Finite Group Databasebeta

← C3×#18132 #10636739

Subgroups — C3×#18132 #10636739

Order 768.

1654 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#534C211yesmaximal, characteristic, complemented
2C3×#3006C212yesmaximal, complemented
3C3×#2404C211yesmaximal, characteristic, complemented
4C3×#1601C214yesmaximal, complemented
5C3×#1595C214yesmaximal, complemented
6C3×#1510C212yesmaximal, complemented
7C3×#1462C211yesmaximal, characteristic, complemented
8#18132C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2211yescharacteristic, complemented
10C23×C3×D4—21no—
11C2×C3×AGL(1,ℤ/8)—22no—
12C3×#534—24no—
13C3×#534—22no—
14C3×#534C2214yes—
15C2×C3×#123—22no—
16C2×C3×#121—21no—
17C2×C3×#121C2212yes—
18C2×C3×#120—22no—
19C2×C3×#120—21no—
20C22×C3×C22⋊C4C2211yescharacteristic
21C3×#449—24no—
22C3×#448C2214yescomplemented
23C3×#447C2214yescomplemented
24C3×#417—22no—
25C3×#407C2212yescomplemented
26C3×#393C2212yescomplemented
27C3×#392C2212yescomplemented
28C3×#386—21no—
29C3×#386—22no—
30C3×#381C2211yescharacteristic, complemented
31C3×#327C2218yescomplemented
32C3×#323C2214yescomplemented
33C2×#534C2×C311yescharacteristic, complemented
34#3006C2×C312yescomplemented
35#2404C2×C311yescharacteristic, complemented
36#1601C2×C314yescomplemented
37#1595C2×C314yescomplemented
38#1510C2×C312yescomplemented
39#1462C2×C311yescharacteristic, complemented
40C25×C3—21no—
41C25×C3D411yescharacteristic
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—44no—
45C22×C3×Q8C2311yescharacteristic, complemented
46C22×C3×D4—22no—
47C22×C3×D4—24no—
48C22×C3×D4—22no—
49C22×C3×D4—24no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4—22no—
53C22×C3×D4—48no—
54C22×C3×D4D412yescomplemented
55C22×C3×D4C2311yescharacteristic, complemented
56C23×C3×C4—21no—
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C3×AGL(1,ℤ/8)—48no—
61C2×C3×SD16—24no—
62C2×C3×D8—24no—
63C2×C3×M16—22no—
64C3×#123—28no—
65C3×#123—28no—
66C3×#121—24no—
67C3×#121—24no—
68C3×#121—22no—
69C3×#121—22no—
70C3×#121D414yescomplemented
71C3×#121C2314yes—
72C3×#120—28no—
73C3×#120—28no—
74C3×#120—44no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D412yescomplemented
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4D412yes—
85C2×C3×C22⋊C4C2312yes—
86C3×#103—24no—
87C3×#102—24no—
88C3×C22⋊C8—22no—
89C3×C22⋊C8C2314yescomplemented
90C3×#95—21no—
91C3×#95—22no—
92C3×#95C2312yescomplemented
93C22×#41C22×C311yescharacteristic, complemented
94C23×D4—21no—
95C2×AGL(1,ℤ/8)—22no—
96#534—24no—
97#534—22no—
98#534C22×C314yes—
99C2×#123—22no—
100C2×#121—21no—
101C2×#121C22×C312yes—
102C2×#120—22no—
103C2×#120—21no—
104C22×C22⋊C4C22×C311yescharacteristic
105#449—24no—
106#448C22×C314yescomplemented
107#447C22×C314yescomplemented
108#417—22no—
109#407C22×C312yescomplemented
110#393C22×C312yescomplemented
111#392C22×C312yescomplemented
112#386—21no—
113#386—22no—
114#381C22×C311yescharacteristic, complemented
115#327C22×C318yescomplemented
116#323C22×C314yescomplemented
117C24×C3—21no—
118C24×C3—21no—
119C24×C3—24no—
120C24×C3—24no—
121C24×C3—22no—
122C24×C3—21no—
123C24×C3—44no—
124C24×C3—42no—
125C24×C3—81no—
126C24×C3—82no—
127C24×C3—82no—
128C24×C3C2×D411yescharacteristic
129C24×C3C2×D412yes—
130C3×#41—28no—
131C3×#41—44no—
132C3×#41—44no—
133C3×#41—44no—
134C2×C3×Q8—22no—
135C2×C3×Q8—42no—
136C2×C3×D4—28no—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—44no—
144C2×C3×D4—42no—
145C2×C3×D4—48no—
146C2×C3×D4—416no—
