Finite Group Databasebeta

← C3×#10589609 #13152405

Subgroups — C3×#10589609 #13152405

Order 1728.

968 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C212yesmaximal, complemented
2C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#63492C211yesmaximal, characteristic, complemented
4C3×#63491C212yesmaximal, complemented
5C3×C4×#611C211yesmaximal, characteristic, complemented
6C3×#63458C212yesmaximal, complemented
7C3×#63457C211yesmaximal, characteristic, complemented
8C3×#63456C212yesmaximal, complemented
9C3×#63455C211yesmaximal, characteristic, complemented
10C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
11C3×#63450C211yesmaximal, characteristic, complemented
12C3×#5507—31nomaximal
13#10589609C311yesmaximal, characteristic, direct product factor
14C22×C3×C4×C9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2212yes—
17C2×C3×#611C2211yescharacteristic
18C3×#3697—28no—
19C2×C3×D36C2212yes—
20C2×C3×D36C2211yescharacteristic
21C2×C3×C4×D9C2212yes—
22C2×C3×C4×D9C2212yes—
23C2×C3×C4×D9C2211yescharacteristic
24C2×C3×Dic18C2212yes—
25C3×C9×C4⋊C4C2211yescharacteristic
26C3×C9×C4⋊C4C2211yescharacteristic
27C3×C9×C4⋊C4C2212yes—
28C3×#3676C2211yescharacteristic
29C3×(C2×D9)⋊C4C2211yescharacteristic
30C3×(C2×D9)⋊C4C2212yes—
31C3×(C2×D9)⋊C4C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C3×C4⋊Dic9C2211yescharacteristic
34C3×Dic9⋊C4C2211yescharacteristic
35C3×Dic9⋊C4C2212yes—
36C3×Dic9⋊C4C2212yes—
37C3×C4×Dic9C2211yescharacteristic
38C2×C3×#235—32no—
39C2×C32×C4⋊C4D311yescharacteristic
40C3×#930—31no—
41C3×#929—32no—
42C3×C4×#67—31no—
43C3×#896—32no—
44C3×#895—31no—
45C3×#894—32no—
46C3×#893—31no—
47C3×D3×C4⋊C4—31no—
48C3×#888—31no—
49C2×#3697C2×C312yescomplemented
50C2×C9×C4⋊C4—21no—
51C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
52#63492C2×C311yescharacteristic, complemented
53#63491C2×C312yescomplemented
54C4×#611C2×C311yescharacteristic, complemented
55#63458C2×C312yescomplemented
56#63457C2×C311yescharacteristic, complemented
57#63456C2×C312yescomplemented
58#63455C2×C311yescharacteristic, complemented
59C4⋊C4×D9C2×C311yescharacteristic, complemented
60#63450C2×C311yescharacteristic, complemented
61C23×C3×C9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9—24no—
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2312yes—
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2311yescharacteristic
70C3×#611—24no—
71C3×#611D414yescomplemented
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C3×D36—24no—
77C3×D36—22no—
78C3×C4×D9—24no—
79C3×C4×D9—22no—
80C3×C4×D9—24no—
81C3×Dic18—24no—
82C3×#547—91no—
83#5507—31no—
84C22×C32×C4C2×D311yescharacteristic
85C22×C32×C4C2×D312yes—
86C2×C3×#67—31no—
87C2×C3×#67—32no—
88C3×#235—68no—
89C2×C3×D12—31no—
90C2×C3×D12—32no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—32no—
94C2×C3×Dic6—32no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C4⋊C4C2×D311yescharacteristic
97C32×C4⋊C4C2×D312yes—
98C3×#217—31no—
99C3×(C2×D3)⋊C4—32no—
100C3×(C2×D3)⋊C4—31no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3—31no—
104C3×Dic3⋊C4—32no—
105C3×Dic3⋊C4—32no—
106C3×Dic3⋊C4—31no—
107C3×C4×Dic3—31no—
108C22×C4×C9—21no—
109C22×C4×C9—22no—
110C22×C4×C9C22×C311yescharacteristic
111C22×C4×C9C22×C312yes—
112C2×#611C22×C312yes—
