Subgroups — C₂×C₃×#2626 #10653162

Order 768.

4142 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#2626	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#399	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#396	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×#393	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#385	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#379	C₂	1	2	yes	maximal
10	C₂×#2626	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂⁶×C₃	C₂²	1	1	yes	characteristic
12	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊Q₈	—	2	8	no	—
16	C₂×C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
17	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
18	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₂²×C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
21	C₂²×C₃×C₂²⋊C₄	—	2	8	no	—
22	C₂²×C₃×C₂²⋊C₄	C₂²	1	8	yes	—
23	C₂²×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂²×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂²×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂²×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂²×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
28	C₂²×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
29	C₃×#399	C₂²	1	16	yes	complemented
30	C₃×#396	C₂²	1	32	yes	complemented
31	C₃×#393	C₂²	1	16	yes	complemented
32	C₃×#385	C₂²	1	16	yes	complemented
33	C₃×#380	C₂²	1	8	yes	complemented
34	C₃×#379	C₂²	1	16	yes	—
35	C₂×C₃×#95	C₂²	1	4	yes	complemented
36	C₂×C₃×#95	C₂²	1	2	yes	complemented
37	C₂²×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂³×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
39	#2626	C₂×C₃	1	16	yes	direct product factor
40	C₂×#399	C₂×C₃	1	2	yes	complemented
41	C₂×#396	C₂×C₃	1	4	yes	complemented
42	C₂×#393	C₂×C₃	1	2	yes	complemented
43	C₂×#385	C₂×C₃	1	2	yes	complemented
44	C₂×#380	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×#379	C₂×C₃	1	2	yes	—
46	C₂⁵×C₃	—	2	4	no	—
47	C₂⁵×C₃	—	2	4	no	—
48	C₂⁵×C₃	—	4	4	no	—
49	C₂⁵×C₃	—	4	4	no	—
50	C₂⁵×C₃	Q₈	1	8	yes	complemented
51	C₂⁵×C₃	C₂³	1	2	yes	—
52	C₂⁵×C₃	C₂³	1	1	yes	characteristic
53	C₂⁵×C₃	C₂³	1	4	yes	—
54	C₂²×C₃×Q₈	—	2	4	no	—
55	C₂²×C₃×Q₈	—	4	4	no	—
56	C₂²×C₃×Q₈	C₂³	1	4	yes	complemented
57	C₂³×C₃×C₄	—	2	4	no	—
58	C₂³×C₃×C₄	—	4	4	no	—
59	C₂³×C₃×C₄	—	4	4	no	—
60	C₂³×C₃×C₄	D₄	1	8	yes	—
61	C₂³×C₃×C₄	C₂³	1	2	yes	—
62	C₂³×C₃×C₄	C₂³	1	2	yes	—
63	C₂³×C₃×C₄	C₂³	1	4	yes	—
64	C₂³×C₃×C₄	C₂³	1	2	yes	—
65	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₂³×C₃×C₄	C₂³	1	4	yes	—
68	C₂³×C₃×C₄	C₂³	1	4	yes	—
69	C₃×C₂²⋊Q₈	—	2	32	no	—
70	C₂×C₃×C₄⋊C₄	—	2	4	no	—
71	C₂×C₃×C₄⋊C₄	—	2	8	no	—
72	C₂×C₃×C₄⋊C₄	—	4	4	no	—
73	C₂×C₃×C₄⋊C₄	—	4	8	no	—
74	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	complemented
75	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	complemented
76	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	complemented
