Finite Group Databasebeta

← C3×#25625 #10644232

Subgroups — C3×#25625 #10644232

Order 768.

1172 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic, complemented
2C3×#3153C211yesmaximal, characteristic, complemented
3C3×#3140C211yesmaximal, characteristic, complemented
4C3×#2544C211yesmaximal, characteristic, complemented
5C3×#2004C212yesmaximal, complemented
6C3×#2002C212yesmaximal, complemented
7C3×#1986C211yesmaximal, characteristic, complemented
8C3×#1980C211yesmaximal, characteristic, complemented
9C3×#1883C212yesmaximal, complemented
10C3×#1882C212yesmaximal, complemented
11C3×C4×#100C211yesmaximal, characteristic, complemented
12#25625C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C2×C3×AGL(1,ℤ/8)C2212yes—
15C3×#559—22no—
16C3×#556—21no—
17C3×#555—21no—
18C3×#553—22no—
19C3×#550—21no—
20C3×#547—21no—
21C3×#538—22no—
22C3×#535C2212yes—
23C3×#533C2211yescharacteristic
24C3×#532C2211yescharacteristic
25C2×C3×#127C2211yescharacteristic
26C2×C3×#124C2211yescharacteristic
27C2×C3×#124C2211yescharacteristic
28C3×#525—22no—
29C3×#520C2212yes—
30C2×C3×C4×D4C2211yescharacteristic
31C3×#497—21no—
32C3×#496—21no—
33C3×#495—22no—
34C3×#489—21no—
35C3×#489—21no—
36C3×#488—22no—
37C2×C3×#104C2212yes—
38C2×C3×#104C2212yes—
39C3×#419C2212yes—
40C3×#418C2212yes—
41C2×C3×#100C2212yes—
42C3×C4×M16C2211yescharacteristic
43C3×C4×M16C2211yescharacteristic
44C3×#394—22no—
45C3×#391C2211yescharacteristic
46C3×#390C2211yescharacteristic
47C3×#386—22no—
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×#343C2212yes—
50C3×#339C2212yes—
51C3×#339C2212yes—
52#3506C2×C311yescharacteristic, complemented
53#3153C2×C311yescharacteristic, complemented
54#3140C2×C311yescharacteristic, complemented
55#2544C2×C311yescharacteristic, complemented
56#2004C2×C312yescomplemented
57#2002C2×C312yescomplemented
58#1986C2×C311yescharacteristic, complemented
59#1980C2×C311yescharacteristic, complemented
60#1883C2×C312yescomplemented
61#1882C2×C312yescomplemented
62C4×#100C2×C311yescharacteristic, complemented
63C2×C3×#41—22no—
64C2×C3×#41C2312yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—22no—
67C22×C3×D4C2311yescharacteristic
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4D412yes—
70C3×#137—24no—
71C3×#137—42no—
72C3×AGL(1,ℤ/8)—24no—
73C3×AGL(1,ℤ/8)—42no—
74C2×C3×Q16—22no—
75C2×C3×SD16—22no—
76C2×C3×SD16—22no—
77C2×C3×D8—22no—
78C2×C3×M16C2312yes—
79C2×C3×M16C2312yes—
80C3×C4⋊Q8D411yescharacteristic, complemented
81C3×C4⋊Q8D411yescharacteristic, complemented
82C3×C4⋊Q8D411yescharacteristic, complemented
83C3×#127—22no—
84C3×#127—42no—
85C3×#127D411yescharacteristic, complemented
86C3×#126—24no—
87C3×#125D412yescomplemented
88C3×#124—24no—
89C3×#124—22no—
90C3×#124—21no—
91C3×#124—21no—
92C3×#124—21no—
93C3×#124—21no—
94C3×#124—42no—
95C3×#124D412yescomplemented
96C3×#123—24no—
97C3×#123—24no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8—21no—
101C3×#121—22no—
102C3×#121—24no—
103C3×#121—21no—
104C3×#121—21no—
105C3×#121—22no—
106C3×#121—22no—
107C3×#120—24no—
108C3×C4×Q8—22no—
109C3×C4×Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×D4—24no—
112C3×C4×D4—22no—
113C3×C4×D4—24no—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4—21no—
119C3×C4×D4—21no—
120C3×#117—22no—
121C3×#117—22no—
122C3×#117C2312yes—
123C2×C3×C4⋊C4—22no—
124C2×C3×C4⋊C4—21no—
125C2×C3×C4⋊C4C2311yescharacteristic
126C2×C3×C22⋊C4—22no—
127C2×C3×C22⋊C4—21no—
128C2×C3×C22⋊C4—21no—
129C2×C3×C22⋊C4—22no—
130C2×C3×C22⋊C4D412yes—
131C2×C3×C42C2311yescharacteristic
132C2×C3×C42C2311yescharacteristic
133C2×C3×C42C2311yescharacteristic
134C3×#104—42no—
135C3×#104—42no—
136C3×#103—22no—
137C3×#103—22no—
138C3×#102—22no—
139C3×#102—22no—
140C3×#100—24no—
141C3×#97—21no—
142C3×#97—21no—
143C3×C4×C8—21no—
144C3×C4×C8—21no—
145C3×#95—21no—
146C3×#95—22no—
147C2×#137C22×C312yes—
148C2×AGL(1,ℤ/8)C22×C312yes—
149#559—22no—
150#556—21no—
151#555—21no—
152#553—22no—
153#550—21no—
154#547—21no—
155#538—22no—
156#535C22×C312yes—
157#533C22×C311yescharacteristic
158#532C22×C311yescharacteristic
159C2×#127C22×C311yescharacteristic
