Finite Group Databasebeta

← C3×#25687 #10644294

Subgroups — C3×#25687 #10644294

Order 768.

1156 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×#3508C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3C3×#3079C211yesmaximal, characteristic, complemented
4C3×#2858C211yesmaximal, characteristic, complemented
5C3×#2037C214yesmaximal, complemented
6C3×#1994C211yesmaximal, characteristic, complemented
7C3×#1965C211yesmaximal, characteristic, complemented
8C3×#1856C214yesmaximal, complemented
9C3×C4×#104C211yesmaximal, characteristic, complemented
10#25687C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#559—21no—
12C3×#559—21no—
13C3×#550—21no—
14C3×#550—21no—
15C3×#546—22no—
16C3×#546—22no—
17C3×#540—22no—
18C3×#535C2212yes—
19C3×#533C2211yescharacteristic
20C3×#532—21no—
21C3×#532—21no—
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×#127C2211yescharacteristic
24C2×C3×#124—22no—
25C3×#518C2212yes—
26C3×C4×#41C2211yescharacteristic
27C3×#501—22no—
28C3×#500—21no—
29C3×#498—21no—
30C3×#477—21no—
31C3×#476—21no—
32C3×#475—21no—
33C3×#474—21no—
34C3×#428C2214yes—
35C3×#428C2212yes—
36C3×#423C2211yescharacteristic
37C2×C3×#104C2212yes—
38C3×#417C2212yes—
39C2×C3×#100C2214yes—
40C3×C4×M16C2211yescharacteristic
41C3×#390—22no—
42C3×#390C2214yes—
43C3×C43C2211yescharacteristic
44C3×#339C2214yes—
45C3×#339C2212yes—
46#3508C2×C311yescharacteristic
47#3165C2×C311yescharacteristic, complemented
48#3079C2×C311yescharacteristic, complemented
49#2858C2×C311yescharacteristic, complemented
50#2037C2×C314yescomplemented
51#1994C2×C311yescharacteristic, complemented
52#1965C2×C311yescharacteristic, complemented
53#1856C2×C314yescomplemented
54C4×#104C2×C311yescharacteristic, complemented
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41C2311yescharacteristic
58C22×C3×Q8—21no—
59C22×C3×D4D412yes—
60C22×C3×D4D412yes—
61C22×C3×D4C2312yes—
62C2×C3×M16C2312yes—
63C2×C3×M16C2312yes—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8D411yescharacteristic
69C3×C4⋊Q8D411yescharacteristic
70C3×#127—22no—
71C3×#127—21no—
72C3×#127—21no—
73C3×#127—21no—
74C3×#127—21no—
75C3×#125D412yes—
76C3×#124—22no—
77C3×#124—22no—
78C3×#124—22no—
79C3×#124—48no—
80C3×#123—24no—
81C3×#123—24no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8—21no—
84C3×#121—22no—
85C3×#121—22no—
86C3×#121—24no—
87C3×#121—24no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#120—24no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8D411yescharacteristic
94C3×C4×Q8D411yescharacteristic
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4D411yescharacteristic
104C3×C4×D4D411yescharacteristic
105C3×#117—21no—
106C3×#117C2311yescharacteristic
107C3×#117C2312yes—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C22⋊C4—24no—
111C2×C3×C22⋊C4—28no—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42—21no—
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#107—24no—
119C3×#106—22no—
120C3×#106—24no—
121C3×C4⋊C8—21no—
122C3×C4⋊C8—21no—
123C3×#104—24no—
124C3×#103—22no—
125C3×#102—22no—
126C3×#100—28no—
127C3×#97—21no—
128C3×C4×C8—21no—
129#559—21no—
130#559—21no—
131#550—21no—
132#550—21no—
133#546—22no—
134#546—22no—
135#540—22no—
136#535C22×C312yes—
137#533C22×C311yescharacteristic
138#532—21no—
139#532—21no—
140C2×C4⋊Q8C22×C311yescharacteristic
141C2×#127C22×C311yescharacteristic
142C2×#124—22no—
143#518C22×C312yes—
144C4×#41C22×C311yescharacteristic
145#501—22no—
146#500—21no—
147#498—21no—
