Subgroups — #72225495

Order 4096.

482 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225490	C₂	1	2	yes	maximal
2	#72225487	C₂	1	2	yes	maximal
3	#72225485	—	2	2	no	—
4	#72225484	—	2	2	no	—
5	#72225402	C₂	1	1	yes	maximal, characteristic
6	#72225398	C₄	1	2	yes	complemented
7	#72225397	C₄	1	2	yes	complemented
8	#72225395	C₂²	1	2	yes	—
9	#72225366	C₂	1	1	yes	maximal, characteristic
10	#72225363	—	2	1	no	—
11	#72225362	C₂²	1	1	yes	characteristic
12	#72225313	C₂²	1	1	yes	characteristic
13	#72216534	—	2	1	no	—
14	#72216301	C₂	1	1	yes	maximal, characteristic
15	#72216296	—	2	1	no	—
16	#72216296	C₂²	1	2	yes	—
17	#72216239	—	2	1	no	—
18	C₁₆×M₆₄	C₂²	1	1	yes	characteristic
19	#153582	—	2	2	no	—
20	#151998	D₄	1	1	yes	characteristic
21	#151989	—	4	1	no	—
22	#151055	—	2	2	no	—
23	#151055	—	2	2	no	—
24	#151055	—	4	1	no	—
25	#151055	—	4	1	no	—
26	#151040	D₄	1	1	yes	characteristic
27	C₂×#7459	—	2	1	no	—
28	C₈×M₆₄	D₄	1	1	yes	characteristic
29	C₈×M₆₄	C₂³	1	1	yes	characteristic
30	#135796	—	2	2	no	—
31	#135796	—	4	1	no	—
32	#135781	—	4	1	no	—
33	#129142	—	2	1	no	—
34	C₂×C₁₆²	D₄	1	1	yes	characteristic
35	#94754	—	4	4	no	—
36	#94748	C₂×C₄	1	4	yes	—
37	#94746	—	2	2	no	—
38	#94053	—	2	1	no	—
39	#94041	—	4	1	no	—
40	#93885	—	2	1	no	—
41	#93885	Q₈	1	2	yes	—
42	#93664	C₂×C₄	1	1	yes	characteristic
43	#93664	Q₈	1	1	yes	characteristic
44	C₁₆×C₃₂	C₂×C₄	1	1	yes	characteristic
45	C₁₆×C₃₂	Q₈	1	1	yes	characteristic
46	#13638	—	2	1	no	—
47	C₂×#1422	—	4	1	no	—
48	#13551	—	8	1	no	—
49	C₄×M₆₄	D₈	1	1	yes	characteristic
50	C₄×M₆₄	D₈	1	1	yes	characteristic
51	C₄×M₆₄	C₂×D₄	1	1	yes	characteristic
52	C₄×M₆₄	#41	1	1	yes	characteristic
53	#12909	—	8	2	no	—
54	#12670	—	2	1	no	—
55	#12664	—	4	1	no	—
56	#12023	—	4	2	no	—
57	#12023	—	4	2	no	—
58	#12023	—	8	2	no	—
59	C₂×#1373	—	2	1	no	—
60	C₂×C₈×C₁₆	—	2	1	no	—
61	C₂×C₈×C₁₆	C₂×D₄	1	1	yes	characteristic
62	#8823	—	4	2	no	—
63	#7544	—	2	4	no	—
64	#7513	—	4	2	no	—
65	#7511	—	4	2	no	—
66	#7476	Q₁₆	1	1	yes	characteristic
67	#7475	Q₁₆	1	1	yes	characteristic
68	#7469	—	8	1	no	—
69	#7464	—	2	1	no	—
70	#7464	D₈	1	2	yes	—
71	#7463	—	4	2	no	—
72	#7463	—	4	2	no	—
73	#7463	—	8	2	no	—
74	#7463	—	8	1	no	—
75	#7463	—	8	1	no	—
76	#7463	—	8	2	no	—
77	#7459	—	2	2	no	—
78	#7459	—	4	1	no	—
79	#7329	Q₁₆	1	1	yes	characteristic
80	#7329	Q₁₆	1	1	yes	characteristic
81	#7329	C₂×Q₈	1	1	yes	characteristic