113C2×#611C22×C311yescharacteristic
114#3697—28no—
115C2×D36C22×C311yescharacteristic
116C2×D36C22×C312yes—
117C2×C4×D9C22×C311yescharacteristic
118C2×C4×D9C22×C312yes—
119C2×C4×D9C22×C312yes—
120C2×Dic18C22×C312yes—
121C9×C4⋊C4—21no—
122C9×C4⋊C4—21no—
123C9×C4⋊C4—22no—
124C9×C4⋊C4C22×C311yescharacteristic
125C9×C4⋊C4C22×C311yescharacteristic
126C9×C4⋊C4C22×C312yes—
127#3676C22×C311yescharacteristic
128(C2×D9)⋊C4C22×C311yescharacteristic
129(C2×D9)⋊C4C22×C312yes—
130(C2×D9)⋊C4C22×C312yes—
131(C2×D9)⋊C4C22×C312yes—
132C4⋊Dic9C22×C311yescharacteristic
133Dic9⋊C4C22×C311yescharacteristic
134Dic9⋊C4C22×C312yes—
135Dic9⋊C4C22×C312yes—
136C4×Dic9C22×C311yescharacteristic
137C22×C3×C9—21no—
138C22×C3×C9—21no—
139C22×C3×C9C2×D412yes—
140C22×C3×C9C2411yescharacteristic
141C2×C3×D9—21no—
142C2×C3×D9—21no—
143C2×C3×D9—22no—
144C2×C3×D9—22no—
145C2×C3×D9—22no—
146C2×C3×D9C2×D412yes—
147C3×C4×C9—22no—
148C3×C4×C9—22no—
149C3×C4×C9—21no—
150C3×C4×C9#4112yescomplemented
151C3×Dic9—22no—
152C3×Dic9—21no—
153C3×Dic9—21no—
154C3×Dic9C2×D412yes—
155C2×C3×#41—92no—
156C2×#235—32no—
157C3×C4⋊Q8—91no—
158C3×#123—92no—
159C3×#123—92no—
160C3×C22⋊Q8—92no—
161C3×#121—91no—
162C3×#121—91no—
163C3×C4×D4—91no—
164C3×C4×D4—91no—
165C2×C3×C4⋊C4—21no—
166C2×C3×C4⋊C4—91no—
167C2×C3×C4⋊C4D911yescharacteristic, complemented
168C2×C3×C4⋊C4C3×D311yescharacteristic
169#930—31no—
170#929—32no—
171C4×#67—31no—
172#896—32no—
173#895—31no—
174#894—32no—
175#893—31no—
176D3×C4⋊C4—31no—
177#888—31no—
178C23×C32C22×D311yescharacteristic
179C22×C3×D3—31no—
180C22×C3×D3—32no—
181C2×C32×C4—22no—
182C2×C32×C4—24no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C3×#67—34no—
188C3×#67—64no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—32no—
193C3×D12—64no—
194C3×D12—62no—
195C3×C4×D3—64no—
196C3×C4×D3—64no—
197C3×C4×D3—62no—
198C3×Dic6—64no—
199C23×C9—21no—
200C23×C9C23×C311yescharacteristic
201C22×D9C23×C312yes—
202C22×D9C23×C311yescharacteristic
203C2×C4×C9—24no—
204C2×C4×C9—24no—
205C2×C4×C9—22no—
206C2×C4×C9—21no—
207C2×C4×C9—21no—
208C2×C4×C9—22no—
209C2×C4×C9—22no—
210C2×C4×C9—44no—
211C2×C4×C9C23×C311yescharacteristic
212C2×C4×C9C23×C312yes—
213C2×C4×C9C23×C312yes—
214C2×C4×C9C23×C311yescharacteristic
215#611—24no—
216#611C3×D414yescomplemented
217C2×Dic9C23×C311yescharacteristic
218C2×Dic9C23×C311yescharacteristic
219C2×Dic9C23×C312yes—
220C2×Dic9C23×C311yescharacteristic
221D36—24no—
222D36—22no—
223C4×D9—24no—
224C4×D9—22no—
225C4×D9—24no—
226Dic18—24no—
227#547—91noSylow
228C2×C3×C9—22no—
229C2×C3×C9C22×D411yescharacteristic
230C2×C3×C9C2×#4111yescharacteristic
231C2×C3×C92−1+411yescharacteristic
232C3×D9—22no—
233C3×D9—42no—
234C3×#41—188no—
235C2×C3×Q8—92no—
236C2×C3×D4—91no—
237C2×C3×D4—92no—
238C2×C3×D4—91no—
239C2×C3×D4—92no—
240C22×C3×C4—21no—
241C22×C3×C4—22no—
242C22×C3×C4—91no—
243C22×C3×C4—92no—
244C22×C3×C4—92no—
245C22×C3×C4C2×D911yescharacteristic
246C22×C3×C4C2×D912yes—
247C22×C3×C4C2×C3×D311yescharacteristic
248C22×C3×C4C2×C3×D312yes—
249C2×#67—31no—
250C2×#67—32no—
251#235—68no—
252C2×D12—31no—
253C2×D12—32no—
254C2×C4×D3—32no—
255C2×C4×D3—31no—
256C2×C4×D3—32no—
257C2×Dic6—32no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—22no—
261C3×C4⋊C4—92no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4—91no—
