Subgroups — #79025

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#235	C₂	1	1	yes	maximal, characteristic
2	#5470	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#5113	C₂	1	2	yes	maximal
5	#5106	C₂	1	2	yes	maximal, complemented
6	#4851	C₂	1	1	yes	maximal, characteristic, complemented
7	#4850	C₂	1	1	yes	maximal, characteristic, complemented
8	#4847	C₂	1	2	yes	maximal, complemented
9	#4846	C₂	1	1	yes	maximal, characteristic, complemented
10	#4842	C₂	1	1	yes	maximal, characteristic, complemented
11	#4829	C₂	1	2	yes	maximal, complemented
12	#3076	—	3	1	no	maximal, Sylow
13	C₂×#235	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	2	yes	—
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₄×#67	—	2	1	no	—
17	C₄×#67	—	2	1	no	—
18	#926	C₂²	1	2	yes	—
19	#926	C₂²	1	1	yes	characteristic
20	#924	—	2	1	no	—
21	#924	—	2	1	no	—
22	C₄×D₁₂	C₂²	1	1	yes	characteristic
23	C₄×D₁₂	C₂²	1	1	yes	characteristic
24	#872	—	2	1	no	—
25	C₄²×D₃	—	2	1	no	—
26	#870	C₂²	1	2	yes	—
27	#869	C₂²	1	1	yes	characteristic
28	#869	C₂²	1	1	yes	characteristic
29	C₄×Dic₆	—	2	1	no	—
30	C₄×Dic₆	—	2	1	no	—
31	C₄×Dic₆	C₂²	1	1	yes	characteristic
32	C₄×Dic₆	C₂²	1	1	yes	characteristic
33	C₃×C₄⋊C₈	C₂²	1	2	yes	—
34	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	#821	C₂²	1	2	yes	—
37	#821	C₂²	1	2	yes	—
38	#818	C₂²	1	2	yes	—
39	#818	C₂²	1	2	yes	—
40	#817	C₂²	1	2	yes	—
41	#817	C₂²	1	2	yes	—
42	#816	C₂²	1	2	yes	—
43	#816	C₂²	1	2	yes	—
44	#533	—	3	1	no	—
45	C₄×#41	—	3	1	no	—
46	#479	—	3	2	no	—
47	#478	—	3	2	no	—
48	#477	—	3	1	no	—
49	#476	—	3	1	no	—
50	#475	—	3	1	no	—
51	#474	—	3	1	no	—
52	#427	—	3	2	no	—
53	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
54	#416	—	3	2	no	—
55	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂×#67	—	2	1	no	—
59	#235	—	2	2	no	—
60	#235	Q₈	1	4	yes	complemented
61	C₂×D₁₂	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₃	—	2	1	no	—
63	C₂×C₄×D₃	—	2	1	no	—
64	C₂×C₄×D₃	—	2	1	no	—
65	C₂×Dic₆	—	2	1	no	—
66	C₂×Dic₆	—	2	1	no	—
67	C₂×Dic₆	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	C₂³	1	2	yes	—
70	C₂×C₃×C₈	C₂³	1	2	yes	—
71	C₃×C₄²	D₄	1	1	yes	characteristic
72	C₃×C₄²	D₄	1	1	yes	characteristic
73	C₃×C₄²	C₂³	1	1	yes	characteristic
74	C₃×C₄²	C₂³	1	1	yes	characteristic
75	#217	—	2	2	no	—
76	#217	—	2	1	no	—
77	(C₂×D₃)⋊C₄	—	2	1	no	—
78	(C₂×D₃)⋊C₄	—	2	1	no	—
79	C₄⋊Dic₃	—	2	1	no	—
80	C₄⋊Dic₃	—	2	1	no	—
81	C₄⋊Dic₃	C₂³	1	2	yes	—
82	C₄⋊Dic₃	C₂³	1	2	yes	—
83	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
85	Dic₃⋊C₄	—	2	2	no	—
86	Dic₃⋊C₄	—	2	2	no	—
87	Dic₃⋊C₄	—	2	1	no	—
88	Dic₃⋊C₄	—	2	1	no	—
