Subgroups — Q₈×#6714 #14326113

Order 1920.

1640 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Q₈×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751088	C₂	1	3	yes	maximal
5	#11751074	C₂	1	3	yes	maximal, complemented
6	C₄×#6714	C₂	1	3	yes	maximal
7	#11751030	C₂	1	3	yes	maximal, complemented
8	Q₈×#685	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×Q₈×D₁₅	—	2	4	no	—
11	C₂×Q₈×D₁₅	—	2	4	no	—
12	C₂×Q₈×D₁₅	C₄	1	8	yes	complemented
13	C₂²×C₄×D₁₅	C₂²	1	3	yes	—
14	C₂²×Dic₃₀	C₂²	1	3	yes	—
15	C₃×C₄×C₅×Q₈	—	2	4	no	—
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
17	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
18	Q₈×Dic₁₅	—	2	4	no	—
19	C₂×#6714	C₂²	1	3	yes	—
20	C₂×#6714	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₅	C₂²	1	3	yes	—
22	C₂×C₄×Dic₁₅	C₂²	1	3	yes	—
23	#91115	C₂²	1	3	yes	—
24	#91115	C₂²	1	6	yes	—
25	#91115	C₂²	1	3	yes	—
26	Q₈×(C₂×D₃)⋊C₄	—	5	1	no	maximal, Hall
27	C₂²×Q₈×D₅	—	3	1	no	—
28	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
29	C₂×Q₈×Dic₅	—	3	1	no	—
30	#65276	—	3	3	no	—
31	#65263	—	3	3	no	—
32	C₄×#685	—	3	3	no	—
33	#65218	—	3	3	no	—
34	C₂³×D₁₅	C₂³	1	1	yes	characteristic
35	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
37	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
41	C₂²×Dic₁₅	C₂³	1	3	yes	—
42	C₂²×Dic₁₅	C₂³	1	3	yes	—
43	C₂²×Dic₁₅	C₂³	1	1	yes	characteristic
44	Q₈×D₁₅	—	2	32	no	—
45	C₂×C₄×D₁₅	—	2	6	no	—
46	C₂×C₄×D₁₅	—	2	6	no	—
47	C₂×C₄×D₁₅	C₂×C₄	1	12	yes	—
48	C₂×Dic₃₀	—	2	6	no	—
49	C₂×Dic₃₀	—	2	6	no	—
50	C₂×Dic₃₀	C₂×C₄	1	12	yes	—
51	C₃×C₅×C₄⋊C₄	—	2	6	no	—
52	C₃×C₄²×C₅	—	2	6	no	—
53	#6714	—	2	6	no	—
54	#6714	Q₈	1	4	yes	direct product factor
55	C₄⋊Dic₁₅	—	2	6	no	—
56	C₄×Dic₁₅	—	2	6	no	—
57	C₂²×D₃×Q₈	—	5	1	no	—
58	C₂×C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
59	C₂×Q₈×Dic₃	—	5	1	no	—
60	#4987	—	5	3	no	—
61	#4973	—	5	3	no	—
62	C₄×(C₂×D₃)⋊C₄	—	5	3	no	—
63	#4929	—	5	3	no	—
64	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
65	C₂×Q₈×D₅	—	3	8	no	—
66	C₂×Q₈×D₅	—	6	4	no	—
67	C₂×Q₈×D₅	—	6	4	no	—
68	C₂²×C₄×D₅	—	3	3	no	—
69	C₂²×Dic₁₀	—	3	3	no	—
70	C₄×C₅×Q₈	—	2	4	no	—
71	C₂×C₅×C₄⋊C₄	C₂×D₃	1	3	yes	—
72	C₂×C₄²×C₅	C₂×D₃	1	3	yes	—
73	Q₈×Dic₅	—	6	4	no	—
74	C₂×#685	—	3	1	no	—
75	C₂×#685	—	3	3	no	—
76	C₂×C₄⋊Dic₅	—	3	3	no	—
77	C₂×C₄×Dic₅	—	3	3	no	—
78	#3963	—	3	3	no	—
79	#3963	—	3	6	no	—
