Subgroups — #72227238

Order 2048.

694 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72227234	C₂	1	2	yes	maximal
2	#72227182	C₂	1	1	yes	maximal, characteristic
3	#72225375	C₂	1	1	yes	maximal, characteristic
4	#72216252	C₂	1	2	yes	maximal
5	#72216161	C₂	1	1	yes	maximal, characteristic
6	#129093	C₂²	1	2	yes	—
7	#129086	C₂²	1	2	yes	—
8	#129054	—	2	2	no	—
9	C₄×#1313	C₄	1	2	yes	—
10	C₄×#1313	C₄	1	2	yes	—
11	#126522	C₂²	1	1	yes	characteristic
12	C₈×#346	C₂²	1	1	yes	characteristic
13	C₂×#7101	C₄	1	1	yes	characteristic
14	C₂×#7100	C₄	1	1	yes	characteristic
15	C₂×#7099	C₄	1	2	yes	—
16	#94857	C₂²	1	1	yes	characteristic
17	#93258	—	2	2	no	—
18	#92913	C₄	1	2	yes	—
19	#92911	C₄	1	2	yes	—
20	#12653	—	2	2	no	—
21	#12653	—	2	2	no	—
22	C₄×#346	C₂×C₄	1	2	yes	—
23	C₄×#346	C₂×C₄	1	1	yes	characteristic
24	C₄×#346	C₂×C₄	1	1	yes	characteristic
25	C₂×#1370	C₂×C₄	1	2	yes	—
26	C₂×#1370	C₂×C₄	1	2	yes	—
27	C₂×#1367	C₂×C₄	1	4	yes	—
28	C₂×#1313	—	2	4	no	—
29	C₂×#1313	—	2	2	no	—
30	C₂×#1313	—	2	2	no	—
31	C₂×#1313	—	4	2	no	—
32	C₂×#1312	—	2	1	no	—
33	C₂×#1312	—	2	1	no	—
34	C₂×#1311	C₂×C₄	1	1	yes	characteristic
35	C₂×#1311	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₄×#322	C₂×C₄	1	2	yes	—
39	C₄×#322	D₄	1	2	yes	—
40	C₄×#322	Q₈	1	2	yes	—
41	C₄×C₈²	C₂³	1	1	yes	characteristic
42	#7151	—	2	2	no	—
43	#7151	—	2	2	no	—
44	#7137	—	4	2	no	—
45	#7101	C₈	1	4	yes	complemented
46	#7100	C₈	1	4	yes	complemented
47	#7099	—	2	4	no	—
48	C₂×C₄⋊C₁₆	—	2	2	no	—
49	C₂×C₄⋊C₁₆	—	2	2	no	—
50	C₂×C₄⋊C₁₆	—	2	4	no	—
51	C₂×#346	—	2	2	no	—
52	C₂×#346	—	2	2	no	—
53	C₂×#346	—	4	2	no	—
54	C₂×#346	—	4	2	no	—
55	C₂×#346	M₁₆	1	2	yes	—
56	C₂×#346	M₁₆	1	2	yes	—
57	C₂×#346	M₁₆	1	4	yes	—
58	C₂×C₄×C₁₆	—	2	2	no	—
59	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
60	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
61	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
62	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
63	C₄²×C₈	C₄⋊C₄	1	4	yes	—
64	C₄²×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₈	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₈	C₂×Q₈	1	1	yes	characteristic
67	C₂×#322	—	2	2	no	—
68	C₂×#322	—	2	2	no	—
69	C₂×#322	—	2	2	no	—
70	C₂×#322	—	2	2	no	—
71	C₂×#322	—	4	2	no	—
72	C₂×C₈²	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈²	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈²	#41	1	1	yes	characteristic
75	C₂×C₈²	#41	1	1	yes	characteristic
76	#1374	—	4	2	no	—
77	#1374	—	4	2	no	—
78	#1370	—	2	2	no	—
79	#1370	—	2	2	no	—
80	#1370	C₂×C₈	1	4	yes	—
81	#1370	C₂×C₈	1	4	yes	—
82	#1367	—	2	4	no	—
83	#1367	—	2	4	no	—
84	#1313	—	4	8	no	—
85	#1313	—	4	4	no	—
86	#1313	—	4	4	no	—
87	#1313	—	8	4	no	—
88	#1312	—	4	1	no	—
89	#1312	—	4	1	no	—
90	#1312	—	4	2	no	—
91	#1311	—	2	2	no	—
92	#1311	M₁₆	1	4	yes	—
93	C₈×C₁₆	—	2	1	no	—
94	C₈×C₁₆	—	2	1	no	—
95	C₈×C₁₆	C₂×C₈	1	2	yes	—
96	C₈×C₁₆	C₂×C₈	1	2	yes	—
97	C₂²×C₁₆	—	2	1	no	—
98	C₂²×C₁₆	—	2	1	no	—
99	C₂²×C₁₆	—	2	4	no	—
100	C₂²×C₁₆	—	4	2	no	—
101	C₂²×C₁₆	—	4	2	no	—
102	C₂²×C₁₆	—	4	4	no	—
103	C₂²×C₁₆	#97	1	1	yes	characteristic
104	C₂²×C₁₆	#97	1	2	yes	—
