Finite Group Databasebeta

← C3×#19088 #10637695

Subgroups — C3×#19088 #10637695

Order 768.

884 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
1C3×#3559C211yesmaximal, characteristic, complemented
2C3×#3234C211yesmaximal, characteristic, complemented
3C3×#3219C211yesmaximal, characteristic, complemented
4C3×#3214C211yesmaximal, characteristic, complemented
5C3×#1664C212yesmaximal, complemented
6C3×#1661C212yesmaximal, complemented
7C3×#1657C212yesmaximal, complemented
8C3×#1656C212yesmaximal, complemented
9C3×#1536C211yesmaximal, characteristic, complemented
10C3×#1535C211yesmaximal, characteristic, complemented
11C3×#1534C211yesmaximal, characteristic, complemented
12#19088C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#563C2212yes—
14C3×#562C2212yes—
15C3×#554—21no—
16C3×#553—21no—
17C3×#553—21no—
18C3×#547—22no—
19C3×#547—21no—
20C3×#544—21no—
21C3×#540—21no—
22C3×#539—22no—
23C3×#539—21no—
24C3×#537—21no—
25C3×#532C2211yescharacteristic
26C3×#532C2211yescharacteristic
27C3×#529C2211yescharacteristic
28C2×C3×#125C2211yescharacteristic
29C3×#501C2212yes—
30C3×#497C2211yescharacteristic
31C3×#496C2212yes—
32C3×#496C2211yescharacteristic
33C3×#491C2211yescharacteristic
34C3×#490C2211yescharacteristic
35C3×#490C2212yes—
36C3×#489C2212yes—
37C3×#481—21no—
38C3×#480—21no—
39C3×#478—22no—
40C3×#464—21no—
41C3×#463—21no—
42C3×#461—21no—
43C3×#459—21no—
44C3×#452—21no—
45C3×#449—22no—
46C3×#449—21no—
47C3×#432C2211yescharacteristic
48C3×#432C2211yescharacteristic
49C3×#432C2211yescharacteristic
50C3×#330C2212yes—
51C3×#330C2212yes—
52C3×#330C2212yes—
53C3×#329C2212yes—
54C3×#329C2212yes—
55C3×#329C2212yes—
56#3559C2×C311yescharacteristic, complemented
57#3234C2×C311yescharacteristic, complemented
58#3219C2×C311yescharacteristic, complemented
59#3214C2×C311yescharacteristic, complemented
60#1664C2×C312yescomplemented
61#1661C2×C312yescomplemented
62#1657C2×C312yescomplemented
63#1656C2×C312yescomplemented
64#1536C2×C311yescharacteristic, complemented
65#1535C2×C311yescharacteristic, complemented
66#1534C2×C311yescharacteristic, complemented
67C2×C3×#41—22no—
68C2×C3×#41—21no—
69C2×C3×#41D412yes—
70C2×C3×Q16—21no—
71C2×C3×SD16—22no—
72C2×C3×SD16—21no—
73C2×C3×SD16—21no—
74C2×C3×D8—22no—
75C2×C3×D8—21no—
76C3×C4⋊Q8—21no—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8D411yescharacteristic
79C3×C4⋊Q8C2311yescharacteristic
80C3×#127D411yescharacteristic
81C3×#127C2311yescharacteristic
82C3×#126—22no—
83C3×#126—22no—
84C3×#126—21no—
85C3×#125—22no—
86C3×#125—21no—
87C3×#125—21no—
88C3×#125—21no—
89C3×#125D412yes—
90C3×#125C2312yes—
91C3×#125C2311yescharacteristic
92C3×#124—21no—
93C3×#124C2311yescharacteristic
94C3×#124C2312yes—
95C3×#123—22no—
96C3×#123—22no—
97C3×#123—22no—
98C3×#123—22no—
99C3×#123—21no—
100C3×#123—22no—
101C3×#123—22no—
102C3×C22⋊Q8—22no—
103C3×C22⋊Q8—22no—
104C3×C22⋊Q8—22no—
105C3×C22⋊Q8—22no—
106C3×C22⋊Q8—22no—
107C3×C22⋊Q8—22no—
108C3×C22⋊Q8D411yescharacteristic