80	#3963	—	3	3	no	—
81	Q₈×C₂²⋊C₄	—	15	1	no	Sylow
82	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
83	C₂²×D₁₅	—	2	1	no	—
84	C₂²×D₁₅	—	2	1	no	—
85	C₂²×D₁₅	—	2	1	no	—
86	C₂²×D₁₅	—	2	1	no	—
87	C₂²×D₁₅	C₂²×C₄	1	2	yes	—
88	C₂²×D₁₅	C₂×Q₈	1	2	yes	—
89	C₂²×D₁₅	#41	1	2	yes	—
90	C₃×C₅×Q₈	C₂²⋊C₄	1	16	yes	complemented
91	C₂×C₃×C₄×C₅	—	2	1	no	—
92	C₂×C₃×C₄×C₅	—	2	1	no	—
93	C₂×C₃×C₄×C₅	—	2	3	no	—
94	C₂×C₃×C₄×C₅	—	2	3	no	—
95	C₂×C₃×C₄×C₅	—	2	3	no	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
97	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
98	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
99	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	complemented
100	C₂×Dic₁₅	—	2	1	no	—
101	C₂×Dic₁₅	—	2	3	no	—
102	C₂×Dic₁₅	—	2	3	no	—
103	C₂×Dic₁₅	—	2	1	no	—
104	C₂×Dic₁₅	—	2	3	no	—
105	C₂×Dic₁₅	—	2	3	no	—
106	C₂×Dic₁₅	—	2	3	no	—
107	C₂×Dic₁₅	C₂²×C₄	1	6	yes	—
108	C₂×Dic₁₅	C₂×Q₈	1	2	yes	complemented
109	C₄×D₁₅	—	2	24	no	—
110	Dic₃₀	—	2	24	no	—
111	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
112	C₂×D₃×Q₈	—	5	8	no	—
113	C₂×D₃×Q₈	—	10	4	no	—
114	C₂×D₃×Q₈	—	10	4	no	—
115	C₂²×C₄×D₃	—	5	3	no	—
116	C₂²×Dic₆	—	5	3	no	—
117	C₃×C₄×Q₈	—	2	4	no	—
118	C₂×C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
119	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
120	Q₈×Dic₃	—	10	4	no	—
121	C₂×(C₂×D₃)⋊C₄	—	5	1	no	—
122	C₂×(C₂×D₃)⋊C₄	—	5	3	no	—
123	C₂×C₄⋊Dic₃	—	5	3	no	—
124	C₂×C₄×Dic₃	—	5	3	no	—
125	C₃⋊#95	—	5	3	no	—
126	C₃⋊#95	—	5	6	no	—
127	C₃⋊#95	—	5	3	no	—
128	C₂³×D₅	—	3	1	no	—
129	C₂×C₅×Q₈	C₄×D₃	1	4	yes	—
130	C₂×C₅×Q₈	D₁₂	1	4	yes	—
131	C₂×C₅×Q₈	#67	1	4	yes	—
132	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
133	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
135	C₂²×Dic₅	—	3	3	no	—
136	C₂²×Dic₅	—	3	1	no	—
137	C₂²×Dic₅	—	3	3	no	—
138	Q₈×D₅	—	6	32	no	—
139	C₂×C₄×D₅	—	3	12	no	—
140	C₂×C₄×D₅	—	6	6	no	—
141	C₂×C₄×D₅	—	6	6	no	—
142	C₂×Dic₁₀	—	3	12	no	—
143	C₂×Dic₁₀	—	6	6	no	—
144	C₂×Dic₁₀	—	6	6	no	—
145	C₅×C₄⋊C₄	—	2	6	no	—
146	C₄²×C₅	—	2	6	no	—
147	#685	—	3	4	no	—
148	#685	—	6	6	no	—
149	C₄⋊Dic₅	—	6	6	no	—
150	C₄×Dic₅	—	6	6	no	—
151	C₂³×Q₈	—	15	1	no	—
152	C₂×C₄×Q₈	—	15	1	no	—
153	C₂×C₄×Q₈	D₁₅	1	1	yes	characteristic, complemented
154	#391	—	15	3	no	—
155	#391	—	15	3	no	—
156	#380	—	15	3	no	—
157	C₄×C₂²⋊C₄	—	15	3	no	—
158	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
