Finite Group Databasebeta

← #13137359

Subgroups — #13137359

Order 1728.

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

normal subgroups only

This group has 380 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#64084⋊C3C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3C4×#6180C211yesmaximal, characteristic, complemented
4#11720318C212yesmaximal
5#11720313C212yesmaximal, complemented
6#10590293—31nomaximal
7#10590274—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×#634⋊C3C2211yescharacteristic
10C42×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2212yes—
12#3752⋊C3—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#3750⋊C3—22no—
15#64084D311yescharacteristic
16#64079D311yescharacteristic, complemented
17C3×#875—31no—
18C3×#870—31no—
19C3×C4×C3⋊C8—31no—
20C3×C4×C3⋊C8—31no—
21C3×C4×C3⋊C8—61no—
22#63567—32no—
23#63566—32no—
24#63564—32no—
25#63563—32no—
26C22×(C3⋊D3)⋊C3—21no—
27C2×C4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊Dic3)⋊C3—21no—
30C2×(C3⋊Dic3)⋊C3—22no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32#634⋊C3—22no—
33#6180—22no—
34#5068—91no—
35#5056—91no—
36#4697—91no—
37C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
38C2×#634C2×D311yescharacteristic
39C2×C3×D12—31no—
40C2×C3×Dic6—31no—
41C32×C42—21no—
42C32×C42C2×D311yescharacteristic
43C32×C42C2×D311yescharacteristic
44#3752—22no—
45C4⋊(C3⋊Dic3)C2×D312yes—
46#3750—22no—
47C3×(C2×D3)⋊C4—62no—
48C3×C4⋊Dic3—32no—
49C3×Dic3⋊C4—62no—
50C2×C3×C3⋊C8—32no—
51C2×C3×C3⋊C8—32no—
52C2×C3×C3⋊C8—62no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×(C3⋊D3)⋊C3—41no—
55C2×(C3⋊D3)⋊C3—42no—
56C4×H(3,𝔽3)—22no—
57C4×H(3,𝔽3)#4112yes—
58(C3⋊Dic3)⋊C3—41no—
59(C3⋊Dic3)⋊C3—42no—
60C3×#125—91no—
61C3×#124—91no—
62#875—31no—
63#875—91no—
64#870—31no—
65#870—91no—
66C3×C4×C8—91no—
67#835—92no—
68#832—92no—
69#821—92no—
70#816—92no—
71#810—92no—
72#809—92no—
73C4×C3⋊C8—31no—
74C4×C3⋊C8—31no—
75C4×C3⋊C8—61no—
76C22×C3⋊D3—21no—
77C22×C3×D3—61no—
78C2×C32×C4—22no—
79C2×C32×C4—21no—
80C2×C32×C4#6712yes—
81C2×C32×C4#6712yes—
82C2×C32×C4C22×D311yescharacteristic
83C2×C32×C4C22×D311yescharacteristic
84C2×C3⋊Dic3—22no—
85C2×C3⋊Dic3—21no—
86C4⋊(C3⋊D3)—22no—
87#634—22no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—61no—
90C3×D12—62no—
91C3×Dic6—62no—
92C3×C3⋊C8—64no—
93C3×C3⋊C8—62no—
94C3×C3⋊C8—62no—
95#488—271noSylow
96C2×H(3,𝔽3)#12411yescharacteristic
97C2×H(3,𝔽3)#13512yes—
98(C3⋊D3)⋊C3—81no—
99C2×C3×Q8—91no—
100C2×C3×D4—91no—
101C2×D12—31no—
102C2×D12—91no—
103C2×Dic6—31no—
104C2×Dic6—91no—
105C2×C3×C8—92no—
106C3×C4⋊C4—92no—
107C3×C4⋊C4—182no—
108C3×C22⋊C4—182no—
109C3×C42—31no—
110C3×C42—31no—
111C3×C42—61no—
112C3×C42D3211yescharacteristic
113(C2×D3)⋊C4—62no—
114(C2×D3)⋊C4—182no—
115C4⋊Dic3—32no—
116C4⋊Dic3—92no—
117Dic3⋊C4—62no—
118Dic3⋊C4—182no—
119C2×C3⋊C8—32no—
