Finite Group Databasebeta

← #13140413

Subgroups — #13140413

Order 1728.

526 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic
2#11721244C212yesmaximal, complemented
3#11720577C211yesmaximal, characteristic, complemented
4C3×#63393C211yesmaximal, characteristic, complemented
5#11720332C212yesmaximal, complemented
6#10591528—31nomaximal
7#10588920—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×C9×#67C411yescharacteristic, complemented
10C2×C9×#67C2211yescharacteristic
11C22×C9×Dic3C411yescharacteristic, complemented
12C22×C9×Dic3C2211yescharacteristic
13C9×#237—22no—
14C9×#237—22no—
15C2×C4×D3×C9C411yescharacteristic, complemented
16C2×C9×Dic6C411yescharacteristic, complemented
17#88551C2212yes—
18C3×#3676C2212yes—
19Dic3⋊Dic9—22no—
20Dic3×Dic9—22no—
21C2×C3×#237D311yescharacteristic
22#63955—32no—
23C2×C9×#41—31no—
24#63661—31no—
25C3×#834—31no—
26#63580—32no—
27#63488—32no—
28#63393—31no—
29#63393—32no—
30#63393—61no—
31#63393D311yescharacteristic, complemented
32C23×C3×C9C2×C411yescharacteristic
33C23×C3×C9C2311yescharacteristic
34C22×D3×C9C2×C411yescharacteristic
35C3×D4×C9—22no—
36C2×C3×C4×C9C2×C411yescharacteristic
37C2×C3⋊Dic9—22no—
38C9×#67—22no—
39C9×#67—22no—
40C2×C9×Dic3—21no—
41C2×C9×Dic3—21no—
42C2×C9×Dic3C2×C411yescharacteristic
43C2×C9×Dic3C2×C411yescharacteristic
44C2×C9×Dic3C2×C411yescharacteristic
45C2×C9×Dic3D412yes—
46C2×C9×Dic3D412yes—
47C2×C3×Dic9—22no—
48C4×D3×C9—21no—
49C4×D3×C9—21no—
50C9×Dic6—22no—
51#5240—91no—
52#4740—91no—
53C2×C32×D4C2×D311yescharacteristic
54C2×C3×#67Dic311yescharacteristic
55C2×C3×#67C2×D311yescharacteristic
56C22×C3×Dic3Dic311yescharacteristic
57C22×C3×Dic3C2×D311yescharacteristic
58C3×#237—22no—
59C3×#237—22no—
60C2×C3×C4×D3Dic311yescharacteristic
61C2×C3×Dic6Dic311yescharacteristic
62#3757—32no—
63C3×#217—32no—
64Dic3⋊Dic3—62no—
65Dic32—62no—
66C9×#41—62no—
67C9×#41—62no—
68C2×Q8×C9—31no—
69C2×D4×C9—21no—
70C2×D4×C9—31no—
71C2×D4×C9—31no—
72C2×D4×C9C2×D311yescharacteristic
73C22×C4×C9—31no—
74C22×C4×C9—31no—
75C22×C4×C9—31no—
76#3676—32no—
77#3676—62no—
78#3676C2×D312yes—
79C4⋊Dic9—62no—
80C4×Dic9—62no—
81C22×C3×C9—21no—
82C22×C3×C9—41no—
83C22×C3×C9—41no—
84C22×C3×C9C22×C411yescharacteristic
85C22×C3×C9C2×D412yes—
86C2×D3×C9—21no—
87C2×D3×C9—41no—
88C3×C4×C9—21no—
89C3⋊Dic9—42no—
90C9×Dic3—21no—
91C9×Dic3C22⋊C412yes—
92C9×Dic3C22⋊C411yescharacteristic
93C9×Dic3C22⋊C411yescharacteristic
94C3×Dic9—42no—
95C2×C3×#41—31no—
96C2×#237D911yescharacteristic
97#926—92no—
98#877—92no—
99C3×#99—91no—
100#834—31no—
101#834—92no—
102#834—91no—
103#834—91no—
104#834—181no—
105#806—92no—
106C23×C32C2×Dic311yescharacteristic
107C23×C32C22×D311yescharacteristic
108C22×C3×D3C2×Dic311yescharacteristic
109C32×D4—22no—
110C2×C32×C4C2×Dic311yescharacteristic
111C2×C3⋊Dic3—62no—
112C3×#67—22no—
113C3×#67—22no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3C2×Dic311yescharacteristic
118C2×C3×Dic3C2×Dic311yescharacteristic
119C2×C3×Dic3C2×Dic311yescharacteristic
