Finite Group Databasebeta

← C3×#10589736 #13152913

Subgroups — C3×#10589736 #13152913

Order 1728.

800 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C3×#63508C212yesmaximal, complemented
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
4C3×#63486C211yesmaximal, characteristic, complemented
5C3×#63459C212yesmaximal, complemented
6C3×#63458C212yesmaximal, complemented
7C3×#63458C212yesmaximal, complemented
8C3×C22⋊C4×D9C212yesmaximal
9C3×#63440C212yesmaximal, complemented
10C3×#5634—31nomaximal
11#10589736C311yesmaximal, characteristic, direct product factor
12C23×C3×D9C2211yescharacteristic
13C2×C3×Q8×C9C2211yescharacteristic, complemented
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C2×C3×C4×D9C2212yes—
17C2×C3×C4×D9C2212yes—
18C2×C3×Dic18C2211yescharacteristic, complemented
19C2×C3×Dic18C2212yescomplemented
20C3×C9×C4⋊C4C2211yescharacteristic, complemented
21C3×C9×C4⋊C4C2212yescomplemented
22C3×C9×C22⋊C4C2212yes—
23C3×#3676C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×C4⋊Dic9C2211yescharacteristic, complemented
29C3×C4⋊Dic9C2212yescomplemented
30C3×Dic9⋊C4C2212yescomplemented
31C3×Dic9⋊C4C2212yescomplemented
32C3×Dic9⋊C4C2212yescomplemented
33C32×C22⋊Q8D311yescharacteristic
34C3×#946—32no—
35C2×C3×(C2×D3)⋊C4—31no—
36C3×#924—31no—
37C3×#897—32no—
38C3×#896—32no—
39C3×#896—32no—
40C3×D3×C22⋊C4—32no—
41C3×#878—32no—
42C9×C22⋊Q8—21no—
43C9×C22⋊Q8C2×C311yescharacteristic, complemented
44#63508C2×C312yescomplemented
45C2×(C2×D9)⋊C4C2×C311yescharacteristic
46#63486C2×C311yescharacteristic, complemented
47#63459C2×C312yescomplemented
48#63458C2×C312yescomplemented
49#63458C2×C312yescomplemented
50C22⋊C4×D9C2×C312yes—
51#63440C2×C312yescomplemented
52C23×C3×C9C2311yescharacteristic
53C22×C3×D9—41no—
54C22×C3×D9—41no—
55C22×C3×D9Q814yescomplemented
56C22×C3×D9C2312yes—
57C3×Q8×C9—41no—
58C2×C3×C4×C9—41no—
59C2×C3×C4×C9C2312yes—
60C2×C3×C4×C9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×Dic9—41no—
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66C2×C3×Dic9C2312yes—
67C3×C4×D9—42no—
68C3×C4×D9—42no—
69C3×Dic18—41no—
70C3×Dic18—42no—
71C3×#543—91no—
72#5634—31no—
73C23×C3×D3—31no—
74C2×C32×Q8C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C3×Dic3—31no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—32no—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—31no—
81C23×D9C22×C311yescharacteristic
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C2×D312yes—
84C32×C22⋊C4C2×D312yes—
85C3×#217—32no—
86C3×(C2×D3)⋊C4—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×C4⋊Dic3—32no—
91C3×C4⋊Dic3—31no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—32no—
95C2×Q8×C9—21no—
96C2×Q8×C9C22×C311yescharacteristic, complemented
97C22×C4×C9—21no—
98C22×C4×C9C22×C311yescharacteristic
99C22×Dic9C22×C311yescharacteristic
100C2×C4×D9C22×C312yes—
101C2×C4×D9C22×C312yes—
102C2×Dic18C22×C312yescomplemented
103C2×Dic18C22×C311yescharacteristic, complemented
104C9×C4⋊C4—21no—
105C9×C4⋊C4—22no—
106C9×C4⋊C4C22×C311yescharacteristic, complemented
