Finite Group Databasebeta

← C3×C3⋊D3×#99 #13162781

Subgroups — C3×C3⋊D3×#99 #13162781

Order 1728.

1468 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
2C3×C22⋊C4×C3⋊D3C212yesmaximal, complemented
3C33×#99C211yesmaximal, characteristic, complemented
4C3×#63661C211yesmaximal, characteristic, complemented
5C3×#63633C212yesmaximal, complemented
6C3⋊D3×#99C311yesmaximal, characteristic, direct product factor
7C3×D3×#99—34nomaximal
8C23×C3×C3⋊D3C411yescharacteristic, complemented
9C23×C3×C3⋊D3C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638C411yescharacteristic, complemented
12C2×C3×#638C2211yescharacteristic
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3—22no—
15C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
16C2×C3×C4×C3⋊D3—22no—
17C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
18C33×C22⋊C4C2212yescomplemented
19C3×#3757C2212yescomplemented
20C3×#3752—22no—
21C2×D4×C3⋊D3C2×C311yescharacteristic
22C2×C3×D3×D4—34no—
23C22⋊C4×C3⋊D3C2×C312yescomplemented
24C3×D3×C22⋊C4—38no—
25C32×#99—24no—
26C32×#99D314yescomplemented
27C32×#99C2×C311yescharacteristic, complemented
28#63661C2×C311yescharacteristic, complemented
29#63633C2×C312yescomplemented
30C3×#834—34no—
31C3×#806—38no—
32C23×C33C2×C411yescharacteristic, complemented
33C23×C33C2311yescharacteristic
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3—21no—
36C22×C3×C3⋊D3—42no—
37C22×C3×C3⋊D3—42no—
38C22×C3×C3⋊D3C2×C411yescharacteristic
39C22×C3×C3⋊D3C2×C411yescharacteristic
40C22×C3×C3⋊D3C2×C411yescharacteristic, complemented
41C22×C3×C3⋊D3D412yes—
42C22×C3×C3⋊D3D412yes—
43C33×D4—22no—
44C2×C33×C4—22no—
45C2×C33×C4C2×C411yescharacteristic, complemented
46C3×#638—22no—
47C3×#638—22no—
48C2×C3×C3⋊Dic3—22no—
49C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
50C3×C4⋊(C3⋊D3)—22no—
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—44no—
54C2×C3×#99—91no—
55D3×#99—34no—
56C23×C3⋊D3C3×C411yescharacteristic, complemented
57C23×C3⋊D3C22×C311yescharacteristic
58C23×C3×D3—34no—
59C23×C3×D3—34no—
60C2×C32×D4—24no—
61C2×C32×D4C2×D314yes—
62C2×C32×D4C22×C311yescharacteristic
63C2×#638C3×C411yescharacteristic, complemented
64C2×#638C22×C311yescharacteristic
65D4×C3⋊D3—22no—
66D4×C3⋊D3—22no—
67C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
68C2×C4×C3⋊D3—22no—
69C2×C4×C3⋊D3C3×C411yescharacteristic, complemented
70C2×C3×#67—34no—
71C2×C3×#67—34no—
72C3×D3×D4—68no—
73C3×D3×D4—68no—
74C2×C3×D12—34no—
75C2×C3×C4×D3—34no—
76C2×C3×C4×D3—68no—
77C32×C22⋊C4—28no—
78C32×C22⋊C4C2×D318yescomplemented
79C32×C22⋊C4C22×C312yescomplemented
80#3757C22×C312yescomplemented
81#3752—22no—
82C3×#217—38no—
83C3×(C2×D3)⋊C4—68no—
84C22×C33—21no—
85C22×C33—41no—
86C22×C33—41no—
87C22×C33C22×C411yescharacteristic
88C22×C33C2×D412yes—
89C2×C3×C3⋊D3—24no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—42no—
94C2×C3×C3⋊D3—42no—
95C2×C3×C3⋊D3—42no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3C22⋊C411yescharacteristic
98C2×C3×C3⋊D3C22⋊C412yes—
99C2×C3×C3⋊D3C22⋊C411yescharacteristic