159	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
160	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
162	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
163	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
164	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
165	C₂×D₁₅	—	2	2	no	—
166	C₂×D₁₅	—	2	2	no	—
167	C₂×D₁₅	—	2	2	no	—
168	C₂×D₁₅	—	2	2	no	—
169	C₂×D₁₅	—	2	2	no	—
170	C₂×D₁₅	—	2	2	no	—
171	C₂×D₁₅	C₄×Q₈	1	4	yes	complemented
172	C₃×C₄×C₅	—	2	2	no	—
173	C₃×C₄×C₅	—	4	3	no	—
174	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
175	Dic₁₅	—	2	6	no	—
176	Dic₁₅	—	2	2	no	—
177	Dic₁₅	—	4	3	no	—
178	C₂³×D₃	—	5	1	no	—
179	C₂×C₃×Q₈	C₄×D₅	1	4	yes	—
180	C₂×C₃×Q₈	D₂₀	1	4	yes	—
181	C₂×C₃×Q₈	C₂²⋊D₅	1	4	yes	—
182	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
183	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
184	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
185	C₂²×Dic₃	—	5	3	no	—
186	C₂²×Dic₃	—	5	1	no	—
187	C₂²×Dic₃	—	5	3	no	—
188	D₃×Q₈	—	10	32	no	—
189	C₂×C₄×D₃	—	5	12	no	—
190	C₂×C₄×D₃	—	10	6	no	—
191	C₂×C₄×D₃	—	10	6	no	—
192	C₂×Dic₆	—	5	12	no	—
193	C₂×Dic₆	—	10	6	no	—
194	C₂×Dic₆	—	10	6	no	—
195	C₃×C₄⋊C₄	—	2	6	no	—
196	C₃×C₄²	—	2	6	no	—
197	(C₂×D₃)⋊C₄	—	5	4	no	—
198	(C₂×D₃)⋊C₄	—	10	6	no	—
199	C₄⋊Dic₃	—	10	6	no	—
200	C₄×Dic₃	—	10	6	no	—
201	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
202	C₂²×D₅	—	3	2	no	—
203	C₂²×D₅	—	3	2	no	—
204	C₂²×D₅	—	3	2	no	—
205	C₂²×D₅	—	6	1	no	—
206	C₂²×D₅	—	6	1	no	—
207	C₂²×D₅	—	6	1	no	—
208	C₂²×D₅	—	6	1	no	—
209	C₅×Q₈	(C₂×D₃)⋊C₄	1	16	yes	complemented
210	C₂×C₄×C₅	—	2	1	no	—
211	C₂×C₄×C₅	—	2	1	no	—
212	C₂×C₄×C₅	—	2	3	no	—
213	C₂×C₄×C₅	—	2	3	no	—
214	C₂×C₄×C₅	—	2	3	no	—
215	C₂×C₄×C₅	C₂×C₄×D₃	1	6	yes	—
216	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
217	C₂×C₄×C₅	D₃×Q₈	1	2	yes	complemented
218	C₂×C₄×C₅	C₂×#67	1	6	yes	—
219	C₂×Dic₅	—	3	6	no	—
220	C₂×Dic₅	—	3	2	no	—
221	C₂×Dic₅	—	6	1	no	—
222	C₂×Dic₅	—	6	3	no	—
223	C₂×Dic₅	—	6	1	no	—
224	C₂×Dic₅	—	6	3	no	—
225	C₂×Dic₅	—	6	3	no	—
226	C₂×Dic₅	—	6	3	no	—
227	C₂×Dic₅	—	6	3	no	—
228	C₄×D₅	—	6	24	no	—
229	Dic₁₀	—	6	24	no	—
230	C₂²×Q₈	—	15	8	no	—
231	C₂²×Q₈	—	15	3	no	—
232	C₂²×Q₈	—	30	4	no	—
233	C₂²×Q₈	—	30	4	no	—
234	C₂²×Q₈	C₂×D₁₅	1	1	yes	characteristic
235	C₂³×C₄	—	15	3	no	—
236	C₄×Q₈	—	2	4	no	—