120C2×C3×Dic3#6712yes—
121C2×C3×Dic3#6712yes—
122C3×C4×D3—21no—
123C3×C4×D3—21no—
124C3×Dic6—22no—
125C23×C9—21no—
126C23×C9—21no—
127C23×C9—31no—
128C23×C9C4×D311yescharacteristic
129C23×C9C22×D311yescharacteristic
130Q8×C9—62no—
131D4×C9—22no—
132D4×C9—42no—
133D4×C9—62no—
134D4×C9—62no—
135C2×C4×C9—21no—
136C2×C4×C9—31no—
137C2×C4×C9—31no—
138C2×C4×C9—32no—
139C2×C4×C9—32no—
140C2×C4×C9—31no—
141C2×C4×C9—61no—
142C2×C4×C9—61no—
143C2×C4×C9—61no—
144C2×C4×C9—61no—
145C2×C4×C9C4×D311yescharacteristic
146C2×Dic9—22no—
147C2×Dic9—62no—
148C2×Dic9—122no—
149#410—271noSylow
150C2×C3×C9—22no—
151C2×C3×C9—21no—
152C2×C3×C9—41no—
153C2×C3×C9C2×C22⋊C411yescharacteristic
154D3×C9—41no—
155C3×#41—62no—
156C3×#41—62no—
157C2×C3×Q8—31no—
158C2×C3×D4—21no—
159C2×C3×D4—31no—
160C2×C3×D4—31no—
161C2×C3×D4C2×D911yescharacteristic
162C2×C3×D4D3211yescharacteristic
163C22×C3×C4—31no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C2×#67Dic911yescharacteristic, complemented
167C2×#67C2×D911yescharacteristic
168C22×Dic3Dic911yescharacteristic, complemented
169C22×Dic3C2×D911yescharacteristic
170#237—22no—
171#237—22no—
172C2×C4×D3Dic911yescharacteristic, complemented
173C2×Dic6Dic911yescharacteristic, complemented
174C3×C22⋊C4—92no—
175#217—32no—
176#217—92no—
177#217—92no—
178#217—182no—
179C4⋊Dic3—182no—
180Dic3⋊C4—182no—
181C4×Dic3—182no—
182C4×Dic3—182no—
183C22×C32—21no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32C22×Dic311yescharacteristic
187C22×C32C2×#6712yes—
188C2×C3×D3—21no—
189C2×C3×D3—41no—
190C32×C4—21no—
191C3⋊Dic3—122no—
192C3×Dic3—21no—
193C3×Dic3—122no—
194C3×Dic3#21712yes—
195C3×Dic3#21711yescharacteristic
196C3×Dic3#21711yescharacteristic
197C22×C9—21no—
198C22×C9—21no—
199C22×C9—22no—
200C22×C9—41no—
201C22×C9—41no—
202C22×C9—41no—
203C22×C9—61no—
204C22×C9—81no—
205C22×C9—81no—
206C22×C9—121no—
207C22×C9C2×C4×D311yescharacteristic
208C22×C9D3×D412yes—
209C4×C9—21no—
210C4×C9—31no—
211C4×C9—31no—
212C4×C9—32no—
213C4×C9—41no—
214C4×C9—61no—
215Dic9—42no—
216Dic9—122no—
217Dic9—242no—
218C2×#41—31no—
219#117—272no—
220#99—91no—
221#99—272no—
222#99—271no—
223C3×C9#41011yescharacteristic, complemented, Sylow
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—31no—
227C23×C3C2×Dic911yescharacteristic
228C23×C3C22×D911yescharacteristic
229C23×C3D3×Dic311yescharacteristic
230C23×C3C2×D3211yescharacteristic
231C22×D3C2×Dic911yescharacteristic
232C3×Q8—62no—
233C3×D4—22no—
234C3×D4—22no—
235C3×D4—42no—
236C3×D4—62no—
237C3×D4—62no—
238C2×C3×C4—21no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—31no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—182no—
249C2×C3×C4C2×Dic911yescharacteristic
250C2×C3×C4D3×Dic311yescharacteristic
251#67—22no—
252#67—22no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—62no—
256C2×Dic3—182no—
257C2×Dic3—182no—
258C2×Dic3—362no—
259C2×Dic3C2×Dic911yescharacteristic
260C2×Dic3C2×Dic911yescharacteristic
261C2×Dic3C2×Dic911yescharacteristic
262C2×Dic3#61112yes—
