Finite Group Databasebeta

← C3×#72094 #11792032

Subgroups — C3×#72094 #11792032

Order 1152.

1105 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C3×#5830C211yesmaximal, characteristic, complemented
3C2×C3×#940C211yesmaximal, characteristic, complemented
4C2×C3×#837C211yesmaximal, characteristic, complemented
5C3×#5215C211yesmaximal, characteristic, complemented
6C2×C3×#823C211yesmaximal, characteristic, complemented
7C3×#4555C211yesmaximal, characteristic, complemented
8C3×#2002—31nomaximal
9#72094C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×D4C2211yescharacteristic
12C22×C3×#67—22no—
13C22×C3×D12C2211yescharacteristic
14C32×AGL(1,ℤ/8)—22no—
15C32×AGL(1,ℤ/8)—22no—
16C2×C32×SD16—21no—
17C2×C32×D8—21no—
18C2×C32×M16C2211yescharacteristic
19C3×#947—21no—
20C3×#940—22no—
21C3×#940—21no—
22C3×#940—22no—
23C3×#937—22no—
24C3×#930—22no—
25C3×#926C2211yescharacteristic
26C2×C3×C4×Dic3C2211yescharacteristic
27C2×C3×#208C2211yescharacteristic
28C3×#837—22no—
29C3×#832—22no—
30C3×#823—22no—
31C2×C3×AGL(1,ℤ/8)—21no—
32C2×C3×AGL(1,ℤ/8)D311yescharacteristic, complemented
33C2×C3×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
34C3×#535—31no—
35C2×C3×#127—31no—
36#5830C2×C311yescharacteristic, complemented
37C2×#940C2×C311yescharacteristic, complemented
38C2×C3×#104—31no—
39C3×#418—31no—
40C2×C3×#100—31no—
41C2×#837C2×C311yescharacteristic, complemented
42#5215C2×C311yescharacteristic, complemented
43C2×#823C2×C311yescharacteristic, complemented
44C3×#339—31no—
45#4555C2×C311yescharacteristic, complemented
46C24×C32—21no—
47C23×C3×D3D412yes—
48C32×#41—21no—
49C32×#41—21no—
50C32×#41—21no—
51C32×#41—21no—
52C2×C32×Q8D411yescharacteristic, complemented
53C2×C32×D4—21no—
54C2×C32×D4—21no—
55C2×C32×D4—21no—
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4D412yescomplemented
60C2×C32×D4D411yescharacteristic, complemented
61C22×C32×C4—21no—
62C22×C32×C4C2311yescharacteristic
63C2×C3×#67—28no—
64C2×C3×#67—24no—
65C2×C3×#67—48no—
66C22×C3×Dic3—21no—
67C2×C3×D12—21no—
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12—21no—
72C2×C3×D12—21no—
73C32×SD16—24no—
74C32×D8—24no—
75C32×M16—21no—
76C32×M16D412yescomplemented
77C2×C32×C8—21no—
78C3×#217—21no—
79C3×(C2×D3)⋊C4—24no—
80C3×C4⋊Dic3D412yescomplemented
81C3×C4×Dic3—21no—
82C3×C4×Dic3—21no—
83C3×C4×Dic3D412yescomplemented
84C3×#208—21no—
85C3×#208—21no—
86C2×C3×C3⋊C8—21no—
87#2002—31noSylow
88C2×C3×#41—21no—
89C2×C3×#41C2×D311yescharacteristic
90C2×C3×#41C22×C311yescharacteristic
91C22×C3×D4—21no—
92C22×C3×D4—31no—
93C22×C3×D4—62no—
94C22×C3×D4C2×D311yescharacteristic
95C22×C3×D4C22×C311yescharacteristic
96C22×#67—22no—
97C22×D12C22×C311yescharacteristic
98C3×AGL(1,ℤ/8)—24no—
99C3×AGL(1,ℤ/8)—24no—
100C3×AGL(1,ℤ/8)—22no—
101C3×AGL(1,ℤ/8)—22no—
102C3×AGL(1,ℤ/8)—22no—
103C3×AGL(1,ℤ/8)—22no—
104C2×C3×SD16—22no—
105C2×C3×SD16—21no—
106C2×C3×SD16—21no—
107C2×C3×D8—22no—
108C2×C3×D8—21no—
109C2×C3×D8—21no—
110C2×C3×M16—21no—
111C2×C3×M16—31no—
112C2×C3×M16C2×D311yescharacteristic
113C2×C3×M16C22×C311yescharacteristic
114C3×#127—62no—
115C3×#127—62no—
116C3×#127—61no—
117C3×#124—61no—
118C3×#121—62no—
119C3×#120—62no—
120C3×#117—31no—
121C2×C3×C42—31no—
122#947—21no—
123#940—22no—
124#940—21no—
125#940—22no—
126#937—22no—
127#930—22no—
128#926C22×C311yescharacteristic
129C2×C4×Dic3C22×C311yescharacteristic
130C2×#208C22×C311yescharacteristic
131C3×#104—62no—
132C3×#102—62no—
133C3×#100—62no—
134#837—22no—
135#832—22no—
