Subgroups — C₂×C₄×#135×C₄₁ #23836352

Order 10496.

2894 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₄×#135×C₄₁	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄×Q₁₆×C₄₁	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×SD₁₆×C₄₁	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×D₈×C₄₁	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₁×#416	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₄₁	C₂²	1	2	yes	—
11	C₂×#135×C₄₁	C₄	1	16	yes	direct product factor
12	C₂×#135×C₄₁	C₂²	1	8	yes	—
13	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
15	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄×#41×C₄₁	—	2	8	no	—
19	C₄×#41×C₄₁	C₂²	1	16	yes	complemented
20	C₂×C₄×Q₈×C₄₁	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₄₁	—	2	4	no	—
22	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
23	C₂×#117×C₄₁	—	2	2	no	—
24	C₂×#117×C₄₁	C₂²	1	2	yes	—
25	C₂²×C₄²×C₄₁	—	2	2	no	—
26	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆×C₄₁	C₂²	1	16	yes	complemented
28	C₄×SD₁₆×C₄₁	C₂²	1	32	yes	complemented
29	C₄×D₈×C₄₁	C₂²	1	16	yes	complemented
30	C₄₁×#427	C₂²	1	8	yes	complemented
31	C₂×#107×C₄₁	C₂²	1	2	yes	—
32	C₂×#106×C₄₁	C₂²	1	2	yes	—
33	C₄₁×#416	C₂²	1	16	yes	complemented
34	C₂×#103×C₄₁	C₂²	1	4	yes	—
35	C₂×#102×C₄₁	C₂²	1	4	yes	—
36	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₄₁	C₂²	1	8	yes	complemented
39	C₂×#41×C₄₁	—	2	4	no	—
40	C₂×#41×C₄₁	—	2	8	no	—
41	C₂×#41×C₄₁	C₂×C₄	1	16	yes	complemented
42	C₂×#41×C₄₁	C₂³	1	8	yes	—
43	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
44	C₂²×D₄×C₄₁	—	2	2	no	—
45	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
46	C₂³×C₄×C₄₁	—	2	2	no	—
47	C₂³×C₄×C₄₁	—	2	4	no	—
48	C₂³×C₄×C₄₁	D₄	1	2	yes	—
49	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
50	#135×C₄₁	C₂×C₄	1	64	yes	direct product factor
51	C₂×Q₁₆×C₄₁	C₂×C₄	1	8	yes	complemented
52	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
53	C₂×SD₁₆×C₄₁	C₂×C₄	1	16	yes	complemented
54	C₂×SD₁₆×C₄₁	C₂³	1	8	yes	—
55	C₂×D₈×C₄₁	C₂×C₄	1	8	yes	complemented
56	C₂×D₈×C₄₁	C₂³	1	4	yes	—
57	C₂²×C₈×C₄₁	—	2	4	no	—
58	C₂²×C₈×C₄₁	C₂×C₄	1	8	yes	complemented
59	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
61	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
64	C₄×Q₈×C₄₁	—	2	8	no	—
65	C₄×Q₈×C₄₁	C₂³	1	16	yes	—
66	C₄×D₄×C₄₁	—	2	32	no	—
67	C₄×D₄×C₄₁	—	2	8	no	—
68	C₄×D₄×C₄₁	C₂³	1	16	yes	—
69	#117×C₄₁	—	2	4	no	—
70	#117×C₄₁	—	2	16	no	—
71	#117×C₄₁	C₂³	1	8	yes	—
72	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
74	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
76	C₂×C₄²×C₄₁	—	2	2	no	—
77	C₂×C₄²×C₄₁	—	2	4	no	—
78	C₂×C₄²×C₄₁	—	2	16	no	—
79	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₄₁	D₄	1	4	yes	—
