Finite Group Databasebeta

← C3×#10589731 #13152910

Subgroups — C3×#10589731 #13152910

Order 1728.

956 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3701C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
4C3×#63508C211yesmaximal, characteristic, complemented
5C3×#63492C211yesmaximal, characteristic, complemented
6C3×#63458C212yesmaximal, complemented
7C3×#63457C211yesmaximal, characteristic, complemented
8C3×#63457C212yesmaximal, complemented
9C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
10C3×#63446C212yesmaximal, complemented
11C3×#63446C212yesmaximal, complemented
12C3×#5629—31nomaximal
13#10589731C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C2×C3×#611C2212yes—
19C3×#3701—24no—
20C3×#3697—24no—
21C2×C3×D36C2211yescharacteristic
22C2×C3×D36C2211yescharacteristic
23C2×C3×D36C2212yes—
24C2×C3×D36C2212yes—
25C2×C3×C4×D9C2211yescharacteristic
26C2×C3×C4×D9C2212yes—
27C2×C3×C4×D9C2212yes—
28C2×C3×C4×D9C2211yescharacteristic
29C2×C3×Dic18C2211yescharacteristic
30C3×C9×C4⋊C4C2211yescharacteristic
31C3×C9×C4⋊C4C2212yes—
32C3×C9×C22⋊C4C2212yes—
33C3×(C2×D9)⋊C4C2212yes—
34C3×(C2×D9)⋊C4C2212yes—
35C3×(C2×D9)⋊C4C2212yes—
36C3×C4⋊Dic9C2211yescharacteristic
37C3×Dic9⋊C4C2212yes—
38C3×Dic9⋊C4C2212yes—
39C2×C3×#239—31no—
40C2×C3×#235—31no—
41C32×C22⋊Q8D311yescharacteristic
42C3×#946—31no—
43C3×#930—31no—
44C3×#896—32no—
45C3×#895—31no—
46C3×#895—32no—
47C3×D3×C4⋊C4—31no—
48C3×#884—32no—
49C3×#884—32no—
50C2×#3701C2×C311yescharacteristic, complemented
51C2×#3697C2×C311yescharacteristic, complemented
52C9×C22⋊Q8—21no—
53C9×C22⋊Q8C2×C311yescharacteristic, complemented
54#63508C2×C311yescharacteristic, complemented
55#63492C2×C311yescharacteristic, complemented
56#63458C2×C312yescomplemented
57#63457C2×C312yescomplemented
58#63457C2×C311yescharacteristic, complemented
59C4⋊C4×D9C2×C311yescharacteristic, complemented
60#63446C2×C312yescomplemented
61#63446C2×C312yescomplemented
62C23×C3×C9C2311yescharacteristic
63C22×C3×D9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C22×C3×D9C2311yescharacteristic
66C3×Q8×C9—22no—
67C2×C3×C4×C9—22no—
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2312yes—
70C2×C3×C4×C9C2312yes—
71C2×C3×C4×C9C2311yescharacteristic
72C3×#611—22no—
73C3×#611—22no—
74C3×#611—42no—
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2311yescharacteristic
78C3×D36—22no—
79C3×D36—24no—
80C3×D36—22no—
81C3×D36—42no—
82C3×C4×D9—22no—
83C3×C4×D9—24no—
84C3×C4×D9—42no—
85C3×C4×D9D414yescomplemented
86C3×Dic18—22no—
87C3×#537—91no—
88#5629—31no—
89C2×C32×Q8C2×D311yescharacteristic
90C22×C32×C4C2×D311yescharacteristic
91C2×C3×#67—31no—
92C2×C3×#67—31no—
93C2×C3×#67—32no—
94C3×#239—64no—
95C3×#235—64no—