89	C₄×Dic₃	—	2	2	no	—
90	C₄×Dic₃	—	2	1	no	—
91	C₄×Dic₃	—	2	1	no	—
92	C₄×Dic₃	—	2	1	no	—
93	C₂×#41	—	3	1	no	—
94	C₂²×C₈	C₂×D₃	1	2	yes	—
95	C₄⋊Q₈	—	3	1	no	—
96	C₄⋊Q₈	—	3	1	no	—
97	#125	—	3	2	no	—
98	C₂²⋊Q₈	—	6	1	no	—
99	C₂²⋊Q₈	—	6	1	no	—
100	C₄×Q₈	—	3	1	no	—
101	C₄×Q₈	—	3	1	no	—
102	C₄×Q₈	—	6	1	no	—
103	C₄×Q₈	—	6	1	no	—
104	C₄×D₄	—	3	1	no	—
105	C₄×D₄	—	3	1	no	—
106	C₄×D₄	—	6	1	no	—
107	C₄×D₄	—	6	1	no	—
108	#117	—	3	1	no	—
109	#117	—	3	2	no	—
110	#117	—	6	1	no	—
111	C₂×C₄²	—	6	1	no	—
112	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
113	#107	—	3	2	no	—
114	#107	—	3	2	no	—
115	#106	—	3	2	no	—
116	#106	—	3	2	no	—
117	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
118	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
119	C₄⋊C₈	C₂×D₃	1	2	yes	—
120	#103	—	3	2	no	—
121	#103	—	3	2	no	—
122	#102	—	3	2	no	—
123	#102	—	3	2	no	—
124	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×D₃	—	2	1	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	4	1	no	—
128	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
134	C₂×C₃×C₄	#41	1	2	yes	—
135	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
136	#67	—	2	4	no	—
137	C₂×Dic₃	—	2	1	no	—
138	C₂×Dic₃	—	2	1	no	—
139	C₂×Dic₃	—	2	1	no	—
140	C₂×Dic₃	—	2	2	no	—
141	C₂×Dic₃	—	2	2	no	—
142	D₁₂	—	2	1	no	—
143	D₁₂	C₂×Q₈	1	2	yes	—
144	C₄×D₃	—	2	4	no	—
145	C₄×D₃	—	4	2	no	—
146	C₄×D₃	—	4	2	no	—
147	Dic₆	—	2	1	no	—
148	Dic₆	—	4	2	no	—
149	Dic₆	—	4	2	no	—
150	Dic₆	C₂×Q₈	1	2	yes	—
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	—	2	2	no	—
153	#41	—	3	4	no	—
154	#41	—	6	2	no	—
155	C₂×Q₈	—	3	1	no	—
156	C₂×Q₈	—	6	1	no	—
157	C₂×Q₈	—	6	1	no	—
158	C₂×D₄	—	3	1	no	—
159	C₂×D₄	—	6	1	no	—
160	C₂²×C₄	—	6	1	no	—
161	C₂²×C₄	—	6	1	no	—
162	C₂²×C₄	—	6	1	no	—
163	C₂²×C₄	D₁₂	1	1	yes	characteristic
164	C₂²×C₄	D₁₂	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	C₂²×D₃	1	2	yes	—
168	C₂×C₈	C₂²×D₃	1	2	yes	—
169	C₄⋊C₄	—	3	2	no	—
170	C₄⋊C₄	—	3	2	no	—
171	C₄⋊C₄	—	3	1	no	—
172	C₄⋊C₄	—	3	1	no	—
173	C₄⋊C₄	—	6	1	no	—
174	C₄⋊C₄	—	6	1	no	—
175	C₄⋊C₄	—	6	2	no	—
176	C₄⋊C₄	—	6	2	no	—
177	C₄⋊C₄	—	6	1	no	—
178	C₄⋊C₄	—	6	1	no	—
179	C₂²⋊C₄	—	6	1	no	—
180	C₂²⋊C₄	—	6	2	no	—
181	C₂²⋊C₄	—	6	1	no	—
182	C₂²⋊C₄	—	6	1	no	—
183	C₄²	—	6	1	no	—
184	C₄²	—	6	1	no	—
185	C₄²	—	6	1	no	—
186	C₄²	—	6	2	no	—
187	C₄²	D₁₂	1	1	yes	characteristic
188	C₄²	D₁₂	1	1	yes	characteristic
189	C₄²	C₂²×D₃	1	1	yes	characteristic
190	C₄²	C₂²×D₃	1	1	yes	characteristic
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
194	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×D₃	—	2	2	no	—
196	C₂×D₃	—	4	1	no	—
