Finite Group Databasebeta

← C3×#25734 #10644341

Subgroups — C3×#25734 #10644341

Order 768.

1758 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3C2×C3×#493C211yesmaximal, characteristic, complemented
4C2×C3×#449C212yesmaximal, complemented
5C2×C3×#447C212yesmaximal, complemented
6C3×#2619C211yesmaximal, characteristic, complemented
7C3×#2012C212yesmaximal, complemented
8C3×#2011C212yesmaximal, complemented
9C3×#1961C211yesmaximal, characteristic, complemented
10C3×#1958C211yesmaximal, characteristic, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25734C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×SD16C2212yescomplemented
16C22×C3×D8C2212yes—
17C22×C3×D8C2212yes—
18C22×C3×D8C2212yescomplemented
19C3×#535—24no—
20C3×#535—24no—
21C2×C3×#127C2211yescharacteristic
22C2×C3×#127C2211yescharacteristic
23C2×C3×#127C2211yescharacteristic, complemented
24C2×C3×#124C2211yescharacteristic
25C2×C3×#124C2211yescharacteristic
26C2×C3×#121—22no—
27C2×C3×#121C2212yes—
28C2×C3×#120—22no—
29C2×C3×#117C2211yescharacteristic
30C3×#496—24no—
31C3×#493—24no—
32C3×#449—28no—
33C3×#447—28no—
34C2×C3×#103C2212yescomplemented
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yescomplemented
38C2×C3×C22⋊C8C2212yes—
39C2×C3×#97C2211yescharacteristic, complemented
40C2×C3×C4×C8C2211yescharacteristic, complemented
41C3×#394—22no—
42C3×#392—22no—
43C3×#390C2212yescomplemented
44C3×#382C2211yescharacteristic, complemented
45C3×#336C2212yescomplemented
46C2×#535C2×C311yescharacteristic, complemented
47C2×#496C2×C311yescharacteristic, complemented
48C2×#493C2×C311yescharacteristic, complemented
49C2×#449C2×C312yescomplemented
50C2×#447C2×C312yescomplemented
51#2619C2×C311yescharacteristic, complemented
52#2012C2×C312yescomplemented
53#2011C2×C312yescomplemented
54#1961C2×C311yescharacteristic, complemented
55#1958C2×C311yescharacteristic, complemented
56#1750C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—24no—
61C2×C3×#41—42no—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4—22no—
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4—24no—
71C22×C3×D4—22no—
72C22×C3×D4—42no—
73C22×C3×D4C2312yes—
74C22×C3×D4C2311yescharacteristic
75C22×C3×D4C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—42no—
82C2×C3×D8—24no—
83C2×C3×D8—24no—
84C2×C3×D8—24no—
85C2×C3×D8—24no—
86C2×C3×D8—24no—
87C2×C3×D8—24no—
88C2×C3×D8—42no—
89C2×C3×D8—42no—
90C2×C3×D8—42no—
91C22×C3×C8C2312yes—
92C22×C3×C8C2312yes—
93C3×#127—22no—
94C3×#127—22no—
95C3×#127—22no—
96C3×#127—22no—
97C3×#127—22no—
98C3×#127D414yescomplemented
99C3×#124—22no—
100C3×#124—22no—
101C3×#124—22no—
102C3×#124D414yescomplemented
103C3×#121—216no—
104C3×#121—24no—
105C3×#121—24no—
106C3×#120—216no—
107C3×#117—22no—
108C3×#117—22no—
109C2×C3×C4⋊C4D412yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—24no—
115C2×C3×C22⋊C4D412yes—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×#103—24no—
119C3×#102—24no—
120C3×#102—24no—
121C3×#102—24no—
122C3×C22⋊C8D418yescomplemented
123C3×#97—22no—
124C3×C4×C8—22no—
125C3×#95—22no—
126C22×#41C22×C311yescharacteristic
127C23×D4C22×C311yescharacteristic
128C22×SD16C22×C312yescomplemented
129C22×D8C22×C312yes—
130C22×D8C22×C312yes—
131C22×D8C22×C312yescomplemented
132#535—24no—
133#535—24no—
134C2×#127C22×C311yescharacteristic
135C2×#127C22×C311yescharacteristic
136C2×#127C22×C311yescharacteristic, complemented
137C2×#124C22×C311yescharacteristic
138C2×#124C22×C311yescharacteristic
139C2×#121—22no—
140C2×#121C22×C312yes—
141C2×#120—22no—
142C2×#117C22×C311yescharacteristic
143#496—24no—
144#493—24no—
145#449—28no—
146#447—28no—
147C2×#103C22×C312yescomplemented
148C2×#102C22×C312yes—
149C2×#102C22×C312yes—
150C2×#102C22×C312yescomplemented
151C2×C22⋊C8C22×C312yes—
152C2×#97C22×C311yescharacteristic, complemented
153C2×C4×C8C22×C311yescharacteristic, complemented
154#394—22no—
155#392—22no—
156#390C22×C312yescomplemented