109C3×#121—22no—
110C3×#121—22no—
111C3×#121—21no—
112C3×#121—21no—
113C3×#121—22no—
114C3×#121—22no—
115C3×#121—21no—
116C3×#121—21no—
117C3×#121—21no—
118C3×#121—21no—
119C3×#121D411yescharacteristic
120C3×C4×Q8D411yescharacteristic
121C3×C4×D4—22no—
122C3×C4×D4—22no—
123C3×C4×D4—21no—
124C3×C4×D4—21no—
125C3×C4×D4—21no—
126C3×C4×D4—21no—
127C3×C4×D4—21no—
128C3×C4×D4—21no—
129C3×C4×D4—21no—
130C3×C4×D4D411yescharacteristic
131C3×#117—21no—
132C3×#117—21no—
133C2×C3×C4⋊C4—22no—
134C2×C3×C4⋊C4—21no—
135C2×C3×C4⋊C4—21no—
136C2×C3×C4⋊C4—21no—
137C2×C3×C4⋊C4D411yescharacteristic
138C2×C3×C4⋊C4D411yescharacteristic
139C2×C3×C42C2311yescharacteristic
140C3×#107—22no—
141C3×#106—22no—
142C3×C4⋊C8—21no—
143C3×C4⋊C8—21no—
144C3×C4⋊C8—21no—
145C3×#103—22no—
146C3×#103—21no—
147C3×#103—21no—
148C3×#102—22no—
149C3×#102—21no—
150C3×#102—21no—
151C3×#102—22no—
152C3×#102—22no—
153C3×C22⋊C8—21no—
154C3×C22⋊C8—21no—
155C3×C22⋊C8—21no—
156C3×#97C2312yes—
157C3×#97C2311yescharacteristic
158C3×#97C2311yescharacteristic
159C3×#97C2312yes—
160#563C22×C312yes—
161#562C22×C312yes—
162#554—21no—
163#553—21no—
164#553—21no—
165#547—22no—
166#547—21no—
167#544—21no—
168#540—21no—
169#539—22no—
170#539—21no—
171#537—21no—
172#532C22×C311yescharacteristic
173#532C22×C311yescharacteristic
174#529C22×C311yescharacteristic
175C2×#125C22×C311yescharacteristic
176#501C22×C312yes—
177#497C22×C311yescharacteristic
178#496C22×C311yescharacteristic
179#496C22×C312yes—
180#491C22×C311yescharacteristic
181#490C22×C312yes—
182#490C22×C311yescharacteristic
183#489C22×C312yes—
184#481—21no—
185#480—21no—
186#478—22no—
187#464—21no—
188#463—21no—
189#461—21no—
190#459—21no—
191#452—21no—
192#449—22no—
193#449—21no—
194#432C22×C311yescharacteristic
195#432C22×C311yescharacteristic
196#432C22×C311yescharacteristic
197#330C22×C312yes—
198#330C22×C312yes—
199#330C22×C312yes—
200#329C22×C312yes—
201#329C22×C312yes—
202#329C22×C312yes—
203C3×#41—84no—
204C3×#41—81no—
205C3×#41—81no—
206C3×#41—82no—
207C2×C3×Q8—22no—
208C2×C3×Q8—21no—
209C2×C3×Q8C2×D412yes—
210C2×C3×Q8C2×D411yescharacteristic
211C2×C3×D4—22no—
212C2×C3×D4—21no—
213C2×C3×D4—22no—
214C2×C3×D4—22no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—41no—
221C2×C3×D4C2×D412yes—
222C2×C3×D4C2×D411yescharacteristic
223C22×C3×C4—21no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—22no—
230C22×C3×C4—41no—
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D411yescharacteristic
234C3×Q16—81no—
235C3×SD16—82no—
236C3×SD16—81no—
237C3×SD16—81no—
238C3×D8—81no—
239C3×D8—82no—
240C2×C3×C8—22no—
241C2×C3×C8—21no—
242C2×C3×C8—22no—
243C2×C3×C8—21no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C4⋊C4C2×D412yes—
265C3×C4⋊C4C2×D412yes—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—21no—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—22no—
276C3×C22⋊C4—22no—
277C3×C22⋊C4—21no—
278C3×C22⋊C4—41no—
279C3×C22⋊C4—41no—
280C3×C42—22no—
281C3×C42—21no—
