Finite Group Databasebeta

← C3×#10588667 #13142586

Subgroups — C3×#10588667 #13142586

Order 1728.

794 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D9C211yesmaximal, characteristic
2C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D9)⋊C4C212yesmaximal
4C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
5C3×C9⋊#95C212yesmaximal, complemented
6C3×#4987—31nomaximal
7#10588667C311yesmaximal, characteristic, direct product factor
8C23×C3×D9C2211yescharacteristic
9C22×C3×C4×C9C2212yes—
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3×Dic9C2212yes—
12C22×C3×Dic9C2211yescharacteristic
13C2×C3×C4×D9—22no—
14C2×C3×C4×D9—22no—
15C2×C3×C4×D9C414yescomplemented
16C3×C9×C4⋊C4—22no—
17C3×(C2×D9)⋊C4—24no—
18C3×C4⋊Dic9—22no—
19C22×C3×C4×D3—31no—
20C2×C32×C4⋊C4D311yescharacteristic
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×C4⋊Dic3—31no—
23C22×C4×D9C2×C311yescharacteristic
24C3×C3⋊#95—32no—
25C2×C9×C4⋊C4—21no—
26C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
27C2×(C2×D9)⋊C4C2×C312yes—
28C2×C4⋊Dic9C2×C311yescharacteristic, complemented
29C9⋊#95C2×C312yescomplemented
30C23×C3×C9C2311yescharacteristic
31C22×C3×D9—21no—
32C22×C3×D9—21no—
33C22×C3×D9—21no—
34C22×C3×D9—21no—
35C22×C3×D9C2×C412yes—
36C22×C3×D9D412yes—
37C22×C3×D9Q812yes—
38C2×C3×C4×C9—22no—
39C2×C3×C4×C9—22no—
40C2×C3×C4×C9—22no—
41C2×C3×C4×C9C2×C412yes—
42C2×C3×C4×C9D412yes—
43C2×C3×C4×C9D412yes—
44C2×C3×Dic9—21no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9—22no—
47C2×C3×Dic9—22no—
48C2×C3×Dic9—22no—
49C2×C3×Dic9C2×C412yes—
50C3×C4×D9—28no—
51C3×#380—91no—
52#4987—31no—
53C23×C3×D3—31no—
54C22×C32×C4C2×D312yes—
55C22×C32×C4C2×D311yescharacteristic
56C22×C3×Dic3—31no—
57C22×C3×Dic3—32no—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—62no—
60C2×C3×C4×D3—62no—
61C23×D9C22×C311yescharacteristic
62C32×C4⋊C4—22no—
63C3×(C2×D3)⋊C4—64no—
64C3×C4⋊Dic3—62no—
65C22×C4×C9—21no—
66C22×C4×C9—22no—
67C22×C4×C9C22×C312yes—
68C22×C4×C9C22×C311yescharacteristic
69C22×Dic9C22×C312yes—
70C22×Dic9C22×C311yescharacteristic
71C2×C4×D9—22no—
72C2×C4×D9—22no—
73C2×C4×D9C3×C414yescomplemented
74C9×C4⋊C4—24no—
75C9×C4⋊C4—22no—
76(C2×D9)⋊C4—24no—
77C4⋊Dic9—22no—
78C22×C3×C9C22×C411yescharacteristic
79C22×C3×C9C2×D411yescharacteristic
80C22×C3×C9C2×D411yescharacteristic
81C22×C3×C9C2×D411yescharacteristic
82C22×C3×C9C2×Q811yescharacteristic
83C22×C3×C9#4111yescharacteristic
84C22×C3×C9#4111yescharacteristic
85C2×C3×D9—22no—
86C2×C3×D9—22no—
87C2×C3×D9—22no—
88C2×C3×D9—22no—
89C2×C3×D9—22no—
90C2×C3×D9—22no—
91C2×C3×D9C4⋊C414yescomplemented
92C3×C4×C9—42no—
93C3×C4×C9C22⋊C414yescomplemented
94C3×Dic9—22no—
95C3×Dic9—42no—
96C23×C3×C4—91no—
97C22×C4×D3—31no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—91no—
100C2×C3×C4⋊C4D911yescharacteristic, complemented
