Finite Group Databasebeta

← C3×#84003 #11868170

Subgroups — C3×#84003 #11868170

Order 1152.

1023 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#575C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic, complemented
3C2×C3×#951C211yesmaximal, characteristic, complemented
4C3×#5819C211yesmaximal, characteristic, complemented
5C3×#5237C211yesmaximal, characteristic, complemented
6C3×#5204C214yesmaximal, complemented
7C3×#5200C214yesmaximal, complemented
8C2×C3×#824C211yesmaximal, characteristic, complemented
9C3×#5124C211yesmaximal, characteristic, complemented
10C3×#3069—31nomaximal
11#84003C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C2×C32×#41C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C32×#137—21no—
16C32×#137C2212yes—
17C32×AGL(1,ℤ/8)—21no—
18C32×AGL(1,ℤ/8)C2212yes—
19C32×#135—22no—
20C2×C32×M16C2211yescharacteristic
21C3×#951—22no—
22C3×#951—22no—
23C3×#951—22no—
24C3×#951C2214yes—
25C3×#948—22no—
26C3×#948C2214yes—
27C2×C3×#216—21no—
28C2×C3×#216C2212yescomplemented
29C2×C3×#214—21no—
30C2×C3×#214C2212yescomplemented
31C2×C3×#208—22no—
32C2×C3×#208C2211yescharacteristic
33C2×C3×#208C2211yescharacteristic
34C2×C3×#208C2211yescharacteristic
35C3×#836C2212yescomplemented
36C3×#833C2212yescomplemented
37C3×#824C2214yescomplemented
38C3×#822C2214yescomplemented
39C3×#575—21no—
40C3×#575D311yescharacteristic
41C3×#575C2×C311yescharacteristic
42C2×C3×#137—31no—
43C2×C3×#137—31no—
44C3×#568—31no—
45C2×#951C2×C311yescharacteristic, complemented
46C2×#951C2×C311yescharacteristic, complemented
47#5819C2×C311yescharacteristic, complemented
48C3×#473—34no—
49C3×#471—34no—
50C3×#430—31no—
51C3×#413—31no—
52C2×C3×#101—31no—
53#5237C2×C311yescharacteristic, complemented
54#5204C2×C314yescomplemented
55#5200C2×C314yescomplemented
56C2×#824C2×C311yescharacteristic, complemented
57#5124C2×C311yescharacteristic, complemented
58C32×#41—21no—
59C32×#41—21no—
60C32×#41—21no—
61C32×#41—21no—
62C32×#41—21no—
63C32×#41—21no—
64C32×#41D412yes—
65C32×#41C2312yes—
66C2×C32×Q8D411yescharacteristic
67C2×C32×Q8C2311yescharacteristic
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4D411yescharacteristic
71C2×C32×D4C2311yescharacteristic
72C22×C32×C4—21no—
73C22×C32×C4—21no—
74C22×C32×C4C2311yescharacteristic
75C22×C3×Dic3—21no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6—21no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6C2312yes—
82C32×Q16—22no—
83C32×SD16—22no—
84C32×SD16—22no—
85C32×D8—22no—
86C32×M16—21no—
87C32×M16C2312yes—
88C2×C32×C8—21no—
89C3×#216—24no—
90C3×#216—24no—
91C3×#214—24no—
92C3×#214—24no—
93C3×#208—21no—
94C3×#208—21no—
95C3×#208—24no—
96C3×#208D412yes—
97C3×#208D412yes—
98C3×#208C2312yes—
99C3×#208C2312yes—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—24no—
103C2×C3×C3⋊C8C2312yes—
104#3069—31noSylow
105C2×C3×#41—21no—
106C2×C3×#41—21no—
107C2×C3×#41C2×D311yescharacteristic
108C2×C3×#41C2×D311yescharacteristic
