Subgroups — C₂²×C₃₁×#554 #21519461

Order 7936.

8454 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#554	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#125	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
5	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₄×C₃₁	C₂	1	2	yes	maximal
8	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
12	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₃₁	C₂²	1	4	yes	—
15	C₃₁×#554	C₂²	1	256	yes	direct product factor
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	24	yes	complemented
17	C₂×C₃₁×#125	C₂²	1	48	yes	complemented
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	96	yes	complemented
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	48	yes	complemented
20	C₂×C₄×Q₈×C₃₁	C₂²	1	24	yes	complemented
21	C₂×C₄×D₄×C₃₁	C₂²	1	48	yes	—
22	C₂×C₄×D₄×C₃₁	C₂²	1	24	yes	—
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
30	C₂²×C₄⋊C₄×C₃₁	C₂²	1	48	yes	complemented
31	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
32	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
35	C₂⁵×C₃₁	C₂³	1	2	yes	—
36	C₂²×Q₈×C₃₁	—	2	8	no	—
37	C₂²×Q₈×C₃₁	—	4	8	no	—
38	C₂²×Q₈×C₃₁	C₂³	1	12	yes	—
39	C₂²×Q₈×C₃₁	C₂³	1	24	yes	—
40	C₂²×D₄×C₃₁	Q₈	1	16	yes	complemented
41	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
42	C₂³×C₄×C₃₁	—	2	16	no	—
43	C₂³×C₄×C₃₁	—	4	16	no	—
44	C₂³×C₄×C₃₁	C₂³	1	24	yes	—
45	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
48	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
49	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
50	C₂³×C₄×C₃₁	C₂³	1	48	yes	—
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	C₂³	1	64	yes	complemented
54	C₃₁×#125	C₂³	1	128	yes	complemented
55	C₃₁×C₂²⋊Q₈	C₂³	1	256	yes	complemented
56	C₃₁×C₂²⋊Q₈	C₂³	1	128	yes	complemented
57	C₄×Q₈×C₃₁	C₂³	1	64	yes	complemented
58	C₄×D₄×C₃₁	C₂³	1	128	yes	—
59	C₄×D₄×C₃₁	C₂³	1	64	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	48	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	48	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	128	yes	complemented
68	C₂×C₂²⋊C₄×C₃₁	C₂³	1	48	yes	—
69	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	24	yes	—
71	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
72	C₂⁴×C₃₁	—	2	8	no	—
73	C₂⁴×C₃₁	—	2	8	no	—
74	C₂⁴×C₃₁	C₂×Q₈	1	16	yes	—
75	C₂⁴×C₃₁	C₂⁴	1	12	yes	—
76	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×Q₈×C₃₁	—	2	48	no	—
78	C₂×Q₈×C₃₁	—	4	48	no	—
79	C₂×Q₈×C₃₁	C₂⁴	1	16	yes	—
80	C₂×Q₈×C₃₁	C₂⁴	1	32	yes	—
81	C₂×D₄×C₃₁	C₂×Q₈	1	96	yes	complemented
82	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
83	C₂²×C₄×C₃₁	—	2	8	no	—
84	C₂²×C₄×C₃₁	—	2	8	no	—
85	C₂²×C₄×C₃₁	—	2	8	no	—
86	C₂²×C₄×C₃₁	—	2	16	no	—
87	C₂²×C₄×C₃₁	—	2	4	no	—
88	C₂²×C₄×C₃₁	—	2	96	no	—
89	C₂²×C₄×C₃₁	—	4	96	no	—
90	C₂²×C₄×C₃₁	C₂×Q₈	1	8	yes	—
91	C₂²×C₄×C₃₁	#41	1	8	yes	—
92	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
93	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
94	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
95	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
96	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
97	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	—
98	C₂²×C₄×C₃₁	C₂⁴	1	64	yes	—
99	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
100	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
101	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
102	C₄⋊C₄×C₃₁	C₂⁴	1	64	yes	—
103	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
104	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
105	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
106	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
107	C₄⋊C₄×C₃₁	C₂⁴	1	64	yes	—
