Subgroups — #225496

Order 512.

664 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#443	C₂	1	1	yes	maximal, characteristic
2	#17858	C₂	1	2	yes	maximal
3	#17789	C₂	1	2	yes	maximal, complemented
4	#9607	C₂	1	2	yes	maximal, complemented
5	#8840	C₂	1	4	yes	maximal, complemented
6	#8807	C₂	1	1	yes	maximal, characteristic, complemented
7	#8605	C₂	1	2	yes	maximal, complemented
8	#8266	C₂	1	1	yes	maximal, characteristic, complemented
9	#2988	C₂²	1	4	yes	—
10	C₂×#443	C₄	1	1	yes	characteristic
11	C₂×#443	C₄	1	1	yes	characteristic
12	C₄×#135	C₂²	1	1	yes	characteristic
13	C₄×#131	C₂²	1	2	yes	—
14	#1831	C₂²	1	2	yes	—
15	C₄×#108	C₂²	1	1	yes	characteristic
16	#1772	C₄	1	1	yes	characteristic
17	#1772	C₄	1	1	yes	characteristic
18	#1764	C₄	1	2	yes	—
19	#1764	C₄	1	2	yes	—
20	C₄×#104	C₂²	1	2	yes	—
21	C₄²×C₈	C₂²	1	1	yes	characteristic
22	#1580	—	2	2	no	—
23	#1579	—	2	2	no	—
24	C₈×SD₁₆	—	2	4	no	—
25	#1565	—	2	4	no	—
26	#1565	C₂²	1	4	yes	—
27	#1564	—	2	2	no	—
28	#1564	C₂²	1	1	yes	characteristic
29	#1564	C₂²	1	1	yes	characteristic
30	C₂×#335	—	2	2	no	—
31	C₂×#335	C₂²	1	2	yes	—
32	C₂×#334	C₂²	1	2	yes	—
33	#1477	C₂²	1	4	yes	—
34	#1450	C₂²	1	2	yes	—
35	#1283	C₂²	1	4	yes	—
36	#1276	C₂²	1	2	yes	—
37	C₂×#135	C₂×C₄	1	1	yes	characteristic
38	C₂×#131	C₂×C₄	1	2	yes	—
39	C₂×#131	C₂×C₄	1	2	yes	—
40	C₄×#41	C₂³	1	2	yes	—
41	C₈×Q₈	—	2	4	no	—
42	#443	—	2	4	no	—
43	#443	C₈	1	8	yes	complemented
44	C₄×Q₁₆	D₄	1	1	yes	characteristic
45	C₄×Q₁₆	Q₈	1	1	yes	characteristic
46	C₄×SD₁₆	D₄	1	2	yes	—
47	C₄×SD₁₆	Q₈	1	2	yes	—
48	C₄×D₈	D₄	1	1	yes	characteristic
49	C₄×D₈	Q₈	1	1	yes	characteristic
50	C₈×D₄	—	2	4	no	—
51	C₈×D₄	—	4	4	no	—
52	#431	—	2	4	no	—
53	#431	C₂³	1	2	yes	—
54	#431	C₂³	1	2	yes	—
55	C₂×#108	C₂×C₄	1	1	yes	characteristic
56	C₂×#108	C₂×C₄	1	1	yes	characteristic
57	#427	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₈	—	2	4	no	—
59	C₂×C₄⋊C₈	C₂³	1	4	yes	—
60	C₂×#104	C₂×C₄	1	2	yes	—
61	C₂×#104	C₂×C₄	1	2	yes	—
62	#416	C₂×C₄	1	2	yes	—
63	#405	—	2	2	no	—
64	#405	—	2	2	no	—
65	#405	C₂×C₄	1	2	yes	—
66	#405	C₂×C₄	1	2	yes	—
67	C₄×M₁₆	—	2	2	no	—
68	C₄×M₁₆	C₂³	1	2	yes	—
69	C₂×C₄×C₈	—	2	2	no	—
70	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
76	C₄³	C₂³	1	1	yes	characteristic
77	#341	C₂×C₄	1	2	yes	—
78	#339	C₂×C₄	1	2	yes	—
79	#339	C₂×C₄	1	2	yes	—
80	#336	—	4	2	no	—
81	#335	—	2	4	no	—
82	#335	—	2	2	no	—
83	#335	—	2	2	no	—
84	#335	—	2	1	no	—
85	#335	D₄	1	2	yes	—
86	#334	—	2	4	no	—
87	#334	—	2	2	no	—
88	#334	—	2	2	no	—
89	#334	—	2	1	no	—
90	#334	D₄	1	2	yes	—
91	Q₈⋊C₈	—	2	4	no	—
92	D₄⋊C₈	—	2	4	no	—
93	#322	—	2	2	no	—
94	C₈²	—	2	2	no	—
95	C₂×#41	C₂²×C₄	1	2	yes	—
96	#135	C₂×C₈	1	4	yes	—
97	#135	M₁₆	1	4	yes	—
98	C₂×Q₁₆	C₄⋊C₄	1	1	yes	characteristic
99	C₂×SD₁₆	C₄⋊C₄	1	2	yes	—
100	C₂×D₈	C₄⋊C₄	1	1	yes	characteristic
101	#131	—	2	4	no	—
102	#131	—	2	4	no	—
103	#131	—	4	2	no	—
104	#131	C₂×C₈	1	8	yes	complemented
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	C₂²×C₄	1	2	yes	—
108	C₂×M₁₆	C₂²×C₄	1	2	yes	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	—	4	4	no	—
113	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₄×Q₈	C₂×D₄	1	2	yes	—
118	C₄×Q₈	C₂×Q₈	1	2	yes	—
119	C₄×D₄	—	2	2	no	—
120	C₄×D₄	—	2	2	no	—
121	C₄×D₄	C₂×D₄	1	2	yes	—
122	C₄×D₄	C₂×Q₈	1	2	yes	—
123	#117	—	2	2	no	—
124	#117	C₂²×C₄	1	2	yes	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄²	C₂⁴	1	1	yes	characteristic
131	#108	—	2	2	no	—
132	#108	C₂×C₈	1	4	yes	complemented
133	#107	C₄⋊C₄	1	1	yes	characteristic
134	#107	C₄⋊C₄	1	1	yes	characteristic
135	#106	C₄⋊C₄	1	2	yes	—
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	4	4	no	—
140	C₄⋊C₈	C₂×D₄	1	4	yes	—
141	#104	—	2	4	no	—
142	#104	C₂×C₈	1	8	yes	complemented
143	#103	C₄⋊C₄	1	2	yes	—
144	#103	C₄⋊C₄	1	2	yes	—
145	#102	C₄⋊C₄	1	2	yes	—
146	#102	C₄⋊C₄	1	2	yes	—
