Finite Group Databasebeta

← C3×#32987 #10651594

Subgroups — C3×#32987 #10651594

Order 768.

1358 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic, complemented
2C2×C3×#481C211yesmaximal, characteristic, complemented
3C2×C3×#482C211yesmaximal, characteristic, complemented
4C3×#2615C211yesmaximal, characteristic
5C3×#1875C212yesmaximal, complemented
6C3×#1874C212yesmaximal, complemented
7C3×#1854C212yesmaximal, complemented
8C3×#1853C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1823C211yesmaximal, characteristic, complemented
11C3×#1780C211yesmaximal, characteristic, complemented
12#32987C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#121C2212yes—
15C2×C3×#121C2211yescharacteristic
16C3×#520—24no—
17C3×#520—24no—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×D4—22no—
20C2×C3×C4×D4—21no—
21C2×C3×#117C2212yes—
22C2×C3×#117C2211yescharacteristic
23C22×C3×C4⋊C4C2212yes—
24C22×C3×C4⋊C4C2211yescharacteristic
25C3×#482—24no—
26C3×#481—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×#106C2212yes—
30C2×C3×#102C2211yescharacteristic
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C3×#393—21no—
38C3×#392—21no—
39C3×#392—22no—
40C3×#380C2211yescharacteristic
41C3×#380C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#520C2×C311yescharacteristic, complemented
47C2×#481C2×C311yescharacteristic, complemented
48C2×#482C2×C311yescharacteristic, complemented
49#2615C2×C311yescharacteristic
50#1875C2×C312yescomplemented
51#1874C2×C312yescomplemented
52#1854C2×C312yescomplemented
53#1853C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1823C2×C311yescharacteristic, complemented
56#1780C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4—22no—
64C22×C3×D4D411yescharacteristic
65C22×C3×D4D412yes—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—22no—
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2312yes—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2312yes—
73C22×C3×C8C2311yescharacteristic
74C3×#121—22no—
75C3×#121—22no—
76C3×#121—24no—
77C3×#121—24no—
78C3×C4×Q8—28no—
79C3×C4×D4—216no—
80C3×C4×D4—28no—
81C3×#117—22no—
82C3×#117—24no—
83C3×#117—24no—
84C3×#117D414yescomplemented
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—24no—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4D414yescomplemented
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C42—22no—
103C2×C3×C42—21no—
104C3×#107—22no—
105C3×#106—24no—
106C3×#106—22no—
107C3×#102—24no—
108C3×#102—22no—
109C3×#102—22no—
110C3×C22⋊C8—22no—
111C3×C22⋊C8—22no—
112C3×C22⋊C8—24no—
113C3×C22⋊C8—24no—
114C3×#95—21no—
115C3×#95—22no—
116C22×#41C22×C311yescharacteristic
117C2×#121C22×C311yescharacteristic
118C2×#121C22×C312yes—
119#520—24no—
120#520—24no—
121C2×C4×Q8—21no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×#117C22×C312yes—
125C2×#117C22×C311yescharacteristic
126C22×C4⋊C4C22×C312yes—
127C22×C4⋊C4C22×C311yescharacteristic
128#482—24no—
129#481—24no—
130C2×#107C22×C311yescharacteristic
131C2×#106C22×C311yescharacteristic
132C2×#106C22×C312yes—
133C2×#102C22×C311yescharacteristic
134C2×#102C22×C311yescharacteristic
135C2×#102C22×C312yes—
136C2×C22⋊C8C22×C311yescharacteristic
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C312yes—
139C2×C22⋊C8C22×C312yes—
140#393—21no—
141#392—21no—
142#392—22no—
143#380C22×C311yescharacteristic
144#380C22×C312yes—
145#340C22×C312yes—
146#340C22×C312yes—
147#340C22×C312yes—
148#340C22×C312yes—
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C24×C3C2×D412yes—
152C3×#41—28no—
153C3×#41—28no—
154C3×#41—216no—
155C2×C3×Q8—22no—
156C2×C3×Q8—24no—
157C2×C3×D4—24no—
158C2×C3×D4—24no—
159C2×C3×D4—24no—
160C2×C3×D4—22no—
161C2×C3×D4—24no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—42no—
167C2×C3×D4—44no—
168C2×C3×D4—82no—
169C2×C3×D4—82no—
170C2×C3×D4—82no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—28no—
181C22×C3×C4—24no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—81no—
185C22×C3×C4—82no—
186C22×C3×C4—81no—
187C22×C3×C4—82no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4#4114yes—
193C22×C3×C4#4114yes—
194C22×C3×C4C2411yescharacteristic
195C2×C3×C8—22no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4C2×D412yes—
212C3×C4⋊C4C2×D414yes—
213C3×C4⋊C4C2×D412yes—
214C3×C22⋊C4—28no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—42no—
217C3×C22⋊C4—44no—
218C3×C22⋊C4—81no—
219C3×C22⋊C4—82no—
220C3×C42—28no—
221C3×C42—24no—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—24no—
225C2×#41—24no—
226C22×Q8C3×D411yescharacteristic
227C22×D4—21no—
228C22×D4—22no—
229C22×D4C3×D411yescharacteristic
230C22×D4C3×D412yes—
231C22×D4C23×C311yescharacteristic
232C23×C4—22no—
233C23×C4—21no—
234C23×C4C23×C311yescharacteristic
235C23×C4C23×C312yes—
236C22×C8C23×C311yescharacteristic
237C22×C8C23×C312yes—
238C22×C8C23×C311yescharacteristic
