Finite Group Databasebeta

← C3×#71729 #11792583

Subgroups — C3×#71729 #11792583

Order 1152.

945 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#124C211yesmaximal, characteristic
2C22×C3×#216C211yesmaximal, characteristic, complemented
3C22×C3×#214C211yesmaximal, characteristic
4C2×C3×#835C211yesmaximal, characteristic, complemented
5C2×C3×#832C211yesmaximal, characteristic
6C3×#4929C211yesmaximal, characteristic, complemented
7C3×#4524C211yesmaximal, characteristic, complemented
8C3×#2015—31nomaximal
9#71729C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C22×C32×D4C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C32×#124—22no—
14C32×#124—22no—
15C2×C32×C22⋊C4—22no—
16C2×C32×C42C2211yescharacteristic
17C2×C3×#216—22no—
18C2×C3×#216—22no—
19C2×C3×#216—22no—
20C2×C3×#214—22no—
21C2×C3×#214—22no—
22C2×C3×#214—22no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C22×C3×C3⋊C8C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#835—22no—
27C3×#832—22no—
28C3×C3⋊#95—22no—
29C22×C3×Q16—31no—
30C22×C3×SD16—31no—
31C2×C3×#124—21no—
32C2×C3×#124D311yescharacteristic
33C2×C3×#124C2×C311yescharacteristic
34C22×#216C2×C311yescharacteristic, complemented
35C22×#214C2×C311yescharacteristic
36C2×C3×#103—31no—
37C2×C3×#102—31no—
38C3×#391—31no—
39C2×#835C2×C311yescharacteristic, complemented
40C2×#832C2×C311yescharacteristic
41#4929C2×C311yescharacteristic, complemented
42C3×#336—31no—
43#4524C2×C311yescharacteristic, complemented
44C24×C32—21no—
45C2×C32×Q8—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8—21no—
50C2×C32×Q8D412yes—
51C2×C32×D4—21no—
52C2×C32×D4—21no—
53C2×C32×D4—21no—
54C2×C32×D4—21no—
55C2×C32×D4—21no—
56C2×C32×D4D412yes—
57C22×C32×C4—21no—
58C22×C32×C4D412yes—
59C22×C32×C4C2311yescharacteristic
60C22×C3×Dic3—21no—
61C22×C3×Dic3—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6D412yes—
68C32×C22⋊C4—28no—
69C32×C42—21no—
70C32×C42—21no—
71C3×#216—28no—
72C3×#214—28no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8D412yes—
80C2×C3×C3⋊C8D412yes—
81#2015—31noSylow
82C22×C3×Q8—21no—
83C22×C3×Q8—31no—
84C22×C3×Q8C2×D311yescharacteristic
85C22×C3×Q8C22×C311yescharacteristic
86C22×C3×D4—21no—
87C22×C3×D4C2×D311yescharacteristic
88C22×C3×D4C22×C311yescharacteristic
89C22×Dic6C22×C311yescharacteristic
90C2×C3×Q16—62no—
91C2×C3×Q16—62no—
92C2×C3×Q16—62no—
93C2×C3×SD16—62no—
94C2×C3×SD16—62no—
95C2×C3×SD16—62no—
96C22×C3×C8—31no—
97C22×C3×C8—31no—
98C3×#124—24no—
99C3×#124—24no—
100C3×#124—22no—
101C3×#124—22no—
102C3×#124—22no—
103C3×#124—22no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C22⋊C4—24no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C42—21no—
109C2×C3×C42C2×D311yescharacteristic
110C2×C3×C42C22×C311yescharacteristic
111C2×#216—22no—
112C2×#216—22no—
113C2×#216—22no—
114C2×#214—22no—
115C2×#214—22no—
116C2×#214—22no—
117C2×C4⋊Dic3C22×C311yescharacteristic
118C22×C3⋊C8C22×C311yescharacteristic
119C22×C3⋊C8C22×C311yescharacteristic
120C3×#103—62no—
121C3×#102—62no—
122C3×#95—62no—
123#835—22no—
124#832—22no—
125C3⋊#95—22no—
126C23×C32—22no—
127C23×C32—41no—
128C23×C32—41no—
129C23×C32—42no—
130C23×C32—42no—
131C23×C32C2×D411yescharacteristic
132C32×Q8—22no—
133C32×Q8—22no—
134C32×Q8—41no—
135C32×Q8—41no—
136C32×D4—22no—
137C32×D4—22no—
138C32×D4—41no—
139C32×D4—41no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—41no—
145C2×C32×C4—41no—
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4#4111yescharacteristic
152C2×C3×Dic3—22no—
153C2×C3×Dic3—41no—
154C2×C3×Dic3—41no—
155C2×C3×Dic3—41no—
156C2×C3×Dic3—42no—
157C3×Dic6—22no—
158C3×Dic6—22no—
159C3×Dic6—41no—
160C3×Dic6—41no—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—22no—
163C22×Q16—31no—
164C22×SD16—31no—
165C2×#124C3×D311yescharacteristic
166C2×#103—31no—
167C2×#102—31no—
168#391—31no—
