Subgroups — #98863

Order 512.

438 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#8784	C₂	1	1	yes	maximal, characteristic
3	#8564	C₂	1	2	yes	maximal
4	C₄×#335	C₂	1	1	yes	maximal, characteristic
5	#7031	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
8	C₄²×Q₈	C₂²	1	1	yes	characteristic
9	#1842	C₂²	1	2	yes	—
10	C₈×C₄⋊C₄	—	2	1	no	—
11	C₈×C₄⋊C₄	—	2	1	no	—
12	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
14	C₄×C₄⋊C₈	—	2	1	no	—
15	C₄×C₄⋊C₈	C₂²	1	2	yes	—
16	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×C₈	—	2	1	no	—
18	C₄²×C₈	C₂²	1	1	yes	characteristic
19	C₂×#335	—	2	1	no	—
20	C₂×Q₈⋊C₈	—	2	2	no	—
21	#1279	—	2	2	no	—
22	#1277	—	2	1	no	—
23	#1277	—	2	1	no	—
24	#1277	—	2	1	no	—
25	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
27	C₂×C₄×Q₈	Q₈	1	1	yes	characteristic
28	C₈×Q₈	—	2	2	no	—
29	C₈×Q₈	—	2	2	no	—
30	C₈×Q₈	—	2	2	no	—
31	C₈×Q₈	C₈	1	4	yes	complemented
32	C₂×C₄⋊C₈	—	2	1	no	—
33	C₂×C₄⋊C₈	—	2	1	no	—
34	C₂×C₄⋊C₈	—	2	2	no	—
35	C₂×C₄⋊C₈	—	2	2	no	—
36	C₂×C₄⋊C₈	—	2	2	no	—
37	C₂×C₄⋊C₈	—	2	2	no	—
38	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	—	2	1	no	—
45	C₂×C₄×C₈	—	2	2	no	—
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	D₄	1	1	yes	characteristic
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₄×C₄⋊C₄	—	2	1	no	—
53	C₄×C₄⋊C₄	—	2	1	no	—
54	C₄×C₄⋊C₄	—	2	1	no	—
55	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
57	C₄×C₄⋊C₄	Q₈	1	1	yes	characteristic
58	C₄³	—	2	1	no	—
59	C₄³	C₂³	1	1	yes	characteristic
60	#336	—	2	1	no	—
61	#336	—	2	1	no	—
62	#336	—	2	1	no	—
63	#336	—	2	2	no	—
64	#335	—	2	4	no	—
65	Q₈⋊C₈	—	4	4	no	—
66	C₂²×Q₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂²×C₈	—	4	2	no	—
70	C₂²×C₈	—	4	2	no	—
71	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₄×Q₈	—	2	2	no	—
76	C₄×Q₈	—	2	2	no	—
77	C₄×Q₈	—	2	1	no	—
78	C₄×Q₈	—	2	1	no	—
79	C₄×Q₈	—	2	1	no	—
80	C₄×Q₈	—	2	1	no	—
81	C₄×Q₈	C₂×C₈	1	2	yes	—
82	C₄×Q₈	M₁₆	1	2	yes	—
83	C₄×Q₈	D₈	1	2	yes	—
84	C₄×Q₈	Q₁₆	1	2	yes	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	2	1	no	—
87	C₂×C₄⋊C₄	—	2	1	no	—
88	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄²	—	2	1	no	—
92	C₂×C₄²	—	2	1	no	—
93	C₂×C₄²	—	2	1	no	—
94	C₂×C₄²	—	2	1	no	—
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
102	C₂×C₄²	#41	1	1	yes	characteristic
103	C₂×C₄²	#41	1	1	yes	characteristic
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	2	4	no	—
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	4	2	no	—
110	C₄⋊C₈	—	4	2	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄⋊C₈	—	8	2	no	—
115	C₄⋊C₈	C₂×C₈	1	2	yes	—
116	C₄⋊C₈	M₁₆	1	2	yes	—
117	C₄×C₈	—	2	4	no	—
118	C₄×C₈	—	2	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	—	4	2	no	—
126	C₄×C₈	—	4	2	no	—
127	C₄×C₈	—	4	2	no	—
128	C₄×C₈	—	4	2	no	—
129	C₄×C₈	—	4	2	no	—
130	C₄×C₈	C₂×C₈	1	2	yes	—
131	C₄×C₈	M₁₆	1	2	yes	—
132	#95	—	2	1	no	—
133	#95	—	2	1	no	—
134	#95	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	C₄⋊C₈	1	2	yes	—
139	C₂×Q₈	#107	1	2	yes	—
140	C₂×Q₈	#108	1	2	yes	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	#117	1	1	yes	characteristic