108	C₂²⋊C₄×C₃₁	C₂⁴	1	64	yes	—
109	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
110	C₄²×C₃₁	C₂⁴	1	32	yes	—
111	C₄²×C₃₁	C₂⁴	1	16	yes	—
112	C₂²×#554	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₃₁	—	2	64	no	—
114	C₂³×C₃₁	—	2	24	no	—
115	C₂³×C₃₁	—	2	24	no	—
116	C₂³×C₃₁	C₂²×Q₈	1	48	yes	—
117	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
118	C₂³×C₃₁	C₂×#41	1	4	yes	—
119	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
120	C₂³×C₃₁	C₂⁵	1	8	yes	—
121	C₂³×C₃₁	C₂⁵	1	3	yes	—
122	Q₈×C₃₁	—	2	32	no	—
123	Q₈×C₃₁	—	4	32	no	—
124	D₄×C₃₁	C₂²×Q₈	1	64	yes	complemented
125	C₂×C₄×C₃₁	—	2	24	no	—
126	C₂×C₄×C₃₁	—	2	24	no	—
127	C₂×C₄×C₃₁	—	2	24	no	—
128	C₂×C₄×C₃₁	—	2	48	no	—
129	C₂×C₄×C₃₁	—	2	12	no	—
130	C₂×C₄×C₃₁	—	2	64	no	—
131	C₂×C₄×C₃₁	—	4	64	no	—
132	C₂×C₄×C₃₁	C₂²×Q₈	1	24	yes	—
133	C₂×C₄×C₃₁	C₂×#41	1	24	yes	—
134	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
135	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
136	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
137	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
138	C₂×C₄×C₃₁	C₂⁵	1	16	yes	—
139	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
140	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
141	C₂×#554	C₂×C₃₁	1	48	yes	direct product factor
142	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₂²×#125	C₂×C₃₁	1	2	yes	complemented
144	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	4	yes	complemented
145	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	complemented
146	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
147	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
148	C₂²×C₄×D₄	C₂×C₃₁	1	2	yes	—
149	C₂³×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
150	C₂²×C₃₁	—	2	8	no	—
151	C₂²×C₃₁	—	2	8	no	—
152	C₂²×C₃₁	—	2	96	no	—
153	C₂²×C₃₁	#554	1	16	yes	direct product factor
154	C₂²×C₃₁	C₂³×Q₈	1	16	yes	—
155	C₂²×C₃₁	C₂³×Q₈	1	6	yes	—
156	C₂²×C₃₁	C₂²×#41	1	6	yes	—
157	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	6	yes	—
158	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
159	C₄×C₃₁	—	2	8	no	—
160	C₄×C₃₁	—	2	8	no	—
161	C₄×C₃₁	—	2	8	no	—
162	C₄×C₃₁	—	2	16	no	—
163	C₄×C₃₁	—	2	4	no	—
164	C₄×C₃₁	C₂³×Q₈	1	8	yes	—
165	C₄×C₃₁	C₂²×#41	1	8	yes	—
166	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
167	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
168	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
169	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
170	C₂⁴×C₄	C₂²×C₃₁	1	4	yes	—
171	#554	C₂²×C₃₁	1	256	yes	direct product factor
172	C₂×C₄⋊Q₈	C₂²×C₃₁	1	24	yes	complemented
173	C₂×#125	C₂²×C₃₁	1	48	yes	complemented
174	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	96	yes	complemented
175	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	48	yes	complemented
176	C₂×C₄×Q₈	C₂²×C₃₁	1	24	yes	complemented
177	C₂×C₄×D₄	C₂²×C₃₁	1	48	yes	—
178	C₂×C₄×D₄	C₂²×C₃₁	1	24	yes	—
179	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
181	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
182	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
183	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
184	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
185	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
186	C₂²×C₄⋊C₄	C₂²×C₃₁	1	48	yes	complemented
187	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
188	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
189	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
190	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×C₃₁	—	2	16	no	—
192	C₂×C₃₁	C₂×#554	1	12	yes	direct product factor
193	C₂×C₃₁	C₂⁴×Q₈	1	1	yes	characteristic
194	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
