Finite Group Databasebeta

← C3×#25520 #10644127

Subgroups — C3×#25520 #10644127

Order 768.

1042 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3493C211yesmaximal, characteristic
2C3×#3056C212yesmaximal, complemented
3C3×#2499C211yesmaximal, characteristic, complemented
4C3×#2061C211yesmaximal, characteristic, complemented
5C3×#2040C211yesmaximal, characteristic, complemented
6C3×#2028C211yesmaximal, characteristic, complemented
7C3×#2007C211yesmaximal, characteristic, complemented
8C3×#1882C212yesmaximal, complemented
9C3×#1871C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C3×#1742C211yesmaximal, characteristic, complemented
12#25520C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C22×C3×M16C2212yes—
15C3×#556—22no—
16C3×#555—22no—
17C3×#553—22no—
18C3×#552—22no—
19C3×#544C2211yescharacteristic
20C3×#538—22no—
21C3×#537C2211yescharacteristic
22C3×#529—22no—
23C3×#525C2212yes—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C3×#520C2212yes—
26C2×C3×#117C2211yescharacteristic
27C3×#470—22no—
28C3×#468—22no—
29C3×#467—22no—
30C3×#465—22no—
31C3×#428C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#419C2212yes—
35C3×#417C2212yes—
36C3×#397—22no—
37C3×#391C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#387—22no—
40C3×#384C2211yescharacteristic
41C3×#382C2211yescharacteristic
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#343C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2212yes—
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3493C2×C311yescharacteristic
49#3056C2×C312yescomplemented
50#2499C2×C311yescharacteristic, complemented
51#2061C2×C311yescharacteristic, complemented
52#2040C2×C311yescharacteristic, complemented
53#2028C2×C311yescharacteristic, complemented
54#2007C2×C311yescharacteristic, complemented
55#1882C2×C312yescomplemented
56#1871C2×C312yescomplemented
57#1859C2×C312yescomplemented
58#1742C2×C311yescharacteristic, complemented
59C2×C3×#41C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C3×#137—24no—
63C3×#137—42no—
64C2×C3×Q16—22no—
65C2×C3×SD16—22no—
66C2×C3×M16—22no—
67C2×C3×M16—24no—
68C2×C3×M16—42no—
69C2×C3×M16C2312yes—
70C2×C3×M16C2312yes—
71C22×C3×C8—22no—
72C3×C4⋊Q8—22no—
73C3×#126—24no—
74C3×#126—24no—
75C3×#125—22no—
76C3×#125—22no—
77C3×#124—22no—
78C3×#124—24no—
79C3×#123—24no—
80C3×#123—22no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8D412yes—
86C3×C22⋊Q8D412yes—
87C3×#121—24no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#121D412yes—
91C3×#121D412yes—
92C3×#120—22no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—24no—
98C3×C4×D4—22no—
99C3×C4×D4—22no—
100C3×#117—24no—
101C3×#117—22no—
102C3×#117—22no—
103C3×#117—22no—
104C3×#117C2312yes—
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4D412yes—
114C2×C3×C42D412yes—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#106—22no—
119C3×#104—24no—
120C3×#104—42no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#102—22no—
125C3×#95—21no—
126C3×#95—22no—
127C3×#95—22no—
128C3×#95—22no—
129C2×#137C22×C312yes—
130C22×M16C22×C312yes—
131#556—22no—
132#555—22no—
133#553—22no—
134#552—22no—
135#544C22×C311yescharacteristic
136#538—22no—
137#537C22×C311yescharacteristic
138#529—22no—
139#525C22×C312yes—
140C2×C22⋊Q8C22×C311yescharacteristic
141#520C22×C312yes—
142C2×#117C22×C311yescharacteristic
143#470—22no—
144#468—22no—
145#467—22no—
146#465—22no—
147#428C22×C312yes—
148C2×#104C22×C312yes—
149C2×#104C22×C312yes—
150#419C22×C312yes—
151#417C22×C312yes—
152#397—22no—
153#391C22×C311yescharacteristic
154#391C22×C311yescharacteristic
155#387—22no—
156#384C22×C311yescharacteristic
157#382C22×C311yescharacteristic
158C4×C22⋊C4C22×C311yescharacteristic
159#343C22×C311yescharacteristic
160#343C22×C312yes—
161#343C22×C311yescharacteristic
162#339C22×C312yes—
163#339C22×C312yes—
164C24×C3C2×D411yescharacteristic
165C3×#41—22no—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8C2×D412yes—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4C2×D412yes—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—24no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2411yescharacteristic
195C3×Q16—44no—
196C3×SD16—44no—
197C3×M16—22no—
198C3×M16—22no—
199C3×M16—22no—
200C3×M16—44no—
201C3×M16#4114yes—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—44no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—44no—
215C3×C4⋊C4—41no—
216C3×C4⋊C4—41no—
217C3×C4⋊C4C2×D412yes—
218C3×C4⋊C4C2×D412yes—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C42—24no—
230C3×C42—22no—
231C3×C42—22no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42—21no—
235C3×C42—21no—
236C2×#41C23×C312yes—
237C22×Q8C23×C311yescharacteristic
