Finite Group Databasebeta

← C3×#32889 #10651496

Subgroups — C3×#32889 #10651496

Order 768.

3606 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×2−1+4C211yesmaximal, characteristic, complemented
2C2×C3×#454C214yesmaximal, complemented
3C3×#3006C212yesmaximal, complemented
4C3×#2436C211yesmaximal, characteristic, complemented
5C3×#1793C214yesmaximal, complemented
6C3×#1789C212yesmaximal, complemented
7C3×#1784C211yesmaximal, characteristic, complemented
8#32889C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×2−1+4—216no—
10C2×C3×2−1+4—24no—
11C2×C3×2−1+4—22no—
12C2×C3×2−1+4C2214yes—
13C22×C3×#41—22no—
14C22×C3×#41—22no—
15C22×C3×#41C2212yes—
16C23×C3×Q8—22no—
17C23×C3×Q8C2211yescharacteristic
18C2×C3×AGL(1,ℤ/8)C2214yescomplemented
19C2×C3×#124C2212yescomplemented
20C2×C3×#121C2211yescharacteristic
21C3×#454—216no—
22C3×#449—24no—
23C3×#448—22no—
24C3×#447—22no—
25C2×C3×#104C2218yescomplemented
26C3×#417C2214yescomplemented
27C2×C3×#101C2214yescomplemented
28C3×#407C2212yes—
29C3×#393—22no—
30C3×#391C2212yescomplemented
31C3×#386—22no—
32C3×C4×C22⋊C4C2211yescharacteristic
33C22×2−1+4C2×C311yescharacteristic, complemented
34C2×#454C2×C314yescomplemented
35#3006C2×C312yescomplemented
36#2436C2×C311yescharacteristic, complemented
37#1793C2×C314yescomplemented
38#1789C2×C312yescomplemented
39#1784C2×C311yescharacteristic, complemented
40C3×2−1+4—232no—
41C3×2−1+4—232no—
42C3×2−1+4—232no—
43C3×2−1+4—216no—
44C3×2−1+4—28no—
45C3×2−1+4D4116yescomplemented
46C2×C3×#41—232no—
47C2×C3×#41—28no—
48C2×C3×#41—28no—
49C2×C3×#41—232no—
50C2×C3×#41—216no—
51C2×C3×#41—28no—
52C2×C3×#41—28no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41D414yes—
56C2×C3×#41C2314yes—
57C22×C3×Q8—22no—
58C22×C3×Q8—232no—
59C22×C3×Q8—28no—
60C22×C3×Q8—28no—
61C22×C3×Q8—28no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8D412yes—
65C22×C3×Q8D412yes—
66C22×C3×Q8C2312yes—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—21no—
70C22×C3×D4D412yes—
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—24no—
73C23×C3×C4—22no—
74C23×C3×C4D412yes—
75C23×C3×C4C2311yescharacteristic
76C3×AGL(1,ℤ/8)—216no—
77C2×C3×SD16—24no—
78C2×C3×D8—24no—
79C2×C3×M16C2314yes—
80C3×#124—28no—
81C3×#121—22no—
82C3×#121—22no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2312yes—
88C3×#104—216no—
89C3×#103—24no—
90C3×#102—24no—
91C3×#101—28no—
92C3×C22⋊C8—22no—
93C3×C22⋊C8—22no—
94C3×#95—21no—
95C3×#95—24no—
96C2×2−1+4—216no—
97C2×2−1+4—24no—
98C2×2−1+4—22no—
99C2×2−1+4C22×C314yes—
100C22×#41—22no—
101C22×#41—22no—
102C22×#41C22×C312yes—
103C23×Q8—22no—
104C23×Q8C22×C311yescharacteristic
105C2×AGL(1,ℤ/8)C22×C314yescomplemented
106C2×#124C22×C312yescomplemented
107C2×#121C22×C311yescharacteristic
108#454—216no—
109#449—24no—
110#448—22no—
111#447—22no—
112C2×#104C22×C318yescomplemented
113#417C22×C314yescomplemented
114C2×#101C22×C314yescomplemented
115#407C22×C312yes—
116#393—22no—
117#391C22×C312yescomplemented
118#386—22no—
119C4×C22⋊C4C22×C311yescharacteristic
120C24×C3—22no—
121C24×C3—21no—
122C24×C3C2×D411yescharacteristic
123C3×#41—264no—
124C3×#41—264no—
125C3×#41—232no—
126C3×#41—216no—
127C3×#41—232no—
128C3×#41—216no—
129C3×#41—28no—
130C3×#41—232no—
131C3×#41—28no—
132C3×#41—232no—
133C3×#41—28no—
134C3×#41C2×D4116yes—
135C2×C3×Q8—216no—
136C2×C3×Q8—216no—
137C2×C3×Q8—28no—
138C2×C3×Q8—264no—
139C2×C3×Q8—232no—