195	C₂×C₃₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
196	C₂⁵	C₂³×C₃₁	1	2	yes	—
197	C₂²×Q₈	—	2	8	no	—
198	C₂²×Q₈	—	4	8	no	—
199	C₂²×Q₈	C₂³×C₃₁	1	12	yes	—
200	C₂²×Q₈	C₂³×C₃₁	1	24	yes	—
201	C₂²×D₄	Q₈×C₃₁	1	16	yes	complemented
202	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
203	C₂³×C₄	—	2	16	no	—
204	C₂³×C₄	—	4	16	no	—
205	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
206	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
207	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
208	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
209	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
210	C₂³×C₄	C₂³×C₃₁	1	24	yes	—
211	C₂³×C₄	C₂³×C₃₁	1	48	yes	—
212	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
213	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
214	C₄⋊Q₈	C₂³×C₃₁	1	64	yes	complemented
215	#125	C₂³×C₃₁	1	128	yes	complemented
216	C₂²⋊Q₈	C₂³×C₃₁	1	256	yes	complemented
217	C₂²⋊Q₈	C₂³×C₃₁	1	128	yes	complemented
218	C₄×Q₈	C₂³×C₃₁	1	64	yes	complemented
219	C₄×D₄	C₂³×C₃₁	1	128	yes	—
220	C₄×D₄	C₂³×C₃₁	1	64	yes	—
221	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
222	C₂×C₄⋊C₄	C₂³×C₃₁	1	48	yes	—
223	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
224	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
225	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
226	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
227	C₂×C₄⋊C₄	C₂³×C₃₁	1	48	yes	—
228	C₂×C₄⋊C₄	C₂³×C₃₁	1	128	yes	complemented
229	C₂×C₂²⋊C₄	C₂³×C₃₁	1	48	yes	—
230	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
231	C₂×C₄²	C₂³×C₃₁	1	24	yes	—
232	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
233	C₃₁	C₂²×#554	1	1	yes	characteristic, direct product factor, Sylow
234	C₂⁴	—	2	8	no	—
235	C₂⁴	—	2	8	no	—
236	C₂⁴	C₂×Q₈×C₃₁	1	16	yes	—
237	C₂⁴	C₂⁴×C₃₁	1	12	yes	—
238	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
239	C₂×Q₈	—	2	48	no	—
240	C₂×Q₈	—	4	48	no	—
241	C₂×Q₈	C₂⁴×C₃₁	1	16	yes	—
242	C₂×Q₈	C₂⁴×C₃₁	1	32	yes	—
243	C₂×D₄	C₂×Q₈×C₃₁	1	96	yes	complemented
244	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	8	no	—
247	C₂²×C₄	—	2	8	no	—
248	C₂²×C₄	—	2	16	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	96	no	—
251	C₂²×C₄	—	4	96	no	—
252	C₂²×C₄	C₂×Q₈×C₃₁	1	8	yes	—
253	C₂²×C₄	#41×C₃₁	1	8	yes	—
254	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
255	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
256	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
257	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
258	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
259	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	—
260	C₂²×C₄	C₂⁴×C₃₁	1	64	yes	—
261	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
262	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
263	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
264	C₄⋊C₄	C₂⁴×C₃₁	1	64	yes	—
265	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
266	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
267	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
268	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
269	C₄⋊C₄	C₂⁴×C₃₁	1	64	yes	—
270	C₂²⋊C₄	C₂⁴×C₃₁	1	64	yes	—
271	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
272	C₄²	C₂⁴×C₃₁	1	32	yes	—
273	C₄²	C₂⁴×C₃₁	1	16	yes	—
274	C₂³	—	2	64	no	—
275	C₂³	—	2	24	no	—
276	C₂³	—	2	24	no	—
277	C₂³	C₂²×Q₈×C₃₁	1	48	yes	—
278	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
279	C₂³	C₂×#41×C₃₁	1	4	yes	—
280	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
281	C₂³	C₂⁵×C₃₁	1	8	yes	—
282	C₂³	C₂⁵×C₃₁	1	3	yes	—
283	Q₈	—	2	32	no	—
284	Q₈	—	4	32	no	—
285	D₄	C₂²×Q₈×C₃₁	1	64	yes	complemented
286	C₂×C₄	—	2	24	no	—
