Finite Group Databasebeta

← C3×#25697 #10644304

Subgroups — C3×#25697 #10644304

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic
2C3×#3166C211yesmaximal, characteristic, complemented
3C3×#3142C211yesmaximal, characteristic, complemented
4C3×#2566C211yesmaximal, characteristic, complemented
5C3×#2037C212yesmaximal, complemented
6C3×#2037C212yesmaximal, complemented
7C3×#1986C211yesmaximal, characteristic, complemented
8C3×#1980C211yesmaximal, characteristic, complemented
9C3×#1860C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C3×C4×#100C211yesmaximal, characteristic, complemented
12#25697C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559—22no—
14C3×#556—21no—
15C3×#555—21no—
16C3×#553—22no—
17C3×#550—21no—
18C3×#547—21no—
19C3×#538—22no—
20C3×#535C2212yes—
21C3×#533C2211yescharacteristic
22C3×#532C2211yescharacteristic
23C2×C3×#125C2211yescharacteristic
24C3×#525—22no—
25C3×#520C2212yes—
26C2×C3×C4×D4C2211yescharacteristic
27C3×#501—21no—
28C3×#501—21no—
29C3×#499—22no—
30C3×#491—21no—
31C3×#490—21no—
32C3×#488—22no—
33C3×#428C2212yes—
34C3×#428C2212yes—
35C2×C3×#104C2212yes—
36C2×C3×#104C2212yes—
37C3×#417C2212yes—
38C3×#417C2212yes—
39C2×C3×#100C2212yes—
40C3×C4×M16C2211yescharacteristic
41C3×C4×M16C2211yescharacteristic
42C3×#396—22no—
43C3×#390C2211yescharacteristic
44C3×#390C2211yescharacteristic
45C3×#385—22no—
46C3×#384C2211yescharacteristic
47C3×#382C2211yescharacteristic
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×#343C2212yes—
50C3×#339C2212yes—
51C3×#339C2212yes—
52#3506C2×C311yescharacteristic
53#3166C2×C311yescharacteristic, complemented
54#3142C2×C311yescharacteristic, complemented
55#2566C2×C311yescharacteristic, complemented
56#2037C2×C312yescomplemented
57#2037C2×C312yescomplemented
58#1986C2×C311yescharacteristic, complemented
59#1980C2×C311yescharacteristic, complemented
60#1860C2×C312yescomplemented
61#1859C2×C312yescomplemented
62C4×#100C2×C311yescharacteristic, complemented
63C2×C3×#41—22no—
64C2×C3×#41C2312yes—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4D412yes—
67C2×C3×M16C2312yes—
68C2×C3×M16C2312yes—
69C3×C4⋊Q8D411yescharacteristic
70C3×C4⋊Q8D411yescharacteristic
71C3×C4⋊Q8D411yescharacteristic
72C3×#127—22no—
73C3×#127D411yescharacteristic
74C3×#126—24no—
75C3×#125—21no—
76C3×#125—21no—
77C3×#125—21no—
78C3×#125—21no—
79C3×#125D412yes—
80C3×#124—24no—
81C3×#124—22no—
82C3×#124D412yes—
83C3×#123—24no—
84C3×#123—24no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—21no—
87C3×C22⋊Q8—21no—
88C3×#121—22no—
89C3×#121—24no—
90C3×#121—21no—
91C3×#121—21no—
92C3×#121—22no—
93C3×#121—22no—
94C3×#120—24no—
95C3×C4×Q8—22no—
96C3×C4×Q8—21no—
97C3×C4×Q8—21no—
98C3×C4×D4—24no—
99C3×C4×D4—22no—
100C3×C4×D4—24no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×C4×D4—21no—
107C3×#117—22no—
108C3×#117—22no—
109C3×#117C2312yes—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4—22no—
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C22⋊C4—22no—
119C2×C3×C22⋊C4—22no—
120C2×C3×C22⋊C4D412yes—
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C2×C3×C42C2311yescharacteristic
124C3×#107—22no—
125C3×#107—22no—
126C3×#106—22no—
127C3×#106—22no—
128C3×#104—24no—
129C3×#104—24no—
130C3×#103—22no—
131C3×#103—22no—
132C3×#102—22no—
133C3×#102—22no—
134C3×#100—24no—
135C3×#97—21no—
136C3×#97—21no—
137C3×C4×C8—21no—
138C3×C4×C8—21no—
139C3×#95—21no—
140C3×#95—22no—
141#559—22no—
142#556—21no—
143#555—21no—
144#553—22no—
145#550—21no—
146#547—21no—