77	C₂×C₃×C₄⋊C₄	C₂³	1	16	yes	complemented
78	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
79	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
80	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
81	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
82	C₂×C₃×C₂²⋊C₄	—	2	32	no	—
83	C₂×C₃×C₂²⋊C₄	—	4	8	no	—
84	C₂×C₃×C₂²⋊C₄	—	4	8	no	—
85	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
86	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
87	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
88	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
89	C₂×C₃×C₂²⋊C₄	C₂³	1	16	yes	—
90	C₂×C₃×C₂²⋊C₄	C₂³	1	16	yes	—
91	C₃×#95	C₂³	1	8	yes	complemented
92	C₃×#95	C₂³	1	16	yes	complemented
93	C₂⁶	C₂²×C₃	1	1	yes	characteristic
94	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
95	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
96	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
97	C₂×C₂²⋊Q₈	—	2	8	no	—
98	C₂×C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
99	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
100	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
101	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
102	C₂²×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
103	C₂²×C₂²⋊C₄	—	2	8	no	—
104	C₂²×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
105	C₂²×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
106	C₂²×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
107	C₂²×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
108	C₂²×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
109	C₂²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
110	C₂²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
111	#399	C₂²×C₃	1	16	yes	complemented
112	#396	C₂²×C₃	1	32	yes	complemented
113	#393	C₂²×C₃	1	16	yes	complemented
114	#385	C₂²×C₃	1	16	yes	complemented
115	#380	C₂²×C₃	1	8	yes	complemented
116	#379	C₂²×C₃	1	16	yes	—
117	C₂×#95	C₂²×C₃	1	4	yes	complemented
118	C₂×#95	C₂²×C₃	1	2	yes	complemented
119	C₂⁴×C₃	—	2	4	no	—
120	C₂⁴×C₃	—	2	4	no	—
121	C₂⁴×C₃	—	2	32	no	—
122	C₂⁴×C₃	—	2	4	no	—
123	C₂⁴×C₃	—	2	8	no	—
124	C₂⁴×C₃	—	2	4	no	—
125	C₂⁴×C₃	—	2	8	no	—
126	C₂⁴×C₃	—	2	8	no	—
127	C₂⁴×C₃	—	2	8	no	—
128	C₂⁴×C₃	—	4	32	no	—
129	C₂⁴×C₃	—	4	8	no	—
130	C₂⁴×C₃	—	4	8	no	—
131	C₂⁴×C₃	—	4	16	no	—
132	C₂⁴×C₃	—	4	16	no	—
133	C₂⁴×C₃	—	4	32	no	—
134	C₂⁴×C₃	C₂×D₄	1	4	yes	—
135	C₂⁴×C₃	C₂×Q₈	1	8	yes	—
136	C₂⁴×C₃	C₂×Q₈	1	4	yes	—
137	C₂⁴×C₃	C₂×Q₈	1	16	yes	complemented
138	C₂⁴×C₃	#41	1	4	yes	—
139	C₂⁴×C₃	C₂⁴	1	4	yes	—
140	C₂⁴×C₃	C₂⁴	1	2	yes	—
141	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
142	C₂×C₃×Q₈	—	2	8	no	—
143	C₂×C₃×Q₈	—	4	8	no	—
144	C₂×C₃×Q₈	—	4	16	no	—
145	C₂²×C₃×C₄	—	2	4	no	—
146	C₂²×C₃×C₄	—	2	4	no	—
147	C₂²×C₃×C₄	—	2	8	no	—
148	C₂²×C₃×C₄	—	2	4	no	—
149	C₂²×C₃×C₄	—	2	4	no	—
150	C₂²×C₃×C₄	—	2	4	no	—
151	C₂²×C₃×C₄	—	2	8	no	—
