Subgroups — #22955

Order 256.

420 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	#1940	C₂	1	1	yes	maximal, characteristic, complemented
6	#1940	C₂	1	1	yes	maximal, characteristic, complemented
7	#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	#1917	C₂	1	1	yes	maximal, characteristic, complemented
9	#1916	C₂	1	1	yes	maximal, characteristic, complemented
10	#1851	C₂	1	1	yes	maximal, characteristic
11	#1851	C₂	1	1	yes	maximal, characteristic
12	#1771	C₂	1	1	yes	maximal, characteristic, complemented
13	#1753	C₂	1	1	yes	maximal, characteristic
14	#1753	C₂	1	1	yes	maximal, characteristic
15	#1749	C₂	1	1	yes	maximal, characteristic
16	C₂²×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈	C₄	1	1	yes	characteristic
19	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
20	C₂²×C₄²	C₂²	1	1	yes	characteristic
21	#446	—	2	4	no	—
22	#423	—	2	4	no	—
23	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×#97	C₂²	1	1	yes	characteristic
34	C₂×#97	C₂²	1	1	yes	characteristic
35	C₂×#97	C₂²	1	1	yes	characteristic
36	C₂×#97	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	#391	C₄	1	1	yes	characteristic
40	#389	C₄	1	1	yes	characteristic
41	#387	C₄	1	1	yes	characteristic
42	#387	C₄	1	1	yes	characteristic
43	#385	C₄	1	1	yes	characteristic
44	#385	C₄	1	1	yes	characteristic
45	#380	C₄	1	1	yes	characteristic
46	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
49	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
51	#336	C₂²	1	1	yes	characteristic
52	#336	C₂²	1	1	yes	characteristic
53	#336	C₂²	1	1	yes	characteristic
54	#336	C₂²	1	1	yes	characteristic
55	#336	C₂²	1	1	yes	characteristic
56	#336	C₂²	1	1	yes	characteristic
57	#336	C₂²	1	1	yes	characteristic
58	#336	C₂²	1	1	yes	characteristic
59	#336	C₂²	1	1	yes	characteristic
60	#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄	C₂³	1	1	yes	characteristic
65	C₂×M₁₆	—	2	2	no	—
66	C₂×M₁₆	—	2	2	no	—
67	C₂×M₁₆	—	2	2	no	—
68	C₂×M₁₆	—	2	2	no	—
69	C₂×M₁₆	—	4	1	no	—
70	C₂×M₁₆	—	4	1	no	—
71	C₂²×C₈	C₂³	1	1	yes	characteristic
72	C₂²×C₈	C₂³	1	1	yes	characteristic
73	C₂²×C₈	C₂³	1	1	yes	characteristic
74	C₂²×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₈	C₂³	1	1	yes	characteristic
78	C₂²×C₈	C₂³	1	1	yes	characteristic
79	C₂²⋊Q₈	—	2	4	no	—
80	C₄×Q₈	—	2	2	no	—
81	C₄×Q₈	—	2	2	no	—
82	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²	—	2	2	no	—
96	C₂×C₄²	—	2	2	no	—
97	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²	C₂³	1	1	yes	characteristic
100	C₂×C₄²	C₂³	1	1	yes	characteristic
101	C₂×C₄²	C₂³	1	1	yes	characteristic
102	C₂×C₄²	C₂³	1	1	yes	characteristic
103	C₂×C₄²	C₂³	1	1	yes	characteristic
104	C₂×C₄²	C₂³	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	4	1	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄⋊C₈	D₄	1	4	yes	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₂²⋊C₈	Q₈	1	4	yes	complemented
115	#97	—	2	2	no	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	#97	—	4	1	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	4	1	no	—
121	#95	C₂×C₄	1	1	yes	characteristic
122	#95	C₂×C₄	1	1	yes	characteristic
123	#95	C₂×C₄	1	1	yes	characteristic
124	#95	C₂×C₄	1	1	yes	characteristic
125	#95	C₂×C₄	1	1	yes	characteristic
126	#95	C₂×C₄	1	1	yes	characteristic
127	#95	C₂×C₄	1	1	yes	characteristic
128	#95	C₂×C₄	1	1	yes	characteristic
129	#95	C₂×C₄	1	1	yes	characteristic
130	#95	C₂×C₄	1	1	yes	characteristic
131	C₂⁴	C₂²×C₄	1	1	yes	characteristic
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	—	4	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	1	no	—
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×Q₈	1	2	yes	—
159	C₂²×C₄	#41	1	2	yes	—
160	C₂²×C₄	C₂⁴	1	1	yes	characteristic
161	M₁₆	—	4	4	no	—
162	M₁₆	—	4	4	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	C₂×D₄	1	2	yes	—
184	C₂×C₈	C₂×D₄	1	2	yes	—
185	C₂×C₈	C₂×Q₈	1	2	yes	—
186	C₂×C₈	C₂×Q₈	1	2	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	4	1	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	M₁₆	1	4	yes	complemented
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	4	1	no	—
199	C₂²⋊C₄	—	4	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	C₂×D₄	1	2	yes	—
213	C₄²	#41	1	2	yes	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	C₄×Q₈	1	2	yes	—
221	C₂³	C₂³×C₄	1	1	yes	characteristic
222	Q₈	—	4	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	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	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₄×D₄	1	2	yes	—
263	C₂×C₄	C₄×D₄	1	2	yes	—
264	C₂×C₄	C₄×Q₈	1	2	yes	—
265	C₂×C₄	C₂×M₁₆	1	2	yes	—
266	C₂×C₄	C₂×M₁₆	1	2	yes	—
267	C₂×C₄	C₂×M₁₆	1	2	yes	—
268	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
270	C₂×C₄	C₂×#41	1	1	yes	characteristic
271	C₂×C₄	C₂×#41	1	1	yes	characteristic
272	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	—	4	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
290	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
291	C₂²	#520	1	1	yes	characteristic
292	C₂²	C₂²×M₁₆	1	1	yes	characteristic
293	C₂²	#568	1	1	yes	characteristic
294	C₂²	#568	1	1	yes	characteristic
295	C₂²	#568	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	#546	1	1	yes	characteristic
310	C₄	D₄×Q₈	1	1	yes	characteristic
311	C₄	#552	1	1	yes	characteristic
312	C₄	#554	1	1	yes	characteristic
313	C₂	—	4	2	no	—
314	C₂	#2360	1	1	yes	characteristic
315	C₂	#2930	1	1	yes	characteristic
316	C₂	D₄×M₁₆	1	1	yes	characteristic
317	C₂	#2961	1	1	yes	characteristic
318	C₂	Q₈×M₁₆	1	1	yes	characteristic
319	C₂	#2989	1	1	yes	characteristic
320	C₂	#2995	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator, socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ 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
1. C₂×#446	C₂	semidirect
4. #1985	C₂	semidirect
5. #1940	C₂	semidirect
6. #1940	C₂	semidirect
7. #1939	C₂	semidirect
8. #1917	C₂	semidirect
9. #1916	C₂	semidirect
12. #1771	C₂	semidirect
114. C₂²⋊C₈	Q₈	semidirect
195. C₄⋊C₄	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446	2
C₂×#423	2
C₄×C₂²⋊Q₈	2
#1985	2
#1940	2
#1939	2
#1917	2
#1916	2
#1851	2
#1771	2
#1753	2
#1749	2