Finite Group Databasebeta

← #13161101

Subgroups — #13161101

Order 1728.

1096 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1096 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×#1004C211yesmaximal, characteristic
2C3×#64145C211yesmaximal, characteristic, complemented
3#11723373C212yesmaximal, complemented
4#11722203C211yesmaximal, characteristic, complemented
5#11722044C212yesmaximal, complemented
6#10592941—31nomaximal
7#10591512—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic, complemented
10C2×C32×#67—22no—
11C32×#237—22no—
12C32×D3×D4—22no—
13C32×#235C2212yes—
14#88855C2212yescomplemented
15C3×#3757—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
17#88798—22no—
18#88792—22no—
19C32×H(5,𝔽2)—31no—
20C3×#1004D314yes—
21#64160—32no—
22#64145D311yescharacteristic, complemented
23C3×#936—34no—
24#63827—38no—
25#63664—32no—
26#63660—31no—
27#63660—34no—
28#63660—64no—
29#63569—38no—
30C23×C33D412yes—
31C22×C32×D3—22no—
32C33×D4—22no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C3×#634—22no—
37C32×#67—24no—
38C32×#67—22no—
39C2×C32×Dic3—22no—
40C32×D12D412yes—
41C32×C4×D3—22no—
42C32×Dic6D412yes—
43C3⋊(C3⋊(C3⋊C8))—22no—
44#5242—94no—
45#5061—91no—
46C32×#41—32no—
47C32×#41—62no—
48C2×C32×D4—24no—
49C2×C32×D4—62no—
50C2×C32×D4—62no—
51C2×C32×D4C2×D314yes—
52C2×C32×D4C2×D311yescharacteristic
53C2×#634C2×D311yescharacteristic, complemented
54C2×C3×#67—28no—
55C3×#237—28no—
56C3×D3×D4—28no—
57C3×#235C2×D318yes—
58C2×C3×Dic6—34no—
59#3757—22no—
60#3756—62no—
61#3754—62no—
62C4×C3⋊Dic3C2×D311yescharacteristic, complemented
63#3748—32no—
64#3748—38no—
65#3748—68no—
66C3×#217—68no—
67C3×C4×Dic3—34no—
68#3718—68no—
69#3715—68no—
70C22×C33—22no—
71C22×C33—42no—
72C22×C33C2×D411yescharacteristic
73C2×C32×D3—22no—
74C2×C32×D3—82no—
75C33×C4C2×D412yes—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3—41no—
78C32×Dic3—22no—
79C3×H(5,𝔽2)—34no—
80#1004C3⋊D311yescharacteristic
81C3×#124—91no—
82#951—98no—
83#942—92no—
84#936—34no—
85#837—98no—
86#833—94no—
87#833—91no—
88#833—184no—
89#805—92no—
90C23×C32—28no—
91C23×C32—62no—
92C23×C32#6718yes—
93C23×C32#6712yes—
94C22×C3×D3—28no—
95C32×Q8—32no—
96C32×D4—22no—
97C32×D4—28no—
98C32×D4—32no—
99C32×D4—48no—
100C32×D4—64no—
101C32×D4—62no—
102C2×C32×C4—24no—
103C2×C32×C4—62no—
104C2×C32×C4—62no—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—21no—
109#634—22no—
110C3×#67—216no—
111C3×#67—28no—
112C2×C3×Dic3—28no—
113C2×C3×Dic3—64no—
114C2×C3×Dic3—64no—
115C3×D12#6718yes—
116C3×C4×D3—28no—
117C3×Dic6—68no—
118C3×Dic6#6718yes—
119C3⋊(C3⋊C8)—62no—
120C3⋊(C3⋊C8)—68no—
121C3⋊(C3⋊C8)—128no—
122#455—271noSylow
123C2×C33—21no—
124C2×C33—42no—
125C2×C33#12011yescharacteristic
126C32×D3—42no—
127C3×#41—38no—
128C3×#41—68no—
129C2×C3×Q8—91no—
130C2×C3×D4—24no—
131C2×C3×D4—68no—
132C2×C3×D4—68no—
133C2×C3×D4D3214yes—
134C2×C3×D4C2×C3⋊D311yescharacteristic
135C2×#67—22no—
136#237—22no—
137D3×D4—22no—
138#235C2×C3⋊D312yes—
139C2×Dic6—34no—
140C3×C22⋊C4—182no—
141C3×C42—91no—
142#217—68no—
143#216—188no—
144#216—182no—
145#215—182no—
146#214—188no—
147C4×Dic3—34no—
148#208—98no—
149#208—92no—
150#208—188no—
151C22×C32—22no—
152C22×C32—24no—
153C22×C32—28no—
154C22×C32—48no—
155C22×C32—42no—
