Subgroups — #72286907

Order 2048.

549 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	#72286896	C₂	1	1	yes	maximal, characteristic
2	#72286496	C₂	1	2	yes	maximal, complemented
3	#72286396	C₂	1	1	yes	maximal, characteristic
4	C₂×#94754	C₂	1	2	yes	maximal, complemented
5	C₂²×#7463	C₂	1	1	yes	maximal, characteristic
6	C₂²×#1397	C₂²	1	1	yes	characteristic
7	C₂²×#1372	—	2	1	no	—
8	#153427	C₄	1	1	yes	characteristic
9	#153427	C₄	1	1	yes	characteristic
10	C₂×#7511	C₂²	1	2	yes	—
11	C₂×#7464	C₂²	1	1	yes	characteristic
12	C₂×#7464	C₂²	1	1	yes	characteristic
13	C₂×#7463	—	2	1	no	—
14	C₂×#7463	—	2	4	no	—
15	C₂×#7463	C₂²	1	2	yes	—
16	#134679	—	2	1	no	—
17	#94756	C₄	1	2	yes	—
18	#94754	—	2	4	no	—
19	#94028	—	2	1	no	—
20	C₂³×C₃₂	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₃₂	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₆	D₄	1	1	yes	characteristic
23	C₂²×#335	—	4	1	no	—
24	C₂×M₁₂₈	C₂×C₄	1	2	yes	—
25	C₂×M₁₂₈	C₂×C₄	1	2	yes	—
26	C₂×#1397	—	2	2	no	—
27	C₂×#1397	—	2	2	no	—
28	C₂×#1397	C₂×C₄	1	1	yes	characteristic
29	C₂×#1397	D₄	1	1	yes	characteristic
30	C₂×#1397	Q₈	1	1	yes	characteristic
31	C₂×#1397	C₂³	1	1	yes	characteristic
32	#12788	—	4	1	no	—
33	C₂×#1373	—	2	1	no	—
34	C₂×#1373	—	2	1	no	—
35	C₂×#1372	—	2	4	no	—
36	C₂×#1372	—	2	2	no	—
37	C₂×#1372	—	4	2	no	—
38	C₂×#1372	—	4	1	no	—
39	C₂²⋊C₆₄	—	2	2	no	—
40	C₂²⋊C₆₄	C₈	1	4	yes	complemented
41	#7511	—	2	2	no	—
42	#7511	—	2	2	no	—
43	#7464	—	2	2	no	—
44	#7464	—	2	2	no	—
45	#7463	—	2	4	no	—
46	#7463	—	2	2	no	—
47	#7463	D₄	1	4	yes	—
48	#7160	—	4	1	no	—
49	C₂³×C₁₆	C₄⋊C₄	1	1	yes	characteristic
50	C₂³×C₁₆	Q₁₆	1	1	yes	characteristic
51	C₂²×C₄⋊C₈	—	8	1	no	—
52	C₂²×C₄×C₈	D₈	1	1	yes	characteristic
53	C₂²×C₃₂	—	2	2	no	—
54	C₂²×C₃₂	—	2	2	no	—
55	C₂²×C₃₂	—	2	2	no	—
56	C₂²×C₃₂	—	2	2	no	—
57	C₂²×C₃₂	—	4	1	no	—
58	C₂²×C₃₂	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₃₂	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₃₂	C₂×C₈	1	2	yes	—
61	C₂²×C₃₂	M₁₆	1	2	yes	—
62	C₂²×C₃₂	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₃₂	C₂×Q₈	1	1	yes	characteristic
64	C₂×#364	—	4	1	no	—
65	C₂×#364	—	4	1	no	—
66	#2112	—	8	1	no	—
67	C₂×C₄×C₁₆	—	2	1	no	—
68	C₂×C₄×C₁₆	—	2	1	no	—
69	C₂×C₄×C₁₆	—	2	1	no	—
70	C₂×C₄×C₁₆	—	2	1	no	—
71	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₆	D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₆	Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
75	C₂×#335	—	4	4	no	—
76	C₂×#335	—	4	2	no	—
77	C₂×#335	—	8	2	no	—
78	C₂×#335	—	8	1	no	—
79	M₁₂₈	—	2	2	no	—
80	M₁₂₈	—	2	2	no	—
81	M₁₂₈	—	2	2	no	—
82	M₁₂₈	—	2	2	no	—
83	C₂×C₆₄	—	2	2	no	—
84	C₂×C₆₄	C₂×C₈	1	4	yes	complemented
85	#1397	—	2	2	no	—
86	#1397	—	2	2	no	—
87	#1397	—	2	1	no	—
88	#1397	—	2	1	no	—
89	#1397	C₂×D₄	1	2	yes	—
90	#1397	#41	1	2	yes	—
91	#1376	—	8	1	no	—
92	#1373	—	4	1	no	—
93	#1373	—	4	1	no	—
94	#1373	—	4	1	no	—
95	#1373	—	4	1	no	—
96	#1372	—	2	4	no	—
97	#1372	—	4	4	no	—
98	#1372	—	4	2	no	—
99	C₂³×C₈	—	16	1	no	—
100	C₂³×C₈	#107	1	1	yes	characteristic
101	C₂³×C₈	SD₃₂	1	1	yes	characteristic