160C2×#124C22×C311yescharacteristic
161C2×#124C22×C311yescharacteristic
162#525—22no—
163#520C22×C312yes—
164C2×C4×D4C22×C311yescharacteristic
165#497—21no—
166#496—21no—
167#495—22no—
168#489—21no—
169#489—21no—
170#488—22no—
171C2×#104C22×C312yes—
172C2×#104C22×C312yes—
173#419C22×C312yes—
174#418C22×C312yes—
175C2×#100C22×C312yes—
176C4×M16C22×C311yescharacteristic
177C4×M16C22×C311yescharacteristic
178#394—22no—
179#391C22×C311yescharacteristic
180#390C22×C311yescharacteristic
181#386—22no—
182C4×C4⋊C4C22×C311yescharacteristic
183#343C22×C312yes—
184#339C22×C312yes—
185#339C22×C312yes—
186C24×C3—21no—
187C24×C3C2×D412yes—
188C3×#41—22no—
189C3×#41—22no—
190C3×#41—22no—
191C3×#41—22no—
192C3×#41—84no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—22no—
198C2×C3×Q8—22no—
199C2×C3×Q8C2×D412yes—
200C2×C3×D4—22no—
201C2×C3×D4—24no—
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—22no—
211C2×C3×D4—22no—
212C2×C3×D4—22no—
213C2×C3×D4—44no—
214C2×C3×D4—44no—
215C2×C3×D4—82no—
216C2×C3×D4C2×D412yes—
217C22×C3×C4—21no—
218C22×C3×C4—24no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—82no—
225C22×C3×C4C2×D412yes—
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C3×Q16—44no—
230C3×SD16—44no—
231C3×SD16—44no—
232C3×D8—44no—
233C3×M16—22no—
234C3×M16—22no—
235C3×M16—22no—
236C3×M16#4114yescomplemented
237C2×C3×C8—22no—
238C2×C3×C8—22no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4C2×D412yes—
250C3×C4⋊C4C2×D412yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—44no—
260C3×C22⋊C4—41no—
261C3×C22⋊C4—41no—
262C3×C22⋊C4—41no—
263C3×C22⋊C4—41no—
264C3×C22⋊C4—82no—
265C3×C42—24no—
266C3×C42—22no—
267C3×C42—22no—
268C3×C42—22no—
269C3×C42C2×D411yescharacteristic
270C3×C42C2×D411yescharacteristic
271C3×C42C2×D411yescharacteristic
272C3×C42C2×D411yescharacteristic
273C2×#41—22no—
274C2×#41C23×C312yes—
275C22×Q8C23×C311yescharacteristic
276C22×D4—22no—
277C22×D4C23×C311yescharacteristic
278C22×D4C23×C311yescharacteristic
279C23×C4C3×D412yes—
280#137—24no—
281#137—42no—
282AGL(1,ℤ/8)—24no—
283AGL(1,ℤ/8)—42no—
284C2×Q16—22no—
285C2×SD16—22no—
286C2×SD16—22no—
287C2×D8—22no—
288C2×M16C23×C312yes—
289C2×M16C23×C312yes—
290C4⋊Q8C3×D411yescharacteristic, complemented
291C4⋊Q8C3×D411yescharacteristic, complemented
292C4⋊Q8C3×D411yescharacteristic, complemented
293#127—22no—
294#127—42no—
295#127C3×D411yescharacteristic, complemented
296#126—24no—
297#125C3×D412yescomplemented
298#124—24no—
299#124—22no—
300#124—21no—
301#124—21no—
302#124—21no—
303#124—21no—
304#124—42no—
305#124C3×D412yescomplemented
306#123—24no—
307#123—24no—
308C22⋊Q8—22no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311#121—22no—
312#121—24no—
313#121—21no—
314#121—21no—
315#121—22no—
316#121—22no—
317#120—24no—
318C4×Q8—22no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—24no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330#117—22no—
331#117—22no—
332#117C23×C312yes—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4C23×C311yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C3×D412yes—
341C2×C42C23×C311yescharacteristic
342C2×C42C23×C311yescharacteristic
343C2×C42C23×C311yescharacteristic
344#104—42no—
345#104—42no—
346#103—22no—
347#103—22no—
348#102—22no—
349#102—22no—
350#100—24no—
351#97—21no—
352#97—21no—
353C4×C8—21no—
354C4×C8—21no—
355#95—21no—
356#95—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—22no—
361C23×C3—42no—
362C23×C3—42no—
363C23×C3—42no—
364C23×C3—42no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3C22×D411yescharacteristic
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—42no—
372C3×Q8—42no—
373C3×Q8—42no—
374C3×Q8—81no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—42no—
378C3×D4—42no—
379C3×D4—42no—
380C3×D4—42no—
381C3×D4—42no—
382C3×D4—42no—
383C3×D4—42no—
384C3×D4—81no—
385C3×D4—84no—
386C3×D4—84no—
387C3×D4—88no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—44no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—42no—
399C2×C3×C4—42no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—82no—
408C2×C3×C4—84no—
409C2×C3×C4#12011yescharacteristic
