Finite Group Databasebeta

Subgroups — #5598

Order 192.

290 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 107 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×#237	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
4	#939	C₂	1	2	yes	maximal, complemented
5	#936	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₃	C₂	1	2	yes	maximal, complemented
7	C₄×#67	C₂	1	1	yes	maximal, characteristic, complemented
8	#897	C₂	1	1	yes	maximal, characteristic, complemented
9	#887	C₂	1	1	yes	maximal, characteristic, complemented
10	#885	C₂	1	2	yes	maximal, complemented
11	#878	C₂	1	2	yes	maximal, complemented
12	#548	—	3	1	no	maximal, Sylow
13	C₂×C₃×D₄	C₂²	1	2	yes	—
14	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂×#67	C₂²	1	2	yes	—
17	C₂×#67	C₂²	1	1	yes	characteristic
18	C₂²×Dic₃	C₂²	1	2	yes	—
19	C₂²×Dic₃	C₂²	1	2	yes	—
20	#237	—	2	4	no	—
21	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
22	C₂×Dic₆	—	2	4	no	—
23	C₂×Dic₆	C₂²	1	2	yes	—
24	C₂×Dic₆	C₂²	1	1	yes	characteristic
25	C₂×Dic₆	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	#217	C₂²	1	2	yes	—
29	#217	C₂²	1	2	yes	—
30	#217	C₂²	1	1	yes	characteristic
31	(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
32	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
33	C₄⋊Dic₃	C₂²	1	2	yes	—
34	Dic₃⋊C₄	C₂²	1	1	yes	characteristic
35	Dic₃⋊C₄	C₂²	1	2	yes	—
36	C₄×Dic₃	C₂²	1	2	yes	—
37	C₄×Dic₃	C₂²	1	1	yes	characteristic
38	C₂×#41	—	3	1	no	—
39	C₂²×Q₈	—	3	1	no	—
40	#124	—	3	1	no	—
41	#124	—	3	2	no	—
42	C₂²⋊Q₈	—	3	1	no	—
43	C₂²⋊Q₈	—	3	2	no	—
44	#121	—	3	2	no	—
45	#121	D₃	1	1	yes	characteristic, complemented
46	C₄×Q₈	—	3	1	no	—
47	C₄×D₄	—	3	2	no	—
48	C₄×D₄	—	3	1	no	—
49	C₂³×C₃	C₂³	1	2	yes	—
50	C₂³×C₃	C₂³	1	1	yes	characteristic
51	C₂²×D₃	C₂³	1	1	yes	characteristic
52	C₃×D₄	—	2	4	no	—
53	C₃×D₄	—	2	2	no	—
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	C₂³	1	2	yes	—
56	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
58	#67	—	2	2	no	—
59	#67	—	2	4	no	—
60	C₂×Dic₃	—	2	4	no	—
61	C₂×Dic₃	—	2	4	no	—
62	C₂×Dic₃	C₂³	1	1	yes	characteristic
63	C₂×Dic₃	C₂³	1	1	yes	characteristic
64	C₂×Dic₃	C₂³	1	1	yes	characteristic
65	C₂×Dic₃	C₂³	1	2	yes	—
66	C₂×Dic₃	C₂³	1	2	yes	—
67	C₄×D₃	—	2	2	no	—
68	Dic₆	—	2	2	no	—
69	Dic₆	—	2	4	no	—
70	Dic₆	D₄	1	4	yes	complemented
71	#41	—	6	4	no	—
72	C₂×Q₈	—	3	1	no	—
73	C₂×Q₈	—	3	2	no	—
74	C₂×Q₈	—	3	1	no	—
75	C₂×Q₈	—	6	4	no	—
76	C₂×D₄	—	3	1	no	—
77	C₂×D₄	—	3	2	no	—
78	C₂×D₄	C₂×D₃	1	2	yes	—
79	C₂×D₄	C₂×D₃	1	1	yes	characteristic
80	C₂²×C₄	—	3	2	no	—
81	C₂²×C₄	—	3	2	no	—
82	C₂²×C₄	—	3	1	no	—
83	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
