Finite Group Databasebeta

← C3×#23926 #10642533

Subgroups — C3×#23926 #10642533

Order 768.

1888 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic, complemented
2C2×C3×#535C211yesmaximal, characteristic
3C2×C3×#448C212yesmaximal, complemented
4C2×C3×#450C212yesmaximal, complemented
5C3×#2000C214yesmaximal, complemented
6C3×#1875C212yesmaximal, complemented
7C3×#1873C212yesmaximal, complemented
8C3×#1780C211yesmaximal, characteristic, complemented
9#23926C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic, complemented
12C23×C3×D4C2211yescharacteristic
13C22×C3×SD16C2214yes—
14C3×#536—28no—
15C3×#535—28no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×#127—21no—
18C2×C3×#124—21no—
19C2×C3×#124—21no—
20C2×C3×#123—21no—
21C2×C3×#123—21no—
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×C22⋊Q8C2212yescomplemented
24C2×C3×#121C2212yes—
25C2×C3×#121C2212yes—
26C2×C3×#120—21no—
27C2×C3×#120—21no—
28C2×C3×#117C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic, complemented
30C3×#450—28no—
31C3×#448—28no—
32C2×C3×#103C2214yescomplemented
33C2×C3×#102C2214yes—
34C2×C3×C22⋊C8C2212yes—
35C2×C3×C22⋊C8C2212yescomplemented
36C2×C3×C22⋊C8C2212yescomplemented
37C3×#340C2214yescomplemented
38C2×#536C2×C311yescharacteristic, complemented
39C2×#535C2×C311yescharacteristic
40C2×#448C2×C312yescomplemented
41C2×#450C2×C312yescomplemented
42#2000C2×C314yescomplemented
43#1875C2×C312yescomplemented
44#1873C2×C312yescomplemented
45#1780C2×C311yescharacteristic, complemented
46C25×C3—21no—
47C2×C3×#41—24no—
48C2×C3×#41—24no—
49C2×C3×#41—42no—
50C22×C3×Q8—24no—
51C22×C3×Q8—21no—
52C22×C3×Q8—24no—
53C22×C3×Q8—42no—
54C22×C3×Q8D411yescharacteristic
55C22×C3×Q8C2312yes—
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4—24no—
59C22×C3×D4—21no—
60C22×C3×D4—24no—
61C22×C3×D4—42no—
62C22×C3×D4D411yescharacteristic
63C22×C3×D4D411yescharacteristic
64C22×C3×D4D411yescharacteristic
65C22×C3×D4C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C2×C3×SD16—28no—
71C2×C3×SD16—28no—
72C2×C3×SD16—44no—
73C22×C3×C8C2314yes—
74C3×C4⋊Q8—28no—
75C3×#127—28no—
76C3×#124—28no—
77C3×#124—28no—
78C3×#123—28no—
79C3×#123—28no—
80C3×C22⋊Q8—24no—
81C3×C22⋊Q8—24no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8D418yescomplemented
84C3×#121—24no—
85C3×#121—24no—
86C3×#121—24no—
87C3×#121D418yes—
88C3×#120—28no—
89C3×#120—28no—
90C3×#117—24no—
91C3×#117—24no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C42—21no—
103C2×C3×C42—21no—
104C3×#103—28no—
105C3×#102—28no—
106C3×C22⋊C8—24no—
107C3×C22⋊C8—24no—
108C3×C22⋊C8—42no—
109C22×#41C22×C311yescharacteristic
110C23×Q8C22×C311yescharacteristic, complemented
111C23×D4C22×C311yescharacteristic
112C22×SD16C22×C314yes—
113#536—28no—
114#535—28no—
115C2×C4⋊Q8—21no—
116C2×#127—21no—
117C2×#124—21no—
118C2×#124—21no—
119C2×#123—21no—
120C2×#123—21no—
121C2×C22⋊Q8C22×C312yes—
122C2×C22⋊Q8C22×C312yescomplemented
123C2×#121C22×C312yes—
124C2×#121C22×C312yes—
125C2×#120—21no—
126C2×#120—21no—
127C2×#117C22×C311yescharacteristic
128C2×#117C22×C311yescharacteristic, complemented
129#450—28no—
130#448—28no—
131C2×#103C22×C314yescomplemented
132C2×#102C22×C314yes—
133C2×C22⋊C8C22×C312yes—
134C2×C22⋊C8C22×C312yescomplemented
135C2×C22⋊C8C22×C312yescomplemented
136#340C22×C314yescomplemented
137C24×C3—24no—
138C24×C3—22no—
139C24×C3—42no—
140C24×C3—42no—
141C24×C3—84no—