410C2×C3×C4#12011yescharacteristic
411C2×C3×C4#12011yescharacteristic
412C2×C3×C4#12011yescharacteristic
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C2×#4112yes—
416C2×C3×C4H(5,𝔽2)12yes—
417C3×C8—24no—
418C3×C8—42no—
419C24—21no—
420C24C2×C3×D412yes—
421#41—22no—
422#41—22no—
423#41—22no—
424#41—22no—
425#41—84no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C2×C3×D412yes—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—82no—
449C2×D4C2×C3×D412yes—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—82no—
458C22×C4C2×C3×D412yes—
459C22×C4C2×C3×D411yescharacteristic
460C22×C4C2×C3×D411yescharacteristic
461C22×C4C24×C311yescharacteristic
462Q16—44no—
463SD16—44no—
464SD16—44no—
465D8—44no—
466M16—22no—
467M16—22no—
468M16—22no—
469M16C3×#4114yescomplemented
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4C2×C3×D412yes—
483C4⋊C4C2×C3×D412yes—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—44no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—82no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42C2×C3×D411yescharacteristic
503C42C2×C3×D411yescharacteristic
504C42C2×C3×D411yescharacteristic
505C42C2×C3×D411yescharacteristic
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—44no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—82no—
516C22×C3—82no—
517C22×C3—82no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3C2×#12011yescharacteristic
522C22×C3#53211yescharacteristic
523C22×C3#53511yescharacteristic
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—42no—
529C3×C4—42no—
530C3×C4—42no—
531C3×C4—42no—
532C3×C4—42no—
533C3×C4—81no—
534C3×C4—81no—
535C3×C4#54612yes—
536C3×C4#55012yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C22×C3×D411yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—81no—
555D4—22no—
556D4—22no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—81no—
565D4—84no—
566D4—84no—
567D4—88no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—82no—
588C2×C4—84no—
589C2×C4C3×#12011yescharacteristic
590C2×C4C3×#12011yescharacteristic
591C2×C4C3×#12011yescharacteristic
592C2×C4C3×#12011yescharacteristic
593C2×C4C22×C3×D411yescharacteristic
594C2×C4C22×C3×D411yescharacteristic
595C2×C4C2×C3×#4112yes—
596C2×C4C3×H(5,𝔽2)12yes—
597C8—24no—
598C8—42no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—82no—
602C2×C3—82no—
603C2×C3—161no—
604C2×C3#243111yescharacteristic
605C2×C3#301811yescharacteristic
606C2×C3#301911yescharacteristic
607C22—21no—
608C22—21no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22C2×C3×#12011yescharacteristic
623C22C3×#53211yescharacteristic
624C22C3×#53511yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—81no—
636C4C3×#54612yes—
637C4C3×#55012yes—
638C3#2562511yescharacteristic, direct product factor, Sylow
639C2—21no—
640C2—21no—
641C2—82no—
642C2—82no—
643C2—161no—
644C2C3×#243111yescharacteristic
645C2C3×#301811yescharacteristic
646C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3506C2semidirect
2. C3×#3153C2semidirect
3. C3×#3140C2semidirect
4. C3×#2544C2semidirect
5. C3×#2004C2semidirect
6. C3×#2002C2semidirect
7. C3×#1986C2semidirect
8. C3×#1980C2semidirect
9. C3×#1883C2semidirect
10. C3×#1882C2semidirect
11. C3×C4×#100C2semidirect
12. #25625C3direct (N × H)
52. #3506C2×C3semidirect
53. #3153C2×C3semidirect
54. #3140C2×C3semidirect
55. #2544C2×C3semidirect
56. #2004C2×C3semidirect
57. #2002C2×C3semidirect
58. #1986C2×C3semidirect
59. #1980C2×C3semidirect
60. #1883C2×C3semidirect
61. #1882C2×C3semidirect
62. C4×#100C2×C3semidirect
80. C3×C4⋊Q8D4semidirect
81. C3×C4⋊Q8D4semidirect
82. C3×C4⋊Q8D4semidirect
85. C3×#127D4semidirect
87. C3×#125D4semidirect
95. C3×#124D4semidirect
236. C3×M16#41semidirect
290. C4⋊Q8C3×D4semidirect
291. C4⋊Q8C3×D4semidirect
292. C4⋊Q8C3×D4semidirect
295. #127C3×D4semidirect
297. #125C3×D4semidirect
305. #124C3×D4semidirect
469. M16C3×#41semidirect
638. C3#25625direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35062
C3×#31532
C3×#31402
C3×#25442
C3×#20042
C3×#20022
C3×#19862
C3×#19802
C3×#18832
C3×#18822
C3×C4×#1002
#256253