Subgroups — #99445

Order 512.

420 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#9602	C₂	1	1	yes	maximal, characteristic
3	#8805	C₂	1	1	yes	maximal, characteristic
4	#8564	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#7031	C₂	1	1	yes	maximal, characteristic
7	#7015	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	—	2	1	no	—
9	C₄²×Q₈	C₂²	1	1	yes	characteristic
10	#1939	—	2	2	no	—
11	#1939	—	2	1	no	—
12	#1842	—	2	1	no	—
13	#1842	C₂²	1	1	yes	characteristic
14	#1842	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₈	—	2	1	no	—
16	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₄²×C₈	C₂²	1	1	yes	characteristic
20	C₂×#334	—	2	2	no	—
21	C₂×#334	—	2	1	no	—
22	C₂×Q₈⋊C₈	—	2	1	no	—
23	C₂×Q₈⋊C₈	C₄	1	1	yes	characteristic
24	C₂×Q₈⋊C₈	C₄	1	1	yes	characteristic
25	#1279	—	2	1	no	—
26	#1279	C₄	1	1	yes	characteristic
27	#1279	C₄	1	1	yes	characteristic
28	#1277	—	2	1	no	—
29	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₄×Q₈	Q₈	1	1	yes	characteristic
32	C₈×Q₈	—	4	2	no	—
33	C₈×Q₈	—	4	2	no	—
34	C₂×C₄⋊C₈	—	2	1	no	—
35	C₂×C₄⋊C₈	—	2	2	no	—
36	C₂×C₄⋊C₈	—	2	1	no	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₄⋊C₈	—	2	1	no	—
40	C₂×C₄⋊C₈	—	2	1	no	—
41	C₂×C₄⋊C₈	—	2	2	no	—
42	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
45	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₈	D₄	1	2	yes	—
47	C₂×C₄×C₈	—	2	1	no	—
48	C₂×C₄×C₈	—	2	2	no	—
49	C₂×C₄×C₈	—	2	1	no	—
50	C₂×C₄×C₈	—	2	1	no	—
51	C₂×C₄×C₈	—	2	1	no	—
52	C₂×C₄×C₈	—	2	1	no	—
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₄⋊C₄	—	2	1	no	—
56	C₄×C₄⋊C₄	—	2	1	no	—
57	C₄×C₄⋊C₄	—	2	1	no	—
58	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
59	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
60	C₄×C₄⋊C₄	Q₈	1	1	yes	characteristic
61	C₄³	—	2	1	no	—
62	C₄³	C₂³	1	1	yes	characteristic
63	#336	—	2	2	no	—
64	#336	—	2	1	no	—
65	#336	—	2	1	no	—
66	#336	—	2	1	no	—
67	#334	—	4	4	no	—
68	#334	—	4	2	no	—
69	Q₈⋊C₈	—	2	2	no	—
70	Q₈⋊C₈	—	4	2	no	—
71	Q₈⋊C₈	C₈	1	4	yes	complemented
72	C₂²×Q₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	—	2	1	no	—
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	—	2	1	no	—
79	C₂²×C₈	—	2	2	no	—
80	C₂²×C₈	—	4	1	no	—
81	C₂²×C₈	—	4	1	no	—
82	C₄×Q₈	—	2	1	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	—	2	1	no	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	4	1	no	—
88	C₄×Q₈	C₂×C₈	1	2	yes	—
89	C₄×Q₈	M₁₆	1	2	yes	—
90	C₄×Q₈	SD₁₆	1	2	yes	—
91	C₄×Q₈	SD₁₆	1	2	yes	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	—	2	1	no	—
94	C₂×C₄⋊C₄	—	2	1	no	—
95	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄²	—	2	1	no	—
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	—	2	1	no	—
101	C₂×C₄²	—	2	1	no	—
102	C₂×C₄²	—	2	1	no	—
103	C₂×C₄²	—	2	1	no	—
104	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₄²	#41	1	1	yes	characteristic
110	C₂×C₄²	#41	1	1	yes	characteristic
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	—	2	1	no	—
116	C₄⋊C₈	—	2	1	no	—
117	C₄⋊C₈	—	2	1	no	—
118	C₄⋊C₈	—	4	2	no	—
119	C₄⋊C₈	—	4	1	no	—
120	C₄⋊C₈	—	4	4	no	—
121	C₄⋊C₈	—	4	2	no	—
122	C₄⋊C₈	—	4	1	no	—
123	C₄⋊C₈	—	4	1	no	—
124	C₄⋊C₈	—	4	1	no	—
125	C₄⋊C₈	—	4	1	no	—
126	C₄⋊C₈	—	4	1	no	—
127	C₄⋊C₈	—	4	1	no	—
128	C₄⋊C₈	—	4	1	no	—
129	C₄⋊C₈	—	4	1	no	—
130	C₄⋊C₈	—	8	2	no	—
131	C₄⋊C₈	C₂×C₈	1	2	yes	—
132	C₄⋊C₈	SD₁₆	1	2	yes	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	4	4	no	—
139	C₄×C₈	—	4	2	no	—
140	C₄×C₈	—	4	1	no	—
141	C₄×C₈	—	4	1	no	—
142	C₄×C₈	—	4	1	no	—
143	C₄×C₈	—	4	1	no	—
144	C₄×C₈	—	4	1	no	—
145	C₄×C₈	—	4	1	no	—
146	C₄×C₈	C₂×C₈	1	2	yes	—
147	#95	—	2	1	no	—
148	#95	—	2	1	no	—
149	#95	—	2	1	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	2	1	no	—
153	C₂×Q₈	C₄⋊C₈	1	2	yes	—
154	C₂×Q₈	#106	1	2	yes	—