82	#7329	#41	1	1	yes	characteristic
83	C₈×C₃₂	C₄⋊C₄	1	1	yes	characteristic
84	C₈×C₃₂	C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₃₂	C₂²×C₄	1	1	yes	characteristic
86	C₈×C₃₂	C₂×Q₈	1	1	yes	characteristic
87	#7179	—	4	1	no	—
88	C₁₆²	—	2	1	no	—
89	C₁₆²	C₄⋊C₄	1	1	yes	characteristic
90	C₁₆²	C₄⋊C₄	1	1	yes	characteristic
91	C₂×M₆₄	—	2	1	no	—
92	C₂×M₆₄	—	2	1	no	—
93	C₂×M₆₄	—	8	1	no	—
94	C₂×M₆₄	SD₃₂	1	1	yes	characteristic
95	C₂×M₆₄	SD₃₂	1	1	yes	characteristic
96	C₂×M₆₄	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×M₆₄	#135	1	1	yes	characteristic
98	C₂×#368	—	2	1	no	—
99	C₂×#368	—	16	4	no	—
100	#2158	—	2	1	no	—
101	C₂×#364	—	4	1	no	—
102	#2108	—	8	1	no	—
103	C₂×C₄×C₁₆	—	2	2	no	—
104	C₂×C₄×C₁₆	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₆	C₂×D₈	1	1	yes	characteristic
106	#1848	—	8	2	no	—
107	#1772	—	4	1	no	—
108	#1567	—	8	2	no	—
109	#1567	—	8	2	no	—
110	C₂×C₈²	AGL(1,ℤ/8)	1	1	yes	characteristic
111	M₁₂₈	—	4	4	no	—
112	#1426	—	4	2	no	—
113	#1422	—	4	2	no	—
114	#1422	—	8	1	no	—
115	#1398	—	16	2	no	—
116	#1397	—	2	1	no	—
117	#1397	—	2	1	no	—
118	#1397	SD₃₂	1	1	yes	characteristic
119	#1397	SD₃₂	1	1	yes	characteristic
120	#1397	C₂×D₈	1	1	yes	characteristic
121	#1397	#135	1	1	yes	characteristic
122	C₄×C₃₂	#107	1	1	yes	characteristic
123	C₄×C₃₂	#107	1	1	yes	characteristic
124	C₄×C₃₂	#107	1	1	yes	characteristic
125	C₄×C₃₂	#107	1	1	yes	characteristic
126	C₄×C₃₂	C₄×D₄	1	1	yes	characteristic
127	C₄×C₃₂	C₄×Q₈	1	1	yes	characteristic
128	C₄×C₃₂	#125	1	1	yes	characteristic
129	C₄×C₃₂	C₄⋊Q₈	1	1	yes	characteristic
130	#1382	—	4	1	no	—
131	#1379	—	8	1	no	—
132	#1373	—	2	2	no	—
133	#1373	—	4	1	no	—
134	#1372	—	8	2	no	—
135	#1372	—	8	2	no	—
136	#1372	—	8	2	no	—
137	#1372	—	8	2	no	—
138	#1372	—	16	4	no	—
139	C₈×C₁₆	—	2	2	no	—
140	C₈×C₁₆	—	2	1	no	—
141	C₈×C₁₆	—	2	1	no	—
142	C₈×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
145	C₈×C₁₆	#137	1	1	yes	characteristic
146	C₂×M₃₂	—	8	1	no	—
147	C₂²×C₁₆	—	2	4	no	—
148	C₂²×C₁₆	#467	1	1	yes	characteristic
149	C₂²×C₁₆	#473	1	1	yes	characteristic
150	C₂²×C₁₆	C₂×SD₃₂	1	1	yes	characteristic
151	C₂²×C₁₆	#510	1	1	yes	characteristic
152	C₂×#108	—	4	1	no	—
153	C₂×#108	—	16	4	no	—
154	#427	—	4	1	no	—
155	#423	—	16	2	no	—
156	#405	—	8	1	no	—
157	C₂×C₄×C₈	—	2	1	no	—
158	C₂×C₄×C₈	#459	1	1	yes	characteristic
159	M₆₄	—	2	2	no	—
160	M₆₄	—	2	2	no	—
161	M₆₄	—	4	1	no	—
162	M₆₄	—	4	1	no	—
