Subgroups — #105404

Order 512.

719 conjugacy classes of subgroups.

The subgroup lattice has 378 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	C₄×#405	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×#405	—	2	1	no	—
12	C₂×#405	—	2	1	no	—
13	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	#2283	C₂²	1	1	yes	characteristic
16	C₄×#117	C₂²	1	1	yes	characteristic
17	#1770	C₄	1	2	yes	complemented
18	C₈×C₄⋊C₄	C₄	1	2	yes	complemented
19	#1752	—	2	1	no	—
20	C₈×C₂²⋊C₄	—	2	1	no	—
21	#1746	—	2	1	no	—
22	#1726	—	2	1	no	—
23	#1726	—	2	1	no	—
24	#1726	—	2	1	no	—
25	C₄×#97	C₄	1	2	yes	complemented
26	C₄²×C₈	C₄	1	2	yes	complemented
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	#405	—	2	8	no	—
61	#405	—	2	8	no	—
62	C₄×M₁₆	—	2	1	no	—
63	C₄×M₁₆	—	2	1	no	—
64	C₄×M₁₆	—	2	1	no	—
65	C₄×M₁₆	C₂×C₄	1	2	yes	—
66	C₄×M₁₆	C₂×C₄	1	2	yes	—
67	C₄×M₁₆	C₂×C₄	1	2	yes	—
68	C₂×#97	—	2	2	no	—
69	C₂×#97	—	2	2	no	—
70	C₂×#97	—	2	1	no	—
71	C₂×#97	—	2	1	no	—
72	C₂×#97	—	2	2	no	—
73	C₂×C₄×C₈	—	2	2	no	—
74	C₂×C₄×C₈	—	2	2	no	—
75	C₂×C₄×C₈	—	2	2	no	—
76	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
77	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄×C₈	D₄	1	1	yes	characteristic
84	C₂×C₄×C₈	D₄	1	1	yes	characteristic
85	C₂×C₄×C₈	D₄	1	1	yes	characteristic
86	C₂×C₄×C₈	D₄	1	1	yes	characteristic
87	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
88	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
89	#383	C₂×C₄	1	2	yes	—
90	#382	—	2	1	no	—
91	#381	C₂×C₄	1	2	yes	—
92	#380	—	2	1	no	—
93	C₄×C₄⋊C₄	—	2	1	no	—
94	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
95	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
96	C₄×C₂²⋊C₄	—	2	1	no	—
97	C₄×C₂²⋊C₄	—	2	1	no	—
98	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
99	#376	—	2	1	no	—
100	#376	—	2	1	no	—
101	C₄³	C₂×C₄	1	2	yes	—
102	#336	—	2	2	no	—
103	#336	—	2	2	no	—
104	#322	—	2	2	no	—
105	#322	—	2	2	no	—
106	#322	—	2	2	no	—
107	C₈²	—	2	2	no	—
108	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
109	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₄	#41	1	1	yes	characteristic
111	C₂×M₁₆	—	2	4	no	—
112	C₂×M₁₆	—	2	4	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	—	2	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
133	#117	—	2	4	no	—
134	#117	—	2	4	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₂²⋊C₄	—	2	2	no	—
138	C₂×C₂²⋊C₄	—	2	2	no	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	C₄²	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
150	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
151	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
152	C₂×C₄²	#41	1	1	yes	characteristic
153	C₂×C₄²	#41	1	1	yes	characteristic
154	C₂×C₄²	#41	1	1	yes	characteristic
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	C₄²	1	4	yes	—
161	C₂²⋊C₈	—	2	2	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₂²⋊C₈	—	2	2	no	—
164	C₂²⋊C₈	—	2	2	no	—
165	C₂²⋊C₈	—	2	2	no	—
166	C₂²⋊C₈	C₄²	1	4	yes	—
167	#97	—	2	8	no	—
168	#97	—	2	8	no	—
169	#97	—	2	2	no	—
170	#97	—	2	2	no	—
171	#97	—	2	2	no	—
172	#97	—	2	2	no	—
173	#97	—	2	2	no	—
174	#97	—	4	4	no	—
175	#97	C₄²	1	4	yes	—
176	C₄×C₈	—	2	8	no	—
177	C₄×C₈	—	2	8	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	4	4	no	—
182	C₄×C₈	C₄²	1	4	yes	—
183	C₄×C₈	C₄²	1	1	yes	characteristic
184	C₄×C₈	C₄²	1	1	yes	characteristic
185	C₄×C₈	C₄²	1	1	yes	characteristic
186	C₄×C₈	C₄²	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
190	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
191	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
192	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
193	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
194	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
195	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
196	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
