Finite Group Databasebeta

← C3×#23948 #10642555

Subgroups — C3×#23948 #10642555

Order 768.

1474 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3492C211yesmaximal, characteristic, complemented
2C3×#3003C212yesmaximal, complemented
3C2×C3×#390C211yesmaximal, characteristic, complemented
4C3×#2027C212yesmaximal, complemented
5C3×#2006C212yesmaximal, complemented
6C3×#1896C212yesmaximal, complemented
7C3×#1883C214yesmaximal, complemented
8C3×#1821C211yesmaximal, characteristic, complemented
9#23948C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C2×C3×AGL(1,ℤ/8)C2214yes—
12C3×#554—22no—
13C3×#553—22no—
14C3×#552—22no—
15C3×#547—22no—
16C3×#544C2211yescharacteristic, complemented
17C3×#539—22no—
18C3×#537C2211yescharacteristic, complemented
19C3×#525—22no—
20C3×#525C2212yes—
21C3×#520C2212yescomplemented
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4—22no—
24C22×C3×C42C2211yescharacteristic
25C3×#450—22no—
26C3×#450—22no—
27C3×#447—22no—
28C3×#447—22no—
29C2×C3×#104C2214yescomplemented
30C3×#418C2214yes—
31C3×#407C2212yes—
32C3×#390—22no—
33C3×#390C2212yes—
34C3×#390C2212yes—
35C3×#343C2212yescomplemented
36C3×#343C2212yescomplemented
37C3×#339C2214yescomplemented
38#3492C2×C311yescharacteristic, complemented
39#3003C2×C312yescomplemented
40C2×#390C2×C311yescharacteristic, complemented
41#2027C2×C312yescomplemented
42#2006C2×C312yescomplemented
43#1896C2×C312yescomplemented
44#1883C2×C314yescomplemented
45#1821C2×C311yescharacteristic, complemented
46C25×C3—21no—
47C2×C3×#41—22no—
48C2×C3×#41C2312yes—
49C22×C3×D4—24no—
50C22×C3×D4—21no—
51C22×C3×D4—22no—
52C22×C3×D4—22no—
53C22×C3×D4—42no—
54C22×C3×D4C2312yes—
55C23×C3×C4—21no—
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C3×AGL(1,ℤ/8)—28no—
59C3×AGL(1,ℤ/8)—44no—
60C2×C3×SD16—24no—
61C2×C3×D8—24no—
62C2×C3×M16C2314yes—
63C3×C4⋊Q8—22no—
64C3×#126—24no—
65C3×#125—24no—
66C3×#125—22no—
67C3×#123—24no—
68C3×#123—24no—
69C3×#123—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8D412yes—
75C3×C22⋊Q8D412yes—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121D412yes—
80C3×#121D412yes—
81C3×#120—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—24no—
84C3×C4×D4—22no—
85C3×C4×D4—24no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117C2312yes—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—24no—
98C2×C3×C22⋊C4—24no—
99C2×C3×C22⋊C4—48no—
100C2×C3×C42—21no—
101C2×C3×C42—42no—
102C2×C3×C42C2312yes—
103C3×#104—44no—
104C3×#102—24no—
105C3×#102—24no—
106C3×C22⋊C8—22no—
107C3×C22⋊C8—22no—
108C23×D4C22×C311yescharacteristic
109C2×AGL(1,ℤ/8)C22×C314yes—
110#554—22no—
111#553—22no—
112#552—22no—
113#547—22no—
114#544C22×C311yescharacteristic, complemented
115#539—22no—
116#537C22×C311yescharacteristic, complemented
117#525—22no—
118#525C22×C312yes—
119#520C22×C312yescomplemented
120C22×C4⋊C4C22×C311yescharacteristic
121C22×C22⋊C4—22no—
122C22×C42C22×C311yescharacteristic
123#450—22no—
124#450—22no—
125#447—22no—
126#447—22no—
127C2×#104C22×C314yescomplemented
128#418C22×C314yes—
129#407C22×C312yes—
130#390—22no—
131#390C22×C312yes—
132#390C22×C312yes—
133#343C22×C312yescomplemented
134#343C22×C312yescomplemented
135#339C22×C314yescomplemented
136C24×C3—24no—
137C24×C3—22no—