155	C₂×Q₈	#108	1	2	yes	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
162	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
163	C₂²×C₄	#120	1	1	yes	characteristic
164	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
165	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
166	C₂²×C₄	#123	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	8	1	no	—
190	C₂×C₈	—	8	1	no	—
191	C₂×C₈	—	8	1	no	—
192	C₂×C₈	—	8	1	no	—
193	C₂×C₈	—	8	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
204	C₄⋊C₄	#106	1	2	yes	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	2	no	—
217	C₄²	—	4	1	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	—	8	1	no	—
222	C₄²	C₂²×C₈	1	1	yes	characteristic
223	C₄²	C₂×M₁₆	1	1	yes	characteristic
224	C₄²	#131	1	1	yes	characteristic
225	C₄²	#131	1	1	yes	characteristic
226	C₄²	C₂×SD₁₆	1	1	yes	characteristic
227	C₄²	C₂×SD₁₆	1	1	yes	characteristic
228	C₄²	C₂×SD₁₆	1	1	yes	characteristic
229	C₄²	#135	1	1	yes	characteristic
230	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₄²	#137	1	1	yes	characteristic
232	C₄²	#137	1	1	yes	characteristic
233	C₄²	#137	1	1	yes	characteristic
234	C₂³	#385	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	2	2	no	—
238	Q₈	#334	1	4	yes	complemented
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	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	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
261	C₂×C₄	#424	1	1	yes	characteristic
262	C₂×C₄	C₂×#106	1	1	yes	characteristic
263	C₂×C₄	#428	1	1	yes	characteristic
264	C₂×C₄	C₂×#108	1	1	yes	characteristic
265	C₂×C₄	#430	1	1	yes	characteristic
266	C₂×C₄	C₈×D₄	1	1	yes	characteristic
267	C₂×C₄	C₈×D₄	1	1	yes	characteristic
268	C₂×C₄	#435	1	1	yes	characteristic
269	C₂×C₄	#435	1	1	yes	characteristic
270	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
271	C₂×C₄	#441	1	1	yes	characteristic
272	C₂×C₄	#443	1	1	yes	characteristic
273	C₂×C₄	#444	1	1	yes	characteristic
274	C₂×C₄	#448	1	1	yes	characteristic
275	C₂×C₄	#448	1	1	yes	characteristic
276	C₂×C₄	#450	1	1	yes	characteristic
277	C₂×C₄	#452	1	1	yes	characteristic
278	C₂×C₄	#453	1	1	yes	characteristic
279	C₂×C₄	#456	1	1	yes	characteristic
280	C₂×C₄	#475	1	1	yes	characteristic
281	C₂×C₄	#479	1	1	yes	characteristic
282	C₂×C₄	#481	1	1	yes	characteristic
283	C₂×C₄	#483	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	4	2	no	—
286	C₈	—	4	2	no	—
287	C₈	—	4	2	no	—
288	C₈	—	8	2	no	—
289	C₈	—	8	1	no	—
290	C₈	—	8	1	no	—
291	C₈	—	8	2	no	—
292	C₈	—	16	1	no	—
293	C₂²	#1850	1	1	yes	characteristic
294	C₂²	#1853	1	1	yes	characteristic
295	C₂²	#1855	1	1	yes	characteristic
296	C₂²	#1859	1	1	yes	characteristic
297	C₂²	#1861	1	1	yes	characteristic
298	C₂²	#1865	1	1	yes	characteristic
299	C₂²	#1866	1	1	yes	characteristic
300	C₄	—	2	4	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	1	no	—
311	C₄	C₂×#334	1	1	yes	characteristic
312	C₄	#1565	1	1	yes	characteristic
313	C₄	#1570	1	1	yes	characteristic
314	C₄	#1573	1	1	yes	characteristic
315	C₄	C₈×SD₁₆	1	1	yes	characteristic
316	C₄	#1579	1	1	yes	characteristic
317	C₄	#1583	1	1	yes	characteristic
318	C₄	#1587	1	1	yes	characteristic
319	C₄	#1623	1	1	yes	characteristic
320	C₄	#1631	1	1	yes	characteristic
321	C₄	#1651	1	1	yes	characteristic
322	C₄	#1653	1	1	yes	characteristic
323	C₂	#8830	1	1	yes	characteristic
324	C₂	#8839	1	1	yes	characteristic
325	C₂	#8846	1	1	yes	characteristic
326	C₂	#8888	1	1	yes	characteristic
327	C₂	#8899	1	1	yes	characteristic
328	C₂	#8904	1	1	yes	characteristic
329	C₂	#8909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#385
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
71. Q₈⋊C₈	C₈	semidirect
238. Q₈	#334	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊C₈	2
#9602	2
#8805	2
#8564	2
C₄×Q₈⋊C₈	2
#7031	2
#7015	2