Finite Group Databasebeta

← C3×#25716 #10644323

Subgroups — C3×#25716 #10644323

Order 768.

1058 conjugacy classes of subgroups.

The subgroup lattice has 544 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×#477C211yesmaximal, characteristic, complemented
4C3×#2547C211yesmaximal, characteristic
5C3×#2036C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic
9C3×#1854C212yesmaximal, complemented
10C3×#1852C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25716C311yesmaximal, 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×#124—21no—
20C2×C3×#123—21no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117—21no—
26C2×C3×#117C2211yescharacteristic
27C22×C3×C4⋊C4C2211yescharacteristic
28C3×#501—24no—
29C3×#477—24no—
30C2×C3×#107C2212yes—
31C2×C3×#107C2212yes—
32C2×C3×#106C2212yes—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C2×C3×#97C2211yescharacteristic
39C3×#397—21no—
40C3×#397—21no—
41C3×#391—21no—
42C3×#380C2212yes—
43C3×#380C2212yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#340C2212yes—
48C2×#552C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#477C2×C311yescharacteristic, complemented
51#2547C2×C311yescharacteristic
52#2036C2×C312yescomplemented
53#2035C2×C312yescomplemented
54#1989C2×C311yescharacteristic, complemented
55#1955C2×C311yescharacteristic
56#1854C2×C312yescomplemented
57#1852C2×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×#124—44no—
74C3×#123—28no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8D414yes—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×#117—28no—
84C3×#117—22no—
85C3×#117—22no—
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—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42C2311yescharacteristic
112C3×#107—24no—
113C3×#107—24no—
114C3×#106—24no—
115C3×C4⋊C8—22no—
116C3×#103—24no—
117C3×#103—24no—
118C3×C22⋊C8—24no—
119C3×C22⋊C8—24no—
120C3×#97—22no—
121C3×#95—21no—
122C3×#95—22no—
123C3×#95—22no—
124#552—24no—
125#552—24no—
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#126—21no—
128C2×#125—21no—
129C2×#125C22×C311yescharacteristic
130C2×#124—21no—
131C2×#123—21no—
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×C22⋊Q8C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×D4C22×C311yescharacteristic
136C2×#117—21no—
137C2×#117C22×C311yescharacteristic
138C22×C4⋊C4C22×C311yescharacteristic
139#501—24no—
140#477—24no—
141C2×#107C22×C312yes—
142C2×#107C22×C312yes—
143C2×#106C22×C312yes—
144C2×C4⋊C8C22×C311yescharacteristic
145C2×#103C22×C312yes—
146C2×#103C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148C2×C22⋊C8C22×C312yes—
149C2×#97C22×C311yescharacteristic
150#397—21no—
151#397—21no—
152#391—21no—
153#380C22×C312yes—
154#380C22×C312yes—
155C4×C4⋊C4C22×C311yescharacteristic
156#340C22×C312yes—
157#340C22×C312yes—
158#340C22×C312yes—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—42no—
165C2×C3×Q8—42no—
166C2×C3×Q8—42no—
167C2×C3×Q8C2×D412yes—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—41no—
171C22×C3×C4—24no—
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—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—42no—
187C22×C3×C4—42no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8#4114yes—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4C2×D412yes—
215C3×C4⋊C4C2×D412yes—
216C3×C4⋊C4C2×D412yes—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—44no—
223C3×C22⋊C4—44no—
224C3×C42—24no—
225C3×C42—24no—
226C3×C42—42no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229C22×Q8—21no—
230C22×Q8C23×C311yescharacteristic
231C22×D4C3×D411yescharacteristic
232C23×C4—21no—
233C23×C4C23×C311yescharacteristic
234C23×C4C23×C311yescharacteristic
235C22×C8C23×C312yes—
236C22×C8C23×C312yes—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239#126—28no—
240#125—28no—
241#125—22no—
242#125C3×D414yes—
243#124—44no—
244#123—28no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C4×Q8—22no—
250C4×Q8C3×D414yes—
251C4×D4—22no—
252C4×D4—22no—
