Finite Group Databasebeta

← #10596012

Subgroups — #10596012

Order 576.

642 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 195 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×C4×D4C211yesmaximal, characteristic
2#64144C212yesmaximal
3#64135C211yesmaximal, characteristic, complemented
4C2×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5#64133C211yesmaximal, characteristic, complemented
6#64133C211yesmaximal, characteristic, complemented
7#64098C212yesmaximal, complemented
8#64088C212yesmaximal, complemented
9#64086C212yesmaximal, complemented
10#64079C211yesmaximal, characteristic, complemented
11C4×#634C211yesmaximal, characteristic, complemented
12#5541—34nomaximal
13C2×C32×D4C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C22×C32×C4C2211yescharacteristic
16C22×C3⋊Dic3C2212yes—
17C2×#634C2211yescharacteristic
18C32×C4⋊C4C2211yescharacteristic
19C32×C22⋊C4C2211yescharacteristic
20C32×C22⋊C4C2211yescharacteristic
21C32×C42C2211yescharacteristic
22#3757C2212yes—
23#3757C2212yes—
24#3757C2212yes—
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26C4⋊(C3⋊Dic3)C2211yescharacteristic
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28C4⋊(C3⋊Dic3)C2212yes—
29#3750C2212yes—
30#3750C2212yes—
31#3750C2212yes—
32#3750C2212yes—
33#3750C2212yes—
34C4×C3⋊Dic3C2212yes—
35C4×C3⋊Dic3C2212yes—
36C3×C4×D4D314yes—
37#935—38no—
38#926—34no—
39C2×C4⋊Dic3—34no—
40#924—34no—
41#924—34no—
42#889—38no—
43#879—38no—
44#877—38no—
45#870—34no—
46C4×Dic6—34no—
47C23×C32C2311yescharacteristic
48C23×C32C2311yescharacteristic
49C32×D4—22no—
50C2×C32×C4—22no—
51C2×C32×C4—22no—
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57C2×C3⋊Dic3—24no—
58C2×C3⋊Dic3C2312yes—
59C2×C3⋊Dic3C2312yes—
60C2×C3⋊Dic3C2312yes—
61C2×C3⋊Dic3C2312yes—
62#634—22no—
63#552—91noSylow
64C2×C3×D4C2×D314yes—
65C22×C3×C4C2×D314yes—
66C22×C3×C4C2×D314yes—
67C22×Dic3—38no—
68C2×Dic6—34no—
69C3×C4⋊C4C2×D314yes—
70C3×C22⋊C4C2×D314yes—
71C3×C22⋊C4C2×D314yes—
72C3×C42C2×D314yes—
73#217—38no—
74#217—38no—
75#217—38no—
76C4⋊Dic3—34no—
77C4⋊Dic3—34no—
78C4⋊Dic3—34no—
79C4⋊Dic3—38no—
80Dic3⋊C4—38no—
81Dic3⋊C4—38no—
82Dic3⋊C4—38no—
83Dic3⋊C4—38no—
84Dic3⋊C4—38no—
85C4×Dic3—38no—
86C4×Dic3—38no—
87C22×C32—21no—
88C22×C32—21no—
89C22×C32—21no—
90C22×C32—21no—
91C22×C32—21no—
92C22×C32#4112yes—
93C22×C32C2411yescharacteristic
94C32×C4—21no—
95C32×C4—21no—
96C32×C4—21no—
97C32×C4—21no—
98C32×C4#4112yes—
99C3⋊Dic3—22no—
100C3⋊Dic3—22no—
101C3⋊Dic3—22no—
102C3⋊Dic3—22no—
103#126—92no—
104#125—92no—
105#125—91no—
106#123—92no—
107C22⋊Q8—91no—
108C22⋊Q8—91no—
109C4×Q8—91no—
110C4×D4C3⋊D311yescharacteristic
111#117—91no—
112#117—92no—
113C2×C4⋊C4—91no—
114C23×C3C22×D314yes—
115C23×C3C22×D314yes—
116C3×D4—28no—
117C2×C3×C4—28no—
118C2×C3×C4—28no—
119C2×C3×C4C22×D314yes—
120C2×C3×C4C22×D314yes—
121C2×C3×C4C22×D314yes—
122C2×C3×C4C22×D314yes—
123C2×C3×C4C22×D314yes—
124C2×Dic3—38no—
125C2×Dic3—38no—
126C2×Dic3—38no—
127C2×Dic3—38no—
128C2×Dic3—616no—
129Dic6—68no—
130C2×C32—22no—
131C2×C32—41no—
132C2×C32C2×#4111yescharacteristic
133C2×C32C2×#4111yescharacteristic
134C2×C322−1+411yescharacteristic
135C2×Q8—91no—
136C2×D4C2×C3⋊D311yescharacteristic
137C22×C4—92no—
138C22×C4C2×C3⋊D311yescharacteristic
139C22×C4C2×C3⋊D311yescharacteristic
140C4⋊C4—91no—
141C4⋊C4—91no—
142C4⋊C4—92no—
143C4⋊C4—92no—
144C4⋊C4—91no—
145C4⋊C4—92no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4—92no—
149C4⋊C4C2×C3⋊D311yescharacteristic
150C22⋊C4—92no—
151C22⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4C2×C3⋊D311yescharacteristic
154C22⋊C4C2×C3⋊D311yescharacteristic
155C42—92no—
156C42—92no—
157C42C2×C3⋊D311yescharacteristic
158C22×C3—24no—
159C22×C3—24no—
160C22×C3—24no—
161C22×C3—24no—
162C22×C3—24no—
163C22×C3#23518yes—
164C22×C3C23×D314yes—
165C3×C4—24no—
166C3×C4—24no—
167C3×C4—24no—
168C3×C4—24no—
169C3×C4#23718yes—
170Dic3—68no—
171Dic3—68no—
172Dic3—68no—
173Dic3—68no—
174C32#55211yescharacteristic, complemented, Sylow
175C23C22×C3⋊D311yescharacteristic
176C23C22×C3⋊D311yescharacteristic
177Q8—182no—
178D4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—92no—
182C2×C4—92no—
183C2×C4—92no—
184C2×C4—92no—
185C2×C4—184no—
186C2×C4C22×C3⋊D311yescharacteristic
187C2×C4C22×C3⋊D311yescharacteristic
188C2×C4C22×C3⋊D311yescharacteristic
189C2×C4C22×C3⋊D311yescharacteristic
190C2×C4C22×C3⋊D311yescharacteristic
191C2×C3—28no—
192C2×C3—44no—
193C2×C3C2×#23514yes—
194C2×C3C2×#23714yes—
195C2×C3#101014yes—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22#382812yes—
202C22C23×C3⋊D311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—182no—
208C4—182no—
209C4—182no—
210C4—182no—
211C4#383012yes—
212C3#554114yescomplemented
213C2—22no—
214C2—41no—
215C2C2×#382811yescharacteristic
216C2C2×#383011yescharacteristic
217C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4×D4C2
prime radicalC2×C32×D4C22
hypercenterC4×D4C3⋊D3
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
3. #64135C2semidirect
4. C2×C4⋊(C3⋊Dic3)C2semidirect
5. #64133C2semidirect
6. #64133C2semidirect
7. #64098C2semidirect
8. #64088C2semidirect
9. #64086C2semidirect
10. #64079C2semidirect
11. C4×#634C2semidirect
174. C32#552semidirect
212. C3#5541semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D42
#641442
#641352
C2×C4⋊(C3⋊Dic3)2
#641332
#640982
#640882
#640862
#640792
C4×#6342
#55413