107C9×C4⋊C4C22×C312yescomplemented
108C9×C22⋊C4—22no—
109C9×C22⋊C4C22×C312yes—
110#3676C22×C312yes—
111(C2×D9)⋊C4C22×C312yes—
112(C2×D9)⋊C4C22×C312yes—
113(C2×D9)⋊C4C22×C312yes—
114(C2×D9)⋊C4C22×C312yes—
115C4⋊Dic9C22×C312yescomplemented
116C4⋊Dic9C22×C311yescharacteristic, complemented
117Dic9⋊C4C22×C312yescomplemented
118Dic9⋊C4C22×C312yescomplemented
119Dic9⋊C4C22×C312yescomplemented
120C22×C3×C9—21no—
121C22×C3×C9—21no—
122C22×C3×C9C2×Q812yes—
123C22×C3×C9C2411yescharacteristic
124C2×C3×D9—22no—
125C2×C3×D9—22no—
126C2×C3×D9—44no—
127C2×C3×D9C2×Q814yes—
128C3×C4×C9—21no—
129C3×C4×C9—22no—
130C3×C4×C9—21no—
131C3×C4×C9—22no—
132C3×Dic9—22no—
133C3×Dic9—22no—
134C3×Dic9—22no—
135C3×C22⋊Q8—21no—
136C3×C22⋊Q8—92no—
137C3×C22⋊Q8—91no—
138C3×C22⋊Q8—92no—
139C3×C22⋊Q8—92no—
140C3×C22⋊Q8—92no—
141C3×C22⋊Q8—92no—
142C3×C22⋊Q8D911yescharacteristic, complemented
143C3×C22⋊Q8C3×D311yescharacteristic
144C2×C3×C22⋊C4—91no—
145C2×C3×C22⋊C4—92no—
146#946—32no—
147C2×(C2×D3)⋊C4—31no—
148#924—31no—
149#897—32no—
150#896—32no—
151#896—32no—
152D3×C22⋊C4—32no—
153#878—32no—
154C23×C32C22×D311yescharacteristic
155C22×C3×D3—32no—
156C22×C3×D3—34no—
157C22×C3×D3—121no—
158C22×C3×D3—121no—
159C32×Q8—41no—
160C2×C32×C4—41no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D312yes—
163C2×C32×C4C22×D312yes—
164C2×C32×C4C22×D311yescharacteristic
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—121no—
169C3×C4×D3—122no—
170C3×C4×D3—122no—
171C3×Dic6—121no—
172C3×Dic6—122no—
173C23×C9—21no—
174C23×C9C23×C311yescharacteristic
175C22×D9—41no—
176C22×D9—41no—
177C22×D9C3×Q814yescomplemented
178C22×D9C23×C312yes—
179Q8×C9—41no—
180Q8×C9—42no—
181C2×C4×C9—21no—
182C2×C4×C9—22no—
183C2×C4×C9—22no—
184C2×C4×C9—21no—
185C2×C4×C9—41no—
186C2×C4×C9—42no—
187C2×C4×C9C23×C311yescharacteristic
188C2×C4×C9C23×C312yes—
189C2×C4×C9C23×C312yes—
190C2×C4×C9C23×C311yescharacteristic
191C2×Dic9—41no—
192C2×Dic9C23×C312yes—
193C2×Dic9C23×C312yes—
194C2×Dic9C23×C312yes—
195C4×D9—42no—
196C4×D9—42no—
197Dic18—41no—
198Dic18—42no—
199#543—91noSylow
200C2×C3×C9—22no—
201C2×C3×C9C22×Q811yescharacteristic
202C2×C3×C9H(5,𝔽2)11yescharacteristic
203C2×C3×C9H(5,𝔽2)11yescharacteristic
204C3×D9—24no—
205C24×C3—91no—
206C23×D3—31no—
207C2×C3×Q8—21no—
208C2×C3×Q8—91no—
209C2×C3×Q8—92no—
210C2×C3×Q8C2×D911yescharacteristic, complemented
211C2×C3×Q8C2×C3×D311yescharacteristic
212C22×C3×C4—21no—
213C22×C3×C4—91no—
214C22×C3×C4—92no—
215C22×C3×C4—92no—
216C22×C3×C4C2×D911yescharacteristic
217C22×C3×C4C2×C3×D311yescharacteristic
218C22×Dic3—31no—
219C2×C4×D3—32no—
220C2×C4×D3—32no—
221C2×Dic6—32no—
222C2×Dic6—31no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4C2×D911yescharacteristic, complemented
231C3×C4⋊C4C2×D912yescomplemented
232C3×C4⋊C4C2×C3×D311yescharacteristic
233C3×C4⋊C4C2×C3×D312yes—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—92no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4—92no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4C2×D912yes—
241C3×C22⋊C4C2×C3×D312yes—
242#217—32no—
243(C2×D3)⋊C4—32no—
244(C2×D3)⋊C4—32no—
