Subgroups — #226372

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22722	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#21339	C₂	1	1	yes	maximal, characteristic
4	#9604	C₂	1	2	yes	maximal
5	#9601	C₂	1	2	yes	maximal
6	#9601	C₂	1	1	yes	maximal, characteristic
7	#9601	C₂	1	1	yes	maximal, characteristic
8	#8810	C₂	1	2	yes	maximal
9	#8806	C₂	1	1	yes	maximal, characteristic
10	#8278	C₂	1	2	yes	maximal
11	C₄×#335	C₂	1	1	yes	maximal, characteristic
12	C₂×#446	—	2	1	no	—
13	C₂×C₈×Q₈	—	2	1	no	—
14	C₄²×Q₈	C₂²	1	1	yes	characteristic
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	—	2	2	no	—
20	#1939	—	2	1	no	—
21	#1939	C₄	1	1	yes	characteristic
22	#1939	C₄	1	1	yes	characteristic
23	#1916	C₄	1	2	yes	—
24	#1916	C₄	1	2	yes	—
25	#1842	C₂²	1	1	yes	characteristic
26	#1842	C₂²	1	1	yes	characteristic
27	#1842	C₂²	1	2	yes	—
28	#1842	C₂²	1	2	yes	—
29	#1767	C₂²	1	2	yes	—
30	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₈	C₂²	1	2	yes	—
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	1	yes	characteristic
38	C₂×#335	C₂²	1	2	yes	—
39	C₂×#335	C₂²	1	2	yes	—
40	C₂×#335	C₂²	1	2	yes	—
41	C₂×#335	C₂²	1	2	yes	—
42	C₂×#335	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	C₈×Q₈	—	4	4	no	—
52	C₂×C₄⋊C₈	—	2	1	no	—
53	C₂×C₄⋊C₈	—	2	2	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	—	2	2	no	—
58	C₂×C₄⋊C₈	—	2	2	no	—
59	C₂×C₄⋊C₈	—	2	1	no	—
60	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₈	C₂³	1	2	yes	—
67	C₂×C₄⋊C₈	C₂³	1	2	yes	—
68	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×#97	—	2	2	no	—
71	C₂×C₄×C₈	—	2	1	no	—
72	C₂×C₄×C₈	—	2	2	no	—
73	C₂×C₄×C₈	—	2	1	no	—
74	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂³	1	2	yes	—
80	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
82	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
84	C₄×C₄⋊C₄	D₄	1	2	yes	—
85	C₄×C₄⋊C₄	Q₈	1	1	yes	characteristic
86	C₄×C₄⋊C₄	Q₈	1	2	yes	—
87	C₄³	C₂³	1	1	yes	characteristic
88	C₄³	C₂³	1	2	yes	—
89	#336	—	2	1	no	—
90	#336	—	2	2	no	—
91	#336	—	2	2	no	—
92	#336	—	2	1	no	—
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	#336	C₂×C₄	1	2	yes	—
100	#335	—	2	2	no	—
101	#335	—	2	4	no	—
102	#335	—	4	2	no	—
103	#335	—	4	2	no	—
104	#335	—	4	2	no	—
105	#335	Q₈	1	4	yes	complemented
106	C₂²×Q₈	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₂²×C₄	1	2	yes	—
114	C₂²×C₈	C₂²×C₄	1	2	yes	—
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	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₄×Q₈	—	2	2	no	—
120	C₄×Q₈	—	2	2	no	—
121	C₄×Q₈	—	4	2	no	—
122	C₄×Q₈	D₈	1	4	yes	—
123	C₄×Q₈	Q₁₆	1	4	yes	—
124	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
126	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
128	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
129	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄²	C₂²×C₄	1	2	yes	—
131	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₄	1	2	yes	—
133	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
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×D₄	1	2	yes	—
138	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
139	C₂×C₄²	C₂×Q₈	1	2	yes	—
140	C₂×C₄²	C₂×Q₈	1	2	yes	—
141	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
142	C₂×C₄²	C₂⁴	1	1	yes	characteristic
143	C₄⋊C₈	—	2	4	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	1	no	—
147	C₄⋊C₈	—	2	1	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	4	2	no	—
157	C₄⋊C₈	—	4	2	no	—
158	C₄⋊C₈	—	4	4	no	—
159	C₄⋊C₈	—	4	2	no	—
160	C₄⋊C₈	—	4	4	no	—
161	C₄⋊C₈	—	8	1	no	—
162	C₄⋊C₈	—	8	2	no	—
