Finite Group Databasebeta

Subgroups — #69256

Order 384.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#912	C₂	1	1	yes	maximal, characteristic
2	D₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
5	#4563	C₂	1	1	yes	maximal, characteristic, complemented
6	#4517	C₂	1	1	yes	maximal, characteristic, complemented
7	#4496	C₂	1	1	yes	maximal, characteristic, complemented
8	#2148	—	3	1	no	maximal, Sylow
9	C₂×#237	—	2	1	no	—
10	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
11	C₂×#214	—	2	1	no	—
12	#912	—	2	1	no	—
13	#912	—	2	1	no	—
14	#912	C₄	1	2	yes	—
15	#912	C₄	1	2	yes	—
16	D₃×M₁₆	—	2	1	no	—
17	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
18	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
19	C₃×M₃₂	C₂²	1	1	yes	characteristic
20	C₃×#108	C₂²	1	1	yes	characteristic
21	#822	—	2	1	no	—
22	#819	C₂²	1	1	yes	characteristic
23	#812	C₂²	1	1	yes	characteristic
24	#798	—	2	1	no	—
25	D₃×C₁₆	—	2	1	no	—
26	C₂×#135	—	3	1	no	—
27	C₂×M₃₂	—	3	1	no	—
28	C₂×#108	—	3	1	no	—
29	#362	—	3	1	no	—
30	#362	—	3	1	no	—
31	#361	—	3	1	no	—
32	#361	D₃	1	1	yes	characteristic, complemented
33	C₂×C₃×D₄	—	2	1	no	—
34	C₂×#67	—	4	1	no	—
35	C₂²×Dic₃	—	4	1	no	—
36	#237	—	2	2	no	—
37	#237	—	2	2	no	—
38	#237	—	4	1	no	—
39	#237	—	4	1	no	—
40	C₂×C₄×D₃	D₄	1	1	yes	characteristic
41	C₂×Dic₆	—	2	1	no	—
42	C₃×D₈	—	2	1	no	—
43	C₃×D₈	C₂×C₄	1	2	yes	—
44	C₃×M₁₆	—	2	1	no	—
45	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
46	#214	—	2	2	no	—
47	#214	—	2	2	no	—
48	#208	—	2	1	no	—
49	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
50	Dic₁₂	—	2	1	no	—
51	Dic₁₂	C₂×C₄	1	2	yes	—
52	ΓL(1,𝔽₂₅)	—	4	1	no	—
53	D₃×C₈	—	4	1	no	—
54	D₃×C₈	C₂×C₄	1	1	yes	characteristic
55	D₃×C₈	C₂×C₄	1	1	yes	characteristic
56	D₃×C₈	D₄	1	1	yes	characteristic
57	D₃×C₈	D₄	1	1	yes	characteristic
58	C₃×C₁₆	—	2	1	no	—
59	C₃⋊C₁₆	—	2	1	no	—
60	C₂×#41	—	6	1	no	—
61	#135	—	3	2	no	—
62	#135	—	3	2	no	—
63	#135	—	6	1	no	—
64	#135	—	6	1	no	—
65	C₂×Q₁₆	—	3	1	no	—
66	C₂×SD₁₆	—	6	1	no	—
67	C₂×D₈	C₂×D₃	1	1	yes	characteristic
68	C₂×M₁₆	—	6	1	no	—
69	C₂²×C₈	—	3	1	no	—
70	M₃₂	—	3	1	no	—
71	M₃₂	—	6	1	no	—
72	M₃₂	C₂×D₃	1	1	yes	characteristic
73	C₂×C₁₆	—	6	1	no	—
74	#108	—	3	1	no	—
75	#108	—	6	1	no	—
76	#108	C₂×D₃	1	1	yes	characteristic
77	C₂³×C₃	—	4	1	no	—
78	C₂²×D₃	SD₁₆	1	1	yes	characteristic
79	C₃×D₄	—	2	2	no	—
80	C₃×D₄	—	4	1	no	—
81	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	#67	—	4	2	no	—
83	#67	—	4	2	no	—
84	C₂×Dic₃	—	4	1	no	—
85	C₂×Dic₃	—	4	2	no	—
86	C₂×Dic₃	—	4	2	no	—
87	C₂×Dic₃	SD₁₆	1	1	yes	characteristic
88	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×D₃	D₈	1	1	yes	characteristic
91	C₄×D₃	D₈	1	1	yes	characteristic
92	Dic₆	—	2	2	no	—
93	Dic₆	—	4	1	no	—
94	C₃×C₈	—	4	1	no	—
95	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₃×C₈	C₂×D₄	1	1	yes	characteristic
97	C₃⋊C₈	—	4	1	no	—
98	C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
