Finite Group Databasebeta

← #13161320

Subgroups — #13161320

Order 1728.

1028 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 164 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×#1007C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic, complemented
3#11723365C212yesmaximal, complemented
4#11722206C211yesmaximal, characteristic, complemented
5#11722044C212yesmaximal, complemented
6#10592944—31nomaximal
7#10591645—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C32×#239—22no—
11C32×D3×Q8—22no—
12C32×#235—22no—
13C32×#235C2212yes—
14#88855C2212yes—
15C3×#3752—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17#88793—22no—
18#88789—22no—
19C32×2−1+4—31no—
20C3×#1007D314yescomplemented
21#64158—32no—
22#64156D311yescharacteristic, complemented
23C3×#947—34no—
24#63823—38no—
25#63664—32no—
26#63663—31no—
27#63663—34no—
28#63663—64no—
29#63569—38no—
30C22×C3×C3⋊D3—21no—
31C33×Q8—22no—
32C2×C33×C4D412yes—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C3×C4⋊(C3⋊D3)—22no—
36C32×#67—22no—
37C32×D12—22no—
38C32×D12D412yes—
39C32×C4×D3—24no—
40C32×C4×D3—22no—
41C32×Dic6—22no—
42C32×Dic6D412yes—
43C3⋊(C3⋊(C3⋊C8))—22no—
44#5245—94no—
45#5062—91no—
46C32×#41—32no—
47C32×#41—62no—
48C32×#41—62no—
49C2×C32×Q8—24no—
50C2×C32×Q8—62no—
51C2×C32×Q8C2×D314yes—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C3×#239—28no—
55C3×D3×Q8—28no—
56C3×#235—28no—
57C3×#235C2×D318yes—
58C2×C3×D12—34no—
59#3755—62no—
60#3753—62no—
61#3752—22no—
62C4×C3⋊Dic3C2×D311yescharacteristic
63#3748—32no—
64#3748—38no—
65#3748—68no—
66C3×(C2×D3)⋊C4—68no—
67C3×C4×Dic3—34no—
68#3715—68no—
69#3713—68no—
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—42no—
72C2×C3×C3⋊D3—41no—
73C2×C32×D3—22no—
74C33×C4—22no—
75C33×C4C2×D412yes—
76C3×C3⋊Dic3—41no—
77C32×Dic3—22no—
78C3×2−1+4—34no—
79#1007C3⋊D311yescharacteristic, complemented
80C3×#124—91no—
81#949—98no—
82#947—34no—
83#932—92no—
84#837—98no—
85#836—94no—
86#836—91no—
87#836—184no—
88#805—92no—
89C22×C3⋊D3—21no—
90C22×C3×D3—64no—
91C32×Q8—22no—
92C32×Q8—28no—
93C32×Q8—32no—
94C32×Q8—48no—
95C32×Q8—62no—
96C32×D4—32no—
97C32×D4—62no—
98C32×D4—62no—
99C2×C32×C4—28no—
100C2×C32×C4—24no—
101C2×C32×C4—64no—
102C2×C32×C4—62no—
103C2×C32×C4#6718yes—
104C2×C32×C4#6712yes—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—21no—
108C4⋊(C3⋊D3)—22no—
109C3×#67—28no—
110C2×C3×Dic3—64no—
111C3×D12—28no—
112C3×D12—68no—
113C3×D12#6718yes—
114C3×C4×D3—216no—
115C3×C4×D3—28no—
116C3×Dic6—28no—
117C3×Dic6#6718yes—
118C3⋊(C3⋊C8)—62no—
119C3⋊(C3⋊C8)—68no—
120C3⋊(C3⋊C8)—128no—
121#454—271noSylow
122C2×C33—21no—
123C2×C33#12011yescharacteristic
124C3×C3⋊D3—81no—
125C32×D3—42no—
126C3×#41—38no—
127C3×#41—68no—
128C3×#41—68no—
129C2×C3×Q8—24no—
130C2×C3×Q8—68no—
131C2×C3×Q8D3214yes—
132C2×C3×Q8C2×C3⋊D311yescharacteristic
133C2×C3×D4—91no—
134#239—22no—
135D3×Q8—22no—
136#235—22no—
137#235C2×C3⋊D312yes—
138C2×D12—34no—
139C3×C22⋊C4—182no—
140C3×C42—91no—
141#215—188no—
142#214—182no—
143#213—188no—
144#213—182no—
145(C2×D3)⋊C4—68no—
146C4×Dic3—34no—
147#208—98no—
148#208—92no—
149#208—188no—
150C22×C32—24no—
151C22×C32—62no—
152C22×C32C2×#6714yes—