148#477—21no—
149#476—21no—
150#475—21no—
151#474—21no—
152#428C22×C312yes—
153#428C22×C314yes—
154#423C22×C311yescharacteristic
155C2×#104C22×C312yes—
156#417C22×C312yes—
157C2×#100C22×C314yes—
158C4×M16C22×C311yescharacteristic
159#390—22no—
160#390C22×C314yes—
161C43C22×C311yescharacteristic
162#339C22×C312yes—
163#339C22×C314yes—
164C24×C3C2×D414yes—
165C3×#41—22no—
166C3×#41—22no—
167C3×#41—84no—
168C3×#41—84no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—44no—
172C2×C3×Q8—41no—
173C2×C3×Q8—41no—
174C2×C3×Q8C2×D411yescharacteristic
175C2×C3×D4—21no—
176C2×C3×D4—24no—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—24no—
185C2×C3×D4—24no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4C2×D411yescharacteristic
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C3×M16—22no—
200C3×M16—24no—
201C3×M16#4114yescomplemented
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—41no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4C2×D411yescharacteristic
215C3×C4⋊C4C2×D412yes—
216C3×C4⋊C4C2×D411yescharacteristic
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—416no—
227C3×C42—24no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—22no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42—42no—
235C3×C42C2×D411yescharacteristic
236C3×C42C2×D411yescharacteristic
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C2×#41—22no—
240C2×#41—22no—
241C2×#41C23×C311yescharacteristic
242C22×Q8—21no—
243C22×D4C3×D412yes—
244C22×D4C3×D412yes—
245C22×D4C23×C312yes—
246C2×M16C23×C312yes—
247C2×M16C23×C312yes—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8C3×D411yescharacteristic
253C4⋊Q8C3×D411yescharacteristic
254#127—22no—
255#127—21no—
256#127—21no—
257#127—21no—
258#127—21no—
259#125C3×D412yes—
260#124—22no—
261#124—22no—
262#124—22no—
263#124—48no—
264#123—24no—
265#123—24no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—22no—
269#121—22no—
270#121—24no—
271#121—24no—
272#121—22no—
273#121—22no—
274#120—24no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8C3×D411yescharacteristic
278C4×Q8C3×D411yescharacteristic
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4C3×D411yescharacteristic
288C4×D4C3×D411yescharacteristic
289#117—21no—
290#117C23×C311yescharacteristic
291#117C23×C312yes—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C23×C312yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42C23×C312yes—
300C2×C42C23×C311yescharacteristic
301#107—22no—
302#107—24no—
303#106—22no—
304#106—24no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307#104—24no—
308#103—22no—
309#102—22no—
310#100—28no—
311#97—21no—
312C4×C8—21no—
313C23×C3—24no—
314C23×C3—21no—
315C23×C3—28no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3C22×D411yescharacteristic
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×Q8—82no—
324C3×Q8—82no—
325C3×D4—21no—
326C3×D4—21no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—44no—
331C3×D4—44no—
332C3×D4—44no—
333C3×D4—82no—
334C3×D4—82no—
335C3×D4—84no—
336C3×D4—84no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—44no—
351C2×C3×C4—42no—
352C2×C3×C4—84no—
353C2×C3×C4—84no—
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4#12011yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C2×#4112yes—
361C2×C3×C42−1+412yes—
362C3×C8—24no—
363C3×C8—42no—
364C24C2×C3×D414yes—
365#41—22no—
366#41—22no—
367#41—84no—
