Finite Group Databasebeta

← C3×#72111 #11792347

Subgroups — C3×#72111 #11792347

Order 1152.

825 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#137C211yesmaximal, characteristic, complemented
2C3×#5822C211yesmaximal, characteristic
3C3×#5816C211yesmaximal, characteristic, complemented
4C3×#5240C211yesmaximal, characteristic, complemented
5C3×#5225C211yesmaximal, characteristic, complemented
6C3×#5139C211yesmaximal, characteristic, complemented
7C3×#4559C211yesmaximal, characteristic, complemented
8C3×#2009—31nomaximal
9#72111C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×Q8C2211yescharacteristic
12C2×C3×#235C2211yescharacteristic
13C32×#137—22no—
14C32×#137—22no—
15C2×C32×Q16—21no—
16C2×C32×SD16—21no—
17C2×C32×M16C2211yescharacteristic
18C3×#951—21no—
19C3×#950—22no—
20C3×#949—21no—
21C3×#947—21no—
22C3×#946—22no—
23C3×#944—21no—
24C3×#942—21no—
25C3×#932—21no—
26C3×#929—22no—
27C3×#926C2211yescharacteristic
28C3×#926C2211yescharacteristic
29C2×C3×#208C2211yescharacteristic
30C3×#835—22no—
31C3×#834—22no—
32C3×#825—22no—
33C3×#575—31no—
34C2×C3×#137—21no—
35C2×C3×#137D311yescharacteristic, complemented
36C2×C3×#137C2×C311yescharacteristic, complemented
37C3×#536—31no—
38#5822C2×C311yescharacteristic
39#5816C2×C311yescharacteristic, complemented
40C3×#421—31no—
41C3×#419—31no—
42C3×#410—31no—
43#5240C2×C311yescharacteristic, complemented
44#5225C2×C311yescharacteristic, complemented
45#5139C2×C311yescharacteristic, complemented
46C3×#344—31no—
47#4559C2×C311yescharacteristic, complemented
48C32×#41—21no—
49C32×#41—21no—
50C32×#41—21no—
51C32×#41—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8—21no—
54C2×C32×Q8—21no—
55C2×C32×Q8—21no—
56C2×C32×Q8—21no—
57C2×C32×Q8D411yescharacteristic
58C2×C32×Q8D412yes—
59C2×C32×D4—21no—
60C2×C32×D4D411yescharacteristic
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4C2311yescharacteristic
64C2×C3×#67—21no—
65C3×#235—21no—
66C3×#235—21no—
67C3×#235—21no—
68C3×#235—21no—
69C2×C3×D12D411yescharacteristic
70C2×C3×C4×D3—21no—
71C2×C3×Dic6D411yescharacteristic
72C32×Q16—24no—
73C32×SD16—24no—
74C32×M16—21no—
75C32×M16D412yes—
76C2×C32×C8—21no—
77C3×#217—21no—
78C3×#217D412yes—
79C3×#216—22no—
80C3×#215—22no—
81C3×#214—22no—
82C3×#213—22no—
83C3×(C2×D3)⋊C4—24no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×Dic3⋊C4—24no—
87C3×C4×Dic3—21no—
88C3×C4×Dic3—21no—
89C3×#208—21no—
90C3×#208—21no—
91C2×C3×C3⋊C8D412yes—
92#2009—31noSylow
93C2×C3×#41—21no—
94C2×C3×#41—31no—
95C2×C3×#41C2×D311yescharacteristic
96C2×C3×#41C22×C311yescharacteristic
97C22×C3×Q8—21no—
98C22×C3×Q8C2×D311yescharacteristic
99C22×C3×Q8C22×C311yescharacteristic
100C2×#235C22×C311yescharacteristic
101C3×#137—24no—
102C3×#137—24no—
103C3×#137—22no—
104C3×#137—22no—
105C3×#137—22no—
106C3×#137—22no—
107C3×#137—61no—
108C3×#137—61no—
109C3×AGL(1,ℤ/8)—61no—
110C3×AGL(1,ℤ/8)—61no—
111C3×#135—62no—
112C2×C3×Q16—22no—
113C2×C3×Q16—21no—
114C2×C3×Q16—21no—
115C2×C3×SD16—22no—
116C2×C3×SD16—21no—
117C2×C3×SD16—21no—
118C2×C3×M16—21no—
119C2×C3×M16—31no—
120C2×C3×M16C2×D311yescharacteristic
121C2×C3×M16C22×C311yescharacteristic
122C3×C4⋊Q8—61no—
