Finite Group Databasebeta

← C3×#72120 #11792344

Subgroups — C3×#72120 #11792344

Order 1152.

827 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#575C211yesmaximal, characteristic
2C3×#5825C211yesmaximal, characteristic, complemented
3C2×C3×#951C211yesmaximal, characteristic, complemented
4C3×#5240C211yesmaximal, characteristic, complemented
5C3×#5236C211yesmaximal, characteristic, complemented
6C3×#5120C211yesmaximal, characteristic, complemented
7C3×#4556C211yesmaximal, characteristic, complemented
8C3×#2009—31nomaximal
9#72120C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C2×C32×#41C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C32×#137—21no—
14C32×#137—21no—
15C32×AGL(1,ℤ/8)—21no—
16C32×AGL(1,ℤ/8)—21no—
17C32×#135—22no—
18C2×C32×M16C2211yescharacteristic
19C3×#951—22no—
20C3×#951—22no—
21C3×#944—21no—
22C2×C3×#216—21no—
23C3×#936—21no—
24C3×#935—22no—
25C2×C3×#214—21no—
26C3×#926C2211yescharacteristic
27C3×#926C2211yescharacteristic
28C3×#924—22no—
29C2×C3×#208C2211yescharacteristic
30C3×#837—22no—
31C3×#834—22no—
32C3×#816—22no—
33C3×#575—21no—
34C3×#575D311yescharacteristic
35C3×#575C2×C311yescharacteristic
36C2×C3×#137—31no—
37C3×#536—31no—
38#5825C2×C311yescharacteristic, complemented
39C2×#951C2×C311yescharacteristic, complemented
40C3×#421—31no—
41C3×#419—31no—
42C3×#410—31no—
43#5240C2×C311yescharacteristic, complemented
44#5236C2×C311yescharacteristic, complemented
45#5120C2×C311yescharacteristic, complemented
46C3×#344—31no—
47#4556C2×C311yescharacteristic, complemented
48C32×#41—21no—
49C32×#41—21no—
50C32×#41—21no—
51C32×#41—21no—
52C32×#41—21no—
53C32×#41—21no—
54C32×#41—21no—
55C32×#41—21no—
56C2×C32×Q8D411yescharacteristic
57C2×C32×Q8D411yescharacteristic
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4D411yescharacteristic
61C2×C32×D4D411yescharacteristic
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4C2311yescharacteristic
65C22×C3×Dic3—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6D412yes—
72C32×Q16—22no—
73C32×SD16—22no—
74C32×SD16—22no—
75C32×D8—22no—
76C32×M16—21no—
77C32×M16—21no—
78C2×C32×C8D412yes—
79C3×#217—24no—
80C3×#217—21no—
81C3×#217D412yes—
82C3×#216—24no—
83C3×#214—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×#208D412yes—
91C2×C3×C3⋊C8—21no—
92#2009—31noSylow
93C2×C3×#41—21no—
94C2×C3×#41—21no—
95C2×C3×#41C2×D311yescharacteristic
96C2×C3×#41C2×D311yescharacteristic
97C2×C3×#41C22×C311yescharacteristic
98C2×C3×#41C22×C311yescharacteristic
99C22×C3×Q8—31no—
100C22×Dic6C22×C311yescharacteristic
101C3×#137—22no—
102C3×#137—22no—
103C3×#137—21no—
104C3×#137—21no—
105C3×#137—21no—
106C3×#137—21no—
107C3×#137—62no—
108C3×#137—62no—
109C3×AGL(1,ℤ/8)—22no—
110C3×AGL(1,ℤ/8)—22no—
111C3×AGL(1,ℤ/8)—21no—
112C3×AGL(1,ℤ/8)—21no—
113C3×AGL(1,ℤ/8)—21no—
114C3×AGL(1,ℤ/8)—21no—
115C3×#135—24no—
116C3×#135—22no—
117C3×#135—22no—
118C2×C3×Q16—61no—
119C2×C3×SD16—61no—
120C2×C3×M16—21no—
121C2×C3×M16—31no—
122C2×C3×M16C2×D311yescharacteristic
123C2×C3×M16C22×C311yescharacteristic
124C3×C4⋊Q8—61no—
125C3×#124—61no—
126C3×#123—62no—
127C3×C22⋊Q8—62no—
