yes	maximal, characteristic
4	C₂×#1387	C₂	1	4	yes	maximal
5	C₂²×M₃₂	C₂²	1	2	yes	—
6	C₂²×#108	—	2	1	no	—
7	C₂²×#108	C₂²	1	2	yes	—
8	C₂²×#107	—	2	1	no	—
9	C₂²×#107	C₂²	1	2	yes	—
10	C₂×#423	—	2	1	no	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#384	—	2	1	no	—
13	C₄×M₃₂	—	2	2	no	—
14	C₄×M₃₂	C₄	1	4	yes	complemented
15	C₂×#346	—	2	1	no	—
16	C₂×C₄×C₁₆	—	2	1	no	—
17	#1848	—	2	2	no	—
18	#1848	C₄	1	4	yes	complemented
19	#1845	—	2	1	no	—
20	#1845	—	2	1	no	—
21	#1845	C₄	1	2	yes	—
22	#1845	C₄	1	2	yes	—
23	#1387	—	2	8	no	—
24	C₂²×M₁₆	—	2	2	no	—
25	C₂³×C₈	D₄	1	1	yes	characteristic
26	C₂³×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₄⋊C₄	—	2	2	no	—
28	C₂²×C₄⋊C₄	—	4	2	no	—
29	C₂²×C₄²	D₄	1	1	yes	characteristic
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	—	2	2	no	—
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	C₂×C₄	1	4	yes	—
34	C₂×M₃₂	D₄	1	4	yes	—
35	C₂×M₃₂	Q₈	1	4	yes	—
36	C₂²×C₁₆	—	2	2	no	—
37	C₂×#108	—	2	4	no	—
38	C₂×#108	—	2	4	no	—
39	C₂×#108	—	2	4	no	—
40	C₂×#108	—	2	2	no	—
41	C₂×#108	—	4	2	no	—
42	C₂×#108	—	4	2	no	—
43	C₂×#108	C₂×C₄	1	4	yes	—
44	C₂×#107	—	2	4	no	—
45	C₂×#107	—	2	4	no	—
46	C₂×#107	—	2	2	no	—
47	C₂×#107	—	2	2	no	—
48	C₂×#107	—	2	2	no	—
49	C₂×#107	—	4	2	no	—
50	C₂×#107	—	4	2	no	—
51	C₂×#107	C₂×C₄	1	4	yes	—
52	#423	—	2	4	no	—
53	#423	—	4	2	no	—
54	C₂×C₄⋊C₈	—	4	2	no	—
55	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
59	C₂×C₄×C₈	D₄	1	1	yes	characteristic
60	C₂×C₄×C₈	D₄	1	1	yes	characteristic
61	C₂×C₄×C₈	D₄	1	2	yes	—
62	C₂×C₄×C₈	Q₈	1	2	yes	—
63	#384	—	2	2	no	—
64	#384	—	2	2	no	—
65	#384	—	4	2	no	—
66	#346	—	2	4	no	—
67	C₄×C₁₆	—	2	4	no	—
68	C₂³×C₄	—	4	2	no	—
69	C₂³×C₄	SD₁₆	1	2	yes	—
70	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×M₁₆	—	2	4	no	—
72	C₂×M₁₆	—	4	2	no	—
73	C₂×M₁₆	—	4	2	no	—
74	C₂×M₁₆	—	4	2	no	—
75	C₂×M₁₆	—	4	2	no	—
76	C₂×M₁₆	—	4	2	no	—
77	C₂²×C₈	—	4	2	no	—
78	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
81	C₂²×C₈	C₄⋊C₄	1	2	yes	—
82	C₂²×C₈	D₈	1	2	yes	—
83	C₂²×C₈	SD₁₆	1	2	yes	—
84	C₂²×C₈	SD₁₆	1	2	yes	—
85	C₂²×C₈	Q₁₆	1	2	yes	—
86	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×D₄	1	2	yes	—
92	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₈	C₂×Q₈	1	2	yes	—
94	C₂²×C₈	#41	1	2	yes	—
95	C₂²×C₈	#41	1	2	yes	—
96	C₂×C₄⋊C₄	—	2	4	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄⋊C₄	—	4	2	no	—
99	C₂×C₄⋊C₄	—	4	2	no	—
100	C₂×C₄⋊C₄	—	4	2	no	—
101	C₂×C₄⋊C₄	—	4	4	no	—
102	C₂×C₄⋊C₄	—	4	4	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₄⋊C₄	—	8	8	no	—
105	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
106	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	C₄⋊C₄	1	2	yes	—