101C2×C3×C4⋊C4C3×D311yescharacteristic
102C2×C3×C22⋊C4—92no—
103C2×(C2×D3)⋊C4—32no—
104C2×C4⋊Dic3—31no—
105C3×#95—92no—
106C3⋊#95—32no—
107C23×C32C22×D311yescharacteristic
108C22×C3×D3—32no—
109C22×C3×D3—32no—
110C22×C3×D3—32no—
111C22×C3×D3—61no—
112C22×C3×D3—61no—
113C22×C3×D3—61no—
114C22×C3×D3—61no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4C4×D312yes—
119C2×C32×C4D1212yes—
120C2×C32×C4#6712yes—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C3×C4×D3—68no—
128C23×C9—21no—
129C23×C9C23×C311yescharacteristic
130C22×D9—21no—
131C22×D9—21no—
132C22×D9—21no—
133C22×D9—21no—
134C22×D9C2×C3×C412yes—
135C22×D9C3×D412yes—
136C22×D9C3×Q812yes—
137C2×C4×C9—24no—
138C2×C4×C9—22no—
139C2×C4×C9—22no—
140C2×C4×C9—22no—
141C2×C4×C9—22no—
142C2×C4×C9—22no—
143C2×C4×C9—22no—
144C2×C4×C9—42no—
145C2×C4×C9—42no—
146C2×C4×C9C2×C3×C412yes—
147C2×C4×C9C3×D412yes—
148C2×C4×C9C3×D412yes—
149C2×Dic9—21no—
150C2×Dic9—21no—
151C2×Dic9—22no—
152C2×Dic9—22no—
153C2×Dic9—22no—
154C2×Dic9C2×C3×C412yes—
155C4×D9—28no—
156#380—91noSylow
157C2×C3×C9C2×C22⋊C411yescharacteristic
158C2×C3×C9C2×C4⋊C411yescharacteristic
159C2×C3×C9#11711yescharacteristic
160C2×C3×C9#12111yescharacteristic
161C2×C3×C9#12111yescharacteristic
162C2×C3×C9C22⋊Q811yescharacteristic
163C2×C3×C9C22⋊Q811yescharacteristic
164C3×D9—24no—
165C24×C3—91no—
166C23×D3—31no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—91no—
170C22×C3×C4—94no—
171C22×C3×C4—92no—
172C22×C3×C4—182no—
173C22×C3×C4—182no—
174C22×C3×C4C2×D912yes—
175C22×C3×C4C2×D911yescharacteristic
176C22×C3×C4C2×C3×D312yes—
177C22×C3×C4C2×C3×D311yescharacteristic
178C22×Dic3—31no—
179C22×Dic3—32no—
180C2×C4×D3—34no—
181C2×C4×D3—62no—
182C2×C4×D3—62no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—182no—
187C3×C22⋊C4—184no—
188(C2×D3)⋊C4—64no—
189C4⋊Dic3—62no—
190C22×C32C2×C4×D311yescharacteristic
191C22×C32C2×D1211yescharacteristic
192C22×C32D3×D411yescharacteristic
193C22×C32#23711yescharacteristic
194C22×C32D3×Q811yescharacteristic
195C22×C32#23911yescharacteristic
196C22×C32C2×#6711yescharacteristic
197C2×C3×D3—34no—
198C2×C3×D3—62no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C2×C3×D3—62no—
202C2×C3×D3—62no—
203C2×C3×D3—62no—
204C32×C4—42no—
205C32×C4(C2×D3)⋊C414yes—
206C3×Dic3—62no—
207C3×Dic3—122no—
208C22×C9—21no—
209C22×C9—21no—
210C22×C9—21no—
211C22×C9—21no—
212C22×C9—21no—
213C22×C9—21no—
214C22×C9—21no—
215C22×C9C22×C3×C411yescharacteristic
216C22×C9C2×C3×D411yescharacteristic
217C22×C9C2×C3×D411yescharacteristic
218C22×C9C2×C3×D411yescharacteristic
219C22×C9C2×C3×Q811yescharacteristic
220C22×C9C3×#4111yescharacteristic
221C22×C9C3×#4111yescharacteristic
222C2×D9—22no—
223C2×D9—22no—
224C2×D9—22no—
225C2×D9—22no—
226C2×D9—22no—
227C2×D9—22no—
228C2×D9C3×C4⋊C414yescomplemented