152	C₂²×C₃×C₄	—	2	8	no	—
153	C₂²×C₃×C₄	—	2	8	no	—
154	C₂²×C₃×C₄	—	2	8	no	—
155	C₂²×C₃×C₄	—	2	8	no	—
156	C₂²×C₃×C₄	—	2	8	no	—
157	C₂²×C₃×C₄	—	4	16	no	—
158	C₂²×C₃×C₄	—	4	8	no	—
159	C₂²×C₃×C₄	—	4	8	no	—
160	C₂²×C₃×C₄	—	4	16	no	—
161	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
162	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
163	C₂²×C₃×C₄	C₂×D₄	1	16	yes	—
164	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
165	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
166	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
167	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
168	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
169	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
170	C₃×C₄⋊C₄	—	2	8	no	—
171	C₃×C₄⋊C₄	—	2	16	no	—
172	C₃×C₄⋊C₄	—	4	8	no	—
173	C₃×C₄⋊C₄	—	4	16	no	—
174	C₃×C₂²⋊C₄	—	2	16	no	—
175	C₃×C₂²⋊C₄	—	2	16	no	—
176	C₃×C₂²⋊C₄	—	2	16	no	—
177	C₃×C₂²⋊C₄	—	2	16	no	—
178	C₃×C₂²⋊C₄	—	4	16	no	—
179	C₃×C₂²⋊C₄	—	4	16	no	—
180	C₂⁵	—	2	4	no	—
181	C₂⁵	—	2	4	no	—
182	C₂⁵	—	4	4	no	—
183	C₂⁵	—	4	4	no	—
184	C₂⁵	C₃×Q₈	1	8	yes	complemented
185	C₂⁵	C₂³×C₃	1	2	yes	—
186	C₂⁵	C₂³×C₃	1	1	yes	characteristic
187	C₂⁵	C₂³×C₃	1	4	yes	—
188	C₂²×Q₈	—	2	4	no	—
189	C₂²×Q₈	—	4	4	no	—
190	C₂²×Q₈	C₂³×C₃	1	4	yes	complemented
191	C₂³×C₄	—	2	4	no	—
192	C₂³×C₄	—	4	4	no	—
193	C₂³×C₄	—	4	4	no	—
194	C₂³×C₄	C₃×D₄	1	8	yes	—
195	C₂³×C₄	C₂³×C₃	1	2	yes	—
196	C₂³×C₄	C₂³×C₃	1	4	yes	—
197	C₂³×C₄	C₂³×C₃	1	2	yes	—
198	C₂³×C₄	C₂³×C₃	1	2	yes	—
199	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
200	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
201	C₂³×C₄	C₂³×C₃	1	4	yes	—
202	C₂³×C₄	C₂³×C₃	1	4	yes	—
203	C₂²⋊Q₈	—	2	32	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	8	no	—
206	C₂×C₄⋊C₄	—	4	4	no	—
207	C₂×C₄⋊C₄	—	4	8	no	—
208	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	complemented
209	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	complemented
210	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	complemented
211	C₂×C₄⋊C₄	C₂³×C₃	1	16	yes	complemented
212	C₂×C₂²⋊C₄	—	2	8	no	—
213	C₂×C₂²⋊C₄	—	2	8	no	—
214	C₂×C₂²⋊C₄	—	2	8	no	—
215	C₂×C₂²⋊C₄	—	2	8	no	—
216	C₂×C₂²⋊C₄	—	2	32	no	—
217	C₂×C₂²⋊C₄	—	4	8	no	—
218	C₂×C₂²⋊C₄	—	4	8	no	—
219	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
220	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
221	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
222	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
223	C₂×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
224	C₂×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
225	#95	C₂³×C₃	1	8	yes	complemented
226	#95	C₂³×C₃	1	16	yes	complemented
227	C₂³×C₃	—	2	8	no	—
228	C₂³×C₃	—	2	8	no	—
229	C₂³×C₃	—	2	16	no	—
230	C₂³×C₃	—	2	16	no	—
231	C₂³×C₃	—	2	4	no	—
232	C₂³×C₃	—	2	4	no	—
233	C₂³×C₃	—	2	16	no	—
234	C₂³×C₃	—	2	32	no	—
235	C₂³×C₃	—	2	32	no	—
236	C₂³×C₃	—	2	4	no	—
