Finite Group Databasebeta

← C3×#25698 #10644305

Subgroups — C3×#25698 #10644305

Order 768.

950 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic
2C3×#3166C211yesmaximal, characteristic, complemented
3C3×#3142C211yesmaximal, characteristic
4C3×#2565C211yesmaximal, characteristic, complemented
5C3×#2038C212yesmaximal, complemented
6C3×#2038C212yesmaximal, complemented
7C3×#1986C211yesmaximal, characteristic, complemented
8C3×#1980C211yesmaximal, characteristic
9C3×#1860C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C3×C4×#101C211yesmaximal, characteristic, complemented
12#25698C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559—21no—
14C3×#557—21no—
15C3×#556—21no—
16C3×#556—22no—
17C3×#554—21no—
18C3×#552—22no—
19C3×#539—22no—
20C3×#536C2212yes—
21C3×#533C2211yescharacteristic
22C3×#532C2211yescharacteristic
23C3×#529—22no—
24C2×C3×#125C2211yescharacteristic
25C3×#520C2212yes—
26C2×C3×C4×Q8C2211yescharacteristic
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×#101C2212yes—
40C3×C4×M16C2211yescharacteristic
41C3×C4×M16C2211yescharacteristic
42C3×#398—22no—
43C3×#391C2211yescharacteristic
44C3×#391C2211yescharacteristic
45C3×#387—22no—
46C3×#384C2211yescharacteristic
47C3×#382C2211yescharacteristic
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×#343C2212yes—
50C3×#339C2212yes—
51C3×#339C2212yes—
52#3507C2×C311yescharacteristic
53#3166C2×C311yescharacteristic, complemented
54#3142C2×C311yescharacteristic
55#2565C2×C311yescharacteristic, complemented
56#2038C2×C312yescomplemented
57#2038C2×C312yescomplemented
58#1986C2×C311yescharacteristic, complemented
59#1980C2×C311yescharacteristic
60#1860C2×C312yescomplemented
61#1859C2×C312yescomplemented
62C4×#101C2×C311yescharacteristic, complemented
63C2×C3×#41C2312yes—
64C22×C3×Q8C2311yescharacteristic
65C2×C3×M16C2312yes—
66C2×C3×M16C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D411yescharacteristic
69C3×C4⋊Q8D411yescharacteristic
70C3×C4⋊Q8D411yescharacteristic
71C3×#127D411yescharacteristic
72C3×#126—24no—
73C3×#126—24no—
74C3×#125—22no—
75C3×#125—24no—
76C3×#125—21no—
77C3×#125—21no—
78C3×#125—21no—
79C3×#125—21no—
80C3×#125D412yes—
81C3×#124—22no—
82C3×#124—22no—
83C3×#124D412yes—
84C3×#123—24no—
85C3×#123—24no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×#121—21no—
92C3×#121—21no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8—21no—
96C3×C4×Q8—21no—
97C3×C4×Q8—21no—
98C3×C4×Q8—21no—
99C3×C4×Q8—21no—
100C3×C4×Q8—21no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4—24no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×#117—24no—
107C3×#117—22no—
108C3×#117C2312yes—
109C2×C3×C4⋊C4—22no—
110C2×C3×C4⋊C4—21no—
111C2×C3×C4⋊C4—22no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4D412yes—
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C42D412yes—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2311yescharacteristic
122C3×#107—22no—
123C3×#107—22no—
124C3×#106—22no—
125C3×#106—22no—
126C3×#104—24no—
127C3×#104—24no—
128C3×#103—22no—
129C3×#103—22no—
130C3×#102—22no—
131C3×#102—22no—
132C3×#101—24no—
133C3×#97—21no—
134C3×#97—21no—
135C3×C4×C8—21no—
136C3×C4×C8—21no—
137C3×#95—21no—
138C3×#95—22no—
139C3×#95—22no—
140C3×#95—22no—
141#559—21no—
142#557—21no—
143#556—21no—
144#556—22no—
145#554—21no—
146#552—22no—
147#539—22no—
148#536C22×C312yes—
149#533C22×C311yescharacteristic
150#532C22×C311yescharacteristic
151#529—22no—
152C2×#125C22×C311yescharacteristic
153#520C22×C312yes—
154C2×C4×Q8C22×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×#101C22×C312yes—
168C4×M16C22×C311yescharacteristic
169C4×M16C22×C311yescharacteristic
170#398—22no—
171#391C22×C311yescharacteristic
172#391C22×C311yescharacteristic
173#387—22no—
174#384C22×C311yescharacteristic
175#382C22×C311yescharacteristic
176C4×C4⋊C4C22×C311yescharacteristic
177#343C22×C312yes—
178#339C22×C312yes—
179#339C22×C312yes—
180C3×#41—22no—
181C3×#41—22no—
182C3×#41—22no—
183C3×#41—22no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8C2×D412yes—
191C2×C3×D4—22no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4C2×D412yes—
195C22×C3×C4—21no—
196C22×C3×C4—24no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C3×M16—22no—
207C3×M16—22no—
208C3×M16—22no—
209C3×M16#4114yes—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—44no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4C2×D412yes—
236C3×C4⋊C4C2×D412yes—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C42—24no—
244C3×C42—22no—
245C3×C42—22no—
246C3×C42—22no—
247C3×C42—22no—
248C3×C42—22no—
249C3×C42C2×D411yescharacteristic