81	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
83	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
84	#107×C₄₁	C₂³	1	8	yes	—
85	#106×C₄₁	C₂³	1	8	yes	—
86	#103×C₄₁	C₂³	1	16	yes	—
87	#102×C₄₁	C₂³	1	16	yes	—
88	C₄×C₈×C₄₁	C₂³	1	8	yes	—
89	C₄×C₈×C₄₁	C₂³	1	8	yes	—
90	C₂⁴×C₄₁	—	2	2	no	—
91	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
92	#41×C₄₁	—	2	32	no	—
93	#41×C₄₁	—	2	16	no	—
94	#41×C₄₁	C₂²×C₄	1	32	yes	complemented
95	C₂×Q₈×C₄₁	—	2	2	no	—
96	C₂×Q₈×C₄₁	—	2	4	no	—
97	C₂×Q₈×C₄₁	C₂²×C₄	1	8	yes	—
98	C₂×Q₈×C₄₁	C₂⁴	1	4	yes	—
99	C₂×D₄×C₄₁	—	2	16	no	—
100	C₂×D₄×C₄₁	—	2	8	no	—
101	C₂×D₄×C₄₁	—	2	2	no	—
102	C₂×D₄×C₄₁	—	2	4	no	—
103	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
104	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
105	C₂²×C₄×C₄₁	—	2	2	no	—
106	C₂²×C₄×C₄₁	—	2	16	no	—
107	C₂²×C₄×C₄₁	—	2	8	no	—
108	C₂²×C₄×C₄₁	—	2	8	no	—
109	C₂²×C₄×C₄₁	—	2	4	no	—
110	C₂²×C₄×C₄₁	—	2	16	no	—
111	C₂²×C₄×C₄₁	—	4	16	no	—
112	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
113	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
115	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₄₁	#41	1	4	yes	—
119	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
120	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
121	Q₁₆×C₄₁	C₂²×C₄	1	16	yes	complemented
122	SD₁₆×C₄₁	C₂²×C₄	1	32	yes	complemented
123	D₈×C₄₁	C₂²×C₄	1	16	yes	complemented
124	C₂×C₈×C₄₁	—	2	4	no	—
125	C₂×C₈×C₄₁	—	2	8	no	—
126	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
127	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
128	C₂×C₈×C₄₁	C₂²×C₄	1	16	yes	complemented
129	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
130	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
131	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
132	C₄⋊C₄×C₄₁	—	2	16	no	—
133	C₄⋊C₄×C₄₁	—	2	4	no	—
134	C₄⋊C₄×C₄₁	C₂⁴	1	8	yes	—
135	C₂²⋊C₄×C₄₁	—	2	8	no	—
136	C₂²⋊C₄×C₄₁	—	2	16	no	—
137	C₄²×C₄₁	—	2	8	no	—
138	C₄²×C₄₁	—	2	16	no	—
139	C₄²×C₄₁	C₂×D₄	1	4	yes	—
140	C₄²×C₄₁	C₂×D₄	1	4	yes	—
141	C₄²×C₄₁	C₂×D₄	1	4	yes	—
142	C₄²×C₄₁	C₂⁴	1	4	yes	—
143	C₂³×C₄₁	—	2	4	no	—
144	C₂³×C₄₁	—	2	8	no	—
145	C₂³×C₄₁	—	2	4	no	—
146	C₂³×C₄₁	—	4	8	no	—
147	C₂³×C₄₁	C₄×D₄	1	4	yes	—
148	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
149	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₄₁	—	2	8	no	—
151	Q₈×C₄₁	—	2	4	no	—
152	Q₈×C₄₁	C₂³×C₄	1	8	yes	—
153	D₄×C₄₁	—	2	32	no	—
154	D₄×C₄₁	—	2	8	no	—
155	D₄×C₄₁	—	2	4	no	—
156	D₄×C₄₁	C₂³×C₄	1	8	yes	—
157	C₂×C₄×C₄₁	—	2	8	no	—
158	C₂×C₄×C₄₁	—	2	4	no	—
159	C₂×C₄×C₄₁	—	2	4	no	—
160	C₂×C₄×C₄₁	—	2	32	no	—
161	C₂×C₄×C₄₁	—	2	16	no	—
162	C₂×C₄×C₄₁	—	2	8	no	—
163	C₂×C₄×C₄₁	—	4	8	no	—
164	C₂×C₄×C₄₁	—	4	32	no	—
165	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