147#538—22no—
148#535C22×C312yes—
149#533C22×C311yescharacteristic
150#532C22×C311yescharacteristic
151C2×#125C22×C311yescharacteristic
152#525—22no—
153#520C22×C312yes—
154C2×C4×D4C22×C311yescharacteristic
155#501—21no—
156#501—21no—
157#499—22no—
158#491—21no—
159#490—21no—
160#488—22no—
161#428C22×C312yes—
162#428C22×C312yes—
163C2×#104C22×C312yes—
164C2×#104C22×C312yes—
165#417C22×C312yes—
166#417C22×C312yes—
167C2×#100C22×C312yes—
168C4×M16C22×C311yescharacteristic
169C4×M16C22×C311yescharacteristic
170#396—22no—
171#390C22×C311yescharacteristic
172#390C22×C311yescharacteristic
173#385—22no—
174#384C22×C311yescharacteristic
175#382C22×C311yescharacteristic
176C4×C4⋊C4C22×C311yescharacteristic
177#343C22×C312yes—
178#339C22×C312yes—
179#339C22×C312yes—
180C24×C3C2×D412yes—
181C3×#41—22no—
182C3×#41—22no—
183C3×#41—22no—
184C3×#41—22no—
185C3×#41—84no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8C2×D412yes—
189C2×C3×D4—22no—
190C2×C3×D4—24no—
191C2×C3×D4—24no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4C2×D412yes—
199C22×C3×C4—21no—
200C22×C3×C4—24no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—82no—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C3×M16—22no—
213C3×M16—22no—
214C3×M16—22no—
215C3×M16#4114yescomplemented
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4—41no—
236C3×C4⋊C4—82no—
237C3×C4⋊C4C2×D412yes—
238C3×C4⋊C4C2×D412yes—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—44no—
248C3×C22⋊C4—82no—
249C3×C42—24no—
250C3×C42—22no—
251C3×C42—22no—
252C3×C42—22no—
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42C2×D411yescharacteristic
256C3×C42C2×D411yescharacteristic
257C2×#41—22no—
258C2×#41C23×C312yes—
259C22×D4C23×C311yescharacteristic
260C23×C4C3×D412yes—
261C2×M16C23×C312yes—
262C2×M16C23×C312yes—
263C4⋊Q8C3×D411yescharacteristic
264C4⋊Q8C3×D411yescharacteristic
265C4⋊Q8C3×D411yescharacteristic
266#127—22no—
267#127C3×D411yescharacteristic
268#126—24no—
269#125—21no—
270#125—21no—
271#125—21no—
272#125—21no—
273#125C3×D412yes—
274#124—24no—
275#124—22no—
276#124C3×D412yes—
277#123—24no—
278#123—24no—
279C22⋊Q8—22no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—22no—
283#121—24no—
284#121—21no—
285#121—21no—
286#121—22no—
287#121—22no—
288#120—24no—
289C4×Q8—22no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—24no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301#117—22no—
302#117—22no—
303#117C23×C312yes—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4C23×C311yescharacteristic
310C2×C4⋊C4C23×C311yescharacteristic
311C2×C4⋊C4C23×C311yescharacteristic
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4C3×D412yes—
315C2×C42C23×C311yescharacteristic
316C2×C42C23×C311yescharacteristic
317C2×C42C23×C311yescharacteristic
318#107—22no—
319#107—22no—
320#106—22no—
321#106—22no—
322#104—24no—
323#104—24no—
324#103—22no—
325#103—22no—
326#102—22no—
327#102—22no—
328#100—24no—
329#97—21no—
330#97—21no—
331C4×C8—21no—
332C4×C8—21no—
333#95—21no—
334#95—22no—
335C23×C3—22no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3C22×D411yescharacteristic
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—42no—
345C3×Q8—81no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—42no—
353C3×D4—81no—
354C3×D4—84no—
355C3×D4—84no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—44no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—42no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—82no—
377C2×C3×C4—84no—
378C2×C3×C4#12011yescharacteristic
