Subgroups — #79413

Order 384.

342 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5541	C₂	1	2	yes	maximal
2	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	#5116	C₂	1	1	yes	maximal, characteristic, complemented
4	#5116	C₂	1	1	yes	maximal, characteristic, complemented
5	#5106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#818	C₂	1	1	yes	maximal, characteristic, complemented
7	#4829	C₂	1	2	yes	maximal, complemented
8	#4723	C₂	1	2	yes	maximal, complemented
9	#4721	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#4683	C₂	1	1	yes	maximal, characteristic, complemented
12	#3336	—	3	1	no	maximal, Sylow
13	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
16	#935	—	2	2	no	—
17	#926	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
19	#924	C₂²	1	2	yes	—
20	#924	C₂²	1	2	yes	—
21	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
22	#889	—	2	2	no	—
23	#879	—	2	2	no	—
24	#877	—	2	2	no	—
25	#870	C₂²	1	2	yes	—
26	C₄×Dic₆	C₂²	1	2	yes	—
27	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
31	#818	C₂²	1	1	yes	characteristic
32	#818	C₂²	1	1	yes	characteristic
33	#818	C₂²	1	1	yes	characteristic
34	#818	C₂²	1	2	yes	—
35	#817	C₂²	1	2	yes	—
36	#817	C₂²	1	2	yes	—
37	#816	C₂²	1	2	yes	—
38	#816	C₂²	1	2	yes	—
39	#816	C₂²	1	2	yes	—
40	#552	—	3	2	no	—
41	#499	—	3	1	no	—
42	#485	—	3	2	no	—
43	#483	—	3	2	no	—
44	#479	—	3	2	no	—
45	#467	—	3	1	no	—
46	#467	—	3	1	no	—
47	C₄×Q₁₆	—	3	1	no	—
48	C₈×D₄	D₃	1	1	yes	characteristic
49	#427	—	3	1	no	—
50	C₂×#107	—	3	1	no	—
51	C₂×C₃×D₄	D₄	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×Dic₃	—	2	2	no	—
55	C₂×Dic₆	C₂³	1	2	yes	—
56	C₂×C₃×C₈	—	2	2	no	—
57	C₂×C₃×C₈	—	2	2	no	—
58	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
64	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
65	C₃×C₄²	C₂³	1	1	yes	characteristic
66	#217	—	2	2	no	—
67	#217	—	2	2	no	—
68	#217	—	2	2	no	—
69	C₄⋊Dic₃	—	2	2	no	—
70	C₄⋊Dic₃	C₂³	1	2	yes	—
71	C₄⋊Dic₃	C₂³	1	2	yes	—
72	C₄⋊Dic₃	C₂³	1	2	yes	—
73	Dic₃⋊C₄	—	2	2	no	—
74	Dic₃⋊C₄	—	2	2	no	—
75	Dic₃⋊C₄	—	2	2	no	—
76	Dic₃⋊C₄	—	2	2	no	—
77	Dic₃⋊C₄	—	2	2	no	—
78	C₄×Dic₃	—	2	2	no	—
79	C₄×Dic₃	—	2	2	no	—
80	Dic₁₂	—	2	2	no	—
81	C₂×Q₁₆	—	3	1	no	—
82	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
84	#126	—	6	2	no	—
85	#125	—	3	2	no	—
86	#125	—	6	2	no	—
87	#123	—	6	2	no	—
88	C₂²⋊Q₈	—	3	2	no	—
89	C₂²⋊Q₈	—	3	2	no	—
90	C₄×Q₈	—	3	2	no	—
91	C₄×D₄	C₂×D₃	1	1	yes	characteristic
92	#117	—	3	2	no	—
93	#117	—	6	2	no	—
94	C₂×C₄⋊C₄	—	3	2	no	—
95	#107	—	3	1	no	—
96	#107	—	3	1	no	—
97	#107	—	3	2	no	—
98	#107	—	3	1	no	—
99	#106	—	3	2	no	—
100	#106	—	3	2	no	—
101	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
102	#103	—	3	2	no	—
103	#103	—	3	2	no	—
104	#103	—	3	2	no	—
105	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
106	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
107	C₄×C₈	C₂×D₃	1	1	yes	characteristic
108	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
110	C₃×D₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	1	no	—
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₃	—	2	2	no	—
121	C₂×Dic₃	—	2	2	no	—