147C2×C3×D4—48no—
148C2×C3×D4—48no—
149C2×C3×D4—42no—
150C2×C3×D4—41no—
151C2×C3×D4—41no—
152C2×C3×D4—48no—
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—82no—
156C2×C3×D4—82no—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—24no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—44no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4—81no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2411yescharacteristic
175C3×SD16—48no—
176C3×D8—48no—
177C3×M16—44no—
178C2×C3×C8—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4C2×D414yes—
182C3×C22⋊C4—28no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—42no—
189C3×C22⋊C4—82no—
190C25—21no—
191C25C3×D411yescharacteristic
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—44no—
195C22×Q8C23×C311yescharacteristic, complemented
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—48no—
204C22×D4C3×D412yescomplemented
205C22×D4C23×C311yescharacteristic, complemented
206C23×C4—21no—
207C23×C4—21no—
208C23×C4C3×D411yescharacteristic
209C23×C4C23×C311yescharacteristic
210AGL(1,ℤ/8)—48no—
211C2×SD16—24no—
212C2×D8—24no—
213C2×M16—22no—
214#123—28no—
215#123—28no—
216#121—24no—
217#121—24no—
218#121—22no—
219#121—22no—
220#121C3×D414yescomplemented
221#121C23×C314yes—
222#120—28no—
223#120—28no—
224#120—44no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C3×D412yescomplemented
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4C3×D412yes—
235C2×C22⋊C4C23×C312yes—
236#103—24no—
237#102—24no—
238C22⋊C8—22no—
239C22⋊C8C23×C314yescomplemented
240#95—21no—
241#95—22no—
242#95C23×C312yescomplemented
243C23×C3—24no—
244C23×C3—24no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—24no—
251C23×C3—41no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—44no—
258C23×C3—48no—
259C23×C3—44no—
260C23×C3—42no—
261C23×C3—41no—
262C23×C3—42no—
263C23×C3—44no—
264C23×C3—81no—
265C23×C3—81no—
266C23×C3—88no—
267C23×C3—84no—
268C23×C3—84no—
269C23×C3—816no—
270C23×C3#12011yescharacteristic
271C23×C3#12011yescharacteristic
272C23×C3AGL(1,ℤ/8)14yescomplemented
273C23×C3C22×D411yescharacteristic
274C3×Q8—24no—
275C3×Q8—42no—
276C3×D4—24no—
277C3×D4—44no—
278C3×D4—42no—
279C3×D4—416no—
280C3×D4—48no—
281C3×D4—44no—
282C3×D4—48no—
283C3×D4—44no—
284C3×D4—82no—
285C3×D4—84no—
286C3×D4—88no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—48no—
295C2×C3×C4—44no—
296C2×C3×C4—48no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—81no—
305C2×C3×C4#12012yes—
306C2×C3×C4C22×D412yes—
307C3×C8—44no—
308C24—21no—
309C24—21no—
310C24—24no—
311C24—24no—
312C24—22no—
313C24—21no—
314C24—44no—
315C24—42no—
316C24—81no—
317C24—82no—
318C24—82no—
319C24C2×C3×D411yescharacteristic
320C24C2×C3×D412yes—
321#41—28no—
322#41—44no—
323#41—44no—
324#41—44no—
325C2×Q8—22no—
326C2×Q8—42no—
327C2×D4—28no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—48no—
337C2×D4—416no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—48no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4C2×C3×D414yes—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—81no—
361C22×C4C2×C3×D412yes—
362C22×C4C2×C3×D412yes—
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D412yes—
365C22×C4C24×C311yescharacteristic
366SD16—48no—
367D8—48no—
368M16—44no—
369C2×C8—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4C2×C3×D414yes—
373C22⋊C4—28no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—82no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—24no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—48no—
