Finite Group Databasebeta

← C3×#10589734 #13152891

Subgroups — C3×#10589734 #13152891

Order 1728.

914 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×D9C211yesmaximal, characteristic, complemented
2C22×C3×Dic18C211yesmaximal, characteristic, complemented
3C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
4C3×#63506C211yesmaximal, characteristic, complemented
5C3×C4×#611C211yesmaximal, characteristic
6C3×#63459C211yesmaximal, characteristic, complemented
7C3×#63458C212yesmaximal, complemented
8C3×#63450C212yesmaximal, complemented
9C3×#63449C211yesmaximal, characteristic, complemented
10C3×#611⋊C4C212yesmaximal
11C3×#63440C212yesmaximal, complemented
12C3×#5632—31nomaximal
13#10589734C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C22×C3×Dic9C2212yes—
18C3×Q8×D9—24no—
19C2×C3×C4×D9C2211yescharacteristic
20C2×C3×C4×D9C2212yes—
21C2×C3×Dic18—24no—
22C2×C3×Dic18C2211yescharacteristic
23C2×C3×Dic18C2212yes—
24C2×C3×Dic18C2212yes—
25C2×C3×Dic18C2211yescharacteristic
26C3×C9×C4⋊C4C2211yescharacteristic
27C3×C9×C4⋊C4C2212yes—
28C3×C9×C22⋊C4C2212yes—
29C3×#3676C2211yescharacteristic
30C3×(C2×D9)⋊C4C2211yescharacteristic
31C3×(C2×D9)⋊C4C2212yes—
32C3×C4⋊Dic9C2212yes—
33C3×Dic9⋊C4C2211yescharacteristic
34C3×Dic9⋊C4C2212yes—
35C3×Dic9⋊C4C2212yes—
36C3×Dic9⋊C4C2212yes—
37C3×C4×Dic9C2211yescharacteristic
38C3×C4×Dic9C2212yes—
39C2×C3×D3×Q8—31no—
40C22×C3×Dic6—31no—
41C32×C22⋊Q8D311yescharacteristic
42C3×#944—31no—
43C3×C4×#67—31no—
44C3×#897—31no—
45C3×#896—32no—
46C3×#888—32no—
47C3×#887—31no—
48C3×#67⋊C4—32no—
49C3×#878—32no—
50C2×Q8×D9C2×C311yescharacteristic, complemented
51C22×Dic18C2×C311yescharacteristic, complemented
52C9×C22⋊Q8—21no—
53C9×C22⋊Q8C2×C311yescharacteristic, complemented
54#63506C2×C311yescharacteristic, complemented
55C4×#611C2×C311yescharacteristic
56#63459C2×C311yescharacteristic, complemented
57#63458C2×C312yescomplemented
58#63450C2×C312yescomplemented
59#63449C2×C311yescharacteristic, complemented
60#611⋊C4C2×C312yes—
61#63440C2×C312yescomplemented
62C23×C3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C3×Q8×C9—22no—
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2312yes—
68C2×C3×C4×C9C2312yes—
69C2×C3×C4×C9C2311yescharacteristic
70C3×#611Q814yescomplemented
71C2×C3×Dic9—24no—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C3×C4×D9—22no—
78C3×C4×D9—24no—
79C3×Dic18—22no—
80C3×Dic18—24no—
81C3×Dic18—24no—
82C3×Dic18D414yescomplemented
83C3×D4×Q8—91no—
84#5632—31no—
85C2×C32×Q8C2×D311yescharacteristic
86C22×C32×C4C2×D311yescharacteristic
87C2×C3×#67—31no—
88C22×C3×Dic3—32no—