237	C₂³×C₃	—	2	4	no	—
238	C₂³×C₃	—	2	8	no	—
239	C₂³×C₃	—	2	8	no	—
240	C₂³×C₃	—	4	32	no	—
241	C₂³×C₃	—	4	128	no	—
242	C₂³×C₃	—	4	16	no	—
243	C₂³×C₃	—	4	16	no	—
244	C₂³×C₃	—	4	32	no	—
245	C₂³×C₃	—	4	8	no	—
246	C₂³×C₃	—	4	8	no	—
247	C₂³×C₃	—	4	16	no	—
248	C₂³×C₃	C₂²⋊Q₈	1	16	yes	—
249	C₂³×C₃	C₂²×D₄	1	4	yes	—
250	C₂³×C₃	C₂²×D₄	1	2	yes	—
251	C₂³×C₃	C₂²×Q₈	1	8	yes	—
252	C₂³×C₃	C₂²×Q₈	1	4	yes	—
253	C₂³×C₃	C₂²×Q₈	1	2	yes	—
254	C₂³×C₃	C₂×#41	1	4	yes	—
255	C₂³×C₃	C₂×#41	1	2	yes	—
256	C₂³×C₃	H(5,𝔽₂)	1	4	yes	—
257	C₂³×C₃	H(5,𝔽₂)	1	4	yes	—
258	C₂³×C₃	C₂⁵	1	1	yes	characteristic
259	C₃×Q₈	—	4	16	no	—
260	C₂×C₃×C₄	—	2	8	no	—
261	C₂×C₃×C₄	—	2	8	no	—
262	C₂×C₃×C₄	—	2	4	no	—
263	C₂×C₃×C₄	—	2	4	no	—
264	C₂×C₃×C₄	—	2	4	no	—
265	C₂×C₃×C₄	—	2	4	no	—
266	C₂×C₃×C₄	—	2	4	no	—
267	C₂×C₃×C₄	—	2	8	no	—
268	C₂×C₃×C₄	—	2	8	no	—
269	C₂×C₃×C₄	—	2	8	no	—
270	C₂×C₃×C₄	—	2	8	no	—
271	C₂×C₃×C₄	—	2	8	no	—
272	C₂×C₃×C₄	—	2	8	no	—
273	C₂×C₃×C₄	—	4	16	no	—
274	C₂×C₃×C₄	—	4	16	no	—
275	C₂×C₃×C₄	—	4	8	no	—
276	C₂×C₃×C₄	—	4	8	no	—
277	C₂×C₃×C₄	—	4	8	no	—
278	C₂×C₃×C₄	—	4	16	no	—
279	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
280	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
281	C₂⁴	—	2	4	no	—
282	C₂⁴	—	2	4	no	—
283	C₂⁴	—	2	32	no	—
284	C₂⁴	—	2	4	no	—
285	C₂⁴	—	2	8	no	—
286	C₂⁴	—	2	4	no	—
287	C₂⁴	—	2	8	no	—
288	C₂⁴	—	2	8	no	—
289	C₂⁴	—	2	8	no	—
290	C₂⁴	—	4	32	no	—
291	C₂⁴	—	4	8	no	—
292	C₂⁴	—	4	8	no	—
293	C₂⁴	—	4	16	no	—
294	C₂⁴	—	4	16	no	—
295	C₂⁴	—	4	32	no	—
296	C₂⁴	C₂×C₃×D₄	1	4	yes	—
297	C₂⁴	C₂×C₃×Q₈	1	8	yes	—
298	C₂⁴	C₂×C₃×Q₈	1	4	yes	—
299	C₂⁴	C₂×C₃×Q₈	1	16	yes	complemented
300	C₂⁴	C₃×#41	1	4	yes	—
301	C₂⁴	C₂⁴×C₃	1	4	yes	—
302	C₂⁴	C₂⁴×C₃	1	2	yes	—
303	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
304	C₂×Q₈	—	2	8	no	—
305	C₂×Q₈	—	4	8	no	—
306	C₂×Q₈	—	4	16	no	—
307	C₂²×C₄	—	2	4	no	—
308	C₂²×C₄	—	2	4	no	—
309	C₂²×C₄	—	2	8	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	4	no	—
313	C₂²×C₄	—	2	8	no	—
314	C₂²×C₄	—	2	8	no	—
315	C₂²×C₄	—	2	8	no	—
316	C₂²×C₄	—	2	8	no	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	4	16	no	—
320	C₂²×C₄	—	4	8	no	—
321	C₂²×C₄	—	4	8	no	—
322	C₂²×C₄	—	4	16	no	—
323	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
324	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
325	C₂²×C₄	C₂×C₃×D₄	1	16	yes	—
326	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
327	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
328	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
329	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
330	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
331	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
332	C₄⋊C₄	—	2	8	no	—
333	C₄⋊C₄	—	2	16	no	—
334	C₄⋊C₄	—	4	8	no	—
335	C₄⋊C₄	—	4	16	no	—
336	C₂²⋊C₄	—	2	16	no	—
337	C₂²⋊C₄	—	2	16	no	—
338	C₂²⋊C₄	—	2	16	no	—
339	C₂²⋊C₄	—	2	16	no	—
340	C₂²⋊C₄	—	4	16	no	—
341	C₂²⋊C₄	—	4	16	no	—
342	C₂²×C₃	—	2	32	no	—
