Finite Group Databasebeta

← C3×#32944 #10651551

Subgroups — C3×#32944 #10651551

Order 768.

1320 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#568C211yesmaximal, characteristic, complemented
2C2×C3×#471C212yesmaximal, complemented
3C2×C3×#468C211yesmaximal, characteristic, complemented
4C3×#3007C211yesmaximal, characteristic, complemented
5C3×#3006C211yesmaximal, characteristic
6C3×#1908C212yesmaximal, complemented
7C3×#1907C212yesmaximal, complemented
8C3×#1901C211yesmaximal, characteristic, complemented
9C3×#1900C211yesmaximal, characteristic, complemented
10C3×#1816C212yesmaximal, complemented
11C3×#1811C211yesmaximal, characteristic, complemented
12#32944C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#137C2212yes—
15C2×C3×AGL(1,ℤ/8)C2212yes—
16C22×C3×SD16C2211yescharacteristic, complemented
17C3×#568—28no—
18C2×C3×#131—22no—
19C2×C3×#131C2212yes—
20C2×C3×#131C2212yes—
21C22×C3×M16—22no—
22C22×C3×M16C2211yescharacteristic
23C22×C3×M16C2211yescharacteristic
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×#121C2211yescharacteristic
26C3×#471—28no—
27C3×#468—22no—
28C3×#468—22no—
29C3×#452—22no—
30C3×#451—21no—
31C3×#450—21no—
32C3×#449—22no—
33C3×#448—21no—
34C3×#447—21no—
35C2×C3×#108C2212yescomplemented
36C2×C3×#107C2211yescharacteristic, complemented
37C3×#417C2212yes—
38C3×#417C2212yes—
39C2×C3×#103C2211yescharacteristic, complemented
40C2×C3×#102C2211yescharacteristic, complemented
41C2×C3×#101C2212yescomplemented
42C2×C3×#100C2212yescomplemented
43C3×#407C2211yescharacteristic
44C3×#407C2211yescharacteristic
45C3×#343C2212yescomplemented
46C3×#343C2212yescomplemented
47C3×#341C2212yescomplemented
48C2×#568C2×C311yescharacteristic, complemented
49C2×#471C2×C312yescomplemented
50C2×#468C2×C311yescharacteristic, complemented
51#3007C2×C311yescharacteristic, complemented
52#3006C2×C311yescharacteristic
53#1908C2×C312yescomplemented
54#1907C2×C312yescomplemented
55#1901C2×C311yescharacteristic, complemented
56#1900C2×C311yescharacteristic, complemented
57#1816C2×C312yescomplemented
58#1811C2×C311yescharacteristic, complemented
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—21no—
62C2×C3×#41—21no—
63C2×C3×#41D412yes—
64C2×C3×#41C2312yes—
65C22×C3×Q8D411yescharacteristic
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4—21no—
69C22×C3×D4D411yescharacteristic
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4C2311yescharacteristic
73C3×#137—24no—
74C3×#137—24no—
75C3×AGL(1,ℤ/8)—24no—
76C3×AGL(1,ℤ/8)—24no—
77C2×C3×Q16—22no—
78C2×C3×SD16—22no—
79C2×C3×SD16—22no—
80C2×C3×SD16—22no—
81C2×C3×SD16—24no—
82C2×C3×D8—22no—
83C3×#131—216no—
84C3×#131—24no—
85C3×#131—24no—
86C3×#131—24no—
87C3×#131D418yescomplemented
88C2×C3×M16—216no—
89C2×C3×M16—21no—
90C2×C3×M16—24no—
91C2×C3×M16—22no—
92C2×C3×M16—22no—
93C2×C3×M16—24no—
94C2×C3×M16D412yes—
95C2×C3×M16D412yes—
96C2×C3×M16C2312yes—
97C2×C3×M16C2312yes—
98C2×C3×M16C2312yes—
99C22×C3×C8—21no—
100C22×C3×C8—22no—
101C22×C3×C8—22no—
102C22×C3×C8C2311yescharacteristic
103C22×C3×C8C2311yescharacteristic
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8—21no—
106C3×C22⋊Q8—21no—
107C3×C22⋊Q8—21no—
108C3×#121—21no—
109C3×#121—21no—
110C3×#121—21no—
111C3×#121—21no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—21no—
117C3×#108—24no—
118C3×#107—22no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×#101—24no—
126C3×#100—24no—
127C3×C22⋊C8—21no—
128C3×C22⋊C8—21no—
129C3×C22⋊C8—21no—
130C3×C22⋊C8—21no—
131C22×#41C22×C311yescharacteristic
132C2×#137C22×C312yes—
133C2×AGL(1,ℤ/8)C22×C312yes—
134C22×SD16C22×C311yescharacteristic, complemented
135#568—28no—
136C2×#131—22no—
137C2×#131C22×C312yes—
138C2×#131C22×C312yes—
139C22×M16—22no—
140C22×M16C22×C311yescharacteristic
141C22×M16C22×C311yescharacteristic
142C2×C22⋊Q8C22×C311yescharacteristic
143C2×#121C22×C311yescharacteristic
144#471—28no—
145#468—22no—
146#468—22no—
147#452—22no—
148#451—21no—
149#450—21no—
150#449—22no—
151#448—21no—
152#447—21no—
153C2×#108C22×C312yescomplemented
154C2×#107C22×C311yescharacteristic, complemented
155#417C22×C312yes—