229C4×C9—24no—
230C4×C9—44no—
231C4×C9—42no—
232C4×C9C3×C22⋊C414yescomplemented
233Dic9—22no—
234Dic9—42no—
235C23×C4—91no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4C3×D911yescharacteristic, complemented
238C2×C22⋊C4—92no—
239#95—92no—
240C3×C9#38011yescharacteristic, complemented, Sylow
241C23×C3—21no—
242C23×C3—92no—
243C23×C3—92no—
244C23×C3—92no—
245C23×C3—181no—
246C23×C3—181no—
247C23×C3—181no—
248C23×C3—181no—
249C23×C3C22×D911yescharacteristic
250C23×C3C22×C3×D311yescharacteristic
251C22×D3—32no—
252C22×D3—32no—
253C22×D3—32no—
254C22×D3—61no—
255C22×D3—61no—
256C22×D3—61no—
257C22×D3—61no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—92no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—188no—
274C2×C3×C4—182no—
275C2×C3×C4—182no—
276C2×C3×C4—182no—
277C2×C3×C4C4×D912yes—
278C2×C3×C4D3612yes—
279C2×C3×C4#61112yes—
280C2×C3×C4C3×C4×D312yes—
281C2×C3×C4C3×D1212yes—
282C2×C3×C4C3×#6712yes—
283C2×Dic3—32no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—62no—
289C4×D3—68no—
290C2×C32D3×C4⋊C411yescharacteristic
291C2×C32#89211yescharacteristic
292C2×C32#89511yescharacteristic
293C2×C32#89711yescharacteristic
294C2×C32C2×(C2×D3)⋊C411yescharacteristic
295C2×C32#93811yescharacteristic
296C2×C32#94611yescharacteristic
297C3×D3—64no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9—21no—
305C2×C9C2×C3×C22⋊C411yescharacteristic
306C2×C9C2×C3×C4⋊C411yescharacteristic
307C2×C9C3×#11711yescharacteristic
308C2×C9C3×#12111yescharacteristic
309C2×C9C3×#12111yescharacteristic
310C2×C9C3×C22⋊Q811yescharacteristic
311C2×C9C3×C22⋊Q811yescharacteristic
312D9—24no—
313C24—91no—
314C22×C4—91no—
315C22×C4—94no—
316C22×C4—92no—
317C22×C4—182no—
318C22×C4—182no—
319C22×C4C2×C3×D912yes—
320C22×C4C2×C3×D911yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—182no—
323C22⋊C4—184no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—94no—
332C22×C3—182no—
333C22×C3—182no—
334C22×C3—182no—
335C22×C3—182no—
336C22×C3—182no—
337C22×C3—182no—
338C22×C3C2×C4×D911yescharacteristic
339C22×C3C2×D3611yescharacteristic
340C22×C3D4×D911yescharacteristic
341C22×C3#369911yescharacteristic
342C22×C3Q8×D911yescharacteristic
343C22×C3#370111yescharacteristic
344C22×C3C2×#61111yescharacteristic
345C22×C3C2×C3×C4×D311yescharacteristic
346C22×C3C2×C3×D1211yescharacteristic
347C22×C3C3×D3×D411yescharacteristic
348C22×C3C3×#23711yescharacteristic
349C22×C3C3×D3×Q811yescharacteristic
350C22×C3C3×#23911yescharacteristic
351C22×C3C2×C3×#6711yescharacteristic
352C2×D3—34no—
353C2×D3—62no—
354C2×D3—62no—
355C2×D3—62no—
356C2×D3—62no—
357C2×D3—62no—
358C2×D3—62no—
359C3×C4—24no—
360C3×C4—44no—
361C3×C4—42no—
362C3×C4—42no—
363C3×C4—182no—
364C3×C4—362no—
365C3×C4(C2×D9)⋊C414yescomplemented
366C3×C4C3×(C2×D3)⋊C414yes—
367Dic3—62no—