102	C₂²×C₄²	SD₃₂	1	1	yes	characteristic
103	C₂×M₃₂	—	8	1	no	—
104	C₂×M₃₂	—	8	1	no	—
105	C₂²×C₁₆	—	2	1	no	—
106	C₂²×C₁₆	—	2	1	no	—
107	C₂²×C₁₆	—	2	1	no	—
108	C₂²×C₁₆	—	2	1	no	—
109	C₂²×C₁₆	—	2	2	no	—
110	C₂²×C₁₆	—	2	1	no	—
111	C₂²×C₁₆	—	2	1	no	—
112	C₂²×C₁₆	—	4	1	no	—
113	C₂²×C₁₆	C₄⋊C₈	1	2	yes	—
114	C₂²×C₁₆	#107	1	1	yes	characteristic
115	C₂²×C₁₆	#107	1	1	yes	characteristic
116	C₂²×C₁₆	#108	1	2	yes	—
117	C₂²×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
119	C₂×C₄⋊C₈	—	8	4	no	—
120	C₂×C₄⋊C₈	—	8	2	no	—
121	C₂×C₄⋊C₈	—	16	1	no	—
122	C₂×C₄⋊C₈	—	16	2	no	—
123	C₂×C₄×C₈	—	2	1	no	—
124	C₂×C₄×C₈	—	2	1	no	—
125	C₂×C₄×C₈	—	2	1	no	—
126	C₂×C₄×C₈	—	2	1	no	—
127	C₂×C₄×C₈	#107	1	1	yes	characteristic
128	C₂×C₄×C₈	SD₃₂	1	1	yes	characteristic
129	C₂×C₄×C₈	SD₃₂	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₈	1	1	yes	characteristic
131	C₂×C₃₂	—	2	1	no	—
132	C₂×C₃₂	—	2	1	no	—
133	C₂×C₃₂	—	2	1	no	—
134	C₂×C₃₂	—	2	1	no	—
135	C₂×C₃₂	—	2	4	no	—
136	C₂×C₃₂	—	2	4	no	—
137	C₂×C₃₂	—	2	1	no	—
138	C₂×C₃₂	—	2	1	no	—
139	C₂×C₃₂	—	2	1	no	—
140	C₂×C₃₂	—	4	4	no	—
141	C₂×C₃₂	C₄⋊C₈	1	2	yes	—
142	C₂×C₃₂	C₄×D₄	1	2	yes	—
143	C₂×C₃₂	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₃₂	C₂²×C₈	1	1	yes	characteristic
145	C₂×C₃₂	C₂×M₁₆	1	1	yes	characteristic
146	C₂×C₃₂	#131	1	2	yes	—
147	#364	—	8	1	no	—
148	#364	—	8	1	no	—
149	#364	—	8	1	no	—
150	#364	—	8	1	no	—
151	C₂²⋊C₁₆	—	16	2	no	—
152	C₄×C₁₆	—	2	1	no	—
153	C₄×C₁₆	—	2	1	no	—
154	C₄×C₁₆	—	2	1	no	—
155	C₄×C₁₆	—	2	1	no	—
156	C₄×C₁₆	—	2	1	no	—
157	C₄×C₁₆	—	2	1	no	—
158	C₄×C₁₆	#120	1	1	yes	characteristic
159	C₄×C₁₆	#123	1	1	yes	characteristic
160	C₄×C₁₆	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₄×C₁₆	#137	1	1	yes	characteristic
162	#335	—	4	4	no	—
163	#335	—	8	4	no	—
164	#335	—	8	2	no	—
165	C₆₄	—	2	4	no	—
166	C₆₄	—	4	2	no	—
167	C₂³×C₄	—	2	1	no	—
168	C₂³×C₄	#367	1	1	yes	characteristic
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	4	1	no	—
177	C₂²×C₈	—	16	4	no	—
178	C₂²×C₈	—	16	2	no	—
179	C₂²×C₈	—	32	4	no	—
180	C₂²×C₈	#335	1	2	yes	—
181	C₂²×C₈	#367	1	1	yes	characteristic
182	C₂²×C₈	#367	1	1	yes	characteristic
183	C₂²×C₈	#368	1	2	yes	—
184	C₂²×C₈	C₂×#107	1	1	yes	characteristic
185	C₂²×C₈	C₂×SD₃₂	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	#367	1	1	yes	characteristic
191	C₂×C₄²	C₂×SD₃₂	1	1	yes	characteristic
192	M₃₂	—	16	1	no	—
193	M₃₂	—	16	1	no	—
194	M₃₂	—	16	1	no	—
195	M₃₂	—	16	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	4	2	no	—
208	C₂×C₁₆	—	4	2	no	—
209	C₂×C₁₆	—	16	1	no	—
210	C₂×C₁₆	—	16	1	no	—
211	C₂×C₁₆	#335	1	2	yes	—
212	C₂×C₁₆	#385	1	1	yes	characteristic
213	C₂×C₁₆	C₂×C₄⋊C₈	1	1	yes	characteristic
214	C₂×C₁₆	#424	1	1	yes	characteristic
215	C₂×C₁₆	#428	1	1	yes	characteristic
216	C₂×C₁₆	C₂×#108	1	1	yes	characteristic
217	C₂×C₁₆	#430	1	1	yes	characteristic
218	C₂×C₁₆	#451	1	1	yes	characteristic
219	C₂×C₁₆	#484	1	1	yes	characteristic
220	C₄⋊C₈	—	8	4	no	—
221	C₄⋊C₈	—	16	4	no	—
222	C₄⋊C₈	—	16	2	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	#447	1	1	yes	characteristic
230	C₄×C₈	#480	1	1	yes	characteristic
