Finite Group Databasebeta

← #10595365

Subgroups — #10595365

Order 576.

658 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×#3797C211yesmaximal, characteristic
3C3×#944C212yesmaximal, complemented
4#63883C214yesmaximal
5C4×#624C211yesmaximal, characteristic
6#63866C212yesmaximal, complemented
7#63840C214yesmaximal, complemented
8D3×C4⋊Q8—32nomaximal
9C2×C32×Q8C2211yescharacteristic
10Q8×C3⋊D3—24no—
11C2×C4×C3⋊D3C2212yes—
12C2×C4×C3⋊D3C2211yescharacteristic
13C2×#634C2212yes—
14C2×#634C2211yescharacteristic
15C2×C3×Dic6C2212yes—
16C2×#627C2212yes—
17C2×#624C2212yes—
18C2×#624C2212yes—
19#3797—24no—
20C3×Dic3⋊C4C2218yes—
21C3×C4×Dic3C2212yes—
22Dic3⋊Dic3C2218yes—
23Dic32C2212yes—
24C2×D3×Q8—32no—
25C2×D3×Q8—32no—
26C2×D3×Q8—61no—
27C3×C4⋊Q8—32no—
28#944—32no—
29#944D312yescomplemented
30D3×C4⋊C4—38no—
31#888—38no—
32C42×D3—32no—
33#869—32no—
34C22×C3⋊D3C2311yescharacteristic
35C32×Q8—22no—
36C2×C32×C4C2312yes—
37C2×C32×C4C2311yescharacteristic
38C2×C3⋊Dic3C2312yes—
39C2×C3⋊Dic3C2311yescharacteristic
40C4×C3⋊D3—24no—
41C4×C3⋊D3D414yes—
42#634—22no—
43#634—24no—
44C2×C3×Dic3C2314yes—
45C2×C3×Dic3C2314yes—
46C3×Dic6—24no—
47#627—24no—
48#624—24no—
49#624Q818yescomplemented
50C2×C4⋊Q8—91noSylow
51C2×C3×Q8—21no—
52C2×C3×Q8—32no—
53C2×C3×Q8C2×D312yes—
54D3×Q8—68no—
55D3×Q8—68no—
56D3×Q8—68no—
57C2×C4×D3—32no—
58C2×C4×D3—34no—
59C2×C4×D3—34no—
60C2×C4×D3—34no—
61C2×C4×D3—61no—
62C2×C4×D3—62no—
63C2×Dic6—32no—
64C2×Dic6—34no—
65C2×Dic6—34no—
66C2×Dic6—62no—
67C2×Dic6—61no—
68C2×Dic6C2×D312yes—
69C3×C4⋊C4—38no—
70C3×C42—32no—
71C4⋊Dic3—38no—
72Dic3⋊C4—38no—
73Dic3⋊C4C2×D318yes—
74C4×Dic3—34no—
75C4×Dic3C2×D312yes—
76C22×C32C2411yescharacteristic
77C2×C3⋊D3C2×D412yes—
78C2×C3⋊D3C2×Q814yes—
79C32×C4—22no—
80C32×C4C2×D412yes—
81C3⋊Dic3—22no—
82C3⋊Dic3C2×D412yes—
83C3×Dic3—24no—
84C3×Dic3C2×Q818yescomplemented
85C22×Q8—91no—
86C22×Q8—91no—
87C4⋊Q8—32no—
88C4⋊Q8—92no—
89C4⋊Q8—94no—
90C2×C4⋊C4—94no—
91C2×C42—91no—
92C22×D3—32no—
93C22×D3—61no—
94C3×Q8—24no—
95C3×Q8—24no—
96C3×Q8—64no—
97C2×C3×C4—22no—
98C2×C3×C4—21no—
99C2×C3×C4—34no—
100C2×C3×C4—34no—
101C2×C3×C4C22×D314yes—
102C2×C3×C4C22×D312yes—
103C2×Dic3—32no—
104C2×Dic3—34no—
105C2×Dic3—61no—
106C2×Dic3—62no—
107C2×Dic3C22×D314yes—
108C2×Dic3C22×D314yes—
109C4×D3—38no—
110C4×D3—316no—
111C4×D3—64no—
112C4×D3—68no—
113C4×D3—68no—
114C4×D3—68no—
115Dic6—24no—
116Dic6—68no—
117Dic6—68no—
118Dic6—68no—
119Dic6—64no—
120Dic6—64no—
121C2×C32C22×D411yescharacteristic
122C2×C32C22×Q812yes—
123C3⋊D3C4⋊Q814yescomplemented
124C2×Q8—32no—
125C2×Q8—92no—
126C2×Q8—92no—
127C2×Q8—91no—
128C2×Q8—184no—
129C2×Q8—184no—
130C2×Q8D3211yescharacteristic
131C22×C4—92no—
132C22×C4—92no—
133C22×C4—91no—
134C22×C4—92no—
135C4⋊C4—38no—
136C4⋊C4—98no—
137C42—32no—
138C42—92no—
139C22×C3—21no—
140C22×C3C23×D312yes—
141C2×D3—34no—
142C2×D3—38no—
143C2×D3—62no—
144C2×D3—64no—
145C3×C4—24no—
146C3×C4—24no—
147C3×C4—22no—
148C3×C4—38no—
149C3×C4—64no—
150C3×C4D3×D414yes—
151Dic3—24no—
152Dic3—34no—
153Dic3—62no—
154Dic3—64no—
155Dic3—64no—
156Dic3D3×Q818yescomplemented
157C32C2×C4⋊Q811yescharacteristic, complemented, Sylow
158C23—91no—
159Q8—22no—
160Q8—64no—
161Q8—184no—
162Q8—184no—
163Q8—182no—
164C2×C4—34no—
165C2×C4—34no—
166C2×C4—98no—
167C2×C4—91no—
168C2×C4—94no—
169C2×C4—92no—
170C2×C4—184no—
171C2×C4—184no—
172C2×C4C2×D3212yes—
173C2×C4C2×D3211yescharacteristic
174C2×C3—22no—
175C2×C3—21no—
176C2×C3C2×D3×D412yes—
177C2×C3C2×D3×Q814yes—
178D3—38no—
179D3—64no—
180C22—94no—
181C22—92no—
182C22C22×D3211yescharacteristic
183C4—22no—
184C4—38no—
185C4—64no—
186C4—92no—
187C4—182no—
188C4#381112yes—
189C3—21no—
190C3D3×C4⋊Q812yescomplemented
191C2—94no—
192C2D32×Q812yes—
193C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radicalC22×C3⋊D3C23
hypercenterC2×Q8D32
hypocenterC32C2×C4⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#944C2semidirect
6. #63866C2semidirect
7. #63840C2semidirect
29. #944D3semidirect
49. #624Q8semidirect
84. C3×Dic3C2×Q8semidirect
123. C3⋊D3C4⋊Q8semidirect
156. Dic3D3×Q8semidirect
157. C32C2×C4⋊Q8semidirect
190. C3D3×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D32
C2×#37972
C3×#9442
#638832
C4×#6242
#638662
#638402
D3×C4⋊Q83