197	C₂×D₃	—	4	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
202	C₃×C₄	#135	1	2	yes	—
203	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
204	C₃×C₄	C₂×#41	1	1	yes	characteristic
205	Dic₃	—	2	2	no	—
206	Dic₃	—	4	2	no	—
207	Dic₃	—	4	1	no	—
208	Dic₃	—	4	2	no	—
209	Dic₃	—	4	1	no	—
210	C₂³	—	6	1	no	—
211	C₂³	C₂×D₁₂	1	1	yes	characteristic
212	Q₈	—	3	2	no	—
213	Q₈	—	6	1	no	—
214	Q₈	—	12	2	no	—
215	Q₈	—	12	2	no	—
216	D₄	—	3	2	no	—
217	D₄	—	6	4	no	—
218	D₄	—	6	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	6	1	no	—
222	C₂×C₄	—	6	4	no	—
223	C₂×C₄	—	6	1	no	—
224	C₂×C₄	—	6	1	no	—
225	C₂×C₄	—	6	2	no	—
226	C₂×C₄	—	6	2	no	—
227	C₂×C₄	—	12	2	no	—
228	C₂×C₄	—	12	2	no	—
229	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
230	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
231	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
232	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
233	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
234	C₂×C₄	#237	1	2	yes	—
235	C₂×C₄	D₃×Q₈	1	2	yes	—
236	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₂×C₃	—	2	2	no	—
240	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
241	C₂×C₃	C₂×#135	1	1	yes	characteristic
242	C₂×C₃	#575	1	1	yes	characteristic
243	D₃	—	4	2	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	6	2	no	—
247	C₂²	—	12	1	no	—
248	C₂²	—	12	1	no	—
249	C₂²	#897	1	2	yes	—
250	C₂²	C₂²×D₁₂	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	6	2	no	—
255	C₄	—	12	1	no	—
256	C₄	—	12	2	no	—
257	C₄	—	12	1	no	—
258	C₄	—	12	2	no	—
259	C₄	#897	1	2	yes	—
260	C₄	#904	1	2	yes	—
261	C₄	C₂×#237	1	1	yes	characteristic
262	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
263	C₃	#3076	1	1	yes	characteristic, complemented, Sylow
264	C₂	—	2	2	no	—
265	C₂	—	12	2	no	—
266	C₂	C₂×#897	1	1	yes	characteristic
267	C₂	C₂×#904	1	1	yes	characteristic
268	C₂	#5748	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄⋊C₈	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#235	C₂²
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#3076
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #5470	C₂	semidirect
3. C₂×C₃×C₄⋊C₈	C₂	semidirect
5. #5106	C₂	semidirect
6. #4851	C₂	semidirect
7. #4850	C₂	semidirect
8. #4847	C₂	semidirect
9. #4846	C₂	semidirect
10. #4842	C₂	semidirect
11. #4829	C₂	semidirect
53. C₂×C₄⋊C₈	D₃	semidirect
60. #235	Q₈	semidirect
263. C₃	#3076	semidirect

Maximal subgroups

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

subgroup	index
C₄×#235	2
#5470	2
C₂×C₃×C₄⋊C₈	2
#5113	2
#5106	2
#4851	2
#4850	2
#4847	2
#4846	2
#4842	2
#4829	2
#3076	3