343	C₂²×C₃	—	2	4	no	—
344	C₂²×C₃	—	2	4	no	—
345	C₂²×C₃	—	2	8	no	—
346	C₂²×C₃	—	2	8	no	—
347	C₂²×C₃	—	2	8	no	—
348	C₂²×C₃	—	2	16	no	—
349	C₂²×C₃	—	4	64	no	—
350	C₂²×C₃	—	4	32	no	—
351	C₂²×C₃	—	4	4	no	—
352	C₂²×C₃	—	4	4	no	—
353	C₂²×C₃	—	4	8	no	—
354	C₂²×C₃	C₂×C₂²⋊Q₈	1	8	yes	—
355	C₂²×C₃	C₂×C₂²⋊Q₈	1	4	yes	—
356	C₂²×C₃	#534	1	4	yes	—
357	C₂²×C₃	#535	1	4	yes	—
358	C₂²×C₃	#538	1	8	yes	—
359	C₂²×C₃	#543	1	8	yes	—
360	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
361	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
362	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
363	C₂²×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
364	C₂²×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
365	C₃×C₄	—	2	4	no	—
366	C₃×C₄	—	2	4	no	—
367	C₃×C₄	—	4	4	no	—
368	C₃×C₄	—	4	4	no	—
369	C₃×C₄	—	4	4	no	—
370	C₃×C₄	—	4	8	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	2	4	no	—
373	C₂³	—	2	8	no	—
374	C₂³	—	2	8	no	—
375	C₂³	—	2	16	no	—
376	C₂³	—	2	16	no	—
377	C₂³	—	2	4	no	—
378	C₂³	—	2	4	no	—
379	C₂³	—	2	8	no	—
380	C₂³	—	2	8	no	—
381	C₂³	—	2	16	no	—
382	C₂³	—	2	32	no	—
383	C₂³	—	2	32	no	—
384	C₂³	—	4	32	no	—
385	C₂³	—	4	128	no	—
386	C₂³	—	4	16	no	—
387	C₂³	—	4	16	no	—
388	C₂³	—	4	32	no	—
389	C₂³	—	4	8	no	—
390	C₂³	—	4	8	no	—
391	C₂³	—	4	16	no	—
392	C₂³	C₃×C₂²⋊Q₈	1	16	yes	—
393	C₂³	C₂²×C₃×D₄	1	4	yes	—
394	C₂³	C₂²×C₃×D₄	1	2	yes	—
395	C₂³	C₂²×C₃×Q₈	1	8	yes	—
396	C₂³	C₂²×C₃×Q₈	1	4	yes	—
397	C₂³	C₂²×C₃×Q₈	1	2	yes	—
398	C₂³	C₂×C₃×#41	1	4	yes	—
399	C₂³	C₂×C₃×#41	1	2	yes	—
400	C₂³	C₃×H(5,𝔽₂)	1	4	yes	—
401	C₂³	C₃×H(5,𝔽₂)	1	4	yes	—
402	C₂³	C₂⁵×C₃	1	1	yes	characteristic
403	Q₈	—	4	16	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	8	no	—
407	C₂×C₄	—	2	8	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	8	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	2	8	no	—
417	C₂×C₄	—	4	16	no	—
418	C₂×C₄	—	4	16	no	—
419	C₂×C₄	—	4	8	no	—
420	C₂×C₄	—	4	8	no	—
421	C₂×C₄	—	4	8	no	—
422	C₂×C₄	—	4	16	no	—
423	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
424	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
425	C₂×C₃	—	2	8	no	—
426	C₂×C₃	—	4	8	no	—
427	C₂×C₃	#2626	1	8	yes	direct product factor
428	C₂×C₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
429	C₂×C₃	C₂×#534	1	1	yes	characteristic
430	C₂×C₃	C₂×#535	1	1	yes	characteristic
431	C₂×C₃	C₂×#538	1	2	yes	—
432	C₂×C₃	C₂×#543	1	2	yes	—
433	C₂²	—	2	32	no	—
434	C₂²	—	2	4	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	2	8	no	—
437	C₂²	—	2	8	no	—
438	C₂²	—	2	8	no	—
439	C₂²	—	2	16	no	—
440	C₂²	—	4	64	no	—
441	C₂²	—	4	32	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	8	no	—
445	C₂²	C₂×C₃×C₂²⋊Q₈	1	8	yes	—
446	C₂²	C₂×C₃×C₂²⋊Q₈	1	4	yes	—
447	C₂²	C₃×#534	1	4	yes	—
448	C₂²	C₃×#535	1	4	yes	—
449	C₂²	C₃×#538	1	8	yes	—
450	C₂²	C₃×#543	1	8	yes	—
451	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
452	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