166	C₂×C₄×C₄₁	C₄×D₄	1	8	yes	—
167	C₂×C₄×C₄₁	#135	1	8	yes	direct product factor
168	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
169	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
170	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
171	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
174	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
175	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
176	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
177	C₈×C₄₁	—	2	4	no	—
178	C₈×C₄₁	C₂³×C₄	1	4	yes	—
179	C₈×C₄₁	C₂³×C₄	1	4	yes	—
180	C₂×C₄×#135	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂²×C₄₁	—	2	8	no	—
182	C₂²×C₄₁	—	2	4	no	—
183	C₂²×C₄₁	—	2	8	no	—
184	C₂²×C₄₁	—	4	16	no	—
185	C₂²×C₄₁	—	4	8	no	—
186	C₂²×C₄₁	C₂×C₄×D₄	1	4	yes	—
187	C₂²×C₄₁	C₂×C₄×D₄	1	2	yes	—
188	C₂²×C₄₁	C₂×#135	1	4	yes	—
189	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
190	C₄×C₄₁	—	2	8	no	—
191	C₄×C₄₁	—	2	4	no	—
192	C₄×C₄₁	—	4	8	no	—
193	C₄×C₄₁	C₂×C₄×D₄	1	4	yes	—
194	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
195	C₄×C₄₁	C₂×#135	1	8	yes	direct product factor
196	C₄×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
197	C₄×C₄₁	C₂²×#41	1	1	yes	characteristic
198	C₂²×#135	C₂×C₄₁	1	1	yes	characteristic
199	C₂×C₄×#41	C₂×C₄₁	1	2	yes	complemented
200	C₄×#135	C₂×C₄₁	1	16	yes	direct product factor
201	C₂×C₄×Q₁₆	C₂×C₄₁	1	2	yes	complemented
202	C₂×C₄×SD₁₆	C₂×C₄₁	1	4	yes	complemented
203	C₂×C₄×D₈	C₂×C₄₁	1	2	yes	complemented
204	C₂×#427	C₂×C₄₁	1	1	yes	characteristic, complemented
205	C₂×#416	C₂×C₄₁	1	2	yes	complemented
206	C₂²×C₄×C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
207	C₂×C₄₁	—	2	4	no	—
208	C₂×C₄₁	—	4	8	no	—
209	C₂×C₄₁	C₄×#135	1	4	yes	direct product factor
210	C₂×C₄₁	C₂²×C₄×D₄	1	1	yes	characteristic
211	C₂×C₄₁	C₂²×#135	1	2	yes	—
212	C₂²×#41	C₂²×C₄₁	1	2	yes	—
213	C₂×#135	C₄×C₄₁	1	16	yes	direct product factor
214	C₂×#135	C₂²×C₄₁	1	8	yes	—
215	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
216	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
217	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
218	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
219	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
220	C₄×#41	—	2	8	no	—
221	C₄×#41	C₂²×C₄₁	1	16	yes	complemented
222	C₂×C₄×Q₈	C₂²×C₄₁	1	4	yes	—
223	C₂×C₄×D₄	—	2	4	no	—
224	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
225	C₂×#117	—	2	2	no	—
226	C₂×#117	C₂²×C₄₁	1	2	yes	—
227	C₂²×C₄²	—	2	2	no	—
228	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
229	C₄×Q₁₆	C₂²×C₄₁	1	16	yes	complemented
230	C₄×SD₁₆	C₂²×C₄₁	1	32	yes	complemented
231	C₄×D₈	C₂²×C₄₁	1	16	yes	complemented
232	#427	C₂²×C₄₁	1	8	yes	complemented
233	C₂×#107	C₂²×C₄₁	1	2	yes	—
