Finite Group Databasebeta

← #10596074

Subgroups — #10596074

Order 576.

648 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 183 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×#121C211yesmaximal, characteristic
2#64145C212yesmaximal, complemented
3#64145C211yesmaximal, characteristic, complemented
4#64144C212yesmaximal, complemented
5D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#64135C211yesmaximal, characteristic, complemented
7#64133C211yesmaximal, characteristic, complemented
8#64097C211yesmaximal, characteristic, complemented
9#64096C211yesmaximal, characteristic, complemented
10#64088C212yesmaximal, complemented
11#64087C212yesmaximal, complemented
12#5603—34nomaximal
13C2×C32×D4C2211yescharacteristic
14C2×C32×D4C2212yes—
15C22×C32×C4C2211yescharacteristic
16C22×C3⋊Dic3C2212yes—
17C2×#634C2211yescharacteristic
18C2×#634C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20C32×C22⋊C4C2212yes—
21#3757C2212yes—
22#3757C2212yes—
23#3757C2212yes—
24#3757C2212yes—
25#3757C2211yescharacteristic
26#3757C2211yescharacteristic
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30#3750C2212yes—
31#3750C2212yes—
32#3750C2212yes—
33C4×C3⋊Dic3C2212yes—
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2211yescharacteristic
36C3×#121D314yes—
37#936—38no—
38#936—34no—
39#935—38no—
40D4×Dic3—34no—
41#926—34no—
42#924—34no—
43#888—34no—
44#887—34no—
45#879—38no—
46#878—38no—
47C23×C32C2311yescharacteristic
48C23×C32C2312yes—
49C32×D4—22no—
50C32×D4—42no—
51C2×C32×C4—22no—
52C2×C32×C4C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55C2×C3⋊Dic3—42no—
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2312yes—
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2312yes—
62#634—22no—
63#634—42no—
64#542—91noSylow
65C2×C3×D4C2×D314yes—
66C2×C3×D4C2×D318yes—
67C22×C3×C4C2×D314yes—
68C22×Dic3—38no—
69C2×Dic6—34no—
70C2×Dic6—38no—
71C3×C4⋊C4C2×D314yes—
72C3×C22⋊C4C2×D318yes—
73#217—38no—
74#217—38no—
75#217—38no—
76#217—38no—
77#217—34no—
78#217—34no—
79C4⋊Dic3—34no—
80C4⋊Dic3—34no—
81C4⋊Dic3—34no—
82Dic3⋊C4—38no—
83Dic3⋊C4—38no—
84Dic3⋊C4—38no—
85C4×Dic3—38no—
86C4×Dic3—34no—
87C4×Dic3—34no—
88C22×C32—21no—
89C22×C32—22no—
90C22×C32—22no—
91C22×C32—21no—
92C22×C32—21no—
93C22×C32—22no—
94C22×C32C2411yescharacteristic
95C32×C4—21no—
96C32×C4—22no—
97C32×C4#4112yes—
98C3⋊Dic3—22no—
99C3⋊Dic3—21no—
100C3⋊Dic3—22no—
101C3⋊Dic3—21no—
102C3⋊Dic3—21no—
103C3⋊Dic3—21no—
104C4⋊Q8—91no—
105#126—92no—
106#124—91no—
107#124—92no—
108#123—92no—
109C22⋊Q8—91no—
110C22⋊Q8—92no—
111#121C3⋊D311yescharacteristic
112C4×Q8—91no—
113C4×D4—91no—
114#117—91no—
115C23×C3C22×D314yes—
116C23×C3C22×D318yes—
117C3×D4—28no—
118C3×D4—48no—
119C2×C3×C4—28no—
120C2×C3×C4C22×D318yes—
121C2×C3×C4C22×D314yes—
122C2×C3×C4C22×D314yes—
123C2×Dic3—34no—
124C2×Dic3—34no—
125C2×Dic3—34no—
126C2×Dic3—38no—
127C2×Dic3—34no—
128C2×Dic3—38no—
129C2×Dic3—128no—
130Dic6—68no—
131Dic6—128no—
132C2×C32—41no—
133C2×C32—42no—
134C2×C32C2×#4111yescharacteristic
135C2×C32H(5,𝔽2)11yescharacteristic
136C2×C322−1+411yescharacteristic
137C2×Q8—91no—
138C2×Q8—92no—
139C2×D4C2×C3⋊D311yescharacteristic
140C2×D4C2×C3⋊D312yes—
141C22×C4—92no—
142C22×C4C2×C3⋊D311yescharacteristic
143C4⋊C4—91no—
144C4⋊C4—91no—
145C4⋊C4—91no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4—92no—
149C4⋊C4C2×C3⋊D311yescharacteristic
150C22⋊C4—91no—
151C22⋊C4—91no—
152C22⋊C4—92no—
153C22⋊C4—92no—
154C22⋊C4—92no—
155C22⋊C4—92no—
156C22⋊C4C2×C3⋊D312yes—
157C42—91no—
158C42—91no—
159C42—92no—
160C22×C3—24no—
161C22×C3—28no—
162C22×C3—28no—
163C22×C3—24no—
164C22×C3—24no—
165C22×C3—28no—
166C22×C3C23×D314yes—
167C3×C4—24no—
168C3×C4—28no—
169C3×C4#23718yes—
170Dic3—68no—
171Dic3—64no—
172Dic3—68no—
173Dic3—64no—
174Dic3—64no—
175Dic3—64no—
176C32#54211yescharacteristic, complemented, Sylow
177C23C22×C3⋊D311yescharacteristic
178C23C22×C3⋊D312yes—
179Q8—182no—
180Q8—362no—
181D4—22no—
182D4—42no—
183C2×C4—22no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4—92no—
187C2×C4—91no—
188C2×C4—91no—
189C2×C4—92no—
190C2×C4—362no—
191C2×C4C22×C3⋊D312yes—
192C2×C4C22×C3⋊D311yescharacteristic
193C2×C4C22×C3⋊D311yescharacteristic
194C2×C3—44no—
195C2×C3—48no—
196C2×C3C2×#23714yes—
197C2×C3#100414yes—
198C2×C3#101014yes—
199C22—21no—
200C22—22no—
201C22—22no—
202C22—21no—
203C22—21no—
204C22—22no—
205C22C23×C3⋊D311yescharacteristic
206C4—21no—
207C4—22no—
208C4—182no—
209C4—181no—
210C4—182no—
211C4—181no—
212C4—181no—
213C4—181no—
214C4#383012yes—
215C3#560314yescomplemented
216C2—41no—
217C2—42no—
218C2C2×#383011yescharacteristic
219C2#6436111yescharacteristic
220C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
Fitting, prime radicalC32×#121C2
hypercenter#121C3⋊D3
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #64145C2semidirect
3. #64145C2semidirect
4. #64144C2semidirect
5. D4×C3⋊Dic3C2semidirect
6. #64135C2semidirect
7. #64133C2semidirect
8. #64097C2semidirect
9. #64096C2semidirect
10. #64088C2semidirect
11. #64087C2semidirect
176. C32#542semidirect
215. C3#5603semidirect

Maximal subgroups

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

subgroupindex
C32×#1212
#641452
#641442
D4×C3⋊Dic32
#641352
#641332
#640972
#640962
#640882
#640872
#56033