Finite Group Databasebeta

← C3×#23933 #10642540

Subgroups — C3×#23933 #10642540

Order 768.

1488 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#450C212yesmaximal
3C2×C3×#390C211yesmaximal, characteristic
4C3×#2023C212yesmaximal, complemented
5C3×#2013C212yesmaximal, complemented
6C3×#1880C214yesmaximal, complemented
7C3×#1872C212yesmaximal, complemented
8C3×#1817C211yesmaximal, characteristic, complemented
9#23933C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C22×C3×SD16C2214yes—
12C3×#533—28no—
13C2×C3×C4⋊Q8C2211yescharacteristic, complemented
14C2×C3×C4⋊Q8C2211yescharacteristic, complemented
15C2×C3×#125—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8C2212yescomplemented
20C2×C3×#117—21no—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4—21no—
23C22×C3×C22⋊C4—21no—
24C22×C3×C42C2211yescharacteristic
25C3×#450—28no—
26C2×C3×#103C2214yescomplemented
27C2×C3×#102C2214yes—
28C2×C3×C22⋊C8C2212yes—
29C3×#390—21no—
30C3×#390—21no—
31C3×#390C2212yes—
32C3×#390C2212yes—
33C3×#340C2214yescomplemented
34C3×#336C2212yescomplemented
35C3×#336C2212yescomplemented
36C2×#533C2×C311yescharacteristic, complemented
37C2×#450C2×C312yes—
38C2×#390C2×C311yescharacteristic
39#2023C2×C312yescomplemented
40#2013C2×C312yescomplemented
41#1880C2×C314yescomplemented
42#1872C2×C312yescomplemented
43#1817C2×C311yescharacteristic, complemented
44C25×C3—21no—
45C22×C3×Q8—21no—
46C22×C3×Q8C2312yes—
47C22×C3×D4—24no—
48C22×C3×D4—21no—
49C22×C3×D4—24no—
50C22×C3×D4—42no—
51C22×C3×D4C2312yes—
52C23×C3×C4—21no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C2×C3×SD16—28no—
57C2×C3×SD16—28no—
58C2×C3×SD16—44no—
59C22×C3×C8C2314yes—
60C3×C4⋊Q8—24no—
61C3×C4⋊Q8—24no—
62C3×#125—28no—
63C3×C22⋊Q8—28no—
64C3×C22⋊Q8—24no—
65C3×C22⋊Q8D418yes—
66C3×C4×Q8—28no—
67C3×C4×Q8—24no—
68C3×C4×Q8—24no—
69C3×#117—28no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—42no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—44no—
85C2×C3×C22⋊C4—44no—
86C2×C3×C42—24no—
87C2×C3×C42—22no—
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#103—44no—
93C3×#102—28no—
94C3×C22⋊C8—24no—
95C23×D4C22×C311yescharacteristic
96C22×SD16C22×C314yes—
97#533—28no—
98C2×C4⋊Q8C22×C311yescharacteristic, complemented
99C2×C4⋊Q8C22×C311yescharacteristic, complemented
100C2×#125—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C22×C312yes—
103C2×C4×Q8—21no—
104C2×C4×Q8C22×C312yescomplemented
105C2×#117—21no—
106C22×C4⋊C4C22×C311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C42C22×C311yescharacteristic
110#450—28no—
111C2×#103C22×C314yescomplemented
112C2×#102C22×C314yes—
113C2×C22⋊C8C22×C312yes—
114#390—21no—
115#390—21no—
116#390C22×C312yes—
117#390C22×C312yes—
118#340C22×C314yescomplemented
119#336C22×C312yescomplemented
120#336C22×C312yescomplemented
121C24×C3—24no—
122C24×C3—22no—
123C24×C3—42no—
124C24×C3—42no—
125C24×C3—42no—
126C24×C3—42no—
127C24×C3—42no—
128C24×C3—42no—
129C24×C3C2×D411yescharacteristic
130C2×C3×Q8—24no—
131C2×C3×Q8—22no—
132C2×C3×Q8—24no—
133C2×C3×Q8—24no—
134C2×C3×Q8—42no—
135C2×C3×Q8—42no—
136C2×C3×Q8C2×D414yes—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—22no—
140C2×C3×D4—24no—
141C2×C3×D4—416no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4#4114yes—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
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—42no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4SD1614yescomplemented
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D414yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C3×SD16—416no—
167C2×C3×C8—24no—
168C2×C3×C8—24no—
169C2×C3×C8—24no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—28no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4C2×D414yes—
179C3×C22⋊C4—28no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—44no—
184C3×C42—28no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—22no—
189C25—21no—
190C22×Q8—21no—
191C22×Q8C23×C312yes—
192C22×D4—24no—
193C22×D4—21no—
194C22×D4—24no—
195C22×D4—42no—
196C22×D4C23×C312yes—
197C23×C4—21no—
198C23×C4C3×D411yescharacteristic
199C23×C4C3×D411yescharacteristic
200C23×C4C23×C311yescharacteristic
201C2×SD16—28no—
202C2×SD16—28no—
203C2×SD16—44no—
204C22×C8C23×C314yes—
205C4⋊Q8—24no—
206C4⋊Q8—24no—
207#125—28no—
208C22⋊Q8—28no—
209C22⋊Q8—24no—
210C22⋊Q8C3×D418yes—
211C4×Q8—28no—
212C4×Q8—24no—
213C4×Q8—24no—
214#117—28no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C23×C312yes—
223C2×C4⋊C4C23×C312yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42C3×D411yescharacteristic
234C2×C42C3×D411yescharacteristic
235C2×C42C23×C311yescharacteristic
236C2×C42C23×C311yescharacteristic
237#103—44no—
238#102—28no—
239C22⋊C8—24no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—24no—
245C23×C3—48no—
246C23×C3—48no—