105	C₂²×C₁₆	#97	1	1	yes	characteristic
106	C₂×C₄×C₈	—	2	4	no	—
107	C₂×C₄×C₈	—	2	4	no	—
108	C₂×C₄×C₈	—	2	2	no	—
109	C₂×C₄×C₈	—	2	4	no	—
110	C₂×C₄×C₈	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₈	#117	1	1	yes	characteristic
112	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₈	C₄×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₈	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₈	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₈	#137	1	2	yes	—
120	C₄³	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₄⋊C₁₆	—	2	4	no	—
122	C₄⋊C₁₆	—	2	4	no	—
123	C₄⋊C₁₆	—	4	4	no	—
124	C₄⋊C₁₆	—	4	2	no	—
125	C₄⋊C₁₆	—	4	2	no	—
126	C₄⋊C₁₆	—	4	4	no	—
127	#346	—	2	8	no	—
128	#346	—	2	4	no	—
129	#346	—	2	4	no	—
130	#346	—	4	4	no	—
131	#346	—	4	2	no	—
132	#346	—	4	2	no	—
133	#346	—	8	2	no	—
134	#346	—	8	2	no	—
135	#346	—	8	2	no	—
136	#346	—	8	2	no	—
137	C₄×C₁₆	—	2	1	no	—
138	C₄×C₁₆	—	2	1	no	—
139	C₄×C₁₆	—	2	1	no	—
140	C₄×C₁₆	—	2	1	no	—
141	C₄×C₁₆	—	4	2	no	—
142	C₄×C₁₆	—	4	2	no	—
143	C₄×C₁₆	C₄×C₈	1	2	yes	—
144	C₄×C₁₆	C₄×C₈	1	2	yes	—
145	C₄×C₁₆	#97	1	2	yes	—
146	C₄×C₁₆	#97	1	2	yes	—
147	#322	—	4	2	no	—
148	#322	—	4	2	no	—
149	#322	—	4	4	no	—
150	#322	—	4	2	no	—
151	#322	—	4	2	no	—
152	#322	—	8	2	no	—
153	#322	—	8	4	no	—
154	C₈²	—	2	1	no	—
155	C₈²	—	2	1	no	—
156	C₈²	—	2	2	no	—
157	C₈²	C₂²×C₈	1	1	yes	characteristic
158	C₈²	C₂²×C₈	1	1	yes	characteristic
159	C₈²	C₂×M₁₆	1	2	yes	—
160	C₈²	#131	1	1	yes	characteristic
161	C₈²	#131	1	1	yes	characteristic
162	C₈²	#131	1	2	yes	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	4	4	no	—
166	C₂²×C₈	C₂×#97	1	1	yes	characteristic
167	C₂²×C₈	C₄×M₁₆	1	1	yes	characteristic
168	C₂²×C₈	#432	1	2	yes	—
169	C₂²×C₈	#440	1	2	yes	—
170	C₂²×C₈	#441	1	1	yes	characteristic
171	C₂²×C₈	#442	1	1	yes	characteristic
172	C₂×C₄²	C₄×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄²	#423	1	2	yes	—
174	C₂×C₄²	#428	1	2	yes	—
175	C₂×C₄²	#430	1	2	yes	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	—	2	4	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	—	4	4	no	—
184	C₂×C₁₆	—	4	4	no	—
185	C₂×C₁₆	—	4	4	no	—
186	C₂×C₁₆	—	4	4	no	—
187	C₂×C₁₆	—	4	4	no	—
188	C₂×C₁₆	—	8	2	no	—
189	C₂×C₁₆	—	8	2	no	—
190	C₂×C₁₆	—	8	2	no	—
191	C₂×C₁₆	—	8	2	no	—
192	C₂×C₁₆	—	8	4	no	—
193	C₂×C₁₆	—	8	4	no	—
194	C₂×C₁₆	—	8	4	no	—
195	C₂×C₁₆	#322	1	2	yes	—
196	C₂×C₁₆	#322	1	2	yes	—
197	C₂×C₁₆	#347	1	4	yes	—
198	C₂×C₁₆	#347	1	4	yes	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	4	2	no	—
209	C₄×C₈	—	4	4	no	—
210	C₄×C₈	—	4	4	no	—
211	C₄×C₈	—	4	4	no	—
212	C₄×C₈	—	4	4	no	—
213	C₄×C₈	—	4	4	no	—
214	C₄×C₈	—	8	4	no	—
215	C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
216	C₄×C₈	C₂×#97	1	1	yes	characteristic
217	C₄×C₈	#405	1	1	yes	characteristic
218	C₄×C₈	#405	1	1	yes	characteristic
219	C₄×C₈	C₈×D₄	1	1	yes	characteristic
220	C₄×C₈	C₈×D₄	1	1	yes	characteristic
221	C₄×C₈	#436	1	2	yes	—
222	C₄×C₈	#444	1	1	yes	characteristic
223	C₄×C₈	#444	1	1	yes	characteristic
224	C₄×C₈	#444	1	2	yes	—
225	C₄×C₈	C₈×Q₈	1	1	yes	characteristic
226	C₄×C₈	C₈×Q₈	1	1	yes	characteristic