245(C2×D3)⋊C4—32no—
246(C2×D3)⋊C4—32no—
247C4⋊Dic3—32no—
248C4⋊Dic3—31no—
249Dic3⋊C4—32no—
250Dic3⋊C4—32no—
251Dic3⋊C4—32no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32D3×Q812yes—
255C22×C32C23×D311yescharacteristic
256C2×C3×D3—34no—
257C2×C3×D3—62no—
258C2×C3×D3—62no—
259C2×C3×D3—124no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—22no—
263C32×C4—22no—
264C3×Dic3—62no—
265C3×Dic3—62no—
266C3×Dic3—62no—
267C22×C9—21no—
268C22×C9—21no—
269C22×C9—21no—
270C22×C9—22no—
271C22×C9—41no—
272C22×C9—41no—
273C22×C9C2×C3×Q812yes—
274C22×C9C24×C311yescharacteristic
275C2×D9—22no—
276C2×D9—22no—
277C2×D9—44no—
278C2×D9C2×C3×Q814yes—
279C4×C9—22no—
280C4×C9—22no—
281C4×C9—21no—
282C4×C9—22no—
283C4×C9—21no—
284C4×C9—42no—
285C4×C9—41no—
286C4×C9—42no—
287Dic9—22no—
288Dic9—22no—
289Dic9—22no—
290C22⋊Q8—92no—
291C22⋊Q8—91no—
292C22⋊Q8—92no—
293C22⋊Q8—92no—
294C22⋊Q8—92no—
295C22⋊Q8—92no—
296C22⋊Q8C3×D911yescharacteristic, complemented
297C2×C22⋊C4—91no—
298C2×C22⋊C4—92no—
299C3×C9#54311yescharacteristic, complemented, Sylow
300C23×C3—21no—
301C23×C3—92no—
302C23×C3—94no—
303C23×C3—361no—
304C23×C3—361no—
305C23×C3C22×D911yescharacteristic
306C23×C3C22×C3×D311yescharacteristic
307C22×D3—32no—
308C22×D3—34no—
309C22×D3—121no—
310C22×D3—121no—
311C3×Q8—41no—
312C3×Q8—41no—
313C3×Q8—42no—
314C3×Q8—361no—
315C3×Q8—362no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—92no—
324C2×C3×C4—92no—
325C2×C3×C4—92no—
326C2×C3×C4—362no—
327C2×C3×C4—362no—
328C2×C3×C4—361no—
329C2×C3×C4C22×D911yescharacteristic
330C2×C3×C4C22×D911yescharacteristic
331C2×C3×C4C22×D912yes—
332C2×C3×C4C22×D912yes—
333C2×C3×C4C22×C3×D311yescharacteristic
334C2×C3×C4C22×C3×D312yes—
335C2×C3×C4C22×C3×D312yes—
336C2×C3×C4C22×C3×D311yescharacteristic
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—121no—
341C4×D3—122no—
342C4×D3—122no—
343Dic6—121no—
344Dic6—122no—
345C2×C32—22no—
346C2×C32#100411yescharacteristic
347C2×C32C2×D3×Q811yescharacteristic
348C2×C32#100911yescharacteristic
349C3×D3—64no—
350C2×C9—22no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—42no—
355C2×C9C22×C3×Q811yescharacteristic
356C2×C9C3×H(5,𝔽2)11yescharacteristic
357C2×C9C3×H(5,𝔽2)11yescharacteristic
358D9—24no—
359C24—91no—
360C2×Q8—91no—
361C2×Q8—92no—
362C2×Q8C2×C3×D911yescharacteristic, complemented
363C22×C4—91no—
364C22×C4—92no—
365C22×C4—92no—
366C22×C4C2×C3×D911yescharacteristic
367C4⋊C4—92no—
368C4⋊C4—92no—
369C4⋊C4—92no—
370C4⋊C4—91no—
371C4⋊C4—92no—
372C4⋊C4C2×C3×D911yescharacteristic, complemented
373C4⋊C4C2×C3×D912yescomplemented
374C22⋊C4—92no—
375C22⋊C4—92no—
376C22⋊C4—92no—
377C22⋊C4—92no—
378C22⋊C4—92no—
379C22⋊C4C2×C3×D912yes—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—94no—
389C22×C3—182no—
390C22×C3—182no—
391C22×C3—364no—
392C22×C3Q8×D912yes—
393C22×C3C23×D911yescharacteristic
394C22×C3C3×D3×Q812yes—
395C22×C3C23×C3×D311yescharacteristic
396C2×D3—34no—
397C2×D3—62no—
398C2×D3—62no—
399C2×D3—124no—
400C3×C4—22no—
401C3×C4—21no—
402C3×C4—22no—
403C3×C4—21no—