89C3×D3×Q8—64no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—31no—
92C2×C3×Dic6—31no—
93C2×C3×Dic6—32no—
94C2×C3×Dic6—32no—
95C2×C3×Dic6—31no—
96C2×C3×Dic6—64no—
97C32×C4⋊C4C2×D311yescharacteristic
98C32×C4⋊C4C2×D312yes—
99C32×C22⋊C4C2×D312yes—
100C3×#217—31no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—31no—
103C3×C4⋊Dic3—32no—
104C3×Dic3⋊C4—32no—
105C3×Dic3⋊C4—32no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—31no—
108C3×C4×Dic3—32no—
109C3×C4×Dic3—31no—
110C2×Q8×C9—21no—
111C2×Q8×C9C22×C311yescharacteristic
112C22×C4×C9—21no—
113C22×C4×C9C22×C311yescharacteristic
114C2×#611C22×C311yescharacteristic
115C22×Dic9C22×C312yes—
116Q8×D9—24no—
117C2×C4×D9C22×C311yescharacteristic
118C2×C4×D9C22×C312yes—
119C2×Dic18—24no—
120C2×Dic18C22×C311yescharacteristic
121C2×Dic18C22×C312yes—
122C2×Dic18C22×C312yes—
123C2×Dic18C22×C311yescharacteristic
124C9×C4⋊C4—21no—
125C9×C4⋊C4—22no—
126C9×C4⋊C4C22×C311yescharacteristic
127C9×C4⋊C4C22×C312yes—
128C9×C22⋊C4—22no—
129C9×C22⋊C4C22×C312yes—
130#3676C22×C311yescharacteristic
131(C2×D9)⋊C4C22×C311yescharacteristic
132(C2×D9)⋊C4C22×C312yes—
133C4⋊Dic9C22×C312yes—
134Dic9⋊C4C22×C311yescharacteristic
135Dic9⋊C4C22×C312yes—
136Dic9⋊C4C22×C312yes—
137Dic9⋊C4C22×C312yes—
138C4×Dic9C22×C311yescharacteristic
139C4×Dic9C22×C312yes—
140C22×C3×C9—21no—
141C22×C3×C9—21no—
142C22×C3×C9C2×Q812yescomplemented
143C22×C3×C9C2411yescharacteristic
144C2×C3×D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9C2×Q812yescomplemented
147C3×C4×C9—21no—
148C3×C4×C9—22no—
149C3×C4×C9—22no—
150C3×C4×C9C2×D412yes—
151C3×Dic9—21no—
152C3×Dic9—22no—
153C3×Dic9—21no—
154C3×Dic9C2×D414yes—
155C3×Dic9C2×Q812yescomplemented
156C22×C3×Q8—91no—
157C22×C3×Q8—91no—
158C2×D3×Q8—31no—
159C22×Dic6—31no—
160C3×C4⋊Q8—91no—
161C3×C4⋊Q8—92no—
162C3×C22⋊Q8—21no—
163C3×C22⋊Q8—92no—
164C3×C22⋊Q8—91no—
165C3×C22⋊Q8—92no—
166C3×C22⋊Q8D911yescharacteristic, complemented
167C3×C22⋊Q8C3×D311yescharacteristic
168C3×C4×Q8—91no—
169C3×C4×D4—91no—
170C3×C4×D4—92no—
171#944—31no—
172C4×#67—31no—
173#897—31no—
174#896—32no—
175#888—32no—
176#887—31no—
177#67⋊C4—32no—
178#878—32no—
179C23×C32C22×D311yescharacteristic
180C22×C3×D3—31no—
181C32×Q8—22no—
182C2×C32×C4—22no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C3×#67—34no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—64no—
194C3×C4×D3—64no—
195C3×C4×D3—62no—
196C3×Dic6—34no—
197C3×Dic6—64no—
198C3×Dic6—64no—
199C3×Dic6—62no—
200C23×C9—21no—
201C23×C9C23×C311yescharacteristic
202C22×D9C23×C311yescharacteristic
203Q8×C9—22no—
204Q8×C9—42no—
205C2×C4×C9—22no—
