Finite Group Databasebeta

← C3×#10596295 #13179584

Subgroups — C3×#10596295 #13179584

Order 1728.

2408 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×#41C211yesmaximal, characteristic, complemented
2C2×C3×#3830C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic, complemented
4C3×#64153C211yesmaximal, characteristic, complemented
5C3×#64148C212yesmaximal, complemented
6C3×#64144C212yesmaximal, complemented
7C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
8C3×#64138C212yesmaximal, complemented
9C3×C4×#638C211yesmaximal, characteristic, complemented
10C3×#64133C211yesmaximal, characteristic, complemented
11C2×C3×#3750C212yesmaximal, complemented
12#10596295C311yesmaximal, characteristic, direct product factor
13C3×#5824—34nomaximal
14C33×#41—24no—
15C2×C33×Q8C2211yescharacteristic
16C2×C33×D4C2212yes—
17C2×C33×D4C2211yescharacteristic
18C22×C33×C4C2212yes—
19C22×C33×C4C2211yescharacteristic
20C2×C3×#638C2211yescharacteristic
21C2×C3×#638C2212yes—
22C22×C3×C3⋊Dic3C2212yes—
23C22×C3×C3⋊Dic3C2212yes—
24C3×#3830—24no—
25C2×C3×C4×C3⋊D3C2211yescharacteristic
26C2×C3×#634C2211yescharacteristic
27C3×#3757C2212yes—
28C3×#3757C2211yescharacteristic
29C3×#3757C2212yes—
30C3×#3752C2211yescharacteristic
31C3×#3752C2212yes—
32C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×#3750C2212yes—
35C3×#3750C2212yes—
36C3×#3750C2212yes—
37C3×#3750C2212yes—
38C3×C4×C3⋊Dic3C2211yescharacteristic
39C2×C32×#41—24no—
40C2×C32×#41D314yescomplemented
41C2×C32×#41C2×C311yescharacteristic, complemented
42C2×#3830C2×C311yescharacteristic, complemented
43C2×C3×#237—34no—
44#64155C2×C311yescharacteristic, complemented
45#64153C2×C311yescharacteristic, complemented
46#64148C2×C312yescomplemented
47#64144C2×C312yescomplemented
48D4×C3⋊Dic3C2×C311yescharacteristic, complemented
49#64138C2×C312yescomplemented
50C4×#638C2×C311yescharacteristic, complemented
51#64133C2×C311yescharacteristic, complemented
52C2×#3750C2×C312yescomplemented
53C3×#946—34no—
54C3×#944—34no—
55C3×#939—38no—
56C3×#935—38no—
57C3×D4×Dic3—34no—
58C3×#929—38no—
59C3×C4×#67—34no—
60C3×#924—34no—
61C2×C3×Dic3⋊C4—38no—
62C23×C33C2311yescharacteristic
63C23×C33C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C33×Q8—22no—
66C33×D4—24no—
67C33×D4D414yescomplemented
68C2×C33×C4—24no—
69C2×C33×C4—22no—
70C2×C33×C4C2311yescharacteristic
71C2×C33×C4C2312yes—
72C2×C33×C4C2311yescharacteristic
73C3×#638—24no—
74C3×#638—22no—
75C2×C3×C3⋊Dic3—24no—
76C2×C3×C3⋊Dic3—24no—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2312yes—
79C2×C3×C3⋊Dic3C2311yescharacteristic
80C2×C3×C3⋊Dic3C2312yes—
81C2×C3×C3⋊Dic3C2311yescharacteristic
82C3×C4×C3⋊D3—22no—
83C3×#634—22no—
84C3×#547—91no—
85#5824—34no—
86C32×#41—232no—
87C32×#41—216no—
88C32×#41—24no—
89C2×C32×Q8—24no—
90C2×C32×Q8C2×D314yes—
91C2×C32×Q8C22×C311yescharacteristic
