Finite Group Databasebeta

← #10596113

Subgroups — #10596113

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×C22⋊Q8C211yesmaximal, characteristic, complemented
2#64155C211yesmaximal, characteristic, complemented
3Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C4×#638C211yesmaximal, characteristic
5C2×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6#64107C212yesmaximal, complemented
7#64101C211yesmaximal, characteristic, complemented
8#64099C212yesmaximal, complemented
9#64099C211yesmaximal, characteristic, complemented
10#64095C212yesmaximal
11#64086C212yesmaximal, complemented
12#5642—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C22×C3⋊Dic3C2212yes—
17C2×C4×C3⋊D3C2211yescharacteristic
18C32×C4⋊C4C2211yescharacteristic
19C32×C4⋊C4C2212yes—
20C32×C22⋊C4C2212yes—
21#3757C2211yescharacteristic
22#3752C2212yes—
23#3752C2212yes—
24#3752C2211yescharacteristic
25C4⋊(C3⋊Dic3)C2212yes—
26C4⋊(C3⋊Dic3)C2212yes—
27C4⋊(C3⋊Dic3)C2212yes—
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30#3750C2212yes—
31#3750C2212yes—
32#3750C2211yescharacteristic
33C4×C3⋊Dic3C2212yes—
34C4×C3⋊Dic3C2212yes—
35C4×C3⋊Dic3C2211yescharacteristic
36C3×C22⋊Q8D314yescomplemented
37#946—34no—
38Q8×Dic3—34no—
39C4×#67—34no—
40C2×C4⋊Dic3—34no—
41#898—38no—
42#892—34no—
43#890—38no—
44#890—34no—
45#886—38no—
46#877—38no—
47C23×C32C2311yescharacteristic
48C22×C3⋊D3C2311yescharacteristic
49C32×Q8—22no—
50C2×C32×C4—22no—
51C2×C32×C4C2312yes—
52C2×C32×C4C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55#638—22no—
56C2×C3⋊Dic3—24no—
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2312yes—
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2312yes—
62C4×C3⋊D3—22no—
63#552—91noSylow
64C2×C3×Q8C2×D314yes—
65C22×C3×C4C2×D314yes—
66C2×#67—34no—
67C22×Dic3—38no—
68C2×C4×D3—34no—
69C3×C4⋊C4C2×D314yes—
70C3×C4⋊C4C2×D318yes—
71C3×C22⋊C4C2×D318yes—
72#217—34no—
73(C2×D3)⋊C4—38no—
74(C2×D3)⋊C4—38no—
75(C2×D3)⋊C4—34no—
76C4⋊Dic3—38no—
77C4⋊Dic3—38no—
78C4⋊Dic3—38no—
79C4⋊Dic3—34no—
80C4⋊Dic3—34no—
81Dic3⋊C4—38no—
82Dic3⋊C4—38no—
83Dic3⋊C4—34no—
84C4×Dic3—38no—
85C4×Dic3—38no—
86C4×Dic3—34no—
87C22×C32—21no—
88C22×C32—21no—
89C22×C32#4112yes—
90C22×C32C2411yescharacteristic
91C2×C3⋊D3—21no—
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C32×C4—21no—
95C32×C4—22no—
96C32×C4—22no—
97C32×C4#4112yes—
98C3⋊Dic3—21no—
99C3⋊Dic3—22no—
100C3⋊Dic3—22no—
101C3⋊Dic3—21no—
102C3⋊Dic3—21no—
103#126—92no—
104#125—92no—
105#125—91no—
106#123—92no—
107C22⋊Q8—91no—
108C22⋊Q8C3⋊D311yescharacteristic, complemented
109C4×Q8—91no—
110C4×D4—91no—
111#117—91no—