156#417C22×C312yes—
157C2×#103C22×C311yescharacteristic, complemented
158C2×#102C22×C311yescharacteristic, complemented
159C2×#101C22×C312yescomplemented
160C2×#100C22×C312yescomplemented
161#407C22×C311yescharacteristic
162#407C22×C311yescharacteristic
163#343C22×C312yescomplemented
164#343C22×C312yescomplemented
165#341C22×C312yescomplemented
166C24×C3—21no—
167C24×C3—21no—
168C24×C3#4111yescharacteristic
169C3×#41—24no—
170C3×#41—28no—
171C3×#41—24no—
172C3×#41—24no—
173C3×#41—24no—
174C3×#41—22no—
175C3×#41—22no—
176C3×#41C2×D414yes—
177C3×#41#4114yes—
178C2×C3×Q8—24no—
179C2×C3×Q8—21no—
180C2×C3×Q8—22no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—22no—
184C2×C3×Q8C2×D412yes—
185C2×C3×D4—28no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—24no—
189C2×C3×D4—21no—
190C2×C3×D4—22no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—22no—
194C2×C3×D4—41no—
195C2×C3×D4—41no—
196C2×C3×D4—82no—
197C2×C3×D4C2×D412yes—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4#4111yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C3×Q16—44no—
215C3×SD16—44no—
216C3×SD16—44no—
217C3×SD16—44no—
218C3×D8—44no—
219C3×M16—24no—
220C3×M16—28no—
221C3×M16—22no—
222C3×M16—24no—
223C3×M16—28no—
224C3×M16—24no—
225C3×M16—22no—
226C3×M16—22no—
227C3×M16—22no—
228C3×M16—22no—
229C3×M16C2×D414yes—
230C2×C3×C8—24no—
231C2×C3×C8—28no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8—28no—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C2×C3×C8C2×D412yes—
238C2×C3×C8C2×D412yes—
239C2×C3×C8C2×D414yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—41no—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C22⋊C4—41no—
249C3×C22⋊C4—41no—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—21no—
253C2×#41—21no—
254C2×#41C3×D412yes—
255C2×#41C23×C312yes—
256C22×Q8C3×D411yescharacteristic
257C22×Q8C23×C311yescharacteristic
258C22×D4—21no—
259C22×D4—21no—
260C22×D4C3×D411yescharacteristic
261C22×D4C23×C311yescharacteristic
262C23×C4—21no—
263C23×C4C23×C311yescharacteristic
264#137—24no—
265#137—24no—
266AGL(1,ℤ/8)—24no—
267AGL(1,ℤ/8)—24no—
268C2×Q16—22no—
269C2×SD16—22no—
270C2×SD16—22no—
271C2×SD16—22no—
272C2×SD16—24no—
273C2×D8—22no—
274#131—216no—
275#131—24no—
276#131—24no—
277#131—24no—
278#131C3×D418yescomplemented
279C2×M16—216no—
280C2×M16—21no—
281C2×M16—24no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—24no—
285C2×M16C3×D412yes—
286C2×M16C3×D412yes—
287C2×M16C23×C312yes—
288C2×M16C23×C312yes—
289C2×M16C23×C312yes—
290C22×C8—21no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8C23×C311yescharacteristic
294C22×C8C23×C311yescharacteristic
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299#121—21no—
300#121—21no—
301#121—21no—
302#121—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C23×C311yescharacteristic
305C2×C4⋊C4C23×C311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308#108—24no—
309#107—22no—
310#103—22no—
311#103—22no—
312#103—22no—
313#102—22no—
314#102—22no—
315#102—22no—
316#101—24no—
317#100—24no—
318C22⋊C8—21no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321C22⋊C8—21no—
322C23×C3—24no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—42no—
329C23×C3—81no—
330C23×C3—81no—
331C23×C3—81no—
332C23×C3—81no—
333C23×C3#12111yescharacteristic
334C23×C3#12111yescharacteristic
335C23×C3C2×#4111yescharacteristic
336C3×Q8—24no—
337C3×Q8—22no—
338C3×Q8—41no—
339C3×Q8—42no—
340C3×Q8—41no—
341C3×Q8#12114yes—
342C3×D4—28no—
343C3×D4—24no—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—22no—
347C3×D4—41no—
348C3×D4—42no—
349C3×D4—41no—
350C3×D4—82no—
351C3×D4—82no—
352C3×D4#12114yes—
353C2×C3×C4—24no—
354C2×C3×C4—28no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—24no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—42no—
369C2×C3×C4—81no—
370C2×C3×C4—81no—
371C2×C3×C4#12111yescharacteristic
372C2×C3×C4#12111yescharacteristic
373C2×C3×C4C22×D412yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4112yes—
377C3×C8—24no—
378C3×C8—22no—
379C3×C8—42no—
380C3×C8—42no—
381C3×C8#12014yes—
382C24—21no—
383C24—21no—