100C33×C4—21no—
101C33×C4—42no—
102C3×C3⋊Dic3—21no—
103C3×C3⋊Dic3—42no—
104C22×C3×D4—91no—
105C2×D3×D4—34no—
106C2×C3×C22⋊C4—92no—
107D3×C22⋊C4—38no—
108C3×#99—24no—
109C3×#99—91no—
110C3×#99—92no—
111C3×#99C3×D314yescomplemented
112C3×#99C3⋊D311yescharacteristic, direct product factor
113#834—34no—
114#806—38no—
115C23×C32—24no—
116C23×C32—24no—
117C23×C32C4×D314yescomplemented
118C23×C32C2×C3×C411yescharacteristic, complemented
119C23×C32C22×D314yes—
120C23×C32C23×C311yescharacteristic
121C22×C3⋊D3—21no—
122C22×C3⋊D3—21no—
123C22×C3⋊D3—42no—
124C22×C3⋊D3—42no—
125C22×C3⋊D3C2×C3×C411yescharacteristic
126C22×C3⋊D3C2×C3×C411yescharacteristic
127C22×C3⋊D3C2×C3×C411yescharacteristic, complemented
128C22×C3⋊D3C3×D412yes—
129C22×C3⋊D3C3×D412yes—
130C22×C3×D3—34no—
131C22×C3×D3—38no—
132C22×C3×D3—34no—
133C22×C3×D3—38no—
134C22×C3×D3—34no—
135C22×C3×D3—64no—
136C22×C3×D3—64no—
137C22×C3×D3—128no—
138C22×C3×D3—128no—
139C32×D4—22no—
140C32×D4—28no—
141C32×D4—48no—
142C2×C32×C4—22no—
143C2×C32×C4—24no—
144C2×C32×C4—28no—
145C2×C32×C4—48no—
146C2×C32×C4C4×D314yescomplemented
147C2×C32×C4C2×C3×C411yescharacteristic, complemented
148#638—22no—
149#638—22no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3C2×C3×C411yescharacteristic, complemented
152C4⋊(C3⋊D3)—22no—
153C4×C3⋊D3—21no—
154C4×C3⋊D3—21no—
155C4×C3⋊D3—44no—
156C3×#67—68no—
157C3×#67—68no—
158C2×C3×Dic3—34no—
159C2×C3×Dic3—68no—
160C3×D12—68no—
161C3×C4×D3—64no—
162C3×C4×D3—64no—
163C3×C4×D3—1216no—
164C2×#99—91noSylow
165C2×C33—22no—
166C2×C33—21no—
167C2×C33—41no—
168C2×C33C2×C22⋊C411yescharacteristic
169C3×C3⋊D3—21no—
170C3×C3⋊D3—22no—
171C3×C3⋊D3—41no—
172C3×C3⋊D3#9912yesdirect product factor
173C24×C3—91no—
174C24×C3—91no—
175C23×D3—34no—
176C23×D3—34no—
177C2×C3×D4—24no—
178C2×C3×D4—91no—
179C2×C3×D4—91no—
180C2×C3×D4—91no—
181C2×C3×D4—182no—
182C2×C3×D4—182no—
183C2×C3×D4C2×C3×D314yes—
184C2×C3×D4C2×C3⋊D311yescharacteristic
185C22×C3×C4—91no—
186C22×C3×C4—182no—
187C2×#67—34no—
188C2×#67—34no—
189D3×D4—68no—
190D3×D4—68no—
191C2×D12—34no—
192C2×C4×D3—34no—
193C2×C4×D3—68no—
194C3×C22⋊C4—28no—
195C3×C22⋊C4—92no—
196C3×C22⋊C4—182no—
197C3×C22⋊C4C2×C3×D318yescomplemented
198C3×C22⋊C4C2×C3⋊D312yescomplemented
199#217—38no—
200(C2×D3)⋊C4—68no—
201C22×C32—21no—
202C22×C32—24no—
203C22×C32—28no—
204C22×C32—24no—
205C22×C32—44no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—44no—
209C22×C32—44no—
210C22×C32—84no—
211C22×C32—84no—
212C22×C32C2×C4×D314yes—
213C22×C32D3×D418yes—
214C22×C32C22×C3×C411yescharacteristic
215C22×C32C2×C3×D412yes—
216C2×C3⋊D3—24no—
217C2×C3⋊D3—22no—
218C2×C3⋊D3—21no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—42no—
221C2×C3⋊D3—42no—
222C2×C3⋊D3—42no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3C3×C22⋊C411yescharacteristic
225C2×C3⋊D3C3×C22⋊C412yes—
226C2×C3⋊D3C3×C22⋊C411yescharacteristic
227C2×C3×D3—34no—
228C2×C3×D3—34no—
229C2×C3×D3—38no—
230C2×C3×D3—616no—
231C2×C3×D3—68no—
232C2×C3×D3—64no—
233C2×C3×D3—128no—
234C2×C3×D3—124no—
235C2×C3×D3—128no—
236C2×C3×D3—128no—