112#117—92no—
113C2×C4⋊C4—91no—
114C23×C3C22×D314yes—
115C22×D3—34no—
116C3×Q8—28no—
117C2×C3×C4—28no—
118C2×C3×C4C22×D318yes—
119C2×C3×C4C22×D318yes—
120C2×C3×C4C22×D314yes—
121C2×C3×C4C22×D314yes—
122#67—68no—
123C2×Dic3—34no—
124C2×Dic3—34no—
125C2×Dic3—38no—
126C2×Dic3—34no—
127C2×Dic3—38no—
128C2×Dic3—616no—
129C4×D3—68no—
130C2×C32—22no—
131C2×C32C2×#4111yescharacteristic
132C2×C32C2×#4111yescharacteristic
133C2×C322−1+411yescharacteristic
134C3⋊D3—41no—
135C2×Q8C2×C3⋊D311yescharacteristic
136C2×D4—91no—
137C22×C4—92no—
138C22×C4—91no—
139C22×C4C2×C3⋊D311yescharacteristic
140C4⋊C4—91no—
141C4⋊C4—91no—
142C4⋊C4—92no—
143C4⋊C4—92no—
144C4⋊C4—92no—
145C4⋊C4—91no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4C2×C3⋊D311yescharacteristic
149C4⋊C4C2×C3⋊D312yes—
150C22⋊C4—91no—
151C22⋊C4—92no—
152C22⋊C4—91no—
153C22⋊C4—92no—
154C22⋊C4C2×C3⋊D312yes—
155C42—92no—
156C42—92no—
157C42—91no—
158C22×C3—24no—
159C22×C3—24no—
160C22×C3#23918yes—
161C22×C3C23×D314yes—
162C2×D3—64no—
163C2×D3—64no—
164C2×D3—64no—
165C3×C4—24no—
166C3×C4—28no—
167C3×C4—28no—
168C3×C4#23718yes—
169Dic3—64no—
170Dic3—68no—
171Dic3—68no—
172Dic3—64no—
173Dic3—64no—
174C32#55211yescharacteristic, complemented, Sylow
175C23—91no—
176C23C22×C3⋊D311yescharacteristic
177Q8—22no—
178D4—182no—
179C2×C4—22no—
180C2×C4—91no—
181C2×C4—91no—
182C2×C4—92no—
183C2×C4—91no—
184C2×C4—92no—
185C2×C4—184no—
186C2×C4—182no—
187C2×C4C22×C3⋊D312yes—
188C2×C4C22×C3⋊D312yes—
189C2×C4C22×C3⋊D311yescharacteristic
190C2×C4C22×C3⋊D311yescharacteristic
191C2×C3—28no—
192C2×C3C2×#23714yes—
193C2×C3C2×#23914yes—
194C2×C3#101014yes—
195D3—124no—
196C22—21no—
197C22—21no—
198C22—181no—
199C22—181no—
200C22—181no—
201C22#383212yes—
202C22C23×C3⋊D311yescharacteristic
203C4—21no—
204C4—22no—
205C4—22no—
206C4—181no—
207C4—182no—
208C4—182no—
209C4—181no—
210C4—181no—
211C4#383012yes—
212C3#564214yescomplemented
213C2—22no—
214C2—361no—
215C2C2×#383011yescharacteristic
216C2C2×#383211yescharacteristic
217C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
prime radicalC2×#638C22
hypercenterC22⋊Q8C3⋊D3
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C22⋊Q8C2semidirect
2. #64155C2semidirect
3. Q8×C3⋊Dic3C2semidirect
5. C2×C4⋊(C3⋊Dic3)C2semidirect
6. #64107C2semidirect
7. #64101C2semidirect
8. #64099C2semidirect
9. #64099C2semidirect
11. #64086C2semidirect
36. C3×C22⋊Q8D3semidirect
108. C22⋊Q8C3⋊D3semidirect
174. C32#552semidirect
212. C3#5642semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊Q82
#641552
Q8×C3⋊Dic32
C4×#6382
C2×C4⋊(C3⋊Dic3)2
#641072
#641012
#640992
#640952
#640862
#56423