148	C₂²×C₄	#121	1	1	yes	characteristic
149	C₂²×C₄	#121	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	8	2	no	—
169	C₂×C₈	—	8	2	no	—
170	C₂×C₈	—	8	2	no	—
171	C₂×C₈	—	8	2	no	—
172	C₂×C₈	—	8	2	no	—
173	C₂×C₈	C₂²⋊C₈	1	2	yes	—
174	C₂×C₈	C₂²⋊C₈	1	2	yes	—
175	C₂×C₈	#104	1	2	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
185	C₄⋊C₄	#107	1	2	yes	—
186	C₄⋊C₄	#108	1	2	yes	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	4	1	no	—
198	C₄²	—	4	2	no	—
199	C₄²	—	4	1	no	—
200	C₄²	—	4	1	no	—
201	C₄²	—	4	2	no	—
202	C₄²	C₂²×C₈	1	1	yes	characteristic
203	C₄²	C₂×M₁₆	1	1	yes	characteristic
204	C₄²	C₂×M₁₆	1	1	yes	characteristic
205	C₄²	C₂×M₁₆	1	1	yes	characteristic
206	C₄²	C₂×D₈	1	1	yes	characteristic
207	C₄²	C₂×Q₁₆	1	1	yes	characteristic
208	C₄²	#135	1	2	yes	—
209	C₄²	#135	1	1	yes	characteristic
210	C₄²	#135	1	1	yes	characteristic
211	C₄²	#137	1	2	yes	—
212	C₂³	#380	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	Q₈	#335	1	4	yes	complemented
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
239	C₂×C₄	#407	1	1	yes	characteristic
240	C₂×C₄	#416	1	1	yes	characteristic
241	C₂×C₄	#419	1	1	yes	characteristic
242	C₂×C₄	C₂×#104	1	1	yes	characteristic
243	C₂×C₄	#421	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
245	C₂×C₄	#423	1	1	yes	characteristic
246	C₂×C₄	C₂×#107	1	1	yes	characteristic
247	C₂×C₄	#427	1	1	yes	characteristic
248	C₂×C₄	C₂×#108	1	1	yes	characteristic
249	C₂×C₄	C₂×#108	1	1	yes	characteristic
250	C₂×C₄	#431	1	1	yes	characteristic
251	C₂×C₄	#432	1	1	yes	characteristic
252	C₂×C₄	#464	1	2	yes	—
253	C₂×C₄	#466	1	1	yes	characteristic
254	C₂×C₄	#466	1	1	yes	characteristic
255	C₂×C₄	#467	1	1	yes	characteristic
256	C₂×C₄	#467	1	1	yes	characteristic
257	C₂×C₄	#471	1	2	yes	—
258	C₂×C₄	#479	1	2	yes	—
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	2	no	—
265	C₈	—	8	4	no	—
266	C₈	—	16	2	no	—
267	C₂²	#1797	1	1	yes	characteristic
268	C₂²	#1805	1	1	yes	characteristic
269	C₂²	#1806	1	1	yes	characteristic
270	C₂²	#1808	1	1	yes	characteristic
271	C₂²	#1810	1	1	yes	characteristic
272	C₂²	#1814	1	1	yes	characteristic
273	C₂²	#1815	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	2	no	—
284	C₄	#1476	1	1	yes	characteristic
285	C₄	#1479	1	1	yes	characteristic
286	C₄	#1488	1	1	yes	characteristic
287	C₄	#1491	1	1	yes	characteristic
288	C₄	C₂×#335	1	1	yes	characteristic
289	C₄	#1564	1	1	yes	characteristic
290	C₄	#1567	1	1	yes	characteristic
291	C₄	#1568	1	1	yes	characteristic
292	C₄	#1682	1	2	yes	—
293	C₄	#1684	1	2	yes	—
294	C₂	#8566	1	1	yes	characteristic
295	C₂	#8596	1	1	yes	characteristic
296	C₂	#8604	1	1	yes	characteristic
297	C₂	#8649	1	1	yes	characteristic
298	C₂	#8661	1	1	yes	characteristic
299	C₂	#8667	1	1	yes	characteristic
300	C₂	#8670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#380
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
31. C₈×Q₈	C₈	semidirect
216. Q₈	#335	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Q₈	2
#8784	2
#8564	2
C₄×#335	2
#7031	2