Finite Group Databasebeta

← #13137364

Subgroups — #13137364

Order 1728.

412 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11721721C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#11720326C212yesmaximal, complemented
5#11720324C212yesmaximal
6#10590278—31nomaximal
7#10590274—31nomaximal
8C2×C4⋊#272C2211yescharacteristic
9C2×#6229C2211yescharacteristic
10C42×H(3,𝔽3)C2211yescharacteristic
11#88542—22no—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#3750⋊C3—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#64079D311yescharacteristic, complemented
16C3×#875—31no—
17C3×#875—31no—
18C3×#875—61no—
19C3×#870—31no—
20C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
21C3×C4×C3⋊C8—31no—
22#63566—32no—
23#63565—32no—
24#63563—32no—
25#63562—32no—
26C22×#272—21no—
27C2×C4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—21no—
30C4⋊#272—22no—
31#6229—22no—
32C2×(C3⋊Dic3)⋊C3—22no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34#5068—91no—
35#5056—91no—
36#4697—91no—
37C2×C3×D12—31no—
38C2×C3×D12—31no—
39C2×C3×D12—61no—
40C2×C3×Dic6—31no—
41C2×C3×Dic6—31no—
42C2×C3×Dic6—61no—
43C32×C42—21no—
44C32×C42C2×D311yescharacteristic
45C32×C42C2×D311yescharacteristic
46C4⋊(C3⋊Dic3)C2×D312yes—
47#3750—22no—
48C2×C3⋊(C3⋊C8)C2×D312yes—
49C3×(C2×D3)⋊C4—64no—
50C3×(C2×D3)⋊C4—62no—
51C3×(C2×D3)⋊C4—62no—
52C3×C4⋊Dic3—32no—
53C3×Dic3⋊C4—62no—
54C2×C3×C3⋊C8—32no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C2×#272—41no—
57C2×#272—42no—
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)#4112yes—
60#1084—41no—
61(C3⋊Dic3)⋊C3—42no—
62C3×#125—91no—
63C3×#124—91no—
64#875—31no—
65#875—31no—
66#875—61no—
67#870—31no—
68#870—91no—
69C3×C4×C8—91no—
70#835—92no—
71#832—92no—
72#821—92no—
73#816—92no—
74#810—92no—
75#809—92no—
76C4×C3⋊C8—31no—
77C4×C3⋊C8—91no—
78C22×C3×D3—62no—
79C22×C3×D3—61no—
80C22×C3×D3—61no—
81C2×C32×C4—22no—
82C2×C32×C4—21no—
83C2×C32×C4D1212yes—
84C2×C32×C4#6712yes—
85C2×C32×C4C22×D311yescharacteristic
86C2×C32×C4C22×D311yescharacteristic
87C2×C3⋊Dic3—22no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C3×D12—62no—
93C3×D12—62no—
94C3×D12—122no—
95C3×Dic6—62no—
96C3×Dic6—62no—
97C3×Dic6—122no—
98C3⋊(C3⋊C8)—22no—
99C3×C3⋊C8—62no—
100#488—271noSylow
101C2×H(3,𝔽3)#12411yescharacteristic
102C2×H(3,𝔽3)#13512yes—
103#272—81no—
104C2×C3×Q8—91no—
105C2×C3×D4—91no—
106C2×D12—31no—
107C2×D12—31no—
108C2×D12—61no—
109C2×Dic6—31no—
110C2×Dic6—31no—
111C2×Dic6—61no—
112C2×C3×C8—92no—
113C3×C4⋊C4—92no—
114C3×C4⋊C4—182no—
115C3×C22⋊C4—182no—
116C3×C42—31no—
117C3×C42—31no—
118C3×C42—61no—
119C3×C42D3211yescharacteristic
120(C2×D3)⋊C4—62no—
121(C2×D3)⋊C4—62no—
122(C2×D3)⋊C4—64no—
123C4⋊Dic3—32no—
124C4⋊Dic3—92no—
125Dic3⋊C4—62no—
126Dic3⋊C4—182no—
127C2×C3⋊C8—32no—
128C2×C3⋊C8—92no—
129C22×C32—21no—
130C22×C32C2×D1211yescharacteristic
131C22×C32C2×#6711yescharacteristic
132C2×C3×D3—122no—