96C2×C3×D12—32no—
97C2×C3×D12—31no—
98C2×C3×D12—31no—
99C2×C3×D12—32no—
100C2×C3×C4×D3—31no—
101C2×C3×C4×D3—32no—
102C2×C3×C4×D3—31no—
103C2×C3×C4×D3—32no—
104C2×C3×Dic6—31no—
105C32×C4⋊C4C2×D311yescharacteristic
106C32×C4⋊C4C2×D312yes—
107C32×C22⋊C4C2×D312yes—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×C4⋊Dic3—31no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C2×Q8×C9—21no—
115C2×Q8×C9C22×C311yescharacteristic
116C22×C4×C9—21no—
117C22×C4×C9C22×C311yescharacteristic
118C2×#611C22×C311yescharacteristic
119C2×#611C22×C311yescharacteristic
120C2×#611C22×C312yes—
121#3701—24no—
122#3697—24no—
123C2×D36C22×C311yescharacteristic
124C2×D36C22×C312yes—
125C2×D36C22×C311yescharacteristic
126C2×D36C22×C312yes—
127C2×C4×D9C22×C311yescharacteristic
128C2×C4×D9C22×C312yes—
129C2×C4×D9C22×C312yes—
130C2×C4×D9C22×C311yescharacteristic
131C2×Dic18C22×C311yescharacteristic
132C9×C4⋊C4—21no—
133C9×C4⋊C4—22no—
134C9×C4⋊C4C22×C311yescharacteristic
135C9×C4⋊C4C22×C312yes—
136C9×C22⋊C4—22no—
137C9×C22⋊C4C22×C312yes—
138(C2×D9)⋊C4C22×C312yes—
139(C2×D9)⋊C4C22×C312yes—
140(C2×D9)⋊C4C22×C312yes—
141C4⋊Dic9C22×C311yescharacteristic
142Dic9⋊C4C22×C312yes—
143Dic9⋊C4C22×C312yes—
144C22×C3×C9—21no—
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9C2411yescharacteristic
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9—22no—
151C2×C3×D9—21no—
152C2×C3×D9—21no—
153C2×C3×D9—22no—
154C2×C3×D9—22no—
155C2×C3×D9—21no—
156C2×C3×D9C2×D412yes—
157C3×C4×C9—21no—
158C3×C4×C9—22no—
159C3×C4×C9—22no—
160C3×C4×C9C2×D412yes—
161C3×Dic9—21no—
162C3×Dic9—22no—
163C3×Dic9C2×D412yes—
164C2×C3×#41—91no—
165C2×C3×#41—91no—
166C2×#239—31no—
167C2×#235—31no—
168C3×C22⋊Q8—21no—
169C3×C22⋊Q8—91no—
170C3×C22⋊Q8—92no—
171C3×C22⋊Q8D911yescharacteristic, complemented
172C3×C22⋊Q8C3×D311yescharacteristic
173C3×#121—91no—
174C3×#121—91no—
175C3×#121—92no—
176C3×#121—92no—
177C3×#121—92no—
178C2×C3×C4⋊C4—91no—
179#946—31no—
180#930—31no—
181#896—32no—
182#895—31no—
183#895—32no—
184D3×C4⋊C4—31no—
185#884—32no—
186#884—32no—
187C23×C32C22×D311yescharacteristic
188C22×C3×D3—31no—
189C22×C3×D3—32no—
190C22×C3×D3—31no—
191C32×Q8—22no—
192C2×C32×C4—22no—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C3×#67—62no—
198C3×#67—62no—
199C3×#67—122no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C3×D12—62no—
204C3×D12—64no—
205C3×D12—62no—
206C3×D12—122no—
207C3×C4×D3—34no—
208C3×C4×D3—64no—
209C3×C4×D3—62no—
210C3×C4×D3—122no—
211C3×Dic6—62no—
212C23×C9—21no—
213C23×C9C23×C311yescharacteristic
214C22×D9C23×C312yes—
215C22×D9C23×C311yescharacteristic
216C22×D9C23×C311yescharacteristic
217Q8×C9—22no—
218Q8×C9—42no—
219C2×C4×C9—22no—
220C2×C4×C9—21no—
