Finite Group Databasebeta

← #13147566

Subgroups — #13147566

Order 1728.

816 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721711C211yesmaximal, characteristic
2C4×C9×Dic6C211yesmaximal, characteristic, complemented
3C3×C4×Dic18C211yesmaximal, characteristic, complemented
4#11720997C214yesmaximal, complemented
5#11720993C212yesmaximal, complemented
6#11720992C212yesmaximal, complemented
7#11720953C212yesmaximal, complemented
8#11720952C212yesmaximal, complemented
9#10593963—31nomaximal
10#10589687—31nomaximal
11C2×C4⋊(C3⋊D9)C2212yes—
12C2×C4⋊(C3⋊D9)C2212yes—
13C2×C4⋊(C3⋊D9)C2212yes—
14C2×C9×Dic6C2211yescharacteristic
15C2×C3×Dic18C2211yescharacteristic
16C2×(C3⋊D9)⋊C4C2214yes—
17C3×C42×C9C2211yescharacteristic
18C9×C4⋊Dic3C2211yescharacteristic
19C9×Dic3⋊C4C2212yes—
20C4×C9×Dic3C2212yes—
21C3×C4⋊Dic9C2211yescharacteristic
22C3×Dic9⋊C4C2212yes—
23C3×C4×Dic9C2212yes—
24#88445C2214yes—
25#88445C2214yes—
26#88445C2214yes—
27#64083—31no—
28C3×C4×Dic6—31no—
29C3×C4×Dic6D311yescharacteristic
30#63897—34no—
31#63894—32no—
32#63894—32no—
33#63864—32no—
34#63864—32no—
35C4×Q8×C9—31no—
36#63509—32no—
37#63456—34no—
38#63455—32no—
39#63437—32no—
40#63436—31no—
41#63436—61no—
42C4×D36—32no—
43C4×Dic18D311yescharacteristic, complemented
44C22×C3⋊D9C2314yes—
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3×C4×C9C2311yescharacteristic
48C4⋊(C3⋊D9)—24no—
49C4⋊(C3⋊D9)—24no—
50C4⋊(C3⋊D9)—42no—
51C2×C9×Dic3C2312yes—
52C2×C9×Dic3C2312yes—
53C2×C3×Dic9C2312yes—
54C2×C3×Dic9C2312yes—
55C9×Dic6—22no—
56C3×Dic18—22no—
57(C3⋊D9)⋊C4—44no—
58#5585—91no—
59#5585—91no—
60C2×C4⋊(C3⋊D3)—32no—
61C2×C4⋊(C3⋊D3)—32no—
62C2×C4⋊(C3⋊D3)—32no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6C2×D311yescharacteristic
65C2×#624—34no—
66C32×C42C2×D311yescharacteristic
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3C2×D311yescharacteristic
69C3×Dic3⋊C4—32no—
70C3×Dic3⋊C4C2×D312yes—
71C3×C4×Dic3—32no—
72C3×C4×Dic3C2×D312yes—
73#3722—34no—
74#3722—34no—
75#3722—34no—
76C2×Q8×C9—31no—
77C2×D36—32no—
78C2×D36—32no—
79C2×D36—32no—
80C2×D36—62no—
81C2×D36—62no—
82C2×D36—62no—
83C2×C4×D9—34no—
84C2×Dic18C2×D311yescharacteristic
85C9×C4⋊C4—32no—
86C9×C4⋊C4—31no—
87C42×C9—21no—
88C42×C9—32no—
89C42×C9C2×D311yescharacteristic
90(C2×D9)⋊C4—34no—
91(C2×D9)⋊C4—34no—
92(C2×D9)⋊C4—34no—
93C4⋊Dic9C2×D311yescharacteristic
94Dic9⋊C4C2×D312yes—
95C4×Dic9C2×D312yes—
96C22×C3×C9C2411yescharacteristic
97C2×C3⋊D9—24no—
98C2×C3⋊D9—24no—
99C2×C3⋊D9—24no—
100C3×C4×C9—21no—
101C3×C4×C9#4112yes—
102C3×C4×C9#4112yes—
103C9×Dic3—22no—
104C9×Dic3—22no—
105C3×Dic9—22no—
106C3×Dic9—22no—
107C3×C4×Q8—31no—
108C3×C4×Q8—91no—
109#947—92no—
110#947—92no—
111#894—94no—
112#894—94no—
113#893—92no—
114#893—92no—
115#875—92no—
116#875—92no—
117#874—91no—
118#874—91no—
119#874—181no—
120C4×D12—92no—
121C4×D12—92no—