234	C₂×#106	C₂²×C₄₁	1	2	yes	—
235	#416	C₂²×C₄₁	1	16	yes	complemented
236	C₂×#103	C₂²×C₄₁	1	4	yes	—
237	C₂×#102	C₂²×C₄₁	1	4	yes	—
238	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
239	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
240	C₂×C₄×C₈	C₂²×C₄₁	1	8	yes	complemented
241	C₄₁	C₂×C₄×#135	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×#41	—	2	4	no	—
243	C₂×#41	—	2	8	no	—
244	C₂×#41	C₂×C₄×C₄₁	1	16	yes	complemented
245	C₂×#41	C₂³×C₄₁	1	8	yes	—
246	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
247	C₂²×D₄	—	2	2	no	—
248	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
249	C₂³×C₄	—	2	2	no	—
250	C₂³×C₄	—	2	4	no	—
251	C₂³×C₄	D₄×C₄₁	1	2	yes	—
252	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
253	#135	C₂×C₄×C₄₁	1	64	yes	direct product factor
254	C₂×Q₁₆	C₂×C₄×C₄₁	1	8	yes	complemented
255	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
256	C₂×SD₁₆	C₂×C₄×C₄₁	1	16	yes	complemented
257	C₂×SD₁₆	C₂³×C₄₁	1	8	yes	—
258	C₂×D₈	C₂×C₄×C₄₁	1	8	yes	complemented
259	C₂×D₈	C₂³×C₄₁	1	4	yes	—
260	C₂²×C₈	—	2	4	no	—
261	C₂²×C₈	C₂×C₄×C₄₁	1	8	yes	complemented
262	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
263	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
264	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
267	C₄×Q₈	—	2	8	no	—
268	C₄×Q₈	C₂³×C₄₁	1	16	yes	—
269	C₄×D₄	—	2	32	no	—
270	C₄×D₄	—	2	8	no	—
271	C₄×D₄	C₂³×C₄₁	1	16	yes	—
272	#117	—	2	4	no	—
273	#117	—	2	16	no	—
274	#117	C₂³×C₄₁	1	8	yes	—
275	C₂×C₄⋊C₄	—	2	4	no	—
276	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₂²⋊C₄	—	2	4	no	—
279	C₂×C₄²	—	2	2	no	—
280	C₂×C₄²	—	2	4	no	—
281	C₂×C₄²	—	2	16	no	—
282	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
283	C₂×C₄²	D₄×C₄₁	1	4	yes	—
284	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
286	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
287	#107	C₂³×C₄₁	1	8	yes	—
288	#106	C₂³×C₄₁	1	8	yes	—
289	#103	C₂³×C₄₁	1	16	yes	—
290	#102	C₂³×C₄₁	1	16	yes	—
291	C₄×C₈	C₂³×C₄₁	1	8	yes	—
292	C₄×C₈	C₂³×C₄₁	1	8	yes	—
293	C₂⁴	—	2	2	no	—
294	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
295	#41	—	2	32	no	—
296	#41	—	2	16	no	—
297	#41	C₂²×C₄×C₄₁	1	32	yes	complemented
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	4	no	—
300	C₂×Q₈	C₂²×C₄×C₄₁	1	8	yes	—
301	C₂×Q₈	C₂⁴×C₄₁	1	4	yes	—
302	C₂×D₄	—	2	16	no	—
303	C₂×D₄	—	2	8	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
307	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	16	no	—
310	C₂²×C₄	—	2	8	no	—
311	C₂²×C₄	—	2	8	no	—
312	C₂²×C₄	—	2	4	no	—
313	C₂²×C₄	—	2	16	no	—
314	C₂²×C₄	—	4	16	no	—
315	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
316	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
318	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
