Finite Group Databasebeta

← #10592755⋊C3 #13142922

Subgroups — #10592755⋊C3 #13142922

Order 1728.

412 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×#124C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3#63662⋊C3C212yesmaximal, complemented
4#63659⋊C3C212yesmaximal
5C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
6#10592755C311yesmaximal, characteristic, complemented
7C3×#5056—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C22⋊C4×H(3,𝔽3)—22no—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#3750⋊C3—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15C32×#124—21no—
16C32×#124D311yescharacteristic
17C32×#124C2×C311yescharacteristic
18#64079C2×C311yescharacteristic, complemented
19C3×#870—31no—
20#63662C2×C312yescomplemented
21#63659C2×C312yes—
22C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
23C3×#835—32no—
24C3×#832—32no—
25C3×C4×C3⋊C8—31no—
26C23×H(3,𝔽3)—21no—
27Q8×H(3,𝔽3)—22no—
28D4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)D412yes—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—22no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34C3×#488—91no—
35#5056—31no—
36#5056—91no—
37C2×C32×Q8—21no—
38C2×C32×Q8C2×D311yescharacteristic
39C2×C32×Q8C22×C311yescharacteristic
40C2×C32×D4—21no—
41C2×C32×D4C2×D311yescharacteristic
42C2×C32×D4C22×C311yescharacteristic
43C32×C22⋊C4—24no—
44C32×C22⋊C4—22no—
45C32×C22⋊C4—22no—
46C32×C42—21no—
47C32×C42C2×D311yescharacteristic
48C32×C42C22×C311yescharacteristic
49C4⋊(C3⋊Dic3)C22×C312yes—
50#3750—22no—
51C2×C3⋊(C3⋊C8)C22×C312yes—
52C3×C4⋊Dic3—32no—
53C3×Dic3⋊C4—62no—
54C2×C3×C3⋊C8—32no—
55C22×H(3,𝔽3)—41no—
56C22×H(3,𝔽3)—42no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)—41no—
60C4×H(3,𝔽3)#4112yes—
61(C3⋊Dic3)⋊C3—42no—
62C3×#125—91no—
63C3×#124—31no—
64C3×#124—31no—
65C3×#124—61no—
66C3×#124C3×D311yescharacteristic
67#870—31no—
68#870—91no—
69C3×#103—92no—
70C3×#102—92no—
71C3×C4×C8—91no—
72#835—32no—
73#835—92no—
74#832—32no—
75#832—92no—
76C4×C3⋊C8—31no—
77C4×C3⋊C8—91no—
78C23×C32—22no—
79C23×C32—21no—
80C23×C32—21no—
81C32×Q8—22no—
82C32×Q8—22no—
83C32×Q8—42no—
84C32×D4—22no—
85C32×D4—22no—
86C32×D4—42no—
87C2×C32×C4—22no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4#6712yes—
93C2×C32×C4C3×D412yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C32×C4C23×C311yescharacteristic
96C2×C3⋊Dic3—22no—
97C2×C3×Dic3—62no—
98C3⋊(C3⋊C8)—22no—
99C3×C3⋊C8—62no—
100#488—91noSylow
101C2×H(3,𝔽3)—81no—
102C2×H(3,𝔽3)#12411yescharacteristic
103C2×H(3,𝔽3)#13512yes—
104C2×C3×Q8—31no—
105C2×C3×Q8—31no—
106C2×C3×Q8—61no—
107C2×C3×Q8C2×C3×D311yescharacteristic
108C2×C3×D4—31no—
109C2×C3×D4—31no—
110C2×C3×D4—61no—
111C2×C3×D4C2×C3×D311yescharacteristic
112C2×C3×C8—92no—
113C3×C4⋊C4—92no—
114C3×C4⋊C4—182no—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—62no—
117C3×C22⋊C4—62no—
118C3×C22⋊C4—64no—
119C3×C42—31no—
120C3×C42—31no—
121C3×C42—61no—
122C3×C42C2×C3×D311yescharacteristic
123C4⋊Dic3—32no—
124C4⋊Dic3—92no—
125Dic3⋊C4—62no—
126Dic3⋊C4—182no—
127C2×C3⋊C8—32no—
128C2×C3⋊C8—92no—
129C22×C32—21no—
130C22×C32—42no—
131C22×C32—44no—
132C22×C32—41no—
133C22×C32—42no—
134C22×C32—41no—
135C22×C32—42no—