138C24×C3—44no—
139C24×C3—44no—
140C24×C3—42no—
141C24×C3—41no—
142C24×C3—41no—
143C24×C3C2×D411yescharacteristic
144C3×#41—24no—
145C3×#41—24no—
146C3×#41—84no—
147C2×C3×Q8—22no—
148C2×C3×Q8C2×D412yes—
149C2×C3×D4—24no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—24no—
157C2×C3×D4—48no—
158C2×C3×D4—48no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4C2×D412yes—
162C2×C3×D4#4114yescomplemented
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4—44no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2411yescharacteristic
180C3×SD16—48no—
181C3×D8—48no—
182C3×M16—24no—
183C3×M16—24no—
184C2×C3×C8—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D412yes—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—48no—
204C3×C22⋊C4—48no—
205C3×C42—24no—
206C3×C42—22no—
207C3×C42—24no—
208C3×C42—42no—
209C25—21no—
210C2×#41—22no—
211C2×#41C23×C312yes—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—42no—
217C22×D4C23×C312yes—
218C23×C4—21no—
219C23×C4C3×D412yes—
220C23×C4C23×C311yescharacteristic
221AGL(1,ℤ/8)—28no—
222AGL(1,ℤ/8)—44no—
223C2×SD16—24no—
224C2×D8—24no—
225C2×M16C23×C314yes—
226C4⋊Q8—22no—
227#126—24no—
228#125—24no—
229#125—22no—
230#123—24no—
231#123—24no—
232#123—22no—
233C22⋊Q8—22no—
234C22⋊Q8—24no—
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237C22⋊Q8C3×D412yes—
238C22⋊Q8C3×D412yes—
239#121—22no—
240#121—22no—
241#121—22no—
242#121C3×D412yes—
243#121C3×D412yes—
244#120—22no—
245C4×Q8—22no—
246C4×D4—24no—
247C4×D4—22no—
248C4×D4—24no—
249C4×D4—22no—
250C4×D4—22no—
251#117—22no—
252#117—22no—
253#117C23×C312yes—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C3×D412yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—48no—
263C2×C42—21no—
264C2×C42—42no—
265C2×C42C23×C312yes—
266#104—44no—
267#102—24no—
268#102—24no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—24no—
277C23×C3—44no—
278C23×C3—44no—
279C23×C3—44no—
280C23×C3—416no—
281C23×C3—44no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—88no—
285C23×C3—82no—
286C23×C3—82no—
287C23×C3—82no—
288C23×C3—82no—
289C23×C3#12011yescharacteristic
290C23×C3#12011yescharacteristic
291C23×C3C22×D411yescharacteristic
292C3×Q8—24no—
293C3×Q8—82no—
294C3×D4—24no—
295C3×D4—28no—
296C3×D4—48no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—44no—
300C3×D4—44no—
301C3×D4—44no—
302C3×D4—82no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—48no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—44no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—82no—
321C2×C3×C4—84no—
322C2×C3×C4#12012yes—
323C2×C3×C4C22×D412yes—
324C2×C3×C4C2×#4112yes—
325C2×C3×C4H(5,𝔽2)12yes—
326C3×C8—44no—
327C24—24no—
328C24—22no—
329C24—44no—
330C24—42no—
331C24—44no—
332C24—41no—
333C24—41no—
334C24C2×C3×D411yescharacteristic
335#41—24no—
336#41—24no—
337#41—84no—
338C2×Q8—22no—
339C2×Q8C2×C3×D412yes—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4—24no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4C2×C3×D412yes—
353C2×D4C3×#4114yescomplemented
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D412yes—
367C22×C4C2×C3×D412yes—
368C22×C4C2×C3×D412yes—
369C22×C4C2×C3×D412yes—
370C22×C4C24×C311yescharacteristic
371SD16—48no—
372D8—48no—
373M16—24no—
374M16—24no—