133C2×C3×D3—124no—
134C2×C3×D3—121no—
135C2×C3×D3—122no—
136C2×C3×D3—121no—
137C2×C3×D3—122no—
138C32×C4—22no—
139C32×C4—22no—
140C32×C4—22no—
141C32×C4—42no—
142C32×C4#23512yes—
143C32×C4#23912yes—
144C3⋊Dic3—42no—
145C3×Dic3—122no—
146C3×Dic3—121no—
147C3×Dic3—122no—
148C3×Dic3—121no—
149#125—91no—
150#124—91no—
151#103—272no—
152#102—272no—
153C4×C8—91no—
154H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
155C23×C3—181no—
156C22×D3—61no—
157C22×D3—61no—
158C22×D3—62no—
159C3×Q8—182no—
160C3×D4—182no—
161C2×C3×C4—32no—
162C2×C3×C4—32no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—62no—
166C2×C3×C4—61no—
167C2×C3×C4—182no—
168C2×C3×C4—181no—
169C2×C3×C4#62612yes—
170C2×C3×C4C2×D3211yescharacteristic
171C2×Dic3—61no—
172C2×Dic3—61no—
173C2×Dic3—62no—
174C2×Dic3—62no—
175C2×Dic3—182no—
176D12—62no—
177D12—62no—
178D12—122no—
179Dic6—62no—
180Dic6—62no—
181Dic6—122no—
182C3×C8—182no—
183C3⋊C8—62no—
184C3⋊C8—182no—
185C2×C32—22no—
186C2×C32—21no—
187C2×C32#87511yescharacteristic
188C2×C32#90412yes—
189C2×C32#94711yescharacteristic
190C2×C32#95012yes—
191C3×D3—242no—
192C3×D3—241no—
193C3×D3—241no—
194C2×Q8—91no—
195C2×D4—91no—
196C2×C8—92no—
197C4⋊C4—92no—
198C4⋊C4—182no—
199C22⋊C4—182no—
200C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—61no—
204C22×C3—362no—
205C22×C3—361no—
206C22×C3C2×#62611yescharacteristic
207C2×D3—121no—
208C2×D3—121no—
209C2×D3—122no—
210C2×D3—122no—
211C2×D3—122no—
212C2×D3—124no—
213C3×C4—22no—
214C3×C4—32no—
215C3×C4—32no—
216C3×C4—62no—
217C3×C4—62no—
218C3×C4—62no—
219C3×C4—122no—
220C3×C4—362no—
221C3×C4—361no—
222C3×C4#379912yes—
223Dic3—121no—
224Dic3—121no—
225Dic3—122no—
226Dic3—122no—
227Dic3—362no—
228C32—21no—
229C32#469711yescharacteristic, complemented
230C32#506811yescharacteristic, complemented
231C23—181no—
232Q8—182no—
233D4—182no—
234C2×C4—181no—
235C2×C4—182no—
236C2×C4#620012yes—
237C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
238C8—182no—
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—31no—
242C2×C3—31no—
243C2×C3—62no—
244C2×C3—61no—
245C2×C3—721no—
246C2×C3#6382912yes—
247C2×C3#6386411yescharacteristic
248D3—241no—
249D3—241no—
250D3—242no—
251C22—362no—
252C22—361no—
253C22C2×#620011yescharacteristic
254C4—22no—
255C4—362no—
256C4—361no—
257C4#8865212yes—
258C3—31no—
259C3—31no—
260C3—61no—
261C3#1059029311yescharacteristic
262C2—721no—
263C2#1172107312yes—
264C2#1172111411yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
3. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. #11720326C2semidirect
15. #64079D3semidirect
20. C4×C3⋊(C3⋊C8)D3semidirect
154. H(3,𝔽3)#488semidirect
229. C32#4697semidirect
230. C32#5068semidirect

Maximal subgroups

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

subgroupindex
#117217212
#64079⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#117203262
#117203242
#105902783
#105902743