247C23×C3—48no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—84no—
255C23×C3—84no—
256C23×C3—84no—
257C23×C3—84no—
258C23×C3#12012yes—
259C23×C3C2×SD1612yes—
260C23×C3#13712yes—
261C23×C3C22×D411yescharacteristic
262C3×Q8—28no—
263C3×Q8—48no—
264C3×Q8—44no—
265C3×D4—28no—
266C3×D4—48no—
267C3×D4—44no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—28no—
277C2×C3×C4—28no—
278C2×C3×C4—21no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—44no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—84no—
292C2×C3×C4#12012yes—
293C2×C3×C4C2×SD1612yes—
294C2×C3×C4C2×SD1612yes—
295C2×C3×C4AGL(1,ℤ/8)12yes—
296C2×C3×C4C22×D412yes—
297C2×C3×C4C2×#4112yes—
298C2×C3×C4H(5,𝔽2)12yes—
299C3×C8—44no—
300C24—24no—
301C24—22no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24C2×C3×D411yescharacteristic
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8C2×C3×D414yes—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—416no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4C3×#4114yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4C3×SD1614yescomplemented
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C2×C3×D414yes—
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C24×C311yescharacteristic
345SD16—416no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—28no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—42no—
357C4⋊C4C2×C3×D414yes—
358C22⋊C4—28no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C42—28no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3—816no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3#44814yes—
383C22×C3C2×#12011yescharacteristic
384C22×C3#52511yescharacteristic
385C22×C3#53511yescharacteristic
386C22×C3C22×SD1611yescharacteristic
387C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
388C22×C3C2×#13711yescharacteristic
389C22×C3#57511yescharacteristic
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—44no—
395C3×C4—41no—
396C3×C4—44no—
397C3×C4—82no—
398C3×C4#54614yes—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—24no—
404C23—48no—
405C23—48no—
406C23—48no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—84no—
414C23—84no—
415C23—84no—
416C23—84no—
417C23C3×#12012yes—
418C23C2×C3×SD1612yes—
419C23C3×#13712yes—
420C23C22×C3×D411yescharacteristic
421Q8—28no—
422Q8—48no—
423Q8—44no—
424D4—28no—
425D4—48no—
426D4—44no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—28no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—28no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—84no—
451C2×C4C3×#12012yes—
452C2×C4C2×C3×SD1612yes—
453C2×C4C2×C3×SD1612yes—
454C2×C4C3×AGL(1,ℤ/8)12yes—
455C2×C4C22×C3×D412yes—
456C2×C4C2×C3×#4112yes—
457C2×C4C3×H(5,𝔽2)12yes—
458C8—44no—
459C2×C3—24no—
460C2×C3—84no—
461C2×C3#240411yescharacteristic
462C2×C3C2×#44811yescharacteristic
463C2×C3#300611yescharacteristic
464C2×C3#310111yescharacteristic
465C2×C3#311011yescharacteristic
466C2×C3#320811yescharacteristic
467C2×C3#321511yescharacteristic
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—816no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22C3×#44814yes—
483C22C2×C3×#12011yescharacteristic
484C22C3×#52511yescharacteristic
485C22C3×#53511yescharacteristic
486C22C22×C3×SD1611yescharacteristic
487C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
488C22C2×C3×#13711yescharacteristic
489C22C3×#57511yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—44no—
495C4—41no—
496C4—44no—
497C4—82no—
498C4C3×#54614yes—
499C3#2393311yescharacteristic, direct product factor, Sylow
500C2—24no—
501C2—84no—
502C2C3×#240411yescharacteristic
503C2C2×C3×#44811yescharacteristic
504C2C3×#300611yescharacteristic
505C2C3×#310111yescharacteristic
506C2C3×#311011yescharacteristic
507C2C3×#320811yescharacteristic
508C2C3×#321511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
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×#533C2semidirect
4. C3×#2023C2semidirect
5. C3×#2013C2semidirect
6. C3×#1880C2semidirect
7. C3×#1872C2semidirect
8. C3×#1817C2semidirect
9. #23933C3direct (N × H)
13. C2×C3×C4⋊Q8C22semidirect
14. C2×C3×C4⋊Q8C22semidirect
19. C2×C3×C4×Q8C22semidirect
26. C2×C3×#103C22semidirect
33. C3×#340C22semidirect
34. C3×#336C22semidirect
35. C3×#336C22semidirect
36. C2×#533C2×C3semidirect
39. #2023C2×C3semidirect
40. #2013C2×C3semidirect
41. #1880C2×C3semidirect
42. #1872C2×C3semidirect
43. #1817C2×C3semidirect
98. C2×C4⋊Q8C22×C3semidirect
99. C2×C4⋊Q8C22×C3semidirect
104. C2×C4×Q8C22×C3semidirect
111. C2×#103C22×C3semidirect
118. #340C22×C3semidirect
119. #336C22×C3semidirect
120. #336C22×C3semidirect
158. C22×C3×C4SD16semidirect
337. C22×C4C3×SD16semidirect
499. C3#23933direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4502
C2×C3×#3902
C3×#20232
C3×#20132
C3×#18802
C3×#18722
C3×#18172
#239333