Finite Group Databasebeta

Subgroups — #71853

Order 384.

362 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₃×#535	C₂	1	1	yes	maximal, characteristic
2	#5798	C₂	1	1	yes	maximal, characteristic, complemented
3	#5236	C₂	1	2	yes	maximal, complemented
4	C₂×#833	C₂	1	1	yes	maximal, characteristic, complemented
5	#4531	C₂	1	2	yes	maximal, complemented
6	#1911	—	3	1	no	maximal, Sylow
7	C₂×C₃×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
9	C₃×#127	C₄	1	2	yes	complemented
10	C₃×#124	C₄	1	2	yes	complemented
11	C₃×#121	—	2	2	no	—
12	C₃×#120	—	2	2	no	—
13	C₃×#117	C₂²	1	1	yes	characteristic
14	C₂×#217	—	2	2	no	—
15	D₄×Dic₃	—	2	4	no	—
16	#926	C₂²	1	2	yes	—
17	C₂×#208	C₂²	1	2	yes	—
18	#837	—	2	4	no	—
19	#833	—	2	2	no	—
20	#535	D₃	1	1	yes	characteristic
21	#518	—	3	1	no	—
22	#421	—	3	2	no	—
23	C₂×#100	—	3	1	no	—
24	H(3,ℤ/4)	—	3	2	no	—
25	C₂⁴×C₃	D₄	1	2	yes	—
26	C₃×#41	—	2	2	no	—
27	C₃×#41	—	2	2	no	—
28	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂×C₃×D₄	—	2	1	no	—
30	C₂×C₃×D₄	—	2	4	no	—
31	C₂×C₃×D₄	—	2	2	no	—
32	C₂×C₃×D₄	—	2	2	no	—
33	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
34	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic, complemented
35	C₂×C₃×D₄	D₄	1	2	yes	—
36	C₂²×C₃×C₄	—	2	1	no	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×Dic₃	—	2	4	no	—
39	C₃×C₄⋊C₄	—	2	1	no	—
40	C₃×C₂²⋊C₄	—	2	4	no	—
41	C₃×C₂²⋊C₄	—	2	1	no	—
42	C₃×C₄²	C₂×C₄	1	2	yes	complemented
43	#217	—	2	4	no	—
44	#217	—	2	2	no	—
45	C₄⋊Dic₃	—	2	2	no	—
46	C₄×Dic₃	D₄	1	4	yes	complemented
47	#208	—	2	2	no	—
48	#208	—	2	2	no	—
49	C₂×C₃⋊C₈	—	2	2	no	—
50	C₂×#41	C₂×D₃	1	1	yes	characteristic
51	C₂²×D₄	C₂×D₃	1	1	yes	characteristic
52	C₂×M₁₆	—	3	2	no	—
53	#127	Dic₃	1	2	yes	complemented
54	#124	Dic₃	1	2	yes	complemented
55	#121	—	2	2	no	—
56	#120	—	2	2	no	—
57	C₄×D₄	—	6	4	no	—
58	#117	—	3	2	no	—
59	#117	C₂×D₃	1	1	yes	characteristic
60	C₂×C₂²⋊C₄	—	6	2	no	—
61	#104	—	6	4	no	—
62	#100	—	6	2	no	—
63	C₂³×C₃	—	2	2	no	—
64	C₂³×C₃	—	2	2	no	—
65	C₂³×C₃	—	2	1	no	—
66	C₂³×C₃	—	8	2	no	—
67	C₂³×C₃	C₂²⋊C₄	1	4	yes	—
68	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
69	C₃×Q₈	—	2	2	no	—
70	C₃×D₄	—	2	4	no	—
71	C₃×D₄	—	2	2	no	—
72	C₃×D₄	—	2	4	no	—
73	C₃×D₄	—	4	1	no	—
74	C₃×D₄	—	4	1	no	—
75	C₃×D₄	—	8	4	no	—
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	—	2	1	no	—
78	C₂×C₃×C₄	—	4	2	no	—
79	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄	#41	1	2	yes	—
83	C₂×Dic₃	—	2	4	no	—
84	C₂×Dic₃	—	8	4	no	—
85	C₃⋊C₈	—	4	2	no	—
86	C₂⁴	#67	1	2	yes	—
87	#41	—	2	2	no	—
88	#41	—	2	2	no	—
89	C₂×Q₈	C₂×Dic₃	1	1	yes	characteristic, complemented
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	2	4	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×Dic₃	1	2	yes	—
95	C₂×D₄	C₂×Dic₃	1	1	yes	characteristic, complemented
96	C₂×D₄	#67	1	2	yes	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	6	4	no	—