263C2×Dic3#61112yes—
264C4×D3—21no—
265C4×D3—21no—
266Dic6—22no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—41no—
270C2×C32C2×#21711yescharacteristic
271C3×D3—41no—
272C2×C9—22no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9—42no—
276C2×C9—41no—
277C2×C9—41no—
278C2×C9—81no—
279C2×C9—121no—
280C2×C9D3×C22⋊C411yescharacteristic
281#41—62no—
282#41—62no—
283C2×Q8—31no—
284C2×D4—31no—
285C2×D4—31no—
286C2×D4D3×D911yescharacteristic
287C22×C4—31no—
288C22×C4—31no—
289C22×C4—31no—
290C4⋊C4—542no—
291C22⋊C4—92no—
292C22⋊C4—272no—
293C42—542no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—22no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—61no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3—121no—
307C22×C3C22×Dic911yescharacteristic
308C22×C3C2×#61112yes—
309C22×C3C2×D3×Dic311yescharacteristic
310C22×C3D3×#6712yes—
311C2×D3—21no—
312C2×D3—41no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—31no—
316C3×C4—31no—
317C3×C4—32no—
318C3×C4—41no—
319C3×C4—61no—
320C3×C4—362no—
321Dic3—21no—
322Dic3—122no—
323Dic3—362no—
324Dic3—362no—
325Dic3—722no—
326Dic3#367612yes—
327Dic3#367611yescharacteristic
328Dic3#367611yescharacteristic
329C32#524011yescharacteristic
330C9—21no—
331C9#474011yescharacteristic, complemented
332C23—31no—
333C23D3×Dic911yescharacteristic
334C23C2×D3×D911yescharacteristic
335Q8—62no—
336D4—22no—
337D4—62no—
338D4—62no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—32no—
342C2×C4—32no—
343C2×C4—31no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—182no—
349C2×C4—542no—
350C2×C4D3×Dic911yescharacteristic
351C2×C3—22no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—42no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—81no—
361C2×C3—121no—
362C2×C3C2×#367611yescharacteristic
363C2×C3D3×#21711yescharacteristic
364D3—41no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—61no—
369C22—121no—
370C22C2×D3×Dic911yescharacteristic
371C22D3×#61112yes—
372C4—21no—
373C4—31no—
374C4—31no—
375C4—32no—
376C4—61no—
377C4—362no—
378C4—1082no—
379C3—21no—
380C3#1058892011yescharacteristic, complemented
381C3#1059152811yescharacteristic
382C2—22no—
383C2—21no—
384C2—41no—
385C2—121no—
386C2D3×#367611yescharacteristic
subgroupG/H
centerC2D3×#3676
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32C2×#217
prime radicalC2×C3×#237D3
cosocleC23×C3×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#410
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721244C2semidirect
3. #11720577C2semidirect
4. C3×#63393C2semidirect
5. #11720332C2semidirect
9. C2×C9×#67C4semidirect
11. C22×C9×Dic3C4semidirect
15. C2×C4×D3×C9C4semidirect
16. C2×C9×Dic6C4semidirect
31. #63393D3semidirect
166. C2×#67Dic9semidirect
168. C22×Dic3Dic9semidirect
172. C2×C4×D3Dic9semidirect
173. C2×Dic6Dic9semidirect
223. C3×C9#410semidirect
331. C9#4740semidirect
380. C3#10588920semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
#117212442
#117205772
C3×#633932
#117203322
#105915283
#105889203