92C2×C32×D4—28no—
93C2×C32×D4—24no—
94C2×C32×D4C2×D318yes—
95C2×C32×D4C2×D314yes—
96C2×C32×D4C22×C312yes—
97C2×C32×D4C22×C311yescharacteristic
98C22×C32×C4—28no—
99C22×C32×C4—24no—
100C22×C32×C4C2×D318yes—
101C22×C32×C4C2×D314yes—
102C22×C32×C4C22×C312yes—
103C22×C32×C4C22×C311yescharacteristic
104C2×#638C22×C312yes—
105C2×#638C22×C311yescharacteristic
106C22×C3⋊Dic3C22×C312yes—
107C22×C3⋊Dic3C22×C312yes—
108#3830—24no—
109C2×C4×C3⋊D3C22×C311yescharacteristic
110C2×#634C22×C311yescharacteristic
111C2×C3×#67—34no—
112C2×C3×#67—38no—
113C22×C3×Dic3—38no—
114C22×C3×Dic3—38no—
115C3×#237—616no—
116C2×C3×C4×D3—34no—
117C2×C3×Dic6—34no—
118#3757C22×C312yes—
119#3757C22×C311yescharacteristic
120#3757C22×C312yes—
121#3752C22×C311yescharacteristic
122#3752C22×C312yes—
123C4⋊(C3⋊Dic3)C22×C311yescharacteristic
124#3750C22×C311yescharacteristic
125#3750C22×C312yes—
126#3750C22×C312yes—
127#3750C22×C312yes—
128#3750C22×C312yes—
129C4×C3⋊Dic3C22×C311yescharacteristic
130C3×#217—38no—
131C3×#217—34no—
132C3×#217—38no—
133C3×(C2×D3)⋊C4—34no—
134C3×(C2×D3)⋊C4—38no—
135C3×C4⋊Dic3—34no—
136C3×Dic3⋊C4—34no—
137C3×Dic3⋊C4—38no—
138C3×Dic3⋊C4—38no—
139C3×Dic3⋊C4—38no—
140C3×Dic3⋊C4—38no—
141C3×C4×Dic3—34no—
142C22×C33—22no—
143C22×C33—22no—
144C22×C33—21no—
145C22×C33—21no—
146C22×C33—21no—
147C22×C33C2×D414yes—
148C22×C33C2411yescharacteristic
149C2×C3×C3⋊D3—21no—
150C2×C3×C3⋊D3—21no—
151C2×C3×C3⋊D3—21no—
152C33×C4—21no—
153C33×C4—22no—
154C33×C4C2×D412yes—
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3—21no—
157C3×C3⋊Dic3—22no—
158C3×C3⋊Dic3—21no—
159C3×C3⋊Dic3#4112yescomplemented
160C2×C3×#41—24no—
161C2×C3×#41—91no—
162C2×C3×#41C3×D314yescomplemented
163C2×C3×#41C3⋊D311yescharacteristic, complemented
164C2×#237—34no—
165C3×C4⋊Q8—91no—
166C3×#123—92no—
167C3×#123—92no—
168C3×C22⋊Q8—91no—
169C3×C22⋊Q8—91no—
170C3×#121—92no—
171C3×C4×D4—91no—
172C3×C4×D4—91no—
173C2×C3×C4⋊C4—92no—
174#946—34no—
175#944—34no—
176#939—38no—
177#935—38no—
178D4×Dic3—34no—
179#929—38no—
180C4×#67—34no—
181#924—34no—
182C2×Dic3⋊C4—38no—
183C23×C32—24no—
184C23×C32—28no—
185C23×C32C22×D314yes—
186C23×C32C22×D318yes—
187C23×C32C23×C311yescharacteristic
188C23×C32C23×C312yes—
189C22×C3⋊D3C23×C311yescharacteristic
190C22×C3×D3—34no—
191C32×Q8—216no—
192C32×Q8—28no—
193C32×Q8—22no—
194C32×D4—232no—
195C32×D4—216no—
196C32×D4—24no—
197C32×D4—216no—
198C32×D4#67116yescomplemented
199C32×D4C3×D414yescomplemented
200C2×C32×C4—232no—
201C2×C32×C4—216no—
202C2×C32×C4—216no—
203C2×C32×C4—24no—
204C2×C32×C4—28no—
205C2×C32×C4—22no—
206C2×C32×C4—24no—
207C2×C32×C4—28no—
208C2×C32×C4—24no—
209C2×C32×C4C22×D314yes—
210C2×C32×C4C22×D318yes—
211C2×C32×C4C22×D314yes—
212C2×C32×C4C23×C311yescharacteristic
213C2×C32×C4C23×C312yes—
214C2×C32×C4C23×C311yescharacteristic
215#638—24no—
