Finite Group Databasebeta

← C3×#29712 #10648319

Subgroups — C3×#29712 #10648319

Order 768.

1368 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#447C212yesmaximal, complemented
2C3×#2856C211yesmaximal, characteristic, complemented
3C2×C3×#390C211yesmaximal, characteristic, complemented
4C3×#2071C212yesmaximal, complemented
5C3×#2054C214yesmaximal, complemented
6C3×#2011C212yesmaximal, complemented
7C3×#1874C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#29712C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C22×C3×D8C2214yes—
12C2×C3×#127C2211yescharacteristic, complemented
13C2×C3×#121—21no—
14C2×C3×#121—21no—
15C2×C3×#121C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4—21no—
18C22×C3×C22⋊C4—21no—
19C22×C3×C42C2211yescharacteristic
20C3×#447—28no—
21C2×C3×C4⋊C8C2212yescomplemented
22C2×C3×#102C2214yes—
23C2×C3×#102C2214yescomplemented
24C2×C3×C22⋊C8C2212yes—
25C3×#390—21no—
26C3×#390—21no—
27C3×#390—21no—
28C3×#390—21no—
29C3×#390C2212yes—
30C3×#390C2212yes—
31C3×#390C2212yescomplemented
32C3×#384C2211yescharacteristic, complemented
33C3×#340C2214yescomplemented
34C3×#336C2212yescomplemented
35C2×#447C2×C312yescomplemented
36#2856C2×C311yescharacteristic, complemented
37C2×#390C2×C311yescharacteristic, complemented
38#2071C2×C312yescomplemented
39#2054C2×C314yescomplemented
40#2011C2×C312yescomplemented
41#1874C2×C312yescomplemented
42#1797C2×C311yescharacteristic, complemented
43C25×C3—21no—
44C22×C3×D4—22no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—22no—
49C22×C3×D4—42no—
50C22×C3×D4—42no—
51C22×C3×D4D414yescomplemented
52C22×C3×D4C2312yes—
53C22×C3×D4C2312yes—
54C23×C3×C4—21no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C2×C3×D8—28no—
59C2×C3×D8—44no—
60C2×C3×D8—44no—
61C22×C3×C8C2314yes—
62C3×#127—42no—
63C3×#121—24no—
64C3×#121—24no—
65C3×#121—44no—
66C3×#121—44no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—42no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—44no—
81C2×C3×C22⋊C4—44no—
82C2×C3×C42—42no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C3×C4⋊C8—42no—
88C3×#102—28no—
89C3×#102—44no—
90C3×C22⋊C8—24no—
91C23×D4C22×C311yescharacteristic
92C22×D8C22×C314yes—
93C2×#127C22×C311yescharacteristic, complemented
94C2×#121—21no—
95C2×#121—21no—
96C2×#121C22×C312yes—
97C22×C4⋊C4C22×C311yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C42C22×C311yescharacteristic
101#447—28no—
102C2×C4⋊C8C22×C312yescomplemented
103C2×#102C22×C314yes—
104C2×#102C22×C314yescomplemented
105C2×C22⋊C8C22×C312yes—
106#390—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#390C22×C312yes—
111#390C22×C312yes—
112#390C22×C312yescomplemented
113#384C22×C311yescharacteristic, complemented
114#340C22×C314yescomplemented
115#336C22×C312yescomplemented
116C24×C3—24no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—42no—
120C24×C3—42no—
121C24×C3—42no—
122C24×C3—42no—
123C24×C3—42no—
124C24×C3—42no—
125C24×C3C2×D411yescharacteristic
126C2×C3×D4—24no—
127C2×C3×D4—22no—
128C2×C3×D4—22no—
129C2×C3×D4—22no—
130C2×C3×D4—24no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—24no—
134C2×C3×D4—48no—
135C2×C3×D4—42no—
136C2×C3×D4—42no—
137C2×C3×D4—48no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—42no—
142C2×C3×D4—44no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4C2×D414yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4D814yescomplemented
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4#4112yes—
168C22×C3×C4C2411yescharacteristic
169C3×D8—416no—
170C2×C3×C8—24no—
171C2×C3×C8—24no—
172C2×C3×C8—24no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—42no—
179C3×C22⋊C4—44no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—44no—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—44no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—41no—
189C3×C42—41no—
190C25—21no—
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4C3×D414yescomplemented
199C22×D4C23×C312yes—
200C22×D4C23×C312yes—
201C23×C4—21no—
202C23×C4C3×D411yescharacteristic
203C23×C4C3×D411yescharacteristic
204C23×C4C23×C311yescharacteristic
205C2×D8—28no—
206C2×D8—44no—
207C2×D8—44no—
208C22×C8C23×C314yes—
209#127—42no—
210#121—24no—
211#121—24no—
212#121—44no—
213#121—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C312yes—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C42—42no—
230C2×C42C3×D411yescharacteristic
231C2×C42C3×D411yescharacteristic
232C2×C42C23×C311yescharacteristic
233C2×C42C23×C311yescharacteristic
234C4⋊C8—42no—
235#102—28no—
236#102—44no—
237C22⋊C8—24no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—24no—
243C23×C3—48no—
244C23×C3—48no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—48no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—84no—
260C23×C3—84no—
261C23×C3—84no—