375C2×C8—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4C2×C3×D412yes—
386C4⋊C4C2×C3×D412yes—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C42—24no—
397C42—22no—
398C42—24no—
399C42—42no—
400C22×C3—24no—
401C22×C3—24no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—48no—
409C22×C3—42no—
410C22×C3—44no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—84no—
414C22×C3—88no—
415C22×C3—88no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3#45312yes—
419C22×C3#45412yes—
420C22×C3C2×#12011yescharacteristic
421C22×C3#52511yescharacteristic
422C22×C3#53511yescharacteristic
423C3×C4—22no—
424C3×C4—44no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4#54614yes—
431C23—22no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—22no—
436C23—24no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—416no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—88no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23C3×#12011yescharacteristic
450C23C3×#12011yescharacteristic
451C23C22×C3×D411yescharacteristic
452Q8—24no—
453Q8—82no—
454D4—24no—
455D4—28no—
456D4—48no—
457D4—44no—
458D4—44no—
459D4—42no—
460D4—42no—
461D4—44no—
462D4—82no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4C3×#12012yes—
483C2×C4C22×C3×D412yes—
484C2×C4C2×C3×#4112yes—
485C2×C4C3×H(5,𝔽2)12yes—
486C8—44no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—22no—
490C2×C3—22no—
491C2×C3—82no—
492C2×C3—84no—
493C2×C3#240411yescharacteristic
494C2×C3C2×#45311yescharacteristic
495C2×C3C2×#45411yescharacteristic
496C22—24no—
497C22—24no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—48no—
505C22—42no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—84no—
510C22—88no—
511C22—88no—
512C22—82no—
513C22—82no—
514C22C3×#45312yes—
515C22C3×#45412yes—
516C22C2×C3×#12011yescharacteristic
517C22C3×#52511yescharacteristic
518C22C3×#53511yescharacteristic
519C4—22no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4C3×#54614yes—
527C3#2394811yescharacteristic, direct product factor, Sylow
528C2—21no—
529C2—21no—
530C2—22no—
531C2—22no—
532C2—84no—
533C2—82no—
534C2C3×#240411yescharacteristic
535C2C2×C3×#45311yescharacteristic
536C2C2×C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3492C2semidirect
2. C3×#3003C2semidirect
3. C2×C3×#390C2semidirect
4. C3×#2027C2semidirect
5. C3×#2006C2semidirect
6. C3×#1896C2semidirect
7. C3×#1883C2semidirect
8. C3×#1821C2semidirect
9. #23948C3direct (N × H)
16. C3×#544C22semidirect
18. C3×#537C22semidirect
21. C3×#520C22semidirect
29. C2×C3×#104C22semidirect
35. C3×#343C22semidirect
36. C3×#343C22semidirect
37. C3×#339C22semidirect
38. #3492C2×C3semidirect
39. #3003C2×C3semidirect
40. C2×#390C2×C3semidirect
41. #2027C2×C3semidirect
42. #2006C2×C3semidirect
43. #1896C2×C3semidirect
44. #1883C2×C3semidirect
45. #1821C2×C3semidirect
114. #544C22×C3semidirect
116. #537C22×C3semidirect
119. #520C22×C3semidirect
127. C2×#104C22×C3semidirect
133. #343C22×C3semidirect
134. #343C22×C3semidirect
135. #339C22×C3semidirect
162. C2×C3×D4#41semidirect
353. C2×D4C3×#41semidirect
527. C3#23948direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34922
C3×#30032
C2×C3×#3902
C3×#20272
C3×#20062
C3×#18962
C3×#18832
C3×#18212
#239483