Finite Group Databasebeta

← C3×#19824 #10638431

Subgroups — C3×#19824 #10638431

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3360C211yesmaximal, characteristic, complemented
2C3×#3335C211yesmaximal, characteristic, complemented
3C3×#3330C211yesmaximal, characteristic
4C3×#3302C211yesmaximal, characteristic, complemented
5C3×#1718C211yesmaximal, characteristic, complemented
6C3×#1703C211yesmaximal, characteristic, complemented
7C3×#1680C211yesmaximal, characteristic, complemented
8C3×#1679C211yesmaximal, characteristic, complemented
9C3×#1671C211yesmaximal, characteristic, complemented
10C3×#1650C211yesmaximal, characteristic, complemented
11C3×#1647C211yesmaximal, characteristic, complemented
12C3×#1644C211yesmaximal, characteristic, complemented
13C3×#1635C211yesmaximal, characteristic, complemented
14C3×#1583C211yesmaximal, characteristic, complemented
15C3×#1580C211yesmaximal, characteristic, complemented
16#19824C311yesmaximal, characteristic, direct product factor, Sylow
17C3×#556C2211yescharacteristic
18C3×#554C2211yescharacteristic
19C3×#547C2211yescharacteristic
20C3×#547C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#498C2211yescharacteristic
23C3×#488—21no—
24C3×#487C2211yescharacteristic
25C3×#487C2211yescharacteristic
26C3×#487C2211yescharacteristic
27C3×#487C2211yescharacteristic
28C3×#485—21no—
29C3×#484—21no—
30C3×#483—21no—
31C3×#481—21no—
32C3×#479—21no—
33C3×#476C2211yescharacteristic
34C3×#476C2211yescharacteristic
35C3×#475C2211yescharacteristic
36C3×#474C2211yescharacteristic
37C3×#474C2211yescharacteristic
38C3×#468—21no—
39C3×#467—21no—
40C3×#465—21no—
41C3×#463—21no—
42C3×#462C2211yescharacteristic
43C3×#461C2211yescharacteristic
44C3×#461C2211yescharacteristic
45C3×#452—21no—
46C3×#449—21no—
47C3×C8×Q8C2211yescharacteristic
48C3×C4×SD16—21no—
49C3×C4×SD16C2211yescharacteristic
50C3×C4×D8C2211yescharacteristic
51C3×#436C2211yescharacteristic
52C3×C8×D4C2211yescharacteristic
53C3×C8×D4C2211yescharacteristic
54C2×C3×#106—21no—
55C3×#417—21no—
56C3×#416—21no—
57C3×#334C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#333C2211yescharacteristic
60C3×#332C2211yescharacteristic
61C3×#332C2211yescharacteristic
62C3×#332C2211yescharacteristic
63C3×Q8⋊C8C2211yescharacteristic
64C3×D4⋊C8C2211yescharacteristic
65C3×D4⋊C8C2211yescharacteristic
66C3×D4⋊C8C2211yescharacteristic
67C3×#322C2211yescharacteristic
68#3360C2×C311yescharacteristic, complemented
69#3335C2×C311yescharacteristic, complemented
70#3330C2×C311yescharacteristic
71#3302C2×C311yescharacteristic, complemented
72#1718C2×C311yescharacteristic, complemented
73#1703C2×C311yescharacteristic, complemented
74#1680C2×C311yescharacteristic, complemented
75#1679C2×C311yescharacteristic, complemented
76#1671C2×C311yescharacteristic, complemented
77#1650C2×C311yescharacteristic, complemented
78#1647C2×C311yescharacteristic, complemented
79#1644C2×C311yescharacteristic, complemented
80#1635C2×C311yescharacteristic, complemented
81#1583C2×C311yescharacteristic, complemented
82#1580C2×C311yescharacteristic, complemented
83C2×C3×#41—21no—
84C2×C3×#41—21no—
85C2×C3×Q16—21no—
86C2×C3×SD16—21no—
87C2×C3×SD16—21no—
88C2×C3×SD16D411yescharacteristic
