Subgroups — #72262918

Order 2048.

At least 6127 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

The subgroup lattice has 307 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262916	C₂	1	2	yes	maximal, complemented
2	#72262915	C₂	1	2	yes	maximal, complemented
3	#72262914	C₂	1	2	yes	maximal, complemented
4	#72262913	C₂	1	2	yes	maximal, complemented
5	#72262827	C₂	1	1	yes	maximal, characteristic
6	#72256111	C₂	1	1	yes	maximal, characteristic, complemented
7	#72237995	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237582	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237486	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237459	C₂	1	1	yes	maximal, characteristic, complemented
11	#72234724	C₂	1	1	yes	maximal, characteristic, complemented
12	#6374128	C₂²	1	1	yes	characteristic
13	#5519715	C₄	1	1	yes	characteristic, complemented
14	#5519571	C₄	1	1	yes	characteristic, complemented
15	#5084903	C₂²	1	1	yes	characteristic
16	#5083335	C₂²	1	1	yes	characteristic
17	#4447413	C₄	1	1	yes	characteristic, complemented
18	#4439977	C₄	1	1	yes	characteristic, complemented
19	#4380237	C₂²	1	1	yes	characteristic
20	#4379716	C₂²	1	1	yes	characteristic
21	#2081109	C₄	1	1	yes	characteristic, complemented
22	#2081074	C₄	1	1	yes	characteristic, complemented
23	#1494104	C₄	1	1	yes	characteristic, complemented
24	#1460307	C₄	1	1	yes	characteristic, complemented
25	#659380	C₂²	1	1	yes	characteristic
26	#650612	C₂²	1	1	yes	characteristic
27	#617710	C₂²	1	1	yes	characteristic
28	#188122	C₂²	1	1	yes	characteristic
29	#188109	C₂²	1	1	yes	characteristic
30	#188109	C₂²	1	1	yes	characteristic
31	#139916	C₂²	1	2	yes	—
32	#139914	C₂²	1	2	yes	—
33	#139912	C₂²	1	2	yes	—
34	#139910	C₂²	1	2	yes	—
35	#139829	C₂²	1	2	yes	—
36	#139717	C₂²	1	2	yes	—
37	#139505	C₂²	1	2	yes	—
38	#139487	C₂²	1	2	yes	—
39	#135473	C₂²	1	2	yes	—
40	#133083	C₂²	1	2	yes	—
41	#119715	C₂²	1	2	yes	—
42	#115179	C₂²	1	2	yes	—
43	C₂²×#534	C₂³	1	1	yes	characteristic
44	C₂³×#127	D₄	1	1	yes	characteristic
45	C₂×#2675	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×#2670	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×#2404	D₄	1	1	yes	characteristic
48	C₂×#2431	D₄	1	1	yes	characteristic
49	C₂×#2410	C₂×C₄	1	1	yes	characteristic
50	C₂×#2403	D₄	1	1	yes	characteristic
51	C₂×#2615	C₂×C₄	1	1	yes	characteristic
52	C₂×#2613	D₄	1	1	yes	characteristic
53	C₂×#2613	D₄	1	1	yes	characteristic
54	C₂×#2613	C₂³	1	1	yes	characteristic
55	#21747	C₂³	1	2	yes	—
56	#17392	C₂³	1	2	yes	—
57	#17105	C₂×C₄	1	1	yes	characteristic, complemented
58	#17102	C₂×C₄	1	1	yes	characteristic, complemented
59	#16214	C₂×C₄	1	1	yes	characteristic, complemented
60	#15984	D₄	1	1	yes	characteristic
61	#15982	C₂³	1	1	yes	characteristic
62	#15950	C₂×C₄	1	1	yes	characteristic
63	#15948	D₄	1	1	yes	characteristic
64	#15948	C₂³	1	1	yes	characteristic
65	#15947	D₄	1	1	yes	characteristic
66	#15668	D₄	1	1	yes	characteristic
67	#15668	C₂³	1	1	yes	characteristic
68	#15636	C₂×C₄	1	1	yes	characteristic, complemented
69	#15622	C₂×C₄	1	1	yes	characteristic, complemented
70	#15420	C₂×C₄	1	1	yes	characteristic, complemented
71	#15412	D₄	1	1	yes	characteristic
72	#15412	C₂³	1	1	yes	characteristic
73	#15310	C₂×C₄	1	1	yes	characteristic
74	#15310	C₂×C₄	1	1	yes	characteristic
75	#15310	D₄	1	1	yes	characteristic
76	#15310	D₄	1	1	yes	characteristic
77	#15267	C₂×C₄	1	1	yes	characteristic, complemented
