Finite Group Databasebeta

Subgroups — #5507

Order 192.

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

normal subgroups only

This group has 292 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×#235	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#930	C₂	1	1	yes	maximal, characteristic, complemented
4	#929	C₂	1	2	yes	maximal, complemented
5	C₄×#67	C₂	1	1	yes	maximal, characteristic, complemented
6	#896	C₂	1	2	yes	maximal, complemented
7	#895	C₂	1	1	yes	maximal, characteristic, complemented
8	#894	C₂	1	2	yes	maximal, complemented
9	#893	C₂	1	1	yes	maximal, characteristic, complemented
10	D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#888	C₂	1	1	yes	maximal, characteristic, complemented
12	#547	—	3	1	no	maximal, Sylow
13	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄	C₂²	1	2	yes	—
15	C₂×#67	C₂²	1	2	yes	—
16	C₂×#67	C₂²	1	1	yes	characteristic
17	#235	—	2	8	no	—
18	C₂×D₁₂	C₂²	1	1	yes	characteristic
19	C₂×D₁₂	C₂²	1	2	yes	—
20	C₂×C₄×D₃	C₂²	1	2	yes	—
21	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₃	C₂²	1	2	yes	—
23	C₂×Dic₆	C₂²	1	2	yes	—
24	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊C₄	C₂²	1	2	yes	—
27	#217	C₂²	1	1	yes	characteristic
28	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
29	(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
30	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
31	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
32	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
33	Dic₃⋊C₄	C₂²	1	2	yes	—
34	Dic₃⋊C₄	C₂²	1	2	yes	—
35	Dic₃⋊C₄	C₂²	1	1	yes	characteristic
36	C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₂×#41	—	3	2	no	—
38	C₄⋊Q₈	—	3	1	no	—
39	#123	—	3	2	no	—
40	#123	—	3	2	no	—
41	C₂²⋊Q₈	—	3	2	no	—
42	#121	—	3	1	no	—
43	#121	—	3	1	no	—
44	C₄×D₄	—	3	1	no	—
45	C₄×D₄	—	3	1	no	—
46	C₂×C₄⋊C₄	—	3	1	no	—
47	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
48	C₂³×C₃	C₂³	1	1	yes	characteristic
49	C₂²×D₃	C₂³	1	2	yes	—
50	C₂²×D₃	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄	—	2	4	no	—
52	C₂×C₃×C₄	—	2	2	no	—
53	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄	C₂³	1	2	yes	—
55	C₂×C₃×C₄	C₂³	1	2	yes	—
56	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
57	#67	—	2	4	no	—
58	#67	D₄	1	4	yes	complemented
59	C₂×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×Dic₃	C₂³	1	2	yes	—
62	C₂×Dic₃	C₂³	1	1	yes	characteristic
63	D₁₂	—	2	4	no	—
64	D₁₂	—	2	2	no	—
65	C₄×D₃	—	2	4	no	—
66	C₄×D₃	—	2	2	no	—
67	C₄×D₃	—	2	4	no	—
68	Dic₆	—	2	4	no	—
69	#41	—	6	8	no	—
70	C₂×Q₈	—	3	2	no	—
71	C₂×D₄	—	3	1	no	—
72	C₂×D₄	—	3	2	no	—
73	C₂×D₄	—	3	1	no	—
74	C₂×D₄	—	3	2	no	—
75	C₂²×C₄	—	3	1	no	—
76	C₂²×C₄	—	3	2	no	—
77	C₂²×C₄	—	3	2	no	—
78	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