239#121—22no—
240#121—22no—
241#121—24no—
242#121—24no—
243C4×Q8—28no—
244C4×D4—216no—
245C4×D4—28no—
246#117—22no—
247#117—24no—
248#117—24no—
249#117C3×D414yescomplemented
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4C3×D414yescomplemented
256C2×C4⋊C4C23×C312yes—
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C312yes—
260C2×C4⋊C4C23×C311yescharacteristic
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C42—22no—
268C2×C42—21no—
269#107—22no—
270#106—24no—
271#106—22no—
272#102—24no—
273#102—22no—
274#102—22no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—24no—
278C22⋊C8—24no—
279#95—21no—
280#95—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—21no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—21no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3#12012yes—
297C23×C3#12314yes—
298C23×C3C22×D411yescharacteristic
299C3×Q8—28no—
300C3×D4—28no—
301C3×D4—216no—
302C3×D4—42no—
303C3×D4—41no—
304C3×D4—41no—
305C3×D4—88no—
306C3×D4—84no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—84no—
331C2×C3×C4—88no—
332C2×C3×C4#12012yes—
333C2×C3×C4#12314yes—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C2×#4112yes—
337C2×C3×C4C2×#4112yes—
338C3×C8—42no—
339C3×C8—41no—
340C3×C8—41no—
341C24—21no—
342C24C2×C3×D411yescharacteristic
343C24C2×C3×D412yes—
344#41—28no—
345#41—28no—
346#41—216no—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—44no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—82no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—81no—
377C22×C4—82no—
378C22×C4—81no—
379C22×C4—82no—
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C2×C3×D412yes—
382C22×C4C2×C3×D412yes—
383C22×C4C2×C3×D412yes—
384C22×C4C3×#4114yes—
385C22×C4C3×#4114yes—
386C22×C4C24×C311yescharacteristic
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4C2×C3×D412yes—
404C4⋊C4C2×C3×D414yes—
405C4⋊C4C2×C3×D412yes—
406C22⋊C4—28no—
407C22⋊C4—24no—
408C22⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—81no—
411C22⋊C4—82no—
412C42—28no—
413C42—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—22no—
417C22×C3—24no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3C2×#12011yescharacteristic
432C22×C3C2×#12312yes—
433C22×C3#57611yescharacteristic
434C22×C3#57712yes—
435C22×C3#57711yescharacteristic
436C3×C4—22no—
437C3×C4—24no—
438C3×C4—42no—
439C3×C4—44no—
440C3×C4—42no—
441C3×C4—82no—
442C3×C4—81no—
443C3×C4#52511yescharacteristic
444C3×C4#52512yes—
445C3×C4#52511yescharacteristic
446C23—21no—
447C23—22no—
448C23—22no—
449C23—21no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23C3×#12012yes—
462C23C3×#12314yes—
463C23C22×C3×D411yescharacteristic
464Q8—28no—
465D4—28no—
466D4—216no—
467D4—42no—
468D4—41no—
469D4—41no—
470D4—84no—
471D4—88no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—88no—
497C2×C4C3×#12012yes—
498C2×C4C3×#12314yes—
499C2×C4C22×C3×D411yescharacteristic
500C2×C4C22×C3×D411yescharacteristic
501C2×C4C2×C3×#4112yes—
502C2×C4C2×C3×#4112yes—
503C8—42no—
504C8—41no—
505C8—41no—
506C2×C3—44no—
507C2×C3—42no—
508C2×C3—161no—
509C2×C3#284911yescharacteristic
510C2×C3#301011yescharacteristic
511C2×C3#301211yescharacteristic
512C2×C3#309612yes—
513C2×C3#309912yes—
514C22—22no—
515C22—22no—
516C22—24no—
517C22—24no—
518C22—22no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—81no—
527C22—81no—
528C22—82no—
529C22—82no—
530C22—81no—
531C22C2×C3×#12011yescharacteristic
532C22C2×C3×#12312yes—
533C22C3×#57611yescharacteristic
534C22C3×#57712yes—
535C22C3×#57711yescharacteristic
536C4—22no—
537C4—24no—
538C4—42no—
539C4—44no—
540C4—42no—
541C4—81no—
542C4—82no—
543C4C3×#52511yescharacteristic
544C4C3×#52512yes—
545C4C3×#52511yescharacteristic
546C3#3298711yescharacteristic, direct product factor, Sylow
547C2—42no—
548C2—44no—
549C2—161no—
550C2C3×#284911yescharacteristic
551C2C3×#301011yescharacteristic
552C2C3×#301211yescharacteristic
553C2C3×#309612yes—
554C2C3×#309912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2615C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#520C2semidirect
2. C2×C3×#481C2semidirect
3. C2×C3×#482C2semidirect
5. C3×#1875C2semidirect
6. C3×#1874C2semidirect
7. C3×#1854C2semidirect
8. C3×#1853C2semidirect
9. C3×#1825C2semidirect
10. C3×#1823C2semidirect
11. C3×#1780C2semidirect
12. #32987C3direct (N × H)
46. C2×#520C2×C3semidirect
47. C2×#481C2×C3semidirect
48. C2×#482C2×C3semidirect
50. #1875C2×C3semidirect
51. #1874C2×C3semidirect
52. #1854C2×C3semidirect
53. #1853C2×C3semidirect
54. #1825C2×C3semidirect
55. #1823C2×C3semidirect
56. #1780C2×C3semidirect
84. C3×#117D4semidirect
90. C2×C3×C4⋊C4D4semidirect
249. #117C3×D4semidirect
255. C2×C4⋊C4C3×D4semidirect
546. C3#32987direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×C3×#4812
C2×C3×#4822
C3×#26152
C3×#18752
C3×#18742
C3×#18542
C3×#18532
C3×#18252
C3×#18232
C3×#17802
#329873