216#638—22no—
217C2×C3⋊Dic3—24no—
218C2×C3⋊Dic3—24no—
219C2×C3⋊Dic3C23×C311yescharacteristic
220C2×C3⋊Dic3C23×C312yes—
221C2×C3⋊Dic3C23×C311yescharacteristic
222C2×C3⋊Dic3C23×C312yes—
223C2×C3⋊Dic3C23×C311yescharacteristic
224C4×C3⋊D3—22no—
225#634—22no—
226C3×#67—68no—
227C3×#67—616no—
228C2×C3×Dic3—34no—
229C2×C3×Dic3—34no—
230C2×C3×Dic3—38no—
231C2×C3×Dic3—34no—
232C2×C3×Dic3—38no—
233C2×C3×Dic3—616no—
234C2×C3×Dic3—616no—
235C3×C4×D3—68no—
236C3×Dic6—68no—
237#547—91noSylow
238C2×C33—24no—
239C2×C33—41no—
240C2×C33C22×D411yescharacteristic
241C2×C33C2×#4111yescharacteristic
242C2×C332−1+411yescharacteristic
243C3×C3⋊D3—41no—
244C3×#41—232no—
245C3×#41—216no—
246C3×#41—24no—
247C3×#41—184no—
248C2×C3×Q8—24no—
249C2×C3×Q8—91no—
250C2×C3×Q8C2×C3×D314yes—
251C2×C3×Q8C2×C3⋊D311yescharacteristic
252C2×C3×D4—28no—
253C2×C3×D4—24no—
254C2×C3×D4—91no—
255C2×C3×D4—92no—
256C2×C3×D4C2×C3×D318yes—
257C2×C3×D4C2×C3×D314yes—
258C2×C3×D4C2×C3⋊D312yes—
259C2×C3×D4C2×C3⋊D311yescharacteristic
260C22×C3×C4—28no—
261C22×C3×C4—24no—
262C22×C3×C4—92no—
263C22×C3×C4—92no—
264C22×C3×C4—91no—
265C22×C3×C4C2×C3×D318yes—
266C22×C3×C4C2×C3×D314yes—
267C22×C3×C4C2×C3⋊D312yes—
268C22×C3×C4C2×C3⋊D311yescharacteristic
269C2×#67—34no—
270C2×#67—38no—
271C22×Dic3—38no—
272C22×Dic3—38no—
273#237—616no—
274C2×C4×D3—34no—
275C2×Dic6—34no—
276C3×C4⋊C4—91no—
277C3×C4⋊C4—92no—
278C3×C4⋊C4—92no—
279C3×C4⋊C4—92no—
280C3×C4⋊C4—92no—
281C3×C4⋊C4—91no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4—91no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4—91no—
286C3×C22⋊C4—92no—
287C3×C42—91no—
288#217—38no—
289#217—34no—
290#217—38no—
291(C2×D3)⋊C4—34no—
292(C2×D3)⋊C4—38no—
293C4⋊Dic3—34no—
294Dic3⋊C4—34no—
295Dic3⋊C4—38no—
296Dic3⋊C4—38no—
297Dic3⋊C4—38no—
298Dic3⋊C4—38no—
299C4×Dic3—34no—
300C22×C32—22no—
301C22×C32—22no—
302C22×C32—28no—
303C22×C32—24no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—21no—
307C22×C32—24no—
308C22×C32—216no—
309C22×C32—28no—
310C22×C32—28no—
311C22×C32—24no—
312C22×C32—24no—
313C22×C32—48no—
314C22×C32—48no—
315C22×C32—44no—
316C22×C32—44no—
317C22×C32C2×#67116yes—
318C22×C32C2×C3×D414yes—
319C22×C32C23×D314yes—
320C22×C32C24×C311yescharacteristic
321C2×C3⋊D3—21no—
322C2×C3⋊D3—21no—
323C2×C3⋊D3—21no—
324C2×C3×D3—64no—
325C2×C3×D3—64no—
326C2×C3×D3—64no—
327C32×C4—28no—
328C32×C4—216no—
329C32×C4—24no—
330C32×C4—28no—
331C32×C4—21no—
332C32×C4—22no—
333C32×C4—28no—
334C32×C4C2×#6718yes—
335C32×C4C2×C3×D412yes—
336C3⋊Dic3—22no—
337C3⋊Dic3—21no—
338C3⋊Dic3—22no—
339C3⋊Dic3—21no—
340C3⋊Dic3C3×#4112yescomplemented
341C3×Dic3—38no—
342C3×Dic3—68no—
343C3×Dic3—64no—
344C3×Dic3—68no—
345C3×Dic3—64no—
346C2×#41—91no—
347C2×#41C3×C3⋊D311yescharacteristic, complemented
348C4⋊Q8—91no—
349#123—92no—
350#123—92no—
351C22⋊Q8—91no—
352C22⋊Q8—91no—
353#121—92no—