89C2×C3×D8D411yescharacteristic
90C2×C3×M16—21no—
91C22×C3×C8—21no—
92C22×C3×C8—21no—
93C3×C4⋊Q8C2311yescharacteristic
94C3×C4⋊Q8C2311yescharacteristic
95C3×C4⋊Q8C2311yescharacteristic
96C3×C4⋊Q8C2311yescharacteristic
97C3×#126—41no—
98C3×#125—21no—
99C3×#124—21no—
100C3×#123—41no—
101C3×#123—41no—
102C3×C22⋊Q8—21no—
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8—21no—
106C3×C22⋊Q8—41no—
107C3×#121—21no—
108C3×#121—21no—
109C3×C4×Q8—21no—
110C3×C4×Q8C2311yescharacteristic
111C3×C4×D4—21no—
112C3×C4×D4C2311yescharacteristic
113C3×C4×D4C2311yescharacteristic
114C3×C4×D4C2311yescharacteristic
115C3×#117—21no—
116C2×C3×C4⋊C4—21no—
117C2×C3×C4⋊C4—21no—
118C2×C3×C4⋊C4—21no—
119C3×#107—21no—
120C3×#107—21no—
121C3×#107—41no—
122C3×#107D411yescharacteristic
123C3×#106—21no—
124C3×#106—21no—
125C3×#106—21no—
126C3×#106—21no—
127C3×#106—21no—
128C3×#106D411yescharacteristic
129C3×C4⋊C8—21no—
130C3×C4⋊C8C2311yescharacteristic
131C3×C4⋊C8C2311yescharacteristic
132C3×C4⋊C8C2311yescharacteristic
133C3×C4⋊C8C2311yescharacteristic
134C3×#103—21no—
135C3×#103—21no—
136C3×#103—21no—
137C3×#103—21no—
138C3×#103—21no—
139C3×#103—21no—
140C3×#103—21no—
141C3×#103—21no—
142C3×#103—21no—
143C3×#103—21no—
144C3×#103—41no—
145C3×#103D411yescharacteristic
146C3×#102—21no—
147C3×#102—21no—
148C3×#102—21no—
149C3×#102—21no—
150C3×#102D411yescharacteristic
151C3×#102D411yescharacteristic
152C3×#102D411yescharacteristic
153C3×C22⋊C8—21no—
154C3×C22⋊C8—21no—
155C3×C22⋊C8—21no—
156C3×C4×C8—21no—
157C3×C4×C8C2311yescharacteristic
158C3×C4×C8C2311yescharacteristic
159C3×C4×C8C2311yescharacteristic
160#556C22×C311yescharacteristic
161#554C22×C311yescharacteristic
162#547C22×C311yescharacteristic
163#547C22×C311yescharacteristic
164#500C22×C311yescharacteristic
165#498C22×C311yescharacteristic
166#488—21no—
167#487C22×C311yescharacteristic
168#487C22×C311yescharacteristic
169#487C22×C311yescharacteristic
170#487C22×C311yescharacteristic
171#485—21no—
172#484—21no—
173#483—21no—
174#481—21no—
175#479—21no—
176#476C22×C311yescharacteristic
177#476C22×C311yescharacteristic
178#475C22×C311yescharacteristic
179#474C22×C311yescharacteristic
180#474C22×C311yescharacteristic
181#468—21no—
182#467—21no—
183#465—21no—
184#463—21no—
185#462C22×C311yescharacteristic
186#461C22×C311yescharacteristic
187#461C22×C311yescharacteristic
188#452—21no—
189#449—21no—
190C8×Q8C22×C311yescharacteristic
191C4×SD16—21no—
192C4×SD16C22×C311yescharacteristic
193C4×D8C22×C311yescharacteristic
194#436C22×C311yescharacteristic
195C8×D4C22×C311yescharacteristic
196C8×D4C22×C311yescharacteristic
197C2×#106—21no—
198#417—21no—
199#416—21no—
200#334C22×C311yescharacteristic
201#334C22×C311yescharacteristic
202#333C22×C311yescharacteristic
203#332C22×C311yescharacteristic
204#332C22×C311yescharacteristic
205#332C22×C311yescharacteristic
206Q8⋊C8C22×C311yescharacteristic
207D4⋊C8C22×C311yescharacteristic
208D4⋊C8C22×C311yescharacteristic
209D4⋊C8C22×C311yescharacteristic
210#322C22×C311yescharacteristic
211C3×#41—42no—
212C3×#41—42no—
213C3×#41—81no—
214C3×#41—81no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8C2×D411yescharacteristic
220C2×C3×D4—41no—
221C2×C3×D4—41no—