206C2×C4×C9—21no—
207C2×C4×C9—22no—
208C2×C4×C9—22no—
209C2×C4×C9—21no—
210C2×C4×C9—42no—
211C2×C4×C9C23×C312yes—
212C2×C4×C9C23×C312yes—
213C2×C4×C9C23×C311yescharacteristic
214C2×C4×C9C23×C311yescharacteristic
215#611C3×Q814yescomplemented
216C2×Dic9—24no—
217C2×Dic9C23×C311yescharacteristic
218C2×Dic9C23×C312yes—
219C2×Dic9C23×C311yescharacteristic
220C2×Dic9C23×C312yes—
221C2×Dic9C23×C311yescharacteristic
222C4×D9—22no—
223C4×D9—24no—
224Dic18—22no—
225Dic18—24no—
226Dic18—24no—
227Dic18C3×D414yescomplemented
228D4×Q8—91noSylow
229C2×C3×C9—22no—
230C2×C3×C9C22×D411yescharacteristic
231C2×C3×C9C22×Q811yescharacteristic
232C2×C3×C92−1+411yescharacteristic
233C3×D9—22no—
234C2×C3×Q8—21no—
235C2×C3×Q8—91no—
236C2×C3×Q8—92no—
237C2×C3×Q8—92no—
238C2×C3×Q8—91no—
239C2×C3×Q8—184no—
240C2×C3×Q8—184no—
241C2×C3×Q8C2×D911yescharacteristic
242C2×C3×Q8C2×C3×D311yescharacteristic
243C2×C3×D4—91no—
244C22×C3×C4—21no—
245C22×C3×C4—92no—
246C22×C3×C4—92no—
247C22×C3×C4—91no—
248C22×C3×C4C2×D911yescharacteristic
249C22×C3×C4C2×C3×D311yescharacteristic
250C2×#67—31no—
251C22×Dic3—32no—
252D3×Q8—64no—
253C2×C4×D3—32no—
254C2×C4×D3—31no—
255C2×Dic6—31no—
256C2×Dic6—32no—
257C2×Dic6—32no—
258C2×Dic6—31no—
259C2×Dic6—64no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—22no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4—92no—
265C3×C4⋊C4—92no—
266C3×C4⋊C4—92no—
267C3×C4⋊C4C2×D911yescharacteristic
268C3×C4⋊C4C2×D912yes—
269C3×C4⋊C4C2×C3×D311yescharacteristic
270C3×C4⋊C4C2×C3×D312yes—
271C3×C22⋊C4—22no—
272C3×C22⋊C4—91no—
273C3×C22⋊C4—91no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4C2×D912yes—
276C3×C22⋊C4C2×C3×D312yes—
277C3×C42—92no—
278C3×C42—91no—
279#217—31no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—31no—
282C4⋊Dic3—32no—
283Dic3⋊C4—32no—
284Dic3⋊C4—32no—
285Dic3⋊C4—32no—
286Dic3⋊C4—31no—
287C4×Dic3—32no—
288C4×Dic3—31no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32D3×Q812yes—
292C22×C32C23×D311yescharacteristic
293C2×C3×D3—32no—
294C2×C3×D3—61no—
295C2×C3×D3—61no—
296C32×C4—21no—
297C32×C4—22no—
298C32×C4—22no—
299C32×C4D3×D412yes—
300C3×Dic3—34no—
301C3×Dic3—32no—
302C3×Dic3—61no—
303C3×Dic3—62no—
304C3×Dic3—61no—
305C22×C9—21no—
306C22×C9—21no—
307C22×C9—21no—
308C22×C9—22no—
309C22×C9—41no—
310C22×C9—41no—
311C22×C9C2×C3×Q812yescomplemented
312C22×C9C24×C311yescharacteristic
313C2×D9—21no—
314C2×D9—21no—
315C2×D9C2×C3×Q812yescomplemented
316C4×C9—21no—
317C4×C9—22no—
318C4×C9—22no—
319C4×C9—22no—
320C4×C9—42no—
321C4×C9—41no—
322C4×C9—42no—
323C4×C9C2×C3×D412yes—
324Dic9—21no—
325Dic9—22no—