354C4×D4—91no—
355C4×D4—91no—
356C2×C4⋊C4—92no—
357C33#54711yescharacteristic, complemented, Sylow
358C23×C3—24no—
359C23×C3—28no—
360C23×C3—91no—
361C23×C3C22×C3×D314yes—
362C23×C3C22×C3×D318yes—
363C23×C3C22×C3⋊D311yescharacteristic
364C23×C3C22×C3⋊D312yes—
365C22×D3—34no—
366C3×Q8—216no—
367C3×Q8—28no—
368C3×Q8—22no—
369C3×Q8—182no—
370C3×D4—232no—
371C3×D4—216no—
372C3×D4—24no—
373C3×D4—216no—
374C3×D4—182no—
375C3×D4—184no—
376C3×D4C3×#67116yescomplemented
377C3×D4#63814yescomplemented
378C2×C3×C4—232no—
379C2×C3×C4—216no—
380C2×C3×C4—216no—
381C2×C3×C4—24no—
382C2×C3×C4—28no—
383C2×C3×C4—22no—
384C2×C3×C4—24no—
385C2×C3×C4—28no—
386C2×C3×C4—24no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4—92no—
390C2×C3×C4—91no—
391C2×C3×C4—92no—
392C2×C3×C4—184no—
393C2×C3×C4—184no—
394C2×C3×C4—182no—
395C2×C3×C4C22×C3×D314yes—
396C2×C3×C4C22×C3×D318yes—
397C2×C3×C4C22×C3×D314yes—
398C2×C3×C4C22×C3⋊D311yescharacteristic
399C2×C3×C4C22×C3⋊D312yes—
400C2×C3×C4C22×C3⋊D311yescharacteristic
401#67—68no—
402#67—616no—
403C2×Dic3—34no—
404C2×Dic3—34no—
405C2×Dic3—38no—
406C2×Dic3—34no—
407C2×Dic3—38no—
408C2×Dic3—616no—
409C2×Dic3—616no—
410C4×D3—68no—
411Dic6—68no—
412C2×C32—24no—
413C2×C32—24no—
414C2×C32—24no—
415C2×C32—24no—
416C2×C32—216no—
417C2×C32—416no—
418C2×C32—48no—
419C2×C32—44no—
420C2×C32—41no—
421C2×C32D3×#4114yes—
422C2×C32#101014yes—
423C2×C32C22×#6714yes—
424C2×C32C22×C3×D411yescharacteristic
425C2×C32C2×C3×#4111yescharacteristic
426C2×C32C3×2−1+411yescharacteristic
427C3⋊D3—41no—
428C3×D3—124no—
429#41—24no—
430#41—184no—
431C2×Q8—91no—
432C2×Q8C2×C3×C3⋊D311yescharacteristic
433C2×D4—91no—
434C2×D4—92no—
435C2×D4C2×C3×C3⋊D312yes—
436C2×D4C2×C3×C3⋊D311yescharacteristic
437C22×C4—92no—
438C22×C4—92no—
439C22×C4—91no—
440C22×C4C2×C3×C3⋊D312yes—
441C22×C4C2×C3×C3⋊D311yescharacteristic
442C4⋊C4—91no—
443C4⋊C4—92no—
444C4⋊C4—92no—
445C4⋊C4—92no—
446C4⋊C4—91no—
447C4⋊C4—92no—
448C22⋊C4—92no—
449C22⋊C4—91no—
450C22⋊C4—92no—
451C22⋊C4—91no—
452C22⋊C4—92no—
453C42—91no—
454C22×C3—28no—
455C22×C3—28no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—28no—
459C22×C3—24no—
460C22×C3—24no—
461C22×C3—24no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—24no—
466C22×C3—216no—
467C22×C3—48no—
468C22×C3—48no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—181no—
472C22×C3—181no—
473C22×C3—181no—
474C22×C3C2×C3×#67116yes—
475C22×C3C2×#63814yes—
476C22×C3C23×C3×D314yes—
477C22×C3C23×C3⋊D311yescharacteristic
478C2×D3—64no—
479C2×D3—64no—
480C2×D3—64no—
481C3×C4—28no—
482C3×C4—216no—
483C3×C4—24no—
484C3×C4—28no—
485C3×C4—21no—
486C3×C4—22no—
487C3×C4—28no—
488C3×C4—92no—
489C3×C4—182no—
490C3×C4—181no—
491C3×C4—181no—
492C3×C4—182no—
493C3×C4C2×C3×#6718yes—
494C3×C4C2×#63812yes—
495Dic3—38no—
496Dic3—68no—
497Dic3—64no—
498Dic3—68no—
499Dic3—64no—
