Subgroups — #72227194

Order 2048.

538 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72227190	C₂	1	2	yes	maximal
2	#72227188	C₂	1	2	yes	maximal
3	#72225407	C₂	1	2	yes	maximal
4	#72216182	C₂	1	1	yes	maximal, characteristic
5	#151042	C₂²	1	4	yes	—
6	#149513	—	2	2	no	—
7	C₈×#346	C₄	1	1	yes	characteristic
8	C₈×#346	C₄	1	1	yes	characteristic
9	C₈×#346	C₂²	1	1	yes	characteristic
10	C₂×#7052	C₄	1	1	yes	characteristic
11	C₂×#7052	C₂²	1	1	yes	characteristic
12	C₂×#7052	C₂²	1	1	yes	characteristic
13	C₂×C₁₆²	C₄	1	1	yes	characteristic
14	#94011	—	2	2	no	—
15	#93876	C₄	1	8	yes	complemented
16	#13640	—	2	4	no	—
17	C₄×#346	—	2	1	no	—
18	C₄×#346	C₈	1	1	yes	characteristic
19	C₄×#346	C₈	1	1	yes	characteristic
20	C₄×#346	D₄	1	1	yes	characteristic
21	C₄×#346	Q₈	1	1	yes	characteristic
22	C₂×#1311	—	2	1	no	—
23	C₂×#1311	C₈	1	1	yes	characteristic
24	C₂×#1311	C₈	1	1	yes	characteristic
25	C₂×#1311	D₄	1	1	yes	characteristic
26	C₂×#1311	Q₈	1	1	yes	characteristic
27	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
33	C₄×C₈²	C₂×C₄	1	1	yes	characteristic
34	#7458	—	2	4	no	—
35	#7455	C₂×C₄	1	8	yes	—
36	#7332	—	4	2	no	—
37	#7330	—	4	2	no	—
38	#7052	—	2	2	no	—
39	#7052	—	2	2	no	—
40	#7052	C₂×C₄	1	4	yes	—
41	C₁₆²	C₈	1	4	yes	—
42	C₂×#346	—	2	4	no	—
43	C₂×#346	—	2	2	no	—
44	C₂×#346	—	2	2	no	—
45	C₂×#346	—	2	2	no	—
46	C₂×#346	—	2	2	no	—
47	C₂×C₄×C₁₆	—	2	1	no	—
48	C₂×C₄×C₁₆	—	2	1	no	—
49	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₆	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₈	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₈	M₁₆	1	1	yes	characteristic
60	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈²	C₂×C₈	1	1	yes	characteristic
62	C₂×C₈²	M₁₆	1	1	yes	characteristic
63	C₂×C₈²	C₂²×C₄	1	1	yes	characteristic
64	C₄⋊C₃₂	—	2	8	no	—
65	C₂²⋊C₃₂	—	4	4	no	—
66	#1397	—	4	4	no	—
67	#1311	—	2	4	no	—
68	#1311	—	2	2	no	—
69	#1311	—	2	1	no	—
70	#1311	—	2	1	no	—
71	#1311	—	2	2	no	—
72	#1311	SD₁₆	1	2	yes	—
73	#1311	SD₁₆	1	2	yes	—
74	C₈×C₁₆	—	2	1	no	—
75	C₈×C₁₆	—	2	1	no	—
76	C₈×C₁₆	—	2	1	no	—
77	C₈×C₁₆	—	2	1	no	—
78	C₈×C₁₆	C₄²	1	2	yes	—
79	C₈×C₁₆	C₄²	1	2	yes	—
80	C₈×C₁₆	C₂×C₈	1	2	yes	—
81	C₈×C₁₆	C₂×C₈	1	2	yes	—
82	C₈×C₁₆	C₂×C₈	1	2	yes	—
83	C₈×C₁₆	M₁₆	1	2	yes	—
84	C₈×C₁₆	C₂²×C₄	1	2	yes	—
85	C₈×C₁₆	#41	1	2	yes	—
86	C₂²×C₁₆	—	2	2	no	—
87	C₂²×C₁₆	—	2	1	no	—
88	C₂²×C₁₆	—	2	1	no	—
89	C₂²×C₁₆	—	2	2	no	—
90	C₂²×C₁₆	—	2	1	no	—
91	C₂²×C₁₆	—	2	1	no	—
92	C₂²×C₁₆	#108	1	2	yes	—
93	C₂²×C₁₆	M₃₂	1	2	yes	—
94	C₂²×C₁₆	M₃₂	1	2	yes	—
95	C₂²×C₁₆	#135	1	2	yes	—
96	C₂×C₄×C₈	—	2	1	no	—
97	C₂×C₄×C₈	—	2	1	no	—