237	C₄×Q₈	—	30	4	no	—
238	C₂×C₄⋊C₄	—	15	3	no	—
239	C₂×C₄⋊C₄	C₂×D₁₅	1	3	yes	—
240	C₂×C₂²⋊C₄	—	15	1	no	—
241	C₂×C₂²⋊C₄	—	15	3	no	—
242	C₂×C₄²	—	15	3	no	—
243	C₂×C₄²	C₂×D₁₅	1	3	yes	—
244	#95	—	15	3	no	—
245	#95	—	15	3	no	—
246	#95	—	15	6	no	—
247	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₃×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
249	C₂×C₃×C₅	#519	1	1	yes	characteristic
250	C₂×C₃×C₅	#548	1	1	yes	characteristic
251	C₂×C₃×C₅	#548	1	1	yes	characteristic
252	C₂×C₃×C₅	D₄×Q₈	1	1	yes	characteristic
253	C₂×C₃×C₅	D₄×Q₈	1	1	yes	characteristic
254	D₁₅	—	2	4	no	—
255	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
256	C₂²×D₃	—	5	2	no	—
257	C₂²×D₃	—	5	2	no	—
258	C₂²×D₃	—	5	2	no	—
259	C₂²×D₃	—	10	1	no	—
260	C₂²×D₃	—	10	1	no	—
261	C₂²×D₃	—	10	1	no	—
262	C₂²×D₃	—	10	1	no	—
263	C₃×Q₈	#685	1	16	yes	complemented
264	C₂×C₃×C₄	—	2	1	no	—
265	C₂×C₃×C₄	—	2	1	no	—
266	C₂×C₃×C₄	—	2	3	no	—
267	C₂×C₃×C₄	—	2	3	no	—
268	C₂×C₃×C₄	—	2	3	no	—
269	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
270	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
271	C₂×C₃×C₄	Q₈×D₅	1	2	yes	complemented
272	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	6	yes	—
273	C₂×Dic₃	—	5	6	no	—
274	C₂×Dic₃	—	5	2	no	—
275	C₂×Dic₃	—	10	1	no	—
276	C₂×Dic₃	—	10	3	no	—
277	C₂×Dic₃	—	10	1	no	—
278	C₂×Dic₃	—	10	3	no	—
279	C₂×Dic₃	—	10	3	no	—
280	C₂×Dic₃	—	10	3	no	—
281	C₂×Dic₃	—	10	3	no	—
282	C₄×D₃	—	10	24	no	—
283	Dic₆	—	10	24	no	—
284	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
285	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
286	C₂²×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
287	C₂²×C₅	#1007	1	1	yes	characteristic
288	C₂²×C₅	D₃×#41	1	1	yes	characteristic
289	C₂²×C₅	#1010	1	1	yes	characteristic
290	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
291	C₂×D₅	—	3	4	no	—
292	C₂×D₅	—	6	2	no	—
293	C₂×D₅	—	6	2	no	—
294	C₂×D₅	—	6	2	no	—
295	C₂×D₅	—	6	2	no	—
296	C₂×D₅	—	6	2	no	—
297	C₂×D₅	—	6	2	no	—
298	C₄×C₅	—	2	2	no	—
299	C₄×C₅	—	4	3	no	—
300	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	12	yes	—
301	Dic₅	—	6	6	no	—
302	Dic₅	—	6	2	no	—
303	Dic₅	—	12	3	no	—
304	C₂⁴	—	15	1	no	—
305	C₂×Q₈	—	15	12	no	—
306	C₂×Q₈	—	30	32	no	—
307	C₂×Q₈	—	30	6	no	—
308	C₂×Q₈	—	30	6	no	—
309	C₂×Q₈	C₄×D₁₅	1	4	yes	—
310	C₂×Q₈	D₆₀	1	4	yes	—
311	C₂×Q₈	#1220	1	4	yes	—
312	C₂²×C₄	—	15	3	no	—
