Subgroups — #72225601

Order 2048.

397 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225599	C₂	1	1	yes	maximal, characteristic
2	#72225363	C₂	1	1	yes	maximal, characteristic
3	#72225352	C₂	1	1	yes	maximal, characteristic
4	#72225342	C₂	1	1	yes	maximal, characteristic
5	#72216560	C₂	1	1	yes	maximal, characteristic
6	#72216298	C₂	1	1	yes	maximal, characteristic
7	#72216273	C₂	1	1	yes	maximal, characteristic
8	#152874	—	2	1	no	—
9	#152873	—	2	1	no	—
10	#152407	—	2	1	no	—
11	#152407	—	2	1	no	—
12	#151989	—	2	1	no	—
13	C₂×#7459	C₂²	1	1	yes	characteristic
14	C₂×#7459	C₂²	1	1	yes	characteristic
15	C₂×#7459	C₂²	1	1	yes	characteristic
16	C₈×M₆₄	C₂²	1	1	yes	characteristic
17	#135781	C₂²	1	1	yes	characteristic
18	#135778	C₂²	1	1	yes	characteristic
19	#129142	C₂²	1	1	yes	characteristic
20	#94366	—	2	1	no	—
21	#94364	—	2	1	no	—
22	#94041	—	2	1	no	—
23	#93882	C₄	1	4	yes	complemented
24	#13638	—	4	1	no	—
25	C₂×#1426	—	4	2	no	—
26	C₂×#1426	—	4	1	no	—
27	C₂×#1426	—	4	1	no	—
28	C₂×#1426	—	4	1	no	—
29	C₂×#1422	—	2	1	no	—
30	C₂×#1422	—	2	1	no	—
31	C₂×#1422	—	2	1	no	—
32	#13551	—	4	1	no	—
33	C₄×M₆₄	D₄	1	1	yes	characteristic
34	C₄×M₆₄	D₄	1	1	yes	characteristic
35	#12909	—	2	1	no	—
36	#12909	—	2	1	no	—
37	#12906	—	2	1	no	—
38	#12906	—	2	1	no	—
39	#12670	D₄	1	1	yes	characteristic
40	#12664	—	2	1	no	—
41	C₂×#1373	D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
43	#8823	—	2	1	no	—
44	#8818	—	2	1	no	—
45	#7479	—	4	1	no	—
46	#7477	—	4	1	no	—
47	#7469	—	4	1	no	—
48	#7459	—	2	1	no	—
49	#7459	—	2	1	no	—
50	#7459	—	2	1	no	—
51	#7459	C₂×C₄	1	2	yes	—
52	#7459	C₂×C₄	1	2	yes	—
53	#7459	Q₈	1	2	yes	—
54	#7329	Q₈	1	2	yes	—
55	C₈×C₃₂	C₂×C₄	1	2	yes	—
56	#7179	—	2	1	no	—
57	C₂×M₆₄	—	2	1	no	—
58	C₂×M₆₄	—	4	1	no	—
59	C₂×M₆₄	—	4	1	no	—
60	C₂×M₆₄	—	4	1	no	—
61	C₂×M₆₄	D₈	1	1	yes	characteristic
62	C₂×M₆₄	D₈	1	1	yes	characteristic
63	C₂×#368	—	4	1	no	—
64	C₂×#368	—	4	1	no	—
65	C₂×#368	—	4	1	no	—
66	C₂×#368	—	4	1	no	—
67	C₂×#368	D₈	1	1	yes	characteristic
68	#2158	D₈	1	1	yes	characteristic
69	C₂×#367	—	4	1	no	—
70	C₂×#367	—	4	1	no	—
71	C₂×#367	—	4	1	no	—
72	C₂×#367	—	4	1	no	—
73	C₂×#364	—	2	1	no	—
74	#2108	—	4	1	no	—
75	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
77	#1848	—	4	1	no	—
78	#1845	—	4	1	no	—
79	#1845	—	8	1	no	—
80	#1772	—	2	1	no	—
81	#1567	—	8	1	no	—
82	C₂×C₈²	#41	1	1	yes	characteristic
83	#1426	—	8	2	no	—
84	#1426	—	8	2	no	—
85	#1426	—	8	4	no	—
86	#1426	—	8	2	no	—
87	#1426	—	8	2	no	—
88	#1422	—	2	2	no	—
89	#1422	—	2	2	no	—
90	#1422	—	2	2	no	—
91	#1422	—	4	1	no	—
92	#1422	—	4	1	no	—
93	#1422	—	4	1	no	—
94	#1398	—	8	2	no	—
95	#1397	—	2	1	no	—
96	#1397	Q₁₆	1	2	yes	—
97	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
98	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
99	#1382	—	2	1	no	—
100	#1379	—	4	1	no	—
101	#1373	—	2	1	no	—
102	#1373	Q₁₆	1	2	yes	—
103	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
104	C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
105	C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
106	C₈×C₁₆	#41	1	1	yes	characteristic
107	C₂×M₃₂	—	4	1	no	—
108	C₂²×C₁₆	C₂×D₈	1	1	yes	characteristic
109	C₂²×C₁₆	C₂×D₈	1	1	yes	characteristic