109C2×C3×#41C22×C311yescharacteristic
110C2×C3×#41C22×C311yescharacteristic
111C22×C3×Q8—31no—
112C22×Dic6C22×C311yescharacteristic
113C3×#137—22no—
114C3×#137—21no—
115C3×#137—21no—
116C3×#137—22no—
117C3×#137—34no—
118C3×#137—62no—
119C3×#137—62no—
120C3×#137—62no—
121C3×#137C2×D312yes—
122C3×#137C22×C312yes—
123C3×AGL(1,ℤ/8)—22no—
124C3×AGL(1,ℤ/8)—21no—
125C3×AGL(1,ℤ/8)—21no—
126C3×AGL(1,ℤ/8)—22no—
127C3×AGL(1,ℤ/8)C2×D312yes—
128C3×AGL(1,ℤ/8)C22×C312yes—
129C3×#135—24no—
130C3×#135—22no—
131C3×#135—22no—
132C2×C3×Q16—32no—
133C2×C3×Q16—61no—
134C2×C3×SD16—32no—
135C2×C3×SD16—61no—
136C3×#131—34no—
137C3×#131—62no—
138C2×C3×M16—21no—
139C2×C3×M16—31no—
140C2×C3×M16—31no—
141C2×C3×M16—31no—
142C2×C3×M16—62no—
143C2×C3×M16C2×D311yescharacteristic
144C2×C3×M16C22×C311yescharacteristic
145#951—22no—
146#951—22no—
147#951—22no—
148#951C22×C314yes—
149#948—22no—
150#948C22×C314yes—
151C2×#216—21no—
152C2×#216C22×C312yescomplemented
153C2×#214—21no—
154C2×#214C22×C312yescomplemented
155C2×#208—22no—
156C2×#208C22×C311yescharacteristic
157C2×#208C22×C311yescharacteristic
158C2×#208C22×C311yescharacteristic
159C3×#108—34no—
160C3×#101—32no—
161C3×#101—34no—
162C3×#100—32no—
163#836C22×C312yescomplemented
164#833C22×C312yescomplemented
165#824C22×C314yescomplemented
166#822C22×C314yescomplemented
167C23×C32—21no—
168C23×C32—21no—
169C23×C32#4111yescharacteristic
170C32×Q8—21no—
171C32×Q8—21no—
172C32×Q8—21no—
173C32×Q8C2×D412yes—
174C32×D4—21no—
175C32×D4—22no—
176C32×D4—22no—
177C32×D4—21no—
178C32×D4—21no—
179C32×D4—41no—
180C32×D4—41no—
181C32×D4C2×D412yes—
182C2×C32×C4—21no—
183C2×C32×C4—22no—
184C2×C32×C4—22no—
185C2×C32×C4—21no—
186C2×C32×C4—41no—
187C2×C32×C4—41no—
188C2×C32×C4C2×D411yescharacteristic
189C2×C32×C4C2×D411yescharacteristic
190C2×C32×C4C2×D411yescharacteristic
191C2×C32×C4C2×D411yescharacteristic
192C2×C32×C4#4111yescharacteristic
193C2×C32×C4C2411yescharacteristic
194C2×C3×Dic3—22no—
195C2×C3×Dic3—41no—
196C2×C3×Dic3—41no—
197C3×Dic6—22no—
198C3×Dic6—22no—
199C3×Dic6—41no—
200C3×Dic6—41no—
201C32×C8—22no—
202C3×C3⋊C8—22no—
203C3×C3⋊C8—22no—
204C3×C3⋊C8C2×D414yes—
205#575C3×D311yescharacteristic
206C2×#137—31no—
207C2×#137—31no—
208#568—31no—
209#473—34no—
210#471—34no—
211#430—31no—
212#413—31no—
213C2×#101—31no—
214C3×#41—21no—
215C3×#41—21no—
216C3×#41—21no—
217C3×#41—21no—
218C3×#41—22no—
219C3×#41—22no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—21no—
224C3×#41—21no—
225C3×#41—21no—
226C3×#41—21no—
227C3×#41—21no—
228C3×#41—22no—
229C3×#41—22no—
230C3×#41—41no—
231C3×#41—41no—
232C3×#41—41no—
233C3×#41—41no—
234C3×#41#6712yes—
235C3×#41C3×D412yes—
236C3×#41C22×D312yes—
237C3×#41C23×C312yes—
238C2×C3×Q8—21no—
239C2×C3×Q8—21no—
240C2×C3×Q8—32no—
241C2×C3×Q8—61no—
242C2×C3×Q8—61no—
243C2×C3×Q8—61no—
244C2×C3×Q8—61no—
245C2×C3×Q8—61no—
246C2×C3×Q8#6711yescharacteristic
247C2×C3×Q8C3×D411yescharacteristic
248C2×C3×Q8C22×D311yescharacteristic
