Finite Group Databasebeta

← #13161333

Subgroups — #13161333

Order 1728.

1436 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1436 conjugacy classes of subgroups — too many to draw. Only its 144 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#11724682C211yesmaximal, characteristic
2C32×#944C211yesmaximal, characteristic, complemented
3#11723381C212yesmaximal, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5#11722048C212yesmaximal, complemented
6#10592794—31nomaximal
7#10591650—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9#89080—22no—
10Q8×C3⋊(C3⋊D3)—22no—
11#89076—22no—
12#89076C2212yes—
13C2×C32×Dic6C2211yescharacteristic
14C3×#3748C2212yes—
15C32×Dic3⋊C4—22no—
16C32×C4×Dic3C2211yescharacteristic
17#88799—22no—
18#88795—22no—
19#64364—31no—
20#64364—34no—
21#64364—64no—
22C32×C4⋊Q8—31no—
23#64151—32no—
24C3×#944D314yescomplemented
25#63833—38no—
26#63663D311yescharacteristic, complemented
27#63632—32no—
28C3×#836—34no—
29#63570—38no—
30C33×Q8—22no—
31C2×C33×C4D412yes—
32C2×C33×C4C2311yescharacteristic
33#6312—22no—
34C4⋊(C3⋊(C3⋊D3))—22no—
35C4⋊(C3⋊(C3⋊D3))D412yes—
36C4×C3⋊(C3⋊D3)—24no—
37C4×C3⋊(C3⋊D3)—22no—
38#6308—22no—
39#6308D412yes—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C32×Dic6—22no—
43C3×C3⋊(C3⋊C8)—22no—
44#5095—94no—
45#4752—91no—
46C2×C32×Q8—24no—
47C2×C32×Q8—31no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C32×Q8C2×D314yes—
50#3832—62no—
51#3832—68no—
52#3832—616no—
53Q8×C3⋊D3—62no—
54Q8×C3⋊D3—68no—
55Q8×C3⋊D3—616no—
56#3828—32no—
57#3828—38no—
58#3828—68no—
59#3828—616no—
60#3828—62no—
61#3828—68no—
62C2×C3×Dic6C2×D314yes—
63C32×C4⋊C4—62no—
64C32×C42—31no—
65#3756—62no—
66#3755—62no—
67#3748C2×D312yes—
68C3×Dic3⋊C4—28no—
69C3×C4×Dic3C2×D314yes—
70C3×#208—38no—
71#3719—68no—
72#3717—68no—
73C22×C33C2×D411yescharacteristic
74C2×C3⋊(C3⋊D3)—22no—
75C33×C4—22no—
76C33×C4C2×D412yes—
77C3⋊(C3⋊Dic3)—22no—
78C32×Dic3—22no—
79C32×Dic3—41no—
80#1007—94no—
81#1007—91no—
82#1007—184no—
83C3×C4⋊Q8—34no—
84#944C3⋊D311yescharacteristic, complemented
85#942—98no—
86#909—92no—
87C3×#101—91no—
88#836—34no—
89#825—92no—
90#805—98no—
91C32×Q8—216no—
92C32×Q8—28no—
93C32×Q8—22no—
94C32×Q8—62no—
95C2×C32×C4—28no—
96C2×C32×C4—24no—
97C2×C32×C4—61no—
98C2×C32×C4—61no—
99C2×C32×C4D1212yes—
100C2×C32×C4#6718yes—
101C2×C32×C4C22×D314yes—
102C2×C32×C4C22×D311yescharacteristic
103#638—616no—
104#638—62no—
105#638—68no—
106C4⋊(C3⋊D3)—32no—
107C4⋊(C3⋊D3)—38no—
108C4⋊(C3⋊D3)—68no—
109C4⋊(C3⋊D3)—616no—
110C4⋊(C3⋊D3)—62no—
111C4⋊(C3⋊D3)—68no—
112C4×C3⋊D3—632no—
113C4×C3⋊D3—64no—
114C4×C3⋊D3—616no—
115C4×C3⋊D3—616no—
116C4×C3⋊D3—68no—
117C4×C3⋊D3—62no—
118#634—32no—
119#634—38no—
120#634—68no—
121#634—616no—
122#634—62no—
123#634—68no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—24no—
126C3×Dic6—28no—
127C3⋊(C3⋊C8)—22no—
128C3×C3⋊C8—68no—
129#456—271noSylow
130C2×C33—21no—
131C2×C33#12011yescharacteristic
132C3⋊(C3⋊D3)—42no—
133C2×C3×Q8—24no—
134C2×C3×Q8—34no—
135C2×C3×Q8D3214yes—
136C2×C3×Q8C2×C3⋊D311yescharacteristic
137#239—188no—
138#239—182no—
139#239—1816no—
140D3×Q8—188no—
141D3×Q8—182no—
142D3×Q8—1816no—
143#235—98no—
144#235—92no—
145#235—188no—
146#235—1816no—
147#235—188no—
148#235—182no—
149C2×Dic6C2×C3⋊D311yescharacteristic
150C3×M16—92no—
151C3×C4⋊C4—68no—
152C3×C42—34no—
153#216—188no—
154#215—188no—
155Dic3⋊C4—22no—