500C32—24no—
501C32#582414yescomplemented
502C32C3×#54711yescharacteristic, complemented
503C23—91no—
504C23C22×C3×C3⋊D311yescharacteristic
505C23C22×C3×C3⋊D312yes—
506Q8—22no—
507Q8—182no—
508D4—24no—
509D4—182no—
510D4—184no—
511D4C3×#63814yescomplemented
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—91no—
515C2×C4—91no—
516C2×C4—92no—
517C2×C4—91no—
518C2×C4—92no—
519C2×C4—184no—
520C2×C4—184no—
521C2×C4—182no—
522C2×C4C22×C3×C3⋊D311yescharacteristic
523C2×C4C22×C3×C3⋊D312yes—
524C2×C4C22×C3×C3⋊D311yescharacteristic
525C2×C3—216no—
526C2×C3—24no—
527C2×C3—24no—
528C2×C3—24no—
529C2×C3—24no—
530C2×C3—416no—
531C2×C3—48no—
532C2×C3—44no—
533C2×C3—41no—
534C2×C3—361no—
535C2×C3C3×D3×#4114yes—
536C2×C3C3×#101014yes—
537C2×C3C22×C3×#6714yes—
538C2×C3#41×C3⋊D311yescharacteristic
539C2×C3#6436711yescharacteristic
540C2×C3C22×#63811yescharacteristic
541D3—124no—
542C22—22no—
543C22—22no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—181no—
548C22—181no—
549C22—181no—
550C22C2×C3×#63814yes—
551C22C23×C3×C3⋊D311yescharacteristic
552C4—21no—
553C4—22no—
554C4—92no—
555C4—182no—
556C4—181no—
557C4—181no—
558C4—182no—
559C4C2×C3×#63812yes—
560C3—24no—
561C3C3×#582414yescomplemented
562C3#1059629511yescharacteristic, direct product factor
563C2—24no—
564C2—41no—
565C2—361no—
566C2C3×#41×C3⋊D311yescharacteristic
567C2C3×#6436711yescharacteristic
568C2C22×C3×#63811yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC2×C33×#41C2
socleC22×C33C24
hypercenterC2×C3×#41C3⋊D3
hypocenterC32C3×#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C2×C3×#3830C2semidirect
3. C3×#64155C2semidirect
4. C3×#64153C2semidirect
5. C3×#64148C2semidirect
6. C3×#64144C2semidirect
7. C3×D4×C3⋊Dic3C2semidirect
8. C3×#64138C2semidirect
9. C3×C4×#638C2semidirect
10. C3×#64133C2semidirect
11. C2×C3×#3750C2semidirect
12. #10596295C3direct (N × H)
40. C2×C32×#41D3semidirect
41. C2×C32×#41C2×C3semidirect
42. C2×#3830C2×C3semidirect
44. #64155C2×C3semidirect
45. #64153C2×C3semidirect
46. #64148C2×C3semidirect
47. #64144C2×C3semidirect
48. D4×C3⋊Dic3C2×C3semidirect
49. #64138C2×C3semidirect
50. C4×#638C2×C3semidirect
51. #64133C2×C3semidirect
52. C2×#3750C2×C3semidirect
67. C33×D4D4semidirect
159. C3×C3⋊Dic3#41semidirect
162. C2×C3×#41C3×D3semidirect
163. C2×C3×#41C3⋊D3semidirect
198. C32×D4#67semidirect
199. C32×D4C3×D4semidirect
340. C3⋊Dic3C3×#41semidirect
347. C2×#41C3×C3⋊D3semidirect
357. C33#547semidirect
376. C3×D4C3×#67semidirect
377. C3×D4#638semidirect
501. C32#5824semidirect
502. C32C3×#547semidirect
511. D4C3×#638semidirect
561. C3C3×#5824semidirect
562. C3#10596295direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×#412
C2×C3×#38302
C3×#641552
C3×#641532
C3×#641482
C3×#641442
C3×D4×C3⋊Dic32
C3×#641382
C3×C4×#6382
C3×#641332
C2×C3×#37502
#105962953
C3×#58243