404C3×C4—22no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—41no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—182no—
413C3×C4—182no—
414C3×C4—182no—
415Dic3—62no—
416Dic3—62no—
417Dic3—62no—
418C32#563411yescharacteristic
419C9—21no—
420C9C3×#54311yescharacteristic, complemented
421C23—92no—
422C23—94no—
423C23—361no—
424C23—361no—
425C23C22×C3×D911yescharacteristic
426Q8—41no—
427Q8—361no—
428Q8—362no—
429C2×C4—41no—
430C2×C4—92no—
431C2×C4—92no—
432C2×C4—92no—
433C2×C4—362no—
434C2×C4—362no—
435C2×C4—361no—
436C2×C4C22×C3×D911yescharacteristic
437C2×C4C22×C3×D911yescharacteristic
438C2×C4C22×C3×D912yes—
439C2×C4C22×C3×D912yes—
440C2×C3—22no—
441C2×C3—22no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—42no—
446C2×C3—184no—
447C2×C3#6371011yescharacteristic
448C2×C3C2×Q8×D911yescharacteristic
449C2×C3#6371511yescharacteristic
450C2×C3C3×#100411yescharacteristic
451C2×C3C2×C3×D3×Q811yescharacteristic
452C2×C3C3×#100911yescharacteristic
453D3—64no—
454C22—21no—
455C22—21no—
456C22—94no—
457C22—182no—
458C22—182no—
459C22—364no—
460C22C3×Q8×D912yes—
461C22C23×C3×D911yescharacteristic
462C4—21no—
463C4—21no—
464C4—22no—
465C4—22no—
466C4—182no—
467C4—182no—
468C4—182no—
469C3—21no—
470C3#1058973611yescharacteristic, direct product factor
471C3C3×#563411yescharacteristic
472C2—22no—
473C2—184no—
474C2C3×#6371011yescharacteristic
475C2C2×C3×Q8×D911yescharacteristic
476C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C23×D3
prime radicalC23×C3×D9C22
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#543
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C9×C22⋊Q8C2semidirect
2. C3×#63508C2semidirect
4. C3×#63486C2semidirect
5. C3×#63459C2semidirect
6. C3×#63458C2semidirect
7. C3×#63458C2semidirect
9. C3×#63440C2semidirect
11. #10589736C3direct (N × H)
13. C2×C3×Q8×C9C22semidirect
18. C2×C3×Dic18C22semidirect
19. C2×C3×Dic18C22semidirect
20. C3×C9×C4⋊C4C22semidirect
21. C3×C9×C4⋊C4C22semidirect
28. C3×C4⋊Dic9C22semidirect
29. C3×C4⋊Dic9C22semidirect
30. C3×Dic9⋊C4C22semidirect
31. C3×Dic9⋊C4C22semidirect
32. C3×Dic9⋊C4C22semidirect
43. C9×C22⋊Q8C2×C3semidirect
44. #63508C2×C3semidirect
46. #63486C2×C3semidirect
47. #63459C2×C3semidirect
48. #63458C2×C3semidirect
49. #63458C2×C3semidirect
51. #63440C2×C3semidirect
55. C22×C3×D9Q8semidirect
96. C2×Q8×C9C22×C3semidirect
102. C2×Dic18C22×C3semidirect
103. C2×Dic18C22×C3semidirect
106. C9×C4⋊C4C22×C3semidirect
107. C9×C4⋊C4C22×C3semidirect
115. C4⋊Dic9C22×C3semidirect
116. C4⋊Dic9C22×C3semidirect
117. Dic9⋊C4C22×C3semidirect
118. Dic9⋊C4C22×C3semidirect
119. Dic9⋊C4C22×C3semidirect
142. C3×C22⋊Q8D9semidirect
177. C22×D9C3×Q8semidirect
210. C2×C3×Q8C2×D9semidirect
230. C3×C4⋊C4C2×D9semidirect
231. C3×C4⋊C4C2×D9semidirect
296. C22⋊Q8C3×D9semidirect
299. C3×C9#543semidirect
362. C2×Q8C2×C3×D9semidirect
372. C4⋊C4C2×C3×D9semidirect
373. C4⋊C4C2×C3×D9semidirect
420. C9C3×#543semidirect
470. C3#10589736direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C3×#635082
C2×C3×(C2×D9)⋊C42
C3×#634862
C3×#634592
C3×#634582
C3×C22⋊C4×D92
C3×#634402
C3×#56343
#105897363