Finite Group Databasebeta

← C3×#29596 #10648203

Subgroups — C3×#29596 #10648203

Order 768.

1786 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic, complemented
3C2×C3×#450C211yesmaximal, characteristic, complemented
4C3×#2848C211yesmaximal, characteristic, complemented
5C3×#2034C214yesmaximal, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1823C211yesmaximal, characteristic, complemented
8C3×#1786C211yesmaximal, characteristic, complemented
9C3×#1779C211yesmaximal, characteristic, complemented
10#29596C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4—21no—
12C23×C3×D4C2211yescharacteristic
13C24×C3×C4C2211yescharacteristic
14C22×C3×SD16C2212yescomplemented
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yescomplemented
17C2×C3×#121—24no—
18C2×C3×#121—22no—
19C2×C3×#121—24no—
20C2×C3×#121C2214yes—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4—21no—
23C3×#483—24no—
24C3×#450—24no—
25C2×C3×#106C2212yescomplemented
26C2×C3×#103C2212yescomplemented
27C2×C3×#102C2212yescomplemented
28C2×C3×#102C2214yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2214yescomplemented
32C3×#385—22no—
33C3×#385—21no—
34C3×#380C2211yescharacteristic
35C3×#380C2212yescomplemented
36C3×#379—21no—
37C3×#340C2214yescomplemented
38C22×#121C2×C311yescharacteristic
39C2×#483C2×C311yescharacteristic, complemented
40C2×#450C2×C311yescharacteristic, complemented
41#2848C2×C311yescharacteristic, complemented
42#2034C2×C314yescomplemented
43#2000C2×C314yescomplemented
44#1823C2×C311yescharacteristic, complemented
45#1786C2×C311yescharacteristic, complemented
46#1779C2×C311yescharacteristic, complemented
47C25×C3—21no—
48C25×C3D411yescharacteristic
49C22×C3×Q8C2311yescharacteristic
50C22×C3×D4—28no—
51C22×C3×D4—24no—
52C22×C3×D4—21no—
53C22×C3×D4—22no—
54C22×C3×D4—44no—
55C22×C3×D4—44no—
56C22×C3×D4—44no—
57C22×C3×D4D414yes—
58C22×C3×D4C2312yes—
59C23×C3×C4—21no—
60C23×C3×C4—22no—
61C23×C3×C4—22no—
62C23×C3×C4—44no—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D412yes—
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C2×C3×SD16—24no—
68C2×C3×SD16—44no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—44no—
74C3×#121—216no—
75C3×#121—28no—
76C3×#121—28no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D414yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—28no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—22no—
91C3×#106—24no—
92C3×#103—24no—
93C3×#102—24no—
94C3×#102—44no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97C3×C22⋊C8—44no—
98C3×#95—21no—
99C3×#95—22no—
100C23×D4—21no—
101C23×D4C22×C311yescharacteristic
102C24×C4C22×C311yescharacteristic
103C22×SD16C22×C312yescomplemented
104C2×C22⋊Q8C22×C311yescharacteristic
105C2×C22⋊Q8C22×C312yescomplemented
106C2×#121—24no—
107C2×#121—22no—
108C2×#121—24no—
109C2×#121C22×C314yes—
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C22⋊C4—21no—
112#483—24no—
113#450—24no—
114C2×#106C22×C312yescomplemented
115C2×#103C22×C312yescomplemented
116C2×#102C22×C312yescomplemented
117C2×#102C22×C314yescomplemented
118C2×C22⋊C8C22×C311yescharacteristic
119C2×C22⋊C8C22×C311yescharacteristic
120C2×C22⋊C8C22×C314yescomplemented
121#385—22no—
122#385—21no—
123#380C22×C311yescharacteristic
124#380C22×C312yescomplemented
125#379—21no—
126#340C22×C314yescomplemented
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—24no—
130C24×C3—22no—
131C24×C3—42no—
132C24×C3—42no—
133C24×C3—44no—
134C24×C3—44no—
135C24×C3—42no—
136C24×C3—42no—
137C24×C3SD1614yescomplemented
138C24×C3C2×D412yes—
139C24×C3C2×D411yescharacteristic
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—24no—
143C2×C3×D4—216no—
144C2×C3×D4—24no—
145C2×C3×D4—22no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—48no—
149C2×C3×D4—44no—
150C2×C3×D4—48no—
151C2×C3×D4—48no—
152C2×C3×D4—432no—
153C2×C3×D4—48no—
154C2×C3×D4C2×D414yes—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4—48no—
171C22×C3×C4—48no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2411yescharacteristic
179C3×SD16—48no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—24no—
183C2×C3×C8—24no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4C2×D414yes—
192C3×C22⋊C4—216no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—42no—
197C25—21no—
198C25C3×D411yescharacteristic
199C22×Q8C23×C311yescharacteristic
200C22×D4—28no—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4C3×D414yes—
208C22×D4C23×C312yes—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—44no—
213C23×C4C3×D411yescharacteristic
214C23×C4C3×D412yes—
215C23×C4C23×C312yes—
216C23×C4C23×C311yescharacteristic
217C2×SD16—24no—
218C2×SD16—44no—
219C22×C8C23×C312yes—
220C22×C8C23×C312yes—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8—44no—
224#121—216no—
225#121—28no—
226#121—28no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4C3×D414yes—
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C4⋊C4C23×C312yes—
234C2×C4⋊C4C23×C312yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241#106—24no—
242#103—24no—
243#102—24no—
244#102—44no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—44no—