122C4×Dic6—31no—
123C4×Dic6D911yescharacteristic, complemented
124C22×C3⋊D3—34no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C4⋊(C3⋊D3)—64no—
129C4⋊(C3⋊D3)—64no—
130C4⋊(C3⋊D3)—122no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3C22×D312yes—
134C2×C3×Dic3C22×D312yes—
135C3×Dic6—22no—
136C3×Dic6—62no—
137#624—124no—
138C22×D9—34no—
139C22×D9—64no—
140Q8×C9—62no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—32no—
145C2×C4×C9—32no—
146C2×C4×C9C22×D311yescharacteristic
147C2×C4×C9C22×D311yescharacteristic
148C2×C4×C9C22×D311yescharacteristic
149C2×Dic9C22×D312yes—
150C2×Dic9C22×D312yes—
151D36—64no—
152D36—64no—
153D36—124no—
154D36—124no—
155D36—122no—
156D36—124no—
157C4×D9—124no—
158Dic18—22no—
159#559—271noSylow
160C2×C3×C9C2×#4111yescharacteristic
161C2×C3×C9C2×#4111yescharacteristic
162C2×C3×C9H(5,𝔽2)11yescharacteristic
163C3⋊D9—44no—
164C2×C3×Q8—31no—
165C2×C3×Q8—91no—
166C2×D12—92no—
167C2×D12—92no—
168C2×D12—92no—
169C2×D12—92no—
170C2×D12—92no—
171C2×D12—92no—
172C2×D12—182no—
173C2×D12—182no—
174C2×D12—182no—
175C2×C4×D3—94no—
176C2×C4×D3—94no—
177C2×Dic6—31no—
178C2×Dic6C2×D911yescharacteristic
179C3×C4⋊C4—31no—
180C3×C4⋊C4—32no—
181C3×C4⋊C4—91no—
182C3×C4⋊C4—92no—
183C3×C42—21no—
184C3×C42—32no—
185C3×C42—92no—
186C3×C42C2×D911yescharacteristic
187C3×C42D3211yescharacteristic
188(C2×D3)⋊C4—94no—
189(C2×D3)⋊C4—94no—
190(C2×D3)⋊C4—94no—
191(C2×D3)⋊C4—94no—
192(C2×D3)⋊C4—94no—
193(C2×D3)⋊C4—94no—
194C4⋊Dic3—31no—
195C4⋊Dic3C2×D911yescharacteristic
196Dic3⋊C4—32no—
197Dic3⋊C4C2×D912yes—
198C4×Dic3—32no—
199C4×Dic3C2×D912yes—
200C22×C32C23×D311yescharacteristic
201C2×C3⋊D3—64no—
202C2×C3⋊D3—64no—
203C2×C3⋊D3—64no—
204C32×C4—21no—
205C32×C4#23512yes—
206C32×C4#23912yes—
207C3×Dic3—22no—
208C3×Dic3—22no—
209C3×Dic3—62no—
210C3×Dic3—62no—
211C22×C9—21no—
212C22×C9C23×D311yescharacteristic
213C2×D9—64no—
214C2×D9—64no—
215C2×D9—64no—
216C2×D9—124no—
217C2×D9—124no—
218C2×D9—124no—
219C4×C9—22no—
220C4×C9—21no—
221C4×C9—22no—
222C4×C9—22no—
223C4×C9—62no—
224C4×C9—62no—
225C4×C9#23512yes—
226C4×C9#23912yes—
227Dic9—22no—
228Dic9—22no—
229#127—271no—
230#124—272no—
231#124—272no—
232#123—274no—
233C4×Q8—31no—
234C4×Q8—91no—
235C4×D4—272no—
236C4×D4—272no—
237C3×C9#55911yescharacteristic, complemented, Sylow
238C22×D3—94no—
239C22×D3—94no—
240C22×D3—184no—
241C3×Q8—62no—
242C3×Q8—182no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—92no—
249C2×C3×C4—92no—
250C2×C3×C4C22×D911yescharacteristic
251C2×C3×C4C22×D911yescharacteristic
252C2×C3×C4C22×D911yescharacteristic
253C2×C3×C4C2×D3211yescharacteristic
254C2×C3×C4C2×D3211yescharacteristic
255C2×C3×C4C2×D3211yescharacteristic
256C2×Dic3—32no—
257C2×Dic3—32no—
258C2×Dic3C22×D912yes—
259C2×Dic3C22×D912yes—
260D12—184no—
261D12—184no—
262D12—184no—
263D12—184no—
264D12—364no—
265D12—364no—
266D12—362no—