237C2×C3×D3—124no—
238C32×C4—21no—
239C32×C4—24no—
240C32×C4—44no—
241C32×C4—42no—
242C32×C4—48no—
243C32×C4—88no—
244C3⋊Dic3—21no—
245C3⋊Dic3—42no—
246C3×Dic3—64no—
247C3×Dic3—128no—
248C22×D4—91no—
249C2×C22⋊C4—92no—
250#99—91no—
251#99—92no—
252#99C3×C3⋊D311yescharacteristic, direct product factor
253C33C2×#9911yescharacteristic, complemented, Sylow
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—91no—
257C23×C3—92no—
258C23×C3—91no—
259C23×C3—92no—
260C23×C3—91no—
261C23×C3—181no—
262C23×C3—181no—
263C23×C3—362no—
264C23×C3—362no—
265C23×C3C3×C4×D314yescomplemented
266C23×C3C4×C3⋊D311yescharacteristic, complemented
267C23×C3C22×C3×D314yes—
268C23×C3C22×C3⋊D311yescharacteristic
269C22×D3—34no—
270C22×D3—38no—
271C22×D3—34no—
272C22×D3—38no—
273C22×D3—34no—
274C22×D3—64no—
275C22×D3—64no—
276C22×D3—128no—
277C22×D3—128no—
278C3×D4—28no—
279C3×D4—22no—
280C3×D4—48no—
281C3×D4—182no—
282C3×D4—182no—
283C3×D4—182no—
284C2×C3×C4—28no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—48no—
288C2×C3×C4—91no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—182no—
292C2×C3×C4—364no—
293C2×C3×C4C3×C4×D314yescomplemented
294C2×C3×C4C4×C3⋊D311yescharacteristic, complemented
295#67—68no—
296#67—68no—
297C2×Dic3—34no—
298C2×Dic3—68no—
299D12—68no—
300C4×D3—64no—
301C4×D3—64no—
302C4×D3—1216no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—24no—
306C2×C32—28no—
307C2×C32—24no—
308C2×C32—48no—
309C2×C32—44no—
310C2×C32—41no—
311C2×C32—44no—
312C2×C32—84no—
313C2×C32D3×C22⋊C414yes—
314C2×C32C2×C3×C22⋊C411yescharacteristic
315C3⋊D3—21no—
316C3⋊D3—22no—
317C3⋊D3—41no—
318C3⋊D3C3×#9912yesdirect product factor
319C3×D3—38no—
320C3×D3—68no—
321C3×D3—64no—
322C3×D3—124no—
323C24—91no—
324C24—91no—
325C2×D4—91no—
326C2×D4—91no—
327C2×D4—91no—
328C2×D4—182no—
329C2×D4—182no—
330C2×D4C2×C3×C3⋊D311yescharacteristic
331C22×C4—91no—
332C22×C4—182no—
333C22⋊C4—92no—
334C22⋊C4—182no—
335C22⋊C4C2×C3×C3⋊D312yescomplemented
336C22×C3—24no—
337C22×C3—21no—
338C22×C3—24no—
339C22×C3—28no—
340C22×C3—44no—
341C22×C3—44no—
342C22×C3—41no—
343C22×C3—44no—
344C22×C3—41no—
345C22×C3—84no—
346C22×C3—84no—
347C22×C3—91no—
348C22×C3—91no—
349C22×C3—92no—
350C22×C3—184no—
351C22×C3—182no—
352C22×C3—181no—
353C22×C3—362no—
354C22×C3—361no—
355C22×C3—361no—
356C22×C3—362no—
357C22×C3—362no—
358C22×C3C2×C3×C4×D314yes—
359C22×C3C3×D3×D418yes—
360C22×C3C2×C4×C3⋊D311yescharacteristic
361C22×C3D4×C3⋊D312yes—
362C2×D3—34no—
363C2×D3—34no—
364C2×D3—38no—
365C2×D3—616no—
366C2×D3—68no—
367C2×D3—64no—
368C2×D3—128no—
369C2×D3—128no—
370C2×D3—124no—
371C2×D3—128no—
372C2×D3—124no—
373C3×C4—24no—
374C3×C4—21no—
375C3×C4—44no—
376C3×C4—48no—
377C3×C4—42no—
378C3×C4—88no—
379C3×C4—181no—
380C3×C4—362no—
381Dic3—64no—
382Dic3—128no—
383C32—24no—
384C32D3×#9914yescomplemented
385C32C2×C3×#9911yescharacteristic, complemented
386C23—91no—
387C23—92no—
388C23—91no—
389C23—92no—
390C23—91no—
391C23—181no—
392C23—181no—
393C23—362no—
394C23—362no—
395C23C3×C4×C3⋊D311yescharacteristic, complemented