142C24×C3C2×D411yescharacteristic
143C24×C3C2×D411yescharacteristic
144C24×C3C2×D411yescharacteristic
145C3×#41—416no—
146C2×C3×Q8—24no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—24no—
151C2×C3×Q8—24no—
152C2×C3×Q8—41no—
153C2×C3×Q8—416no—
154C2×C3×Q8—42no—
155C2×C3×Q8—42no—
156C2×C3×Q8C2×D414yes—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—28no—
163C2×C3×D4—28no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—24no—
167C2×C3×D4—22no—
168C2×C3×D4—24no—
169C2×C3×D4—416no—
170C2×C3×D4—41no—
171C2×C3×D4—41no—
172C2×C3×D4—41no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4—84no—
176C2×C3×D4—84no—
177C2×C3×D4C2×D414yes—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2411yescharacteristic
197C3×SD16—416no—
198C2×C3×C8—24no—
199C2×C3×C8—24no—
200C2×C3×C8—24no—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C2×D414yes—
206C3×C4⋊C4C2×D414yes—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—28no—
213C3×C42—24no—
214C3×C42—24no—
215C25—21no—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—42no—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—42no—
223C22×Q8C3×D411yescharacteristic
224C22×Q8C23×C312yes—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—24no—
228C22×D4—21no—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4C3×D411yescharacteristic
232C22×D4C3×D411yescharacteristic
233C22×D4C3×D411yescharacteristic
234C22×D4C23×C312yes—
235C23×C4—21no—
236C23×C4C23×C311yescharacteristic
237C23×C4C23×C311yescharacteristic
238C23×C4C23×C311yescharacteristic
239C2×SD16—28no—
240C2×SD16—28no—
241C2×SD16—44no—
242C22×C8C23×C314yes—
243C4⋊Q8—28no—
244#127—28no—
245#124—28no—
246#124—28no—
247#123—28no—
248#123—28no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8C3×D418yescomplemented
253#121—24no—
254#121—24no—
255#121—24no—
256#121C3×D418yes—
257#120—28no—
258#120—28no—
259#117—24no—
260#117—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4C23×C312yes—
264C2×C4⋊C4C23×C312yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C42—21no—
272C2×C42—21no—
273#103—28no—
274#102—28no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277C22⋊C8—42no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—21no—
285C23×C3—24no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—24no—
289C23×C3—48no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—44no—
293C23×C3—44no—
294C23×C3—88no—
295C23×C3—88no—
296C23×C3—88no—
297C23×C3#12012yes—
298C23×C3#12012yes—
299C23×C3C22×D411yescharacteristic
300C3×Q8—28no—
301C3×Q8—44no—
302C3×Q8—48no—
303C3×Q8—44no—
304C3×D4—28no—
305C3×D4—44no—
306C3×D4—44no—
307C3×D4—48no—
308C3×D4—44no—
309C3×D4—44no—
310C3×D4—88no—
311C3×D4—88no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—28no—
317C2×C3×C4—28no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—21no—
321C2×C3×C4—24no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—48no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—44no—
328C2×C3×C4—44no—
329C2×C3×C4—44no—
330C2×C3×C4#12012yes—
331C2×C3×C4#12012yes—
332C2×C3×C4C22×D412yes—
333C2×C3×C4C22×D412yes—
334C2×C3×C4H(5,𝔽2)12yes—
335C3×C8—44no—
336C24—24no—
337C24—22no—
338C24—42no—
339C24—42no—
340C24—84no—
341C24C2×C3×D411yescharacteristic
342C24C2×C3×D411yescharacteristic
343C24C2×C3×D411yescharacteristic
344#41—416no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—41no—
