Finite Group Databasebeta

← #13139015

Subgroups — #13139015

Order 1728.

515 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#88530C211yesmaximal, characteristic, complemented
2C2×#3745⋊C3C211yesmaximal, characteristic
3#11721110C212yesmaximal, complemented
4C2×#88438C212yesmaximal, complemented
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#10590869—31nomaximal
7#10590865—31nomaximal
8C2×#6229C2212yes—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88530—22no—
12C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#3745⋊C3—22no—
15#88450—24no—
16#88438—24no—
17C2×#3745D311yescharacteristic
18C2×C3×Dic12—31no—
19C2×C3×Dic12—31no—
20C2×C3×Dic12—61no—
21C2×C3×D24—31no—
22#63864—32no—
23#63863—32no—
24C2×#3717—32no—
25C2×#3715—32no—
26C8×C3⋊Dic3D311yescharacteristic, complemented
27C3×C8×Dic3—31no—
28C22×(C3⋊D3)⋊C3—22no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—22no—
31#6229—22no—
32#6229—22no—
33C2×(C3⋊Dic3)⋊C3D412yes—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36C8×H(3,𝔽3)D412yes—
37(C3⋊(C3⋊C8))⋊C3D412yes—
38#5191—91no—
39#5155—91no—
40#4696—91no—
41C2×C4⋊(C3⋊D3)C2×D312yes—
42C2×C3×D12—32no—
43C2×C3×Dic6—32no—
44C2×C3×Dic6—32no—
45C2×C3×Dic6—62no—
46C2×C32×C8—21no—
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C2×D311yescharacteristic
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
51#3745—22no—
52C3×C4×Dic3—31no—
53C2×C3×C3⋊C8—31no—
54C3×Dic12—64no—
55C3×Dic12—62no—
56C3×Dic12—62no—
57C3×D24—62no—
58#3722—64no—
59(C2×D3)⋊Dic3—64no—
60#3717—64no—
61#3715—64no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×(C3⋊D3)⋊C3—42no—
64C2×(C3⋊D3)⋊C3—44no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)C2×D411yescharacteristic
67#1084—42no—
68(C3⋊Dic3)⋊C3—22no—
69C2×C3×Q16—91no—
70C2×C3×D8—91no—
71#947—92no—
72C2×#215—92no—
73#936—92no—
74C2×#214—92no—
75C2×Dic12—31no—
76C2×Dic12—31no—
77C2×Dic12—61no—
78C2×D24—31no—
79C2×D24—91no—
80C2×#204—92no—
81#875—92no—
82C3×C4×C8—91no—
83C8×Dic3—31no—
84C8×Dic3—91no—
85C22×C3⋊D3—22no—
86C22×C3×D3—62no—
87C2×C32×C4—21no—
88C2×C32×C4C22×D311yescharacteristic
89C2×C32×C4C22×D311yescharacteristic
90C2×C3⋊Dic3D1212yes—
91C4⋊(C3⋊D3)—22no—
92C4⋊(C3⋊D3)—22no—
93C2×C3×Dic3—32no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—122no—
97C3×D12—62no—
98C3×D12—62no—
99C3×Dic6—64no—
100C3×Dic6—62no—
101C3×Dic6—62no—
102C3×Dic6—62no—
103C3×Dic6—62no—
104C3×Dic6—122no—
105C32×C8—22no—
106C32×C8D1212yes—
107C32×C8#6712yes—
108C3⋊(C3⋊C8)D1212yes—
109C3×C3⋊C8—32no—
110#495—271noSylow
111C2×H(3,𝔽3)#12711yescharacteristic
112C2×H(3,𝔽3)#13512yes—
113(C3⋊D3)⋊C3—82no—
114C2×C3×Q8—92no—
115C2×C3×D4—92no—
116C2×D12—32no—
117C2×D12—92no—
118C2×Dic6—32no—
119C2×Dic6—32no—
120C2×Dic6—62no—
121C3×Q16—182no—
122C3×D8—182no—
123C2×C3×C8—31no—
124C2×C3×C8—31no—
125C2×C3×C8—61no—
126C2×C3×C8—91no—
127C2×C3×C8D3211yescharacteristic
128C3×C42—91no—
129#217—184no—
130#215—184no—
131#214—184no—
132(C2×D3)⋊C4—184no—
133(C2×D3)⋊C4—184no—
134C4×Dic3—31no—
135C4×Dic3—91no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—91no—
138Dic12—62no—
139Dic12—62no—
140Dic12—64no—
141D24—62no—
142D24—182no—
143#204—184no—
144C22×C32—21no—