123C3×#124—61no—
124C3×#123—62no—
125C3×C22⋊Q8—62no—
126C3×#117—31no—
127C3×#117—31no—
128#951—21no—
129#950—22no—
130#949—21no—
131#947—21no—
132#946—22no—
133#944—21no—
134#942—21no—
135#932—21no—
136#929—22no—
137#926C22×C311yescharacteristic
138#926C22×C311yescharacteristic
139C2×#208C22×C311yescharacteristic
140C3×#104—62no—
141C3×#103—62no—
142C3×#99—62no—
143#835—22no—
144#834—22no—
145#825—22no—
146C23×C32—21no—
147C23×C32C2×D411yescharacteristic
148C22×C3×D3—21no—
149C32×Q8—22no—
150C32×Q8—21no—
151C32×Q8—21no—
152C32×Q8—22no—
153C32×Q8—41no—
154C32×Q8—41no—
155C32×D4—21no—
156C32×D4—21no—
157C32×D4—41no—
158C32×D4—41no—
159C2×C32×C4—22no—
160C2×C32×C4—21no—
161C2×C32×C4—41no—
162C2×C32×C4—41no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4#4111yescharacteristic
171C3×#67—41no—
172C3×#67—41no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—41no—
176C3×D12—21no—
177C3×D12—21no—
178C3×C4×D3—41no—
179C3×C4×D3—41no—
180C3×Dic6—21no—
181C3×Dic6—21no—
182C32×C8—22no—
183C3×C3⋊C8—22no—
184#575—31no—
185C2×#137C3×D311yescharacteristic, complemented
186#536—31no—
187#421—31no—
188#419—31no—
189#410—31no—
190#344—31no—
191C3×#41—22no—
192C3×#41—22no—
193C3×#41—21no—
194C3×#41—21no—
195C3×#41—21no—
196C3×#41—21no—
197C3×#41—21no—
198C3×#41—21no—
199C3×#41—21no—
200C3×#41—21no—
201C3×#41—41no—
202C3×#41—41no—
203C3×#41—61no—
204C3×#41—61no—
205C3×#41—61no—
206C3×#41—61no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—22no—
210C2×C3×Q8—22no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—22no—
221C2×C3×Q8—31no—
222C2×C3×Q8—41no—
223C2×C3×Q8—41no—
224C2×C3×Q8—41no—
225C2×C3×Q8#6712yes—
226C2×C3×Q8#6711yescharacteristic
227C2×C3×Q8C3×D411yescharacteristic
228C2×C3×Q8C3×D412yes—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—21no—
232C2×C3×D4—31no—
233C2×C3×D4—41no—
234C2×C3×D4—61no—
235C2×C3×D4#6711yescharacteristic
236C2×C3×D4C3×D411yescharacteristic
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—41no—
243C22×C3×C4—41no—
244C22×C3×C4—61no—
245C22×C3×C4C22×D311yescharacteristic
246C22×C3×C4C23×C311yescharacteristic
247C2×#67—21no—
248#235—21no—
249#235—21no—
250#235—21no—
251#235—21no—
252C2×D12C3×D411yescharacteristic
253C2×C4×D3—21no—
254C2×Dic6C3×D411yescharacteristic
255C3×Q16—28no—
256C3×Q16—24no—
257C3×Q16—24no—
258C3×Q16—62no—
259C3×SD16—28no—
260C3×SD16—24no—
261C3×SD16—24no—
262C3×SD16—62no—
263C3×SD16—62no—
264C3×D8—62no—
265C3×M16—22no—
266C3×M16—21no—
267C3×M16—21no—
268C3×M16—22no—
269C3×M16—61no—
270C3×M16—61no—
271C3×M16#6712yes—
272C3×M16C3×D412yes—
273C2×C3×C8—22no—
274C2×C3×C8—21no—
275C2×C3×C8—21no—
276C2×C3×C8—32no—
277C3×C4⋊C4—64no—
278C3×C4⋊C4—61no—
279C3×C4⋊C4—61no—
280C3×C22⋊C4—32no—
281C3×C22⋊C4—64no—
282C3×C22⋊C4—61no—
283C3×C42—61no—
284C3×C42—61no—
285#217—21no—
286#217C3×D412yes—
287#216—22no—
288#215—22no—
289#214—22no—
290#213—22no—
291(C2×D3)⋊C4—24no—
292C4⋊Dic3—21no—
293C4⋊Dic3—21no—
294Dic3⋊C4—24no—
295C4×Dic3—21no—
296C4×Dic3—21no—
297#208—21no—
298#208—21no—
299C2×C3⋊C8C3×D412yes—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—81no—
303C22×C32#12011yescharacteristic
304C22×C32#12111yescharacteristic
