Subgroups — #22284

Order 256.

398 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3154	C₂	1	1	yes	maximal, characteristic
2	#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	#2949	C₂	1	1	yes	maximal, characteristic, complemented
4	#1977	C₂	1	1	yes	maximal, characteristic, complemented
5	#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	#1912	C₂	1	1	yes	maximal, characteristic, complemented
7	#1911	C₂	1	1	yes	maximal, characteristic, complemented
8	#1910	C₂	1	1	yes	maximal, characteristic, complemented
9	#1909	C₂	1	1	yes	maximal, characteristic, complemented
10	#1905	C₂	1	1	yes	maximal, characteristic, complemented
11	#1904	C₂	1	1	yes	maximal, characteristic, complemented
12	#1903	C₂	1	1	yes	maximal, characteristic, complemented
13	#1902	C₂	1	1	yes	maximal, characteristic, complemented
14	#1833	C₂	1	1	yes	maximal, characteristic, complemented
15	#1747	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137	C₂²	1	1	yes	characteristic
17	C₂×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
18	C₂×#135	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆	C₂²	1	1	yes	characteristic
20	#568	C₂²	1	1	yes	characteristic
21	#568	C₂²	1	1	yes	characteristic
22	#536	C₂²	1	1	yes	characteristic
23	#535	C₂²	1	1	yes	characteristic
24	#519	C₂²	1	1	yes	characteristic
25	#518	C₂²	1	1	yes	characteristic
26	#497	C₄	1	2	yes	complemented
27	#496	C₄	1	2	yes	complemented
28	#495	C₄	1	2	yes	complemented
29	#492	C₄	1	2	yes	complemented
30	#444	—	2	2	no	—
31	#443	—	2	2	no	—
32	#440	—	2	2	no	—
33	C₄×SD₁₆	—	2	2	no	—
34	C₂×#108	C₂²	1	1	yes	characteristic
35	#427	C₂²	1	1	yes	characteristic
36	#421	C₂²	1	1	yes	characteristic
37	#421	C₂²	1	1	yes	characteristic
38	#421	C₂²	1	1	yes	characteristic
39	#421	C₂²	1	1	yes	characteristic
40	#419	C₂²	1	1	yes	characteristic
41	#419	C₂²	1	1	yes	characteristic
42	#418	C₂²	1	1	yes	characteristic
43	#418	C₂²	1	1	yes	characteristic
44	C₂×#101	C₂²	1	1	yes	characteristic
45	C₂×#100	C₂²	1	1	yes	characteristic
46	#410	C₂²	1	1	yes	characteristic
47	#410	C₂²	1	1	yes	characteristic
48	#405	C₂²	1	1	yes	characteristic
49	#405	C₂²	1	1	yes	characteristic
50	#405	C₂²	1	1	yes	characteristic
51	#344	C₂²	1	1	yes	characteristic
52	#344	C₂²	1	1	yes	characteristic
53	#338	C₂²	1	1	yes	characteristic
54	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
55	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
56	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
57	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
58	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
59	C₂×#41	C₂³	1	1	yes	characteristic
60	C₂×#41	C₂³	1	1	yes	characteristic
61	C₂²×Q₈	C₂³	1	1	yes	characteristic
62	C₂²×D₄	C₂³	1	1	yes	characteristic
63	#137	—	4	1	no	—
64	#137	—	4	1	no	—
65	AGL(1,ℤ/8)	—	4	1	no	—
66	AGL(1,ℤ/8)	—	4	1	no	—
67	#135	—	2	1	no	—
68	#135	—	2	1	no	—
69	#135	—	2	1	no	—
70	#135	—	2	1	no	—
71	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
72	C₂×Q₁₆	C₂×C₄	1	2	yes	complemented