163	C₄⋊C₈	—	8	2	no	—
164	C₄⋊C₈	—	8	1	no	—
165	C₄⋊C₈	C₂×Q₈	1	2	yes	—
166	C₄⋊C₈	C₂×Q₈	1	2	yes	—
167	#97	—	4	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	—	4	1	no	—
176	C₄×C₈	—	4	2	no	—
177	C₄×C₈	—	4	4	no	—
178	C₄×C₈	—	4	2	no	—
179	C₄×C₈	C₂×Q₈	1	2	yes	—
180	C₄×C₈	#41	1	2	yes	—
181	#95	C₄⋊C₄	1	2	yes	—
182	#95	C₄⋊C₄	1	2	yes	—
183	#95	C₄⋊C₄	1	2	yes	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	#107	1	4	yes	—
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	2	yes	—
190	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
193	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
194	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	1	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	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	C₄×Q₈	1	2	yes	—
232	C₂×C₈	C₄×Q₈	1	2	yes	—
233	C₂×C₈	#131	1	2	yes	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	#107	1	8	yes	—
239	C₄⋊C₄	#107	1	4	yes	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	1	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	2	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	4	2	no	—
254	C₄²	#131	1	2	yes	—
255	C₄²	#131	1	2	yes	—
256	C₄²	C₂×D₈	1	2	yes	—
257	C₄²	C₂×D₈	1	4	yes	—
258	C₄²	C₂×Q₁₆	1	4	yes	—
259	C₄²	C₂×Q₁₆	1	2	yes	—
260	C₄²	C₂²×Q₈	1	1	yes	characteristic
261	C₄²	C₂×#41	1	1	yes	characteristic
262	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
263	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
264	C₂³	C₂²×C₄⋊C₄	1	1	yes	characteristic
265	Q₈	—	2	8	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	#424	1	2	yes	—
286	C₂×C₄	#424	1	2	yes	—
287	C₂×C₄	C₂×#107	1	2	yes	—
288	C₂×C₄	C₂×#107	1	4	yes	—
289	C₂×C₄	C₂×#107	1	4	yes	—
290	C₂×C₄	C₂×#107	1	2	yes	—
291	C₂×C₄	#430	1	2	yes	—
292	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
293	C₂×C₄	#520	1	1	yes	characteristic
294	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
295	C₂×C₄	#550	1	1	yes	characteristic
296	C₂×C₄	#557	1	1	yes	characteristic
297	C₂×C₄	Q₈²	1	1	yes	characteristic
298	C₂×C₄	C₂×#131	1	1	yes	characteristic
299	C₂×C₄	C₂×#131	1	1	yes	characteristic
300	C₂×C₄	C₂×#131	1	1	yes	characteristic
301	C₂×C₄	#568	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
304	C₂×C₄	#576	1	1	yes	characteristic
305	C₂×C₄	#578	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	8	1	no	—
315	C₈	—	8	2	no	—
316	C₈	—	8	1	no	—
317	C₈	—	8	2	no	—
318	C₂²	Q₈×C₄⋊C₄	1	1	yes	characteristic
319	C₂²	C₂×#424	1	1	yes	characteristic
320	C₂²	#2906	1	1	yes	characteristic
321	C₂²	C₂²×#107	1	1	yes	characteristic
322	C₂²	#2913	1	1	yes	characteristic
323	C₂²	C₂×#430	1	1	yes	characteristic
324	C₂²	#2916	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	4	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	2	no	—
334	C₄	#1571	1	4	yes	—
335	C₄	#2960	1	1	yes	characteristic
336	C₄	#2962	1	1	yes	characteristic
337	C₄	#2993	1	1	yes	characteristic
338	C₄	#2994	1	1	yes	characteristic
339	C₄	Q₈×D₈	1	1	yes	characteristic
340	C₄	Q₈×Q₁₆	1	1	yes	characteristic
341	C₄	#3391	1	1	yes	characteristic
342	C₄	#3395	1	1	yes	characteristic
343	C₂	C₂×#1571	1	1	yes	characteristic
344	C₂	#17812	1	1	yes	characteristic
345	C₂	#17828	1	1	yes	characteristic
346	C₂	#17839	1	1	yes	characteristic
347	C₂	#22721	1	1	yes	characteristic
348	C₂	Q₈×#107	1	1	yes	characteristic
349	C₂	#22733	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
105. #335	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#22722	2
Q₈×C₄⋊C₈	2
#21339	2
#9604	2
#9601	2
#8810	2
#8806	2
#8278	2
C₄×#335	2