267D12—362no—
268D12—364no—
269C4×D3—364no—
270C4×D3—364no—
271Dic6—22no—
272Dic6—62no—
273C2×C32C2×#23511yescharacteristic
274C2×C32C2×#23911yescharacteristic
275C2×C32#100911yescharacteristic
276C3⋊D3—124no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9—21no—
280C2×C9C2×#23511yescharacteristic
281C2×C9C2×#23911yescharacteristic
282C2×C9#100911yescharacteristic
283D9—124no—
284D9—244no—
285C2×Q8—31no—
286C2×Q8—91no—
287C2×D4—272no—
288C2×D4—272no—
289C2×D4—272no—
290C22×C4—274no—
291C4⋊C4—31no—
292C4⋊C4—32no—
293C4⋊C4—91no—
294C4⋊C4—92no—
295C22⋊C4—274no—
296C22⋊C4—274no—
297C22⋊C4—274no—
298C42—32no—
299C42—92no—
300C42D3×D911yescharacteristic
301C22×C3—21no—
302C22×C3C23×D911yescharacteristic
303C22×C3C22×D3211yescharacteristic
304C2×D3—184no—
305C2×D3—184no—
306C2×D3—184no—
307C2×D3—184no—
308C2×D3—184no—
309C2×D3—184no—
310C2×D3—364no—
311C2×D3—364no—
312C2×D3—364no—
313C3×C4—22no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—62no—
319C3×C4—62no—
320C3×C4—182no—
321C3×C4—182no—
322C3×C4#369712yes—
323C3×C4#370112yes—
324C3×C4#379912yes—
325C3×C4#379912yes—
326Dic3—22no—
327Dic3—22no—
328Dic3—62no—
329Dic3—62no—
330C32#558511yescharacteristic
331C9—21no—
332C9#558511yescharacteristic, complemented
333C23—274no—
334Q8—62no—
335Q8—182no—
336D4—544no—
337D4—544no—
338D4—1082no—
339C2×C4—32no—
340C2×C4—32no—
341C2×C4—92no—
342C2×C4—92no—
343C2×C4—1084no—
344C2×C4C2×D3×D911yescharacteristic
345C2×C4C2×D3×D911yescharacteristic
346C2×C4C2×D3×D911yescharacteristic
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3C2×#369711yescharacteristic
351C2×C3C2×#370111yescharacteristic
352C2×C3#6371511yescharacteristic
353C2×C3C2×#379911yescharacteristic
354C2×C3C2×#379911yescharacteristic
355C2×C3#6430811yescharacteristic
356D3—364no—
357D3—364no—
358D3—724no—
359C22—544no—
360C22—544no—
361C22—544no—
362C22C22×D3×D911yescharacteristic
363C4—21no—
364C4—62no—
365C4—62no—
366C4—182no—
367C4—182no—
368C4#8863512yes—
369C4#8864212yes—
370C3—21no—
371C3#1058968711yescharacteristic, complemented
372C3#1059396311yescharacteristic
373C2—1084no—
374C2C2×#8863511yescharacteristic
375C2C2×#8864211yescharacteristic
376C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721711C2
hypercenterC42D3×D9
hypocenterC3×C9#559
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×C9×Dic6C2semidirect
3. C3×C4×Dic18C2semidirect
4. #11720997C2semidirect
5. #11720993C2semidirect
6. #11720992C2semidirect
7. #11720953C2semidirect
8. #11720952C2semidirect
43. C4×Dic18D3semidirect
123. C4×Dic6D9semidirect
237. C3×C9#559semidirect
332. C9#5585semidirect
371. C3#10589687semidirect

Maximal subgroups

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

subgroupindex
#117217112
C4×C9×Dic62
C3×C4×Dic182
#117209972
#117209932
#117209922
#117209532
#117209522
#105939633
#105896873