Subgroups — #146325

Order 512.

569 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2110	C₂	1	1	yes	maximal, characteristic
2	C₂×#2110	C₂	1	1	yes	maximal, characteristic
3	#33324	C₂	1	1	yes	maximal, characteristic
4	#12554	C₂	1	1	yes	maximal, characteristic
5	C₂×#1378	C₂	1	1	yes	maximal, characteristic
6	#12549	C₂	1	1	yes	maximal, characteristic
7	#10711	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×#405	C₂²	1	1	yes	characteristic
13	C₂×#405	C₂²	1	1	yes	characteristic
14	C₂×#405	C₂²	1	1	yes	characteristic
15	#2110	—	2	2	no	—
16	#2110	—	2	2	no	—
17	#2110	—	2	2	no	—
18	#2110	C₄	1	4	yes	complemented
19	C₂×#347	—	2	2	no	—
20	C₂×#347	C₄	1	4	yes	complemented
21	#2108	—	2	2	no	—
22	#2108	C₄	1	4	yes	complemented
23	C₄×M₃₂	—	2	2	no	—
24	C₂×#344	C₄	1	1	yes	characteristic
25	C₂×#344	C₄	1	1	yes	characteristic
26	#1742	C₄	1	1	yes	characteristic
27	#1740	C₄	1	1	yes	characteristic
28	#1379	C₄	1	4	yes	complemented
29	#1378	C₄	1	4	yes	complemented
30	#1376	C₄	1	4	yes	complemented
31	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₈	C₂³	1	1	yes	characteristic
35	C₂×#117	C₂×C₄	1	1	yes	characteristic
36	C₂×#117	C₂×C₄	1	1	yes	characteristic
37	C₂×#117	C₂×C₄	1	1	yes	characteristic
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	—	2	1	no	—
41	C₂×M₃₂	—	2	1	no	—
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	—	2	1	no	—
44	C₂×M₃₂	—	2	1	no	—
45	C₂×M₃₂	—	2	1	no	—
46	C₂×M₃₂	—	2	1	no	—
47	C₂×M₃₂	—	2	1	no	—
48	C₂×M₃₂	C₂×C₄	1	2	yes	—
49	C₂×M₃₂	C₂×C₄	1	2	yes	—
50	C₂×M₃₂	C₂×C₄	1	2	yes	—
51	C₂×M₃₂	C₂×C₄	1	2	yes	—
52	C₂×M₃₂	C₂×C₄	1	2	yes	—
53	C₂×M₃₂	C₂×C₄	1	2	yes	—
54	C₂×M₃₂	C₂×C₄	1	2	yes	—
55	C₂×M₃₂	C₂×C₄	1	2	yes	—
56	C₂×M₃₂	C₂×C₄	1	2	yes	—
57	C₂×M₃₂	C₂×C₄	1	2	yes	—
58	C₂×M₃₂	D₄	1	2	yes	—
59	C₂×M₃₂	D₄	1	2	yes	—
60	C₂×M₃₂	D₄	1	2	yes	—
61	C₂×M₃₂	Q₈	1	2	yes	—
62	C₂²×C₁₆	—	2	1	no	—
63	C₂²×C₁₆	—	2	1	no	—
64	C₂²×C₁₆	C₂×C₄	1	2	yes	—
65	C₂²×C₁₆	C₂×C₄	1	2	yes	—
66	#405	—	2	1	no	—
67	#405	—	2	1	no	—
68	#405	—	2	1	no	—
69	#405	—	2	1	no	—
70	#405	—	2	1	no	—
71	#405	—	2	1	no	—
72	#405	—	2	1	no	—
73	#405	—	2	1	no	—
74	#405	C₂×C₄	1	2	yes	—
75	#405	C₂×C₄	1	2	yes	—
76	#405	C₂×C₄	1	2	yes	—
77	#405	C₂×C₄	1	2	yes	—
78	C₂×#97	—	2	1	no	—
79	C₂×#97	—	2	1	no	—
80	C₂×#97	C₂×C₄	1	2	yes	—
81	C₂×C₄×C₈	—	2	1	no	—
82	C₂×C₄×C₈	—	2	1	no	—
83	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
84	#347	—	2	8	no	—
85	#347	—	2	8	no	—
86	#346	—	2	4	no	—
87	C₄×C₁₆	—	2	4	no	—
88	#344	—	2	2	no	—
89	#344	—	2	2	no	—
90	#343	—	2	2	no	—
91	#342	—	2	2	no	—
92	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂×M₁₆	—	2	1	no	—
105	C₂×M₁₆	—	2	1	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	C₄²	1	2	yes	—
108	C₂×M₁₆	C₄²	1	2	yes	—
109	C₂×M₁₆	M₁₆	1	2	yes	—
110	C₂×M₁₆	M₁₆	1	2	yes	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	C₄²	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	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
124	C₂²×C₈	#41	1	1	yes	characteristic
125	C₂²×C₈	#41	1	1	yes	characteristic
126	C₂²×C₈	#41	1	1	yes	characteristic
127	C₂²×C₈	#41	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	#117	—	2	1	no	—
130	#117	—	2	1	no	—
131	#117	—	2	1	no	—
132	#117	—	2	1	no	—
133	#117	—	2	1	no	—
134	#117	—	2	1	no	—
135	#117	—	2	1	no	—
136	#117	—	2	1	no	—
137	#117	C₄²	1	2	yes	—
138	#117	C₄²	1	2	yes	—
139	#117	M₁₆	1	2	yes	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	C₂×C₂²⋊C₄	—	2	1	no	—
144	C₂×C₂²⋊C₄	—	2	1	no	—
145	C₂×C₂²⋊C₄	M₁₆	1	2	yes	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	C₄²	1	2	yes	—
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	M₃₂	—	2	2	no	—
154	M₃₂	—	2	2	no	—
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	2	no	—
158	M₃₂	—	2	2	no	—
159	M₃₂	C₄²	1	4	yes	—
160	M₃₂	C₄²	1	4	yes	—
161	M₃₂	C₂²⋊C₄	1	4	yes	—
162	M₃₂	C₂²⋊C₄	1	4	yes	—
163	M₃₂	C₄⋊C₄	1	4	yes	—
164	M₃₂	C₄⋊C₄	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	C₄²	1	4	yes	—
174	C₂×C₁₆	C₄²	1	4	yes	—
175	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
176	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
177	#97	—	2	2	no	—
178	#97	—	2	2	no	—
179	#97	—	2	2	no	—
180	#97	—	2	2	no	—
181	#97	—	2	2	no	—
182	#97	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
196	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