320	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
321	C₂²×C₄	#41×C₄₁	1	4	yes	—
322	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
323	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
324	Q₁₆	C₂²×C₄×C₄₁	1	16	yes	complemented
325	SD₁₆	C₂²×C₄×C₄₁	1	32	yes	complemented
326	D₈	C₂²×C₄×C₄₁	1	16	yes	complemented
327	C₂×C₈	—	2	4	no	—
328	C₂×C₈	—	2	8	no	—
329	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
330	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
331	C₂×C₈	C₂²×C₄×C₄₁	1	16	yes	complemented
332	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
333	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
334	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
335	C₄⋊C₄	—	2	16	no	—
336	C₄⋊C₄	—	2	4	no	—
337	C₄⋊C₄	C₂⁴×C₄₁	1	8	yes	—
338	C₂²⋊C₄	—	2	8	no	—
339	C₂²⋊C₄	—	2	16	no	—
340	C₄²	—	2	8	no	—
341	C₄²	—	2	16	no	—
342	C₄²	C₂×D₄×C₄₁	1	4	yes	—
343	C₄²	C₂×D₄×C₄₁	1	4	yes	—
344	C₄²	C₂×D₄×C₄₁	1	4	yes	—
345	C₄²	C₂⁴×C₄₁	1	4	yes	—
346	C₂³	—	2	4	no	—
347	C₂³	—	2	8	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	4	8	no	—
350	C₂³	C₄×D₄×C₄₁	1	4	yes	—
351	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
352	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
353	Q₈	—	2	8	no	—
354	Q₈	—	2	4	no	—
355	Q₈	C₂³×C₄×C₄₁	1	8	yes	—
356	D₄	—	2	32	no	—
357	D₄	—	2	8	no	—
358	D₄	—	2	4	no	—
359	D₄	C₂³×C₄×C₄₁	1	8	yes	—
360	C₂×C₄	—	2	8	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	32	no	—
364	C₂×C₄	—	2	16	no	—
365	C₂×C₄	—	2	8	no	—
366	C₂×C₄	—	4	8	no	—
367	C₂×C₄	—	4	32	no	—
368	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
369	C₂×C₄	C₄×D₄×C₄₁	1	8	yes	—
370	C₂×C₄	#135×C₄₁	1	8	yes	direct product factor
371	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
372	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
373	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
374	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
375	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
377	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
378	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
379	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
380	C₈	—	2	4	no	—
381	C₈	C₂³×C₄×C₄₁	1	4	yes	—
382	C₈	C₂³×C₄×C₄₁	1	4	yes	—
383	C₂²	—	2	4	no	—
384	C₂²	—	2	8	no	—
385	C₂²	—	2	8	no	—
386	C₂²	—	4	8	no	—
387	C₂²	—	4	16	no	—
388	C₂²	C₂×C₄×D₄×C₄₁	1	4	yes	—
389	C₂²	C₂×C₄×D₄×C₄₁	1	2	yes	—
390	C₂²	C₂×#135×C₄₁	1	4	yes	—
391	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
392	C₄	—	2	4	no	—
393	C₄	—	2	8	no	—
394	C₄	—	4	8	no	—
395	C₄	C₂×C₄×D₄×C₄₁	1	4	yes	—
396	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
397	C₄	C₂×#135×C₄₁	1	8	yes	direct product factor
398	C₄	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