120C2×C3⋊C8—32no—
121C2×C3⋊C8—62no—
122C22×C32—21no—
123C22×C32C2×#6711yescharacteristic
124C22×C32C2×#6711yescharacteristic
125C2×C3⋊D3—41no—
126C2×C3⋊D3—42no—
127C2×C3×D3—121no—
128C2×C3×D3—122no—
129C32×C4—24no—
130C32×C4—22no—
131C32×C4—22no—
132C32×C4—22no—
133C32×C4#23712yes—
134C32×C4#23912yes—
135C3⋊Dic3—41no—
136C3⋊Dic3—42no—
137C3×Dic3—121no—
138C3×Dic3—122no—
139#125—91no—
140#124—91no—
141#103—272no—
142#102—272no—
143C4×C8—91no—
144H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
145C23×C3—181no—
146C22×D3—61no—
147C22×D3—181no—
148C3×Q8—182no—
149C3×D4—182no—
150C2×C3×C4—32no—
151C2×C3×C4—32no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—62no—
155C2×C3×C4—61no—
156C2×C3×C4—181no—
157C2×C3×C4—182no—
158C2×C3×C4#62512yes—
159C2×C3×C4C2×D3211yescharacteristic
160C2×Dic3—62no—
161C2×Dic3—61no—
162C2×Dic3—182no—
163C2×Dic3—181no—
164D12—62no—
165D12—182no—
166Dic6—62no—
167Dic6—182no—
168C3×C8—182no—
169C3⋊C8—62no—
170C3⋊C8—62no—
171C3⋊C8—64no—
172C2×C32—22no—
173C2×C32—21no—
174C2×C32#93611yescharacteristic
175C2×C32#94711yescharacteristic
176C2×C32#95012yes—
177C2×C32#95012yes—
178C3⋊D3—81no—
179C3×D3—241no—
180C2×Q8—91no—
181C2×D4—91no—
182C2×C8—92no—
183C4⋊C4—92no—
184C4⋊C4—182no—
185C22⋊C4—182no—
186C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
187C22×C3—31no—
188C22×C3—31no—
189C22×C3—61no—
190C22×C3—361no—
191C22×C3—362no—
192C22×C3C2×#62511yescharacteristic
193C2×D3—122no—
194C2×D3—121no—
195C2×D3—361no—
196C2×D3—362no—
197C3×C4—22no—
198C3×C4—32no—
199C3×C4—32no—
200C3×C4—62no—
201C3×C4—62no—
202C3×C4—64no—
203C3×C4—62no—
204C3×C4—361no—
205C3×C4—362no—
206C3×C4#379612yes—
207Dic3—122no—
208Dic3—121no—
209Dic3—361no—
210Dic3—362no—
211C32—21no—
212C32#505611yescharacteristic, complemented
213C32#506811yescharacteristic, complemented
214C23—181no—
215Q8—182no—
216D4—182no—
217C2×C4—181no—
218C2×C4—182no—
219C2×C4#619812yes—
220C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
221C8—182no—
222C2×C3—32no—
223C2×C3—32no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—62no—
227C2×C3—61no—
228C2×C3—721no—
229C2×C3#6382212yes—
230C2×C3#6386311yescharacteristic
231D3—241no—
232D3—721no—
233C22—362no—
234C22—361no—
235C22C2×#619811yescharacteristic
236C4—22no—
237C4—362no—
238C4—361no—
239C4#8864912yes—
240C3—31no—
241C3—31no—
242C3—61no—
243C3#1059027811yescharacteristic
244C2—721no—
245C2#1172106012yes—
246C2#1172111311yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
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×#6180C2semidirect
5. #11720313C2semidirect
16. #64079D3semidirect
144. H(3,𝔽3)#488semidirect
212. C32#5056semidirect
213. C32#5068semidirect

Maximal subgroups

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

subgroupindex
#64084⋊C32
#64079⋊C32
C4×#61802
#117203182
#117203132
#105902933
#105902743