313	C₂²×C₄	—	15	1	no	—
314	C₂²×C₄	—	15	12	no	—
315	C₂²×C₄	—	15	3	no	—
316	C₂²×C₄	—	30	6	no	—
317	C₂²×C₄	—	30	6	no	—
318	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
319	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
320	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
321	C₄⋊C₄	—	2	6	no	—
322	C₄⋊C₄	—	30	6	no	—
323	C₂²⋊C₄	—	15	4	no	—
324	C₂²⋊C₄	—	30	6	no	—
325	C₄²	—	2	6	no	—
326	C₄²	—	30	6	no	—
327	C₃×C₅	Q₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
328	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
329	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
330	C₂²×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
331	C₂²×C₃	#4147	1	1	yes	characteristic
332	C₂²×C₃	D₅×#41	1	1	yes	characteristic
333	C₂²×C₃	#4150	1	1	yes	characteristic
334	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
335	C₂×D₃	—	5	4	no	—
336	C₂×D₃	—	10	2	no	—
337	C₂×D₃	—	10	2	no	—
338	C₂×D₃	—	10	2	no	—
339	C₂×D₃	—	10	2	no	—
340	C₂×D₃	—	10	2	no	—
341	C₂×D₃	—	10	2	no	—
342	C₃×C₄	—	2	2	no	—
343	C₃×C₄	—	4	3	no	—
344	C₃×C₄	C₂×#685	1	12	yes	—
345	Dic₃	—	10	6	no	—
346	Dic₃	—	10	2	no	—
347	Dic₃	—	20	3	no	—
348	C₂×C₅	C₄×D₃×Q₈	1	1	yes	characteristic
349	C₂×C₅	#5572	1	1	yes	characteristic
350	C₂×C₅	Q₈×D₁₂	1	1	yes	characteristic
351	C₂×C₅	#5576	1	1	yes	characteristic
352	C₂×C₅	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic
353	C₂×C₅	Q₈×#67	1	1	yes	characteristic
354	C₂×C₅	#5831	1	1	yes	characteristic
355	D₅	—	6	4	no	—
356	C₂³	—	15	2	no	—
357	C₂³	—	15	2	no	—
358	C₂³	—	15	2	no	—
359	C₂³	—	30	1	no	—
360	C₂³	—	30	1	no	—
361	C₂³	—	30	1	no	—
362	C₂³	—	30	1	no	—
363	C₂³	C₂³×D₁₅	1	1	yes	characteristic
364	Q₈	—	30	24	no	—
365	Q₈	#6714	1	16	yes	direct product factor
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	3	no	—
369	C₂×C₄	—	2	3	no	—
370	C₂×C₄	—	2	3	no	—
371	C₂×C₄	—	15	6	no	—
372	C₂×C₄	—	15	2	no	—
373	C₂×C₄	—	30	1	no	—
374	C₂×C₄	—	30	3	no	—
375	C₂×C₄	—	30	1	no	—
376	C₂×C₄	—	30	3	no	—
377	C₂×C₄	—	30	24	no	—
378	C₂×C₄	—	30	3	no	—
379	C₂×C₄	—	30	3	no	—
380	C₂×C₄	—	30	3	no	—
381	C₂×C₄	C₂×C₄×D₁₅	1	6	yes	—
382	C₂×C₄	C₂×D₆₀	1	6	yes	—
383	C₂×C₄	Q₈×D₁₅	1	2	yes	complemented
384	C₂×C₄	C₂×#1220	1	6	yes	—
385	C₂×C₃	C₄×Q₈×D₅	1	1	yes	characteristic
386	C₂×C₃	#65906	1	1	yes	characteristic
387	C₂×C₃	Q₈×D₂₀	1	1	yes	characteristic
388	C₂×C₃	#65910	1	1	yes	characteristic