396C23C22×C3×C3⋊D311yescharacteristic
397D4—22no—
398D4—182no—
399D4—182no—
400D4—182no—
401C2×C4—22no—
402C2×C4—91no—
403C2×C4—181no—
404C2×C4—181no—
405C2×C4—182no—
406C2×C4—364no—
407C2×C4C3×C4×C3⋊D311yescharacteristic, complemented
408C2×C3—28no—
409C2×C3—24no—
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—24no—
413C2×C3—48no—
414C2×C3—44no—
415C2×C3—44no—
416C2×C3—41no—
417C2×C3—84no—
418C2×C3—92no—
419C2×C3—181no—
420C2×C3—182no—
421C2×C3—361no—
422C2×C3C3×D3×C22⋊C414yes—
423C2×C3C22⋊C4×C3⋊D311yescharacteristic
424D3—38no—
425D3—64no—
426D3—68no—
427D3—124no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—91no—
432C22—91no—
433C22—92no—
434C22—184no—
435C22—182no—
436C22—181no—
437C22—362no—
438C22—361no—
439C22—361no—
440C22—362no—
441C22—362no—
442C22C2×C3×C4×C3⋊D311yescharacteristic
443C22C3×D4×C3⋊D312yes—
444C4—21no—
445C4—42no—
446C4—181no—
447C4—362no—
448C3—24no—
449C3C3×D3×#9914yescomplemented
450C3C3⋊D3×#9911yescharacteristic, direct product factor
451C2—22no—
452C2—21no—
453C2—41no—
454C2—92no—
455C2—181no—
456C2—182no—
457C2—361no—
458C2C3×C22⋊C4×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3C22⋊C4×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC33×#99C2
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC23×C32C23×C3
hypercenterC3×#99C3⋊D3
hypocenterC32C2×C3×#99
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
2. C3×C22⋊C4×C3⋊D3C2semidirect
3. C33×#99C2semidirect
4. C3×#63661C2semidirect
5. C3×#63633C2semidirect
6. C3⋊D3×#99C3direct (N × H)
8. C23×C3×C3⋊D3C4semidirect
11. C2×C3×#638C4semidirect
15. C2×C3×C4⋊(C3⋊D3)C4semidirect
17. C2×C3×C4×C3⋊D3C4semidirect
18. C33×C22⋊C4C22semidirect
19. C3×#3757C22semidirect
23. C22⋊C4×C3⋊D3C2×C3semidirect
26. C32×#99D3semidirect
27. C32×#99C2×C3semidirect
28. #63661C2×C3semidirect
29. #63633C2×C3semidirect
32. C23×C33C2×C4semidirect
40. C22×C3×C3⋊D3C2×C4semidirect
45. C2×C33×C4C2×C4semidirect
49. C2×C3×C3⋊Dic3C2×C4semidirect
56. C23×C3⋊D3C3×C4semidirect
63. C2×#638C3×C4semidirect
67. C2×C4⋊(C3⋊D3)C3×C4semidirect
69. C2×C4×C3⋊D3C3×C4semidirect
78. C32×C22⋊C4C2×D3semidirect
79. C32×C22⋊C4C22×C3semidirect
80. #3757C22×C3semidirect
111. C3×#99C3×D3semidirect
112. C3×#99C3⋊D3direct (N × H)
117. C23×C32C4×D3semidirect
118. C23×C32C2×C3×C4semidirect
127. C22×C3⋊D3C2×C3×C4semidirect
146. C2×C32×C4C4×D3semidirect
147. C2×C32×C4C2×C3×C4semidirect
151. C2×C3⋊Dic3C2×C3×C4semidirect
172. C3×C3⋊D3#99direct (N × H)
197. C3×C22⋊C4C2×C3×D3semidirect
198. C3×C22⋊C4C2×C3⋊D3semidirect
252. #99C3×C3⋊D3direct (N × H)
253. C33C2×#99semidirect
265. C23×C3C3×C4×D3semidirect
266. C23×C3C4×C3⋊D3semidirect
293. C2×C3×C4C3×C4×D3semidirect
294. C2×C3×C4C4×C3⋊D3semidirect
318. C3⋊D3C3×#99direct (N × H)
335. C22⋊C4C2×C3×C3⋊D3semidirect
384. C32D3×#99semidirect
385. C32C2×C3×#99semidirect
395. C23C3×C4×C3⋊D3semidirect
407. C2×C4C3×C4×C3⋊D3semidirect
449. C3C3×D3×#99semidirect
450. C3C3⋊D3×#99direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C3×C22⋊C4×C3⋊D32
C33×#992
C3×#636612
C3×#636332
C3⋊D3×#993
C3×D3×#993