156C22×C32—48no—
157C22×C32—62no—
158C22×C32—88no—
159C22×C32—242no—
160C22×C32C2×#6714yes—
161C22×C32C2×#6711yescharacteristic
162C2×C3×D3—28no—
163C2×C3×D3—88no—
164C32×C4—28no—
165C32×C4—62no—
166C32×C4D3×D412yes—
167C32×C4C2×#6718yes—
168C3⋊Dic3—41no—
169C3⋊Dic3—41no—
170C3×Dic3—28no—
171C3×Dic3—124no—
172C3×Dic3—124no—
173H(5,𝔽2)—31no—
174#137—272no—
175#124—91no—
176#104—272no—
177#100—271no—
178C33#45511yescharacteristic, complemented, Sylow
179C23×C3—28no—
180C23×C3—68no—
181C23×C3#62518yes—
182C23×C3#63812yes—
183C22×D3—22no—
184C3×Q8—38no—
185C3×Q8—182no—
186C3×D4—28no—
187C3×D4—22no—
188C3×D4—38no—
189C3×D4—48no—
190C3×D4—616no—
191C3×D4—68no—
192C2×C3×C4—24no—
193C2×C3×C4—68no—
194C2×C3×C4—68no—
195C2×C3×C4—181no—
196C2×C3×C4—181no—
197C2×C3×C4C2×D3214yes—
198C2×C3×C4C22×C3⋊D311yescharacteristic
199#67—24no—
200#67—22no—
201C2×Dic3—22no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204D12#63812yes—
205C4×D3—22no—
206Dic6—68no—
207Dic6#63812yes—
208C3⋊C8—188no—
209C3⋊C8—182no—
210C3⋊C8—368no—
211C2×C32—21no—
212C2×C32—24no—
213C2×C32—24no—
214C2×C32—44no—
215C2×C32—42no—
216C2×C32—48no—
217C2×C32—88no—
218C2×C32—122no—
219C2×C32#93711yescharacteristic
220C2×C32#95314yes—
221C3×D3—48no—
222#41—32no—
223#41—62no—
224C2×Q8—91no—
225C2×D4—62no—
226C2×D4—62no—
227C2×D4D3×C3⋊D311yescharacteristic
228Q16—542no—
229SD16—542no—
230M16—272no—
231C22⋊C4—182no—
232C42—91no—
233C22×C3—22no—
234C22×C3—28no—
235C22×C3—24no—
236C22×C3—48no—
237C22×C3—48no—
238C22×C3—42no—
239C22×C3—68no—
240C22×C3—88no—
241C22×C3—248no—
242C22×C3C2×#62514yes—
243C22×C3C2×#63811yescharacteristic
244C2×D3—22no—
245C2×D3—82no—
246C3×C4—28no—
247C3×C4—68no—
248C3×C4—361no—
249C3×C4—361no—
250C3×C4D3×#6718yes—
251C3×C4C2×#63812yes—
252Dic3—22no—
253Dic3—124no—
254Dic3—124no—
255C32—24no—
256C32#506111yescharacteristic, complemented
257C32#524214yescomplemented
258C23—62no—
259C23#629012yes—
260Q8—32no—
261Q8—182no—
262D4—22no—
263D4—32no—
264D4—64no—
265D4—62no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4C2×D3×C3⋊D311yescharacteristic
271C8—542no—
272C2×C3—24no—
273C2×C3—21no—
274C2×C3—24no—
275C2×C3—44no—
276C2×C3—48no—
277C2×C3—42no—
278C2×C3—88no—
279C2×C3—128no—
280C2×C3#6397614yes—
281C2×C3#6416211yescharacteristic
282D3—42no—
283C22—22no—
284C22—42no—
285C22—62no—
286C22—242no—
287C22C2×#629011yescharacteristic
288C4—62no—
289C4—361no—
290C4—361no—
291C4D3×#63812yes—
292C3—24no—
293C3#1059151214yescomplemented
294C3#1059294111yescharacteristic, complemented
295C2—21no—
296C2—42no—
297C2—122no—
298C2#1172362911yescharacteristic
subgroupG/H
centerC2#11723629
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#120
prime radicalC32×#1004C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64145C2semidirect
3. #11723373C2semidirect
4. #11722203C2semidirect
5. #11722044C2semidirect
9. C2×C3×#634C22semidirect
14. #88855C22semidirect
16. C3×C4×C3⋊Dic3C22semidirect
22. #64145D3semidirect
53. C2×#634C2×D3semidirect
62. C4×C3⋊Dic3C2×D3semidirect
178. C33#455semidirect
256. C32#5061semidirect
257. C32#5242semidirect
293. C3#10591512semidirect
294. C3#10592941semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
C3×#641452
#117233732
#117222032
#117220442
#105929413
#105915123