249C2×C3×Q8C23×C311yescharacteristic
250C2×C3×D4—21no—
251C2×C3×D4—21no—
252C2×C3×D4—21no—
253C2×C3×D4—21no—
254C2×C3×D4—21no—
255C2×C3×D4—21no—
256C2×C3×D4—41no—
257C2×C3×D4—41no—
258C2×C3×D4#6711yescharacteristic
259C2×C3×D4C3×D411yescharacteristic
260C2×C3×D4C22×D311yescharacteristic
261C2×C3×D4C23×C311yescharacteristic
262C22×C3×C4—21no—
263C22×C3×C4—21no—
264C22×C3×C4—21no—
265C22×C3×C4—21no—
266C22×C3×C4—21no—
267C22×C3×C4—41no—
268C22×C3×C4—41no—
269C22×C3×C4—61no—
270C22×C3×C4C22×D311yescharacteristic
271C22×C3×C4C23×C311yescharacteristic
272C22×Dic3—21no—
273C2×Dic6—21no—
274C2×Dic6—21no—
275C2×Dic6—21no—
276C2×Dic6—21no—
277C2×Dic6—21no—
278C2×Dic6C23×C312yes—
279C3×Q16—24no—
280C3×Q16—22no—
281C3×Q16—22no—
282C3×Q16—64no—
283C3×Q16—64no—
284C3×SD16—24no—
285C3×SD16—24no—
286C3×SD16—22no—
287C3×SD16—22no—
288C3×SD16—22no—
289C3×SD16—22no—
290C3×SD16—64no—
291C3×SD16—64no—
292C3×D8—24no—
293C3×D8—22no—
294C3×D8—22no—
295C3×M16—22no—
296C3×M16—21no—
297C3×M16—21no—
298C3×M16—22no—
299C3×M16—32no—
300C3×M16—32no—
301C3×M16—32no—
302C3×M16—32no—
303C3×M16—64no—
304C3×M16—61no—
305C3×M16—61no—
306C3×M16C22×D312yes—
307C3×M16C23×C312yes—
308C2×C3×C8—22no—
309C2×C3×C8—21no—
310C2×C3×C8—21no—
311C2×C3×C8—32no—
312C2×C3×C8—64no—
313C2×C3×C8—61no—
314C2×C3×C8—61no—
315#216—24no—
316#216—24no—
317#214—24no—
318#214—24no—
319#208—21no—
320#208—21no—
321#208—24no—
322#208C3×D412yes—
323#208C3×D412yes—
324#208C23×C312yes—
325#208C23×C312yes—
326C2×C3⋊C8—21no—
327C2×C3⋊C8—21no—
328C2×C3⋊C8—24no—
329C2×C3⋊C8C23×C312yes—
330C22×C32—22no—
331C22×C32—41no—
332C22×C32—41no—
333C22×C32—81no—
334C22×C32—81no—
335C22×C32#12111yescharacteristic
336C22×C32#12111yescharacteristic
337C22×C32C2×#4111yescharacteristic
338C32×C4—22no—
339C32×C4—41no—
340C32×C4#12111yescharacteristic
341C32×C4#12111yescharacteristic
342C32×C4C22×D411yescharacteristic
343C32×C4C22×D411yescharacteristic
344C3×Dic3—42no—
345C2×#41C2×C3×D311yescharacteristic
346C2×#41C2×C3×D311yescharacteristic
347C22×Q8—31no—
348#137—21no—
349#137—34no—
350#137—62no—
351#137—62no—
352#137—62no—
353#137C2×C3×D312yes—
354AGL(1,ℤ/8)—21no—
355AGL(1,ℤ/8)C2×C3×D312yes—
356#135—22no—
357C2×Q16—32no—
358C2×Q16—61no—
359C2×SD16—32no—
360C2×SD16—61no—
361#131—34no—
362#131—62no—
363C2×M16—31no—
364C2×M16—31no—
365C2×M16—31no—
366C2×M16—62no—
367C2×M16C2×C3×D311yescharacteristic
368#108—34no—
369#101—32no—
370#101—34no—
371#100—32no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3#23711yescharacteristic
380C23×C3C3×#4111yescharacteristic
381C3×Q8—21no—
382C3×Q8—21no—
383C3×Q8—22no—
384C3×Q8—21no—
385C3×Q8—21no—
386C3×Q8—21no—
387C3×Q8—21no—
388C3×Q8—22no—
389C3×Q8—41no—
390C3×Q8—41no—
391C3×Q8—62no—
392C3×Q8—62no—
393C3×Q8—121no—
394C3×Q8—121no—
395C3×Q8C2×#6712yes—
396C3×Q8C2×C3×D412yes—
397C3×D4—22no—
398C3×D4—22no—
399C3×D4—22no—
400C3×D4—22no—
401C3×D4—22no—
402C3×D4—21no—
403C3×D4—21no—
404C3×D4—21no—
405C3×D4—21no—
406C3×D4—21no—
407C3×D4—21no—
