Finite Group Databasebeta

← C3×#25666 #10644273

Subgroups — C3×#25666 #10644273

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2C3×#3141C211yesmaximal, characteristic, complemented
3C3×#3079C211yesmaximal, characteristic, complemented
4C3×#2868C211yesmaximal, characteristic, complemented
5C3×#2039C214yesmaximal, complemented
6C3×#1997C211yesmaximal, characteristic, complemented
7C3×#1962C211yesmaximal, characteristic, complemented
8C3×#1859C214yesmaximal, complemented
9C3×C4×#104C211yesmaximal, characteristic, complemented
10#25666C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#556—21no—
12C3×#555—21no—
13C3×#554—21no—
14C3×#553—22no—
15C3×#548—22no—
16C3×#547—21no—
17C3×#542—22no—
18C3×#537C2212yes—
19C3×#533—21no—
20C3×#532—21no—
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8—22no—
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#520C2212yes—
26C3×C4×#41C2211yescharacteristic
27C3×#489—22no—
28C3×#487—21no—
29C3×#486—21no—
30C3×#477—21no—
31C3×#476—21no—
32C3×#475—21no—
33C3×#474—21no—
34C3×#428C2212yes—
35C3×#423C2211yescharacteristic
36C2×C3×#104C2212yes—
37C2×C3×#104C2214yes—
38C3×#417C2214yes—
39C3×#417C2212yes—
40C3×C4×M16C2211yescharacteristic
41C3×#384—22no—
42C3×#384C2214yes—
43C3×C43C2211yescharacteristic
44C3×#343C2214yes—
45C3×#339C2212yes—
46#3511C2×C311yescharacteristic
47#3141C2×C311yescharacteristic, complemented
48#3079C2×C311yescharacteristic, complemented
49#2868C2×C311yescharacteristic, complemented
50#2039C2×C314yescomplemented
51#1997C2×C311yescharacteristic, complemented
52#1962C2×C311yescharacteristic, complemented
53#1859C2×C314yescomplemented
54C4×#104C2×C311yescharacteristic, complemented
55C2×C3×#41—22no—
56C2×C3×#41C2311yescharacteristic
57C2×C3×#41C2312yes—
58C22×C3×Q8—21no—
59C22×C3×Q8D412yes—
60C2×C3×M16C2312yes—
61C2×C3×M16C2312yes—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—48no—
71C3×C4⋊Q8D411yescharacteristic
72C3×#127D411yescharacteristic
73C3×#126—24no—
74C3×#125—22no—
75C3×#124—22no—
76C3×#124—24no—
77C3×#124D412yes—
78C3×#123—24no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—24no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×#121—22no—
84C3×#121—22no—
85C3×#121—22no—
86C3×#121—21no—
87C3×#121—21no—
88C3×#121—24no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8D411yescharacteristic
92C3×C4×Q8D411yescharacteristic
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—24no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4D411yescharacteristic
101C3×C4×D4D411yescharacteristic
102C3×#117—22no—
103C3×#117—22no—
104C3×#117—21no—
105C3×#117C2311yescharacteristic
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—28no—
109C2×C3×C4⋊C4D412yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42—21no—
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#106—22no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8—21no—
121C3×#104—24no—
122C3×#104—28no—
123C3×#103—22no—
124C3×#103—24no—
125C3×#102—22no—
126C3×#102—24no—
127C3×#97—21no—
128C3×C4×C8—21no—
129#556—21no—
130#555—21no—
131#554—21no—
132#553—22no—
133#548—22no—
134#547—21no—
135#542—22no—
136#537C22×C312yes—
137#533—21no—
138#532—21no—
139#532C22×C311yescharacteristic
140C2×C4⋊Q8—22no—
141C2×C4⋊Q8C22×C311yescharacteristic
142C2×C4⋊Q8C22×C311yescharacteristic
143#520C22×C312yes—
144C4×#41C22×C311yescharacteristic
145#489—22no—
146#487—21no—
147#486—21no—
148#477—21no—
149#476—21no—
150#475—21no—
151#474—21no—
152#428C22×C312yes—
153#423C22×C311yescharacteristic
154C2×#104C22×C314yes—
155C2×#104C22×C312yes—
156#417C22×C312yes—
157#417C22×C314yes—
158C4×M16C22×C311yescharacteristic
159#384—22no—
160#384C22×C314yes—
161C43C22×C311yescharacteristic
162#343C22×C314yes—
163#339C22×C312yes—
164C3×#41—24no—
165C3×#41—22no—
166C3×#41—24no—
167C3×#41—22no—
168C3×#41—84no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—44no—
172C2×C3×Q8—44no—
173C2×C3×Q8—41no—
174C2×C3×Q8—41no—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8C2×D411yescharacteristic
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—24no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4C2×D412yes—
183C2×C3×D4C2×D411yescharacteristic
184C22×C3×C4—22no—
185C22×C3×C4—24no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2411yescharacteristic
195C3×M16—24no—
196C3×M16—22no—
197C3×M16—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—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—416no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4C2×D411yescharacteristic
216C3×C4⋊C4C2×D411yescharacteristic
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C42—24no—
226C3×C42—24no—
227C3×C42—24no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—22no—
232C3×C42C2×D411yescharacteristic
233C3×C42C2×D411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2×D411yescharacteristic
236C3×C42#4114yescomplemented
237C2×#41—22no—
238C2×#41C23×C311yescharacteristic
239C2×#41C23×C312yes—
240C22×Q8—21no—
241C22×Q8C3×D412yes—
242C2×M16C23×C312yes—
243C2×M16C23×C312yes—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—48no—
253C4⋊Q8C3×D411yescharacteristic
254#127C3×D411yescharacteristic
255#126—24no—
256#125—22no—
