Finite Group Databasebeta

← C3×#29610 #10648217

Subgroups — C3×#29610 #10648217

Order 768.

1594 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic, complemented
2C3×#3094C211yesmaximal, characteristic, complemented
3C3×#3006C211yesmaximal, characteristic, complemented
4C3×#2433C211yesmaximal, characteristic, complemented
5C3×#2037C214yesmaximal, complemented
6C3×#2002C214yesmaximal, complemented
7C3×#1826C211yesmaximal, characteristic, complemented
8C3×#1789C211yesmaximal, characteristic, complemented
9C3×#1783C211yesmaximal, characteristic, complemented
10#29610C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C2×C3×AGL(1,ℤ/8)C2212yescomplemented
14C3×#535—22no—
15C3×#535—22no—
16C3×#535—22no—
17C3×#535C2214yes—
18C2×C3×#127—21no—
19C2×C3×#127C2212yescomplemented
20C2×C3×#124—21no—
21C2×C3×#121—21no—
22C2×C3×#121—21no—
23C2×C3×#121C2211yescharacteristic
24C2×C3×#120—22no—
25C2×C3×#120—22no—
26C2×C3×#117C2211yescharacteristic
27C3×#482—22no—
28C3×#481—21no—
29C3×#480—21no—
30C3×#449—22no—
31C3×#448—21no—
32C3×#447—21no—
33C3×#428C2212yescomplemented
34C2×C3×#104C2214yescomplemented
35C3×#418C2212yescomplemented
36C3×#417C2212yescomplemented
37C2×C3×#100C2214yescomplemented
38C3×#407C2211yescharacteristic
39C3×#407C2211yescharacteristic
40C3×#392—22no—
41C3×#390C2212yescomplemented
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#339C2214yescomplemented
44C2×#535C2×C311yescharacteristic, complemented
45#3094C2×C311yescharacteristic, complemented
46#3006C2×C311yescharacteristic, complemented
47#2433C2×C311yescharacteristic, complemented
48#2037C2×C314yescomplemented
49#2002C2×C314yescomplemented
50#1826C2×C311yescharacteristic, complemented
51#1789C2×C311yescharacteristic, complemented
52#1783C2×C311yescharacteristic, complemented
53C25×C3D412yes—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41—44no—
57C2×C3×#41D412yescomplemented
58C2×C3×#41C2312yes—
59C22×C3×Q8D411yescharacteristic, complemented
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—24no—
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4—21no—
68C22×C3×D4—21no—
69C22×C3×D4—44no—
70C22×C3×D4D412yes—
71C22×C3×D4D411yescharacteristic, complemented
72C22×C3×D4C2311yescharacteristic
73C22×C3×D4C2312yes—
74C23×C3×C4—21no—
75C23×C3×C4C2311yescharacteristic
76C3×AGL(1,ℤ/8)—44no—
77C2×C3×SD16—22no—
78C2×C3×D8—22no—
79C2×C3×M16C2312yes—
80C2×C3×M16C2312yes—
81C3×#127—24no—
82C3×#127—24no—
83C3×#127—44no—
84C3×#124—24no—
85C3×#124—24no—
86C3×#121—24no—
87C3×#121—24no—
88C3×#121—24no—
89C3×#121—24no—
90C3×#121—21no—
91C3×#121—21no—
92C3×#121—21no—
93C3×#121—21no—
94C3×#120—28no—
95C3×#120—28no—
96C3×#120—48no—
97C3×#117—21no—
98C3×#117—21no—
99C3×#117D412yescomplemented
100C3×#117C2312yes—
101C2×C3×C4⋊C4D411yescharacteristic, complemented
102C2×C3×C4⋊C4D411yescharacteristic, complemented
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—24no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C42—21no—
111C2×C3×C42C2312yes—
112C3×#107—22no—
113C3×#106—22no—
114C3×#104—44no—
115C3×#103—22no—
116C3×#102—22no—
117C3×#102—22no—
118C3×#102—22no—
119C3×#100—44no—
120C3×C22⋊C8—21no—
121C3×C22⋊C8—21no—
122C3×C22⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×#95—21no—
125C22×#41C22×C311yescharacteristic
126C23×D4C22×C311yescharacteristic
127C2×AGL(1,ℤ/8)C22×C312yescomplemented
128#535—22no—
129#535—22no—
130#535—22no—
131#535C22×C314yes—
132C2×#127—21no—
133C2×#127C22×C312yescomplemented
134C2×#124—21no—
135C2×#121—21no—
136C2×#121—21no—
137C2×#121C22×C311yescharacteristic
138C2×#120—22no—
139C2×#120—22no—
140C2×#117C22×C311yescharacteristic
141#482—22no—
142#481—21no—
143#480—21no—
144#449—22no—
145#448—21no—
146#447—21no—
147#428C22×C312yescomplemented
148C2×#104C22×C314yescomplemented
149#418C22×C312yescomplemented
150#417C22×C312yescomplemented
151C2×#100C22×C314yescomplemented
152#407C22×C311yescharacteristic
153#407C22×C311yescharacteristic
154#392—22no—
155#390C22×C312yescomplemented
156C4×C22⋊C4C22×C311yescharacteristic
157#339C22×C314yescomplemented
158C24×C3—22no—
159C24×C3—21no—
160C24×C3—22no—