169#336—31no—
170C24×C3—22no—
171C24×C3—21no—
172C24×C3—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—22no—
185C2×C3×Q8—32no—
186C2×C3×Q8—41no—
187C2×C3×Q8—41no—
188C2×C3×Q8—41no—
189C2×C3×Q8—41no—
190C2×C3×Q8—61no—
191C2×C3×Q8—61no—
192C2×C3×Q8—61no—
193C2×C3×Q8—61no—
194C2×C3×Q8—61no—
195C2×C3×Q8#6712yes—
196C2×C3×Q8C3×D412yes—
197C2×C3×D4—22no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—22no—
209C2×C3×D4—41no—
210C2×C3×D4—41no—
211C2×C3×D4—41no—
212C2×C3×D4—41no—
213C2×C3×D4#6712yes—
214C2×C3×D4C3×D412yes—
215C22×C3×C4—22no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—22no—
219C22×C3×C4—21no—
220C22×C3×C4—61no—
221C22×C3×C4—61no—
222C22×C3×C4#6712yes—
223C22×C3×C4C3×D412yes—
224C22×C3×C4C22×D311yescharacteristic
225C22×C3×C4C23×C311yescharacteristic
226C22×Dic3—21no—
227C22×Dic3—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6C3×D412yes—
234C3×Q16—68no—
235C3×SD16—68no—
236C2×C3×C8—32no—
237C2×C3×C8—32no—
238C2×C3×C8—61no—
239C2×C3×C8—61no—
240C2×C3×C8—61no—
241C2×C3×C8—61no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—61no—
244C3×C22⋊C4—216no—
245C3×C22⋊C4—28no—
246C3×C22⋊C4—28no—
247C3×C42—22no—
248C3×C42—22no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253#216—28no—
254#214—28no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8C3×D412yes—
262C2×C3⋊C8C3×D412yes—
263C22×C32—42no—
264C22×C32—44no—
265C22×C32—82no—
266C22×C32—81no—
267C22×C32—81no—
268C22×C32#12011yescharacteristic
269C22×C32#12111yescharacteristic
270C22×C32#12711yescharacteristic
271C22×C32#13512yes—
272C22×C32#13712yes—
273C32×C4—42no—
274C32×C4—42no—
275C32×C4#12011yescharacteristic
276C32×C4#12011yescharacteristic
277C32×C4#12111yescharacteristic
278C32×C4#12111yescharacteristic
279C3×Dic3—42no—
280C3×Dic3—81no—
281C22×Q8—31no—
282C22×Q8C2×C3×D311yescharacteristic
283C22×D4C2×C3×D311yescharacteristic
284C2×Q16—62no—
285C2×Q16—62no—
286C2×Q16—62no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C22×C8—31no—
291C22×C8—31no—
292#124—22no—
293#124—22no—
294C2×C4⋊C4—31no—
295C2×C22⋊C4—22no—
296C2×C42C2×C3×D311yescharacteristic
297#103—62no—
298#102—62no—
299#95—62no—
300C23×C3—24no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C2×#6711yescharacteristic
317C23×C3C2×C3×D411yescharacteristic
318C3×Q8—22no—
319C3×Q8—22no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—42no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—42no—
326C3×Q8—41no—
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×Q8—62no—
331C3×Q8—62no—
332C3×Q8—121no—
333C3×Q8—121no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—42no—
342C3×D4—41no—
343C3×D4—41no—
344C3×D4—41no—
345C3×D4—41no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—62no—
371C2×C3×C4—121no—
372C2×C3×C4—121no—
373C2×C3×C4—121no—
374C2×C3×C4—122no—
375C2×C3×C4D3×D411yescharacteristic
376C2×C3×C4D3×D411yescharacteristic
377C2×C3×C4D3×D411yescharacteristic
378C2×C3×C4#23711yescharacteristic
379C2×C3×C4C2×#6711yescharacteristic
380C2×C3×C4C2×#6711yescharacteristic
381C2×C3×C4C2×C3×D411yescharacteristic
382C2×C3×C4C2×C3×D411yescharacteristic
383C2×C3×C4C2×C3×D411yescharacteristic
384C2×C3×C4C2×C3×D411yescharacteristic
385C2×C3×C4C2×C3×D411yescharacteristic
386C2×C3×C4C3×#4111yescharacteristic
387C2×Dic3—22no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—42no—
392Dic6—22no—
393Dic6—22no—
394Dic6—41no—
395Dic6—41no—
396C3×C8—62no—
397C3×C8—62no—
398C3⋊C8—22no—
399C3⋊C8—22no—
400C2×C32—82no—
401C2×C32#39211yescharacteristic
402C2×C32#45212yes—
403C2×C32#46412yes—
404C2×C32#49511yescharacteristic
405C2×C32#49711yescharacteristic
406C24—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—32no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8C3×#6712yes—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C3×#6712yes—