78	#15267	C₂×C₄	1	1	yes	characteristic, complemented
79	#15261	D₄	1	1	yes	characteristic
80	#15261	D₄	1	1	yes	characteristic
81	#15259	C₂×C₄	1	1	yes	characteristic
82	#15259	D₄	1	1	yes	characteristic
83	#15258	C₂×C₄	1	1	yes	characteristic
84	#15258	C₂×C₄	1	1	yes	characteristic
85	#15258	C₂×C₄	1	1	yes	characteristic
86	#15258	D₄	1	1	yes	characteristic
87	#15257	C₂×C₄	1	1	yes	characteristic
88	#15257	D₄	1	1	yes	characteristic
89	#14706	D₄	1	1	yes	characteristic
90	#14706	C₂³	1	1	yes	characteristic
91	#14659	D₄	1	1	yes	characteristic
92	#14657	C₂×C₄	1	1	yes	characteristic
93	#14651	C₂×C₄	1	1	yes	characteristic, complemented
94	#14651	C₂×C₄	1	1	yes	characteristic, complemented
95	#14335	D₄	1	1	yes	characteristic
96	#14290	C₂×C₄	1	1	yes	characteristic
97	#14255	D₄	1	1	yes	characteristic
98	#14232	C₂×C₄	1	1	yes	characteristic, complemented
99	#14207	C₂×C₄	1	1	yes	characteristic, complemented
100	#13903	D₄	1	1	yes	characteristic
101	#13867	D₄	1	1	yes	characteristic
102	#13823	C₂×C₄	1	1	yes	characteristic, complemented
103	C₂²⋊C₄²	C₂×C₄	1	1	yes	characteristic, complemented
104	#8553	C₂³	1	2	yes	—
105	#8548	C₂³	1	2	yes	—
106	C₂³×#41	C₂²⋊C₄	1	1	yes	characteristic
107	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
108	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
109	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
110	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
111	C₂²×#127	C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×#127	C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×#127	C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×#127	C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×#127	C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×#124	C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×#123	C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×#123	C₂×D₄	1	1	yes	characteristic
119	C₂²×#123	C₂×D₄	1	1	yes	characteristic
120	C₂²×#123	C₂×D₄	1	1	yes	characteristic
121	C₂²×#123	C₂⁴	1	1	yes	characteristic
122	C₂²×#121	C₂²×C₄	1	1	yes	characteristic
123	C₂²×#121	C₂×D₄	1	1	yes	characteristic
124	C₂²×#121	C₂×D₄	1	1	yes	characteristic
125	C₂²×#121	C₂×D₄	1	1	yes	characteristic
126	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
130	C₂²×#120	C₂×D₄	1	1	yes	characteristic
131	C₂²×#120	C₂×D₄	1	1	yes	characteristic
132	C₂²×#120	C₂×D₄	1	1	yes	characteristic
133	C₂²×#120	C₂×D₄	1	1	yes	characteristic
134	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
135	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
136	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
137	C₂×#394	C₂²⋊C₄	1	1	yes	characteristic
138	C₂×#394	C₂²⋊C₄	1	1	yes	characteristic
139	C₂×#394	C₂×D₄	1	1	yes	characteristic
140	C₂×#394	C₂×D₄	1	1	yes	characteristic
141	C₂×#393	C₂²⋊C₄	1	1	yes	characteristic
142	C₂×#393	C₂²⋊C₄	1	1	yes	characteristic
143	C₂×#393	C₂²×C₄	1	1	yes	characteristic
144	C₂×#393	C₂×D₄	1	1	yes	characteristic
145	C₂×#392	C₂²⋊C₄	1	1	yes	characteristic
146	C₂×#392	C₂²⋊C₄	1	1	yes	characteristic
147	C₂×#392	C₂²⋊C₄	1	1	yes	characteristic
148	C₂×#392	C₂²⋊C₄	1	1	yes	characteristic
149	C₂×#392	C₂²×C₄	1	1	yes	characteristic
150	C₂×#392	C₂²×C₄	1	1	yes	characteristic
151	C₂×#392	C₂²×C₄	1	1	yes	characteristic
152	C₂×#392	C₂×D₄	1	1	yes	characteristic
153	C₂×#391	C₂×D₄	1	1	yes	characteristic
154	C₂×#390	C₂×D₄	1	1	yes	characteristic
155	C₂×#386	C₂²⋊C₄	1	1	yes	characteristic