136#823—22no—
137C23×C32—22no—
138C23×C32—21no—
139C23×C32—41no—
140C23×C32—41no—
141C23×C32—44no—
142C23×C32C2×D411yescharacteristic
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3—22no—
146C22×C3×D3—44no—
147C32×Q8—21no—
148C32×Q8—21no—
149C32×D4—21no—
150C32×D4—22no—
151C32×D4—22no—
152C32×D4—21no—
153C32×D4—41no—
154C32×D4—41no—
155C32×D4—41no—
156C32×D4—41no—
157C2×C32×C4—21no—
158C2×C32×C4—41no—
159C2×C32×C4—41no—
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D411yescharacteristic
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4#4111yescharacteristic
166C3×#67—416no—
167C3×#67—84no—
168C2×C3×Dic3—24no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C3×D12—41no—
172C3×D12—41no—
173C3×D12—41no—
174C3×D12—41no—
175C32×C8—22no—
176C3×C3⋊C8—41no—
177C2×AGL(1,ℤ/8)C3×D311yescharacteristic, complemented
178#535—31no—
179C2×#127—31no—
180C2×#104—31no—
181#418—31no—
182C2×#100—31no—
183#339—31no—
184C24×C3—21no—
185C24×C3—21no—
186C24×C3—32no—
187C24×C3—41no—
188C23×D3C3×D412yes—
189C3×#41—22no—
190C3×#41—22no—
191C3×#41—21no—
192C3×#41—21no—
193C3×#41—21no—
194C3×#41—21no—
195C3×#41—21no—
196C3×#41—21no—
197C3×#41—21no—
198C3×#41—21no—
199C3×#41—41no—
200C3×#41—41no—
201C2×C3×Q8—21no—
202C2×C3×Q8#6711yescharacteristic, complemented
203C2×C3×Q8C3×D411yescharacteristic, complemented
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—22no—
220C2×C3×D4—41no—
221C2×C3×D4—41no—
222C2×C3×D4—41no—
223C2×C3×D4—41no—
224C2×C3×D4—68no—
225C2×C3×D4—64no—
226C2×C3×D4—61no—
227C2×C3×D4—61no—
228C2×C3×D4—61no—
229C2×C3×D4—61no—
230C2×C3×D4—61no—
231C2×C3×D4—61no—
232C2×C3×D4—128no—
233C2×C3×D4#6711yescharacteristic, complemented
234C2×C3×D4#6712yescomplemented
235C2×C3×D4C3×D411yescharacteristic, complemented
236C2×C3×D4C3×D412yescomplemented
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—41no—
241C22×C3×C4—61no—
242C22×C3×C4C22×D311yescharacteristic
243C22×C3×C4C23×C311yescharacteristic
244C2×#67—28no—
245C2×#67—24no—
246C2×#67—48no—
247C22×Dic3—21no—
248C2×D12—21no—
249C2×D12—21no—
250C2×D12—21no—
251C2×D12—21no—
252C2×D12—21no—
253C2×D12—21no—
254C3×SD16—28no—
255C3×SD16—24no—
256C3×SD16—24no—
257C3×D8—28no—
258C3×D8—24no—
259C3×D8—24no—
260C3×M16—22no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16—22no—
264C3×M16—61no—
265C3×M16—61no—
266C3×M16#6712yescomplemented
267C3×M16C3×D412yescomplemented
268C2×C3×C8—22no—
269C2×C3×C8—21no—
270C2×C3×C8—21no—
271C2×C3×C8—61no—
272C3×C4⋊C4—32no—
273C3×C22⋊C4—64no—
274C3×C22⋊C4—61no—
275C3×C42—32no—
276C3×C42—61no—
277C3×C42—61no—
278#217—21no—
279(C2×D3)⋊C4—24no—
280C4⋊Dic3C3×D412yescomplemented
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3C3×D412yescomplemented
284#208—21no—
285#208—21no—
286C2×C3⋊C8—21no—
287C22×C32—22no—
288C22×C32—41no—
289C22×C32—42no—
290C22×C32—44no—
291C22×C32—42no—
292C22×C32—82no—
293C22×C32—82no—
294C22×C32#12011yescharacteristic
295C22×C32#12011yescharacteristic
296C22×C32#12111yescharacteristic
297C2×C3×D3—44no—
298C2×C3×D3—42no—
299C2×C3×D3—44no—
300C2×C3×D3—82no—
301C32×C4—41no—
302C32×C4#12011yescharacteristic
303C32×C4#12111yescharacteristic
304C32×C4#12111yescharacteristic
305C32×C4#12711yescharacteristic
306C3×Dic3—24no—
307C3×Dic3—42no—
308C2×#41C2×C3×D311yescharacteristic
309C22×D4—31no—
310C22×D4—62no—
311C22×D4C2×C3×D311yescharacteristic
312AGL(1,ℤ/8)—22no—
313AGL(1,ℤ/8)—22no—
314C2×SD16—21no—
315C2×D8—21no—
316C2×M16—31no—
317C2×M16C2×C3×D311yescharacteristic
