Subgroups — #105405

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20476	C₂	1	1	yes	maximal, characteristic
2	#20366	C₂	1	1	yes	maximal, characteristic
3	#17323	C₂	1	1	yes	maximal, characteristic
4	#7993	C₂	1	1	yes	maximal, characteristic
5	#7973	C₂	1	1	yes	maximal, characteristic
6	#7949	C₂	1	1	yes	maximal, characteristic
7	#7941	C₂	1	1	yes	maximal, characteristic
8	#2920	C₂²	1	1	yes	characteristic
9	#2920	C₂²	1	1	yes	characteristic
10	C₂×#432	C₂²	1	1	yes	characteristic
11	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#2283	C₂²	1	1	yes	characteristic
14	C₄×#117	C₂²	1	1	yes	characteristic
15	#1770	C₄	1	2	yes	complemented
16	C₈×C₄⋊C₄	C₄	1	2	yes	complemented
17	#1752	—	2	1	no	—
18	C₈×C₂²⋊C₄	—	2	1	no	—
19	#1746	—	2	1	no	—
20	#1746	—	2	1	no	—
21	#1746	—	2	1	no	—
22	#1744	C₄	1	2	yes	complemented
23	#1744	C₄	1	2	yes	complemented
24	#1729	—	2	1	no	—
25	#1729	—	2	1	no	—
26	#1726	—	2	1	no	—
27	#1452	—	2	1	no	—
28	#1451	—	2	1	no	—
29	#1451	C₄	1	2	yes	complemented
30	#1450	—	2	1	no	—
31	C₈×M₁₆	—	2	1	no	—
32	C₈×M₁₆	C₄	1	2	yes	complemented
33	#1291	C₄	1	4	yes	complemented
34	#1288	C₄	1	4	yes	complemented
35	#1285	C₄	1	4	yes	complemented
36	#1284	C₄	1	4	yes	—
37	C₂²×M₁₆	—	2	1	no	—
38	C₂³×C₈	D₄	1	1	yes	characteristic
39	C₂³×C₈	D₄	1	1	yes	characteristic
40	C₂×#117	—	2	1	no	—
41	C₂²×C₄²	C₂³	1	1	yes	characteristic
42	#432	—	2	1	no	—
43	#432	—	2	1	no	—
44	#432	—	2	1	no	—
45	#432	C₂×C₄	1	2	yes	—
46	#432	C₂×C₄	1	2	yes	—
47	#432	C₂×C₄	1	2	yes	—
48	#432	C₂×C₄	1	2	yes	—
49	#432	C₂×C₄	1	2	yes	—
50	#431	—	2	1	no	—
51	#431	—	2	1	no	—
52	#431	—	2	1	no	—
53	#431	C₂×C₄	1	2	yes	—
54	#423	—	2	1	no	—
55	#423	—	2	1	no	—
56	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	#407	—	2	1	no	—
58	#407	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₈	—	2	1	no	—
60	C₄×M₁₆	—	2	1	no	—
61	C₄×M₁₆	—	2	1	no	—
62	C₄×M₁₆	—	2	1	no	—
63	C₄×M₁₆	C₂×C₄	1	2	yes	—
64	C₄×M₁₆	C₂×C₄	1	2	yes	—
65	C₄×M₁₆	C₂×C₄	1	2	yes	—
66	C₂×#97	—	2	1	no	—
67	C₂×#97	—	2	1	no	—
68	C₂×#97	—	2	2	no	—
69	C₂×C₄×C₈	—	2	2	no	—
70	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈	D₄	1	1	yes	characteristic
78	C₂×C₄×C₈	D₄	1	1	yes	characteristic
79	C₂×C₄×C₈	D₄	1	1	yes	characteristic
80	C₂×C₄×C₈	D₄	1	1	yes	characteristic
81	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
82	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
83	#383	C₂×C₄	1	2	yes	—
84	#382	—	2	1	no	—
85	#381	C₂×C₄	1	2	yes	—
86	#380	—	2	1	no	—
87	C₄×C₄⋊C₄	—	2	1	no	—
88	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
89	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
90	C₄×C₂²⋊C₄	—	2	1	no	—
91	C₄×C₂²⋊C₄	—	2	1	no	—
92	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
93	#376	—	2	1	no	—
94	#376	—	2	1	no	—
95	C₄³	C₂×C₄	1	2	yes	—
96	#336	—	2	2	no	—
97	#336	—	2	2	no	—
98	#336	—	2	2	no	—
99	#336	—	2	2	no	—
100	#336	—	2	2	no	—
101	#336	—	2	2	no	—
102	#322	—	2	2	no	—
103	#322	—	2	2	no	—
104	#322	—	2	2	no	—
105	C₈²	—	2	2	no	—
106	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
107	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂³×C₄	#41	1	1	yes	characteristic
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	2	2	no	—
113	C₂×M₁₆	—	4	2	no	—
114	C₂×M₁₆	—	4	2	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	4	2	no	—
118	C₂²×C₈	—	4	2	no	—
119	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
131	#117	—	4	2	no	—
132	#117	—	4	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₂²⋊C₄	—	2	2	no	—
136	C₂×C₂²⋊C₄	—	2	2	no	—
137	C₂×C₄²	—	2	2	no	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₄²	1	4	yes	—
141	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
149	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
150	C₂×C₄²	#41	1	1	yes	characteristic
151	C₂×C₄²	#41	1	1	yes	characteristic
152	C₂×C₄²	#41	1	1	yes	characteristic
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	C₄²	1	4	yes	—
159	C₂²⋊C₈	—	2	2	no	—
160	C₂²⋊C₈	—	2	2	no	—
161	C₂²⋊C₈	—	2	2	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₂²⋊C₈	—	2	2	no	—
164	C₂²⋊C₈	C₄²	1	4	yes	—
165	#97	—	2	2	no	—
166	#97	—	2	2	no	—
167	#97	—	2	2	no	—
168	#97	—	2	2	no	—
169	#97	—	2	2	no	—