140C2×C3×Q8—216no—
141C2×C3×Q8—28no—
142C2×C3×Q8—232no—
143C2×C3×Q8—22no—
144C2×C3×Q8—24no—
145C2×C3×Q8—28no—
146C2×C3×Q8—24no—
147C2×C3×Q8—24no—
148C2×C3×Q8C2×D414yes—
149C2×C3×Q8C2×D418yes—
150C2×C3×D4—216no—
151C2×C3×D4—216no—
152C2×C3×D4—28no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—28no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—24no—
159C2×C3×D4—44no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4C2×D414yes—
163C22×C3×C4—232no—
164C22×C3×C4—216no—
165C22×C3×C4—216no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—28no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2411yescharacteristic
180C3×SD16—48no—
181C3×D8—48no—
182C3×M16—28no—
183C2×C3×C8—24no—
184C3×C4⋊C4—22no—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—48no—
188C3×C42—24no—
1892−1+4—232no—
1902−1+4—232no—
1912−1+4—232no—
1922−1+4—216no—
1932−1+4—28no—
1942−1+4C3×D4116yescomplemented
195C2×#41—232no—
196C2×#41—28no—
197C2×#41—28no—
198C2×#41—232no—
199C2×#41—216no—
200C2×#41—28no—
201C2×#41—28no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41C3×D414yes—
205C2×#41C23×C314yes—
206C22×Q8—22no—
207C22×Q8—232no—
208C22×Q8—28no—
209C22×Q8—28no—
210C22×Q8—28no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8C3×D412yes—
214C22×Q8C3×D412yes—
215C22×Q8C23×C312yes—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4C3×D412yes—
220C22×D4C23×C311yescharacteristic
221C23×C4—24no—
222C23×C4—22no—
223C23×C4C3×D412yes—
224C23×C4C23×C311yescharacteristic
225AGL(1,ℤ/8)—216no—
226C2×SD16—24no—
227C2×D8—24no—
228C2×M16C23×C314yes—
229#124—28no—
230#121—22no—
231#121—22no—
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—24no—
236C2×C42C23×C312yes—
237#104—216no—
238#103—24no—
239#102—24no—
240#101—28no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243#95—21no—
244#95—24no—
245C23×C3—28no—
246C23×C3—22no—
247C23×C3—24no—
248C23×C3—22no—
249C23×C3—41no—
250C23×C3—44no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—44no—
254C23×C3—42no—
255C23×C3—41no—
256C23×C3#12011yescharacteristic
257C23×C3#12011yescharacteristic
258C23×C3C22×D411yescharacteristic
259C3×Q8—232no—
260C3×Q8—28no—
261C3×Q8—216no—
262C3×Q8—24no—
263C3×Q8—216no—
264C3×Q8#12018yes—
265C3×D4—216no—
266C3×D4—232no—
267C3×D4—28no—
268C3×D4—216no—
269C3×D4—24no—
270C3×D4—44no—
271C3×D4—82no—
272C3×D4—82no—
273C3×D4#12018yes—
274C2×C3×C4—28no—
275C2×C3×C4—28no—
276C2×C3×C4—264no—
277C2×C3×C4—216no—
278C2×C3×C4—28no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—216no—
285C2×C3×C4—24no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4#12018yes—
292C2×C3×C4#12012yes—
293C2×C3×C4C22×D414yes—
294C2×C3×C4C22×D412yes—
295C3×C8—44no—
296C24—22no—
297C24—21no—
298C24C2×C3×D411yescharacteristic
299#41—264no—
300#41—264no—
301#41—232no—
302#41—216no—
303#41—232no—
304#41—216no—
305#41—28no—
306#41—232no—
307#41—28no—
308#41—232no—
309#41—28no—
310#41C2×C3×D4116yes—
311C2×Q8—216no—
312C2×Q8—216no—
313C2×Q8—28no—
314C2×Q8—264no—
315C2×Q8—232no—
316C2×Q8—216no—
317C2×Q8—28no—
318C2×Q8—232no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—28no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8C2×C3×D414yes—
325C2×Q8C2×C3×D418yes—
326C2×D4—216no—
327C2×D4—216no—
328C2×D4—28no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—28no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—44no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4C2×C3×D414yes—