227	C₄×C₈	#446	1	2	yes	—
228	C₂²×C₄	#1569	1	4	yes	—
229	C₂²×C₄	#1767	1	1	yes	characteristic
230	C₂²×C₄	#1776	1	1	yes	characteristic
231	C₂²×C₄	#1778	1	1	yes	characteristic
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	2	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	—	4	2	no	—
243	C₂×C₈	—	8	4	no	—
244	C₂×C₈	—	8	4	no	—
245	C₂×C₈	—	8	4	no	—
246	C₂×C₈	C₂×#322	1	1	yes	characteristic
247	C₂×C₈	C₈×M₁₆	1	1	yes	characteristic
248	C₂×C₈	#1451	1	1	yes	characteristic
249	C₂×C₈	#1452	1	1	yes	characteristic
250	C₂×C₈	#1579	1	1	yes	characteristic
251	C₂×C₈	#1579	1	1	yes	characteristic
252	C₂×C₈	#1580	1	1	yes	characteristic
253	C₂×C₈	#1580	1	1	yes	characteristic
254	C₂×C₈	#1592	1	4	yes	—
255	C₂×C₈	C₂×#347	1	2	yes	—
256	C₂×C₈	C₂×#347	1	2	yes	—
257	C₂×C₈	#2165	1	4	yes	—
258	C₂×C₈	#2180	1	2	yes	—
259	C₂×C₈	AGL(1,ℤ/16)	1	2	yes	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	2	no	—
265	C₄²	C₈×C₄⋊C₄	1	1	yes	characteristic
266	C₄²	#1770	1	1	yes	characteristic
267	C₄²	#1773	1	1	yes	characteristic
268	C₄²	#1773	1	1	yes	characteristic
269	C₁₆	—	2	4	no	—
270	C₁₆	—	2	4	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	2	4	no	—
273	C₁₆	—	8	4	no	—
274	C₁₆	—	8	4	no	—
275	C₁₆	—	8	4	no	—
276	C₁₆	—	16	4	no	—
277	C₁₆	#1313	1	4	yes	complemented
278	C₂³	#8276	1	1	yes	characteristic
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	#8261	1	1	yes	characteristic
284	C₂×C₄	#8269	1	1	yes	characteristic
285	C₂×C₄	#8284	1	1	yes	characteristic
286	C₂×C₄	#8286	1	1	yes	characteristic
287	C₂×C₄	#8294	1	1	yes	characteristic
288	C₂×C₄	#8295	1	1	yes	characteristic
289	C₂×C₄	#12658	1	2	yes	—
290	C₂×C₄	#12659	1	2	yes	—
291	C₂×C₄	#12677	1	2	yes	—
292	C₂×C₄	#12678	1	2	yes	—
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	4	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	4	no	—
301	C₈	—	4	2	no	—
302	C₈	—	16	4	no	—
303	C₈	C₂×#1313	1	1	yes	characteristic
304	C₈	#11404	1	1	yes	characteristic
305	C₈	#11415	1	2	yes	—
306	C₈	#12065	1	2	yes	—
307	C₈	#12066	1	1	yes	characteristic
308	C₈	#12067	1	1	yes	characteristic
309	C₂²	#96894	1	1	yes	characteristic
310	C₂²	#129123	1	2	yes	—
311	C₂²	#129124	1	2	yes	—
312	C₂²	#129125	1	2	yes	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	4	no	—
316	C₄	#129054	1	1	yes	characteristic
317	C₄	#129055	1	1	yes	characteristic
318	C₄	#129084	1	2	yes	—
319	C₄	#129165	1	1	yes	characteristic
320	C₄	#129166	1	1	yes	characteristic
321	C₄	#129174	1	2	yes	—
322	C₂	#72216225	1	1	yes	characteristic
323	C₂	#72216252	1	1	yes	characteristic
324	C₂	#72225332	1	1	yes	characteristic
325	C₂	#72227236	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	#1776
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×C₈²	C₂³
socle, prime radical	C₂³	#8276
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #94857 ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
45. #7101	C₈	semidirect
46. #7100	C₈	semidirect
277. C₁₆	#1313	semidirect

Maximal subgroups

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

subgroup	index
#72227234	2
#72227182	2
#72225375	2
#72216252	2
#72216161	2