109	C₂×C₄²	D₈	1	2	yes	—
110	C₂×C₄²	SD₁₆	1	2	yes	—
111	C₂×C₄²	Q₁₆	1	2	yes	—
112	M₃₂	—	2	4	no	—
113	M₃₂	—	2	4	no	—
114	M₃₂	—	2	4	no	—
115	M₃₂	C₄⋊C₄	1	8	yes	complemented
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	2	4	no	—
119	#108	—	2	8	no	—
120	#108	—	4	8	no	—
121	#108	—	4	4	no	—
122	#107	—	2	8	no	—
123	#107	—	4	4	no	—
124	#107	—	4	4	no	—
125	#107	—	4	4	no	—
126	C₄⋊C₈	—	4	8	no	—
127	C₄×C₈	C₄²	1	2	yes	—
128	C₄×C₈	C₄²	1	2	yes	—
129	C₄×C₈	C₂²⋊C₄	1	2	yes	—
130	C₄×C₈	C₂²⋊C₄	1	2	yes	—
131	C₄×C₈	C₄⋊C₄	1	2	yes	—
132	C₄×C₈	C₄⋊C₄	1	2	yes	—
133	C₄×C₈	C₄⋊C₄	1	2	yes	—
134	C₄×C₈	C₄⋊C₄	1	2	yes	—
135	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	—	8	8	no	—
140	C₂²×C₄	—	8	2	no	—
141	C₂²×C₄	—	8	2	no	—
142	C₂²×C₄	#102	1	2	yes	—
143	C₂²×C₄	#102	1	2	yes	—
144	C₂²×C₄	#103	1	4	yes	—
145	C₂²×C₄	#106	1	4	yes	—
146	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	#124	1	1	yes	characteristic
150	C₂²×C₄	#124	1	1	yes	characteristic
151	C₂²×C₄	#127	1	1	yes	characteristic
152	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
157	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
158	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
159	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
160	M₁₆	—	4	4	no	—
161	M₁₆	—	4	4	no	—
162	M₁₆	—	8	2	no	—
163	M₁₆	—	8	2	no	—
164	C₂×C₈	—	4	4	no	—
165	C₂×C₈	—	8	2	no	—
166	C₂×C₈	—	8	2	no	—
167	C₂×C₈	#95	1	2	yes	—
168	C₂×C₈	#95	1	2	yes	—
169	C₂×C₈	#102	1	2	yes	—
170	C₂×C₈	#102	1	2	yes	—
171	C₂×C₈	#102	1	4	yes	—
172	C₂×C₈	#103	1	2	yes	—
173	C₂×C₈	#103	1	2	yes	—
174	C₂×C₈	#103	1	4	yes	—
175	C₂×C₈	#106	1	4	yes	—
176	C₂×C₈	#107	1	4	yes	—
177	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
180	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₈	#117	1	2	yes	—
183	C₂×C₈	C₄×D₄	1	2	yes	—
184	C₂×C₈	C₄×D₄	1	2	yes	—
185	C₂×C₈	C₄×Q₈	1	2	yes	—
186	C₂×C₈	C₄×Q₈	1	2	yes	—
187	C₂×C₈	#121	1	2	yes	—
188	C₂×C₈	#121	1	2	yes	—
189	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
190	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
191	C₂×C₈	#125	1	2	yes	—
192	C₂×C₈	C₄⋊Q₈	1	2	yes	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	8	8	no	—
196	C₄⋊C₄	—	8	8	no	—
197	C₄⋊C₄	—	8	2	no	—
198	C₄⋊C₄	—	8	2	no	—
199	C₄²	#95	1	2	yes	—
200	C₄²	#95	1	2	yes	—
201	C₄²	#102	1	4	yes	—
202	C₄²	#103	1	2	yes	—
203	C₄²	#103	1	2	yes	—
204	C₄²	#107	1	4	yes	—
205	C₁₆	—	2	8	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	C₂×#102	1	1	yes	characteristic
209	C₂³	C₂×#103	1	1	yes	characteristic
210	C₂³	C₂×#106	1	1	yes	characteristic
211	C₂³	#486	1	1	yes	characteristic
212	C₂³	#487	1	1	yes	characteristic
213	C₂³	#492	1	1	yes	characteristic
214	C₂³	#498	1	1	yes	characteristic