161C24×C3—21no—
162C24×C3—44no—
163C24×C3—44no—
164C24×C3—82no—
165C24×C3C2×D411yescharacteristic
166C24×C3C2×D414yes—
167C3×#41—24no—
168C3×#41—24no—
169C3×#41—42no—
170C3×#41—42no—
171C3×#41—44no—
172C3×#41—44no—
173C2×C3×Q8—21no—
174C2×C3×Q8—42no—
175C2×C3×Q8C2×D412yes—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—28no—
179C2×C3×D4—28no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—24no—
188C2×C3×D4—24no—
189C2×C3×D4—24no—
190C2×C3×D4—42no—
191C2×C3×D4—44no—
192C2×C3×D4—44no—
193C2×C3×D4—44no—
194C2×C3×D4—44no—
195C2×C3×D4—44no—
196C2×C3×D4—48no—
197C2×C3×D4—44no—
198C2×C3×D4—44no—
199C2×C3×D4—48no—
200C2×C3×D4—41no—
201C2×C3×D4—41no—
202C2×C3×D4—84no—
203C2×C3×D4—84no—
204C2×C3×D4—84no—
205C2×C3×D4C2×D412yes—
206C22×C3×C4—24no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—21no—
214C22×C3×C4—44no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C3×SD16—44no—
222C3×D8—44no—
223C3×M16—24no—
224C3×M16—24no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—28no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—48no—
239C3×C22⋊C4—41no—
240C3×C22⋊C4—41no—
241C3×C42—22no—
242C3×C42—22no—
243C3×C42—24no—
244C25C3×D412yes—
245C2×#41—21no—
246C2×#41—21no—
247C2×#41—44no—
248C2×#41C3×D412yescomplemented
249C2×#41C23×C312yes—
250C22×Q8C3×D411yescharacteristic, complemented
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4—44no—
261C22×D4C3×D412yes—
262C22×D4C3×D411yescharacteristic, complemented
263C22×D4C23×C311yescharacteristic
264C22×D4C23×C312yes—
265C23×C4—21no—
266C23×C4C23×C311yescharacteristic
267AGL(1,ℤ/8)—44no—
268C2×SD16—22no—
269C2×D8—22no—
270C2×M16C23×C312yes—
271C2×M16C23×C312yes—
272#127—24no—
273#127—24no—
274#127—44no—
275#124—24no—
276#124—24no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—24no—
281#121—21no—
282#121—21no—
283#121—21no—
284#121—21no—
285#120—28no—
286#120—28no—
287#120—48no—
288#117—21no—
289#117—21no—
290#117C3×D412yescomplemented
291#117C23×C312yes—
292C2×C4⋊C4C3×D411yescharacteristic, complemented
293C2×C4⋊C4C3×D411yescharacteristic, complemented
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—21no—
301C2×C42—21no—
302C2×C42C23×C312yes—
303#107—22no—
304#106—22no—
305#104—44no—
306#103—22no—
307#102—22no—
308#102—22no—
309#102—22no—
310#100—44no—
311C22⋊C8—21no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315#95—21no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—24no—
319C23×C3—22no—
320C23×C3—24no—
321C23×C3—24no—
322C23×C3—41no—
323C23×C3—44no—
324C23×C3—44no—
325C23×C3—42no—
326C23×C3—44no—
327C23×C3—44no—
328C23×C3—42no—
329C23×C3—44no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—44no—
334C23×C3—41no—
335C23×C3—42no—
336C23×C3—82no—
337C23×C3—82no—
338C23×C3—816no—
339C23×C3#12011yescharacteristic
340C23×C3#12011yescharacteristic
341C23×C3C22×D411yescharacteristic
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—42no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—44no—
348C3×D4—44no—
349C3×D4—42no—
350C3×D4—44no—
351C3×D4—44no—
352C3×D4—44no—
353C3×D4—42no—
354C3×D4—44no—
355C3×D4—42no—
356C3×D4—88no—
357C3×D4—88no—
358C3×D4—816no—
359C3×D4—82no—
360C3×D4—82no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—44no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—44no—
377C2×C3×C4—42no—
378C2×C3×C4—41no—
379C2×C3×C4#12011yescharacteristic
380C2×C3×C4#12011yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4H(5,𝔽2)14yes—
384C3×C8—42no—
385C3×C8—42no—
386C24—22no—
387C24—21no—
388C24—22no—
389C24—21no—
390C24—44no—
391C24—44no—
392C24—82no—
393C24C2×C3×D411yescharacteristic
394C24C2×C3×D414yes—
395#41—24no—
396#41—24no—
397#41—42no—
398#41—42no—
399#41—44no—
400#41—44no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8C2×C3×D412yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—48no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4C2×C3×D412yes—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—44no—
443C22×C4C2×C3×D411yescharacteristic
444C22×C4C2×C3×D411yescharacteristic
445C22×C4C2×C3×D411yescharacteristic
446C22×C4C2×C3×D411yescharacteristic