248#95—21no—
249#95—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—28no—
259C23×C3—24no—
260C23×C3—48no—
261C23×C3—44no—
262C23×C3—48no—
263C23×C3—44no—
264C23×C3—48no—
265C23×C3—48no—
266C23×C3—44no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—44no—
270C23×C3—48no—
271C23×C3—88no—
272C23×C3—88no—
273C23×C3#12012yes—
274C23×C3C2×SD1614yes—
275C23×C3C2×SD1612yes—
276C23×C3AGL(1,ℤ/8)12yes—
277C23×C3C22×D411yescharacteristic
278C3×Q8—42no—
279C3×Q8—42no—
280C3×D4—24no—
281C3×D4—24no—
282C3×D4—48no—
283C3×D4—432no—
284C3×D4—44no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—48no—
293C2×C3×C4—44no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—48no—
297C2×C3×C4—44no—
298C2×C3×C4—44no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4#12012yes—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4H(5,𝔽2)14yes—
310C3×C8—42no—
311C3×C8—42no—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—22no—
316C24—42no—
317C24—42no—
318C24—44no—
319C24—44no—
320C24—42no—
321C24—42no—
322C24C3×SD1614yescomplemented
323C24C2×C3×D412yes—
324C24C2×C3×D411yescharacteristic
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—24no—
328C2×D4—216no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—48no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—432no—
338C2×D4—48no—
339C2×D4C2×C3×D414yes—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D412yes—
362C22×C4C2×C3×D412yes—
363C22×C4C24×C311yescharacteristic
364SD16—48no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—24no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4C2×C3×D414yes—
377C22⋊C4—216no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22×C3—22no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—22no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—24no—
389C22×C3—416no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—816no—
398C22×C3—84no—
399C22×C3—84no—
400C22×C3#45014yes—
401C22×C3C2×#12011yescharacteristic
402C22×C3#53412yes—
403C22×C3C22×SD1611yescharacteristic
404C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
405C22×C3#57512yes—
406C3×C4—22no—
407C3×C4—42no—
408C3×C4—44no—
409C3×C4—81no—
410C3×C4—82no—
411C3×C4—81no—
412C3×C4#53412yes—
413C3×C4#53512yes—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—28no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—48no—
425C23—44no—
426C23—48no—
427C23—44no—
428C23—48no—
429C23—48no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—48no—
434C23—44no—
435C23—88no—
436C23—88no—
437C23C3×#12012yes—
438C23C2×C3×SD1614yes—
439C23C2×C3×SD1612yes—
440C23C3×AGL(1,ℤ/8)12yes—
441C23C22×C3×D411yescharacteristic
442Q8—42no—
443Q8—42no—
444D4—24no—
445D4—24no—
446D4—48no—
447D4—432no—
448D4—44no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—48no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4C3×#12012yes—
471C2×C4C22×C3×D411yescharacteristic
472C2×C4C22×C3×D411yescharacteristic
473C2×C4C3×H(5,𝔽2)14yes—
474C8—42no—
475C8—42no—
476C2×C3—24no—
477C2×C3—44no—
478C2×C3—84no—
479C2×C3#261311yescharacteristic
480C2×C3C2×#45011yescharacteristic
481C2×C3#300511yescharacteristic
482C2×C3#318712yes—
483C2×C3#319412yes—
484C22—24no—
485C22—24no—
486C22—22no—
487C22—22no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—416no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—816no—
500C22—84no—
501C22—84no—
502C22C3×#45014yes—
503C22C2×C3×#12011yescharacteristic
504C22C3×#53412yes—
505C22C22×C3×SD1611yescharacteristic
506C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
507C22C3×#57512yes—
508C4—22no—
509C4—44no—
510C4—42no—
511C4—82no—
512C4—81no—
513C4—81no—
514C4C3×#53412yes—
515C4C3×#53512yes—
516C3#2959611yescharacteristic, direct product factor, Sylow
517C2—24no—
518C2—44no—
519C2—84no—
520C2C3×#261311yescharacteristic
521C2C2×C3×#45011yescharacteristic
522C2C3×#300511yescharacteristic
523C2C3×#318712yes—
524C2C3×#319412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#483C2semidirect
3. C2×C3×#450C2semidirect
4. C3×#2848C2semidirect
5. C3×#2034C2semidirect
6. C3×#2000C2semidirect
7. C3×#1823C2semidirect
8. C3×#1786C2semidirect
9. C3×#1779C2semidirect
10. #29596C3direct (N × H)
14. C22×C3×SD16C22semidirect
16. C2×C3×C22⋊Q8C22semidirect
25. C2×C3×#106C22semidirect
26. C2×C3×#103C22semidirect
27. C2×C3×#102C22semidirect
28. C2×C3×#102C22semidirect
31. C2×C3×C22⋊C8C22semidirect
35. C3×#380C22semidirect
37. C3×#340C22semidirect
39. C2×#483C2×C3semidirect
40. C2×#450C2×C3semidirect
41. #2848C2×C3semidirect
42. #2034C2×C3semidirect
43. #2000C2×C3semidirect
44. #1823C2×C3semidirect
45. #1786C2×C3semidirect
46. #1779C2×C3semidirect
103. C22×SD16C22×C3semidirect
105. C2×C22⋊Q8C22×C3semidirect
114. C2×#106C22×C3semidirect
115. C2×#103C22×C3semidirect
116. C2×#102C22×C3semidirect
117. C2×#102C22×C3semidirect
120. C2×C22⋊C8C22×C3semidirect
124. #380C22×C3semidirect
126. #340C22×C3semidirect
137. C24×C3SD16semidirect
322. C24C3×SD16semidirect
516. C3#29596direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C2×C3×#4832
C2×C3×#4502
C3×#28482
C3×#20342
C3×#20002
C3×#18232
C3×#17862
C3×#17792
#295963