257#124—22no—
258#124—24no—
259#124C3×D412yes—
260#123—24no—
261C22⋊Q8—22no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265#121—22no—
266#121—22no—
267#121—22no—
268#121—21no—
269#121—21no—
270#121—24no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8C3×D411yescharacteristic
274C4×Q8C3×D411yescharacteristic
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—24no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4C3×D411yescharacteristic
283C4×D4C3×D411yescharacteristic
284#117—22no—
285#117—22no—
286#117—21no—
287#117C23×C311yescharacteristic
288C2×C4⋊C4—22no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C3×D412yes—
292C2×C4⋊C4C23×C312yes—
293C2×C4⋊C4C23×C312yes—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42C23×C312yes—
298C2×C42C23×C311yescharacteristic
299#107—22no—
300#106—22no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303#104—28no—
304#104—24no—
305#103—22no—
306#103—24no—
307#102—22no—
308#102—24no—
309#97—21no—
310C4×C8—21no—
311C23×C3—21no—
312C23×C3—22no—
313C23×C3C22×D411yescharacteristic
314C3×Q8—21no—
315C3×Q8—21no—
316C3×Q8—24no—
317C3×Q8—44no—
318C3×Q8—82no—
319C3×Q8—82no—
320C3×Q8—82no—
321C3×D4—21no—
322C3×D4—21no—
323C3×D4—24no—
324C3×D4—44no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—84no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—24no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—44no—
345C2×C3×C4—44no—
346C2×C3×C4—42no—
347C2×C3×C4—84no—
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4#12011yescharacteristic
350C2×C3×C4#12011yescharacteristic
351C2×C3×C4#12011yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C2×#4112yes—
355C2×C3×C42−1+412yes—
356C3×C8—42no—
357C3×C8—42no—
358#41—24no—
359#41—22no—
360#41—24no—
361#41—22no—
362#41—84no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8C2×C3×D412yes—
370C2×Q8C2×C3×D411yescharacteristic
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4C2×C3×D412yes—
377C2×D4C2×C3×D411yescharacteristic
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D411yescharacteristic
387C22×C4C2×C3×D414yes—
388C22×C4C24×C311yescharacteristic
389M16—24no—
390M16—22no—
391M16—22no—
392C2×C8—22no—
393C2×C8—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—416no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4C2×C3×D411yescharacteristic
410C4⋊C4C2×C3×D411yescharacteristic
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—22no—
426C42C2×C3×D411yescharacteristic
427C42C2×C3×D411yescharacteristic
428C42C2×C3×D411yescharacteristic
429C42C2×C3×D411yescharacteristic
430C42C3×#4114yescomplemented
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—42no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—42no—
439C22×C3C2×#12011yescharacteristic
440C22×C3#54812yes—
441C3×C4—28no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—42no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—44no—
448C3×C4—41no—
449C3×C4—82no—
450C3×C4#45312yes—
451C3×C4#45612yes—
452C3×C4#53211yescharacteristic
453C3×C4#53611yescharacteristic
454C3×C4#54712yes—
455C23—21no—
456C23—22no—
457C23C22×C3×D411yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—24no—
461Q8—44no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465D4—21no—
466D4—21no—
467D4—24no—
468D4—44no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—84no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—84no—
492C2×C4C3×#12011yescharacteristic
493C2×C4C3×#12011yescharacteristic
494C2×C4C3×#12011yescharacteristic
495C2×C4C3×#12011yescharacteristic
496C2×C4C22×C3×D411yescharacteristic
497C2×C4C22×C3×D411yescharacteristic
498C2×C4C2×C3×#4112yes—
499C2×C4C3×2−1+412yes—
500C8—42no—
501C8—42no—
502C2×C3—22no—
503C2×C3—81no—
504C2×C3—82no—
505C2×C3#243211yescharacteristic
506C2×C3C2×#45311yescharacteristic
507C2×C3C2×#45611yescharacteristic
508C22—21no—
509C22—21no—
510C22—42no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—41no—
515C22—42no—
516C22C2×C3×#12011yescharacteristic
517C22C3×#54812yes—
518C4—28no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—41no—
523C4—41no—
524C4—44no—
525C4—41no—
526C4—82no—
527C4C3×#45312yes—
528C4C3×#45612yes—
529C4C3×#53211yescharacteristic
530C4C3×#53611yescharacteristic
531C4C3×#54712yes—
532C3#2566611yescharacteristic, direct product factor, Sylow
533C2—22no—
534C2—81no—
535C2—82no—
536C2C3×#243211yescharacteristic
537C2C2×C3×#45311yescharacteristic
538C2C2×C3×#45611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3511C2
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×#3141C2semidirect
3. C3×#3079C2semidirect
4. C3×#2868C2semidirect
5. C3×#2039C2semidirect
6. C3×#1997C2semidirect
7. C3×#1962C2semidirect
8. C3×#1859C2semidirect
9. C3×C4×#104C2semidirect
10. #25666C3direct (N × H)
47. #3141C2×C3semidirect
48. #3079C2×C3semidirect
49. #2868C2×C3semidirect
50. #2039C2×C3semidirect
51. #1997C2×C3semidirect
52. #1962C2×C3semidirect
53. #1859C2×C3semidirect
54. C4×#104C2×C3semidirect
236. C3×C42#41semidirect
430. C42C3×#41semidirect
532. C3#25666direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35112
C3×#31412
C3×#30792
C3×#28682
C3×#20392
C3×#19972
C3×#19622
C3×#18592
C3×C4×#1042
#256663