79	C₂²×C₄	C₂×D₃	1	2	yes	—
80	C₄⋊C₄	—	3	1	no	—
81	C₄⋊C₄	—	3	2	no	—
82	C₄⋊C₄	—	3	1	no	—
83	C₄⋊C₄	—	3	2	no	—
84	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
85	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
86	C₄⋊C₄	C₂×D₃	1	2	yes	—
87	C₂²⋊C₄	—	3	1	no	—
88	C₂²⋊C₄	—	3	1	no	—
89	C₂²⋊C₄	—	3	2	no	—
90	C₂²⋊C₄	—	3	2	no	—
91	C₂²⋊C₄	—	3	2	no	—
92	C₄²	—	3	1	no	—
93	C₂²×C₃	—	2	1	no	—
94	C₂²×C₃	—	2	1	no	—
95	C₂²×C₃	C₂×D₄	1	2	yes	—
96	C₂²×C₃	C₂⁴	1	1	yes	characteristic
97	C₂×D₃	—	2	1	no	—
98	C₂×D₃	—	2	1	no	—
99	C₂×D₃	—	2	2	no	—
100	C₂×D₃	—	2	2	no	—
101	C₂×D₃	—	2	2	no	—
102	C₂×D₃	C₂×D₄	1	2	yes	—
103	C₃×C₄	—	2	2	no	—
104	C₃×C₄	—	2	2	no	—
105	C₃×C₄	—	2	1	no	—
106	C₃×C₄	#41	1	2	yes	complemented
107	Dic₃	—	2	2	no	—
108	Dic₃	—	2	1	no	—
109	Dic₃	—	2	1	no	—
110	Dic₃	C₂×D₄	1	2	yes	—
111	C₂³	—	3	1	no	—
112	C₂³	—	3	2	no	—
113	C₂³	C₂²×D₃	1	1	yes	characteristic
114	Q₈	—	6	4	no	—
115	D₄	—	3	4	no	—
116	D₄	—	6	4	no	—
117	D₄	—	6	4	no	—
118	D₄	—	6	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	3	1	no	—
122	C₂×C₄	—	3	1	no	—
123	C₂×C₄	—	3	1	no	—
124	C₂×C₄	—	3	2	no	—
125	C₂×C₄	—	6	4	no	—
126	C₂×C₄	—	6	4	no	—
127	C₂×C₄	—	6	2	no	—
128	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
129	C₂×C₄	C₂²×D₃	1	2	yes	—
130	C₂×C₄	C₂²×D₃	1	2	yes	—
131	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₃	—	2	2	no	—
133	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
136	D₃	—	2	2	no	—
137	D₃	—	4	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	3	2	no	—
141	C₂²	—	6	1	no	—
142	C₂²	—	6	1	no	—
143	C₂²	—	6	2	no	—
144	C₂²	—	6	2	no	—
145	C₂²	—	6	2	no	—
146	C₂²	D₃×D₄	1	2	yes	—
147	C₂²	C₂³×D₃	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	3	2	no	—
152	C₄	—	6	1	no	—
153	C₄	—	6	2	no	—
154	C₄	—	6	1	no	—
155	C₄	#235	1	2	yes	complemented
156	C₃	#547	1	1	yes	characteristic, complemented, Sylow
157	C₂	—	2	2	no	—
158	C₂	—	6	2	no	—
159	C₂	—	12	2	no	—
160	C₂	C₂×#235	1	1	yes	characteristic
161	C₂	C₂×D₃×D₄	1	1	yes	characteristic
162	C₂	#1007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₂×C₃×C₄⋊C₄	C₂
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#547
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#235	C₂	semidirect
2. C₂×C₃×C₄⋊C₄	C₂	semidirect
3. #930	C₂	semidirect
4. #929	C₂	semidirect
5. C₄×#67	C₂	semidirect
6. #896	C₂	semidirect
7. #895	C₂	semidirect
8. #894	C₂	semidirect
9. #893	C₂	semidirect
10. D₃×C₄⋊C₄	C₂	semidirect
11. #888	C₂	semidirect
47. C₂×C₄⋊C₄	D₃	semidirect
58. #67	D₄	semidirect
106. C₃×C₄	#41	semidirect
155. C₄	#235	semidirect
156. C₃	#547	semidirect

Maximal subgroups

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

subgroup	index
C₂×#235	2
C₂×C₃×C₄⋊C₄	2
#930	2
#929	2
C₄×#67	2
#896	2
#895	2
#894	2
#893	2
D₃×C₄⋊C₄	2
#888	2
#547	3