305C22×C32#12711yescharacteristic
306C2×C3×D3—42no—
307C2×C3×D3—41no—
308C32×C4—41no—
309C32×C4—42no—
310C32×C4#12011yescharacteristic
311C32×C4#12011yescharacteristic
312C32×C4#12111yescharacteristic
313C32×C4#12111yescharacteristic
314C3×Dic3—42no—
315C3×Dic3—41no—
316C3×Dic3—81no—
317C2×#41—31no—
318C2×#41C2×C3×D311yescharacteristic
319C22×Q8C2×C3×D311yescharacteristic
320#137—22no—
321#137—22no—
322#137—61no—
323#137—61no—
324AGL(1,ℤ/8)—61no—
325AGL(1,ℤ/8)—61no—
326#135—62no—
327C2×Q16—21no—
328C2×SD16—21no—
329C2×M16—31no—
330C2×M16C2×C3×D311yescharacteristic
331C4⋊Q8—61no—
332#124—61no—
333#123—62no—
334C22⋊Q8—62no—
335#117—31no—
336#117—31no—
337#104—62no—
338#103—62no—
339#99—62no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—41no—
344C23×C3—61no—
345C23×C3D3×D411yescharacteristic
346C23×C3C2×C3×D411yescharacteristic
347C22×D3—21no—
348C3×Q8—24no—
349C3×Q8—22no—
350C3×Q8—21no—
351C3×Q8—22no—
352C3×Q8—22no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×Q8—22no—
356C3×Q8—21no—
357C3×Q8—22no—
358C3×Q8—42no—
359C3×Q8—41no—
360C3×Q8—42no—
361C3×Q8—42no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—61no—
367C3×Q8—61no—
368C3×D4—22no—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—41no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—61no—
381C3×D4—61no—
382C3×D4—121no—
383C3×D4—121no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—41no—
395C2×C3×C4—42no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4—42no—
399C2×C3×C4—42no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—62no—
409C2×C3×C4—61no—
410C2×C3×C4—121no—
411C2×C3×C4—121no—
412C2×C3×C4—121no—
413C2×C3×C4D3×D411yescharacteristic
414C2×C3×C4D3×D411yescharacteristic
415C2×C3×C4#23711yescharacteristic
416C2×C3×C4C2×#6711yescharacteristic
417C2×C3×C4C2×#6711yescharacteristic
418C2×C3×C4C2×#6711yescharacteristic
419C2×C3×C4C2×C3×D411yescharacteristic
420C2×C3×C4C2×C3×D411yescharacteristic
421C2×C3×C4C2×C3×D411yescharacteristic
422C2×C3×C4C2×C3×D411yescharacteristic
423C2×C3×C4C2×C3×D411yescharacteristic
424C2×C3×C4C3×#4111yescharacteristic
425#67—41no—
426#67—41no—
427C2×Dic3—22no—
428C2×Dic3—21no—
429C2×Dic3—41no—
430D12—21no—
431D12—21no—
432C4×D3—41no—
433C4×D3—41no—
434Dic6—21no—
435Dic6—21no—
436C3×C8—24no—
437C3×C8—22no—
438C3×C8—22no—
439C3×C8—62no—
440C3⋊C8—22no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—81no—
445C2×C32#39211yescharacteristic
446C3×D3—81no—
447#41—21no—
448#41—21no—
449#41—21no—
450#41—21no—
451#41—61no—
452#41—61no—
453#41—61no—
454#41—61no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—31no—
461C2×Q8C3×#6712yes—
462C2×Q8C3×#6711yescharacteristic
463C2×D4—21no—
464C2×D4—31no—
465C2×D4—61no—
466C2×D4C3×#6711yescharacteristic
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—61no—
470C22×C4C22×C3×D311yescharacteristic
471Q16—24no—
472Q16—62no—
473SD16—24no—
474SD16—62no—
475SD16—62no—
476D8—62no—
477M16—21no—
478M16—61no—
479M16—61no—
480M16C3×#6712yes—
481C2×C8—21no—
482C2×C8—32no—
483C4⋊C4—64no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C22⋊C4—32no—
487C22⋊C4—64no—
488C22⋊C4—61no—
489C42—61no—