73	C₂×SD₁₆	—	2	1	no	—
74	C₂×SD₁₆	—	2	1	no	—
75	C₂×SD₁₆	—	2	1	no	—
76	C₂×SD₁₆	—	2	1	no	—
77	C₂×SD₁₆	C₂×C₄	1	2	yes	—
78	C₂×SD₁₆	C₂×C₄	1	2	yes	—
79	C₂×SD₁₆	C₂×C₄	1	2	yes	—
80	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic, complemented
81	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic, complemented
82	C₂×SD₁₆	C₂×C₄	1	2	yes	complemented
83	C₂×D₈	C₂×C₄	1	2	yes	—
84	C₂×D₈	C₂×C₄	1	1	yes	characteristic, complemented
85	#131	—	2	2	no	—
86	#131	—	2	2	no	—
87	#131	—	4	1	no	—
88	#131	—	4	1	no	—
89	C₂×M₁₆	—	4	1	no	—
90	C₂×M₁₆	—	4	1	no	—
91	C₂×M₁₆	C₂³	1	1	yes	characteristic
92	C₂×M₁₆	C₂³	1	1	yes	characteristic
93	C₂×M₁₆	C₂³	1	1	yes	characteristic
94	C₂×M₁₆	C₂³	1	1	yes	characteristic
95	C₂×M₁₆	C₂³	1	1	yes	characteristic
96	C₂²×C₈	C₂³	1	1	yes	characteristic
97	C₄⋊Q₈	C₂×C₄	1	2	yes	complemented
98	#127	C₂×C₄	1	2	yes	—
99	#124	C₂×C₄	1	2	yes	—
100	#124	C₂×C₄	1	2	yes	complemented
101	#123	—	4	1	no	—
102	C₂²⋊Q₈	—	4	1	no	—
103	#121	—	4	1	no	—
104	#120	—	4	1	no	—
105	C₄×Q₈	—	2	2	no	—
106	C₄×Q₈	—	4	1	no	—
107	C₄×D₄	—	2	2	no	—
108	C₄×D₄	—	4	1	no	—
109	#117	—	4	1	no	—
110	#117	C₂³	1	1	yes	characteristic
111	#117	C₂³	1	1	yes	characteristic
112	#117	C₂³	1	1	yes	characteristic
113	#117	C₂³	1	1	yes	characteristic
114	#117	C₂³	1	1	yes	characteristic
115	C₂×C₂²⋊C₄	—	4	1	no	—
116	#108	—	2	1	no	—
117	#108	—	2	1	no	—
118	#107	—	2	1	no	—
119	#106	—	2	1	no	—
120	#104	—	2	2	no	—
121	#104	—	2	2	no	—
122	#104	—	4	1	no	—
123	#104	—	4	1	no	—
124	#103	—	2	2	no	—
125	#103	—	4	1	no	—
126	#102	—	2	2	no	—
127	#102	—	4	1	no	—
128	#101	—	4	1	no	—
129	#100	—	4	1	no	—
130	#99	—	4	1	no	—
131	#99	—	4	1	no	—
132	#97	—	2	1	no	—
133	#97	C₂×C₄	1	2	yes	complemented
134	#97	D₄	1	2	yes	complemented
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	C₂×C₄	1	2	yes	complemented
137	C₄×C₈	D₄	1	2	yes	complemented
138	C₂⁴	—	2	1	no	—
139	#41	—	2	1	no	—
140	#41	—	2	1	no	—
141	#41	—	2	1	no	—
142	#41	—	2	1	no	—
143	#41	—	2	1	no	—
144	#41	—	2	1	no	—
145	#41	—	2	1	no	—
146	#41	—	2	1	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
153	C₂×Q₈	C₂²×C₄	1	2	yes	—
154	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	C₂²×C₄	1	2	yes	—
164	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
165	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	C₂⁴	1	1	yes	characteristic
171	Q₁₆	—	2	1	no	—
172	Q₁₆	—	2	1	no	—
173	Q₁₆	—	4	2	no	—
174	SD₁₆	—	2	2	no	—
175	SD₁₆	—	2	2	no	—
176	SD₁₆	—	2	2	no	—
177	SD₁₆	—	2	2	no	—
178	SD₁₆	—	2	1	no	—
179	SD₁₆	—	2	1	no	—
180	SD₁₆	—	2	1	no	—
181	SD₁₆	—	2	1	no	—
182	SD₁₆	—	4	2	no	—
183	SD₁₆	—	4	2	no	—
184	D₈	—	2	1	no	—
185	D₈	—	2	1	no	—
186	D₈	—	4	2	no	—
187	M₁₆	—	2	1	no	—
188	M₁₆	—	2	1	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	2	1	no	—
192	M₁₆	—	2	1	no	—
193	M₁₆	—	2	1	no	—
194	M₁₆	—	2	1	no	—
195	M₁₆	—	2	1	no	—
196	M₁₆	—	2	1	no	—
197	M₁₆	—	4	2	no	—
198	M₁₆	—	4	2	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	C₂²×C₄	1	2	yes	—
205	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