98	C₂×C₄×C₈	—	2	1	no	—
99	C₂×C₄×C₈	—	2	1	no	—
100	C₂×C₄×C₈	—	2	2	no	—
101	C₂×C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×M₁₆	1	1	yes	characteristic
107	C₄³	C₄⋊C₈	1	1	yes	characteristic
108	C₂×C₃₂	—	4	8	no	—
109	#346	—	4	4	no	—
110	#346	—	4	8	no	—
111	#346	—	4	4	no	—
112	#346	—	4	4	no	—
113	#346	—	4	4	no	—
114	C₄×C₁₆	—	2	2	no	—
115	C₄×C₁₆	—	2	2	no	—
116	C₄×C₁₆	—	2	2	no	—
117	C₄×C₁₆	—	2	1	no	—
118	C₄×C₁₆	—	2	1	no	—
119	C₄×C₁₆	—	2	1	no	—
120	C₄×C₁₆	—	2	1	no	—
121	C₄×C₁₆	—	2	2	no	—
122	C₄×C₁₆	—	2	1	no	—
123	C₄×C₁₆	—	2	1	no	—
124	C₄×C₁₆	C₄×C₈	1	2	yes	—
125	C₄×C₁₆	C₄×C₈	1	2	yes	—
126	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
127	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
128	C₄×C₁₆	#106	1	1	yes	characteristic
129	C₄×C₁₆	#106	1	1	yes	characteristic
130	C₄×C₁₆	M₃₂	1	2	yes	—
131	C₄×C₁₆	M₃₂	1	1	yes	characteristic
132	C₄×C₁₆	M₃₂	1	1	yes	characteristic
133	C₄×C₁₆	C₄×D₄	1	2	yes	—
134	C₄×C₁₆	C₄×Q₈	1	2	yes	—
135	C₄×C₁₆	C₂×SD₁₆	1	2	yes	—
136	C₈²	—	2	2	no	—
137	C₈²	—	2	1	no	—
138	C₈²	—	2	1	no	—
139	C₈²	#106	1	2	yes	—
140	C₈²	#108	1	2	yes	—
141	C₈²	C₂×C₄²	1	1	yes	characteristic
142	C₈²	#117	1	1	yes	characteristic
143	C₈²	C₂²×C₈	1	1	yes	characteristic
144	C₈²	C₂×M₁₆	1	1	yes	characteristic
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	#427	1	1	yes	characteristic
150	C₂²×C₈	C₂×#108	1	1	yes	characteristic
151	C₂²×C₈	C₂×M₃₂	1	1	yes	characteristic
152	C₂²×C₈	C₂×M₃₂	1	1	yes	characteristic
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	#364	1	2	yes	—
156	C₂×C₄²	C₂×C₄⋊C₈	1	1	yes	characteristic
157	C₂×C₁₆	—	2	4	no	—
158	C₂×C₁₆	—	2	8	no	—
159	C₂×C₁₆	—	2	4	no	—
160	C₂×C₁₆	—	2	4	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	—	2	4	no	—
164	C₂×C₁₆	—	2	4	no	—
165	C₂×C₁₆	—	2	4	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	4	2	no	—
170	C₂×C₁₆	—	4	4	no	—
171	C₂×C₁₆	—	4	2	no	—
172	C₂×C₁₆	—	4	2	no	—
173	C₂×C₁₆	—	4	2	no	—
174	C₂×C₁₆	#334	1	2	yes	—
175	C₂×C₁₆	#334	1	2	yes	—
176	C₂×C₁₆	#346	1	2	yes	—
177	C₂×C₁₆	#346	1	2	yes	—
178	C₂×C₁₆	C₄×SD₁₆	1	4	yes	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	4	4	no	—
184	C₄×C₈	—	4	2	no	—
185	C₄×C₈	—	4	2	no	—
186	C₄×C₈	—	4	2	no	—
187	C₄×C₈	—	4	2	no	—
188	C₄×C₈	#334	1	2	yes	—
189	C₄×C₈	#334	1	2	yes	—
190	C₄×C₈	#364	1	2	yes	—
191	C₄×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
192	C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
193	C₄×C₈	C₄×M₁₆	1	1	yes	characteristic
194	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
195	C₄×C₈	C₂×#106	1	1	yes	characteristic