326Dic9—21no—
327Dic9C2×C3×D414yes—
328Dic9C2×C3×Q812yescomplemented
329C22×Q8—91no—
330C22×Q8—91no—
331C4⋊Q8—91no—
332C4⋊Q8—92no—
333C22⋊Q8—92no—
334C22⋊Q8—91no—
335C22⋊Q8—92no—
336C22⋊Q8C3×D911yescharacteristic, complemented
337C4×Q8—91no—
338C4×D4—91no—
339C4×D4—92no—
340C3×C9D4×Q811yescharacteristic, complemented, Sylow
341C23×C3—21no—
342C23×C3—91no—
343C23×C3C22×D911yescharacteristic
344C23×C3C22×C3×D311yescharacteristic
345C22×D3—31no—
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—42no—
349C3×Q8—94no—
350C3×Q8—184no—
351C3×Q8—184no—
352C3×Q8—182no—
353C3×D4—94no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—42no—
361C2×C3×C4—91no—
362C2×C3×C4—92no—
363C2×C3×C4—91no—
364C2×C3×C4—92no—
365C2×C3×C4—91no—
366C2×C3×C4—184no—
367C2×C3×C4—184no—
368C2×C3×C4—182no—
369C2×C3×C4C22×D912yes—
370C2×C3×C4C22×D911yescharacteristic
371C2×C3×C4C22×D912yes—
372C2×C3×C4C22×D911yescharacteristic
373C2×C3×C4C22×C3×D312yes—
374C2×C3×C4C22×C3×D312yes—
375C2×C3×C4C22×C3×D311yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377#67—34no—
378C2×Dic3—31no—
379C2×Dic3—32no—
380C2×Dic3—31no—
381C2×Dic3—32no—
382C2×Dic3—31no—
383C2×Dic3—64no—
384C4×D3—64no—
385C4×D3—62no—
386Dic6—34no—
387Dic6—64no—
388Dic6—64no—
389Dic6—62no—
390C2×C32—22no—
391C2×C32C2×D3×D411yescharacteristic
392C2×C32C2×D3×Q811yescharacteristic
393C2×C32#101011yescharacteristic
394C3×D3—62no—
395C2×C9—22no—
396C2×C9—21no—
397C2×C9—21no—
398C2×C9—21no—
399C2×C9—42no—
400C2×C9C22×C3×D411yescharacteristic
401C2×C9C22×C3×Q811yescharacteristic
402C2×C9C3×2−1+411yescharacteristic
403D9—22no—
404C2×Q8—91no—
405C2×Q8—92no—
406C2×Q8—92no—
407C2×Q8—91no—
408C2×Q8—184no—
409C2×Q8—184no—
410C2×Q8C2×C3×D911yescharacteristic
411C2×D4—91no—
412C22×C4—92no—
413C22×C4—92no—
414C22×C4—91no—
415C22×C4C2×C3×D911yescharacteristic
416C4⋊C4—91no—
417C4⋊C4—92no—
418C4⋊C4—92no—
419C4⋊C4—92no—
420C4⋊C4—92no—
421C4⋊C4C2×C3×D911yescharacteristic
422C4⋊C4C2×C3×D912yes—
423C22⋊C4—91no—
424C22⋊C4—91no—
425C22⋊C4—92no—
426C22⋊C4C2×C3×D912yes—
427C42—92no—
428C42—91no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—92no—
438C22×C3—181no—
439C22×C3—181no—
440C22×C3Q8×D912yescomplemented
441C22×C3C23×D911yescharacteristic
442C22×C3C3×D3×Q812yes—
443C22×C3C23×C3×D311yescharacteristic
444C2×D3—32no—
445C2×D3—61no—
446C2×D3—61no—
447C3×C4—21no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—41no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—94no—
458C3×C4—92no—
459C3×C4—182no—
460C3×C4—181no—
461C3×C4—181no—
462C3×C4D4×D912yes—
463C3×C4C3×D3×D412yes—
464Dic3—34no—