262C23×C3—84no—
263C23×C3#12112yes—
264C23×C3C2×D812yes—
265C23×C3AGL(1,ℤ/8)12yes—
266C23×C3C22×D411yescharacteristic
267C3×D4—28no—
268C3×D4—48no—
269C3×D4—44no—
270C3×D4—44no—
271C3×D4—44no—
272C3×D4—84no—
273C3×D4—84no—
274C3×D4—88no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—44no—
289C2×C3×C4—44no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—84no—
297C2×C3×C4#12112yes—
298C2×C3×C4C2×D812yes—
299C2×C3×C4C2×D812yes—
300C2×C3×C4AGL(1,ℤ/8)12yes—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4H(5,𝔽2)12yes—
304C2×C3×C4H(5,𝔽2)12yes—
305C3×C8—44no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C2×C3×D411yescharacteristic
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—48no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—48no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4C2×C3×D414yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4C3×D814yescomplemented
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D412yes—
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C3×#4112yes—
358C22×C4C24×C311yescharacteristic
359D8—416no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C42—22no—
377C42—22no—
378C42—41no—
379C42—41no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—44no—
387C22×C3—44no—
388C22×C3—44no—
389C22×C3—816no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3#45914yes—
402C22×C3C2×#12111yescharacteristic
403C22×C3#53411yescharacteristic
404C22×C3#53511yescharacteristic
405C22×C3C22×D811yescharacteristic
406C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
407C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
408C22×C3#57511yescharacteristic
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—41no—
414C3×C4—82no—
415C3×C4—82no—
416C3×C4#53412yes—
417C3×C4#53812yes—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—24no—
423C23—48no—
424C23—48no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—48no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—84no—
440C23—84no—
441C23—84no—
442C23—84no—
443C23C3×#12112yes—
444C23C2×C3×D812yes—
445C23C3×AGL(1,ℤ/8)12yes—
446C23C22×C3×D411yescharacteristic
447D4—28no—
448D4—48no—
449D4—44no—
450D4—44no—
451D4—44no—
452D4—84no—
453D4—84no—
454D4—88no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—84no—
477C2×C4C3×#12112yes—
478C2×C4C2×C3×D812yes—
479C2×C4C2×C3×D812yes—
480C2×C4C3×AGL(1,ℤ/8)12yes—
481C2×C4C22×C3×D411yescharacteristic
482C2×C4C2×C3×#4111yescharacteristic
483C2×C4C3×H(5,𝔽2)12yes—
484C2×C4C3×H(5,𝔽2)12yes—
485C8—44no—
486C2×C3—24no—
487C2×C3—84no—
488C2×C3—162no—
489C2×C3#261711yescharacteristic
490C2×C3C2×#45911yescharacteristic
491C2×C3#303611yescharacteristic
492C2×C3#318611yescharacteristic
493C2×C3#319211yescharacteristic
494C2×C3#320511yescharacteristic
495C2×C3#321311yescharacteristic
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—816no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22C3×#45914yes—
518C22C2×C3×#12111yescharacteristic
519C22C3×#53411yescharacteristic
520C22C3×#53511yescharacteristic
521C22C22×C3×D811yescharacteristic
522C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
523C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
524C22C3×#57511yescharacteristic
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—41no—
530C4—82no—
531C4—82no—
532C4C3×#53412yes—
533C4C3×#53812yes—
534C3#2971211yescharacteristic, direct product factor, Sylow
535C2—24no—
536C2—84no—
537C2—162no—
538C2C3×#261711yescharacteristic
539C2C2×C3×#45911yescharacteristic
540C2C3×#303611yescharacteristic
541C2C3×#318611yescharacteristic
542C2C3×#319211yescharacteristic
543C2C3×#320511yescharacteristic
544C2C3×#321311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C2×C3×#447C2semidirect
2. C3×#2856C2semidirect
3. C2×C3×#390C2semidirect
4. C3×#2071C2semidirect
5. C3×#2054C2semidirect
6. C3×#2011C2semidirect
7. C3×#1874C2semidirect
8. C3×#1797C2semidirect
9. #29712C3direct (N × H)
12. C2×C3×#127C22semidirect
21. C2×C3×C4⋊C8C22semidirect
23. C2×C3×#102C22semidirect
31. C3×#390C22semidirect
32. C3×#384C22semidirect
33. C3×#340C22semidirect
34. C3×#336C22semidirect
35. C2×#447C2×C3semidirect
36. #2856C2×C3semidirect
37. C2×#390C2×C3semidirect
38. #2071C2×C3semidirect
39. #2054C2×C3semidirect
40. #2011C2×C3semidirect
41. #1874C2×C3semidirect
42. #1797C2×C3semidirect
51. C22×C3×D4D4semidirect
93. C2×#127C22×C3semidirect
102. C2×C4⋊C8C22×C3semidirect
104. C2×#102C22×C3semidirect
112. #390C22×C3semidirect
113. #384C22×C3semidirect
114. #340C22×C3semidirect
115. #336C22×C3semidirect
160. C22×C3×C4D8semidirect
198. C22×D4C3×D4semidirect
350. C22×C4C3×D8semidirect
534. C3#29712direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4472
C3×#28562
C2×C3×#3902
C3×#20712
C3×#20542
C3×#20112
C3×#18742
C3×#17972
#297123