Finite Group Databasebeta

← #10596029

Subgroups — #10596029

Order 576.

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

normal subgroups only

This group has 714 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×C4×D4C211yesmaximal, characteristic, complemented
2#64147C211yesmaximal, characteristic, complemented
3#64147C211yesmaximal, characteristic, complemented
4#64138C212yesmaximal, complemented
5C4×#638C212yesmaximal, complemented
6#64101C211yesmaximal, characteristic, complemented
7C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
8#64093C212yesmaximal, complemented
9#64088C212yesmaximal, complemented
10C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11#64079C211yesmaximal, characteristic, complemented
12#5558—34nomaximal
13C2×C32×D4C2211yescharacteristic
14C22×C32×C4C2212yes—
15C2×#638C2212yes—
16C2×#638C2212yes—
17C2×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C4×C3⋊D3C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×C4×C3⋊D3C2212yes—
21C32×C4⋊C4C2211yescharacteristic
22C32×C22⋊C4C2212yes—
23C32×C42C2211yescharacteristic
24#3757C2212yes—
25#3757C2212yes—
26#3752C2211yescharacteristic
27#3752C2211yescharacteristic
28#3752C2212yes—
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2211yescharacteristic
31C4⋊(C3⋊Dic3)C2211yescharacteristic
32#3750C2212yes—
33#3750C2212yes—
34#3750C2212yes—
35C4×C3⋊Dic3C2212yes—
36C3×C4×D4D314yescomplemented
37#938—34no—
38#938—34no—
39#929—38no—
40C4×#67—38no—
41#892—34no—
42D3×C4⋊C4—34no—
43#884—38no—
44#879—38no—
45C4×D12—34no—
46#870—34no—
47C23×C32C2312yes—
48C22×C3⋊D3C2311yescharacteristic
49C22×C3⋊D3C2311yescharacteristic
50C32×D4—22no—
51C2×C32×C4—42no—
52C2×C32×C4C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2311yescharacteristic
56#638—24no—
57#638—42no—
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2312yes—
60C2×C3⋊Dic3C2312yes—
61C2×C3⋊Dic3C2311yescharacteristic
62C4⋊(C3⋊D3)—22no—
63C4×C3⋊D3—22no—
64C4×C3⋊D3—22no—
65C4×C3⋊D3—42no—
66#553—91noSylow
67C2×C3×D4C2×D314yes—
68C22×C3×C4C2×D318yes—
69C2×#67—38no—
70C2×#67—38no—
71C2×D12—34no—
72C2×C4×D3—34no—
73C2×C4×D3—34no—
74C2×C4×D3—38no—
75C3×C4⋊C4C2×D314yes—
76C3×C22⋊C4C2×D318yes—
77C3×C42C2×D314yes—
78#217—38no—
79#217—38no—
80(C2×D3)⋊C4—34no—
81(C2×D3)⋊C4—34no—
82(C2×D3)⋊C4—38no—
83C4⋊Dic3—34no—
84C4⋊Dic3—34no—
85C4⋊Dic3—34no—
86Dic3⋊C4—38no—
87Dic3⋊C4—38no—
88Dic3⋊C4—38no—
89C4×Dic3—38no—
90C22×C32—22no—
91C22×C32—22no—
92C22×C32—22no—
93C22×C32C2411yescharacteristic
94C2×C3⋊D3—21no—
95C2×C3⋊D3—21no—
96C2×C3⋊D3—21no—
97C2×C3⋊D3—21no—
98C2×C3⋊D3—21no—
99C2×C3⋊D3#4112yes—
100C32×C4—21no—
101C32×C4—22no—
102C32×C4—21no—
103C32×C4#4112yes—
104C3⋊Dic3—21no—
105C3⋊Dic3—21no—
106C3⋊Dic3—22no—
107C3⋊Dic3—22no—
108#126—92no—
109#125—91no—
110#123—92no—
111#121—91no—
112#121—91no—
113#121—92no—
114C4×D4—92no—
115C4×D4—91no—