287	C₂×C₄	—	2	24	no	—
288	C₂×C₄	—	2	24	no	—
289	C₂×C₄	—	2	48	no	—
290	C₂×C₄	—	2	12	no	—
291	C₂×C₄	—	2	64	no	—
292	C₂×C₄	—	4	64	no	—
293	C₂×C₄	C₂²×Q₈×C₃₁	1	24	yes	—
294	C₂×C₄	C₂×#41×C₃₁	1	24	yes	—
295	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
296	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
297	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
298	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
299	C₂×C₄	C₂⁵×C₃₁	1	16	yes	—
300	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
301	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
302	C₂²	—	2	8	no	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	96	no	—
305	C₂²	C₃₁×#554	1	16	yes	direct product factor
306	C₂²	C₂³×Q₈×C₃₁	1	16	yes	—
307	C₂²	C₂³×Q₈×C₃₁	1	6	yes	—
308	C₂²	C₂²×#41×C₃₁	1	6	yes	—
309	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	6	yes	—
310	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
311	C₄	—	2	8	no	—
312	C₄	—	2	8	no	—
313	C₄	—	2	8	no	—
314	C₄	—	2	16	no	—
315	C₄	—	2	4	no	—
316	C₄	C₂³×Q₈×C₃₁	1	8	yes	—
317	C₄	C₂²×#41×C₃₁	1	8	yes	—
318	C₂	—	2	16	no	—
319	C₂	C₂×C₃₁×#554	1	12	yes	direct product factor
320	C₂	C₂⁴×Q₈×C₃₁	1	1	yes	characteristic
321	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
322	C₂	C₂²×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#554	C₂	direct (N × H)
2. C₂²×C₃₁×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₃₁×#125	C₂	semidirect
4. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
5. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
6. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
9. C₂³×C₄⋊C₄×C₃₁	C₂	semidirect
15. C₃₁×#554	C₂²	direct (N × H)
16. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
17. C₂×C₃₁×#125	C₂²	semidirect
18. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
19. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
20. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
40. C₂²×D₄×C₃₁	Q₈	semidirect
53. C₃₁×C₄⋊Q₈	C₂³	semidirect
54. C₃₁×#125	C₂³	semidirect
55. C₃₁×C₂²⋊Q₈	C₂³	semidirect
56. C₃₁×C₂²⋊Q₈	C₂³	semidirect
57. C₄×Q₈×C₃₁	C₂³	semidirect
67. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
81. C₂×D₄×C₃₁	C₂×Q₈	semidirect
112. C₂²×#554	C₃₁	direct (N × H)
124. D₄×C₃₁	C₂²×Q₈	semidirect
141. C₂×#554	C₂×C₃₁	direct (N × H)
142. C₂²×C₄⋊Q₈	C₂×C₃₁	semidirect
143. C₂²×#125	C₂×C₃₁	semidirect
144. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
145. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
146. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
149. C₂³×C₄⋊C₄	C₂×C₃₁	semidirect
153. C₂²×C₃₁	#554	direct (N × H)
171. #554	C₂²×C₃₁	direct (N × H)
172. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
173. C₂×#125	C₂²×C₃₁	semidirect
174. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
175. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
176. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
186. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
192. C₂×C₃₁	C₂×#554	direct (N × H)
201. C₂²×D₄	Q₈×C₃₁	semidirect
214. C₄⋊Q₈	C₂³×C₃₁	semidirect
215. #125	C₂³×C₃₁	semidirect
216. C₂²⋊Q₈	C₂³×C₃₁	semidirect
217. C₂²⋊Q₈	C₂³×C₃₁	semidirect
218. C₄×Q₈	C₂³×C₃₁	semidirect
228. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
233. C₃₁	C₂²×#554	direct (N × H)
243. C₂×D₄	C₂×Q₈×C₃₁	semidirect
285. D₄	C₂²×Q₈×C₃₁	semidirect
305. C₂²	C₃₁×#554	direct (N × H)
319. C₂	C₂×C₃₁×#554	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#554	2
C₂²×C₃₁×C₄⋊Q₈	2
C₂²×C₃₁×#125	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₄×Q₈×C₃₁	2
C₂²×C₄×D₄×C₃₁	2
C₂³×C₄⋊C₄×C₃₁	2
C₂²×#554	31

Subgroups — C22×C31×#554 #21519461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#554 #21519461