238C23×C4C23×C311yescharacteristic
239#137—24no—
240#137—42no—
241C2×Q16—22no—
242C2×SD16—22no—
243C2×M16—22no—
244C2×M16—24no—
245C2×M16—42no—
246C2×M16C23×C312yes—
247C2×M16C23×C312yes—
248C22×C8—22no—
249C4⋊Q8—22no—
250#126—24no—
251#126—24no—
252#125—22no—
253#125—22no—
254#124—22no—
255#124—24no—
256#123—24no—
257#123—22no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C22⋊Q8C3×D412yes—
263C22⋊Q8C3×D412yes—
264#121—24no—
265#121—22no—
266#121—22no—
267#121C3×D412yes—
268#121C3×D412yes—
269#120—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—22no—
277#117—24no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117C23×C312yes—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4C3×D412yes—
291C2×C42C3×D412yes—
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C311yescharacteristic
294#107—22no—
295#106—22no—
296#104—24no—
297#104—42no—
298#103—22no—
299#103—22no—
300#103—22no—
301#102—22no—
302#95—21no—
303#95—22no—
304#95—22no—
305#95—22no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—22no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3#12012yes—
312C23×C3C22×D411yescharacteristic
313C3×Q8—22no—
314C3×Q8—22no—
315C3×Q8—42no—
316C3×Q8—42no—
317C3×Q8—82no—
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—82no—
323C3×D4—82no—
324C3×D4—82no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—84no—
352C2×C3×C4#12011yescharacteristic
353C2×C3×C4#12011yescharacteristic
354C2×C3×C4C22×D412yes—
355C2×C3×C4C2×#4112yes—
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358C3×C8—24no—
359C3×C8—42no—
360C24C2×C3×D411yescharacteristic
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8C2×C3×D412yes—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4C2×C3×D412yes—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D412yes—
387C22×C4C2×C3×D412yes—
388C22×C4C2×C3×D412yes—
389C22×C4C2×C3×D412yes—
390C22×C4C24×C311yescharacteristic
391Q16—44no—
392SD16—44no—
393M16—22no—
394M16—22no—
395M16—22no—
396M16—44no—
397M16C3×#4114yes—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—44no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—44no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4C2×C3×D412yes—
414C4⋊C4C2×C3×D412yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C42—24no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—21no—
431C42—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3C2×#12011yescharacteristic
443C22×C3#53211yescharacteristic
444C22×C3#53711yescharacteristic
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—42no—
448C3×C4—42no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4—81no—
454C3×C4—81no—
455C3×C4—81no—
456C3×C4#54612yes—
457C3×C4#54712yes—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23C3×#12012yes—
464C23C22×C3×D411yescharacteristic
465Q8—22no—
466Q8—22no—
467Q8—42no—
468Q8—42no—
469Q8—82no—
470D4—22no—
471D4—22no—
472D4—42no—
473D4—42no—
474D4—82no—
475D4—82no—
476D4—82no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—84no—
504C2×C4C3×#12011yescharacteristic
505C2×C4C3×#12011yescharacteristic
506C2×C4C22×C3×D412yes—
507C2×C4C2×C3×#4112yes—
508C2×C4C3×H(5,𝔽2)11yescharacteristic
509C2×C4C3×2−1+411yescharacteristic
510C8—24no—
511C8—42no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—42no—
515C2×C3—82no—
516C2×C3#243011yescharacteristic
517C2×C3#301811yescharacteristic
518C2×C3#302211yescharacteristic
519C22—21no—
520C22—21no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—81no—
528C22—81no—
529C22C2×C3×#12011yescharacteristic
530C22C3×#53211yescharacteristic
531C22C3×#53711yescharacteristic
532C4—21no—
533C4—21no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4C3×#54612yes—
544C4C3×#54712yes—
545C3#2552011yescharacteristic, direct product factor, Sylow
546C2—21no—
547C2—21no—
548C2—42no—
549C2—82no—
550C2C3×#243011yescharacteristic
551C2C3×#301811yescharacteristic
552C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3493C2
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
2. C3×#3056C2semidirect
3. C3×#2499C2semidirect
4. C3×#2061C2semidirect
5. C3×#2040C2semidirect
6. C3×#2028C2semidirect
7. C3×#2007C2semidirect
8. C3×#1882C2semidirect
9. C3×#1871C2semidirect
10. C3×#1859C2semidirect
11. C3×#1742C2semidirect
12. #25520C3direct (N × H)
49. #3056C2×C3semidirect
50. #2499C2×C3semidirect
51. #2061C2×C3semidirect
52. #2040C2×C3semidirect
53. #2028C2×C3semidirect
54. #2007C2×C3semidirect
55. #1882C2×C3semidirect
56. #1871C2×C3semidirect
57. #1859C2×C3semidirect
58. #1742C2×C3semidirect
545. C3#25520direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34932
C3×#30562
C3×#24992
C3×#20612
C3×#20402
C3×#20282
C3×#20072
C3×#18822
C3×#18712
C3×#18592
C3×#17422
#255203