Finite Group Databasebeta

← C3×#29602 #10648209

Subgroups — C3×#29602 #10648209

Order 768.

1518 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic, complemented
2C2×C3×#482C211yesmaximal, characteristic, complemented
3C2×C3×#447C211yesmaximal, characteristic, complemented
4C3×#2615C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1999C212yesmaximal, complemented
8C3×#1999C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1786C211yesmaximal, characteristic, complemented
11C3×#1780C211yesmaximal, characteristic, complemented
12#29602C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×D8C2212yes—
16C3×#535—24no—
17C3×#535—24no—
18C2×C3×#127—21no—
19C2×C3×#124—21no—
20C2×C3×#121C2212yes—
21C2×C3×#121C2212yes—
22C2×C3×#121C2211yescharacteristic
23C2×C3×#121C2212yescomplemented
24C2×C3×#120—22no—
25C2×C3×#117C2211yescharacteristic
26C3×#482—24no—
27C3×#447—24no—
28C2×C3×#107C2211yescharacteristic, complemented
29C2×C3×#106C2211yescharacteristic, complemented
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2211yescharacteristic, complemented
34C2×C3×#102C2211yescharacteristic, complemented
35C2×C3×C22⋊C8C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2212yescomplemented
38C2×C3×C22⋊C8C2212yescomplemented
39C3×#393—21no—
40C3×#392—21no—
41C3×#392—22no—
42C3×#380C2211yescharacteristic
43C3×#380C2212yescomplemented
44C3×#340C2212yescomplemented
45C3×#340C2212yescomplemented
46C2×#535C2×C311yescharacteristic, complemented
47C2×#482C2×C311yescharacteristic, complemented
48C2×#447C2×C311yescharacteristic, complemented
49#2615C2×C311yescharacteristic, complemented
50#2034C2×C312yescomplemented
51#2033C2×C312yescomplemented
52#1999C2×C312yescomplemented
53#1999C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1786C2×C311yescharacteristic, complemented
56#1780C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—42no—
61C2×C3×#41—42no—
62C22×C3×Q8D411yescharacteristic
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—22no—
69C22×C3×D4—42no—
70C22×C3×D4—42no—
71C22×C3×D4D412yes—
72C22×C3×D4D411yescharacteristic
73C22×C3×D4D414yescomplemented
74C22×C3×D4C2311yescharacteristic
75C22×C3×D4C2312yes—
76C23×C3×C4C2312yes—
77C23×C3×C4C2311yescharacteristic
78C2×C3×D8—24no—
79C2×C3×D8—42no—
80C2×C3×D8—42no—
81C22×C3×C8C2312yes—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#127—28no—
85C3×#124—28no—
86C3×#121—24no—
87C3×#121—24no—
88C3×#121—24no—
89C3×#121—24no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—42no—
93C3×#121—42no—
94C3×#120—216no—
95C3×#117—22no—
96C3×#117D414yescomplemented
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C42—21no—
109C3×#107—22no—
110C3×#106—22no—
111C3×#102—24no—
112C3×#102—22no—
113C3×#102—22no—
114C3×#102—42no—
115C3×#102—42no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—22no—
118C3×C22⋊C8—42no—
119C3×C22⋊C8—42no—
120C3×#95—21no—
121C3×#95—22no—
122C22×#41C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C22×D8C22×C312yes—
125#535—24no—
126#535—24no—
127C2×#127—21no—
128C2×#124—21no—
129C2×#121C22×C311yescharacteristic
130C2×#121C22×C312yes—
131C2×#121C22×C312yes—
132C2×#121C22×C312yescomplemented
133C2×#120—22no—
134C2×#117C22×C311yescharacteristic
135#482—24no—
136#447—24no—
137C2×#107C22×C311yescharacteristic, complemented
138C2×#106C22×C311yescharacteristic, complemented
139C2×#102C22×C312yes—
140C2×#102C22×C312yes—
141C2×#102C22×C312yes—
142C2×#102C22×C311yescharacteristic, complemented
143C2×#102C22×C311yescharacteristic, complemented
144C2×C22⋊C8C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C312yescomplemented