128C3×#117—31no—
129C3×#117—31no—
130#951—22no—
131#951—22no—
132#944—21no—
133C2×#216—21no—
134#936—21no—
135#935—22no—
136C2×#214—21no—
137#926C22×C311yescharacteristic
138#926C22×C311yescharacteristic
139#924—22no—
140C2×#208C22×C311yescharacteristic
141C3×#104—62no—
142C3×#103—62no—
143C3×#99—62no—
144#837—22no—
145#834—22no—
146#816—22no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32C2×D411yescharacteristic
150C32×Q8—21no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—21no—
154C32×D4—21no—
155C32×D4—21no—
156C32×D4—21no—
157C32×D4—21no—
158C32×D4—41no—
159C32×D4—41no—
160C32×D4—41no—
161C32×D4—41no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—41no—
165C2×C32×C4—41no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4#4111yescharacteristic
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—41no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3—41no—
179C3×Dic6—22no—
180C3×Dic6—22no—
181C3×Dic6—41no—
182C3×Dic6—41no—
183C32×C8—22no—
184C3×C3⋊C8—22no—
185#575C3×D311yescharacteristic
186C2×#137—31no—
187#536—31no—
188#421—31no—
189#419—31no—
190#410—31no—
191#344—31no—
192C3×#41—22no—
193C3×#41—22no—
194C3×#41—22no—
195C3×#41—22no—
196C3×#41—21no—
197C3×#41—21no—
198C3×#41—21no—
199C3×#41—21no—
200C3×#41—21no—
201C3×#41—21no—
202C3×#41—21no—
203C3×#41—21no—
204C3×#41—21no—
205C3×#41—21no—
206C3×#41—21no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—21no—
210C3×#41—21no—
211C3×#41—21no—
212C3×#41—41no—
213C3×#41—41no—
214C3×#41—41no—
215C3×#41—41no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—32no—
219C2×C3×Q8—61no—
220C2×C3×Q8—61no—
221C2×C3×Q8—61no—
222C2×C3×Q8—61no—
223C2×C3×Q8—61no—
224C2×C3×Q8#6711yescharacteristic
225C2×C3×Q8#6711yescharacteristic
226C2×C3×Q8C3×D411yescharacteristic
227C2×C3×Q8C3×D411yescharacteristic
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—21no—
234C2×C3×D4—41no—
235C2×C3×D4—41no—
236C2×C3×D4#6711yescharacteristic
237C2×C3×D4#6711yescharacteristic
238C2×C3×D4C3×D411yescharacteristic
239C2×C3×D4C3×D411yescharacteristic
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—21no—
243C22×C3×C4—21no—
244C22×C3×C4—21no—
245C22×C3×C4—41no—
246C22×C3×C4—41no—
247C22×C3×C4—61no—
248C22×C3×C4C22×D311yescharacteristic
249C22×C3×C4C23×C311yescharacteristic
250C22×Dic3—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6—21no—
256C2×Dic6C3×D412yes—
257C3×Q16—24no—
258C3×Q16—22no—
259C3×Q16—22no—
260C3×Q16—64no—
261C3×SD16—24no—
262C3×SD16—24no—
263C3×SD16—22no—
264C3×SD16—22no—
265C3×SD16—22no—
266C3×SD16—22no—
267C3×SD16—64no—
268C3×D8—24no—
269C3×D8—22no—
270C3×D8—22no—
271C3×M16—22no—
272C3×M16—22no—
273C3×M16—21no—
274C3×M16—21no—
275C3×M16—21no—
276C3×M16—21no—
277C3×M16—32no—
278C3×M16—61no—
279C2×C3×C8—22no—
280C2×C3×C8—61no—
281C2×C3×C8#6712yes—
282C2×C3×C8C3×D412yes—
283C3×C4⋊C4—64no—
284C3×C4⋊C4—61no—
285C3×C4⋊C4—61no—
286C3×C22⋊C4—32no—
287C3×C22⋊C4—64no—
288C3×C22⋊C4—61no—
289C3×C42—61no—
290C3×C42—61no—
291#217—24no—
292#217—21no—
293#217C3×D412yes—
294#216—24no—
295#214—24no—
296C4⋊Dic3—21no—
297C4⋊Dic3—21no—
298Dic3⋊C4—24no—