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	8	8	no	—
220	C₂×C₄	—	8	4	no	—
221	C₂×C₄	—	16	4	no	—
222	C₂×C₄	—	16	4	no	—
223	C₂×C₄	#340	1	4	yes	—
224	C₂×C₄	#340	1	4	yes	—
225	C₂×C₄	#361	1	4	yes	—
226	C₂×C₄	#362	1	4	yes	—
227	C₂×C₄	C₂×#95	1	1	yes	characteristic
228	C₂×C₄	#382	1	1	yes	characteristic
229	C₂×C₄	#384	1	1	yes	characteristic
230	C₂×C₄	#390	1	1	yes	characteristic
231	C₂×C₄	#390	1	1	yes	characteristic
232	C₂×C₄	#391	1	1	yes	characteristic
233	C₂×C₄	#391	1	1	yes	characteristic
234	C₂×C₄	C₂×#102	1	1	yes	characteristic
235	C₂×C₄	C₂×#102	1	1	yes	characteristic
236	C₂×C₄	C₂×#102	1	1	yes	characteristic
237	C₂×C₄	C₂×#103	1	1	yes	characteristic
238	C₂×C₄	C₂×#103	1	1	yes	characteristic
239	C₂×C₄	C₂×#103	1	1	yes	characteristic
240	C₂×C₄	C₂×#106	1	1	yes	characteristic
241	C₂×C₄	C₂×#107	1	1	yes	characteristic
242	C₂×C₄	C₂×#107	1	1	yes	characteristic
243	C₂×C₄	#486	1	1	yes	characteristic
244	C₂×C₄	#486	1	1	yes	characteristic
245	C₂×C₄	#486	1	1	yes	characteristic
246	C₂×C₄	#487	1	1	yes	characteristic
247	C₂×C₄	#487	1	1	yes	characteristic
248	C₂×C₄	#487	1	1	yes	characteristic
249	C₂×C₄	#492	1	1	yes	characteristic
250	C₂×C₄	#493	1	1	yes	characteristic
251	C₂×C₄	#494	1	1	yes	characteristic
252	C₂×C₄	#498	1	1	yes	characteristic
253	C₂×C₄	#500	1	1	yes	characteristic
254	C₂×C₄	#500	1	1	yes	characteristic
255	C₈	—	8	4	no	—
256	C₈	#340	1	4	yes	—
257	C₈	#340	1	4	yes	—
258	C₈	C₄×C₄⋊C₄	1	2	yes	—
259	C₈	#380	1	2	yes	—
260	C₈	#389	1	2	yes	—
261	C₈	#389	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	C₂×#340	1	1	yes	characteristic
271	C₂²	#1834	1	1	yes	characteristic
272	C₂²	#1844	1	1	yes	characteristic
273	C₂²	#1956	1	1	yes	characteristic
274	C₂²	#1957	1	1	yes	characteristic
275	C₂²	#1990	1	1	yes	characteristic
276	C₂²	#1990	1	1	yes	characteristic
277	C₂²	C₂×#361	1	2	yes	—
278	C₂²	C₂×#362	1	2	yes	—
279	C₄	—	2	4	no	—
280	C₄	—	16	4	no	—
281	C₄	#1387	1	8	yes	complemented
282	C₄	C₄⋊#95	1	1	yes	characteristic
283	C₄	C₂×#340	1	1	yes	characteristic
284	C₄	#1834	1	1	yes	characteristic
285	C₄	#1845	1	1	yes	characteristic
286	C₄	#1956	1	1	yes	characteristic
287	C₄	#1957	1	1	yes	characteristic
288	C₄	#1990	1	1	yes	characteristic
289	C₄	#1990	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	#10841	1	1	yes	characteristic
295	C₂	C₂×#1387	1	2	yes	—
296	C₂	#13212	1	1	yes	characteristic
297	C₂	#13213	1	1	yes	characteristic
298	C₂	#13290	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. C₄×M₃₂	C₄	semidirect
18. #1848	C₄	semidirect
115. M₃₂	C₄⋊C₄	semidirect
281. C₄	#1387	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
C₂×#1848	2
C₂×#1845	2
C₂×#1387	2