352C2×Q8—416no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8C2×C3×D414yes—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—416no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4C2×C3×D414yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D414yes—
394C22×C4C2×C3×D414yes—
395C22×C4C24×C311yescharacteristic
396SD16—416no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4C2×C3×D414yes—
405C4⋊C4C2×C3×D414yes—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—28no—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C42—24no—
413C42—24no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—41no—
423C22×C3—44no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—42no—
429C22×C3—816no—
430C22×C3—84no—
431C22×C3—84no—
432C22×C3C2×#12011yescharacteristic
433C22×C3C2×#12011yescharacteristic
434C22×C3#53411yescharacteristic
435C22×C3#57714yes—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—41no—
439C3×C4—44no—
440C3×C4—44no—
441C3×C4—44no—
442C3×C4D4214yes—
443C23—24no—
444C23—24no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—24no—
450C23—21no—
451C23—24no—
452C23—21no—
453C23—21no—
454C23—48no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—88no—
460C23—88no—
461C23—88no—
462C23C3×#12012yes—
463C23C3×#12012yes—
464C23C22×C3×D411yescharacteristic
465Q8—28no—
466Q8—44no—
467Q8—48no—
468Q8—44no—
469D4—28no—
470D4—44no—
471D4—44no—
472D4—48no—
473D4—44no—
474D4—44no—
475D4—88no—
476D4—88no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4C3×#12012yes—
496C2×C4C3×#12012yes—
497C2×C4C22×C3×D412yes—
498C2×C4C22×C3×D412yes—
499C2×C4C3×H(5,𝔽2)12yes—
500C8—44no—
501C2×C3—42no—
502C2×C3—42no—
503C2×C3—81no—
504C2×C3—84no—
505C2×C3#240311yescharacteristic
506C2×C3#300912yes—
507C2×C3#301012yes—
508C2×C3#320012yes—
509C22—22no—
510C22—22no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—41no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—44no—
522C22—44no—
523C22—42no—
524C22—816no—
525C22—84no—
526C22—84no—
527C22C2×C3×#12011yescharacteristic
528C22C2×C3×#12011yescharacteristic
529C22C3×#53411yescharacteristic
530C22C3×#57714yes—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—44no—
536C4—44no—
537C4C3×D4214yes—
538C3#2392611yescharacteristic, direct product factor, Sylow
539C2—42no—
540C2—42no—
541C2—81no—
542C2—84no—
543C2C3×#240311yescharacteristic
544C2C3×#300912yes—
545C2C3×#301012yes—
546C2C3×#320012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#536C2semidirect
3. C2×C3×#448C2semidirect
4. C2×C3×#450C2semidirect
5. C3×#2000C2semidirect
6. C3×#1875C2semidirect
7. C3×#1873C2semidirect
8. C3×#1780C2semidirect
9. #23926C3direct (N × H)
11. C23×C3×Q8C22semidirect
23. C2×C3×C22⋊Q8C22semidirect
29. C2×C3×#117C22semidirect
32. C2×C3×#103C22semidirect
35. C2×C3×C22⋊C8C22semidirect
36. C2×C3×C22⋊C8C22semidirect
37. C3×#340C22semidirect
38. C2×#536C2×C3semidirect
40. C2×#448C2×C3semidirect
41. C2×#450C2×C3semidirect
42. #2000C2×C3semidirect
43. #1875C2×C3semidirect
44. #1873C2×C3semidirect
45. #1780C2×C3semidirect
83. C3×C22⋊Q8D4semidirect
110. C23×Q8C22×C3semidirect
122. C2×C22⋊Q8C22×C3semidirect
128. C2×#117C22×C3semidirect
131. C2×#103C22×C3semidirect
134. C2×C22⋊C8C22×C3semidirect
135. C2×C22⋊C8C22×C3semidirect
136. #340C22×C3semidirect
252. C22⋊Q8C3×D4semidirect
538. C3#23926direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#5352
C2×C3×#4482
C2×C3×#4502
C3×#20002
C3×#18752
C3×#18732
C3×#17802
#239263