197	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
198	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	#95	C₄²	1	4	yes	—
205	C₂⁴	C₄×D₄	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	#117	1	1	yes	characteristic
219	C₂²×C₄	#117	1	1	yes	characteristic
220	C₂²×C₄	#117	1	1	yes	characteristic
221	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
226	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
227	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
228	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
229	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
230	M₁₆	—	2	8	no	—
231	M₁₆	—	2	8	no	—
232	M₁₆	—	4	4	no	—
233	M₁₆	—	4	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
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	C₂×C₈	#95	1	1	yes	characteristic
257	C₂×C₈	#95	1	1	yes	characteristic
258	C₂×C₈	#95	1	1	yes	characteristic
259	C₂×C₈	#95	1	1	yes	characteristic
260	C₂×C₈	#104	1	4	yes	—
261	C₂×C₈	#104	1	4	yes	—
262	C₄⋊C₄	—	2	8	no	—
263	C₄⋊C₄	—	4	4	no	—
264	C₂²⋊C₄	—	2	8	no	—
265	C₂²⋊C₄	—	4	4	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	4	4	no	—
269	C₄²	—	4	4	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×C₄²	1	1	yes	characteristic
273	C₄²	C₂×C₄²	1	1	yes	characteristic
274	C₄²	C₂×C₄²	1	1	yes	characteristic
275	C₄²	C₂×C₄²	1	1	yes	characteristic
276	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
277	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
278	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
279	C₄²	#117	1	1	yes	characteristic
280	C₄²	#117	1	1	yes	characteristic
281	C₄²	#117	1	1	yes	characteristic
282	C₄²	C₄×D₄	1	1	yes	characteristic
283	C₄²	C₄×D₄	1	1	yes	characteristic
284	C₄²	C₄×D₄	1	1	yes	characteristic
285	C₄²	C₄×D₄	1	1	yes	characteristic
286	C₄²	C₄×Q₈	1	1	yes	characteristic
287	C₄²	C₄×Q₈	1	1	yes	characteristic
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	4	1	no	—
291	C₂³	C₄×C₂²⋊C₄	1	1	yes	characteristic
292	C₂³	C₄×C₂²⋊C₄	1	1	yes	characteristic
293	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	8	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	C₄³	1	1	yes	characteristic
320	C₂×C₄	C₂×#95	1	1	yes	characteristic
321	C₂×C₄	#376	1	1	yes	characteristic
322	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
323	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
324	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
325	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
327	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
328	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
329	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
330	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
331	C₂×C₄	C₂×#104	1	2	yes	—
332	C₂×C₄	C₂×#104	1	2	yes	—
333	C₈	—	2	8	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	4	4	no	—
339	C₈	—	4	4	no	—
340	C₈	—	4	4	no	—
341	C₈	#339	1	2	yes	—
342	C₈	#339	1	2	yes	—
343	C₈	#339	1	2	yes	—
344	C₈	#339	1	2	yes	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	4	4	no	—
353	C₂²	C₄×#95	1	1	yes	characteristic
354	C₂²	#1757	1	2	yes	—
355	C₂²	C₄×#104	1	2	yes	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	4	8	no	—
366	C₄	—	8	4	no	—
367	C₄	C₂×#339	1	1	yes	characteristic
368	C₄	C₂×#339	1	1	yes	characteristic
369	C₄	#1747	1	2	yes	—
370	C₄	C₄×#104	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	2	2	no	—
373	C₂	—	4	1	no	—
374	C₂	#10699	1	1	yes	characteristic
375	C₂	C₄×#339	1	1	yes	characteristic
376	C₂	#10711	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#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×C₈ ◁ 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
17. #1770	C₄	semidirect
18. C₈×C₄⋊C₄	C₄	semidirect
25. C₄×#97	C₄	semidirect
26. C₄²×C₈	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
C₄×#405	2
#17323	2
#7993	2
#7973	2
#7949	2
#7941	2