196	C₄×C₈	C₂×#108	1	1	yes	characteristic
197	C₄×C₈	#431	1	1	yes	characteristic
198	C₄×C₈	#431	1	1	yes	characteristic
199	C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
200	C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
201	C₃₂	—	8	8	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	#1564	1	1	yes	characteristic
204	C₂²×C₄	#2150	1	1	yes	characteristic
205	C₂²×C₄	C₂×#364	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂×#334	1	1	yes	characteristic
222	C₂×C₈	#1564	1	1	yes	characteristic
223	C₂×C₈	C₄×#106	1	1	yes	characteristic
224	C₂×C₈	C₄×#108	1	1	yes	characteristic
225	C₂×C₈	C₂×#346	1	1	yes	characteristic
226	C₂×C₈	C₄×M₃₂	1	1	yes	characteristic
227	C₂×C₈	#2163	1	1	yes	characteristic
228	C₂×C₈	#2163	1	1	yes	characteristic
229	C₄²	—	4	2	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	4	1	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	C₂×#334	1	1	yes	characteristic
235	C₄²	C₄×C₄⋊C₈	1	1	yes	characteristic
236	C₄²	#2150	1	1	yes	characteristic
237	C₄²	C₂×#364	1	1	yes	characteristic
238	C₁₆	—	2	8	no	—
239	C₁₆	—	2	4	no	—
240	C₁₆	—	2	4	no	—
241	C₁₆	—	2	4	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	4	4	no	—
245	C₁₆	—	4	2	no	—
246	C₁₆	—	4	2	no	—
247	C₁₆	—	4	2	no	—
248	C₁₆	#1369	1	4	yes	—
249	C₁₆	#1398	1	4	yes	—
250	C₂³	#12013	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₄×#334	1	1	yes	characteristic
260	C₂×C₄	#11996	1	1	yes	characteristic
261	C₂×C₄	#11996	1	1	yes	characteristic
262	C₂×C₄	#12013	1	1	yes	characteristic
263	C₂×C₄	#12653	1	1	yes	characteristic
264	C₂×C₄	C₄×#364	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	2	no	—
274	C₈	C₂×#1369	1	2	yes	—
275	C₈	#12668	1	2	yes	—
276	C₈	C₂×#1398	1	2	yes	—
277	C₈	#13642	1	2	yes	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	#129086	1	1	yes	characteristic
281	C₂²	#151045	1	2	yes	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	8	2	no	—
287	C₄	#129088	1	1	yes	characteristic
288	C₄	C₄×#1369	1	1	yes	characteristic
289	C₄	#151979	1	1	yes	characteristic
290	C₄	#151981	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	#72216247	1	1	yes	characteristic
293	C₂	#72227192	1	1	yes	characteristic
294	C₂	#72227193	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×#106
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂²	#129086
prime radical	C₂³	#12013
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#1311 ◁ C₈×#346 ◁ 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
15. #93876	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72227190	2
#72227188	2
#72225407	2
#72216182	2