157#382C22×C311yescharacteristic, complemented
158#336C22×C312yescomplemented
159C24×C3—24no—
160C24×C3—21no—
161C24×C3—21no—
162C24×C3—21no—
163C24×C3—22no—
164C24×C3—42no—
165C24×C3—42no—
166C24×C3—84no—
167C24×C3C2×D411yescharacteristic
168C3×#41—48no—
169C3×#41—48no—
170C2×C3×Q8—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8C2×D412yes—
174C2×C3×D4—24no—
175C2×C3×D4—24no—
176C2×C3×D4—28no—
177C2×C3×D4—28no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—48no—
194C2×C3×D4—48no—
195C2×C3×D4—42no—
196C2×C3×D4—42no—
197C2×C3×D4—42no—
198C2×C3×D4—44no—
199C2×C3×D4—44no—
200C2×C3×D4—42no—
201C2×C3×D4—84no—
202C2×C3×D4—84no—
203C2×C3×D4—84no—
204C2×C3×D4—84no—
205C2×C3×D4C2×D412yes—
206C2×C3×D4C2×D412yes—
207C2×C3×D4C2×D412yes—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—22no—
211C22×C3×C4—24no—
212C22×C3×C4—21no—
213C22×C3×C4—22no—
214C22×C3×C4—42no—
215C22×C3×C4—42no—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D414yes—
220C22×C3×C4C2411yescharacteristic
221C3×SD16—48no—
222C3×D8—48no—
223C3×D8—48no—
224C3×D8—48no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8C2×D414yes—
230C2×C3×C8C2×D414yes—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C22⋊C4—28no—
235C3×C22⋊C4—28no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—44no—
238C3×C22⋊C4—84no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42C2×D412yes—
242C3×C42C2×D412yes—
243C25—21no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—24no—
247C2×#41—42no—
248C22×Q8C23×C311yescharacteristic
249C22×D4—21no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—24no—
257C22×D4—22no—
258C22×D4—42no—
259C22×D4C23×C312yes—
260C22×D4C23×C311yescharacteristic
261C22×D4C23×C311yescharacteristic
262C22×D4C23×C311yescharacteristic
263C23×C4—21no—
264C23×C4C23×C311yescharacteristic
265C2×SD16—24no—
266C2×SD16—24no—
267C2×SD16—42no—
268C2×D8—24no—
269C2×D8—24no—
270C2×D8—24no—
271C2×D8—24no—
272C2×D8—24no—
273C2×D8—24no—
274C2×D8—42no—
275C2×D8—42no—
276C2×D8—42no—
277C22×C8C23×C312yes—
278C22×C8C23×C312yes—
279#127—22no—
280#127—22no—
281#127—22no—
282#127—22no—
283#127—22no—
284#127C3×D414yescomplemented
285#124—22no—
286#124—22no—
287#124—22no—
288#124C3×D414yescomplemented
289#121—216no—
290#121—24no—
291#121—24no—
292#120—216no—
293#117—22no—
294#117—22no—
295C2×C4⋊C4C3×D412yes—
296C2×C4⋊C4C23×C312yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4C3×D412yes—
302C2×C42C23×C311yescharacteristic
303C2×C42C23×C311yescharacteristic
304#103—24no—
305#102—24no—
306#102—24no—
307#102—24no—
308C22⋊C8C3×D418yescomplemented
309#97—22no—
310C4×C8—22no—
311#95—22no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—48no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—44no—
326C23×C3—44no—
327C23×C3—44no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—44no—
332C23×C3—44no—
333C23×C3—88no—
334C23×C3—88no—
335C23×C3—88no—
336C23×C3#12712yes—
337C23×C3C22×D411yescharacteristic
338C3×Q8—24no—
339C3×Q8—42no—
340C3×D4—24no—
341C3×D4—24no—
342C3×D4—24no—
343C3×D4—48no—
344C3×D4—48no—
345C3×D4—42no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—88no—
353C3×D4—88no—
354C3×D4—88no—
355C3×D4—88no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—48no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—44no—
369C2×C3×C4#12014yes—
370C2×C3×C4#12712yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D412yes—
374C2×C3×C4H(5,𝔽2)12yes—
375C3×C8—24no—
376C3×C8—24no—
377C24—24no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—84no—
385C24C2×C3×D411yescharacteristic
386#41—48no—
387#41—48no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8C2×C3×D412yes—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
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—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—84no—
422C2×D4—84no—
423C2×D4C2×C3×D412yes—
424C2×D4C2×C3×D412yes—
425C2×D4C2×C3×D412yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4C2×C3×D412yes—