490C42—61no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3—82no—
503C22×C3—122no—
504C22×C3—121no—
505C22×C3#93711yescharacteristic
506C22×C3#93911yescharacteristic
507C22×C3#94011yescharacteristic
508C22×C3C3×#12011yescharacteristic
509C22×C3C3×#12111yescharacteristic
510C22×C3C3×#12711yescharacteristic
511C2×D3—42no—
512C2×D3—41no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—42no—
518C3×C4—44no—
519C3×C4—41no—
520C3×C4—42no—
521C3×C4—41no—
522C3×C4—42no—
523C3×C4—122no—
524C3×C4—121no—
525C3×C4—241no—
526C3×C4#93711yescharacteristic
527C3×C4#93811yescharacteristic
528C3×C4#93911yescharacteristic
529C3×C4#95311yescharacteristic
530C3×C4C3×#12011yescharacteristic
531C3×C4C3×#12011yescharacteristic
532C3×C4C3×#12111yescharacteristic
533C3×C4C3×#12111yescharacteristic
534Dic3—42no—
535Dic3—41no—
536Dic3—81no—
537C32#200911yescharacteristic, complemented, Sylow
538C23—21no—
539C23—61no—
540C23C3×D3×D411yescharacteristic
541Q8—21no—
542Q8—22no—
543Q8—22no—
544Q8—21no—
545Q8—41no—
546Q8—41no—
547Q8—61no—
548Q8—61no—
549D4—21no—
550D4—21no—
551D4—41no—
552D4—41no—
553D4—61no—
554D4—61no—
555D4—121no—
556D4—121no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—62no—
564C2×C4—61no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4C3×D3×D411yescharacteristic
569C2×C4C3×D3×D411yescharacteristic
570C2×C4C3×#23711yescharacteristic
571C2×C4C2×C3×#6711yescharacteristic
572C2×C4C2×C3×#6711yescharacteristic
573C2×C4C2×C3×#6711yescharacteristic
574C8—22no—
575C8—62no—
576C2×C3—22no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—41no—
585C2×C3—41no—
586C2×C3—82no—
587C2×C3—81no—
588C2×C3—81no—
589C2×C3—241no—
590C2×C3#522311yescharacteristic
591C2×C3C3×#39211yescharacteristic
592D3—81no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—122no—
597C22—121no—
598C22C3×#93711yescharacteristic
599C22C3×#93911yescharacteristic
600C22C3×#94011yescharacteristic
601C4—41no—
602C4—42no—
603C4—122no—
604C4—121no—
605C4—241no—
606C4C3×#93711yescharacteristic
607C4C3×#93811yescharacteristic
608C4C3×#93911yescharacteristic
609C4C3×#95311yescharacteristic
610C3—21no—
611C3#7211111yescharacteristic, direct product factor
612C3C3×#200911yescharacteristic, complemented
613C2—21no—
614C2—21no—
615C2—21no—
616C2—81no—
617C2—241no—
618C2C3×#522311yescharacteristic
subgroupG/H
centerC2×C3#5223
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#137C2
socleC2×C32#392
prime radicalC3×#5822C2
hypercenterC2×C3×#137D3
hypocenterC3C3×#2009
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
1. C2×C32×#137C2semidirect
3. C3×#5816C2semidirect
4. C3×#5240C2semidirect
5. C3×#5225C2semidirect
6. C3×#5139C2semidirect
7. C3×#4559C2semidirect
9. #72111C3direct (N × H)
35. C2×C3×#137D3semidirect
36. C2×C3×#137C2×C3semidirect
39. #5816C2×C3semidirect
43. #5240C2×C3semidirect
44. #5225C2×C3semidirect
45. #5139C2×C3semidirect
47. #4559C2×C3semidirect
185. C2×#137C3×D3semidirect
537. C32#2009semidirect
611. C3#72111direct (N × H)
612. C3C3×#2009semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1372
C3×#58222
C3×#58162
C3×#52402
C3×#52252
C3×#51392
C3×#45592
C3×#20093
#721113