389	C₂×C₃	C₂²×#685	1	1	yes	characteristic
390	C₂×C₃	Q₈×C₂²⋊D₅	1	1	yes	characteristic
391	C₂×C₃	#66165	1	1	yes	characteristic
392	D₃	—	10	4	no	—
393	C₅	Q₈×(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented, Sylow
394	C₂²	—	15	4	no	—
395	C₂²	—	30	2	no	—
396	C₂²	—	30	2	no	—
397	C₂²	—	30	2	no	—
398	C₂²	—	30	2	no	—
399	C₂²	—	30	2	no	—
400	C₂²	—	30	2	no	—
401	C₂²	C₂²×C₄×D₁₅	1	1	yes	characteristic
402	C₂²	C₂²×D₆₀	1	1	yes	characteristic
403	C₂²	C₂×Q₈×D₁₅	1	1	yes	characteristic
404	C₂²	#92098	1	1	yes	characteristic
405	C₂²	#41×D₁₅	1	1	yes	characteristic
406	C₂²	#92101	1	1	yes	characteristic
407	C₂²	C₂²×#1220	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	4	3	no	—
410	C₄	—	30	6	no	—
411	C₄	—	30	2	no	—
412	C₄	—	60	3	no	—
413	C₄	C₂×#6714	1	12	yes	—
414	C₃	Q₈×#685	1	1	yes	characteristic, complemented, Sylow
415	C₂	—	30	4	no	—
416	C₂	C₄×Q₈×D₁₅	1	1	yes	characteristic
417	C₂	#11756331	1	1	yes	characteristic
418	C₂	Q₈×D₆₀	1	1	yes	characteristic
419	C₂	#11756335	1	1	yes	characteristic
420	C₂	C₂²×#6714	1	1	yes	characteristic
421	C₂	Q₈×#1220	1	1	yes	characteristic
422	C₂	#11756590	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅
commutator	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle, cosocle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂³×D₁₅	C₂³
hypercenter	C₂×C₄×Q₈	D₁₅
hypocenter	C₃×C₅	Q₈×C₂²⋊C₄
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₅×Q₈	C₂	semidirect
3. C₂×Q₈×Dic₁₅	C₂	semidirect
5. #11751074	C₂	semidirect
7. #11751030	C₂	semidirect
12. C₂×Q₈×D₁₅	C₄	semidirect
28. C₂×C₄×C₅×Q₈	D₃	semidirect
54. #6714	Q₈	direct (N × H)
58. C₂×C₃×C₄×Q₈	D₅	semidirect
90. C₃×C₅×Q₈	C₂²⋊C₄	semidirect
99. C₂×C₃×C₄×C₅	C₂×Q₈	semidirect
108. C₂×Dic₁₅	C₂×Q₈	semidirect
153. C₂×C₄×Q₈	D₁₅	semidirect
171. C₂×D₁₅	C₄×Q₈	semidirect
209. C₅×Q₈	(C₂×D₃)⋊C₄	semidirect
217. C₂×C₄×C₅	D₃×Q₈	semidirect
263. C₃×Q₈	#685	semidirect
271. C₂×C₃×C₄	Q₈×D₅	semidirect
327. C₃×C₅	Q₈×C₂²⋊C₄	semidirect
365. Q₈	#6714	direct (N × H)
383. C₂×C₄	Q₈×D₁₅	semidirect
393. C₅	Q₈×(C₂×D₃)⋊C₄	semidirect
414. C₃	Q₈×#685	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₁₅	2
C₂×C₃×C₄×C₅×Q₈	2
C₂×Q₈×Dic₁₅	2
#11751088	2
#11751074	2
C₄×#6714	2
#11751030	2
Q₈×#685	3
Q₈×(C₂×D₃)⋊C₄	5

Subgroups — Q8×#6714 #14326113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6714 #14326113