299C4×Dic3—21no—
300C4×Dic3—21no—
301#208—21no—
302#208C3×D412yes—
303C2×C3⋊C8—21no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—42no—
308C22×C32—81no—
309C22×C32#12011yescharacteristic
310C22×C32#12111yescharacteristic
311C22×C32#12711yescharacteristic
312C32×C4—41no—
313C32×C4—41no—
314C32×C4#12011yescharacteristic
315C32×C4#12011yescharacteristic
316C32×C4#12111yescharacteristic
317C32×C4#12111yescharacteristic
318C3×Dic3—42no—
319C3×Dic3—42no—
320C3×Dic3—81no—
321C2×#41C2×C3×D311yescharacteristic
322C2×#41C2×C3×D311yescharacteristic
323C22×Q8—31no—
324#137—21no—
325#137—21no—
326#137—62no—
327#137—62no—
328AGL(1,ℤ/8)—21no—
329AGL(1,ℤ/8)—21no—
330#135—22no—
331C2×Q16—61no—
332C2×SD16—61no—
333C2×M16—31no—
334C2×M16C2×C3×D311yescharacteristic
335C4⋊Q8—61no—
336#124—61no—
337#123—62no—
338C22⋊Q8—62no—
339#117—31no—
340#117—31no—
341#104—62no—
342#103—62no—
343#99—62no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3D3×D411yescharacteristic
352C23×C3C2×C3×D411yescharacteristic
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×Q8—21no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—62no—
366C3×Q8—62no—
367C3×Q8—121no—
368C3×Q8—121no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—21no—
374C3×D4—21no—
375C3×D4—21no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—21no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—42no—
382C3×D4—42no—
383C3×D4—42no—
384C3×D4—42no—
385C3×D4—41no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—41no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—42no—
408C2×C3×C4—42no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—62no—
418C2×C3×C4—62no—
419C2×C3×C4—121no—
420C2×C3×C4—121no—
421C2×C3×C4—121no—
422C2×C3×C4D3×D411yescharacteristic
423C2×C3×C4D3×D411yescharacteristic
424C2×C3×C4#23711yescharacteristic
425C2×C3×C4C2×#6711yescharacteristic
426C2×C3×C4C2×#6711yescharacteristic
427C2×C3×C4C2×#6711yescharacteristic
428C2×C3×C4C2×C3×D411yescharacteristic
429C2×C3×C4C2×C3×D411yescharacteristic
430C2×C3×C4C2×C3×D411yescharacteristic
431C2×C3×C4C2×C3×D411yescharacteristic
432C2×C3×C4C2×C3×D411yescharacteristic
433C2×C3×C4C3×#4111yescharacteristic
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439Dic6—22no—
440Dic6—22no—
441Dic6—41no—
442Dic6—41no—
443C3×C8—24no—
444C3×C8—22no—
445C3×C8—22no—
446C3×C8—62no—
447C3⋊C8—22no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—81no—
452C2×C32—81no—
453C2×C32#39211yescharacteristic
454#41—21no—
455#41—21no—
456#41—21no—
457#41—21no—
458#41—21no—
459#41—21no—
460#41—21no—
461#41—21no—
462C2×Q8—32no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×Q8C3×#6711yescharacteristic
469C2×Q8C3×#6711yescharacteristic
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C3×#6711yescharacteristic
473C2×D4C3×#6711yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—61no—
477C22×C4C22×C3×D311yescharacteristic
478Q16—22no—
479Q16—64no—
480SD16—22no—
481SD16—22no—
482SD16—64no—
483D8—22no—
484M16—21no—
485M16—21no—
486M16—32no—
487M16—61no—
488C2×C8—61no—
489C2×C8C3×#6712yes—