318#127—62no—
319#127—62no—
320#127—61no—
321#124—61no—
322#121—62no—
323#120—62no—
324#117—31no—
325C2×C42—31no—
326#104—62no—
327#102—62no—
328#100—62no—
329C23×C3—22no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—41no—
335C23×C3—42no—
336C23×C3—44no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—44no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—62no—
345C23×C3—62no—
346C23×C3—62no—
347C23×C3—84no—
348C23×C3—124no—
349C23×C3D3×D411yescharacteristic
350C23×C3C2×C3×D411yescharacteristic
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—22no—
354C22×D3—44no—
355C3×Q8—22no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—41no—
361C3×D4—22no—
362C3×D4—24no—
363C3×D4—22no—
364C3×D4—21no—
365C3×D4—21no—
366C3×D4—22no—
367C3×D4—21no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—21no—
371C3×D4—42no—
372C3×D4—41no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—1216no—
386C3×D4—121no—
387C3×D4—121no—
388C3×D4—121no—
389C3×D4—121no—
390C3×D4—244no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
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—41no—
400C2×C3×C4—42no—
401C2×C3×C4—42no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—64no—
407C2×C3×C4—62no—
408C2×C3×C4—62no—
409C2×C3×C4D3×D411yescharacteristic
410C2×C3×C4D3×D411yescharacteristic
411C2×C3×C4#23711yescharacteristic
412C2×C3×C4C2×#6711yescharacteristic
413C2×C3×C4C2×#6711yescharacteristic
414C2×C3×C4C2×#6711yescharacteristic
415C2×C3×C4C2×C3×D411yescharacteristic
416C2×C3×C4C2×C3×D411yescharacteristic
417C2×C3×C4C2×C3×D411yescharacteristic
418C2×C3×C4C2×C3×D411yescharacteristic
419C2×C3×C4C2×C3×D411yescharacteristic
420C2×C3×C4C3×#4111yescharacteristic
421#67—416no—
422#67—84no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426D12—41no—
427D12—41no—
428D12—41no—
429D12—41no—
430C3×C8—24no—
431C3×C8—22no—
432C3×C8—22no—
433C3×C8—121no—
434C3⋊C8—41no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—81no—
438C2×C32—82no—
439C2×C32#39211yescharacteristic
440C2×C32#45312yes—
441C3×D3—82no—
442C24—21no—
443C24—32no—
444#41—21no—
445#41—21no—
446#41—21no—
447#41—21no—
448C2×Q8C3×#6711yescharacteristic, complemented
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—68no—
456C2×D4—64no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—128no—
464C2×D4C3×#6711yescharacteristic, complemented
465C2×D4C3×#6712yescomplemented
466C22×C4—21no—
467C22×C4—61no—
468C22×C4C22×C3×D311yescharacteristic
469SD16—24no—
470D8—24no—
471M16—21no—
472M16—61no—
473M16—61no—
474M16C3×#6712yescomplemented
475C2×C8—21no—
476C2×C8—61no—
477C4⋊C4—32no—
478C22⋊C4—64no—
479C22⋊C4—61no—
480C42—32no—
481C42—61no—
482C42—61no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—42no—
489C22×C3—44no—
490C22×C3—42no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—42no—
494C22×C3—41no—
495C22×C3—42no—
496C22×C3—41no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—82no—
500C22×C3—84no—
501C22×C3—82no—
502C22×C3—82no—
503C22×C3—84no—
504C22×C3—84no—
505C22×C3—82no—
506C22×C3—82no—
507C22×C3—124no—
508C22×C3—124no—
509C22×C3—122no—
510C22×C3—242no—
511C22×C3#93711yescharacteristic
512C22×C3#93711yescharacteristic
513C22×C3#93811yescharacteristic
514C22×C3C3×#12011yescharacteristic
515C22×C3C3×#12011yescharacteristic
516C22×C3C3×#12111yescharacteristic
517C2×D3—44no—
518C2×D3—42no—
519C2×D3—44no—
520C2×D3—82no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—42no—
526C3×C4—41no—