153C22×C32C2×#6711yescharacteristic
154C2×C3⋊D3—42no—
155C2×C3⋊D3—41no—
156C2×C3×D3—28no—
157C2×C3×D3—128no—
158C2×C3×D3—124no—
159C32×C4—22no—
160C32×C4—28no—
161C32×C4—28no—
162C32×C4—48no—
163C32×C4—62no—
164C32×C4D3×D412yes—
165C32×C4C2×#6718yes—
166C3⋊Dic3—41no—
167C3×Dic3—28no—
168C3×Dic3—124no—
1692−1+4—31no—
170AGL(1,ℤ/8)—272no—
171#124—91no—
172#104—272no—
173#101—271no—
174C33#45411yescharacteristic, complemented, Sylow
175C23×C3—181no—
176C22×D3—64no—
177C3×Q8—28no—
178C3×Q8—22no—
179C3×Q8—38no—
180C3×Q8—48no—
181C3×Q8—68no—
182C3×D4—38no—
183C3×D4—68no—
184C3×D4—68no—
185C3×D4—182no—
186C2×C3×C4—28no—
187C2×C3×C4—24no—
188C2×C3×C4—616no—
189C2×C3×C4—68no—
190C2×C3×C4—181no—
191C2×C3×C4#62518yes—
192C2×C3×C4#63812yes—
193C2×C3×C4C2×D3214yes—
194C2×C3×C4C22×C3⋊D311yescharacteristic
195#67—22no—
196C2×Dic3—64no—
197D12—22no—
198D12—68no—
199D12#63812yes—
200C4×D3—24no—
201C4×D3—22no—
202Dic6—22no—
203Dic6#63812yes—
204C3⋊C8—188no—
205C3⋊C8—182no—
206C3⋊C8—368no—
207C2×C32—21no—
208C2×C32—24no—
209C2×C32—24no—
210C2×C32—44no—
211C2×C32—122no—
212C2×C32#93711yescharacteristic
213C2×C32#95314yes—
214C3⋊D3—81no—
215C3×D3—48no—
216C3×D3—244no—
217#41—32no—
218#41—62no—
219#41—62no—
220C2×Q8—62no—
221C2×Q8D3×C3⋊D311yescharacteristic
222C2×D4—91no—
223SD16—542no—
224D8—542no—
225M16—272no—
226C22⋊C4—182no—
227C42—91no—
228C22×C3—24no—
229C22×C3—68no—
230C22×C3—362no—
231C22×C3—361no—
232C22×C3C2×#62514yes—
233C22×C3C2×#63811yescharacteristic
234C2×D3—22no—
235C2×D3—128no—
236C2×D3—124no—
237C3×C4—22no—
238C3×C4—28no—
239C3×C4—28no—
240C3×C4—48no—
241C3×C4—68no—
242C3×C4—361no—
243C3×C4D3×#6718yes—
244C3×C4C2×#63812yes—
245Dic3—22no—
246Dic3—124no—
247C32—24no—
248C32#506211yescharacteristic, complemented
249C32#524514yescomplemented
250C23—181no—
251Q8—22no—
252Q8—32no—
253Q8—62no—
254D4—32no—
255D4—62no—
256D4—62no—
257D4—182no—
258C2×C4—64no—
259C2×C4—62no—
260C2×C4—181no—
261C2×C4#629012yes—
262C2×C4C2×D3×C3⋊D311yescharacteristic
263C8—542no—
264C2×C3—24no—
265C2×C3—21no—
266C2×C3—24no—
267C2×C3—44no—
268C2×C3—128no—
269C2×C3—721no—
270C2×C3#6397614yes—
271C2×C3#6416211yescharacteristic
272D3—42no—
273D3—244no—
274C22—62no—
275C22—362no—
276C22—361no—
277C22C2×#629011yescharacteristic
278C4—22no—
279C4—62no—
280C4—361no—
281C4D3×#63812yes—
282C3—24no—
283C3#1059164514yescomplemented
284C3#1059294411yescharacteristic, complemented
285C2—21no—
286C2—122no—
287C2—721no—
288C2#1172362911yescharacteristic
subgroupG/H
centerC2#11723629
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#120
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#454
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C32×#1007C2semidirect
2. C3×#64156C2semidirect
3. #11723365C2semidirect
4. #11722206C2semidirect
5. #11722044C2semidirect
20. C3×#1007D3semidirect
22. #64156D3semidirect
79. #1007C3⋊D3semidirect
174. C33#454semidirect
248. C32#5062semidirect
249. C32#5245semidirect
283. C3#10591645semidirect
284. C3#10592944semidirect

Maximal subgroups

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

subgroupindex
C32×#10072
C3×#641562
#117233652
#117222062
#117220442
#105929443
#105916453