147C2×C22⋊C8C22×C312yescomplemented
148#393—21no—
149#392—21no—
150#392—22no—
151#380C22×C311yescharacteristic
152#380C22×C312yescomplemented
153#340C22×C312yescomplemented
154#340C22×C312yescomplemented
155C24×C3—22no—
156C24×C3—24no—
157C24×C3—21no—
158C24×C3—44no—
159C24×C3—81no—
160C24×C3—82no—
161C24×C3—81no—
162C24×C3C2×D412yes—
163C24×C3C2×D411yescharacteristic
164C3×#41—48no—
165C3×#41—44no—
166C3×#41—44no—
167C2×C3×Q8—22no—
168C2×C3×Q8—42no—
169C2×C3×D4—24no—
170C2×C3×D4—28no—
171C2×C3×D4—28no—
172C2×C3×D4—24no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—48no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—42no—
192C2×C3×D4—82no—
193C2×C3×D4—82no—
194C2×C3×D4—82no—
195C2×C3×D4—82no—
196C2×C3×D4—82no—
197C2×C3×D4—82no—
198C2×C3×D4—82no—
199C2×C3×D4C2×D414yes—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2411yescharacteristic
217C3×D8—48no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—28no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—44no—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—81no—
239C3×C22⋊C4—82no—
240C3×C42—24no—
241C25—21no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—42no—
245C2×#41—42no—
246C22×Q8C3×D411yescharacteristic
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—42no—
254C22×D4—42no—
255C22×D4C3×D412yes—
256C22×D4C3×D411yescharacteristic
257C22×D4C3×D414yescomplemented
258C22×D4C23×C311yescharacteristic
259C22×D4C23×C312yes—
260C23×C4C23×C312yes—
261C23×C4C23×C311yescharacteristic
262C2×D8—24no—
263C2×D8—42no—
264C2×D8—42no—
265C22×C8C23×C312yes—
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268#127—28no—
269#124—28no—
270#121—24no—
271#121—24no—
272#121—24no—
273#121—24no—
274#121—22no—
275#121—22no—
276#121—42no—
277#121—42no—
278#120—216no—
279#117—22no—
280#117C3×D414yescomplemented
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C312yes—
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C42—21no—
293#107—22no—
294#106—22no—
295#102—24no—
296#102—22no—
297#102—22no—
298#102—42no—
299#102—42no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304#95—21no—
305#95—22no—
306C23×C3—22no—
307C23×C3—21no—
308C23×C3—22no—
309C23×C3—21no—
310C23×C3—24no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—48no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—41no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—84no—
331C23×C3—84no—
332C23×C3—88no—
333C23×C3—88no—
334C23×C3#12012yes—
335C23×C3C22×D411yescharacteristic
336C3×Q8—44no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—48no—
340C3×D4—44no—
341C3×D4—48no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—42no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4—88no—
348C3×D4—88no—
349C3×D4—84no—
350C3×D4—88no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4#12012yes—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4H(5,𝔽2)12yes—
376C2×C3×C4H(5,𝔽2)12yes—
377C3×C8—42no—
378C3×C8—41no—
379C3×C8—41no—
380C24—22no—
381C24—24no—
382C24—21no—
383C24—44no—
384C24—81no—
385C24—82no—
386C24—81no—
387C24C2×C3×D412yes—
388C24C2×C3×D411yescharacteristic
389#41—48no—
390#41—44no—
391#41—44no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4C2×C3×D414yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4C2×C3×D412yes—
438C22×C4C2×C3×D411yescharacteristic
439C22×C4C2×C3×D412yes—
440C22×C4C2×C3×D412yes—
441C22×C4C24×C311yescharacteristic
442D8—48no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4C2×C3×D412yes—
457C4⋊C4C2×C3×D412yes—
458C22⋊C4—28no—
459C22⋊C4—28no—
460C22⋊C4—24no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—81no—