368#41—84no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—44no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8C2×C3×D411yescharacteristic
375C2×D4—21no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4C2×C3×D411yescharacteristic
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C24×C311yescharacteristic
399M16—22no—
400M16—24no—
401M16C3×#4114yescomplemented
402C2×C8—22no—
403C2×C8—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4C2×C3×D411yescharacteristic
415C4⋊C4C2×C3×D412yes—
416C4⋊C4C2×C3×D411yescharacteristic
417C4⋊C4C2×C3×D412yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—21no—
423C22⋊C4—22no—
424C22⋊C4—21no—
425C22⋊C4—21no—
426C22⋊C4—416no—
427C42—24no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—42no—
435C42C2×C3×D411yescharacteristic
436C42C2×C3×D411yescharacteristic
437C42C2×C3×D411yescharacteristic
438C42C2×C3×D411yescharacteristic
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—41no—
442C22×C3—44no—
443C22×C3—41no—
444C22×C3—44no—
445C22×C3—41no—
446C22×C3—44no—
447C22×C3—88no—
448C22×C3C2×#12011yescharacteristic
449C22×C3#54712yes—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—44no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—42no—
456C3×C4—41no—
457C3×C4—42no—
458C3×C4—82no—
459C3×C4#45312yes—
460C3×C4#45512yes—
461C3×C4#53211yescharacteristic
462C3×C4#53611yescharacteristic
463C3×C4#54812yes—
464C23—24no—
465C23—21no—
466C23—28no—
467C23—44no—
468C23—44no—
469C23C22×C3×D411yescharacteristic
470Q8—21no—
471Q8—21no—
472Q8—42no—
473Q8—42no—
474Q8—82no—
475Q8—82no—
476D4—21no—
477D4—21no—
478D4—42no—
479D4—44no—
480D4—44no—
481D4—42no—
482D4—42no—
483D4—44no—
484D4—82no—
485D4—82no—
486D4—84no—
487D4—84no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4C3×#12011yescharacteristic
506C2×C4C3×#12011yescharacteristic
507C2×C4C3×#12011yescharacteristic
508C2×C4C3×#12011yescharacteristic
509C2×C4C22×C3×D411yescharacteristic
510C2×C4C22×C3×D411yescharacteristic
511C2×C4C2×C3×#4112yes—
512C2×C4C3×2−1+412yes—
513C8—24no—
514C8—42no—
515C2×C3—22no—
516C2×C3—81no—
517C2×C3—84no—
518C2×C3#243211yescharacteristic
519C2×C3C2×#45311yescharacteristic
520C2×C3C2×#45511yescharacteristic
521C22—21no—
522C22—21no—
523C22—41no—
524C22—44no—
525C22—41no—
526C22—44no—
527C22—41no—
528C22—44no—
529C22—88no—
530C22C2×C3×#12011yescharacteristic
531C22C3×#54712yes—
532C4—21no—
533C4—21no—
534C4—44no—
535C4—41no—
536C4—41no—
537C4—42no—
538C4—41no—
539C4—42no—
540C4—82no—
541C4C3×#45312yes—
542C4C3×#45512yes—
543C4C3×#53211yescharacteristic
544C4C3×#53611yescharacteristic
545C4C3×#54812yes—
546C3#2568711yescharacteristic, direct product factor, Sylow
547C2—22no—
548C2—84no—
549C2—81no—
550C2C3×#243211yescharacteristic
551C2C2×C3×#45311yescharacteristic
552C2C2×C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3508C2
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×#3165C2semidirect
3. C3×#3079C2semidirect
4. C3×#2858C2semidirect
5. C3×#2037C2semidirect
6. C3×#1994C2semidirect
7. C3×#1965C2semidirect
8. C3×#1856C2semidirect
9. C3×C4×#104C2semidirect
10. #25687C3direct (N × H)
47. #3165C2×C3semidirect
48. #3079C2×C3semidirect
49. #2858C2×C3semidirect
50. #2037C2×C3semidirect
51. #1994C2×C3semidirect
52. #1965C2×C3semidirect
53. #1856C2×C3semidirect
54. C4×#104C2×C3semidirect
201. C3×M16#41semidirect
401. M16C3×#41semidirect
546. C3#25687direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35082
C3×#31652
C3×#30792
C3×#28582
C3×#20372
C3×#19942
C3×#19652
C3×#18562
C3×C4×#1042
#256873