156	C₂×#386	C₂²⋊C₄	1	1	yes	characteristic
157	C₂×#386	C₂×D₄	1	1	yes	characteristic
158	C₂×#385	C₂²×C₄	1	1	yes	characteristic
159	C₂×#381	C₂²⋊C₄	1	1	yes	characteristic
160	C₂×#381	C₂×D₄	1	1	yes	characteristic
161	C₂×#381	C₂×D₄	1	1	yes	characteristic
162	C₂×#381	C₂×D₄	1	1	yes	characteristic
163	C₂×#380	C₂²⋊C₄	1	1	yes	characteristic
164	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
165	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
166	C₂×#379	C₂²×C₄	1	1	yes	characteristic
167	C₂×#379	C₂×D₄	1	1	yes	characteristic
168	C₂×C₄×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
169	#1445	C₂×D₄	1	1	yes	characteristic
170	#1445	C₂×D₄	1	1	yes	characteristic
171	#1440	C₂²×C₄	1	1	yes	characteristic
172	#1440	C₂²×C₄	1	1	yes	characteristic
173	#1440	C₂×D₄	1	1	yes	characteristic
174	#1440	C₂×D₄	1	1	yes	characteristic
175	#1438	C₂²⋊C₄	1	1	yes	characteristic
176	#1438	C₂²⋊C₄	1	1	yes	characteristic
177	#1438	C₂²⋊C₄	1	1	yes	characteristic
178	#1438	C₂²⋊C₄	1	1	yes	characteristic
179	#1438	C₂²⋊C₄	1	1	yes	characteristic
180	#1438	C₂²⋊C₄	1	1	yes	characteristic
181	#1438	C₂²⋊C₄	1	1	yes	characteristic
182	#1438	C₂²⋊C₄	1	1	yes	characteristic
183	#1438	C₂²⋊C₄	1	1	yes	characteristic
184	#1438	C₂²⋊C₄	1	1	yes	characteristic
185	#1438	C₂²⋊C₄	1	1	yes	characteristic
186	#1438	C₂²⋊C₄	1	1	yes	characteristic
187	#1438	C₂×D₄	1	1	yes	characteristic
188	#1438	C₂×D₄	1	1	yes	characteristic
189	#1437	C₂²⋊C₄	1	1	yes	characteristic
190	#1437	C₂²⋊C₄	1	1	yes	characteristic
191	#1437	C₂²⋊C₄	1	1	yes	characteristic
192	#1437	C₂²⋊C₄	1	1	yes	characteristic
193	#1437	C₂²⋊C₄	1	1	yes	characteristic
194	#1437	C₂²⋊C₄	1	1	yes	characteristic
195	#1437	C₂²×C₄	1	1	yes	characteristic
196	#1437	C₂²×C₄	1	1	yes	characteristic
197	#1437	C₂²×C₄	1	1	yes	characteristic
198	#1437	C₂²×C₄	1	1	yes	characteristic
199	#1437	C₂×D₄	1	1	yes	characteristic
200	#1437	C₂×D₄	1	1	yes	characteristic
201	#1436	C₂²⋊C₄	1	1	yes	characteristic
202	#1436	C₂²⋊C₄	1	1	yes	characteristic
203	#1436	C₂×D₄	1	1	yes	characteristic
204	#1436	C₂×D₄	1	1	yes	characteristic
205	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
206	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	C₂⁶	#120	1	1	yes	characteristic
208	C₂⁶	#120	1	1	yes	characteristic
209	C₂³×Q₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂³×D₄	#120	1	1	yes	characteristic
220	C₂³×D₄	#120	1	1	yes	characteristic
221	C₂³×D₄	#120	1	1	yes	characteristic
222	C₂³×D₄	#120	1	1	yes	characteristic
223	C₂³×D₄	#120	1	1	yes	characteristic
224	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
225	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	C₂⁴×C₄	#120	1	1	yes	characteristic
228	C₂⁴×C₄	#120	1	1	yes	characteristic
229	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
230	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
232	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
233	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
234	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
239	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
240	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
241	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
244	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
245	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
246	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
247	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