282C3×C42C2×D411yescharacteristic
283C3×C42C2×D411yescharacteristic
284C3×C42C2×D411yescharacteristic
285C3×C42C2411yescharacteristic
286C2×#41—22no—
287C2×#41—21no—
288C2×#41C3×D412yes—
289C2×Q16—21no—
290C2×SD16—22no—
291C2×SD16—21no—
292C2×SD16—21no—
293C2×D8—22no—
294C2×D8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—22no—
297C4⋊Q8C3×D411yescharacteristic
298C4⋊Q8C23×C311yescharacteristic
299#127C3×D411yescharacteristic
300#127C23×C311yescharacteristic
301#126—22no—
302#126—22no—
303#126—21no—
304#125—22no—
305#125—21no—
306#125—21no—
307#125—21no—
308#125C3×D412yes—
309#125C23×C312yes—
310#125C23×C311yescharacteristic
311#124—21no—
312#124C23×C311yescharacteristic
313#124C23×C312yes—
314#123—22no—
315#123—22no—
316#123—22no—
317#123—22no—
318#123—21no—
319#123—22no—
320#123—22no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8C3×D411yescharacteristic
328#121—22no—
329#121—22no—
330#121—21no—
331#121—21no—
332#121—22no—
333#121—22no—
334#121—21no—
335#121—21no—
336#121—21no—
337#121—21no—
338#121C3×D411yescharacteristic
339C4×Q8C3×D411yescharacteristic
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4C3×D411yescharacteristic
350#117—21no—
351#117—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4C3×D411yescharacteristic
357C2×C4⋊C4C3×D411yescharacteristic
358C2×C42C23×C311yescharacteristic
359#107—22no—
360#106—22no—
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364#103—22no—
365#103—21no—
366#103—21no—
367#102—22no—
368#102—21no—
369#102—21no—
370#102—22no—
371#102—22no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375#97C23×C312yes—
376#97C23×C311yescharacteristic
377#97C23×C311yescharacteristic
378#97C23×C312yes—
379C23×C3—22no—
380C23×C3—21no—
381C23×C3—41no—
382C23×C3#12011yescharacteristic
383C3×Q8—41no—
384C3×Q8—42no—
385C3×Q8—41no—
386C3×Q8—41no—
387C3×Q8—82no—
388C3×D4—41no—
389C3×D4—42no—
390C3×D4—42no—
391C3×D4—41no—
392C3×D4—41no—
393C3×D4—41no—
394C3×D4—41no—
395C3×D4—42no—
396C3×D4—44no—
397C3×D4—42no—
398C3×D4—81no—
399C3×D4—82no—
400C3×D4—82no—
401C3×D4—82no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—21no—
407C2×C3×C4—22no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—42no—
412C2×C3×C4—42no—
413C2×C3×C4—41no—
414C2×C3×C4—82no—
415C2×C3×C4—82no—
416C2×C3×C4—81no—
417C2×C3×C4—82no—
418C2×C3×C4—82no—
419C2×C3×C4—82no—
420C2×C3×C4#12011yescharacteristic
421C2×C3×C4#12011yescharacteristic
422C2×C3×C4#12011yescharacteristic
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C22×D411yescharacteristic
426C3×C8—42no—
427C3×C8—42no—
428C3×C8—41no—
429C3×C8—41no—
430#41—84no—
431#41—81no—
432#41—81no—
433#41—82no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8C2×C3×D412yes—
437C2×Q8C2×C3×D411yescharacteristic
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—41no—
448C2×D4C2×C3×D412yes—
449C2×D4C2×C3×D411yescharacteristic
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4C2×C3×D411yescharacteristic