206	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄	1	2	yes	—
208	C₂×C₈	C₂×D₄	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄	1	1	yes	characteristic
210	C₂×C₈	#41	1	1	yes	characteristic
211	C₂×C₈	#41	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂²×C₄	1	2	yes	—
232	C₄²	C₂×D₄	1	2	yes	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	—	16	1	no	—
239	C₂³	H(5,𝔽₂)	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	2	1	no	—
242	Q₈	—	2	1	no	—
243	Q₈	—	2	1	no	—
244	Q₈	—	2	2	no	—
245	Q₈	—	2	1	no	—
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	2	1	no	—
249	D₄	—	2	1	no	—
250	D₄	—	2	2	no	—
251	D₄	—	2	1	no	—
252	D₄	—	2	2	no	—
253	D₄	—	2	1	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	1	no	—
257	D₄	—	4	1	no	—
258	D₄	—	4	1	no	—
259	D₄	—	4	1	no	—
260	D₄	—	16	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	16	2	no	—
276	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂×#41	1	1	yes	characteristic
279	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	2	2	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	4	1	no	—
287	C₈	C₄×D₄	1	1	yes	characteristic
288	C₈	C₄×D₄	1	1	yes	characteristic
289	C₈	C₄×D₄	1	1	yes	characteristic
290	C₈	C₄×D₄	1	1	yes	characteristic
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	8	1	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	1	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	16	1	no	—
300	C₂²	#518	1	1	yes	characteristic
301	C₂²	#535	1	1	yes	characteristic
302	C₂²	#540	1	1	yes	characteristic
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	1	no	—
310	C₄	—	8	1	no	—
311	C₄	C₂×C₄×D₄	1	1	yes	characteristic
312	C₄	#518	1	1	yes	characteristic
313	C₄	#535	1	1	yes	characteristic
314	C₄	#540	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	8	1	no	—
319	C₂	—	8	1	no	—
320	C₂	—	8	2	no	—
321	C₂	#2328	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2328
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#3154	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2957	C₂	semidirect
3. #2949	C₂	semidirect
4. #1977	C₂	semidirect
5. #1976	C₂	semidirect
6. #1912	C₂	semidirect
7. #1911	C₂	semidirect
8. #1910	C₂	semidirect
9. #1909	C₂	semidirect
10. #1905	C₂	semidirect
11. #1904	C₂	semidirect
12. #1903	C₂	semidirect
13. #1902	C₂	semidirect
14. #1833	C₂	semidirect
15. #1747	C₂	semidirect
26. #497	C₄	semidirect
27. #496	C₄	semidirect
28. #495	C₄	semidirect
29. #492	C₄	semidirect
71. C₂×Q₁₆	C₂×C₄	semidirect
72. C₂×Q₁₆	C₂×C₄	semidirect
80. C₂×SD₁₆	C₂×C₄	semidirect
81. C₂×SD₁₆	C₂×C₄	semidirect
82. C₂×SD₁₆	C₂×C₄	semidirect
84. C₂×D₈	C₂×C₄	semidirect
97. C₄⋊Q₈	C₂×C₄	semidirect
100. #124	C₂×C₄	semidirect
133. #97	C₂×C₄	semidirect
134. #97	D₄	semidirect
136. C₄×C₈	C₂×C₄	semidirect
137. C₄×C₈	D₄	semidirect

Maximal subgroups

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

subgroup	index
#3154	2
#2957	2
#2949	2
#1977	2
#1976	2
#1912	2
#1911	2
#1910	2
#1909	2
#1905	2
#1904	2
#1903	2
#1902	2
#1833	2
#1747	2