Finite Group Databasebeta

← C3×#25715 #10644322

Subgroups — C3×#25715 #10644322

Order 768.

1038 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
1C2×C3×#552C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#475C211yesmaximal, characteristic, complemented
4C3×#2563C211yesmaximal, characteristic
5C3×#2036C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic
9C3×#1854C212yesmaximal, complemented
10C3×#1853C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25715C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#552—24no—
14C3×#552—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—21no—
17C2×C3×#125—21no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C2×C3×#117C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#501—24no—
28C3×#475—24no—
29C2×C3×#107C2212yes—
30C2×C3×#106C2212yes—
31C2×C3×#106C2212yes—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C2×C3×#97C2211yescharacteristic
38C3×#393—21no—
39C3×#393—21no—
40C3×#390—21no—
41C3×#382—21no—
42C3×#380C2212yes—
43C3×#380C2212yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#340C2212yes—
48C2×#552C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#475C2×C311yescharacteristic, complemented
51#2563C2×C311yescharacteristic
52#2036C2×C312yescomplemented
53#2035C2×C312yescomplemented
54#1989C2×C311yescharacteristic, complemented
55#1955C2×C311yescharacteristic
56#1854C2×C312yescomplemented
57#1853C2×C312yescomplemented
58#1763C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4D411yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×#126—28no—
70C3×#125—28no—
71C3×#125—22no—
72C3×#125D414yes—
73C3×#123—28no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8D414yes—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×#117—28no—
83C3×#117—22no—
84C3×#117—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C42—21no—
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110C2×C3×C42C2311yescharacteristic
111C3×#107—24no—
112C3×#106—24no—
113C3×#106—24no—
114C3×C4⋊C8—22no—
115C3×#103—24no—
116C3×#103—24no—
117C3×C22⋊C8—24no—
118C3×C22⋊C8—24no—
119C3×#97—22no—
120C3×#95—21no—
121C3×#95—22no—
122C3×#95—22no—
123#552—24no—
124#552—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×#126—21no—
127C2×#125—21no—
128C2×#125C22×C311yescharacteristic
129C2×#123—21no—
130C2×C22⋊Q8C22×C311yescharacteristic
131C2×C22⋊Q8C22×C311yescharacteristic
132C2×C4×Q8C22×C311yescharacteristic
133C2×C4×D4C22×C311yescharacteristic
134C2×#117—21no—
135C2×#117C22×C311yescharacteristic
136C22×C4⋊C4C22×C311yescharacteristic
137#501—24no—
138#475—24no—
139C2×#107C22×C312yes—
140C2×#106C22×C312yes—
141C2×#106C22×C312yes—
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×C22⋊C8C22×C312yes—
146C2×C22⋊C8C22×C312yes—
147C2×#97C22×C311yescharacteristic
148#393—21no—
149#393—21no—
150#390—21no—
151#382—21no—
152#380C22×C312yes—
153#380C22×C312yes—
154C4×C4⋊C4C22×C311yescharacteristic
155#340C22×C312yes—
156#340C22×C312yes—
157#340C22×C312yes—
158C24×C3C2×D411yescharacteristic
159C24×C3C2×D411yescharacteristic
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—42no—
164C2×C3×Q8—82no—
165C2×C3×Q8C2×D412yes—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—41no—
169C22×C3×C4—24no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8#4114yes—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4C2×D412yes—
213C3×C4⋊C4C2×D412yes—
214C3×C4⋊C4C2×D412yes—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—82no—
221C3×C22⋊C4—82no—
222C3×C42—24no—
223C3×C42—24no—
224C3×C42—42no—
225C3×C42C2×D412yes—
226C3×C42C2×D412yes—
227C22×Q8—21no—
228C22×Q8C23×C311yescharacteristic
229C22×D4C3×D411yescharacteristic
230C23×C4—21no—
231C23×C4C23×C311yescharacteristic
232C23×C4C23×C311yescharacteristic
233C22×C8C23×C312yes—
234C22×C8C23×C312yes—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237#126—28no—
238#125—28no—
239#125—22no—
240#125C3×D414yes—
241#123—28no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C4×Q8—22no—
247C4×Q8C3×D414yes—
248C4×D4—22no—
249C4×D4—22no—
250#117—28no—
251#117—22no—
252#117—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C3×D411yescharacteristic
263C2×C4⋊C4C23×C312yes—