136C22×C32C2×#6711yescharacteristic
137C22×C32C2×C3×D411yescharacteristic
138C32×C4—22no—
139C32×C4—22no—
140C32×C4—22no—
141C32×C4—42no—
142C32×C4—42no—
143C32×C4—41no—
144C32×C4—41no—
145C32×C4#23712yes—
146C32×C4C3×#4112yes—
147C3⋊Dic3—42no—
148C3×Dic3—122no—
149#125—91no—
150#124(C3⋊D3)⋊C311yescharacteristic
151#103—92no—
152#102—92no—
153C4×C8—91no—
154H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
155C23×C3—21no—
156C23×C3—61no—
157C23×C3—61no—
158C23×C3—62no—
159C3×Q8—22no—
160C3×Q8—62no—
161C3×Q8—62no—
162C3×Q8—122no—
163C3×D4—22no—
164C3×D4—62no—
165C3×D4—62no—
166C3×D4—122no—
167C2×C3×C4—21no—
168C2×C3×C4—32no—
169C2×C3×C4—32no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—62no—
175C2×C3×C4—62no—
176C2×C3×C4—61no—
177C2×C3×C4—182no—
178C2×C3×C4C3×#6712yes—
179C2×C3×C4C22×C3×D311yescharacteristic
180C2×Dic3—62no—
181C2×Dic3—182no—
182C3×C8—182no—
183C3⋊C8—62no—
184C3⋊C8—182no—
185C2×C32—21no—
186C2×C32—22no—
187C2×C32—82no—
188C2×C32—81no—
189C2×C32—81no—
190C2×C32#93611yescharacteristic
191C2×C32#95012yes—
192C2×C32C3×#12411yescharacteristic
193C2×C32C3×#13512yes—
194C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
195C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
196C2×C8—92no—
197C4⋊C4—92no—
198C4⋊C4—182no—
199C22⋊C4—22no—
200C42C2×(C3⋊D3)⋊C311yescharacteristic
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—41no—
204C22×C3—42no—
205C22×C3—61no—
206C22×C3—121no—
207C22×C3—121no—
208C22×C3—122no—
209C22×C3—122no—
210C22×C3—122no—
211C22×C3—124no—
212C22×C3C2×C3×#6711yescharacteristic
213C3×C4—22no—
214C3×C4—32no—
215C3×C4—32no—
216C3×C4—41no—
217C3×C4—62no—
218C3×C4—62no—
219C3×C4—62no—
220C3×C4—122no—
221C3×C4—121no—
222C3×C4—121no—
223C3×C4—122no—
224C3×C4—362no—
225C3×C4C3×#23712yes—
226Dic3—122no—
227Dic3—362no—
228C32—21no—
229C32#505611yescharacteristic, complemented
230C32C3×#48811yescharacteristic, complemented
231C23—21no—
232Q8—22no—
233D4—22no—
234C2×C4—21no—
235C2×C4—182no—
236C2×C4#638⋊C312yes—
237C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
238C8—182no—
239C2×C3—31no—
240C2×C3—31no—
241C2×C3—32no—
242C2×C3—32no—
243C2×C3—61no—
244C2×C3—62no—
245C2×C3—81no—
246C2×C3—241no—
247C2×C3—241no—
248C2×C3—242no—
249C2×C3C3×#93611yescharacteristic
250C2×C3C3×#95012yes—
251C22—41no—
252C22—42no—
253C22C2×#638⋊C311yescharacteristic
254C4—22no—
255C4—41no—
256C4—362no—
257C4#3830⋊C312yes—
258C3—31no—
259C3—31no—
260C3—61no—
261C3C3×#505611yescharacteristic
262C2—81no—
263C2#64145⋊C311yescharacteristic
264C2#64159⋊C312yes—
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#124C2
socleC22×C3C2×C3×#67
prime radicalC2×D4×H(3,𝔽3)C22
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#488
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
3. #63662⋊C3C2semidirect
5. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
6. #10592755C3semidirect
18. #64079C2×C3semidirect
20. #63662C2×C3semidirect
22. C4×C3⋊(C3⋊C8)C2×C3semidirect
154. H(3,𝔽3)#488semidirect
229. C32#5056semidirect
230. C32C3×#488semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
#64079⋊C32
#63662⋊C32
#63659⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#105927553
C3×#50563