116C4×D4C3⋊D311yescharacteristic, complemented
117#117—91no—
118C2×C4⋊C4—91no—
119C23×C3C22×D318yes—
120C22×D3—34no—
121C22×D3—34no—
122C3×D4—28no—
123C2×C3×C4—48no—
124C2×C3×C4C22×D318yes—
125C2×C3×C4C22×D314yes—
126C2×C3×C4C22×D314yes—
127C2×C3×C4C22×D314yes—
128#67—616no—
129#67—128no—
130C2×Dic3—34no—
131C2×Dic3—38no—
132C2×Dic3—38no—
133C2×Dic3—34no—
134D12—68no—
135C4×D3—68no—
136C4×D3—68no—
137C4×D3—128no—
138C2×C32—42no—
139C2×C32C2×#4111yescharacteristic
140C2×C32C2×#4111yescharacteristic
141C2×C32H(5,𝔽2)11yescharacteristic
142C3⋊D3—22no—
143C3⋊D3—41no—
144C2×D4—92no—
145C2×D4—92no—
146C2×D4—91no—
147C2×D4C2×C3⋊D311yescharacteristic
148C22×C4—91no—
149C22×C4—91no—
150C22×C4—92no—
151C22×C4C2×C3⋊D312yes—
152C4⋊C4—91no—
153C4⋊C4—92no—
154C4⋊C4—91no—
155C4⋊C4—92no—
156C4⋊C4—91no—
157C4⋊C4—92no—
158C4⋊C4C2×C3⋊D311yescharacteristic
159C22⋊C4—92no—
160C22⋊C4—92no—
161C22⋊C4—92no—
162C22⋊C4—91no—
163C22⋊C4—91no—
164C22⋊C4C2×C3⋊D312yes—
165C42—92no—
166C42C2×C3⋊D311yescharacteristic
167C22×C3—28no—
168C22×C3—28no—
169C22×C3—28no—
170C22×C3C23×D314yes—
171C2×D3—38no—
172C2×D3—64no—
173C2×D3—64no—
174C2×D3—64no—
175C2×D3—64no—
176C2×D3—64no—
177C3×C4—28no—
178C3×C4—24no—
179C3×C4—24no—
180C3×C4#23518yes—
181Dic3—64no—
182Dic3—64no—
183Dic3—68no—
184Dic3—68no—
185C32#55311yescharacteristic, complemented, Sylow
186C23—91no—
187C23—91no—
188C23C22×C3⋊D312yes—
189D4—22no—
190D4—182no—
191D4—184no—
192D4—362no—
193C2×C4—42no—
194C2×C4—91no—
195C2×C4—92no—
196C2×C4—91no—
197C2×C4—92no—
198C2×C4—182no—
199C2×C4—182no—
200C2×C4—362no—
201C2×C4C22×C3⋊D312yes—
202C2×C4C22×C3⋊D311yescharacteristic
203C2×C4C22×C3⋊D311yescharacteristic
204C2×C4C22×C3⋊D311yescharacteristic
205C2×C3—48no—
206C2×C3C2×#23514yes—
207C2×C3#100414yes—
208C2×C3D3×#4114yes—
209D3—68no—
210D3—124no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—92no—
215C22—181no—
216C22—181no—
217C22—181no—
218C22—181no—
219C22—181no—
220C22C23×C3⋊D311yescharacteristic
221C4—21no—
222C4—22no—
223C4—21no—
224C4—181no—
225C4—181no—
226C4—182no—
227C4—182no—
228C4#382812yes—
229C3#555814yescomplemented
230C2—42no—
231C2—182no—
232C2—361no—
233C2C2×#382811yescharacteristic
234C2#6436111yescharacteristic
235C2#41×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4×D4C2
hypercenterC4×D4C3⋊D3
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C32×C4×D4C2semidirect
2. #64147C2semidirect
3. #64147C2semidirect
4. #64138C2semidirect
5. C4×#638C2semidirect
6. #64101C2semidirect
7. C4⋊C4×C3⋊D3C2semidirect
8. #64093C2semidirect
9. #64088C2semidirect
10. C4×C4⋊(C3⋊D3)C2semidirect
11. #64079C2semidirect
36. C3×C4×D4D3semidirect
116. C4×D4C3⋊D3semidirect
185. C32#553semidirect
229. C3#5558semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D42
#641472
#641382
C4×#6382
#641012
C4⋊C4×C3⋊D32
#640932
#640882
C4×C4⋊(C3⋊D3)2
#640792
#55583