464C22⋊C4—82no—
465C42—24no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—44no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—44no—
479C22×C3—42no—
480C22×C3—44no—
481C22×C3—816no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—81no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3C2×#12011yescharacteristic
492C22×C3#53412yes—
493C22×C3#57612yes—
494C22×C3#57611yescharacteristic
495C22×C3#57711yescharacteristic
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—82no—
501C3×C4—81no—
502C3×C4#53412yes—
503C3×C4#53511yescharacteristic
504C3×C4#53511yescharacteristic
505C23—21no—
506C23—22no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—48no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—84no—
530C23—84no—
531C23—88no—
532C23—88no—
533C23C3×#12012yes—
534C23C22×C3×D411yescharacteristic
535Q8—44no—
536D4—24no—
537D4—24no—
538D4—48no—
539D4—44no—
540D4—48no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—41no—
545D4—41no—
546D4—88no—
547D4—88no—
548D4—84no—
549D4—88no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4C3×#12012yes—
572C2×C4C22×C3×D411yescharacteristic
573C2×C4C22×C3×D411yescharacteristic
574C2×C4C3×H(5,𝔽2)12yes—
575C2×C4C3×H(5,𝔽2)12yes—
576C8—42no—
577C8—41no—
578C8—41no—
579C2×C3—42no—
580C2×C3—82no—
581C2×C3—84no—
582C2×C3—161no—
583C2×C3#261311yescharacteristic
584C2×C3#300811yescharacteristic
585C2×C3#301211yescharacteristic
586C2×C3#319812yes—
587C2×C3#319912yes—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—816no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22C2×C3×#12011yescharacteristic
614C22C3×#53412yes—
615C22C3×#57612yes—
616C22C3×#57611yescharacteristic
617C22C3×#57711yescharacteristic
618C4—22no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—82no—
623C4—81no—
624C4C3×#53412yes—
625C4C3×#53511yescharacteristic
626C4C3×#53511yescharacteristic
627C3#2960211yescharacteristic, direct product factor, Sylow
628C2—42no—
629C2—82no—
630C2—84no—
631C2—161no—
632C2C3×#261311yescharacteristic
633C2C3×#300811yescharacteristic
634C2C3×#301211yescharacteristic
635C2C3×#319812yes—
636C2C3×#319912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. C2×C3×#482C2semidirect
3. C2×C3×#447C2semidirect
4. C3×#2615C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#1999C2semidirect
8. C3×#1999C2semidirect
9. C3×#1825C2semidirect
10. C3×#1786C2semidirect
11. C3×#1780C2semidirect
12. #29602C3direct (N × H)
23. C2×C3×#121C22semidirect
28. C2×C3×#107C22semidirect
29. C2×C3×#106C22semidirect
33. C2×C3×#102C22semidirect
34. C2×C3×#102C22semidirect
37. C2×C3×C22⋊C8C22semidirect
38. C2×C3×C22⋊C8C22semidirect
43. C3×#380C22semidirect
44. C3×#340C22semidirect
45. C3×#340C22semidirect
46. C2×#535C2×C3semidirect
47. C2×#482C2×C3semidirect
48. C2×#447C2×C3semidirect
49. #2615C2×C3semidirect
50. #2034C2×C3semidirect
51. #2033C2×C3semidirect
52. #1999C2×C3semidirect
53. #1999C2×C3semidirect
54. #1825C2×C3semidirect
55. #1786C2×C3semidirect
56. #1780C2×C3semidirect
73. C22×C3×D4D4semidirect
96. C3×#117D4semidirect
132. C2×#121C22×C3semidirect
137. C2×#107C22×C3semidirect
138. C2×#106C22×C3semidirect
142. C2×#102C22×C3semidirect
143. C2×#102C22×C3semidirect
146. C2×C22⋊C8C22×C3semidirect
147. C2×C22⋊C8C22×C3semidirect
152. #380C22×C3semidirect
153. #340C22×C3semidirect
154. #340C22×C3semidirect
257. C22×D4C3×D4semidirect
280. #117C3×D4semidirect
627. C3#29602direct (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
C2×C3×#4822
C2×C3×#4472
C3×#26152
C3×#20342
C3×#20332
C3×#19992
C3×#18252
C3×#17862
C3×#17802
#296023