222C2×C3×D4C2×D411yescharacteristic
223C2×C3×D4C2×D411yescharacteristic
224C2×C3×D4C2×D411yescharacteristic
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—41no—
229C22×C3×C4—41no—
230C22×C3×C4—41no—
231C3×Q16—81no—
232C3×SD16—41no—
233C3×SD16—42no—
234C3×SD16—81no—
235C3×D8—22no—
236C3×M16—81no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—42no—
243C2×C3×C8—81no—
244C2×C3×C8C2×D411yescharacteristic
245C2×C3×C8C2×D411yescharacteristic
246C2×C3×C8C2×D411yescharacteristic
247C2×C3×C8C2×D411yescharacteristic
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—41no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4—41no—
260C3×C4⋊C4—41no—
261C3×C4⋊C4—41no—
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C4⋊C4C2×D411yescharacteristic
265C3×C4⋊C4C2×D411yescharacteristic
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—41no—
270C3×C22⋊C4—41no—
271C3×C22⋊C4—41no—
272C3×C22⋊C4—41no—
273C3×C22⋊C4—41no—
274C3×C42—21no—
275C3×C42—41no—
276C3×C42C2411yescharacteristic
277C2×#41—21no—
278C2×#41—21no—
279C2×Q16—21no—
280C2×SD16—21no—
281C2×SD16—21no—
282C2×SD16C3×D411yescharacteristic
283C2×D8C3×D411yescharacteristic
284C2×M16—21no—
285C22×C8—21no—
286C22×C8—21no—
287C4⋊Q8C23×C311yescharacteristic
288C4⋊Q8C23×C311yescharacteristic
289C4⋊Q8C23×C311yescharacteristic
290C4⋊Q8C23×C311yescharacteristic
291#126—41no—
292#125—21no—
293#124—21no—
294#123—41no—
295#123—41no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—41no—
301#121—21no—
302#121—21no—
303C4×Q8—21no—
304C4×Q8C23×C311yescharacteristic
305C4×D4—21no—
306C4×D4C23×C311yescharacteristic
307C4×D4C23×C311yescharacteristic
308C4×D4C23×C311yescharacteristic
309#117—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313#107—21no—
314#107—21no—
315#107—41no—
316#107C3×D411yescharacteristic
317#106—21no—
318#106—21no—
319#106—21no—
320#106—21no—
321#106—21no—
322#106C3×D411yescharacteristic
323C4⋊C8—21no—
324C4⋊C8C23×C311yescharacteristic
325C4⋊C8C23×C311yescharacteristic
326C4⋊C8C23×C311yescharacteristic
327C4⋊C8C23×C311yescharacteristic
328#103—21no—
329#103—21no—
330#103—21no—
331#103—21no—
332#103—21no—
333#103—21no—
334#103—21no—
335#103—21no—
336#103—21no—
337#103—21no—
338#103—41no—
339#103C3×D411yescharacteristic
340#102—21no—
341#102—21no—
342#102—21no—
343#102—21no—
344#102C3×D411yescharacteristic
345#102C3×D411yescharacteristic
346#102C3×D411yescharacteristic
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8—21no—
350C4×C8—21no—
351C4×C8C23×C311yescharacteristic
352C4×C8C23×C311yescharacteristic
353C4×C8C23×C311yescharacteristic
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—81no—
366C3×D4—21no—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4—82no—
372C3×D4—82no—
373C3×D4#13512yes—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—82no—
390C2×C3×C4—82no—
391C2×C3×C4—161no—
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395C3×C8—21no—
396C3×C8—21no—
397C3×C8—21no—
398C3×C8—21no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8—41no—
402C3×C8—41no—
403C3×C8—41no—
404#41—42no—
405#41—42no—
406#41—81no—
407#41—81no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×C3×D411yescharacteristic