265C3×C4⋊C4C2×D911yescharacteristic
266C3×C4⋊C4C2×D911yescharacteristic
267C3×C4⋊C4C2×D912yes—
268C3×C4⋊C4C2×C3×D311yescharacteristic
269C3×C4⋊C4C2×C3×D311yescharacteristic
270C3×C4⋊C4C2×C3×D312yes—
271C3×C22⋊C4—91no—
272C3×C22⋊C4—91no—
273C3×C22⋊C4—92no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4—92no—
276C3×C42—91no—
277#217—31no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—31no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—32no—
282C4⋊Dic3—31no—
283Dic3⋊C4—32no—
284Dic3⋊C4—32no—
285Dic3⋊C4—31no—
286C4×Dic3—31no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32D3×D412yes—
290C22×C32C23×D311yescharacteristic
291C2×C3×D3—32no—
292C2×C3×D3—61no—
293C2×C3×D3—61no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C32×C4—21no—
298C32×C4—22no—
299C32×C4—22no—
300C32×C4#23512yes—
301C3×Dic3—32no—
302C3×Dic3—61no—
303C3×Dic3—62no—
304C3×Dic3—61no—
305C22×C9—22no—
306C22×C9—22no—
307C22×C9—21no—
308C22×C9—21no—
309C22×C9—21no—
310C22×C9—22no—
311C22×C9C2×C3×D412yes—
312C22×C9C24×C311yescharacteristic
313C2×D9—21no—
314C2×D9—21no—
315C2×D9—22no—
316C2×D9—22no—
317C2×D9—22no—
318C2×D9C2×C3×D412yes—
319C4×C9—22no—
320C4×C9—22no—
321C4×C9—22no—
322C4×C9—21no—
323C4×C9—22no—
324C4×C9—42no—
325C4×C9—42no—
326C4×C9C3×#4112yescomplemented
327Dic9—22no—
328Dic9—21no—
329Dic9—21no—
330Dic9C2×C3×D412yes—
331C2×#41—92no—
332C4⋊Q8—91no—
333#123—92no—
334#123—92no—
335C22⋊Q8—92no—
336#121—91no—
337#121—91no—
338C4×D4—91no—
339C4×D4—91no—
340C2×C4⋊C4—91no—
341C2×C4⋊C4C3×D911yescharacteristic, complemented
342C3×C9#54711yescharacteristic, complemented, Sylow
343C23×C3—21no—
344C23×C3—91no—
345C23×C3—92no—
346C23×C3C22×D911yescharacteristic
347C23×C3C22×C3×D311yescharacteristic
348C22×D3—31no—
349C22×D3—32no—
350C3×Q8—184no—
351C3×D4—94no—
352C3×D4—184no—
353C3×D4—184no—
354C3×D4—182no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—44no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—92no—
369C2×C3×C4—184no—
370C2×C3×C4—184no—
371C2×C3×C4—182no—
372C2×C3×C4C22×D911yescharacteristic
373C2×C3×C4C22×D912yes—
374C2×C3×C4C22×D912yes—
375C2×C3×C4C22×D911yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×C3×C4C22×C3×D312yes—
378C2×C3×C4C22×C3×D312yes—
379C2×C3×C4C22×C3×D311yescharacteristic
380#67—34no—
381#67—64no—
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—31no—
385C2×Dic3—32no—
386D12—64no—
387D12—62no—
388C4×D3—64no—
389C4×D3—64no—
390C4×D3—62no—
391Dic6—64no—
392C2×C32—22no—
393C2×C32C2×#23511yescharacteristic
394C2×C32C2×D3×D411yescharacteristic
395C2×C32#100711yescharacteristic
396C3×D3—62no—
397C3×D3—122no—
398C2×C9—24no—
399C2×C9—22no—
400C2×C9—21no—
401C2×C9—21no—
402C2×C9—21no—
403C2×C9C22×C3×D411yescharacteristic
404C2×C9C2×C3×#4111yescharacteristic
405C2×C9C3×2−1+411yescharacteristic
406D9—22no—
407D9—42no—
408#41—188no—
409C2×Q8—92no—
410C2×D4—91no—
411C2×D4—92no—
412C2×D4—91no—
413C2×D4—92no—
414C22×C4—91no—
415C22×C4—92no—
416C22×C4—92no—
417C22×C4C2×C3×D911yescharacteristic
418C22×C4C2×C3×D912yes—
419C4⋊C4—91no—
420C4⋊C4—92no—
421C4⋊C4—91no—
422C4⋊C4—92no—
423C4⋊C4C2×C3×D911yescharacteristic
424C4⋊C4C2×C3×D911yescharacteristic
425C4⋊C4C2×C3×D912yes—
426C22⋊C4—91no—
427C22⋊C4—91no—
428C22⋊C4—92no—
429C22⋊C4—92no—
430C22⋊C4—92no—
431C42—91no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—22no—