399	C₄	C₂²×#41×C₄₁	1	1	yes	characteristic
400	C₂	—	2	4	no	—
401	C₂	—	4	8	no	—
402	C₂	C₄×#135×C₄₁	1	4	yes	direct product factor
403	C₂	C₂²×C₄×D₄×C₄₁	1	1	yes	characteristic
404	C₂	C₂²×#135×C₄₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₄₁	D₄
commutator	C₄	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
socle	C₂³×C₄₁	C₂²×D₄
prime radical	C₂²×#135×C₄₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#41×C₄₁	C₂	semidirect
3. C₄×#135×C₄₁	C₂	direct (N × H)
4. C₂×C₄×Q₁₆×C₄₁	C₂	semidirect
5. C₂×C₄×SD₁₆×C₄₁	C₂	semidirect
6. C₂×C₄×D₈×C₄₁	C₂	semidirect
7. C₂×C₄₁×#427	C₂	semidirect
8. C₂×C₄₁×#416	C₂	semidirect
9. C₂²×C₄×C₈×C₄₁	C₂	semidirect
11. C₂×#135×C₄₁	C₄	direct (N × H)
19. C₄×#41×C₄₁	C₂²	semidirect
27. C₄×Q₁₆×C₄₁	C₂²	semidirect
28. C₄×SD₁₆×C₄₁	C₂²	semidirect
29. C₄×D₈×C₄₁	C₂²	semidirect
30. C₄₁×#427	C₂²	semidirect
33. C₄₁×#416	C₂²	semidirect
38. C₂×C₄×C₈×C₄₁	C₂²	semidirect
41. C₂×#41×C₄₁	C₂×C₄	semidirect
50. #135×C₄₁	C₂×C₄	direct (N × H)
51. C₂×Q₁₆×C₄₁	C₂×C₄	semidirect
53. C₂×SD₁₆×C₄₁	C₂×C₄	semidirect
55. C₂×D₈×C₄₁	C₂×C₄	semidirect
58. C₂²×C₈×C₄₁	C₂×C₄	semidirect
94. #41×C₄₁	C₂²×C₄	semidirect
121. Q₁₆×C₄₁	C₂²×C₄	semidirect
122. SD₁₆×C₄₁	C₂²×C₄	semidirect
123. D₈×C₄₁	C₂²×C₄	semidirect
128. C₂×C₈×C₄₁	C₂²×C₄	semidirect
167. C₂×C₄×C₄₁	#135	direct (N × H)
180. C₂×C₄×#135	C₄₁	direct (N × H)
195. C₄×C₄₁	C₂×#135	direct (N × H)
199. C₂×C₄×#41	C₂×C₄₁	semidirect
200. C₄×#135	C₂×C₄₁	direct (N × H)
201. C₂×C₄×Q₁₆	C₂×C₄₁	semidirect
202. C₂×C₄×SD₁₆	C₂×C₄₁	semidirect
203. C₂×C₄×D₈	C₂×C₄₁	semidirect
204. C₂×#427	C₂×C₄₁	semidirect
205. C₂×#416	C₂×C₄₁	semidirect
206. C₂²×C₄×C₈	C₂×C₄₁	semidirect
209. C₂×C₄₁	C₄×#135	direct (N × H)
213. C₂×#135	C₄×C₄₁	direct (N × H)
221. C₄×#41	C₂²×C₄₁	semidirect
229. C₄×Q₁₆	C₂²×C₄₁	semidirect
230. C₄×SD₁₆	C₂²×C₄₁	semidirect
231. C₄×D₈	C₂²×C₄₁	semidirect
232. #427	C₂²×C₄₁	semidirect
235. #416	C₂²×C₄₁	semidirect
240. C₂×C₄×C₈	C₂²×C₄₁	semidirect
241. C₄₁	C₂×C₄×#135	direct (N × H)
244. C₂×#41	C₂×C₄×C₄₁	semidirect
253. #135	C₂×C₄×C₄₁	direct (N × H)
254. C₂×Q₁₆	C₂×C₄×C₄₁	semidirect
256. C₂×SD₁₆	C₂×C₄×C₄₁	semidirect
258. C₂×D₈	C₂×C₄×C₄₁	semidirect
261. C₂²×C₈	C₂×C₄×C₄₁	semidirect
297. #41	C₂²×C₄×C₄₁	semidirect
324. Q₁₆	C₂²×C₄×C₄₁	semidirect
325. SD₁₆	C₂²×C₄×C₄₁	semidirect
326. D₈	C₂²×C₄×C₄₁	semidirect
331. C₂×C₈	C₂²×C₄×C₄₁	semidirect
370. C₂×C₄	#135×C₄₁	direct (N × H)
397. C₄	C₂×#135×C₄₁	direct (N × H)
402. C₂	C₄×#135×C₄₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#135×C₄₁	2
C₂×C₄×#41×C₄₁	2
C₄×#135×C₄₁	2
C₂×C₄×Q₁₆×C₄₁	2
C₂×C₄×SD₁₆×C₄₁	2
C₂×C₄×D₈×C₄₁	2
C₂×C₄₁×#427	2
C₂×C₄₁×#416	2
C₂²×C₄×C₈×C₄₁	2
C₂×C₄×#135	41

Subgroups — C2×C4×#135×C41 #23836352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#135×C₄₁ #23836352