253#117—28no—
254#117—22no—
255#117—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C312yes—
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4C3×D411yescharacteristic
277C2×C22⋊C4C3×D411yescharacteristic
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42C23×C311yescharacteristic
282#107—24no—
283#107—24no—
284#106—24no—
285C4⋊C8—22no—
286#103—24no—
287#103—24no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290#97—22no—
291#95—21no—
292#95—22no—
293#95—22no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—22no—
303C23×C3AGL(1,ℤ/8)12yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—42no—
308C3×Q8—84no—
309C3×D4—44no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—48no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4#12012yes—
341C2×C3×C4#12012yes—
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C2×#4112yes—
345C2×C3×C42−1+412yes—
346C3×C8—24no—
347C3×C8—42no—
348C24C2×C3×D411yescharacteristic
349C24C2×C3×D411yescharacteristic
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8C2×C3×D412yes—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—41no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C24×C311yescharacteristic
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8C3×#4114yes—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4C2×C3×D412yes—
404C4⋊C4C2×C3×D412yes—
405C4⋊C4C2×C3×D412yes—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C42—24no—
414C42—24no—
415C42—42no—
416C42C2×C3×D412yes—
417C42C2×C3×D412yes—
418C22×C3—24no—
419C22×C3—22no—
420C22×C3—42no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3C2×#12011yescharacteristic
429C22×C3#54712yes—
430C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C3#57511yescharacteristic
432C22×C3#57812yes—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—42no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—82no—
442C3×C4—82no—
443C3×C4#53211yescharacteristic
444C3×C4#53611yescharacteristic
445C3×C4#54812yes—
446C23—21no—
447C23—22no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—22no—
453C23—21no—
454C23—22no—
455C23C3×AGL(1,ℤ/8)12yes—
456C23C22×C3×D411yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—42no—
460Q8—84no—
461D4—44no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—48no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4C3×#12012yes—
493C2×C4C3×#12012yes—
494C2×C4C22×C3×D411yescharacteristic
495C2×C4C22×C3×D411yescharacteristic
496C2×C4C2×C3×#4112yes—
497C2×C4C3×2−1+412yes—
498C8—24no—
499C8—42no—
500C2×C3—42no—
501C2×C3—81no—
502C2×C3#243211yescharacteristic
503C2×C3#300611yescharacteristic
504C2×C3#301111yescharacteristic
505C2×C3#334312yes—
506C2×C3#334912yes—
507C22—22no—
508C22—24no—
509C22—41no—
510C22—41no—
511C22—42no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22C2×C3×#12011yescharacteristic
518C22C3×#54712yes—
519C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
520C22C3×#57511yescharacteristic
521C22C3×#57812yes—
522C4—22no—
523C4—22no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—82no—
531C4—82no—
532C4C3×#53211yescharacteristic
533C4C3×#53611yescharacteristic
534C4C3×#54812yes—
535C3#2571611yescharacteristic, direct product factor, Sylow
536C2—42no—
537C2—81no—
538C2C3×#243211yescharacteristic
539C2C3×#300611yescharacteristic
540C2C3×#301111yescharacteristic
541C2C3×#334312yes—
542C2C3×#334912yes—
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×#477C2semidirect
5. C3×#2036C2semidirect
6. C3×#2035C2semidirect
7. C3×#1989C2semidirect
9. C3×#1854C2semidirect
10. C3×#1852C2semidirect
11. C3×#1763C2semidirect
12. #25716C3direct (N × H)
49. C2×#501C2×C3semidirect
50. C2×#477C2×C3semidirect
52. #2036C2×C3semidirect
53. #2035C2×C3semidirect
54. #1989C2×C3semidirect
56. #1854C2×C3semidirect
57. #1852C2×C3semidirect
58. #1763C2×C3semidirect
535. C3#25716direct (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×#4772
C3×#25472
C3×#20362
C3×#20352
C3×#19892
C3×#19552
C3×#18542
C3×#18522
C3×#17632
#257163