122	C₂×Dic₃	—	2	2	no	—
123	C₂×Dic₃	—	2	2	no	—
124	C₂×Dic₃	—	4	4	no	—
125	Dic₆	—	2	2	no	—
126	Dic₆	—	2	2	no	—
127	C₃×C₈	—	2	1	no	—
128	C₃×C₈	—	2	1	no	—
129	C₃×C₈	—	2	1	no	—
130	C₃×C₈	#41	1	2	yes	—
131	C₂×Q₈	—	3	2	no	—
132	C₂×D₄	D₁₂	1	1	yes	characteristic
133	C₂²×C₄	—	6	2	no	—
134	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
136	Q₁₆	—	6	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
143	C₄⋊C₄	—	3	2	no	—
144	C₄⋊C₄	—	3	2	no	—
145	C₄⋊C₄	—	3	2	no	—
146	C₄⋊C₄	—	6	2	no	—
147	C₄⋊C₄	—	6	2	no	—
148	C₄⋊C₄	—	6	2	no	—
149	C₄⋊C₄	—	6	2	no	—
150	C₄⋊C₄	—	6	2	no	—
151	C₄⋊C₄	—	6	2	no	—
152	C₄⋊C₄	D₁₂	1	1	yes	characteristic
153	C₂²⋊C₄	—	6	2	no	—
154	C₂²⋊C₄	—	6	2	no	—
155	C₂²⋊C₄	—	6	2	no	—
156	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
157	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
158	C₄²	—	6	2	no	—
159	C₄²	—	6	2	no	—
160	C₄²	C₂²×D₃	1	1	yes	characteristic
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	#135	1	2	yes	—
167	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	C₂×#41	1	1	yes	characteristic
173	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
174	Dic₃	—	4	2	no	—
175	Dic₃	—	4	2	no	—
176	Dic₃	—	4	2	no	—
177	Dic₃	—	4	2	no	—
178	C₂³	C₂×D₁₂	1	1	yes	characteristic
179	C₂³	C₂×D₁₂	1	1	yes	characteristic
180	Q₈	—	6	2	no	—
181	Q₈	—	6	2	no	—
182	D₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	6	2	no	—
188	C₂×C₄	—	6	2	no	—
189	C₂×C₄	—	6	2	no	—
190	C₂×C₄	—	6	2	no	—
191	C₂×C₄	—	12	4	no	—
192	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
193	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
194	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
195	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
196	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
197	C₈	—	2	1	no	—
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	#237	1	2	yes	—
201	C₂×C₃	—	2	2	no	—
202	C₂×C₃	—	4	1	no	—
203	C₂×C₃	#547	1	1	yes	characteristic
204	C₂×C₃	C₂×#135	1	1	yes	characteristic
205	C₂×C₃	#578	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#904	1	2	yes	—
212	C₂²	C₂²×D₁₂	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	12	2	no	—
218	C₄	—	12	2	no	—
219	C₄	—	12	2	no	—
220	C₄	—	12	2	no	—
221	C₄	C₂×#237	1	1	yes	characteristic
222	C₄	#1010	1	1	yes	characteristic
223	C₃	#3336	1	1	yes	characteristic, complemented, Sylow
224	C₂	—	2	2	no	—
225	C₂	—	4	1	no	—
226	C₂	#5555	1	1	yes	characteristic
227	C₂	C₂×#904	1	1	yes	characteristic
228	C₂	#5753	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₈×D₄	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×D₄	D₄
hypercenter	C₈×D₄	D₃
hypocenter	C₃	#3336
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
3. #5116	C₂	semidirect
4. #5116	C₂	semidirect
5. #5106	C₂	semidirect
6. C₂×#818	C₂	semidirect
7. #4829	C₂	semidirect
8. #4723	C₂	semidirect
9. #4721	C₂	semidirect
10. C₄×Dic₁₂	C₂	semidirect
11. #4683	C₂	semidirect
223. C₃	#3336	semidirect

Maximal subgroups

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

subgroup	index
#5541	2
C₃×C₈×D₄	2
#5116	2
#5106	2
C₂×#818	2
#4829	2
#4723	2
#4721	2
C₄×Dic₁₂	2
#4683	2
#3336	3