221C2×C4×C9—22no—
222C2×C4×C9—22no—
223C2×C4×C9—21no—
224C2×C4×C9—42no—
225C2×C4×C9C23×C312yes—
226C2×C4×C9C23×C312yes—
227C2×C4×C9C23×C311yescharacteristic
228C2×C4×C9C23×C311yescharacteristic
229#611—22no—
230#611—22no—
231#611—42no—
232C2×Dic9C23×C312yes—
233C2×Dic9C23×C311yescharacteristic
234C2×Dic9C23×C311yescharacteristic
235D36—22no—
236D36—24no—
237D36—22no—
238D36—42no—
239C4×D9—22no—
240C4×D9—24no—
241C4×D9—42no—
242C4×D9C3×D414yescomplemented
243Dic18—22no—
244#537—91noSylow
245C2×C3×C9—41no—
246C2×C3×C9C22×D411yescharacteristic
247C2×C3×C9H(5,𝔽2)11yescharacteristic
248C2×C3×C92−1+411yescharacteristic
249C3×D9—22no—
250C3×D9—42no—
251C3×D9—41no—
252C3×#41—184no—
253C3×#41—184no—
254C2×C3×Q8—21no—
255C2×C3×Q8—91no—
256C2×C3×Q8C2×D911yescharacteristic
257C2×C3×Q8C2×C3×D311yescharacteristic
258C2×C3×D4—91no—
259C2×C3×D4—91no—
260C2×C3×D4—92no—
261C2×C3×D4—91no—
262C2×C3×D4—92no—
263C2×C3×D4—91no—
264C2×C3×D4—92no—
265C22×C3×C4—21no—
266C22×C3×C4—91no—
267C22×C3×C4—92no—
268C22×C3×C4—91no—
269C22×C3×C4—92no—
270C22×C3×C4C2×D911yescharacteristic
271C22×C3×C4C2×C3×D311yescharacteristic
272C2×#67—31no—
273C2×#67—31no—
274C2×#67—32no—
275#239—64no—
276#235—64no—
277C2×D12—32no—
278C2×D12—31no—
279C2×D12—31no—
280C2×D12—32no—
281C2×C4×D3—31no—
282C2×C4×D3—32no—
283C2×C4×D3—31no—
284C2×C4×D3—32no—
285C2×Dic6—31no—
286C3×C4⋊C4—21no—
287C3×C4⋊C4—22no—
288C3×C4⋊C4—92no—
289C3×C4⋊C4—92no—
290C3×C4⋊C4—91no—
291C3×C4⋊C4C2×D911yescharacteristic
292C3×C4⋊C4C2×D912yes—
293C3×C4⋊C4C2×C3×D311yescharacteristic
294C3×C4⋊C4C2×C3×D312yes—
295C3×C22⋊C4—22no—
296C3×C22⋊C4—92no—
297C3×C22⋊C4—92no—
298C3×C22⋊C4—92no—
299C3×C22⋊C4C2×D912yes—
300C3×C22⋊C4C2×C3×D312yes—
301(C2×D3)⋊C4—32no—
302(C2×D3)⋊C4—32no—
303(C2×D3)⋊C4—32no—
304C4⋊Dic3—31no—
305Dic3⋊C4—32no—
306Dic3⋊C4—32no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32C23×D311yescharacteristic
311C2×C3×D3—32no—
312C2×C3×D3—61no—
313C2×C3×D3—61no—
314C2×C3×D3—62no—
315C2×C3×D3—61no—
316C2×C3×D3—62no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—62no—
320C32×C4—21no—
321C32×C4—22no—
322C32×C4—22no—
323C32×C4D3×D412yes—
324C3×Dic3—32no—
325C3×Dic3—61no—
326C3×Dic3—62no—
327C22×C9—22no—
328C22×C9—21no—
329C22×C9—21no—
330C22×C9—21no—
331C22×C9—21no—
332C22×C9—41no—
333C22×C9—41no—
334C22×C9C24×C311yescharacteristic
335C2×D9—21no—
336C2×D9—21no—
337C2×D9—22no—
338C2×D9—21no—
339C2×D9—21no—
340C2×D9—22no—
341C2×D9—22no—
342C2×D9—21no—
343C2×D9C2×C3×D412yes—
344C4×C9—21no—
345C4×C9—22no—
346C4×C9—22no—
347C4×C9—22no—
348C4×C9—42no—
349C4×C9—41no—
350C4×C9—42no—
351C4×C9C2×C3×D412yes—
352Dic9—21no—
353Dic9—22no—
354Dic9C2×C3×D412yes—
355C2×#41—91no—
356C2×#41—91no—
357C22⋊Q8—91no—
