Subgroups — #226377

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22821	C₂	1	1	yes	maximal, characteristic
2	#22722	C₂	1	1	yes	maximal, characteristic
3	#21339	C₂	1	1	yes	maximal, characteristic
4	#9606	C₂	1	2	yes	maximal
5	#9602	C₂	1	1	yes	maximal, characteristic
6	#9601	C₂	1	2	yes	maximal
7	#9601	C₂	1	1	yes	maximal, characteristic
8	#8812	C₂	1	2	yes	maximal
9	#8806	C₂	1	1	yes	maximal, characteristic
10	#8279	C₂	1	2	yes	maximal
11	C₄×#334	C₂	1	1	yes	maximal, characteristic
12	C₂×#446	—	2	1	no	—
13	C₄²×Q₈	C₂²	1	1	yes	characteristic
14	#1984	—	2	1	no	—
15	#1984	C₄	1	1	yes	characteristic
16	#1984	C₄	1	1	yes	characteristic
17	#1940	—	2	2	no	—
18	#1939	—	2	1	no	—
19	#1939	C₄	1	1	yes	characteristic
20	#1939	C₄	1	1	yes	characteristic
21	#1916	—	2	2	no	—
22	#1916	C₄	1	2	yes	—
23	#1916	C₄	1	2	yes	—
24	#1842	C₂²	1	1	yes	characteristic
25	#1842	C₂²	1	1	yes	characteristic
26	#1842	C₂²	1	2	yes	—
27	#1831	C₂²	1	2	yes	—
28	C₈×C₄⋊C₄	—	2	1	no	—
29	#1767	C₂²	1	2	yes	—
30	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₈	C₂²	1	2	yes	—
32	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₈	C₂²	1	2	yes	—
35	#1744	C₂²	1	2	yes	—
36	C₄²×C₈	C₂²	1	1	yes	characteristic
37	C₂×#335	C₂²	1	2	yes	—
38	C₂×#335	C₂²	1	2	yes	—
39	C₂×#335	C₂²	1	2	yes	—
40	C₂×#334	C₂²	1	1	yes	characteristic
41	C₂×#334	C₂²	1	2	yes	—
42	C₂×#334	C₂²	1	1	yes	characteristic
43	#1277	C₂²	1	1	yes	characteristic
44	#1277	C₂²	1	1	yes	characteristic
45	#1277	C₂²	1	2	yes	—
46	#1277	C₂²	1	2	yes	—
47	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
49	C₂×C₄×Q₈	Q₈	1	1	yes	characteristic
50	#446	—	4	4	no	—
51	C₂×C₄⋊C₈	—	2	1	no	—
52	C₂×C₄⋊C₈	—	2	2	no	—
53	C₂×C₄⋊C₈	—	2	1	no	—
54	C₂×C₄⋊C₈	—	2	1	no	—
55	C₂×C₄⋊C₈	—	2	2	no	—
56	C₂×C₄⋊C₈	—	2	1	no	—
57	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₈	C₂³	1	2	yes	—
64	C₂×C₄⋊C₈	C₂³	1	2	yes	—
65	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×#97	—	2	2	no	—
68	C₂×C₄×C₈	—	2	1	no	—
69	C₂×C₄×C₈	—	2	1	no	—
70	C₂×C₄×C₈	—	2	2	no	—
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	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂³	1	2	yes	—
76	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
79	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₄×C₄⋊C₄	D₄	1	2	yes	—
81	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
82	C₄×C₄⋊C₄	Q₈	1	1	yes	characteristic
83	C₄×C₄⋊C₄	Q₈	1	2	yes	—
84	C₄³	C₂³	1	2	yes	—
85	C₄³	C₂³	1	1	yes	characteristic
86	#336	—	2	2	no	—
87	#336	—	2	2	no	—
88	#336	—	2	1	no	—
89	#336	—	2	2	no	—
90	#336	—	2	2	no	—
91	#336	—	2	1	no	—
92	#336	C₂×C₄	1	2	yes	—
93	#336	C₂×C₄	1	2	yes	—
94	#336	C₂×C₄	1	2	yes	—
95	#336	C₂×C₄	1	2	yes	—
96	#336	C₂×C₄	1	2	yes	—
97	#336	C₂×C₄	1	2	yes	—
98	#336	C₂×C₄	1	2	yes	—
99	#335	—	2	4	no	—
100	#335	—	4	2	no	—
101	#335	—	4	2	no	—
102	#334	—	2	2	no	—
103	#334	—	4	2	no	—
104	#334	Q₈	1	4	yes	complemented
105	C₂²×Q₈	C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	C₂²×C₄	1	2	yes	—
113	C₂²×C₈	C₂²×C₄	1	2	yes	—
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	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₄×Q₈	—	2	4	no	—
119	C₄×Q₈	—	2	2	no	—
120	C₄×Q₈	—	2	2	no	—
121	C₄×Q₈	—	2	2	no	—
122	C₄×Q₈	—	2	2	no	—
123	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
125	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
127	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
128	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄²	C₂²×C₄	1	2	yes	—
130	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄²	C₂²×C₄	1	2	yes	—
132	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄²	C₂×D₄	1	2	yes	—
134	C₂×C₄²	C₂×D₄	1	2	yes	—
135	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
138	C₂×C₄²	C₂×Q₈	1	2	yes	—
139	C₂×C₄²	C₂×Q₈	1	2	yes	—