147	C₂²⋊C₈	—	2	2	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	C₂²⋊C₈	—	4	4	no	—
150	#97	—	2	2	no	—
151	#97	—	2	2	no	—
152	#97	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	4	1	no	—
161	C₄×C₈	—	4	4	no	—
162	C₄×C₈	C₂×C₈	1	4	yes	complemented
163	C₄×C₈	C₂×D₄	1	1	yes	characteristic
164	C₄×C₈	C₂×D₄	1	1	yes	characteristic
165	C₄×C₈	C₂×D₄	1	1	yes	characteristic
166	C₄×C₈	C₂×D₄	1	1	yes	characteristic
167	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
168	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
169	C₄×C₈	#41	1	1	yes	characteristic
170	C₄×C₈	#41	1	1	yes	characteristic
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	#41	C₂²×C₈	1	4	yes	—
174	#41	C₂×M₁₆	1	4	yes	—
175	C₂×Q₈	C₂×C₄⋊C₄	1	2	yes	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	C₂×C₄⋊C₄	1	2	yes	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
184	C₂²×C₄	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂²×C₄	2₋¹⁺⁴	1	1	yes	characteristic
186	Q₁₆	C₄⋊C₈	1	4	yes	—
187	SD₁₆	C₄⋊C₈	1	8	yes	—
188	D₈	C₄⋊C₈	1	4	yes	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	8	no	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	2	2	no	—
193	M₁₆	—	4	4	no	—
194	M₁₆	C₂²×C₈	1	4	yes	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	4	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	8	4	no	—
207	C₂×C₈	—	16	4	no	—
208	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₈	C₄×D₄	1	1	yes	characteristic
213	C₂×C₈	C₄×D₄	1	1	yes	characteristic
214	C₂×C₈	C₄×D₄	1	1	yes	characteristic
215	C₂×C₈	C₄×D₄	1	1	yes	characteristic
216	C₂×C₈	C₂²×C₈	1	2	yes	—
217	C₂×C₈	C₂²×C₈	1	2	yes	—
218	C₂×C₈	C₂×M₁₆	1	2	yes	—
219	C₂×C₈	#131	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	C₂×C₄⋊C₄	1	2	yes	—
223	C₄⋊C₄	C₂×C₄⋊C₄	1	2	yes	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	C₂²×C₈	1	2	yes	—
236	C₄²	C₂²×D₄	1	1	yes	characteristic
237	C₄²	C₂²×D₄	1	1	yes	characteristic
238	C₄²	C₂²×Q₈	1	1	yes	characteristic
239	C₄²	C₂×#41	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	#520	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	C₂×C₄⋊C₈	1	4	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	4	no	—
246	D₄	—	2	2	no	—
247	D₄	C₂×C₄⋊C₈	1	4	yes	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
266	C₂×C₄	D₄²	1	1	yes	characteristic
267	C₂×C₄	#547	1	1	yes	characteristic
268	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
269	C₂×C₄	#554	1	1	yes	characteristic
270	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
271	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
272	C₂×C₄	C₂×#131	1	1	yes	characteristic
273	C₂×C₄	#568	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	4	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	4	2	no	—
280	C₈	—	4	4	no	—
281	C₈	—	8	2	no	—
282	C₈	—	8	4	no	—
283	C₈	C₂×C₄⋊C₈	1	2	yes	—
284	C₈	C₂×C₄⋊C₈	1	2	yes	—
285	C₈	C₈×D₄	1	2	yes	—
286	C₈	C₈×D₄	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	D₄×C₄⋊C₄	1	1	yes	characteristic
293	C₂²	#2988	1	1	yes	characteristic
294	C₂²	#2990	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	2	no	—
303	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
304	C₄	#2906	1	1	yes	characteristic
305	C₄	C₂×C₈×D₄	1	1	yes	characteristic
306	C₄	D₄×M₁₆	1	1	yes	characteristic
307	C₄	#3290	1	2	yes	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	4	4	no	—
311	C₂	D₄×C₄⋊C₈	1	1	yes	characteristic
312	C₂	#22691	1	1	yes	characteristic
313	C₂	#22694	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #17789	C₂	semidirect
4. #9607	C₂	semidirect
5. #8840	C₂	semidirect
6. #8807	C₂	semidirect
7. #8605	C₂	semidirect
8. #8266	C₂	semidirect
43. #443	C₈	semidirect
104. #131	C₂×C₈	semidirect
132. #108	C₂×C₈	semidirect
142. #104	C₂×C₈	semidirect
162. C₄×C₈	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#443	2
#17858	2
#17789	2
#9607	2
#8840	2
#8807	2
#8605	2
#8266	2