250C3×C42C2×D411yescharacteristic
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C2×#41C23×C312yes—
254C22×Q8C23×C311yescharacteristic
255C2×M16C23×C312yes—
256C2×M16C23×C312yes—
257C4⋊Q8—22no—
258C4⋊Q8C3×D411yescharacteristic
259C4⋊Q8C3×D411yescharacteristic
260C4⋊Q8C3×D411yescharacteristic
261#127C3×D411yescharacteristic
262#126—24no—
263#126—24no—
264#125—22no—
265#125—24no—
266#125—21no—
267#125—21no—
268#125—21no—
269#125—21no—
270#125C3×D412yes—
271#124—22no—
272#124—22no—
273#124C3×D412yes—
274#123—24no—
275#123—24no—
276C22⋊Q8—24no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281#121—21no—
282#121—21no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—24no—
294C4×D4—21no—
295C4×D4—21no—
296#117—24no—
297#117—22no—
298#117C23×C312yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C3×D412yes—
305C2×C4⋊C4C23×C311yescharacteristic
306C2×C4⋊C4C23×C311yescharacteristic
307C2×C4⋊C4C23×C311yescharacteristic
308C2×C42C3×D412yes—
309C2×C42C23×C311yescharacteristic
310C2×C42C23×C311yescharacteristic
311C2×C42C23×C311yescharacteristic
312#107—22no—
313#107—22no—
314#106—22no—
315#106—22no—
316#104—24no—
317#104—24no—
318#103—22no—
319#103—22no—
320#102—22no—
321#102—22no—
322#101—24no—
323#97—21no—
324#97—21no—
325C4×C8—21no—
326C4×C8—21no—
327#95—21no—
328#95—22no—
329#95—22no—
330#95—22no—
331C23×C3—22no—
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—42no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×Q8—81no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—81no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—84no—
368C2×C3×C4#12011yescharacteristic
369C2×C3×C4#12011yescharacteristic
370C2×C3×C4#12011yescharacteristic
371C2×C3×C4#12011yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4112yes—
375C2×C3×C42−1+412yes—
376C3×C8—24no—
377C3×C8—42no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8C2×C3×D412yes—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4C2×C3×D412yes—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4C2×C3×D412yes—
400C22×C4C2×C3×D412yes—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C24×C311yescharacteristic
404M16—22no—
405M16—22no—
406M16—22no—
407M16C3×#4114yes—
408C2×C8—22no—
409C2×C8—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4C2×C3×D412yes—
434C4⋊C4C2×C3×D412yes—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C42—24no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42C2×C3×D411yescharacteristic
448C42C2×C3×D411yescharacteristic
449C42C2×C3×D411yescharacteristic
450C42C2×C3×D411yescharacteristic
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#53211yescharacteristic
458C22×C3#53611yescharacteristic
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4—81no—
472C3×C4#54712yes—
473C3×C4#54812yes—
474C23—22no—
475C23C22×C3×D411yescharacteristic
476Q8—22no—
477Q8—22no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—81no—
482D4—22no—
483D4—22no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—81no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—84no—
511C2×C4C3×#12011yescharacteristic
512C2×C4C3×#12011yescharacteristic
513C2×C4C3×#12011yescharacteristic
514C2×C4C3×#12011yescharacteristic
515C2×C4C22×C3×D411yescharacteristic
516C2×C4C22×C3×D411yescharacteristic
517C2×C4C2×C3×#4112yes—
518C2×C4C3×2−1+412yes—
519C8—24no—
520C8—42no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—82no—
524C2×C3#243211yescharacteristic
525C2×C3#301811yescharacteristic
526C2×C3#302011yescharacteristic
527C22—21no—
528C22—21no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22C2×C3×#12011yescharacteristic
533C22C3×#53211yescharacteristic
534C22C3×#53611yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4C3×#54712yes—
549C4C3×#54812yes—
550C3#2569811yescharacteristic, direct product factor, Sylow
551C2—21no—
552C2—21no—
553C2—82no—
554C2C3×#243211yescharacteristic
555C2C3×#301811yescharacteristic
556C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#532C22
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
4. C3×#2565C2semidirect
5. C3×#2038C2semidirect
6. C3×#2038C2semidirect
7. C3×#1986C2semidirect
9. C3×#1860C2semidirect
10. C3×#1859C2semidirect
11. C3×C4×#101C2semidirect
12. #25698C3direct (N × H)
53. #3166C2×C3semidirect
55. #2565C2×C3semidirect
56. #2038C2×C3semidirect
57. #2038C2×C3semidirect
58. #1986C2×C3semidirect
60. #1860C2×C3semidirect
61. #1859C2×C3semidirect
62. C4×#101C2×C3semidirect
550. C3#25698direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35072
C3×#31662
C3×#31422
C3×#25652
C3×#20382
C3×#19862
C3×#19802
C3×#18602
C3×#18592
C3×C4×#1012
#256983