140	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
141	C₂×C₄²	C₂⁴	1	1	yes	characteristic
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	1	no	—
146	C₄⋊C₈	—	2	1	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	1	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	4	2	no	—
156	C₄⋊C₈	—	4	4	no	—
157	C₄⋊C₈	—	4	2	no	—
158	C₄⋊C₈	—	4	2	no	—
159	C₄⋊C₈	—	8	2	no	—
160	C₄⋊C₈	—	8	1	no	—
161	C₄⋊C₈	—	8	1	no	—
162	C₄⋊C₈	C₂×Q₈	1	2	yes	—
163	C₄⋊C₈	C₂×Q₈	1	2	yes	—
164	#97	—	4	4	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	4	1	no	—
173	C₄×C₈	—	4	2	no	—
174	C₄×C₈	—	4	2	no	—
175	C₄×C₈	—	8	2	no	—
176	C₄×C₈	C₂×Q₈	1	2	yes	—
177	C₄×C₈	#41	1	2	yes	—
178	#95	C₄⋊C₄	1	2	yes	—
179	#95	C₄⋊C₄	1	2	yes	—
180	#95	C₄⋊C₄	1	2	yes	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
185	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
186	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
188	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
191	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
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	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	C₄×Q₈	1	2	yes	—
229	C₂×C₈	C₄×Q₈	1	2	yes	—
230	C₂×C₈	#131	1	2	yes	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	#108	1	4	yes	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	4	2	no	—
253	C₄²	#131	1	2	yes	—
254	C₄²	#131	1	2	yes	—
255	C₄²	AGL(1,ℤ/8)	1	2	yes	—
256	C₄²	#137	1	2	yes	—
257	C₄²	C₂²×Q₈	1	1	yes	characteristic
258	C₄²	C₂×#41	1	1	yes	characteristic
259	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
260	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
261	C₂³	C₂²×C₄⋊C₄	1	1	yes	characteristic
262	Q₈	—	4	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	#424	1	2	yes	—
284	C₂×C₄	#424	1	2	yes	—
285	C₂×C₄	#428	1	2	yes	—
286	C₂×C₄	#428	1	2	yes	—
287	C₂×C₄	C₂×#108	1	4	yes	—
288	C₂×C₄	C₂×#108	1	2	yes	—
289	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
290	C₂×C₄	#520	1	1	yes	characteristic
291	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
292	C₂×C₄	#550	1	1	yes	characteristic
293	C₂×C₄	#557	1	1	yes	characteristic
294	C₂×C₄	Q₈²	1	1	yes	characteristic
295	C₂×C₄	C₂×#131	1	1	yes	characteristic
296	C₂×C₄	C₂×#131	1	1	yes	characteristic
297	C₂×C₄	C₂×#131	1	1	yes	characteristic
298	C₂×C₄	#568	1	1	yes	characteristic
299	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₂×C₄	C₂×#137	1	1	yes	characteristic
301	C₂×C₄	#576	1	1	yes	characteristic
302	C₂×C₄	#578	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	8	1	no	—
312	C₈	—	8	1	no	—
313	C₈	—	8	2	no	—
314	C₈	—	8	2	no	—
315	C₂²	Q₈×C₄⋊C₄	1	1	yes	characteristic
316	C₂²	C₂×#424	1	1	yes	characteristic
317	C₂²	#2906	1	1	yes	characteristic
318	C₂²	C₂×#428	1	1	yes	characteristic
319	C₂²	#2913	1	1	yes	characteristic
320	C₂²	C₂²×#108	1	1	yes	characteristic
321	C₂²	#2916	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	4	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	1	no	—
331	C₄	#1574	1	4	yes	—
332	C₄	#2960	1	1	yes	characteristic
333	C₄	#2962	1	1	yes	characteristic
334	C₄	#2993	1	1	yes	characteristic
335	C₄	#2994	1	1	yes	characteristic
336	C₄	#3385	1	1	yes	characteristic
337	C₄	#3386	1	1	yes	characteristic
338	C₄	#3398	1	1	yes	characteristic
339	C₄	#3399	1	1	yes	characteristic
340	C₂	C₂×#1574	1	1	yes	characteristic
341	C₂	#17812	1	1	yes	characteristic
342	C₂	#17830	1	1	yes	characteristic
343	C₂	#17836	1	1	yes	characteristic
344	C₂	#22721	1	1	yes	characteristic
345	C₂	#22728	1	1	yes	characteristic
346	C₂	Q₈×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂²×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
104. #334	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#22821	2
#22722	2
#21339	2
#9606	2
#9602	2
#9601	2
#8812	2
#8806	2
#8279	2
C₄×#334	2