110	C₂²×C₁₆	#135	1	1	yes	characteristic
111	C₂²×C₁₆	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×#108	—	2	1	no	—
113	C₂×#108	—	8	1	no	—
114	C₂×#108	—	8	1	no	—
115	#427	—	2	1	no	—
116	C₂×#107	—	8	1	no	—
117	C₂×#107	—	8	1	no	—
118	C₂×#107	—	16	2	no	—
119	#423	—	8	1	no	—
120	#405	—	4	1	no	—
121	C₂×C₄×C₈	—	2	1	no	—
122	C₂×C₄×C₈	#121	1	1	yes	characteristic
123	#384	—	8	1	no	—
124	M₆₄	—	2	2	no	—
125	M₆₄	—	2	2	no	—
126	M₆₄	—	2	1	no	—
127	M₆₄	—	2	1	no	—
128	M₆₄	—	4	2	no	—
129	M₆₄	—	4	2	no	—
130	M₆₄	—	4	2	no	—
131	M₆₄	—	8	1	no	—
132	M₆₄	—	8	1	no	—
133	M₆₄	—	8	1	no	—
134	M₆₄	SD₃₂	1	2	yes	—
135	M₆₄	SD₃₂	1	2	yes	—
136	C₂×C₃₂	—	2	2	no	—
137	C₂×C₃₂	—	8	1	no	—
138	C₂×C₃₂	—	8	1	no	—
139	C₂×C₃₂	—	8	1	no	—
140	C₂×C₃₂	#107	1	2	yes	—
141	C₂×C₃₂	#107	1	2	yes	—
142	#368	—	2	1	no	—
143	#368	—	2	1	no	—
144	#368	—	8	2	no	—
145	#368	—	8	1	no	—
146	#368	—	8	1	no	—
147	#368	—	8	1	no	—
148	#368	—	8	1	no	—
149	#368	—	16	1	no	—
150	#367	—	2	1	no	—
151	#367	—	8	2	no	—
152	#367	—	8	1	no	—
153	#367	—	8	1	no	—
154	#367	—	8	1	no	—
155	#367	—	8	1	no	—
156	#367	—	16	1	no	—
157	#366	—	2	1	no	—
158	#364	—	2	2	no	—
159	#364	—	4	1	no	—
160	#346	—	8	1	no	—
161	C₄×C₁₆	—	8	1	no	—
162	C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
164	C₄×C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
165	C₄×C₁₆	C₄⋊Q₈	1	1	yes	characteristic
166	C₄×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
167	C₄×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
168	C₄×C₁₆	#135	1	1	yes	characteristic
169	C₄×C₁₆	#137	1	1	yes	characteristic
170	#339	—	4	1	no	—
171	#335	—	16	2	no	—
172	C₈²	—	2	1	no	—
173	C₈²	C₄×Q₈	1	1	yes	characteristic
174	C₈²	#125	1	1	yes	characteristic
175	C₂×M₁₆	—	4	1	no	—
176	C₂×M₁₆	—	16	1	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	#459	1	1	yes	characteristic
179	C₂²×C₈	#466	1	1	yes	characteristic
180	#117	—	4	1	no	—
181	C₂×C₄⋊C₄	—	16	2	no	—
182	C₂×C₄⋊C₄	—	16	1	no	—
183	C₂×C₄²	#472	1	1	yes	characteristic
184	M₃₂	—	4	2	no	—
185	M₃₂	—	8	1	no	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	8	1	no	—
191	C₂×C₁₆	C₂×#107	1	1	yes	characteristic
192	C₂×C₁₆	#428	1	1	yes	characteristic
193	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
194	C₂×C₁₆	C₄×Q₁₆	1	1	yes	characteristic
195	C₂×C₁₆	#474	1	1	yes	characteristic
196	C₂×C₁₆	#477	1	1	yes	characteristic
197	C₂×C₁₆	#499	1	1	yes	characteristic
198	C₂×C₁₆	#500	1	1	yes	characteristic
199	C₂×C₁₆	C₂×SD₃₂	1	1	yes	characteristic
200	C₂×C₁₆	C₂×SD₃₂	1	1	yes	characteristic
201	C₂×C₁₆	#509	1	1	yes	characteristic
202	C₂×C₁₆	#509	1	1	yes	characteristic
203	C₂×C₁₆	#509	1	1	yes	characteristic
204	C₂×C₁₆	#510	1	1	yes	characteristic
205	C₂×C₁₆	#510	1	1	yes	characteristic
206	C₂×C₁₆	#510	1	1	yes	characteristic
207	#108	—	4	1	no	—
208	#108	—	4	1	no	—
209	#108	—	16	1	no	—
210	#108	—	16	1	no	—
211	#108	—	16	1	no	—
212	#108	—	16	1	no	—
213	#107	—	4	1	no	—
214	#107	—	16	1	no	—
215	#107	—	16	1	no	—
216	#107	—	16	1	no	—
217	#107	—	16	1	no	—
218	#107	—	32	4	no	—
219	#106	—	4	1	no	—
220	C₄⋊C₈	—	16	2	no	—
221	#97	—	8	1	no	—
222	C₄×C₈	—	2	2	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	8	1	no	—