408C3×D4—22no—
409C3×D4—41no—
410C3×D4—41no—
411C3×D4—42no—
412C3×D4—42no—
413C3×D4—42no—
414C3×D4—42no—
415C3×D4—41no—
416C3×D4—41no—
417C3×D4—41no—
418C3×D4—41no—
419C3×D4C2×#6712yes—
420C3×D4C2×C3×D412yes—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—22no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—21no—
430C2×C3×C4—21no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—21no—
434C2×C3×C4—21no—
435C2×C3×C4—42no—
436C2×C3×C4—42no—
437C2×C3×C4—41no—
438C2×C3×C4—41no—
439C2×C3×C4—42no—
440C2×C3×C4—42no—
441C2×C3×C4—41no—
442C2×C3×C4—41no—
443C2×C3×C4—41no—
444C2×C3×C4—41no—
445C2×C3×C4—62no—
446C2×C3×C4—121no—
447C2×C3×C4—121no—
448C2×C3×C4D3×D411yescharacteristic
449C2×C3×C4D3×D411yescharacteristic
450C2×C3×C4#23711yescharacteristic
451C2×C3×C4C2×#6711yescharacteristic
452C2×C3×C4C2×#6711yescharacteristic
453C2×C3×C4C2×C3×D411yescharacteristic
454C2×C3×C4C2×C3×D411yescharacteristic
455C2×C3×C4C2×C3×D411yescharacteristic
456C2×C3×C4C2×C3×D411yescharacteristic
457C2×C3×C4C3×#4111yescharacteristic
458C2×C3×C4C23×D311yescharacteristic
459C2×C3×C4C24×C311yescharacteristic
460C2×Dic3—22no—
461C2×Dic3—41no—
462C2×Dic3—41no—
463Dic6—22no—
464Dic6—22no—
465Dic6—41no—
466Dic6—41no—
467C3×C8—24no—
468C3×C8—22no—
469C3×C8—22no—
470C3×C8—34no—
471C3×C8—62no—
472C3×C8—62no—
473C3⋊C8—22no—
474C3⋊C8—22no—
475C3⋊C8C2×C3×D414yes—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—42no—
480C2×C32—81no—
481C2×C32C2×#12111yescharacteristic
482#41—21no—
483#41—21no—
484#41—21no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41C3×#6712yes—
489#41C22×C3×D312yes—
490C2×Q8—32no—
491C2×Q8—61no—
492C2×Q8—61no—
493C2×Q8—61no—
494C2×Q8—61no—
495C2×Q8—61no—
496C2×Q8C3×#6711yescharacteristic
497C2×Q8C22×C3×D311yescharacteristic
498C2×D4—21no—
499C2×D4—21no—
500C2×D4C3×#6711yescharacteristic
501C2×D4C22×C3×D311yescharacteristic
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—61no—
505C22×C4C22×C3×D311yescharacteristic
506Q16—22no—
507Q16—64no—
508Q16—64no—
509SD16—22no—
510SD16—22no—
511SD16—64no—
512SD16—64no—
513D8—22no—
514M16—21no—
515M16—32no—
516M16—32no—
517M16—32no—
518M16—32no—
519M16—64no—
520M16—61no—
521M16—61no—
522M16C22×C3×D312yes—
523C2×C8—21no—
524C2×C8—32no—
525C2×C8—64no—
526C2×C8—61no—
527C2×C8—61no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—42no—
534C22×C3—41no—
535C22×C3—41no—
536C22×C3—42no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—82no—
545C22×C3—82no—
546C22×C3#93911yescharacteristic
547C22×C3#93911yescharacteristic
548C22×C3C3×#12111yescharacteristic
549C22×C3C3×#12111yescharacteristic
550C22×C3C2×#23711yescharacteristic
551C22×C3C2×C3×#4111yescharacteristic
552C3×C4—22no—
553C3×C4—22no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—42no—
559C3×C4—42no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4—122no—
563C3×C4#93911yescharacteristic
564C3×C4#93911yescharacteristic
565C3×C4C3×#12111yescharacteristic