201	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
203	C₂²×C₄	#131	1	1	yes	characteristic
204	C₂²×C₄	#131	1	1	yes	characteristic
205	C₂²×C₄	#131	1	1	yes	characteristic
206	C₂²×C₄	#131	1	1	yes	characteristic
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	—	4	1	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	#97	1	4	yes	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
232	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₈	#117	1	1	yes	characteristic
239	C₂×C₈	#117	1	1	yes	characteristic
240	C₂×C₈	#117	1	1	yes	characteristic
241	C₂×C₈	#117	1	1	yes	characteristic
242	C₂×C₈	#117	1	1	yes	characteristic
243	C₂×C₈	#117	1	1	yes	characteristic
244	C₂×C₈	C₄×D₄	1	1	yes	characteristic
245	C₂×C₈	C₄×D₄	1	1	yes	characteristic
246	C₂×C₈	C₄×D₄	1	1	yes	characteristic
247	C₂×C₈	C₄×D₄	1	1	yes	characteristic
248	C₂×C₈	C₄×D₄	1	1	yes	characteristic
249	C₂×C₈	C₄×D₄	1	1	yes	characteristic
250	C₂×C₈	C₄×D₄	1	1	yes	characteristic
251	C₂×C₈	C₄×D₄	1	1	yes	characteristic
252	C₂×C₈	C₄×D₄	1	1	yes	characteristic
253	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
254	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
255	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	#97	1	4	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	1	no	—
274	C₁₆	—	2	4	no	—
275	C₁₆	—	2	4	no	—
276	C₁₆	—	2	4	no	—
277	C₁₆	#95	1	8	yes	—
278	C₂³	—	8	1	no	—
279	C₂³	C₂×#97	1	1	yes	characteristic
280	C₂³	C₄×M₁₆	1	1	yes	characteristic
281	C₂³	C₄×M₁₆	1	1	yes	characteristic
282	C₂³	#435	1	1	yes	characteristic
283	C₂³	#435	1	1	yes	characteristic
284	C₂³	#436	1	1	yes	characteristic
285	C₂³	#446	1	1	yes	characteristic
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	C₄³	1	1	yes	characteristic
293	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
298	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₂×#97	1	1	yes	characteristic
300	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
301	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
302	C₂×C₄	#405	1	1	yes	characteristic
303	C₂×C₄	#405	1	1	yes	characteristic
304	C₂×C₄	#405	1	1	yes	characteristic
305	C₂×C₄	#405	1	1	yes	characteristic
306	C₂×C₄	#405	1	1	yes	characteristic
307	C₂×C₄	#405	1	1	yes	characteristic
308	C₂×C₄	#435	1	1	yes	characteristic
309	C₂×C₄	#435	1	1	yes	characteristic
310	C₂×C₄	#435	1	1	yes	characteristic
311	C₂×C₄	#435	1	1	yes	characteristic
312	C₂×C₄	#435	1	1	yes	characteristic
313	C₂×C₄	#435	1	1	yes	characteristic
314	C₂×C₄	#436	1	1	yes	characteristic
315	C₂×C₄	#436	1	1	yes	characteristic
316	C₂×C₄	#436	1	1	yes	characteristic
317	C₂×C₄	#446	1	1	yes	characteristic
318	C₂×C₄	#446	1	1	yes	characteristic
319	C₂×C₄	#446	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	2	no	—
323	C₈	C₂×#95	1	1	yes	characteristic
324	C₈	#376	1	1	yes	characteristic
325	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
326	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
327	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
328	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
329	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
330	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	C₄×#97	1	1	yes	characteristic
339	C₂²	#1752	1	1	yes	characteristic
340	C₂²	#1753	1	1	yes	characteristic
341	C₂²	#1753	1	1	yes	characteristic
342	C₂²	#1770	1	1	yes	characteristic
343	C₂²	#1771	1	1	yes	characteristic
344	C₂²	#1771	1	1	yes	characteristic
345	C₄	—	4	4	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₄×#95	1	1	yes	characteristic
349	C₄	#1726	1	1	yes	characteristic
350	C₄	#1752	1	1	yes	characteristic
351	C₄	#1753	1	1	yes	characteristic
352	C₄	#1753	1	1	yes	characteristic
353	C₄	#1770	1	1	yes	characteristic
354	C₄	#1771	1	1	yes	characteristic
355	C₄	#1771	1	1	yes	characteristic
356	C₂	—	2	1	no	—
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	#10696	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10696
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
18. #2110	C₄	semidirect
20. C₂×#347	C₄	semidirect
22. #2108	C₄	semidirect
28. #1379	C₄	semidirect
29. #1378	C₄	semidirect
30. #1376	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2110	2
#33324	2
#12554	2
C₂×#1378	2
#12549	2
#10711	2