145C22×C32C2×D1211yescharacteristic
146C22×C32D3×D411yescharacteristic
147C2×C3⋊D3—44no—
148C2×C3⋊D3—42no—
149C2×C3×D3—122no—
150C2×C3×D3—124no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4C2×D1211yescharacteristic
154C32×C4C2×D1211yescharacteristic
155C32×C4D3×D411yescharacteristic
156C32×C4C2×#6711yescharacteristic
157C3⋊Dic3—22no—
158C3×Dic3—62no—
159C3×Dic3—124no—
160C3×Dic3—122no—
161C3×Dic3—122no—
162C2×Q16—91no—
163C2×SD16—272no—
164C2×D8—91no—
165#124—272no—
166C4×C8—91no—
167H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
168C23×C3—182no—
169C22×D3—62no—
170C22×D3—182no—
171C3×Q8—182no—
172C3×Q8—182no—
173C3×D4—182no—
174C3×D4—182no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—61no—
178C2×C3×C4—92no—
179C2×C3×C4—182no—
180C2×C3×C4C2×D3211yescharacteristic
181C2×Dic3—32no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—92no—
185C2×Dic3—122no—
186D12—62no—
187D12—62no—
188D12—182no—
189D12—182no—
190Dic6—62no—
191Dic6—62no—
192Dic6—62no—
193Dic6—62no—
194Dic6—64no—
195Dic6—122no—
196C3×C8—32no—
197C3×C8—32no—
198C3×C8—62no—
199C3×C8—92no—
200C3×C8#62612yes—
201C3⋊C8—32no—
202C3⋊C8—92no—
203C2×C32—22no—
204C2×C32—21no—
205C2×C32#87411yescharacteristic
206C2×C32#90412yes—
207C2×C32#91212yes—
208C2×C32#94011yescharacteristic
209C3⋊D3—82no—
210C3×D3—242no—
211C2×Q8—92no—
212C2×D4—92no—
213Q16—182no—
214SD16—544no—
215D8—182no—
216C2×C8—91no—
217C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
218C22⋊C4—544no—
219C42—91no—
220C22×C3—31no—
221C22×C3—31no—
222C22×C3—61no—
223C22×C3—362no—
224C22×C3—364no—
225C22×C3D3×D1211yescharacteristic
226C2×D3—124no—
227C2×D3—122no—
228C2×D3—362no—
229C2×D3—364no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—61no—
235C3×C4—61no—
236C3×C4—182no—
237C3×C4—362no—
238C3×C4D3×D1211yescharacteristic
239C3×C4C2×#62611yescharacteristic
240Dic3—62no—
241Dic3—122no—
242Dic3—122no—
243Dic3—124no—
244Dic3—182no—
245C32—21no—
246C32#469611yescharacteristic, complemented
247C32#515511yescharacteristic, complemented
248C23—182no—
249Q8—182no—
250Q8—182no—
251D4—182no—
252D4—182no—
253C2×C4—92no—
254C2×C4—182no—
255C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
256C8—92no—
257C8#620012yes—
258C2×C3—32no—
259C2×C3—32no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—62no—
263C2×C3—61no—
264C2×C3—722no—
265C2×C3#6380712yes—
266C2×C3#6391111yescharacteristic
267D3—242no—
268D3—722no—
269C22—364no—
270C22—362no—
271C22#8865411yescharacteristic
272C4—182no—
273C4—362no—
274C4#8865411yescharacteristic
275C4C2×#620011yescharacteristic
276C3—31no—
277C3—31no—
278C3—61no—
279C3#1059086411yescharacteristic
280C2—722no—
281C2#1172085512yes—
282C2#1172115811yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#3745⋊C3C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#495
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88530C2semidirect
3. #11721110C2semidirect
4. C2×#88438C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
26. C8×C3⋊Dic3D3semidirect
167. H(3,𝔽3)#495semidirect
246. C32#4696semidirect
247. C32#5155semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
C2×#3745⋊C32
#117211102
C2×#884382
C8×(C3⋊Dic3)⋊C32
#105908693
#105908653