226	C₄×C₈	#389	1	1	yes	characteristic
227	C₄×C₈	#462	1	1	yes	characteristic
228	C₄×C₈	#467	1	1	yes	characteristic
229	C₄×C₈	#473	1	1	yes	characteristic
230	C₃₂	—	2	4	no	—
231	C₃₂	—	2	2	no	—
232	C₃₂	—	8	2	no	—
233	C₃₂	—	8	2	no	—
234	C₃₂	—	8	2	no	—
235	C₃₂	#367	1	4	yes	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	32	1	no	—
238	C₂²×C₄	#1667	1	1	yes	characteristic
239	M₁₆	—	8	1	no	—
240	M₁₆	—	8	1	no	—
241	M₁₆	—	32	1	no	—
242	M₁₆	—	32	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	4	no	—
246	C₂×C₈	—	8	1	no	—
247	C₂×C₈	—	32	1	no	—
248	C₂×C₈	#1669	1	1	yes	characteristic
249	C₂×C₈	#1674	1	1	yes	characteristic
250	C₂×C₈	#1942	1	1	yes	characteristic
251	C₂×C₈	#1947	1	1	yes	characteristic
252	C₂×C₈	#2207	1	1	yes	characteristic
253	C₂×C₈	#2208	1	1	yes	characteristic
254	C₂×C₈	#2213	1	1	yes	characteristic
255	C₂×C₈	#2214	1	1	yes	characteristic
256	C₂×C₈	#2220	1	1	yes	characteristic
257	C₂×C₈	#2220	1	1	yes	characteristic
258	C₂×C₈	#2223	1	1	yes	characteristic
259	C₂×C₈	#2223	1	1	yes	characteristic
260	C₄⋊C₄	—	8	1	no	—
261	C₄⋊C₄	—	32	4	no	—
262	C₄⋊C₄	—	32	1	no	—
263	C₄⋊C₄	—	32	1	no	—
264	C₂²⋊C₄	—	8	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	8	1	no	—
267	C₄²	#1676	1	1	yes	characteristic
268	C₄²	#1952	1	1	yes	characteristic
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	—	2	1	no	—
273	C₁₆	—	8	2	no	—
274	C₁₆	C₂×#367	1	1	yes	characteristic
275	C₁₆	#2159	1	1	yes	characteristic
276	C₁₆	#2176	1	1	yes	characteristic
277	C₁₆	#2177	1	1	yes	characteristic
278	C₁₆	#2225	1	1	yes	characteristic
279	C₁₆	#2226	1	1	yes	characteristic
280	C₁₆	#2255	1	1	yes	characteristic
281	C₁₆	#2255	1	1	yes	characteristic
282	C₂³	#12438	1	1	yes	characteristic
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	16	1	no	—
287	C₂×C₄	—	64	2	no	—
288	C₂×C₄	—	64	1	no	—
289	C₂×C₄	#9614	1	1	yes	characteristic
290	C₂×C₄	#12425	1	1	yes	characteristic
291	C₂×C₄	#12426	1	1	yes	characteristic
292	C₂×C₄	#12435	1	1	yes	characteristic
293	C₂×C₄	#12435	1	1	yes	characteristic
294	C₂×C₄	#12438	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	4	2	no	—
297	C₈	—	16	1	no	—
298	C₈	—	64	1	no	—
299	C₈	#12446	1	1	yes	characteristic
300	C₈	#12447	1	1	yes	characteristic
301	C₈	#12462	1	1	yes	characteristic
302	C₈	#12462	1	1	yes	characteristic
303	C₈	#13168	1	1	yes	characteristic
304	C₈	#13172	1	1	yes	characteristic
305	C₈	#13177	1	1	yes	characteristic
306	C₈	#13181	1	1	yes	characteristic
307	C₂²	—	4	1	no	—
308	C₂²	#123388	1	1	yes	characteristic
309	C₂²	#123393	1	1	yes	characteristic
310	C₂²	#140437	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	—	32	1	no	—
314	C₄	—	128	1	no	—
315	C₄	#123388	1	1	yes	characteristic
316	C₄	#123393	1	1	yes	characteristic
317	C₄	#140429	1	1	yes	characteristic
318	C₄	#140433	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	#72225600	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72225600
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂³	#12438
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
23. #93882	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72225599	2
#72225363	2
#72225352	2
#72225342	2
#72216560	2
#72216298	2
#72216273	2