435C22×C4C2×C3×D412yes—
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C2×C3×D414yes—
438C22×C4C24×C311yescharacteristic
439SD16—48no—
440D8—48no—
441D8—48no—
442D8—48no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8C2×C3×D414yes—
448C2×C8C2×C3×D414yes—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—24no—
455C22⋊C4—44no—
456C22⋊C4—84no—
457C42—21no—
458C42—21no—
459C42C2×C3×D412yes—
460C42C2×C3×D412yes—
461C22×C3—24no—
462C22×C3—22no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—816no—
472C22×C3—82no—
473C22×C3—82no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—82no—
481C22×C3—84no—
482C22×C3—84no—
483C22×C3C2×#12011yescharacteristic
484C22×C3C2×#12711yescharacteristic
485C22×C3#53511yescharacteristic
486C22×C3#57614yes—
487C3×C4—22no—
488C3×C4—24no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4—82no—
492C3×C4D4212yes—
493C3×C4D4212yes—
494C23—24no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—48no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—88no—
516C23—88no—
517C23—88no—
518C23C3×#12712yes—
519C23C22×C3×D411yescharacteristic
520Q8—24no—
521Q8—42no—
522D4—24no—
523D4—24no—
524D4—24no—
525D4—48no—
526D4—48no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—88no—
535D4—88no—
536D4—88no—
537D4—88no—
538C2×C4—24no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—48no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4C3×#12014yes—
552C2×C4C3×#12712yes—
553C2×C4C22×C3×D411yescharacteristic
554C2×C4C22×C3×D411yescharacteristic
555C2×C4C22×C3×D412yes—
556C2×C4C3×H(5,𝔽2)12yes—
557C8—24no—
558C8—24no—
559C2×C3—42no—
560C2×C3—82no—
561C2×C3—82no—
562C2×C3—82no—
563C2×C3—162no—
564C2×C3#243311yescharacteristic
565C2×C3#300812yes—
566C2×C3#315512yes—
567C2×C3#322012yes—
568C22—24no—
569C22—22no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—816no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—84no—
589C22—84no—
590C22C2×C3×#12011yescharacteristic
591C22C2×C3×#12711yescharacteristic
592C22C3×#53511yescharacteristic
593C22C3×#57614yes—
594C4—22no—
595C4—24no—
596C4—42no—
597C4—42no—
598C4—82no—
599C4C3×D4212yes—
600C4C3×D4212yes—
601C3#2573411yescharacteristic, direct product factor, Sylow
602C2—42no—
603C2—82no—
604C2—82no—
605C2—82no—
606C2—162no—
607C2C3×#243311yescharacteristic
608C2C3×#300812yes—
609C2C3×#315512yes—
610C2C3×#322012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. C2×C3×#496C2semidirect
3. C2×C3×#493C2semidirect
4. C2×C3×#449C2semidirect
5. C2×C3×#447C2semidirect
6. C3×#2619C2semidirect
7. C3×#2012C2semidirect
8. C3×#2011C2semidirect
9. C3×#1961C2semidirect
10. C3×#1958C2semidirect
11. C3×#1750C2semidirect
12. #25734C3direct (N × H)
15. C22×C3×SD16C22semidirect
18. C22×C3×D8C22semidirect
23. C2×C3×#127C22semidirect
34. C2×C3×#103C22semidirect
37. C2×C3×#102C22semidirect
39. C2×C3×#97C22semidirect
40. C2×C3×C4×C8C22semidirect
43. C3×#390C22semidirect
44. C3×#382C22semidirect
45. C3×#336C22semidirect
46. C2×#535C2×C3semidirect
47. C2×#496C2×C3semidirect
48. C2×#493C2×C3semidirect
49. C2×#449C2×C3semidirect
50. C2×#447C2×C3semidirect
51. #2619C2×C3semidirect
52. #2012C2×C3semidirect
53. #2011C2×C3semidirect
54. #1961C2×C3semidirect
55. #1958C2×C3semidirect
56. #1750C2×C3semidirect
98. C3×#127D4semidirect
102. C3×#124D4semidirect
122. C3×C22⋊C8D4semidirect
128. C22×SD16C22×C3semidirect
131. C22×D8C22×C3semidirect
136. C2×#127C22×C3semidirect
147. C2×#103C22×C3semidirect
150. C2×#102C22×C3semidirect
152. C2×#97C22×C3semidirect
153. C2×C4×C8C22×C3semidirect
156. #390C22×C3semidirect
157. #382C22×C3semidirect
158. #336C22×C3semidirect
284. #127C3×D4semidirect
288. #124C3×D4semidirect
308. C22⋊C8C3×D4semidirect
601. C3#25734direct (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×#4962
C2×C3×#4932
C2×C3×#4492
C2×C3×#4472
C3×#26192
C3×#20122
C3×#20112
C3×#19612
C3×#19582
C3×#17502
#257343