163	M₆₄	—	4	2	no	—
164	M₆₄	—	4	2	no	—
165	M₆₄	—	8	2	no	—
166	M₆₄	—	16	1	no	—
167	C₂×C₃₂	—	2	1	no	—
168	C₂×C₃₂	—	2	1	no	—
169	C₂×C₃₂	—	2	1	no	—
170	C₂×C₃₂	—	2	1	no	—
171	C₂×C₃₂	—	16	1	no	—
172	C₂×C₃₂	#367	1	1	yes	characteristic
173	C₂×C₃₂	#367	1	1	yes	characteristic
174	C₂×C₃₂	#367	1	1	yes	characteristic
175	C₂×C₃₂	#367	1	1	yes	characteristic
176	C₂×C₃₂	C₄×D₈	1	1	yes	characteristic
177	C₂×C₃₂	C₄×Q₁₆	1	1	yes	characteristic
178	C₂×C₃₂	#499	1	1	yes	characteristic
179	C₂×C₃₂	#500	1	1	yes	characteristic
180	#368	—	4	1	no	—
181	#368	—	4	1	no	—
182	#368	—	32	8	no	—
183	#367	—	4	1	no	—
184	#366	—	4	1	no	—
185	#364	—	4	2	no	—
186	#364	—	8	1	no	—
187	#346	—	16	1	no	—
188	C₄×C₁₆	—	2	2	no	—
189	C₄×C₁₆	—	2	2	no	—
190	C₄×C₁₆	—	2	4	no	—
191	C₄×C₁₆	—	16	1	no	—
192	C₄×C₁₆	#389	1	1	yes	characteristic
193	C₄×C₁₆	C₂×#107	1	1	yes	characteristic
194	C₄×C₁₆	#462	1	1	yes	characteristic
195	C₄×C₁₆	#466	1	1	yes	characteristic
196	C₄×C₁₆	#472	1	1	yes	characteristic
197	C₄×C₁₆	#474	1	1	yes	characteristic
198	C₄×C₁₆	C₂×SD₃₂	1	1	yes	characteristic
199	C₄×C₁₆	#509	1	1	yes	characteristic
200	#339	—	8	1	no	—
201	#335	—	16	4	no	—
202	#335	—	16	2	no	—
203	#335	—	16	2	no	—
204	C₈²	—	2	1	no	—
205	C₈²	#428	1	1	yes	characteristic
206	C₈²	#477	1	1	yes	characteristic
207	C₆₄	—	8	4	no	—
208	C₂×M₁₆	—	8	1	no	—
209	C₂×M₁₆	—	32	2	no	—
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	#1669	1	1	yes	characteristic
212	C₂²×C₈	#2208	1	1	yes	characteristic
213	#117	—	8	1	no	—
214	C₂×C₄²	#2213	1	1	yes	characteristic
215	M₃₂	—	8	2	no	—
216	M₃₂	—	16	1	no	—
217	C₂×C₁₆	—	2	4	no	—
218	C₂×C₁₆	—	2	1	no	—
219	C₂×C₁₆	—	2	1	no	—
220	C₂×C₁₆	—	2	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	2	4	no	—
223	C₂×C₁₆	—	4	1	no	—
224	C₂×C₁₆	—	4	1	no	—
225	C₂×C₁₆	—	4	8	no	—
226	C₂×C₁₆	—	16	1	no	—
227	C₂×C₁₆	#1674	1	1	yes	characteristic
228	C₂×C₁₆	#1676	1	1	yes	characteristic
229	C₂×C₁₆	#1942	1	1	yes	characteristic
230	C₂×C₁₆	#1952	1	1	yes	characteristic
231	C₂×C₁₆	C₂×#367	1	1	yes	characteristic
232	C₂×C₁₆	#2159	1	1	yes	characteristic
233	C₂×C₁₆	#2225	1	1	yes	characteristic
234	C₂×C₁₆	#2226	1	1	yes	characteristic
235	#108	—	8	1	no	—
236	#108	—	8	1	no	—
237	#108	—	32	8	no	—
238	#107	—	8	1	no	—
239	#106	—	8	1	no	—
240	C₄⋊C₈	—	32	2	no	—
241	C₄⋊C₈	—	32	2	no	—
242	#97	—	16	1	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	2	1	no	—
246	C₄×C₈	—	16	1	no	—