447C22×C4C2×C3×D411yescharacteristic
448C22×C4C24×C311yescharacteristic
449SD16—44no—
450D8—44no—
451M16—24no—
452M16—24no—
453C2×C8—22no—
454C2×C8—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4C2×C3×D412yes—
460C4⋊C4C2×C3×D412yes—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—24no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—48no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C42—22no—
470C42—22no—
471C42—24no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—44no—
480C22×C3—48no—
481C22×C3—42no—
482C22×C3—44no—
483C22×C3—42no—
484C22×C3—44no—
485C22×C3—42no—
486C22×C3—44no—
487C22×C3—84no—
488C22×C3—84no—
489C22×C3—88no—
490C22×C3—84no—
491C22×C3—88no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3#45312yes—
496C22×C3C2×#12011yescharacteristic
497C22×C3#53412yes—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—44no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4—42no—
504C3×C4—81no—
505C3×C4#53412yes—
506C3×C4#53512yes—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—41no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—41no—
526C23—42no—
527C23—82no—
528C23—82no—
529C23—816no—
530C23C3×#12011yescharacteristic
531C23C3×#12011yescharacteristic
532C23C22×C3×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—42no—
536D4—22no—
537D4—22no—
538D4—44no—
539D4—44no—
540D4—42no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—42no—
545D4—44no—
546D4—42no—
547D4—88no—
548D4—88no—
549D4—816no—
550D4—82no—
551D4—82no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4C3×#12011yescharacteristic
571C2×C4C3×#12011yescharacteristic
572C2×C4C22×C3×D411yescharacteristic
573C2×C4C22×C3×D411yescharacteristic
574C2×C4C3×H(5,𝔽2)14yes—
575C8—42no—
576C8—42no—
577C2×C3—22no—
578C2×C3—22no—
579C2×C3—41no—
580C2×C3—84no—
581C2×C3—82no—
582C2×C3—161no—
583C2×C3#261311yescharacteristic
584C2×C3C2×#45311yescharacteristic
585C2×C3#301911yescharacteristic
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—44no—
594C22—48no—
595C22—42no—
596C22—44no—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—84no—
602C22—88no—
603C22—84no—
604C22—84no—
605C22—88no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22C3×#45312yes—
610C22C2×C3×#12011yescharacteristic
611C22C3×#53412yes—
612C4—21no—
613C4—21no—
614C4—44no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—81no—
619C4C3×#53412yes—
620C4C3×#53512yes—
621C3#2961011yescharacteristic, direct product factor, Sylow
622C2—22no—
623C2—22no—
624C2—41no—
625C2—84no—
626C2—82no—
627C2—161no—
628C2C3×#261311yescharacteristic
629C2C2×C3×#45311yescharacteristic
630C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. C3×#3094C2semidirect
3. C3×#3006C2semidirect
4. C3×#2433C2semidirect
5. C3×#2037C2semidirect
6. C3×#2002C2semidirect
7. C3×#1826C2semidirect
8. C3×#1789C2semidirect
9. C3×#1783C2semidirect
10. #29610C3direct (N × H)
13. C2×C3×AGL(1,ℤ/8)C22semidirect
19. C2×C3×#127C22semidirect
33. C3×#428C22semidirect
34. C2×C3×#104C22semidirect
35. C3×#418C22semidirect
36. C3×#417C22semidirect
37. C2×C3×#100C22semidirect
41. C3×#390C22semidirect
43. C3×#339C22semidirect
44. C2×#535C2×C3semidirect
45. #3094C2×C3semidirect
46. #3006C2×C3semidirect
47. #2433C2×C3semidirect
48. #2037C2×C3semidirect
49. #2002C2×C3semidirect
50. #1826C2×C3semidirect
51. #1789C2×C3semidirect
52. #1783C2×C3semidirect
57. C2×C3×#41D4semidirect
59. C22×C3×Q8D4semidirect
71. C22×C3×D4D4semidirect
99. C3×#117D4semidirect
101. C2×C3×C4⋊C4D4semidirect
102. C2×C3×C4⋊C4D4semidirect
127. C2×AGL(1,ℤ/8)C22×C3semidirect
133. C2×#127C22×C3semidirect
147. #428C22×C3semidirect
148. C2×#104C22×C3semidirect
149. #418C22×C3semidirect
150. #417C22×C3semidirect
151. C2×#100C22×C3semidirect
155. #390C22×C3semidirect
157. #339C22×C3semidirect
248. C2×#41C3×D4semidirect
250. C22×Q8C3×D4semidirect
262. C22×D4C3×D4semidirect
290. #117C3×D4semidirect
292. C2×C4⋊C4C3×D4semidirect
293. C2×C4⋊C4C3×D4semidirect
621. C3#29610direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C3×#30942
C3×#30062
C3×#24332
C3×#20372
C3×#20022
C3×#18262
C3×#17892
C3×#17832
#296103