566C3×C4C3×#12111yescharacteristic
567C3×C4C2×D3×D411yescharacteristic
568C3×C4C22×#6711yescharacteristic
569C3×C4C22×C3×D411yescharacteristic
570C3×C4C22×C3×D411yescharacteristic
571Dic3—42no—
572C32#306911yescharacteristic, complemented, Sylow
573C23—21no—
574C23—21no—
575C23C3×#23711yescharacteristic
576Q8—21no—
577Q8—21no—
578Q8—21no—
579Q8—62no—
580Q8—62no—
581Q8—121no—
582Q8—121no—
583Q8C2×C3×#6712yes—
584D4—22no—
585D4—22no—
586D4—21no—
587D4—21no—
588D4—21no—
589D4—41no—
590D4—41no—
591D4C2×C3×#6712yes—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—62no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4C3×D3×D411yescharacteristic
602C2×C4C3×D3×D411yescharacteristic
603C2×C4C3×#23711yescharacteristic
604C2×C4C2×C3×#6711yescharacteristic
605C2×C4C2×C3×#6711yescharacteristic
606C2×C4C23×C3×D311yescharacteristic
607C8—22no—
608C8—34no—
609C8—62no—
610C8—62no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—41no—
619C2×C3—41no—
620C2×C3—41no—
621C2×C3—42no—
622C2×C3—42no—
623C2×C3—82no—
624C2×C3—82no—
625C2×C3—81no—
626C2×C3—81no—
627C2×C3C2×#93911yescharacteristic
628C2×C3C2×C3×#12111yescharacteristic
629C22—22no—
630C22—41no—
631C22—41no—
632C22—81no—
633C22—81no—
634C22C3×#93911yescharacteristic
635C22C3×#93911yescharacteristic
636C22C2×C3×#23711yescharacteristic
637C4—22no—
638C4—41no—
639C4—122no—
640C4C3×#93911yescharacteristic
641C4C3×#93911yescharacteristic
642C4C2×C3×D3×D411yescharacteristic
643C4C22×C3×#6711yescharacteristic
644C3—21no—
645C3#8400311yescharacteristic, direct product factor
646C3C3×#306911yescharacteristic, complemented
647C2—21no—
648C2—21no—
649C2—21no—
650C2—42no—
651C2—81no—
652C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC2×C3C2×#939
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC32×#575C2
socleC2×C32C2×#121
hypercenterC3×#575D3
hypocenterC3C3×#3069
LCSG ▷ C2×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. C2×C3×#951C2semidirect
3. C2×C3×#951C2semidirect
4. C3×#5819C2semidirect
5. C3×#5237C2semidirect
6. C3×#5204C2semidirect
7. C3×#5200C2semidirect
8. C2×C3×#824C2semidirect
9. C3×#5124C2semidirect
11. #84003C3direct (N × H)
28. C2×C3×#216C22semidirect
30. C2×C3×#214C22semidirect
35. C3×#836C22semidirect
36. C3×#833C22semidirect
37. C3×#824C22semidirect
38. C3×#822C22semidirect
45. C2×#951C2×C3semidirect
46. C2×#951C2×C3semidirect
47. #5819C2×C3semidirect
53. #5237C2×C3semidirect
54. #5204C2×C3semidirect
55. #5200C2×C3semidirect
56. C2×#824C2×C3semidirect
57. #5124C2×C3semidirect
152. C2×#216C22×C3semidirect
154. C2×#214C22×C3semidirect
163. #836C22×C3semidirect
164. #833C22×C3semidirect
165. #824C22×C3semidirect
166. #822C22×C3semidirect
572. C32#3069semidirect
645. C3#84003direct (N × H)
646. C3C3×#3069semidirect

Maximal subgroups

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

subgroupindex
C32×#5752
C2×C3×#9512
C3×#58192
C3×#52372
C3×#52042
C3×#52002
C2×C3×#8242
C3×#51242
C3×#30693
#840033