170	#97	—	4	4	no	—
171	#97	C₄²	1	4	yes	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	—	4	4	no	—
176	C₄×C₈	C₄²	1	4	yes	—
177	C₄×C₈	C₄²	1	1	yes	characteristic
178	C₄×C₈	C₄²	1	1	yes	characteristic
179	C₄×C₈	C₄²	1	1	yes	characteristic
180	C₄×C₈	C₄²	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
185	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
186	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
188	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
189	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
190	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
191	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
192	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	C₄²	1	4	yes	—
199	C₂⁴	C₄×D₄	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	#117	1	1	yes	characteristic
213	C₂²×C₄	#117	1	1	yes	characteristic
214	C₂²×C₄	#117	1	1	yes	characteristic
215	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
219	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
223	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
224	M₁₆	—	4	4	no	—
225	M₁₆	—	4	4	no	—
226	M₁₆	—	4	4	no	—
227	M₁₆	—	4	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	#95	1	1	yes	characteristic
249	C₂×C₈	#95	1	1	yes	characteristic
250	C₂×C₈	#95	1	1	yes	characteristic
251	C₂×C₈	#95	1	1	yes	characteristic
252	C₂×C₈	#95	1	1	yes	characteristic
253	C₂×C₈	#95	1	1	yes	characteristic
254	C₂×C₈	#95	1	1	yes	characteristic
255	C₂×C₈	#95	1	1	yes	characteristic
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	—	4	4	no	—
260	C₄²	—	4	4	no	—
261	C₄²	—	4	4	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₂×C₄²	1	1	yes	characteristic
267	C₄²	C₂×C₄²	1	1	yes	characteristic
268	C₄²	C₂×C₄²	1	1	yes	characteristic
269	C₄²	C₂×C₄²	1	1	yes	characteristic
270	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
272	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
273	C₄²	#117	1	1	yes	characteristic
274	C₄²	#117	1	1	yes	characteristic
275	C₄²	#117	1	1	yes	characteristic
276	C₄²	C₄×D₄	1	1	yes	characteristic
277	C₄²	C₄×D₄	1	1	yes	characteristic
278	C₄²	C₄×D₄	1	1	yes	characteristic
279	C₄²	C₄×D₄	1	1	yes	characteristic
280	C₄²	C₄×Q₈	1	1	yes	characteristic
281	C₄²	C₄×Q₈	1	1	yes	characteristic
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	C₄×C₂²⋊C₄	1	1	yes	characteristic
286	C₂³	C₄×C₂²⋊C₄	1	1	yes	characteristic
287	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	C₄³	1	1	yes	characteristic
314	C₂×C₄	C₂×#95	1	1	yes	characteristic
315	C₂×C₄	#376	1	1	yes	characteristic
316	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
317	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
318	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
319	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
321	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
322	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
323	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
324	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
325	C₂×C₄	#421	1	2	yes	—
326	C₂×C₄	#421	1	2	yes	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	4	4	no	—
336	C₈	—	4	4	no	—
337	C₈	—	4	4	no	—
338	C₈	—	4	4	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	4	4	no	—
347	C₂²	C₄×#95	1	1	yes	characteristic
348	C₂²	#1757	1	2	yes	—
349	C₂²	#1759	1	2	yes	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	8	4	no	—
360	C₄	—	8	4	no	—
361	C₄	#1733	1	1	yes	characteristic
362	C₄	#1733	1	1	yes	characteristic
363	C₄	#1747	1	2	yes	—
364	C₄	#1759	1	2	yes	—
365	C₂	—	2	2	no	—
366	C₂	—	2	2	no	—
367	C₂	—	4	1	no	—
368	C₂	#10699	1	1	yes	characteristic
369	C₂	#10701	1	1	yes	characteristic
370	C₂	#10711	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄×#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
15. #1770	C₄	semidirect
16. C₈×C₄⋊C₄	C₄	semidirect
22. #1744	C₄	semidirect
23. #1744	C₄	semidirect
29. #1451	C₄	semidirect
32. C₈×M₁₆	C₄	semidirect
33. #1291	C₄	semidirect
34. #1288	C₄	semidirect
35. #1285	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20476	2
#20366	2
#17323	2
#7993	2
#7973	2
#7949	2
#7941	2