384C24C3×#4111yescharacteristic
385#41—24no—
386#41—28no—
387#41—24no—
388#41—24no—
389#41—24no—
390#41—22no—
391#41—22no—
392#41C2×C3×D414yes—
393#41C3×#4114yes—
394C2×Q8—24no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8C2×C3×D412yes—
401C2×D4—28no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—82no—
413C2×D4C2×C3×D412yes—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C3×#4111yescharacteristic
429C22×C4C24×C311yescharacteristic
430Q16—44no—
431SD16—44no—
432SD16—44no—
433SD16—44no—
434D8—44no—
435M16—24no—
436M16—28no—
437M16—22no—
438M16—24no—
439M16—28no—
440M16—24no—
441M16—22no—
442M16—22no—
443M16—22no—
444M16—22no—
445M16C2×C3×D414yes—
446C2×C8—24no—
447C2×C8—28no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8C2×C3×D412yes—
454C2×C8C2×C3×D412yes—
455C2×C8C2×C3×D414yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—24no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—82no—
478C22×C3—84no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—162no—
483C22×C3C2×#12111yescharacteristic
484C22×C3#52512yes—
485C3×C4—21no—
486C3×C4—24no—
487C3×C4—21no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4C2×#12011yescharacteristic
492C3×C4C2×#12112yes—
493C3×C4#52511yescharacteristic
494C23—24no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23C3×#12111yescharacteristic
506C23C3×#12111yescharacteristic
507C23C2×C3×#4111yescharacteristic
508Q8—24no—
509Q8—22no—
510Q8—41no—
511Q8—42no—
512Q8—41no—
513Q8C3×#12114yes—
514D4—28no—
515D4—24no—
516D4—24no—
517D4—24no—
518D4—22no—
519D4—41no—
520D4—42no—
521D4—41no—
522D4—82no—
523D4—82no—
524D4C3×#12114yes—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4C3×#12111yescharacteristic
544C2×C4C3×#12111yescharacteristic
545C2×C4C22×C3×D412yes—
546C2×C4C22×C3×D411yescharacteristic
547C2×C4C22×C3×D411yescharacteristic
548C2×C4C2×C3×#4112yes—
549C8—24no—
550C8—22no—
551C8—42no—
552C8—42no—
553C8C3×#12014yes—
554C2×C3—22no—
555C2×C3—41no—
556C2×C3—41no—
557C2×C3—44no—
558C2×C3—161no—
559C2×C3#284711yescharacteristic
560C2×C3#306811yescharacteristic
561C2×C3#306911yescharacteristic
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—24no—
571C22—42no—
572C22—42no—
573C22—82no—
574C22—84no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—162no—
579C22C2×C3×#12111yescharacteristic
580C22C3×#52512yes—
581C4—21no—
582C4—24no—
583C4—21no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4C2×C3×#12011yescharacteristic
588C4C2×C3×#12112yes—
589C4C3×#52511yescharacteristic
590C3#3294411yescharacteristic, direct product factor, Sylow
591C2—22no—
592C2—41no—
593C2—41no—
594C2—44no—
595C2—161no—
596C2C3×#284711yescharacteristic
597C2C3×#306811yescharacteristic
598C2C3×#306911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#568C2semidirect
2. C2×C3×#471C2semidirect
3. C2×C3×#468C2semidirect
4. C3×#3007C2semidirect
6. C3×#1908C2semidirect
7. C3×#1907C2semidirect
8. C3×#1901C2semidirect
9. C3×#1900C2semidirect
10. C3×#1816C2semidirect
11. C3×#1811C2semidirect
12. #32944C3direct (N × H)
16. C22×C3×SD16C22semidirect
35. C2×C3×#108C22semidirect
36. C2×C3×#107C22semidirect
39. C2×C3×#103C22semidirect
40. C2×C3×#102C22semidirect
41. C2×C3×#101C22semidirect
42. C2×C3×#100C22semidirect
45. C3×#343C22semidirect
46. C3×#343C22semidirect
47. C3×#341C22semidirect
48. C2×#568C2×C3semidirect
49. C2×#471C2×C3semidirect
50. C2×#468C2×C3semidirect
51. #3007C2×C3semidirect
53. #1908C2×C3semidirect
54. #1907C2×C3semidirect
55. #1901C2×C3semidirect
56. #1900C2×C3semidirect
57. #1816C2×C3semidirect
58. #1811C2×C3semidirect
87. C3×#131D4semidirect
134. C22×SD16C22×C3semidirect
153. C2×#108C22×C3semidirect
154. C2×#107C22×C3semidirect
157. C2×#103C22×C3semidirect
158. C2×#102C22×C3semidirect
159. C2×#101C22×C3semidirect
160. C2×#100C22×C3semidirect
163. #343C22×C3semidirect
164. #343C22×C3semidirect
165. #341C22×C3semidirect
278. #131C3×D4semidirect
590. C3#32944direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5682
C2×C3×#4712
C2×C3×#4682
C3×#30072
C3×#30062
C3×#19082
C3×#19072
C3×#19012
C3×#19002
C3×#18162
C3×#18112
#329443