392C22×C3—44no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—88no—
402C22×C3—84no—
403C22×C3—816no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3B(4,𝔽2)12yes—
407C22×C3C2×#12011yescharacteristic
408C22×C3C2×AGL(1,ℤ/8)12yes—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—44no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—82no—
415C23—24no—
416C23—24no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—44no—
430C23—48no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—41no—
436C23—81no—
437C23—81no—
438C23—88no—
439C23—84no—
440C23—84no—
441C23—816no—
442C23C3×#12011yescharacteristic
443C23C3×#12011yescharacteristic
444C23C3×AGL(1,ℤ/8)14yescomplemented
445C23C22×C3×D411yescharacteristic
446Q8—24no—
447Q8—42no—
448D4—24no—
449D4—44no—
450D4—42no—
451D4—416no—
452D4—48no—
453D4—44no—
454D4—48no—
455D4—44no—
456D4—82no—
457D4—84no—
458D4—88no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—81no—
477C2×C4C3×#12012yes—
478C2×C4C22×C3×D412yes—
479C8—44no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—41no—
483C2×C3—44no—
484C2×C3—82no—
485C2×C3—84no—
486C2×C3#300311yescharacteristic
487C2×C3#301911yescharacteristic
488C2×C3C2×B(4,𝔽2)11yescharacteristic
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—24no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—44no—
498C22—44no—
499C22—48no—
500C22—44no—
501C22—42no—
502C22—42no—
503C22—44no—
504C22—44no—
505C22—42no—
506C22—44no—
507C22—82no—
508C22—88no—
509C22—82no—
510C22—84no—
511C22—816no—
512C22—81no—
513C22—81no—
514C22C3×B(4,𝔽2)12yes—
515C22C2×C3×#12011yescharacteristic
516C22C2×C3×AGL(1,ℤ/8)12yes—
517C4—22no—
518C4—22no—
519C4—44no—
520C4—42no—
521C4—42no—
522C4—82no—
523C3#1813211yescharacteristic, direct product factor, Sylow
524C2—22no—
525C2—22no—
526C2—41no—
527C2—44no—
528C2—84no—
529C2—82no—
530C2C3×#300311yescharacteristic
531C2C3×#301911yescharacteristic
532C2C2×C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#534C2semidirect
2. C3×#3006C2semidirect
3. C3×#2404C2semidirect
4. C3×#1601C2semidirect
5. C3×#1595C2semidirect
6. C3×#1510C2semidirect
7. C3×#1462C2semidirect
8. #18132C3direct (N × H)
9. C22×C3×#41C22semidirect
22. C3×#448C22semidirect
23. C3×#447C22semidirect
25. C3×#407C22semidirect
26. C3×#393C22semidirect
27. C3×#392C22semidirect
30. C3×#381C22semidirect
31. C3×#327C22semidirect
32. C3×#323C22semidirect
33. C2×#534C2×C3semidirect
34. #3006C2×C3semidirect
35. #2404C2×C3semidirect
36. #1601C2×C3semidirect
37. #1595C2×C3semidirect
38. #1510C2×C3semidirect
39. #1462C2×C3semidirect
45. C22×C3×Q8C23semidirect
54. C22×C3×D4D4semidirect
55. C22×C3×D4C23semidirect
70. C3×#121D4semidirect
76. C2×C3×C4⋊C4D4semidirect
89. C3×C22⋊C8C23semidirect
92. C3×#95C23semidirect
93. C22×#41C22×C3semidirect
106. #448C22×C3semidirect
107. #447C22×C3semidirect
109. #407C22×C3semidirect
110. #393C22×C3semidirect
111. #392C22×C3semidirect
114. #381C22×C3semidirect
115. #327C22×C3semidirect
116. #323C22×C3semidirect
195. C22×Q8C23×C3semidirect
204. C22×D4C3×D4semidirect
205. C22×D4C23×C3semidirect
220. #121C3×D4semidirect
226. C2×C4⋊C4C3×D4semidirect
239. C22⋊C8C23×C3semidirect
242. #95C23×C3semidirect
272. C23×C3AGL(1,ℤ/8)semidirect
444. C23C3×AGL(1,ℤ/8)semidirect
523. C3#18132direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5342
C3×#30062
C3×#24042
C3×#16012
C3×#15952
C3×#15102
C3×#14622
#181323