379C2×C3×C4#12011yescharacteristic
380C2×C3×C4#12011yescharacteristic
381C2×C3×C4#12011yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4112yes—
385C2×C3×C42−1+412yes—
386C3×C8—24no—
387C3×C8—42no—
388C24C2×C3×D412yes—
389#41—22no—
390#41—22no—
391#41—22no—
392#41—22no—
393#41—84no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8C2×C3×D412yes—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4C2×C3×D412yes—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—82no—
416C22×C4C2×C3×D412yes—
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C24×C311yescharacteristic
420M16—22no—
421M16—22no—
422M16—22no—
423M16C3×#4114yescomplemented
424C2×C8—22no—
425C2×C8—22no—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—42no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—82no—
445C4⋊C4C2×C3×D412yes—
446C4⋊C4C2×C3×D412yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—44no—
456C22⋊C4—82no—
457C42—24no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42C2×C3×D411yescharacteristic
462C42C2×C3×D411yescharacteristic
463C42C2×C3×D411yescharacteristic
464C42C2×C3×D411yescharacteristic
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—44no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—82no—
475C22×C3C2×#12011yescharacteristic
476C22×C3#53211yescharacteristic
477C22×C3#53611yescharacteristic
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4#54712yes—
491C3×C4#54812yes—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23C22×C3×D411yescharacteristic
499Q8—22no—
500Q8—22no—
501Q8—42no—
502Q8—81no—
503D4—22no—
504D4—22no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—81no—
511D4—84no—
512D4—84no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4C3×#12011yescharacteristic
536C2×C4C3×#12011yescharacteristic
537C2×C4C3×#12011yescharacteristic
538C2×C4C3×#12011yescharacteristic
539C2×C4C22×C3×D411yescharacteristic
540C2×C4C22×C3×D411yescharacteristic
541C2×C4C2×C3×#4112yes—
542C2×C4C3×2−1+412yes—
543C8—24no—
544C8—42no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—82no—
548C2×C3—82no—
549C2×C3#243211yescharacteristic
550C2×C3#301811yescharacteristic
551C2×C3#301911yescharacteristic
552C22—21no—
553C22—21no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22C2×C3×#12011yescharacteristic
563C22C3×#53211yescharacteristic
564C22C3×#53611yescharacteristic
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4C3×#54712yes—
578C4C3×#54812yes—
579C3#2569711yescharacteristic, direct product factor, Sylow
580C2—21no—
581C2—21no—
582C2—82no—
583C2—82no—
584C2C3×#243211yescharacteristic
585C2C3×#301811yescharacteristic
586C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3506C2
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
2. C3×#3166C2semidirect
3. C3×#3142C2semidirect
4. C3×#2566C2semidirect
5. C3×#2037C2semidirect
6. C3×#2037C2semidirect
7. C3×#1986C2semidirect
8. C3×#1980C2semidirect
9. C3×#1860C2semidirect
10. C3×#1859C2semidirect
11. C3×C4×#100C2semidirect
12. #25697C3direct (N × H)
53. #3166C2×C3semidirect
54. #3142C2×C3semidirect
55. #2566C2×C3semidirect
56. #2037C2×C3semidirect
57. #2037C2×C3semidirect
58. #1986C2×C3semidirect
59. #1980C2×C3semidirect
60. #1860C2×C3semidirect
61. #1859C2×C3semidirect
62. C4×#100C2×C3semidirect
215. C3×M16#41semidirect
423. M16C3×#41semidirect
579. C3#25697direct (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×#31662
C3×#31422
C3×#25662
C3×#20372
C3×#19862
C3×#19802
C3×#18602
C3×#18592
C3×C4×#1002
#256973