248	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
249	C₂²×C₂²⋊C₄	C₂³×C₄	1	1	yes	characteristic
250	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
251	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
252	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂⁵	#379	1	1	yes	characteristic
265	C₂⁵	#379	1	1	yes	characteristic
266	C₂⁵	#379	1	1	yes	characteristic
267	C₂⁵	#379	1	1	yes	characteristic
268	C₂⁵	#379	1	1	yes	characteristic
269	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
270	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
271	C₂⁵	#458	1	1	yes	characteristic
272	C₂⁵	#458	1	1	yes	characteristic
273	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
274	C₂⁵	C₂×#120	1	1	yes	characteristic
275	C₂³×C₄	#379	1	1	yes	characteristic
276	C₂³×C₄	#379	1	1	yes	characteristic
277	C₂³×C₄	#379	1	1	yes	characteristic
278	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
279	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
280	C₂³×C₄	C₂×#120	1	1	yes	characteristic
281	C₂³×C₄	C₂×#120	1	1	yes	characteristic
282	C₂³×C₄	C₂×#120	1	1	yes	characteristic
283	C₂⁴	#1790	1	1	yes	characteristic
284	C₂⁴	#1790	1	1	yes	characteristic
285	C₂⁴	#1790	1	1	yes	characteristic
286	C₂⁴	#1790	1	1	yes	characteristic
287	C₂⁴	C₂×#379	1	1	yes	characteristic
288	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
289	C₂⁴	C₂×#458	1	1	yes	characteristic
290	C₂²×C₄	#2883	1	2	yes	—
291	C₂²×C₄	#3026	1	2	yes	—
292	C₂³	#20585	1	1	yes	characteristic
293	C₂³	C₂×#1790	1	1	yes	characteristic
294	C₂³	#20609	1	1	yes	characteristic
295	C₂³	#33551	1	2	yes	—
296	C₂²	#187934	1	1	yes	characteristic
297	C₂²	#493492	1	1	yes	characteristic
298	C₂²	#493496	1	1	yes	characteristic
299	C₂²	#493562	1	1	yes	characteristic
300	C₂²	#493582	1	1	yes	characteristic
301	C₂²	#493641	1	1	yes	characteristic
302	C₂²	#493641	1	1	yes	characteristic
303	C₂	#72262280	1	1	yes	characteristic
304	C₂	#72262886	1	1	yes	characteristic
305	C₂	#72262888	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#187934
commutator	C₂²×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂²×#123	C₂⁴
prime radical	#72262827	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#534 ◁ G
LECS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72262916	C₂	semidirect
2. #72262915	C₂	semidirect
3. #72262914	C₂	semidirect
4. #72262913	C₂	semidirect
6. #72256111	C₂	semidirect
7. #72237995	C₂	semidirect
8. #72237582	C₂	semidirect
9. #72237486	C₂	semidirect
10. #72237459	C₂	semidirect
11. #72234724	C₂	semidirect
13. #5519715	C₄	semidirect
14. #5519571	C₄	semidirect
17. #4447413	C₄	semidirect
18. #4439977	C₄	semidirect
21. #2081109	C₄	semidirect
22. #2081074	C₄	semidirect
23. #1494104	C₄	semidirect
24. #1460307	C₄	semidirect
45. C₂×#2675	C₂×C₄	semidirect
46. C₂×#2670	C₂×C₄	semidirect
57. #17105	C₂×C₄	semidirect
58. #17102	C₂×C₄	semidirect
59. #16214	C₂×C₄	semidirect
68. #15636	C₂×C₄	semidirect
69. #15622	C₂×C₄	semidirect
70. #15420	C₂×C₄	semidirect
77. #15267	C₂×C₄	semidirect
78. #15267	C₂×C₄	semidirect
93. #14651	C₂×C₄	semidirect
94. #14651	C₂×C₄	semidirect
98. #14232	C₂×C₄	semidirect
99. #14207	C₂×C₄	semidirect
102. #13823	C₂×C₄	semidirect
103. C₂²⋊C₄²	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#72262916	2
#72262915	2
#72262914	2
#72262913	2
#72262827	2
#72256111	2
#72237995	2
#72237582	2
#72237486	2
#72237459	2
#72234724	2