368Dic3—122no—
369C32#498711yescharacteristic
370C9—21no—
371C9C3×#38011yescharacteristic, complemented
372C23—92no—
373C23—92no—
374C23—92no—
375C23—181no—
376C23—181no—
377C23—181no—
378C23—181no—
379C23C22×C3×D911yescharacteristic
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—92no—
384C2×C4—181no—
385C2×C4—181no—
386C2×C4—188no—
387C2×C4—182no—
388C2×C4—182no—
389C2×C4—182no—
390C2×C4C3×C4×D912yes—
391C2×C4C3×D3612yes—
392C2×C4C3×#61112yes—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—184no—
401C2×C3C4⋊C4×D911yescharacteristic
402C2×C3#6345411yescharacteristic
403C2×C3#6345711yescharacteristic
404C2×C3#6345911yescharacteristic
405C2×C3C2×(C2×D9)⋊C411yescharacteristic
406C2×C3#6350011yescharacteristic
407C2×C3#6350811yescharacteristic
408C2×C3C3×D3×C4⋊C411yescharacteristic
409C2×C3C3×#89211yescharacteristic
410C2×C3C3×#89511yescharacteristic
411C2×C3C3×#89711yescharacteristic
412C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
413C2×C3C3×#93811yescharacteristic
414C2×C3C3×#94611yescharacteristic
415D3—64no—
416C22—94no—
417C22—182no—
418C22—182no—
419C22—182no—
420C22—182no—
421C22—182no—
422C22—182no—
423C22C2×C3×C4×D911yescharacteristic
424C22C2×C3×D3611yescharacteristic
425C22C3×D4×D911yescharacteristic
426C22C3×#369911yescharacteristic
427C22C3×Q8×D911yescharacteristic
428C22C3×#370111yescharacteristic
429C22C2×C3×#61111yescharacteristic
430C4—42no—
431C4—182no—
432C4—362no—
433C4C3×(C2×D9)⋊C414yescomplemented
434C3—21no—
435C3#1058866711yescharacteristic, direct product factor
436C3C3×#498711yescharacteristic
437C2—184no—
438C2C3×C4⋊C4×D911yescharacteristic
439C2C3×#6345411yescharacteristic
440C2C3×#6345711yescharacteristic
441C2C3×#6345911yescharacteristic
442C2C2×C3×(C2×D9)⋊C411yescharacteristic
443C2C3×#6350011yescharacteristic
444C2C3×#6350811yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutatorC22×C9C22×C3×C4
FrattiniC23×C3C22×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
cosocleC23×C9C23×C3
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#380
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C9×C4⋊C4C2semidirect
4. C2×C3×C4⋊Dic9C2semidirect
5. C3×C9⋊#95C2semidirect
7. #10588667C3direct (N × H)
15. C2×C3×C4×D9C4semidirect
26. C2×C9×C4⋊C4C2×C3semidirect
28. C2×C4⋊Dic9C2×C3semidirect
29. C9⋊#95C2×C3semidirect
73. C2×C4×D9C3×C4semidirect
91. C2×C3×D9C4⋊C4semidirect
93. C3×C4×C9C22⋊C4semidirect
100. C2×C3×C4⋊C4D9semidirect
228. C2×D9C3×C4⋊C4semidirect
232. C4×C9C3×C22⋊C4semidirect
237. C2×C4⋊C4C3×D9semidirect
240. C3×C9#380semidirect
365. C3×C4(C2×D9)⋊C4semidirect
371. C9C3×#380semidirect
433. C4C3×(C2×D9)⋊C4semidirect
435. C3#10588667direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C2×C3×C9×C4⋊C42
C2×C3×(C2×D9)⋊C42
C2×C3×C4⋊Dic92
C3×C9⋊#952
C3×#49873
#105886673