156C4×Dic3C2×C3⋊D311yescharacteristic
157#208—38no—
158Dic12—182no—
159#204—182no—
160C22×C32—24no—
161C22×C32C2×D1211yescharacteristic
162C22×C32C2×#6714yes—
163C2×C3⋊D3—616no—
164C2×C3⋊D3—62no—
165C2×C3⋊D3—68no—
166C32×C4—216no—
167C32×C4—22no—
168C32×C4—28no—
169C32×C4—28no—
170C32×C4—62no—
171C32×C4—121no—
172C32×C4D3×D418yes—
173C32×C4D3×D412yes—
174C3⋊Dic3—616no—
175C3⋊Dic3—62no—
176C3⋊Dic3—68no—
177C3×Dic3—28no—
178C3×Dic3—44no—
1792−1+4—271no—
180#137—272no—
181C4⋊Q8—31no—
182#104—272no—
183#101—91no—
184C33#45611yescharacteristic, complemented, Sylow
185C3×Q8—216no—
186C3×Q8—28no—
187C3×Q8—22no—
188C3×Q8—68no—
189C2×C3×C4—28no—
190C2×C3×C4—24no—
191C2×C3×C4—64no—
192C2×C3×C4—64no—
193C2×C3×C4#62618yes—
194C2×C3×C4#63812yes—
195C2×C3×C4C2×D3214yes—
196C2×C3×C4C22×C3⋊D311yescharacteristic
197#67—1816no—
198#67—188no—
199#67—182no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202D12—98no—
203D12—92no—
204D12—188no—
205D12—1816no—
206D12—182no—
207D12—188no—
208C4×D3—1832no—
209C4×D3—184no—
210C4×D3—1816no—
211C4×D3—1816no—
212C4×D3—188no—
213C4×D3—182no—
214Dic6—22no—
215Dic6—98no—
216Dic6—92no—
217Dic6—188no—
218Dic6—1816no—
219Dic6—182no—
220Dic6—188no—
221C3×C8—182no—
222C3⋊C8—68no—
223C2×C32—21no—
224C2×C32—24no—
225C2×C32—24no—
226C2×C32—44no—
227C2×C32#88211yescharacteristic
228C2×C32#93714yes—
229C3⋊D3—1216no—
230C3⋊D3—122no—
231C3⋊D3—128no—
232#41—272no—
233#41—542no—
234#41—542no—
235C2×Q8—31no—
236C2×Q8—542no—
237C2×Q8D3×C3⋊D311yescharacteristic
238Q16—542no—
239SD16—542no—
240M16—92no—
241C4⋊C4—62no—
242C42—31no—
243C22×C3—24no—
244C22×C3C2×#62614yes—
245C22×C3C2×#63811yescharacteristic
246C2×D3—1816no—
247C2×D3—182no—
248C2×D3—188no—
249C3×C4—216no—
250C3×C4—22no—
251C3×C4—28no—
252C3×C4—28no—
253C3×C4—68no—
254C3×C4—124no—
255C3×C4#381118yes—
256C3×C4D4×C3⋊D312yes—
257Dic3—22no—
258Dic3—41no—
259Dic3—1816no—
260Dic3—182no—
261Dic3—188no—
262C32—24no—
263C32#475211yescharacteristic, complemented
264C32#509514yescomplemented
265Q8—22no—
266Q8—62no—
267Q8—272no—
268Q8—542no—
269D4—272no—
270D4—542no—
271D4—542no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—544no—
275C2×C4—542no—
276C2×C4#629112yes—
277C2×C4C2×D3×C3⋊D311yescharacteristic
278C8—182no—
279C2×C3—21no—
280C2×C3—24no—
281C2×C3—24no—
282C2×C3—44no—
283C2×C3#6398114yes—
284C2×C3#6414611yescharacteristic
285D3—3616no—
286D3—362no—
287D3—368no—
288C22—542no—
289C22C2×#629111yescharacteristic
290C4—22no—
291C4—62no—
292C4—121no—
293C4—542no—
294C4#8903912yes—
295C3—24no—
296C3#1059165014yescomplemented
297C3#1059279411yescharacteristic, complemented
298C2—21no—
299C2—1082no—
300C2#1172363811yescharacteristic
subgroupG/H
centerC2#11723638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#120
prime radical#11724682C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#456
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
2. C32×#944C2semidirect
3. #11723381C2semidirect
4. C3×#63663C2semidirect
5. #11722048C2semidirect
24. C3×#944D3semidirect
26. #63663D3semidirect
84. #944C3⋊D3semidirect
184. C33#456semidirect
263. C32#4752semidirect
264. C32#5095semidirect
296. C3#10591650semidirect
297. C3#10592794semidirect

Maximal subgroups

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

subgroupindex
#117246822
C32×#9442
#117233812
C3×#636632
#117220482
#105927943
#105916503