Finite Group Databasebeta

← C3×#29742 #10648349

Subgroups — C3×#29742 #10648349

Order 768.

1246 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#469C211yesmaximal, characteristic, complemented
2C2×C3×#407C211yesmaximal, characteristic, complemented
3C3×#2847C212yesmaximal, complemented
4C3×#2077C212yesmaximal, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1999C212yesmaximal, complemented
8C3×#1895C212yesmaximal, complemented
9C3×#1809C211yesmaximal, characteristic, complemented
10#29742C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×D8C2211yescharacteristic, complemented
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×C3×#121C2212yes—
17C2×C3×#121C2212yes—
18C2×C3×#120—22no—
19C3×#469—24no—
20C2×C3×#107C2211yescharacteristic, complemented
21C2×C3×#106C2211yescharacteristic, complemented
22C2×C3×#106C2211yescharacteristic, complemented
23C2×C3×#102C2212yescomplemented
24C2×C3×#102C2212yescomplemented
25C2×C3×#102C2212yescomplemented
26C3×#407—24no—
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C3×#386—22no—
30C3×#386—22no—
31C3×#386—22no—
32C3×#380C2212yes—
33C3×#380C2212yes—
34C3×#380C2212yes—
35C3×#340C2212yescomplemented
36C3×#340C2212yescomplemented
37C3×#340C2212yescomplemented
38C2×#469C2×C311yescharacteristic, complemented
39C2×#407C2×C311yescharacteristic, complemented
40#2847C2×C312yescomplemented
41#2077C2×C312yescomplemented
42#2034C2×C312yescomplemented
43#2033C2×C312yescomplemented
44#1999C2×C312yescomplemented
45#1895C2×C312yescomplemented
46#1809C2×C311yescharacteristic, complemented
47C25×C3D411yescharacteristic
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4C2312yes—
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—41no—
55C23×C3×C4—41no—
56C23×C3×C4—41no—
57C23×C3×C4—41no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D412yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C2×C3×D8—22no—
63C2×C3×D8—41no—
64C2×C3×D8—41no—
65C2×C3×M16—22no—
66C2×C3×M16—41no—
67C2×C3×M16—41no—
68C2×C3×M16—41no—
69C2×C3×M16—41no—
70C2×C3×M16D414yescomplemented
71C22×C3×C8C2312yes—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×#121—24no—
75C3×#121—24no—
76C3×#121—42no—
77C3×#121—42no—
78C3×#121—42no—
79C3×#121—42no—
80C3×#120—48no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C3×#107—41no—
91C3×#106—22no—
92C3×#106—41no—
93C3×#102—24no—
94C3×#102—42no—
95C3×#102—42no—
96C3×C22⋊C8—24no—
97C3×C22⋊C8—24no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C24×C4C22×C311yescharacteristic
102C22×D8C22×C311yescharacteristic, complemented
103C22×M16C22×C311yescharacteristic
104C22×M16C22×C311yescharacteristic
105C2×#121C22×C312yes—
106C2×#121C22×C312yes—
107C2×#121C22×C312yes—
108C2×#120—22no—
109#469—24no—
110C2×#107C22×C311yescharacteristic, complemented
111C2×#106C22×C311yescharacteristic, complemented
112C2×#106C22×C311yescharacteristic, complemented
113C2×#102C22×C312yescomplemented
114C2×#102C22×C312yescomplemented
115C2×#102C22×C312yescomplemented
116#407—24no—
117C2×C22⋊C8C22×C312yes—
118C2×C22⋊C8C22×C312yes—
119#386—22no—
120#386—22no—
121#386—22no—
122#380C22×C312yes—
123#380C22×C312yes—
124#380C22×C312yes—
125#340C22×C312yescomplemented
126#340C22×C312yescomplemented
127#340C22×C312yescomplemented
128C24×C3—22no—
129C24×C3—24no—
130C24×C3—22no—
131C24×C3—42no—
132C24×C3—41no—
133C24×C3—41no—
134C24×C3C2×D411yescharacteristic
135C24×C3C2×D412yes—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—44no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—24no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—44no—
171C22×C3×C4—44no—
172C22×C3×C4—44no—
173C22×C3×C4—44no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4#4112yes—
178C22×C3×C4C2411yescharacteristic
179C3×D8—44no—
180C3×M16—44no—
181C3×M16—44no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8C2×D412yes—
190C2×C3×C8C2×D412yes—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—44no—
199C3×C22⋊C4—44no—
200C3×C22⋊C4—44no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—44no—
203C25C3×D411yescharacteristic
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4C23×C312yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—41no—
213C23×C4—41no—
214C23×C4C3×D411yescharacteristic
215C23×C4C3×D412yes—
216C23×C4C23×C312yes—
217C23×C4C23×C311yescharacteristic
218C2×D8—22no—
219C2×D8—41no—
220C2×D8—41no—
221C2×M16—22no—
222C2×M16—41no—
223C2×M16—41no—
224C2×M16—41no—
225C2×M16—41no—
226C2×M16C3×D414yescomplemented
227C22×C8C23×C312yes—
228C22×C8C23×C311yescharacteristic
229C22×C8C23×C311yescharacteristic
230#121—24no—
231#121—24no—
232#121—42no—
233#121—42no—
234#121—42no—
235#121—42no—
236#120—48no—
237C2×C4⋊C4C23×C312yes—
238C2×C4⋊C4C23×C312yes—
239C2×C4⋊C4C23×C312yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246#107—41no—
247#106—22no—
248#106—41no—
249#102—24no—
250#102—42no—
251#102—42no—