247	C₄×C₈	#1667	1	1	yes	characteristic
248	C₄×C₈	#1947	1	1	yes	characteristic
249	C₄×C₈	#2207	1	1	yes	characteristic
250	C₄×C₈	#2214	1	1	yes	characteristic
251	C₃₂	—	2	2	no	—
252	C₃₂	—	2	2	no	—
253	C₃₂	—	2	1	no	—
254	C₃₂	—	2	1	no	—
255	C₃₂	—	2	1	no	—
256	C₃₂	—	2	1	no	—
257	C₃₂	—	2	1	no	—
258	C₃₂	—	2	1	no	—
259	C₃₂	—	4	1	no	—
260	C₃₂	—	4	1	no	—
261	C₃₂	—	16	2	no	—
262	C₃₂	#1426	1	2	yes	—
263	C₃₂	#2179	1	2	yes	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	#12447	1	1	yes	characteristic
266	M₁₆	—	16	1	no	—
267	M₁₆	—	16	1	no	—
268	M₁₆	—	64	4	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	4	no	—
276	C₂×C₈	—	16	1	no	—
277	C₂×C₈	—	64	2	no	—
278	C₂×C₈	#9614	1	1	yes	characteristic
279	C₂×C₈	#12426	1	1	yes	characteristic
280	C₂×C₈	#12446	1	1	yes	characteristic
281	C₂×C₈	#13177	1	1	yes	characteristic
282	C₄⋊C₄	—	16	1	no	—
283	C₂²⋊C₄	—	16	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	16	1	no	—
286	C₄²	#12425	1	1	yes	characteristic
287	C₄²	#13181	1	1	yes	characteristic
288	C₁₆	—	2	8	no	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₁₆	—	2	1	no	—
295	C₁₆	—	4	4	no	—
296	C₁₆	—	16	2	no	—
297	C₁₆	#13170	1	1	yes	characteristic
298	C₁₆	#13173	1	1	yes	characteristic
299	C₁₆	C₂×#1426	1	1	yes	characteristic
300	C₁₆	#13638	1	1	yes	characteristic
301	C₂³	#151659	1	1	yes	characteristic
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	32	1	no	—
308	C₂×C₄	#140433	1	1	yes	characteristic
309	C₂×C₄	#151659	1	1	yes	characteristic
310	C₈	—	2	4	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	4	2	no	—
314	C₈	—	32	1	no	—
315	C₈	—	128	2	no	—
316	C₈	#140431	1	1	yes	characteristic
317	C₈	#140438	1	1	yes	characteristic
318	C₈	#152871	1	1	yes	characteristic
319	C₈	#152874	1	1	yes	characteristic
320	C₂²	—	2	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	#72216273	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	64	1	no	—
327	C₄	#72225492	1	1	yes	characteristic
328	C₄	#72225493	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	4	1	no	—
331	C₂	#72225494	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72225494
commutator	C₈×C₃₂	C₂²×C₄
Frattini, cosocle	C₈×M₆₄	C₂³
prime radical	C₂³	#151659
LCS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
6. #72225398	C₄	semidirect
7. #72225397	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72225490	2
#72225487	2
#72225402	2
#72225366	2
#72216301	2