Finite Group Databasebeta

← C3×#27646 #10646253

Subgroups — C3×#27646 #10646253

Order 768.

894 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
1C2×C3×#482C212yesmaximal
2C3×C4×#121C211yesmaximal, characteristic
3C3×#2285C211yesmaximal, characteristic, complemented
4C3×#2085C212yesmaximal, complemented
5C3×#2073C212yesmaximal, complemented
6C3×#1948C211yesmaximal, characteristic, complemented
7C3×#1947C211yesmaximal, characteristic, complemented
8C3×#1825C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic, complemented
10C3×#1802C211yesmaximal, characteristic, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#27646C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×#117C2212yes—
17C22×C3×C42C2211yescharacteristic
18C3×#482—28no—
19C2×C3×#107C2212yes—
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×C22⋊C8C2212yes—
26C3×#390C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#388—21no—
29C3×#386—21no—
30C3×#384C2211yescharacteristic
31C3×#384C2211yescharacteristic
32C3×#382C2212yes—
33C3×#380—21no—
34C3×#380—21no—
35C3×#380C2211yescharacteristic
36C3×C4×C4⋊C4—21no—
37C3×C4×C4⋊C4—21no—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×C4×C22⋊C4—21no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C2×#482C2×C312yes—
46C4×#121C2×C311yescharacteristic
47#2285C2×C311yescharacteristic, complemented
48#2085C2×C312yescomplemented
49#2073C2×C312yescomplemented
50#1948C2×C311yescharacteristic, complemented
51#1947C2×C311yescharacteristic, complemented
52#1825C2×C312yescomplemented
53#1802C2×C311yescharacteristic, complemented
54#1802C2×C311yescharacteristic, complemented
55#1797C2×C311yescharacteristic, complemented
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×C3×C8C2312yes—
63C22×C3×C8C2312yes—
64C3×#121—22no—
65C3×#121D414yes—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×C4×D4—44no—
69C3×#117—24no—
70C3×#117—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42—21no—
90C2×C3×C42—22no—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C3×#107—24no—
96C3×#106—24no—
97C3×C4⋊C8—22no—
98C3×C4⋊C8—22no—
99C3×#102—24no—
100C3×#102—24no—
101C3×C22⋊C8—24no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—22no—
105C3×#95—22no—
106C3×#95—21no—
107C2×#121C22×C311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4C22×C311yescharacteristic
110C2×#117C22×C312yes—
111C22×C42C22×C311yescharacteristic
112#482—28no—
113C2×#107C22×C312yes—
114C2×#106C22×C312yes—
115C2×C4⋊C8C22×C311yescharacteristic
116C2×C4⋊C8C22×C311yescharacteristic
117C2×#102C22×C312yes—
118C2×#102C22×C312yes—
119C2×C22⋊C8C22×C312yes—
120#390C22×C311yescharacteristic
121#389C22×C311yescharacteristic
122#388—21no—
123#386—21no—
124#384C22×C311yescharacteristic
125#384C22×C311yescharacteristic
126#382C22×C312yes—
127#380—21no—
128#380—21no—
129#380C22×C311yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C22×C311yescharacteristic
133C4×C22⋊C4—21no—
134#340C22×C312yes—
135#340C22×C312yes—
136#340C22×C312yes—
137#340C22×C312yes—
138#336C22×C312yes—
139C24×C3—21no—
140C24×C3C2×D411yescharacteristic
141C2×C3×D4—24no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4C2×D412yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—41no—
164C22×C3×C4—41no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4C2×D412yes—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—42no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—21no—
206C3×C42—21no—
207C3×C42—44no—
208C3×C42—42no—
209C3×C42#4112yes—
210C3×C42#4112yes—
211C22×D4—21no—
212C22×D4C23×C311yescharacteristic
213C23×C4—21no—
214C23×C4C3×D411yescharacteristic
215C23×C4C3×D411yescharacteristic
216C23×C4C23×C311yescharacteristic
217C22×C8C23×C312yes—
218C22×C8C23×C312yes—
219#121—22no—
220#121C3×D414yes—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—44no—
224#117—24no—
225#117—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C23×C312yes—
232C2×C4⋊C4C23×C312yes—
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C311yescharacteristic
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42C3×D411yescharacteristic
247C2×C42C3×D411yescharacteristic
248C2×C42C23×C311yescharacteristic
249C2×C42C23×C311yescharacteristic
250#107—24no—
251#106—24no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254#102—24no—