252C22⋊C8—24no—
253C22⋊C8—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—44no—
268C23×C3—48no—
269C23×C3—44no—
270C23×C3—48no—
271C23×C3—44no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3#12112yes—
280C23×C3AGL(1,ℤ/8)14yes—
281C23×C3C22×D411yescharacteristic
282C3×D4—42no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4—42no—
286C3×D4—88no—
287C3×D4—88no—
288C3×D4—88no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—48no—
299C2×C3×C4—44no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4#12112yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318C2×C3×C4H(5,𝔽2)11yescharacteristic
319C2×C3×C4H(5,𝔽2)11yescharacteristic
320C3×C8—22no—
321C3×C8—22no—
322C3×C8—42no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—41no—
327C24—41no—
328C24—42no—
329C24C2×C3×D411yescharacteristic
330C24C2×C3×D412yes—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4C2×C3×D412yes—
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D412yes—
372C22×C4C3×#4112yes—
373C22×C4C24×C311yescharacteristic
374D8—44no—
375M16—44no—
376M16—44no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8C2×C3×D412yes—
385C2×C8C2×C3×D412yes—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22×C3—22no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—416no—
404C22×C3—42no—
405C22×C3—41no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—41no—
410C22×C3—42no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3—82no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3C2×#12111yescharacteristic
419C22×C3#53411yescharacteristic
420C22×C3#53411yescharacteristic
421C22×C3C2×AGL(1,ℤ/8)12yes—
422C22×C3#57512yes—
423C3×C4—22no—
424C3×C4—42no—
425C3×C4—82no—
426C3×C4—82no—
427C3×C4—82no—
428C3×C4#53411yescharacteristic
429C3×C4#53511yescharacteristic
430C3×C4#53811yescharacteristic
431C3×C4#53811yescharacteristic
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—21no—
441C23—21no—
442C23—44no—
443C23—48no—
444C23—44no—
445C23—48no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23C3×#12112yes—
455C23C3×AGL(1,ℤ/8)14yes—
456C23C22×C3×D411yescharacteristic
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—88no—
462D4—88no—
463D4—88no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—48no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
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×#12112yes—
489C2×C4C22×C3×D411yescharacteristic
490C2×C4C2×C3×#4111yescharacteristic
491C2×C4C3×H(5,𝔽2)11yescharacteristic
492C2×C4C3×H(5,𝔽2)11yescharacteristic
493C2×C4C3×H(5,𝔽2)11yescharacteristic
494C2×C4C3×H(5,𝔽2)11yescharacteristic
495C8—22no—
496C8—22no—
497C8—42no—
498C2×C3—42no—
499C2×C3—44no—
500C2×C3—162no—
501C2×C3#261711yescharacteristic
502C2×C3#305512yes—
503C2×C3#319211yescharacteristic
504C2×C3#319412yes—
505C2×C3#319711yescharacteristic
506C22—22no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—416no—
512C22—42no—
513C22—41no—
514C22—42no—
515C22—42no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22C2×C3×#12111yescharacteristic
527C22C3×#53411yescharacteristic
528C22C3×#53411yescharacteristic
529C22C2×C3×AGL(1,ℤ/8)12yes—
530C22C3×#57512yes—
531C4—22no—
532C4—42no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4C3×#53411yescharacteristic
537C4C3×#53511yescharacteristic
538C4C3×#53811yescharacteristic
539C4C3×#53811yescharacteristic
540C3#2974211yescharacteristic, direct product factor, Sylow
541C2—42no—
542C2—44no—
543C2—162no—
544C2C3×#261711yescharacteristic
545C2C3×#305512yes—
546C2C3×#319211yescharacteristic
547C2C3×#319412yes—
548C2C3×#319711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#469C2semidirect
2. C2×C3×#407C2semidirect
3. C3×#2847C2semidirect
4. C3×#2077C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#1999C2semidirect
8. C3×#1895C2semidirect
9. C3×#1809C2semidirect
10. #29742C3direct (N × H)
12. C22×C3×D8C22semidirect
20. C2×C3×#107C22semidirect
21. C2×C3×#106C22semidirect
22. C2×C3×#106C22semidirect
23. C2×C3×#102C22semidirect
24. C2×C3×#102C22semidirect
25. C2×C3×#102C22semidirect
35. C3×#340C22semidirect
36. C3×#340C22semidirect
37. C3×#340C22semidirect
38. C2×#469C2×C3semidirect
39. C2×#407C2×C3semidirect
40. #2847C2×C3semidirect
41. #2077C2×C3semidirect
42. #2034C2×C3semidirect
43. #2033C2×C3semidirect
44. #1999C2×C3semidirect
45. #1895C2×C3semidirect
46. #1809C2×C3semidirect
70. C2×C3×M16D4semidirect
102. C22×D8C22×C3semidirect
110. C2×#107C22×C3semidirect
111. C2×#106C22×C3semidirect
112. C2×#106C22×C3semidirect
113. C2×#102C22×C3semidirect
114. C2×#102C22×C3semidirect
115. C2×#102C22×C3semidirect
125. #340C22×C3semidirect
126. #340C22×C3semidirect
127. #340C22×C3semidirect
226. C2×M16C3×D4semidirect
540. C3#29742direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4692
C2×C3×#4072
C3×#28472
C3×#20772
C3×#20342
C3×#20332
C3×#19992
C3×#18952
C3×#18092
#297423