Finite Group Databasebeta

← #13150256

Subgroups — #13150256

Order 1728.

1094 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic
2C2×C4×C9×Dic3C211yesmaximal, characteristic, complemented
3C3×#63488C211yesmaximal, characteristic, complemented
4#11721240C214yesmaximal, complemented
5#11720993C212yesmaximal, complemented
6C4×(C3⋊D9)⋊C4C212yesmaximal, complemented
7#11720931C212yesmaximal, complemented
8#11720912C212yesmaximal, complemented
9#10594954—31nomaximal
10C4×#3697—31nomaximal
11C22×C3×C4×C9C2211yescharacteristic
12C2×#6233C2212yes—
13#88736C418yescomplemented
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×#6226C2211yescharacteristic
17C22×C9×Dic3C2212yes—
18C2×(C3⋊D9)⋊C4C2214yes—
19C3×#3676C2212yes—
20C4×C9×Dic3C2211yescharacteristic
21C4×C9×Dic3C2212yes—
22C4×C9×Dic3C2211yescharacteristic
23C3×C4⋊Dic9C2212yes—
24C3×C4×Dic9C2212yes—
25#88445C2214yes—
26#88442C2214yes—
27Dic3×Dic9C2214yes—
28C2×#3828—31no—
29C3×#926—31no—
30C2×C3×C4×Dic3D311yescharacteristic
31#63952—34no—
32#63894—32no—
33C4×#624—32no—
34#63874—32no—
35#63844—32no—
36C2×#3697—31no—
37C2×#3697—61no—
38C2×C42×C9—31no—
39C4×#611—34no—
40#63488D311yescharacteristic, complemented
41C4×D36—32no—
42#63434—32no—
43C42×D9—32no—
44C4×Dic18—32no—
45C23×C3×C9C2311yescharacteristic
46C22×C3⋊D9C2312yes—
47C2×C3×C4×C9C2×C414yescomplemented
48C2×C3×C4×C9C2311yescharacteristic
49C2×C3×C4×C9C2311yescharacteristic
50#6233C2×C418yescomplemented
51C2×C3⋊Dic9C2312yes—
52C4⋊(C3⋊D9)C2×C414yescomplemented
53C4×C3⋊D9C2×C418yescomplemented
54#6226C2×C414yescomplemented
55C2×C9×Dic3—24no—
56C2×C9×Dic3C2312yes—
57C2×C9×Dic3C2312yes—
58C2×C3×Dic9C2314yes—
59(C3⋊D9)⋊C4—28no—
60#5522—91no—
61C4×#235—91no—
62C22×C32×C4C2×D311yescharacteristic
63C2×#638—32no—
64#3828—38no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4×C3⋊D3—32no—
67C2×#634—31no—
68C22×C3×Dic3C2×D312yes—
69C2×#624—34no—
70C3×#217—32no—
71C3×C4⋊Dic3—32no—
72C3×C4×Dic3—32no—
73C3×C4×Dic3C2×D312yes—
74C3×C4×Dic3C2×D311yescharacteristic
75C3×C4×Dic3C2×D311yescharacteristic
76#3724—34no—
77#3722—34no—
78Dic32—34no—
79C22×C4×C9—21no—
80C22×C4×C9—32no—
81C22×C4×C9C2×D311yescharacteristic
82C2×#611—32no—
83C2×#611—62no—
84#3697—38no—
85#3697—68no—
86C2×D36—31no—
87C2×D36—61no—
88C2×C4×D9—32no—
89C2×C4×D9—34no—
90C2×C4×D9—62no—
91C2×Dic18—31no—
92C2×Dic18—61no—
93C42×C9—32no—
94C42×C9—31no—
95C42×C9—31no—
96#3676C2×D312yes—
97(C2×D9)⋊C4—34no—
98C4⋊Dic9C2×D312yes—
99Dic9⋊C4—34no—
100C4×Dic9—34no—
101C4×Dic9C2×D312yes—
102C22×C3×C9—22no—
103C22×C3×C9C22×C412yes—
104C22×C3×C9C2411yescharacteristic
105C2×C3⋊D9—24no—
106C2×C3⋊D9C22×C414yes—
107C3×C4×C9C22×C412yes—
108C3×C4×C9C22×C412yes—
109C3⋊Dic9C22×C414yes—
110C9×Dic3—22no—
111C9×Dic3#4114yescomplemented
112C3×Dic9—24no—
113C2×#235—91no—
114C2×#235—91no—
115C2×#235—181no—
116C3×#117—91no—
117C2×C3×C42—31no—
118C4×#67—94no—
119#926—31no—
120C2×C4×Dic3D911yescharacteristic, complemented
121#893—92no—
122#887—92no—
123#67⋊C4—94no—
124C4×D12—92no—
125#872—92no—
126#872—92no—
127C42×D3—92no—
128C42×D3—92no—
129C4×Dic6—92no—
130C23×C32C22×D311yescharacteristic