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C2×C3×D411yescharacteristic
416C2×D4C2×C3×D411yescharacteristic
417C2×D4C2×C3×D411yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424Q16—81no—
425SD16—41no—
426SD16—42no—
427SD16—81no—
428D8—22no—
429M16—81no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—42no—
436C2×C8—81no—
437C2×C8C2×C3×D411yescharacteristic
438C2×C8C2×C3×D411yescharacteristic
439C2×C8C2×C3×D411yescharacteristic
440C2×C8C2×C3×D411yescharacteristic
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4C2×C3×D411yescharacteristic
456C4⋊C4C2×C3×D411yescharacteristic
457C4⋊C4C2×C3×D411yescharacteristic
458C4⋊C4C2×C3×D411yescharacteristic
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C42—21no—
468C42—41no—
469C42C24×C311yescharacteristic
470C22×C3—22no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3D4211yescharacteristic
480C3×C4—21no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4C2×#13511yescharacteristic
491C3×C4#57511yescharacteristic
492C3×C4#57711yescharacteristic
493C3×C4#57711yescharacteristic
494C23—21no—
495C23—21no—
496C23—21no—
497Q8—21no—
498Q8—21no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—81no—
506D4—21no—
507D4—21no—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—82no—
512D4—82no—
513D4C3×#13512yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—161no—
532C2×C4C22×C3×D411yescharacteristic
533C2×C4C22×C3×D411yescharacteristic
534C2×C4C3×H(5,𝔽2)11yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—41no—
544C2×C3—42no—
545C2×C3—81no—
546C2×C3—81no—
547C2×C3#326711yescharacteristic
548C2×C3#328211yescharacteristic
549C2×C3#328911yescharacteristic
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22C3×D4211yescharacteristic
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4C2×C3×#13511yescharacteristic
571C4C3×#57511yescharacteristic
572C4C3×#57711yescharacteristic
573C4C3×#57711yescharacteristic
574C3#1982411yescharacteristic, direct product factor, Sylow
575C2—42no—
576C2—81no—
577C2—81no—
578C2C3×#326711yescharacteristic
579C2C3×#328211yescharacteristic
580C2C3×#328911yescharacteristic
subgroupG/H
center, socleC22×C3D42
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3330C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3360C2semidirect
2. C3×#3335C2semidirect
4. C3×#3302C2semidirect
5. C3×#1718C2semidirect
6. C3×#1703C2semidirect
7. C3×#1680C2semidirect
8. C3×#1679C2semidirect
9. C3×#1671C2semidirect
10. C3×#1650C2semidirect
11. C3×#1647C2semidirect
12. C3×#1644C2semidirect
13. C3×#1635C2semidirect
14. C3×#1583C2semidirect
15. C3×#1580C2semidirect
16. #19824C3direct (N × H)
68. #3360C2×C3semidirect
69. #3335C2×C3semidirect
71. #3302C2×C3semidirect
72. #1718C2×C3semidirect
73. #1703C2×C3semidirect
74. #1680C2×C3semidirect
75. #1679C2×C3semidirect
76. #1671C2×C3semidirect
77. #1650C2×C3semidirect
78. #1647C2×C3semidirect
79. #1644C2×C3semidirect
80. #1635C2×C3semidirect
81. #1583C2×C3semidirect
82. #1580C2×C3semidirect
574. C3#19824direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#33602
C3×#33352
C3×#33302
C3×#33022
C3×#17182
C3×#17032
C3×#16802
C3×#16792
C3×#16712
C3×#16502
C3×#16472
C3×#16442
C3×#16352
C3×#15832
C3×#15802
#198243