425C22×C4—21no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4C3×#6712yes—
429C22×C4C22×C3×D311yescharacteristic
430Q16—68no—
431SD16—68no—
432C2×C8—32no—
433C2×C8—32no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C22⋊C4—28no—
441C42—21no—
442C42—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—21no—
448C22×C3—44no—
449C22×C3—48no—
450C22×C3—42no—
451C22×C3—44no—
452C22×C3—42no—
453C22×C3—44no—
454C22×C3—84no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—82no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—82no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3#93811yescharacteristic
464C22×C3#94011yescharacteristic
465C22×C3#95012yes—
466C22×C3#95112yes—
467C22×C3#95311yescharacteristic
468C22×C3C3×#12011yescharacteristic
469C22×C3C3×#12111yescharacteristic
470C22×C3C3×#12711yescharacteristic
471C22×C3C3×#13512yes—
472C22×C3C3×#13712yes—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—44no—
478C3×C4—44no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4—122no—
484C3×C4—241no—
485C3×C4#93711yescharacteristic
486C3×C4#93711yescharacteristic
487C3×C4#93911yescharacteristic
488C3×C4#93911yescharacteristic
489C3×C4C3×#12011yescharacteristic
490C3×C4C3×#12011yescharacteristic
491C3×C4C3×#12111yescharacteristic
492C3×C4C3×#12111yescharacteristic
493Dic3—42no—
494Dic3—81no—
495C32#201511yescharacteristic, complemented, Sylow
496C23—22no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23C2×C3×#6711yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506Q8—62no—
507Q8—62no—
508Q8—121no—
509Q8—121no—
510D4—22no—
511D4—22no—
512D4—41no—
513D4—41no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—62no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—122no—
525C2×C4C3×D3×D411yescharacteristic
526C2×C4C3×D3×D411yescharacteristic
527C2×C4C3×D3×D411yescharacteristic
528C2×C4C3×#23711yescharacteristic
529C2×C4C2×C3×#6711yescharacteristic
530C2×C4C2×C3×#6711yescharacteristic
531C8—62no—
532C8—62no—
533C2×C3—21no—
534C2×C3—22no—
535C2×C3—22no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—84no—
539C2×C3—82no—
540C2×C3—82no—
541C2×C3#505412yes—
542C2×C3#505911yescharacteristic
543C2×C3#506011yescharacteristic
544C2×C3#522311yescharacteristic
545C2×C3#523912yes—
546C2×C3C3×#39211yescharacteristic
547C2×C3C3×#45212yes—
548C2×C3C3×#46412yes—
549C2×C3C3×#49511yescharacteristic
550C2×C3C3×#49711yescharacteristic
551C22—42no—
552C22—44no—
553C22—82no—
554C22—81no—
555C22—81no—
556C22C3×#93811yescharacteristic
557C22C3×#94011yescharacteristic
558C22C3×#95012yes—
559C22C3×#95112yes—
560C22C3×#95311yescharacteristic
561C4—42no—
562C4—42no—
563C4—122no—
564C4—241no—
565C4C3×#93711yescharacteristic
566C4C3×#93711yescharacteristic
567C4C3×#93911yescharacteristic
568C4C3×#93911yescharacteristic
569C3—21no—
570C3#7172911yescharacteristic, direct product factor
571C3C3×#201511yescharacteristic, complemented
572C2—82no—
573C2C3×#505412yes—
574C2C3×#505911yescharacteristic
575C2C3×#506011yescharacteristic
576C2C3×#522311yescharacteristic
577C2C3×#523912yes—
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#124C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
hypercenterC2×C3×#124D3
hypocenterC3C3×#2015
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×#124

Complements

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

NH ≅ G/Ntype
2. C22×C3×#216C2semidirect
4. C2×C3×#835C2semidirect
6. C3×#4929C2semidirect
7. C3×#4524C2semidirect
9. #71729C3direct (N × H)
34. C22×#216C2×C3semidirect
39. C2×#835C2×C3semidirect
41. #4929C2×C3semidirect
43. #4524C2×C3semidirect
495. C32#2015semidirect
570. C3#71729direct (N × H)
571. C3C3×#2015semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1242
C22×C3×#2162
C22×C3×#2142
C2×C3×#8352
C2×C3×#8322
C3×#49292
C3×#45242
C3×#20153
#717293