358C22⋊Q8—92no—
359C22⋊Q8C3×D911yescharacteristic, complemented
360#121—91no—
361#121—91no—
362#121—92no—
363#121—92no—
364#121—92no—
365C2×C4⋊C4—91no—
366C3×C9#53711yescharacteristic, complemented, Sylow
367C23×C3—21no—
368C23×C3—91no—
369C23×C3—91no—
370C23×C3—92no—
371C23×C3C22×D911yescharacteristic
372C23×C3C22×C3×D311yescharacteristic
373C22×D3—31no—
374C22×D3—32no—
375C22×D3—31no—
376C3×Q8—22no—
377C3×Q8—22no—
378C3×Q8—42no—
379C3×Q8—182no—
380C3×D4—182no—
381C3×D4—184no—
382C3×D4—182no—
383C3×D4—182no—
384C3×D4—182no—
385C3×D4—362no—
386C3×D4—362no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—42no—
394C2×C3×C4—91no—
395C2×C3×C4—94no—
396C2×C3×C4—92no—
397C2×C3×C4—91no—
398C2×C3×C4—184no—
399C2×C3×C4—182no—
400C2×C3×C4—362no—
401C2×C3×C4C22×D911yescharacteristic
402C2×C3×C4C22×D912yes—
403C2×C3×C4C22×D912yes—
404C2×C3×C4C22×D911yescharacteristic
405C2×C3×C4C22×C3×D312yes—
406C2×C3×C4C22×C3×D312yes—
407C2×C3×C4C22×C3×D311yescharacteristic
408C2×C3×C4C22×C3×D311yescharacteristic
409#67—62no—
410#67—62no—
411#67—122no—
412C2×Dic3—31no—
413C2×Dic3—32no—
414C2×Dic3—31no—
415D12—62no—
416D12—64no—
417D12—62no—
418D12—122no—
419C4×D3—34no—
420C4×D3—64no—
421C4×D3—62no—
422C4×D3—122no—
423Dic6—62no—
424C2×C32—41no—
425C2×C32C2×D3×D411yescharacteristic
426C2×C32#100711yescharacteristic
427C2×C32#100911yescharacteristic
428C3×D3—62no—
429C3×D3—122no—
430C3×D3—121no—
431C2×C9—21no—
432C2×C9—21no—
433C2×C9—21no—
434C2×C9—42no—
435C2×C9—41no—
436C2×C9C22×C3×D411yescharacteristic
437C2×C9C3×H(5,𝔽2)11yescharacteristic
438C2×C9C3×2−1+411yescharacteristic
439D9—22no—
440D9—42no—
441D9—41no—
442#41—184no—
443#41—184no—
444C2×Q8—91no—
445C2×Q8C2×C3×D911yescharacteristic
446C2×D4—91no—
447C2×D4—91no—
448C2×D4—92no—
449C2×D4—91no—
450C2×D4—92no—
451C2×D4—91no—
452C2×D4—92no—
453C22×C4—91no—
454C22×C4—92no—
455C22×C4—91no—
456C22×C4—92no—
457C22×C4C2×C3×D911yescharacteristic
458C4⋊C4—92no—
459C4⋊C4—91no—
460C4⋊C4—92no—
461C4⋊C4C2×C3×D911yescharacteristic
462C4⋊C4C2×C3×D912yes—
463C22⋊C4—92no—
464C22⋊C4—92no—
465C22⋊C4—92no—
466C22⋊C4C2×C3×D912yes—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—92no—
478C22×C3—181no—
479C22×C3—181no—
480C22×C3—182no—
481C22×C3—181no—
482C22×C3—181no—
483C22×C3—182no—
484C22×C3—181no—
485C22×C3—182no—
486C22×C3C23×D911yescharacteristic
487C22×C3C23×C3×D311yescharacteristic
488C2×D3—32no—
489C2×D3—61no—
490C2×D3—61no—
491C2×D3—62no—
492C2×D3—61no—
493C2×D3—62no—
494C2×D3—61no—
495C2×D3—61no—
496C2×D3—62no—
497C3×C4—21no—
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—21no—
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—41no—
505C3×C4—42no—
506C3×C4—42no—
507C3×C4—92no—
508C3×C4—181no—