100	#41	—	6	2	no	—
101	#41	—	6	2	no	—
102	#41	—	12	1	no	—
103	#41	—	12	1	no	—
104	C₂×Q₈	—	6	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	12	1	no	—
107	C₂²×C₄	—	3	1	no	—
108	C₂²×C₄	—	12	1	no	—
109	Q₁₆	—	3	2	no	—
110	Q₁₆	—	6	1	no	—
111	SD₁₆	—	6	2	no	—
112	SD₁₆	—	6	2	no	—
113	D₈	—	2	1	no	—
114	D₈	C₄×D₃	1	2	yes	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	6	1	no	—
117	M₁₆	—	12	1	no	—
118	C₂×C₈	—	3	1	no	—
119	C₂×C₈	—	3	1	no	—
120	C₂×C₈	—	3	1	no	—
121	C₂×C₈	—	3	1	no	—
122	C₂×C₈	—	3	1	no	—
123	C₂×C₈	—	12	1	no	—
124	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	6	1	no	—
127	C₂²×C₃	—	4	2	no	—
128	C₂²×C₃	—	8	2	no	—
129	C₂²×C₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×D₃	—	4	1	no	—
131	C₂×D₃	#102	1	1	yes	characteristic
132	C₂×D₃	#102	1	1	yes	characteristic
133	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₃×C₄	C₂×D₈	1	1	yes	characteristic
135	Dic₃	—	4	2	no	—
136	Dic₃	#102	1	1	yes	characteristic
137	Dic₃	#102	1	1	yes	characteristic
138	C₂³	—	3	1	no	—
139	C₂³	—	4	1	no	—
140	Q₈	—	6	2	no	—
141	Q₈	—	12	1	no	—
142	D₄	—	2	2	no	—
143	D₄	—	4	1	no	—
144	D₄	—	12	2	no	—
145	D₄	—	12	2	no	—
146	C₂×C₄	—	3	1	no	—
147	C₂×C₄	—	3	1	no	—
148	C₂×C₄	—	3	1	no	—
149	C₂×C₄	—	3	1	no	—
150	C₂×C₄	—	3	1	no	—
151	C₂×C₄	—	12	2	no	—
152	C₂×C₄	—	12	2	no	—
153	C₂×C₄	—	12	1	no	—
154	C₂×C₄	D₃×D₄	1	1	yes	characteristic
155	C₈	—	3	1	no	—
156	C₈	—	3	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	12	1	no	—
159	C₈	C₂×C₄×D₃	1	1	yes	characteristic
160	C₈	D₃×D₄	1	1	yes	characteristic
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	8	2	no	—
163	C₂×C₃	C₂×#102	1	1	yes	characteristic
164	D₃	—	2	1	no	—
165	D₃	—	2	1	no	—
166	C₂²	—	3	1	no	—
167	C₂²	—	3	1	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	—	12	1	no	—
171	C₂²	D₃×SD₁₆	1	1	yes	characteristic
172	C₄	—	3	1	no	—
173	C₄	—	3	1	no	—
174	C₄	—	12	2	no	—
175	C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
176	C₄	D₃×D₈	1	1	yes	characteristic
177	C₃	#2148	1	1	yes	characteristic, complemented, Sylow
178	C₂	—	2	1	no	—
179	C₂	—	6	1	no	—
180	C₂	—	6	1	no	—
181	C₂	—	8	2	no	—
182	C₂	D₃×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×#102
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#361	C₂
socle	C₂×C₃	C₂×#102
prime radical	C₂×#912	C₂
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#361	D₃
hypocenter	C₃	#2148
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

Complements

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

N	H ≅ G/N	type
2. D₃×M₃₂	C₂	semidirect
3. D₃×#108	C₂	semidirect
4. C₃×#361	C₂	semidirect
5. #4563	C₂	semidirect
6. #4517	C₂	semidirect
7. #4496	C₂	semidirect
32. #361	D₃	semidirect
177. C₃	#2148	semidirect

Maximal subgroups

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

subgroup	index
C₂×#912	2
D₃×M₃₂	2
D₃×#108	2
C₃×#361	2
#4563	2
#4517	2
#4496	2
#2148	3