339C22×C4—232no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4C2×C3×D414yes—
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D412yes—
354C22×C4C2×C3×D412yes—
355C22×C4C24×C311yescharacteristic
356SD16—48no—
357D8—48no—
358M16—28no—
359C2×C8—24no—
360C4⋊C4—22no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—48no—
364C42—24no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—28no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—44no—
374C22×C3—416no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3—84no—
378C22×C3—81no—
379C22×C3—81no—
380C22×C3—81no—
381C22×C3#45412yescomplemented
382C22×C3C2×#12012yes—
383C22×C3C2×#12011yescharacteristic
384C3×C4—28no—
385C3×C4—28no—
386C3×C4—22no—
387C3×C4—82no—
388C3×C4—81no—
389C3×C4C2×#12014yes—
390C23—28no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—41no—
395C23—44no—
396C23—42no—
397C23—42no—
398C23—44no—
399C23—42no—
400C23—41no—
401C23C3×#12011yescharacteristic
402C23C3×#12011yescharacteristic
403C23C22×C3×D411yescharacteristic
404Q8—232no—
405Q8—28no—
406Q8—216no—
407Q8—24no—
408Q8—216no—
409Q8C3×#12018yes—
410D4—216no—
411D4—232no—
412D4—28no—
413D4—216no—
414D4—24no—
415D4—44no—
416D4—82no—
417D4—82no—
418D4C3×#12018yes—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—264no—
422C2×C4—216no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—216no—
430C2×C4—24no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4C3×#12012yes—
437C2×C4C3×#12018yes—
438C2×C4C22×C3×D414yes—
439C2×C4C22×C3×D412yes—
440C8—44no—
441C2×C3—22no—
442C2×C3—22no—
443C2×C3—41no—
444C2×C3—48no—
445C2×C3—161no—
446C2×C3#284611yescharacteristic
447C2×C3C2×#45411yescharacteristic
448C2×C3#302011yescharacteristic
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—28no—
457C22—44no—
458C22—416no—
459C22—44no—
460C22—44no—
461C22—84no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22C3×#45412yescomplemented
466C22C2×C3×#12012yes—
467C22C2×C3×#12011yescharacteristic
468C4—28no—
469C4—28no—
470C4—22no—
471C4—82no—
472C4—81no—
473C4C2×C3×#12014yes—
474C3#3288911yescharacteristic, direct product factor, Sylow
475C2—22no—
476C2—22no—
477C2—41no—
478C2—48no—
479C2—161no—
480C2C3×#284611yescharacteristic
481C2C2×C3×#45411yescharacteristic
482C2C3×#302011yescharacteristic
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. C22×C3×2−1+4C2semidirect
2. C2×C3×#454C2semidirect
3. C3×#3006C2semidirect
4. C3×#2436C2semidirect
5. C3×#1793C2semidirect
6. C3×#1789C2semidirect
7. C3×#1784C2semidirect
8. #32889C3direct (N × H)
18. C2×C3×AGL(1,ℤ/8)C22semidirect
19. C2×C3×#124C22semidirect
25. C2×C3×#104C22semidirect
26. C3×#417C22semidirect
27. C2×C3×#101C22semidirect
30. C3×#391C22semidirect
33. C22×2−1+4C2×C3semidirect
34. C2×#454C2×C3semidirect
35. #3006C2×C3semidirect
36. #2436C2×C3semidirect
37. #1793C2×C3semidirect
38. #1789C2×C3semidirect
39. #1784C2×C3semidirect
45. C3×2−1+4D4semidirect
105. C2×AGL(1,ℤ/8)C22×C3semidirect
106. C2×#124C22×C3semidirect
112. C2×#104C22×C3semidirect
113. #417C22×C3semidirect
114. C2×#101C22×C3semidirect
117. #391C22×C3semidirect
194. 2−1+4C3×D4semidirect
381. C22×C3#454semidirect
465. C22C3×#454semidirect
474. C3#32889direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×2−1+42
C2×C3×#4542
C3×#30062
C3×#24362
C3×#17932
C3×#17892
C3×#17842
#328893