509C3×C4—182no—
510C3×C4D4×D912yes—
511C3×C4C3×D3×D412yes—
512Dic3—32no—
513Dic3—61no—
514Dic3—62no—
515C32#562911yescharacteristic
516C9—21no—
517C9C3×#53711yescharacteristic, complemented
518C23—91no—
519C23—91no—
520C23—92no—
521C23C22×C3×D911yescharacteristic
522Q8—22no—
523Q8—182no—
524D4—182no—
525D4—184no—
526D4—182no—
527D4—182no—
528D4—182no—
529D4—362no—
530D4—362no—
531C2×C4—22no—
532C2×C4—91no—
533C2×C4—94no—
534C2×C4—92no—
535C2×C4—91no—
536C2×C4—184no—
537C2×C4—182no—
538C2×C4—362no—
539C2×C4C22×C3×D911yescharacteristic
540C2×C4C22×C3×D912yes—
541C2×C4C22×C3×D912yes—
542C2×C4C22×C3×D911yescharacteristic
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—42no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—182no—
550C2×C3—362no—
551C2×C3—361no—
552C2×C3C2×D4×D911yescharacteristic
553C2×C3#6371311yescharacteristic
554C2×C3#6371511yescharacteristic
555C2×C3C2×C3×D3×D411yescharacteristic
556C2×C3C3×#100711yescharacteristic
557C2×C3C3×#100911yescharacteristic
558D3—62no—
559D3—122no—
560D3—121no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—92no—
565C22—181no—
566C22—181no—
567C22—182no—
568C22—181no—
569C22—181no—
570C22—182no—
571C22—181no—
572C22—182no—
573C22C23×C3×D911yescharacteristic
574C4—21no—
575C4—22no—
576C4—22no—
577C4—92no—
578C4—181no—
579C4—182no—
580C4C3×D4×D912yes—
581C3—21no—
582C3#1058973111yescharacteristic, direct product factor
583C3C3×#562911yescharacteristic
584C2—41no—
585C2—182no—
586C2—362no—
587C2—361no—
588C2C2×C3×D4×D911yescharacteristic
589C2C3×#6371311yescharacteristic
590C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C23×D3
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#537
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×#3701C2semidirect
2. C2×C3×#3697C2semidirect
3. C3×C9×C22⋊Q8C2semidirect
4. C3×#63508C2semidirect
5. C3×#63492C2semidirect
6. C3×#63458C2semidirect
7. C3×#63457C2semidirect
8. C3×#63457C2semidirect
9. C3×C4⋊C4×D9C2semidirect
10. C3×#63446C2semidirect
11. C3×#63446C2semidirect
13. #10589731C3direct (N × H)
50. C2×#3701C2×C3semidirect
51. C2×#3697C2×C3semidirect
53. C9×C22⋊Q8C2×C3semidirect
54. #63508C2×C3semidirect
55. #63492C2×C3semidirect
56. #63458C2×C3semidirect
57. #63457C2×C3semidirect
58. #63457C2×C3semidirect
59. C4⋊C4×D9C2×C3semidirect
60. #63446C2×C3semidirect
61. #63446C2×C3semidirect
85. C3×C4×D9D4semidirect
171. C3×C22⋊Q8D9semidirect
242. C4×D9C3×D4semidirect
359. C22⋊Q8C3×D9semidirect
366. C3×C9#537semidirect
517. C9C3×#537semidirect
582. C3#10589731direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37012
C2×C3×#36972
C3×C9×C22⋊Q82
C3×#635082
C3×#634922
C3×#634582
C3×#634572
C3×C4⋊C4×D92
C3×#634462
C3×#56293
#105897313