231	C₄×C₈	#509	1	1	yes	characteristic
232	C₄×C₈	#510	1	1	yes	characteristic
233	C₃₂	—	2	2	no	—
234	C₃₂	—	2	2	no	—
235	C₃₂	—	4	1	no	—
236	C₃₂	C₈×D₄	1	2	yes	—
237	C₃₂	#435	1	2	yes	—
238	C₂⁴	#1426	1	1	yes	characteristic
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	#1372	1	2	yes	—
249	C₂²×C₄	#1426	1	1	yes	characteristic
250	C₂²×C₄	#1426	1	2	yes	—
251	C₂²×C₄	C₂×#367	1	1	yes	characteristic
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	16	4	no	—
266	C₂×C₈	—	32	8	no	—
267	C₂×C₈	—	32	4	no	—
268	C₂×C₈	#1372	1	2	yes	—
269	C₂×C₈	C₂×#335	1	1	yes	characteristic
270	C₂×C₈	#1571	1	1	yes	characteristic
271	C₂×C₈	#1852	1	1	yes	characteristic
272	C₂×C₈	#2159	1	1	yes	characteristic
273	C₂×C₈	C₂×#368	1	1	yes	characteristic
274	C₂×C₈	#2161	1	1	yes	characteristic
275	C₂×C₈	#2185	1	1	yes	characteristic
276	C₂×C₈	#2233	1	1	yes	characteristic
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	#1426	1	2	yes	—
282	C₄²	#2187	1	1	yes	characteristic
283	C₄²	#2235	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	2	no	—
287	C₁₆	—	4	1	no	—
288	C₁₆	—	32	1	no	—
289	C₁₆	—	32	1	no	—
290	C₁₆	#1565	1	1	yes	characteristic
291	C₁₆	#1573	1	1	yes	characteristic
292	C₁₆	#1850	1	1	yes	characteristic
293	C₁₆	#1855	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	—	2	4	no	—
296	C₂³	—	2	1	no	—
297	C₂³	#7463	1	2	yes	—
298	C₂³	C₂×#1426	1	1	yes	characteristic
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	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	#7463	1	2	yes	—
313	C₂×C₄	C₂×#1372	1	1	yes	characteristic
314	C₂×C₄	#12033	1	1	yes	characteristic
315	C₂×C₄	#12920	1	1	yes	characteristic
316	C₂×C₄	C₂×#1426	1	1	yes	characteristic
317	C₂×C₄	C₂×#1426	1	1	yes	characteristic
318	C₂×C₄	#13638	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	2	no	—
322	C₈	—	4	1	no	—
323	C₈	—	32	4	no	—
324	C₈	#8829	1	1	yes	characteristic
325	C₈	#12009	1	1	yes	characteristic
326	C₈	#12035	1	1	yes	characteristic
327	C₈	#12922	1	1	yes	characteristic
328	C₂²	—	2	4	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	#94754	1	4	yes	complemented
334	C₂²	C₂×#7463	1	1	yes	characteristic
335	C₂²	#151066	1	1	yes	characteristic
336	C₂²	#152427	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	1	no	—
341	C₄	#135814	1	1	yes	characteristic
342	C₄	#151038	1	1	yes	characteristic
343	C₄	#151068	1	1	yes	characteristic
344	C₄	#152426	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	2	4	no	—
348	C₂	C₂×#94754	1	2	yes	—
349	C₂	#72286713	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#7463
commutator	C₂×C₃₂	C₂²×C₈
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂⁴	#1426
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ C₂²×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72286496	C₂	semidirect
4. C₂×#94754	C₂	semidirect
40. C₂²⋊C₆₄	C₈	semidirect
84. C₂×C₆₄	C₂×C₈	semidirect
333. C₂²	#94754	semidirect

Maximal subgroups

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

subgroup	index
#72286896	2
#72286496	2
#72286396	2
C₂×#94754	2
C₂²×#7463	2