131C22×C3⋊D3—32no—
132C2×C32×C4C4×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135#638—38no—
136C2×C3⋊Dic3—32no—
137C4⋊(C3⋊D3)—34no—
138C4×C3⋊D3—38no—
139#634—34no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3C22×D312yes—
143C2×C3×Dic3C22×D312yes—
144#624—68no—
145C23×C9—21no—
146C23×C9C22×D311yescharacteristic
147C22×D9—32no—
148C22×D9—62no—
149C2×C4×C9—24no—
150C2×C4×C9—21no—
151C2×C4×C9—21no—
152C2×C4×C9—32no—
153C2×C4×C9—32no—
154C2×C4×C9—64no—
155C2×C4×C9C4×D314yescomplemented
156C2×C4×C9C22×D311yescharacteristic
157C2×C4×C9C22×D311yescharacteristic
158#611—38no—
159#611—68no—
160C2×Dic9—32no—
161C2×Dic9—62no—
162C2×Dic9C22×D314yes—
163D36—34no—
164D36—64no—
165C4×D9—38no—
166C4×D9—68no—
167C4×D9—68no—
168Dic18—34no—
169Dic18—64no—
170C4×#41—271noSylow
171C2×C3×C9—22no—
172C2×C3×C9C23×C411yescharacteristic
173C2×C3×C9C2×#4112yes—
174C3⋊D9—24no—
175C22×C3×C4—21no—
176C22×C3×C4—32no—
177C22×C3×C4C2×D911yescharacteristic
178C22×C3×C4D3211yescharacteristic
179C2×#67—92no—
180C2×#67—92no—
181C2×#67—182no—
182C22×Dic3C2×D912yes—
183#235—98no—
184#235—98no—
185#235—188no—
186C2×D12—91no—
187C2×D12—91no—
188C2×D12—181no—
189C2×C4×D3—92no—
190C2×C4×D3—92no—
191C2×C4×D3—94no—
192C2×C4×D3—94no—
193C2×C4×D3—182no—
194C2×Dic6—91no—
195C2×Dic6—91no—
196C2×Dic6—181no—
197C3×C4⋊C4—92no—
198C3×C22⋊C4—92no—
199C3×C42—32no—
200C3×C42—31no—
201C3×C42—31no—
202C3×C42—92no—
203#217—32no—
204(C2×D3)⋊C4—94no—
205(C2×D3)⋊C4—94no—
206C4⋊Dic3—32no—
207Dic3⋊C4—94no—
208Dic3⋊C4—94no—
209C4×Dic3—32no—
210C4×Dic3—94no—
211C4×Dic3—94no—
212C4×Dic3C2×D911yescharacteristic
213C4×Dic3C2×D911yescharacteristic
214C4×Dic3C2×D912yes—
215C22×C32—22no—
216C22×C32C2×C4×D312yes—
217C22×C32C23×D311yescharacteristic
218C2×C3⋊D3—34no—
219C2×C3⋊D3—64no—
220C32×C4C2×C4×D312yes—
221C32×C4C2×C4×D312yes—
222C3⋊Dic3—34no—
223C3×Dic3—22no—
224C3×Dic3—64no—
225C3×Dic3#23514yes—
226C22×C9—22no—
227C22×C9—21no—
228C22×C9—22no—
229C22×C9—42no—
230C22×C9C2×C4×D312yes—
231C22×C9C23×D311yescharacteristic
232C2×D9—34no—
233C2×D9—64no—
234C2×D9—64no—
235C2×D9—124no—
236C4×C9—22no—
237C4×C9—22no—
238C4×C9—34no—
239C4×C9—62no—
240C4×C9C2×C4×D312yes—
241C4×C9C2×C4×D312yes—
242Dic9—24no—
243Dic9—34no—
244Dic9—64no—
245C2×#41—271no—
246C4×Q8—272no—
247C4×D4—272no—
248C4×D4—274no—
249#117—91no—
250#117—272no—
251C2×C42—31no—
252C2×C42—272no—
253C3×C9C4×#4111yescharacteristic, complemented, Sylow
254C23×C3—21no—
255C23×C3C22×D911yescharacteristic
256C23×C3C2×D3211yescharacteristic
257C22×D3—92no—
258C22×D3—92no—
259C22×D3—182no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—32no—
264C2×C3×C4—32no—
265C2×C3×C4—64no—
266C2×C3×C4—94no—
267C2×C3×C4C4×D914yescomplemented
268C2×C3×C4C22×D911yescharacteristic
269C2×C3×C4C22×D911yescharacteristic
270C2×C3×C4#62414yes—
271C2×C3×C4C2×D3211yescharacteristic
272C2×C3×C4C2×D3211yescharacteristic
273#67—98no—
274#67—98no—
275#67—188no—
276C2×Dic3—24no—
277C2×Dic3—34no—
278C2×Dic3—92no—
279C2×Dic3—92no—
280C2×Dic3—182no—
281C2×Dic3C22×D912yes—
282C2×Dic3C22×D912yes—
283D12—94no—
284D12—94no—
285D12—184no—
286C4×D3—98no—