453	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
454	C₂²	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
455	C₂²	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
456	C₄	—	2	4	no	—
457	C₄	—	2	4	no	—
458	C₄	—	4	4	no	—
459	C₄	—	4	4	no	—
460	C₄	—	4	4	no	—
461	C₄	—	4	8	no	—
462	C₃	C₂×#2626	1	1	yes	characteristic, direct product factor, Sylow
463	C₂	—	2	8	no	—
464	C₂	—	4	8	no	—
465	C₂	C₃×#2626	1	8	yes	direct product factor
466	C₂	C₂²×C₃×C₂²⋊Q₈	1	1	yes	characteristic
467	C₂	C₂×C₃×#534	1	1	yes	characteristic
468	C₂	C₂×C₃×#535	1	1	yes	characteristic
469	C₂	C₂×C₃×#538	1	2	yes	—
470	C₂	C₂×C₃×#543	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂⁶×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₂²⋊Q₈	C₂	semidirect
3. C₃×#2626	C₂	direct (N × H)
4. C₂×C₃×#399	C₂	semidirect
5. C₂×C₃×#396	C₂	semidirect
6. C₂×C₃×#393	C₂	semidirect
7. C₂×C₃×#385	C₂	semidirect
8. C₂×C₃×#380	C₂	semidirect
10. C₂×#2626	C₃	direct (N × H)
12. C₂³×C₃×Q₈	C₂²	semidirect
16. C₂×C₃×C₂²⋊Q₈	C₂²	semidirect
17. C₂²×C₃×C₄⋊C₄	C₂²	semidirect
18. C₂²×C₃×C₄⋊C₄	C₂²	semidirect
19. C₂²×C₃×C₄⋊C₄	C₂²	semidirect
20. C₂²×C₃×C₄⋊C₄	C₂²	semidirect
29. C₃×#399	C₂²	semidirect
30. C₃×#396	C₂²	semidirect
31. C₃×#393	C₂²	semidirect
32. C₃×#385	C₂²	semidirect
33. C₃×#380	C₂²	semidirect
35. C₂×C₃×#95	C₂²	semidirect
36. C₂×C₃×#95	C₂²	semidirect
37. C₂²×C₂²⋊Q₈	C₂×C₃	semidirect
39. #2626	C₂×C₃	direct (N × H)
40. C₂×#399	C₂×C₃	semidirect
41. C₂×#396	C₂×C₃	semidirect
42. C₂×#393	C₂×C₃	semidirect
43. C₂×#385	C₂×C₃	semidirect
44. C₂×#380	C₂×C₃	semidirect
50. C₂⁵×C₃	Q₈	semidirect
56. C₂²×C₃×Q₈	C₂³	semidirect
74. C₂×C₃×C₄⋊C₄	C₂³	semidirect
75. C₂×C₃×C₄⋊C₄	C₂³	semidirect
76. C₂×C₃×C₄⋊C₄	C₂³	semidirect
77. C₂×C₃×C₄⋊C₄	C₂³	semidirect
91. C₃×#95	C₂³	semidirect
92. C₃×#95	C₂³	semidirect
94. C₂³×Q₈	C₂²×C₃	semidirect
98. C₂×C₂²⋊Q₈	C₂²×C₃	semidirect
99. C₂²×C₄⋊C₄	C₂²×C₃	semidirect
100. C₂²×C₄⋊C₄	C₂²×C₃	semidirect
101. C₂²×C₄⋊C₄	C₂²×C₃	semidirect
102. C₂²×C₄⋊C₄	C₂²×C₃	semidirect
111. #399	C₂²×C₃	semidirect
112. #396	C₂²×C₃	semidirect
113. #393	C₂²×C₃	semidirect
114. #385	C₂²×C₃	semidirect
115. #380	C₂²×C₃	semidirect
117. C₂×#95	C₂²×C₃	semidirect
118. C₂×#95	C₂²×C₃	semidirect
137. C₂⁴×C₃	C₂×Q₈	semidirect
184. C₂⁵	C₃×Q₈	semidirect
190. C₂²×Q₈	C₂³×C₃	semidirect
208. C₂×C₄⋊C₄	C₂³×C₃	semidirect
209. C₂×C₄⋊C₄	C₂³×C₃	semidirect
210. C₂×C₄⋊C₄	C₂³×C₃	semidirect
211. C₂×C₄⋊C₄	C₂³×C₃	semidirect
225. #95	C₂³×C₃	semidirect
226. #95	C₂³×C₃	semidirect
299. C₂⁴	C₂×C₃×Q₈	semidirect
427. C₂×C₃	#2626	direct (N × H)
462. C₃	C₂×#2626	direct (N × H)
465. C₂	C₃×#2626	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×C₃×C₂²⋊Q₈	2
C₂³×C₃×C₂²⋊C₄	2
C₃×#2626	2
C₂×C₃×#399	2
C₂×C₃×#396	2
C₂×C₃×#393	2
C₂×C₃×#385	2
C₂×C₃×#380	2
C₂×C₃×#379	2
C₂×#2626	3

Subgroups — C2×C3×#2626 #10653162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2626 #10653162