Finite Group Databasebeta

← #10596142

Subgroups — #10596142

Order 576.

636 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C32×#124C211yesmaximal, characteristic
2Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#64153C211yesmaximal, characteristic, complemented
4#64145C211yesmaximal, characteristic
5D4×C3⋊Dic3C211yesmaximal, characteristic
6#64088C212yesmaximal, complemented
7#64088C212yesmaximal, complemented
8#64087C212yesmaximal, complemented
9#64086C212yesmaximal, complemented
10C4×#634C211yesmaximal, characteristic, complemented
11C4×#634C211yesmaximal, characteristic, complemented
12#5671—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C2×C32×D4C2211yescharacteristic
15C22×C3⋊Dic3C2212yes—
16C2×#634C2211yescharacteristic
17C2×#634C2211yescharacteristic
18C32×C22⋊C4C2212yes—
19C32×C22⋊C4C2212yes—
20C32×C42C2211yescharacteristic
21#3757C2212yes—
22#3757C2212yes—
23#3757C2212yes—
24C4⋊(C3⋊Dic3)C2211yescharacteristic
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26C4⋊(C3⋊Dic3)C2212yes—
27#3750C2211yescharacteristic
28#3750C2212yes—
29#3750C2211yescharacteristic
30#3750C2212yes—
31#3750C2212yes—
32C4×C3⋊Dic3C2212yes—
33C4×C3⋊Dic3C2211yescharacteristic
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2212yes—
36C3×#124D314yes—
37Q8×Dic3—34no—
38#944—34no—
39#936—34no—
40D4×Dic3—34no—
41#879—38no—
42#879—38no—
43#878—38no—
44#877—38no—
45C4×Dic6—34no—
46C4×Dic6—34no—
47C23×C32C2312yes—
48C32×Q8—22no—
49C32×D4—22no—
50C2×C32×C4C2311yescharacteristic
51C2×C32×C4C2312yes—
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2311yescharacteristic
54C2×C3⋊Dic3—24no—
55C2×C3⋊Dic3C2311yescharacteristic
56C2×C3⋊Dic3C2312yes—
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2312yes—
59C2×C3⋊Dic3C2311yescharacteristic
60C2×C3⋊Dic3C2311yescharacteristic
61#634—22no—
62#634—22no—
63#556—91noSylow
64C2×C3×Q8C2×D314yes—
65C2×C3×D4C2×D314yes—
66C22×Dic3—38no—
67C2×Dic6—34no—
68C2×Dic6—34no—
69C3×C22⋊C4C2×D318yes—
70C3×C22⋊C4C2×D318yes—
71C3×C42C2×D314yes—
72#217—38no—
73#217—38no—
74#217—38no—
75C4⋊Dic3—34no—
76C4⋊Dic3—34no—
77C4⋊Dic3—38no—
78Dic3⋊C4—34no—
79Dic3⋊C4—38no—
80Dic3⋊C4—34no—
81Dic3⋊C4—38no—
82Dic3⋊C4—38no—
83C4×Dic3—38no—
84C4×Dic3—34no—
85C4×Dic3—34no—
86C4×Dic3—38no—
87C22×C32—22no—
88C22×C32—22no—
89C22×C32—22no—
90C22×C32C2411yescharacteristic
91C32×C4—21no—
92C32×C4—22no—
93C32×C4—21no—
94C32×C4#4112yes—
95C3⋊Dic3—22no—
96C3⋊Dic3—21no—
97C3⋊Dic3—22no—
98C3⋊Dic3—21no—
99C3⋊Dic3—21no—
100C3⋊Dic3#4112yes—
101C4⋊Q8—91no—
102#126—92no—
103#126—92no—
104#124—91no—
105#124C3⋊D311yescharacteristic
106C22⋊Q8—92no—
107C4×Q8—91no—
108C4×Q8—91no—
109C4×Q8—91no—
110C4×D4—91no—
111#117—92no—
112C23×C3C22×D318yes—
113C3×Q8—28no—
114C3×D4—28no—
115C2×C3×C4C22×D314yes—
116C2×C3×C4C22×D318yes—
117C2×C3×C4C22×D314yes—
118C2×C3×C4C22×D314yes—
119C2×Dic3—34no—
120C2×Dic3—34no—
121C2×Dic3—38no—
122C2×Dic3—34no—
123C2×Dic3—38no—
124C2×Dic3—34no—
125C2×Dic3—616no—
126Dic6—68no—
127Dic6—68no—
128C2×C32—42no—
129C2×C32C2×#4111yescharacteristic
130C2×C32C2×#4111yescharacteristic
131C2×C322−1+411yescharacteristic
132C2×Q8—91no—
133C2×Q8—91no—
134C2×Q8C2×C3⋊D311yescharacteristic
135C2×D4C2×C3⋊D311yescharacteristic
136C22×C4—92no—
137C4⋊C4—91no—
138C4⋊C4—92no—
139C4⋊C4—91no—
140C4⋊C4—92no—
141C4⋊C4—92no—
142C4⋊C4—91no—
143C4⋊C4—91no—
144C4⋊C4—92no—
145C22⋊C4—92no—
146C22⋊C4—92no—
147C22⋊C4—92no—
148C22⋊C4C2×C3⋊D312yes—
149C22⋊C4C2×C3⋊D312yes—
150C42—92no—
151C42—92no—
152C42—91no—
153C42—91no—
154C42C2×C3⋊D311yescharacteristic
155C22×C3—28no—
156C22×C3—28no—
157C22×C3—28no—
158C22×C3C23×D314yes—
159C3×C4—28no—
160C3×C4—24no—
161C3×C4—24no—
162C3×C4#23718yes—
163Dic3—38no—
164Dic3—68no—
165Dic3—64no—
166Dic3—68no—
167Dic3—64no—
168Dic3—64no—
169C32#55611yescharacteristic, complemented, Sylow
170C23C22×C3⋊D312yes—
171Q8—22no—
172Q8—182no—
173Q8—182no—
174D4—22no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4—92no—
178C2×C4—91no—
179C2×C4—92no—
180C2×C4—91no—
181C2×C4—184no—
182C2×C4C22×C3⋊D311yescharacteristic
183C2×C4C22×C3⋊D312yes—
184C2×C4C22×C3⋊D311yescharacteristic
185C2×C4C22×C3⋊D311yescharacteristic
186C2×C3—48no—
187C2×C3C2×#23714yes—
188C2×C3D3×#4114yes—
189C2×C3#101014yes—
190C22—22no—
191C22—22no—
192C22—22no—
193C22C23×C3⋊D311yescharacteristic
194C4—22no—
195C4—21no—
196C4—21no—
197C4—92no—
198C4—182no—
199C4—182no—
200C4—181no—
201C4—181no—
202C4—181no—
203C4#383012yes—
204C3#567114yescomplemented
205C2—42no—
206C2C2×#383011yescharacteristic
207C2#41×C3⋊D311yescharacteristic
208C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#124C2
prime radicalC2×C32×D4C22
hypercenter#124C3⋊D3
hypocenterC32#556
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
2. Q8×C3⋊Dic3C2semidirect
3. #64153C2semidirect
6. #64088C2semidirect
7. #64088C2semidirect
8. #64087C2semidirect
9. #64086C2semidirect
10. C4×#634C2semidirect
11. C4×#634C2semidirect
169. C32#556semidirect
204. C3#5671semidirect

Maximal subgroups

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

subgroupindex
C32×#1242
Q8×C3⋊Dic32
#641532
#641452
D4×C3⋊Dic32
#640882
#640872
#640862
C4×#6342
#56713