287C4×D3—98no—
288C4×D3—188no—
289C4×D3—188no—
290C4×D3—188no—
291Dic6—94no—
292Dic6—94no—
293Dic6—184no—
294C2×C32—22no—
295C2×C32C22×C4×D311yescharacteristic
296C2×C32C2×#23512yes—
297C3⋊D3—64no—
298C2×C9—22no—
299C2×C9—22no—
300C2×C9—21no—
301C2×C9—42no—
302C2×C9C22×C4×D311yescharacteristic
303C2×C9D3×#4112yes—
304D9—64no—
305D9—124no—
306#41—278no—
307C2×Q8—271no—
308C2×D4—272no—
309C2×D4—271no—
310C22×C4—32no—
311C22×C4—274no—
312C22×C4—272no—
313C22×C4D3×D911yescharacteristic
314C4⋊C4—92no—
315C4⋊C4—274no—
316C22⋊C4—92no—
317C22⋊C4—274no—
318C42—32no—
319C42—31no—
320C42—31no—
321C42—92no—
322C42—274no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—42no—
328C22×C3C2×C4×D912yes—
329C22×C3C23×D911yescharacteristic
330C22×C3C2×#62412yes—
331C22×C3C22×D3211yescharacteristic
332C2×D3—94no—
333C2×D3—94no—
334C2×D3—184no—
335C2×D3—184no—
336C2×D3—184no—
337C2×D3—364no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—34no—
341C3×C4—62no—
342C3×C4—184no—
343C3×C4C2×C4×D912yes—
344C3×C4C2×C4×D912yes—
345C3×C4C2×#62412yes—
346C3×C4C2×#62412yes—
347Dic3—22no—
348Dic3—64no—
349Dic3—94no—
350Dic3—94no—
351Dic3—184no—
352Dic3#369714yescomplemented
353C32C4×#23511yescharacteristic
354C9—21no—
355C9#552211yescharacteristic, complemented
356C23—272no—
357C23C2×D3×D911yescharacteristic
358Q8—274no—
359D4—278no—
360D4—274no—
361C2×C4—32no—
362C2×C4—32no—
363C2×C4—64no—
364C2×C4—94no—
365C2×C4—272no—
366C2×C4—278no—
367C2×C4—548no—
368C2×C4(C3⋊D9)⋊C414yescomplemented
369C2×C4C2×D3×D911yescharacteristic
370C2×C4C2×D3×D911yescharacteristic
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—42no—
376C2×C3C22×C4×D911yescharacteristic
377C2×C3C2×#369712yes—
378C2×C3D3×#23512yes—
379C2×C3C22×#62411yescharacteristic
380D3—184no—
381D3—184no—
382D3—364no—
383C22—22no—
384C22—274no—
385C22—544no—
386C22C2×(C3⋊D9)⋊C412yes—
387C22C22×D3×D911yescharacteristic
388C4—34no—
389C4—62no—
390C4—184no—
391C4—274no—
392C4C2×(C3⋊D9)⋊C412yes—
393C4C2×(C3⋊D9)⋊C412yes—
394C3—21no—
395C3C4×#369711yescharacteristic, complemented
396C3#1059495411yescharacteristic
397C2—22no—
398C2—544no—
399C2D3×#369712yes—
400C2C22×(C3⋊D9)⋊C411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×#88736C2
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C4×#41
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C9×Dic3C2semidirect
3. C3×#63488C2semidirect
4. #11721240C2semidirect
5. #11720993C2semidirect
6. C4×(C3⋊D9)⋊C4C2semidirect
7. #11720931C2semidirect
8. #11720912C2semidirect
13. #88736C4semidirect
40. #63488D3semidirect
47. C2×C3×C4×C9C2×C4semidirect
50. #6233C2×C4semidirect
52. C4⋊(C3⋊D9)C2×C4semidirect
53. C4×C3⋊D9C2×C4semidirect
54. #6226C2×C4semidirect
111. C9×Dic3#41semidirect
120. C2×C4×Dic3D9semidirect
155. C2×C4×C9C4×D3semidirect
253. C3×C9C4×#41semidirect
267. C2×C3×C4C4×D9semidirect
352. Dic3#3697semidirect
355. C9#5522semidirect
368. C2×C4(C3⋊D9)⋊C4semidirect
395. C3C4×#3697semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×C4×C9×Dic32
C3×#634882
#117212402
#117209932
C4×(C3⋊D9)⋊C42
#117209312
#117209122
#105949543
C4×#36973