459C22×C4C2×C3×D411yescharacteristic
460C22×C4C2×C3×D411yescharacteristic
461Q16—81no—
462SD16—82no—
463SD16—81no—
464SD16—81no—
465D8—81no—
466D8—82no—
467C2×C8—22no—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4C2×C3×D411yescharacteristic
490C4⋊C4C2×C3×D411yescharacteristic
491C4⋊C4C2×C3×D412yes—
492C4⋊C4C2×C3×D412yes—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—21no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C42—22no—
508C42—21no—
509C42C2×C3×D411yescharacteristic
510C42C2×C3×D411yescharacteristic
511C42C2×C3×D411yescharacteristic
512C42C24×C311yescharacteristic
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—42no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—41no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3C2×#12011yescharacteristic
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—42no—
533C3×C4—42no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—42no—
537C3×C4—41no—
538C3×C4—81no—
539C23—22no—
540C23—21no—
541C23—41no—
542C23C3×#12011yescharacteristic
543Q8—41no—
544Q8—42no—
545Q8—41no—
546Q8—41no—
547Q8—82no—
548D4—41no—
549D4—42no—
550D4—42no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—42no—
556D4—44no—
557D4—42no—
558D4—81no—
559D4—82no—
560D4—82no—
561D4—82no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4C3×#12011yescharacteristic
581C2×C4C3×#12011yescharacteristic
582C2×C4C3×#12011yescharacteristic
583C2×C4C22×C3×D411yescharacteristic
584C2×C4C22×C3×D411yescharacteristic
585C2×C4C22×C3×D411yescharacteristic
586C8—42no—
587C8—41no—
588C8—41no—
589C8—42no—
590C2×C3—41no—
591C2×C3—82no—
592C2×C3—81no—
593C2×C3—161no—
594C2×C3#301911yescharacteristic
595C2×C3#302011yescharacteristic
596C2×C3#302111yescharacteristic
597C22—21no—
598C22—21no—
599C22—21no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—41no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22C2×C3×#12011yescharacteristic
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—81no—
623C3#1908811yescharacteristic, direct product factor, Sylow
624C2—41no—
625C2—82no—
626C2—81no—
627C2—161no—
628C2C3×#301911yescharacteristic
629C2C3×#302011yescharacteristic
630C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3559C2semidirect
2. C3×#3234C2semidirect
3. C3×#3219C2semidirect
4. C3×#3214C2semidirect
5. C3×#1664C2semidirect
6. C3×#1661C2semidirect
7. C3×#1657C2semidirect
8. C3×#1656C2semidirect
9. C3×#1536C2semidirect
10. C3×#1535C2semidirect
11. C3×#1534C2semidirect
12. #19088C3direct (N × H)
56. #3559C2×C3semidirect
57. #3234C2×C3semidirect
58. #3219C2×C3semidirect
59. #3214C2×C3semidirect
60. #1664C2×C3semidirect
61. #1661C2×C3semidirect
62. #1657C2×C3semidirect
63. #1656C2×C3semidirect
64. #1536C2×C3semidirect
65. #1535C2×C3semidirect
66. #1534C2×C3semidirect
623. C3#19088direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35592
C3×#32342
C3×#32192
C3×#32142
C3×#16642
C3×#16612
C3×#16572
C3×#16562
C3×#15362
C3×#15352
C3×#15342
#190883