440C22×C3—92no—
441C22×C3—181no—
442C22×C3—181no—
443C22×C3—182no—
444C22×C3—182no—
445C22×C3—182no—
446C22×C3D4×D912yes—
447C22×C3C23×D911yescharacteristic
448C22×C3C3×D3×D412yes—
449C22×C3C23×C3×D311yescharacteristic
450C2×D3—32no—
451C2×D3—61no—
452C2×D3—61no—
453C2×D3—62no—
454C2×D3—62no—
455C2×D3—62no—
456C3×C4—22no—
457C3×C4—21no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—21no—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—92no—
467C3×C4—181no—
468C3×C4—182no—
469C3×C4—181no—
470C3×C4#369712yescomplemented
471C3×C4C3×#23512yes—
472Dic3—32no—
473Dic3—61no—
474Dic3—62no—
475Dic3—61no—
476C32#550711yescharacteristic
477C9—21no—
478C9C3×#54711yescharacteristic, complemented
479C23—91no—
480C23—92no—
481C23C22×C3×D911yescharacteristic
482Q8—184no—
483D4—94no—
484D4—184no—
485D4—184no—
486D4—182no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—91no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—92no—
493C2×C4—184no—
494C2×C4—184no—
495C2×C4—182no—
496C2×C4C22×C3×D911yescharacteristic
497C2×C4C22×C3×D912yes—
498C2×C4C22×C3×D912yes—
499C2×C4C22×C3×D911yescharacteristic
500C2×C3—24no—
501C2×C3—22no—
502C2×C3—22no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—182no—
507C2×C3—362no—
508C2×C3C2×#369711yescharacteristic
509C2×C3C2×D4×D911yescharacteristic
510C2×C3#6371311yescharacteristic
511C2×C3C2×C3×#23511yescharacteristic
512C2×C3C2×C3×D3×D411yescharacteristic
513C2×C3C3×#100711yescharacteristic
514D3—62no—
515D3—122no—
516C22—21no—
517C22—21no—
518C22—92no—
519C22—181no—
520C22—181no—
521C22—182no—
522C22—182no—
523C22—182no—
524C22C3×D4×D912yes—
525C22C23×C3×D911yescharacteristic
526C4—21no—
527C4—22no—
528C4—22no—
529C4—92no—
530C4—181no—
531C4—182no—
532C4—181no—
533C4C3×#369712yescomplemented
534C3—21no—
535C3#1058960911yescharacteristic, direct product factor
536C3C3×#550711yescharacteristic
537C2—22no—
538C2—182no—
539C2—362no—
540C2C2×C3×#369711yescharacteristic
541C2C2×C3×D4×D911yescharacteristic
542C2C3×#6371311yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC22×C32C23×D3
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#547
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3697C2semidirect
2. C2×C3×C9×C4⋊C4C2semidirect
3. C3×#63492C2semidirect
4. C3×#63491C2semidirect
5. C3×C4×#611C2semidirect
6. C3×#63458C2semidirect
7. C3×#63457C2semidirect
8. C3×#63456C2semidirect
9. C3×#63455C2semidirect
10. C3×C4⋊C4×D9C2semidirect
11. C3×#63450C2semidirect
13. #10589609C3direct (N × H)
49. C2×#3697C2×C3semidirect
51. C2×C9×C4⋊C4C2×C3semidirect
52. #63492C2×C3semidirect
53. #63491C2×C3semidirect
54. C4×#611C2×C3semidirect
55. #63458C2×C3semidirect
56. #63457C2×C3semidirect
57. #63456C2×C3semidirect
58. #63455C2×C3semidirect
59. C4⋊C4×D9C2×C3semidirect
60. #63450C2×C3semidirect
71. C3×#611D4semidirect
150. C3×C4×C9#41semidirect
167. C2×C3×C4⋊C4D9semidirect
216. #611C3×D4semidirect
326. C4×C9C3×#41semidirect
341. C2×C4⋊C4C3×D9semidirect
342. C3×C9#547semidirect
470. C3×C4#3697semidirect
478. C9C3×#547semidirect
533. C4C3×#3697semidirect
535. C3#10589609direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×C3×C9×C4⋊C42
C3×#634922
C3×#634912
C3×C4×#6112
C3×#634582
C3×#634572
C3×#634562
C3×#634552
C3×C4⋊C4×D92
C3×#634502
C3×#55073
#105896093