264C2×C4⋊C4C23×C312yes—
265C2×C4⋊C4C23×C311yescharacteristic
266C2×C4⋊C4C23×C311yescharacteristic
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4C3×D411yescharacteristic
274C2×C22⋊C4C3×D411yescharacteristic
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42C23×C311yescharacteristic
279#107—24no—
280#106—24no—
281#106—24no—
282C4⋊C8—22no—
283#103—24no—
284#103—24no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287#97—22no—
288#95—21no—
289#95—22no—
290#95—22no—
291C23×C3—21no—
292C23×C3—22no—
293C23×C3—21no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3#13712yes—
301C23×C3C22×D411yescharacteristic
302C3×Q8—22no—
303C3×Q8—22no—
304C3×Q8—42no—
305C3×Q8—84no—
306C3×D4—44no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—48no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4#12012yes—
338C2×C3×C4#12012yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C2×#4112yes—
342C2×C3×C42−1+412yes—
343C3×C8—24no—
344C3×C8—42no—
345C24C2×C3×D411yescharacteristic
346C24C2×C3×D411yescharacteristic
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—42no—
351C2×Q8—82no—
352C2×Q8C2×C3×D412yes—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—41no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C2×C3×D411yescharacteristic
376C22×C4C2×C3×D411yescharacteristic
377C22×C4C24×C311yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8C3×#4114yes—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—42no—
399C4⋊C4C2×C3×D412yes—
400C4⋊C4C2×C3×D412yes—
401C4⋊C4C2×C3×D412yes—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C42—24no—
410C42—24no—
411C42—42no—
412C42C2×C3×D412yes—
413C42C2×C3×D412yes—
414C22×C3—24no—
415C22×C3—22no—
416C22×C3—42no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3C2×#12011yescharacteristic
425C22×C3#54712yes—
426C22×C3C2×#13711yescharacteristic
427C22×C3#57511yescharacteristic
428C22×C3#57712yes—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—41no—
432C3×C4—41no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—82no—
438C3×C4—82no—
439C3×C4#53211yescharacteristic
440C3×C4#53611yescharacteristic
441C3×C4#54812yes—
442C23—21no—
443C23—22no—
444C23—21no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—22no—
449C23—21no—
450C23—22no—
451C23C3×#13712yes—
452C23C22×C3×D411yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—42no—
456Q8—84no—
457D4—44no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—48no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4C3×#12012yes—
489C2×C4C3×#12012yes—
490C2×C4C22×C3×D411yescharacteristic
491C2×C4C22×C3×D411yescharacteristic
492C2×C4C2×C3×#4112yes—
493C2×C4C3×2−1+412yes—
494C8—24no—
495C8—42no—
496C2×C3—42no—
497C2×C3—81no—
498C2×C3#243211yescharacteristic
499C2×C3#300711yescharacteristic
500C2×C3#301011yescharacteristic
501C2×C3#334412yes—
502C2×C3#334812yes—
503C22—22no—
504C22—24no—
505C22—41no—
506C22—41no—
507C22—42no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—42no—
512C22—42no—
513C22C2×C3×#12011yescharacteristic
514C22C3×#54712yes—
515C22C2×C3×#13711yescharacteristic
516C22C3×#57511yescharacteristic
517C22C3×#57712yes—
518C4—22no—
519C4—22no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—82no—
527C4—82no—
528C4C3×#53211yescharacteristic
529C4C3×#53611yescharacteristic
530C4C3×#54812yes—
531C3#2571511yescharacteristic, direct product factor, Sylow
532C2—42no—
533C2—81no—
534C2C3×#243211yescharacteristic
535C2C3×#300711yescharacteristic
536C2C3×#301011yescharacteristic
537C2C3×#334412yes—
538C2C3×#334812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
3. C2×C3×#475C2semidirect
5. C3×#2036C2semidirect
6. C3×#2035C2semidirect
7. C3×#1989C2semidirect
9. C3×#1854C2semidirect
10. C3×#1853C2semidirect
11. C3×#1763C2semidirect
12. #25715C3direct (N × H)
49. C2×#501C2×C3semidirect
50. C2×#475C2×C3semidirect
52. #2036C2×C3semidirect
53. #2035C2×C3semidirect
54. #1989C2×C3semidirect
56. #1854C2×C3semidirect
57. #1853C2×C3semidirect
58. #1763C2×C3semidirect
531. C3#25715direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5522
C2×C3×#5012
C2×C3×#4752
C3×#25632
C3×#20362
C3×#20352
C3×#19892
C3×#19552
C3×#18542
C3×#18532
C3×#17632
#257153