84	C₄⋊C₄	—	3	1	no	—
85	C₄⋊C₄	—	3	2	no	—
86	C₄⋊C₄	—	3	2	no	—
87	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
88	C₂²⋊C₄	—	3	1	no	—
89	C₂²⋊C₄	—	3	2	no	—
90	C₂²⋊C₄	—	3	2	no	—
91	C₂²⋊C₄	—	3	1	no	—
92	C₂²⋊C₄	—	3	2	no	—
93	C₂²⋊C₄	C₂×D₃	1	2	yes	—
94	C₄²	—	3	2	no	—
95	C₄²	—	3	1	no	—
96	C₂²×C₃	—	2	1	no	—
97	C₂²×C₃	—	2	1	no	—
98	C₂²×C₃	—	2	2	no	—
99	C₂²×C₃	—	2	2	no	—
100	C₂²×C₃	—	2	2	no	—
101	C₂²×C₃	#41	1	2	yes	complemented
102	C₂²×C₃	C₂⁴	1	1	yes	characteristic
103	C₂×D₃	—	2	1	no	—
104	C₂×D₃	—	2	1	no	—
105	C₂×D₃	—	2	1	no	—
106	C₃×C₄	—	2	1	no	—
107	C₃×C₄	—	2	2	no	—
108	C₃×C₄	C₂×D₄	1	2	yes	—
109	Dic₃	—	2	2	no	—
110	Dic₃	—	2	1	no	—
111	Dic₃	—	2	1	no	—
112	Dic₃	—	2	1	no	—
113	Dic₃	C₂×D₄	1	4	yes	—
114	C₂³	—	3	1	no	—
115	C₂³	C₂²×D₃	1	2	yes	—
116	C₂³	C₂²×D₃	1	1	yes	characteristic
117	Q₈	—	3	4	no	—
118	Q₈	—	6	4	no	—
119	Q₈	—	6	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	2	no	—
122	D₄	—	6	4	no	—
123	D₄	—	6	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	3	2	no	—
126	C₂×C₄	—	3	1	no	—
127	C₂×C₄	—	3	1	no	—
128	C₂×C₄	—	3	2	no	—
129	C₂×C₄	—	3	1	no	—
130	C₂×C₄	—	6	4	no	—
131	C₂×C₄	—	6	4	no	—
132	C₂×C₄	—	6	2	no	—
133	C₂×C₄	C₂²×D₃	1	2	yes	—
134	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₃	—	2	2	no	—
137	C₂×C₃	—	4	2	no	—
138	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
141	D₃	—	4	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	6	1	no	—
148	C₂²	—	6	1	no	—
149	C₂²	—	6	1	no	—
150	C₂²	#237	1	2	yes	complemented
151	C₂²	C₂³×D₃	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	2	no	—
154	C₄	—	3	4	no	—
155	C₄	—	6	1	no	—
156	C₄	—	6	2	no	—
157	C₄	—	6	1	no	—
158	C₄	—	6	1	no	—
159	C₄	D₃×D₄	1	2	yes	—
160	C₃	#548	1	1	yes	characteristic, complemented, Sylow
161	C₂	—	2	2	no	—
162	C₂	—	4	2	no	—
163	C₂	—	12	1	no	—
164	C₂	C₂×D₃×D₄	1	1	yes	characteristic
165	C₂	C₂×#237	1	1	yes	characteristic
166	C₂	#1010	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#121	C₂
hypercenter	#121	D₃
hypocenter	C₃	#548
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂×#237	C₂	semidirect
2. C₂²×Dic₆	C₂	semidirect
3. C₃×#121	C₂	semidirect
4. #939	C₂	semidirect
5. #936	C₂	semidirect
6. D₄×Dic₃	C₂	semidirect
7. C₄×#67	C₂	semidirect
8. #897	C₂	semidirect
9. #887	C₂	semidirect
10. #885	C₂	semidirect
11. #878	C₂	semidirect
45. #121	D₃	semidirect
70. Dic₆	D₄	semidirect
101. C₂²×C₃	#41	semidirect
150. C₂²	#237	semidirect
160. C₃	#548	semidirect

Maximal subgroups

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

subgroup	index
C₂×#237	2
C₂²×Dic₆	2
C₃×#121	2
#939	2
#936	2
D₄×Dic₃	2
C₄×#67	2
#897	2
#887	2
#885	2
#878	2
#548	3