465Dic3—32no—
466Dic3—61no—
467Dic3—62no—
468Dic3—61no—
469C32#563211yescharacteristic
470C9—21no—
471C9C3×D4×Q811yescharacteristic, complemented
472C23—91no—
473C23C22×C3×D911yescharacteristic
474Q8—22no—
475Q8—94no—
476Q8—184no—
477Q8—184no—
478Q8—182no—
479D4—94no—
480C2×C4—22no—
481C2×C4—91no—
482C2×C4—92no—
483C2×C4—91no—
484C2×C4—91no—
485C2×C4—92no—
486C2×C4—184no—
487C2×C4—184no—
488C2×C4—182no—
489C2×C4C22×C3×D912yes—
490C2×C4C22×C3×D911yescharacteristic
491C2×C4C22×C3×D912yes—
492C2×C4C22×C3×D911yescharacteristic
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—42no—
499C2×C3—182no—
500C2×C3C2×D4×D911yescharacteristic
501C2×C3C2×Q8×D911yescharacteristic
502C2×C3#6371611yescharacteristic
503C2×C3C2×C3×D3×D411yescharacteristic
504C2×C3C2×C3×D3×Q811yescharacteristic
505C2×C3C3×#101011yescharacteristic
506D3—62no—
507C22—21no—
508C22—21no—
509C22—92no—
510C22—181no—
511C22—181no—
512C22C3×Q8×D912yescomplemented
513C22C23×C3×D911yescharacteristic
514C4—21no—
515C4—22no—
516C4—22no—
517C4—94no—
518C4—92no—
519C4—182no—
520C4—181no—
521C4—181no—
522C4C3×D4×D912yes—
523C3—21no—
524C3#1058973411yescharacteristic, direct product factor
525C3C3×#563211yescharacteristic
526C2—22no—
527C2—182no—
528C2C2×C3×D4×D911yescharacteristic
529C2C2×C3×Q8×D911yescharacteristic
530C2C3×#6371611yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×D4×Q8
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. C2×C3×Q8×D9C2semidirect
2. C22×C3×Dic18C2semidirect
3. C3×C9×C22⋊Q8C2semidirect
4. C3×#63506C2semidirect
6. C3×#63459C2semidirect
7. C3×#63458C2semidirect
8. C3×#63450C2semidirect
9. C3×#63449C2semidirect
11. C3×#63440C2semidirect
13. #10589734C3direct (N × H)
50. C2×Q8×D9C2×C3semidirect
51. C22×Dic18C2×C3semidirect
53. C9×C22⋊Q8C2×C3semidirect
54. #63506C2×C3semidirect
56. #63459C2×C3semidirect
57. #63458C2×C3semidirect
58. #63450C2×C3semidirect
59. #63449C2×C3semidirect
61. #63440C2×C3semidirect
70. C3×#611Q8semidirect
82. C3×Dic18D4semidirect
142. C22×C3×C9C2×Q8semidirect
146. C2×C3×D9C2×Q8semidirect
155. C3×Dic9C2×Q8semidirect
166. C3×C22⋊Q8D9semidirect
215. #611C3×Q8semidirect
227. Dic18C3×D4semidirect
311. C22×C9C2×C3×Q8semidirect
315. C2×D9C2×C3×Q8semidirect
328. Dic9C2×C3×Q8semidirect
336. C22⋊Q8C3×D9semidirect
340. C3×C9D4×Q8semidirect
440. C22×C3Q8×D9semidirect
471. C9C3×D4×Q8semidirect
512. C22C3×Q8×D9semidirect
524. C3#10589734direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D92
C22×C3×Dic182
C3×C9×C22⋊Q82
C3×#635062
C3×C4×#6112
C3×#634592
C3×#634582
C3×#634502
C3×#634492
C3×#611⋊C42
C3×#634402
C3×#56323
#105897343