255#102—24no—
256C22⋊C8—24no—
257#95—21no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—22no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3#12112yes—
276C23×C3C22×D411yescharacteristic
277C3×D4—22no—
278C3×D4—22no—
279C3×D4—48no—
280C3×D4—41no—
281C3×D4—41no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—44no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—44no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—84no—
314C2×C3×C4#12112yes—
315C2×C3×C4#12312yes—
316C2×C3×C4#12312yes—
317C2×C3×C4#13512yes—
318C2×C3×C4AGL(1,ℤ/8)12yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C2×C3×C42−1+411yescharacteristic
325C3×C8—42no—
326C3×C8—42no—
327C24—21no—
328C24C2×C3×D411yescharacteristic
329C2×D4—24no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4C2×C3×D412yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C3×#4112yes—
363C22×C4C24×C311yescharacteristic
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4C2×C3×D412yes—
385C22⋊C4—24no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—42no—
391C42—22no—
392C42—22no—
393C42—21no—
394C42—21no—
395C42—44no—
396C42—42no—
397C42C3×#4112yes—
398C42C3×#4112yes—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3C2×#12111yescharacteristic
414C22×C3C2×#12311yescharacteristic
415C22×C3#53611yescharacteristic
416C22×C3C2×#13511yescharacteristic
417C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
418C22×C3#57511yescharacteristic
419C22×C3#57511yescharacteristic
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—42no—
425C3×C4—81no—
426C3×C4—81no—
427C3×C4—82no—
428C3×C4—82no—
429C3×C4#48214yescomplemented
430C3×C4#54711yescharacteristic
431C3×C4#54711yescharacteristic
432C3×C4#55211yescharacteristic
433C3×C4#55211yescharacteristic
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23C3×#12112yes—
448C23C22×C3×D411yescharacteristic
449D4—22no—
450D4—22no—
451D4—48no—
452D4—41no—
453D4—41no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—84no—
486C2×C4C3×#12112yes—
487C2×C4C3×#12312yes—
488C2×C4C3×#12312yes—
489C2×C4C3×#13512yes—
490C2×C4C3×AGL(1,ℤ/8)12yes—
491C2×C4C22×C3×D411yescharacteristic
492C2×C4C2×C3×#4111yescharacteristic
493C2×C4C2×C3×#4111yescharacteristic
494C2×C4C2×C3×#4111yescharacteristic
495C2×C4C3×2−1+411yescharacteristic
496C2×C4C3×2−1+411yescharacteristic
497C8—42no—
498C8—42no—
499C2×C3—42no—
500C2×C3—82no—
501C2×C3#248611yescharacteristic
502C2×C3#303511yescharacteristic
503C2×C3#303611yescharacteristic
504C2×C3C2×#48211yescharacteristic
505C2×C3#309311yescharacteristic
506C2×C3#323311yescharacteristic
507C2×C3#323511yescharacteristic
508C22—22no—
509C22—22no—
510C22—22no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22C2×C3×#12111yescharacteristic
523C22C2×C3×#12311yescharacteristic
524C22C3×#53611yescharacteristic
525C22C2×C3×#13511yescharacteristic
526C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
527C22C3×#57511yescharacteristic
528C22C3×#57511yescharacteristic
529C4—22no—
530C4—22no—
531C4—42no—
532C4—41no—
533C4—41no—
534C4—81no—
535C4—82no—
536C4—82no—
537C4—81no—
538C4C3×#48214yescomplemented
539C4C3×#54711yescharacteristic
540C4C3×#54711yescharacteristic
541C4C3×#55211yescharacteristic
542C4C3×#55211yescharacteristic
543C3#2764611yescharacteristic, direct product factor, Sylow
544C2—42no—
545C2—82no—
546C2C3×#248611yescharacteristic
547C2C3×#303511yescharacteristic
548C2C3×#303611yescharacteristic
549C2C2×C3×#48211yescharacteristic
550C2C3×#309311yescharacteristic
551C2C3×#323311yescharacteristic
552C2C3×#323511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2285C2semidirect
4. C3×#2085C2semidirect
5. C3×#2073C2semidirect
6. C3×#1948C2semidirect
7. C3×#1947C2semidirect
8. C3×#1825C2semidirect
9. C3×#1802C2semidirect
10. C3×#1802C2semidirect
11. C3×#1797C2semidirect
12. #27646C3direct (N × H)
47. #2285C2×C3semidirect
48. #2085C2×C3semidirect
49. #2073C2×C3semidirect
50. #1948C2×C3semidirect
51. #1947C2×C3semidirect
52. #1825C2×C3semidirect
53. #1802C2×C3semidirect
54. #1802C2×C3semidirect
55. #1797C2×C3semidirect
429. C3×C4#482semidirect
538. C4C3×#482semidirect
543. C3#27646direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4822
C3×C4×#1212
C3×#22852
C3×#20852
C3×#20732
C3×#19482
C3×#19472
C3×#18252
C3×#18022
C3×#17972
#276463