99	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
100	M₁₆	—	6	2	no	—
101	M₁₆	—	6	2	no	—
102	C₂×C₈	—	6	2	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	6	2	no	—
105	C₂²⋊C₄	—	2	4	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	6	4	no	—
108	C₂²⋊C₄	—	6	2	no	—
109	C₄²	—	3	4	no	—
110	C₄²	C₂×Dic₃	1	2	yes	complemented
111	C₂²×C₃	—	2	4	no	—
112	C₂²×C₃	—	4	1	no	—
113	C₂²×C₃	—	4	4	no	—
114	C₂²×C₃	—	4	4	no	—
115	C₂²×C₃	—	4	2	no	—
116	C₂²×C₃	—	4	1	no	—
117	C₂²×C₃	—	8	2	no	—
118	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₃	#120	1	1	yes	characteristic
120	C₂²×C₃	#123	1	1	yes	characteristic
121	C₃×C₄	—	4	2	no	—
122	C₃×C₄	—	4	1	no	—
123	C₃×C₄	C₄×D₄	1	2	yes	—
124	C₃×C₄	#121	1	2	yes	—
125	Dic₃	—	4	4	no	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	2	1	no	—
129	C₂³	—	8	2	no	—
130	C₂³	#217	1	4	yes	—
131	C₂³	C₂×#67	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	D₄	—	2	4	no	—
134	D₄	—	2	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	6	4	no	—
143	C₂×C₄	—	24	4	no	—
144	C₂×C₄	D₃×D₄	1	2	yes	—
145	C₂×C₄	#237	1	2	yes	—
146	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
147	C₂×C₄	C₂×#67	1	1	yes	characteristic
148	C₈	—	12	2	no	—
149	C₂×C₃	—	2	1	no	—
150	C₂×C₃	—	2	1	no	—
151	C₂×C₃	—	2	1	no	—
152	C₂×C₃	—	4	4	no	—
153	C₂×C₃	—	8	1	no	—
154	C₂×C₃	#386	1	1	yes	characteristic
155	C₂²	—	2	4	no	—
156	C₂²	—	4	1	no	—
157	C₂²	—	4	4	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	8	2	no	—
162	C₂²	#935	1	1	yes	characteristic
163	C₂²	#937	1	1	yes	characteristic
164	C₂²	C₂×#217	1	1	yes	characteristic
165	C₄	—	4	2	no	—
166	C₄	—	4	1	no	—
167	C₄	—	12	4	no	—
168	C₄	D₄×Dic₃	1	2	yes	—
169	C₄	#939	1	2	yes	—
170	C₃	#1911	1	1	yes	characteristic, complemented, Sylow
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	—	4	4	no	—
175	C₂	—	8	1	no	—
176	C₂	#5220	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#5220
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting, prime radical	C₃×#535	C₂
socle	C₂×C₃	#386
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#535	D₃
hypocenter	C₃	#1911
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

Complements

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

N	H ≅ G/N	type
2. #5798	C₂	semidirect
3. #5236	C₂	semidirect
4. C₂×#833	C₂	semidirect
5. #4531	C₂	semidirect
9. C₃×#127	C₄	semidirect
10. C₃×#124	C₄	semidirect
28. C₂×C₃×Q₈	C₂×C₄	semidirect
34. C₂×C₃×D₄	C₂×C₄	semidirect
42. C₃×C₄²	C₂×C₄	semidirect
46. C₄×Dic₃	D₄	semidirect
53. #127	Dic₃	semidirect
54. #124	Dic₃	semidirect
89. C₂×Q₈	C₂×Dic₃	semidirect
95. C₂×D₄	C₂×Dic₃	semidirect
110. C₄²	C₂×Dic₃	semidirect
170. C₃	#1911	semidirect

Maximal subgroups

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

subgroup	index
C₃×#535	2
#5798	2
#5236	2
C₂×#833	2
#4531	2
#1911	3