490C4⋊C4—64no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C22⋊C4—32no—
494C22⋊C4—64no—
495C22⋊C4—61no—
496C42—61no—
497C42—61no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—81no—
512C22×C3—82no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3—82no—
516C22×C3#93711yescharacteristic
517C22×C3#93911yescharacteristic
518C22×C3#94011yescharacteristic
519C22×C3C3×#12011yescharacteristic
520C22×C3C3×#12111yescharacteristic
521C22×C3C3×#12711yescharacteristic
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—122no—
533C3×C4—122no—
534C3×C4—241no—
535C3×C4#93711yescharacteristic
536C3×C4#93811yescharacteristic
537C3×C4#93911yescharacteristic
538C3×C4#95311yescharacteristic
539C3×C4C3×#12011yescharacteristic
540C3×C4C3×#12011yescharacteristic
541C3×C4C3×#12111yescharacteristic
542C3×C4C3×#12111yescharacteristic
543Dic3—42no—
544Dic3—42no—
545Dic3—81no—
546C32#200911yescharacteristic, complemented, Sylow
547C23—21no—
548C23—21no—
549C23C3×D3×D411yescharacteristic
550Q8—21no—
551Q8—21no—
552Q8—21no—
553Q8—21no—
554Q8—62no—
555Q8—62no—
556Q8—121no—
557Q8—121no—
558D4—21no—
559D4—21no—
560D4—21no—
561D4—21no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4C3×D3×D411yescharacteristic
578C2×C4C3×D3×D411yescharacteristic
579C2×C4C3×#23711yescharacteristic
580C2×C4C2×C3×#6711yescharacteristic
581C2×C4C2×C3×#6711yescharacteristic
582C2×C4C2×C3×#6711yescharacteristic
583C8—22no—
584C8—62no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—41no—
593C2×C3—41no—
594C2×C3—41no—
595C2×C3—82no—
596C2×C3—82no—
597C2×C3—81no—
598C2×C3—81no—
599C2×C3—81no—
600C2×C3—81no—
601C2×C3#522311yescharacteristic
602C2×C3C3×#39211yescharacteristic
603C22—41no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—81no—
608C22C3×#93711yescharacteristic
609C22C3×#93911yescharacteristic
610C22C3×#94011yescharacteristic
611C4—41no—
612C4—41no—
613C4—122no—
614C4—122no—
615C4—241no—
616C4C3×#93711yescharacteristic
617C4C3×#93811yescharacteristic
618C4C3×#93911yescharacteristic
619C4C3×#95311yescharacteristic
620C3—21no—
621C3#7212011yescharacteristic, direct product factor
622C3C3×#200911yescharacteristic, complemented
623C2—21no—
624C2—21no—
625C2—21no—
626C2—81no—
627C2—81no—
628C2C3×#522311yescharacteristic
subgroupG/H
centerC2×C3#5223
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC32×#575C2
socleC2×C32#392
hypercenterC3×#575D3
hypocenterC3C3×#2009
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#575

Complements

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

NH ≅ G/Ntype
2. C3×#5825C2semidirect
3. C2×C3×#951C2semidirect
4. C3×#5240C2semidirect
5. C3×#5236C2semidirect
6. C3×#5120C2semidirect
7. C3×#4556C2semidirect
9. #72120C3direct (N × H)
38. #5825C2×C3semidirect
39. C2×#951C2×C3semidirect
43. #5240C2×C3semidirect
44. #5236C2×C3semidirect
45. #5120C2×C3semidirect
47. #4556C2×C3semidirect
546. C32#2009semidirect
621. C3#72120direct (N × H)
622. C3C3×#2009semidirect

Maximal subgroups

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

subgroupindex
C32×#5752
C3×#58252
C2×C3×#9512
C3×#52402
C3×#52362
C3×#51202
C3×#45562
C3×#20093
#721203