527C3×C4—41no—
528C3×C4—64no—
529C3×C4—122no—
530C3×C4#93911yescharacteristic
531C3×C4#93911yescharacteristic
532C3×C4#94011yescharacteristic
533C3×C4#95311yescharacteristic
534C3×C4C3×#12011yescharacteristic
535C3×C4C3×#12111yescharacteristic
536C3×C4C3×#12111yescharacteristic
537C3×C4C3×#12711yescharacteristic
538Dic3—24no—
539Dic3—42no—
540C32#200211yescharacteristic, complemented, Sylow
541C23—21no—
542C23—22no—
543C23—44no—
544C23—41no—
545C23—41no—
546C23—62no—
547C23—62no—
548C23—62no—
549C23—124no—
550C23C3×D3×D411yescharacteristic
551Q8—21no—
552Q8—21no—
553D4—22no—
554D4—21no—
555D4—21no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—1216no—
562D4—121no—
563D4—121no—
564D4—121no—
565D4—121no—
566D4—244no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—64no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4C3×D3×D411yescharacteristic
574C2×C4C3×D3×D411yescharacteristic
575C2×C4C3×#23711yescharacteristic
576C2×C4C2×C3×#6711yescharacteristic
577C2×C4C2×C3×#6711yescharacteristic
578C2×C4C2×C3×#6711yescharacteristic
579C8—22no—
580C8—121no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—22no—
586C2×C3—21no—
587C2×C3—41no—
588C2×C3—41no—
589C2×C3—84no—
590C2×C3—82no—
591C2×C3—82no—
592C2×C3—81no—
593C2×C3—82no—
594C2×C3—81no—
595C2×C3—242no—
596C2×C3#519812yes—
597C2×C3#522311yescharacteristic
598C2×C3C3×#39211yescharacteristic
599C2×C3C3×#45312yes—
600D3—82no—
601C22—22no—
602C22—44no—
603C22—42no—
604C22—41no—
605C22—42no—
606C22—82no—
607C22—82no—
608C22—124no—
609C22—124no—
610C22—122no—
611C22—242no—
612C22C3×#93711yescharacteristic
613C22C3×#93711yescharacteristic
614C22C3×#93811yescharacteristic
615C4—41no—
616C4—64no—
617C4—122no—
618C4C3×#93911yescharacteristic
619C4C3×#93911yescharacteristic
620C4C3×#94011yescharacteristic
621C4C3×#95311yescharacteristic
622C3—21no—
623C3#7209411yescharacteristic, direct product factor
624C3C3×#200211yescharacteristic, complemented
625C2—21no—
626C2—21no—
627C2—82no—
628C2—81no—
629C2—242no—
630C2C3×#519812yes—
631C2C3×#522311yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC2×C32×AGL(1,ℤ/8)C2
socleC22×C32#120
hypercenterC2×C3×AGL(1,ℤ/8)D3
hypocenterC3C3×#2002
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C32×AGL(1,ℤ/8)C2semidirect
2. C3×#5830C2semidirect
3. C2×C3×#940C2semidirect
4. C2×C3×#837C2semidirect
5. C3×#5215C2semidirect
6. C2×C3×#823C2semidirect
7. C3×#4555C2semidirect
9. #72094C3direct (N × H)
32. C2×C3×AGL(1,ℤ/8)D3semidirect
33. C2×C3×AGL(1,ℤ/8)C2×C3semidirect
36. #5830C2×C3semidirect
37. C2×#940C2×C3semidirect
41. C2×#837C2×C3semidirect
42. #5215C2×C3semidirect
43. C2×#823C2×C3semidirect
45. #4555C2×C3semidirect
52. C2×C32×Q8D4semidirect
59. C2×C32×D4D4semidirect
60. C2×C32×D4D4semidirect
76. C32×M16D4semidirect
80. C3×C4⋊Dic3D4semidirect
83. C3×C4×Dic3D4semidirect
177. C2×AGL(1,ℤ/8)C3×D3semidirect
202. C2×C3×Q8#67semidirect
203. C2×C3×Q8C3×D4semidirect
233. C2×C3×D4#67semidirect
234. C2×C3×D4#67semidirect
235. C2×C3×D4C3×D4semidirect
236. C2×C3×D4C3×D4semidirect
266. C3×M16#67semidirect
267. C3×M16C3×D4semidirect
280. C4⋊Dic3C3×D4semidirect
283. C4×Dic3C3×D4semidirect
448. C2×Q8C3×#67semidirect
464. C2×D4C3×#67semidirect
465. C2×D4C3×#67semidirect
474. M16C3×#67semidirect
540. C32#2002semidirect
623. C3#72094direct (N × H)
624. C3C3×#2002semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
C3×#58302
C2×C3×#9402
C2×C3×#8372
C3×#52152
C2×C3×#8232
C3×#45552
C3×#20023
#720943