Finite Group Databasebeta

← C3×#29598 #10648205

Subgroups — C3×#29598 #10648205

Order 768.

1786 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic, complemented
3C2×C3×#450C211yesmaximal, characteristic, complemented
4C3×#2848C211yesmaximal, characteristic, complemented
5C3×#2033C214yesmaximal, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic, complemented
8C3×#1786C211yesmaximal, characteristic, complemented
9C3×#1779C211yesmaximal, characteristic, complemented
10#29598C311yesmaximal, 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×#484—24no—
24C3×#450—24no—
25C2×C3×#107C2212yescomplemented
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×#484C2×C311yescharacteristic, complemented
40C2×#450C2×C311yescharacteristic, complemented
41#2848C2×C311yescharacteristic, complemented
42#2033C2×C314yescomplemented
43#2000C2×C314yescomplemented
44#1824C2×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—48no—
56C22×C3×D4D414yes—
57C22×C3×D4C2312yes—
58C23×C3×C4—21no—
59C23×C3×C4—22no—
60C23×C3×C4—22no—
61C23×C3×C4—44no—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D412yes—
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C2×C3×SD16—24no—
67C2×C3×SD16—44no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—44no—
73C3×#121—216no—
74C3×#121—28no—
75C3×#121—28no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4D414yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—28no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—22no—
90C3×#107—24no—
91C3×#103—24no—
92C3×#102—24no—
93C3×#102—44no—
94C3×C22⋊C8—22no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—44no—
97C3×#95—21no—
98C3×#95—22no—
99C23×D4—21no—
100C23×D4C22×C311yescharacteristic
101C24×C4C22×C311yescharacteristic
102C22×SD16C22×C312yescomplemented
103C2×C22⋊Q8C22×C311yescharacteristic
104C2×C22⋊Q8C22×C312yescomplemented
105C2×#121—24no—
106C2×#121—22no—
107C2×#121—24no—
108C2×#121C22×C314yes—
109C22×C4⋊C4C22×C311yescharacteristic
110C22×C22⋊C4—21no—
111#484—24no—
112#450—24no—
113C2×#107C22×C312yescomplemented
114C2×#103C22×C312yescomplemented
115C2×#102C22×C312yescomplemented
116C2×#102C22×C314yescomplemented
117C2×C22⋊C8C22×C311yescharacteristic
118C2×C22⋊C8C22×C311yescharacteristic
119C2×C22⋊C8C22×C314yescomplemented
120#385—22no—
121#385—21no—
122#380C22×C311yescharacteristic
123#380C22×C312yescomplemented
124#379—21no—
125#340C22×C314yescomplemented
126C24×C3—22no—
127C24×C3—24no—
128C24×C3—24no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—42no—
133C24×C3—44no—
134C24×C3—44no—
135C24×C3—42no—
136C24×C3—42no—
137C24×C3C2×D412yes—
138C24×C3C2×D411yescharacteristic
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—24no—
142C2×C3×D4—216no—
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—48no—
148C2×C3×D4—44no—
149C2×C3×D4—48no—
150C2×C3×D4—432no—
151C2×C3×D4—48no—
152C2×C3×D4—48no—
153C2×C3×D4C2×D414yes—
154C22×C3×C4—24no—
155C22×C3×C4—22no—
156C22×C3×C4—24no—
157C22×C3×C4—22no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4—48no—
170C22×C3×C4—48no—
171C22×C3×C4—44no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2411yescharacteristic
177C3×SD16—48no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—24no—
181C2×C3×C8—24no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—44no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4C2×D414yes—
190C3×C22⋊C4—216no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—44no—
195C25—21no—
196C25C3×D411yescharacteristic
197C22×Q8C23×C311yescharacteristic
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—21no—
201C22×D4—22no—
202C22×D4—44no—
203C22×D4—48no—
204C22×D4C3×D414yes—
205C22×D4C23×C312yes—
206C23×C4—21no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—44no—
210C23×C4C3×D411yescharacteristic
211C23×C4C3×D412yes—
212C23×C4C23×C312yes—
213C23×C4C23×C311yescharacteristic
214C2×SD16—24no—
215C2×SD16—44no—
216C22×C8C23×C312yes—
217C22×C8C23×C312yes—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—44no—
221#121—216no—
222#121—28no—
223#121—28no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4C3×D414yes—
229C2×C4⋊C4C23×C311yescharacteristic
230C2×C4⋊C4C23×C312yes—
231C2×C4⋊C4C23×C312yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238#107—24no—
239#103—24no—
240#102—24no—
241#102—44no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—44no—
245#95—21no—
246#95—22no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—28no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—24no—
257C23×C3—48no—
258C23×C3—48no—
259C23×C3—44no—
260C23×C3—48no—
261C23×C3—48no—
262C23×C3—44no—
263C23×C3—44no—
264C23×C3—42no—
265C23×C3—44no—
266C23×C3—42no—
267C23×C3—48no—
268C23×C3—88no—
269C23×C3—88no—
270C23×C3#12012yes—
271C23×C3#13512yes—
272C23×C3AGL(1,ℤ/8)14yes—
273C23×C3#13712yes—
274C23×C3C22×D411yescharacteristic
275C3×Q8—42no—
276C3×Q8—42no—
277C3×D4—24no—
278C3×D4—24no—
279C3×D4—48no—
280C3×D4—432no—
281C3×D4—44no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—48no—
290C2×C3×C4—44no—
291C2×C3×C4—48no—
292C2×C3×C4—44no—
293C2×C3×C4—44no—
294C2×C3×C4—44no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4#12012yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4H(5,𝔽2)14yes—
306C3×C8—42no—
307C3×C8—42no—
308C24—22no—
309C24—24no—
310C24—24no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—42no—
315C24—44no—
316C24—44no—
317C24—42no—
318C24—42no—
319C24C2×C3×D412yes—
320C24C2×C3×D411yescharacteristic
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—24no—
324C2×D4—216no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—48no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—432no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4C2×C3×D414yes—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—44no—
354C22×C4C2×C3×D412yes—
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×D412yes—
357C22×C4C2×C3×D412yes—
358C22×C4C24×C311yescharacteristic
359SD16—48no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—24no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4C2×C3×D414yes—
372C22⋊C4—216no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22×C3—22no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—22no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—416no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—816no—
393C22×C3—84no—
394C22×C3—84no—
395C22×C3#45214yes—
396C22×C3C2×#12011yescharacteristic
397C22×C3#53412yes—
398C22×C3C2×#13511yescharacteristic
399C22×C3C2×AGL(1,ℤ/8)12yes—
400C22×C3C2×#13711yescharacteristic
401C3×C4—22no—
402C3×C4—42no—
403C3×C4—44no—
404C3×C4—81no—
405C3×C4—82no—
406C3×C4—81no—
407C3×C4#53412yes—
408C3×C4#53512yes—
409C23—22no—
410C23—22no—
411C23—28no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—48no—
420C23—48no—
421C23—44no—
422C23—48no—
423C23—48no—
424C23—44no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—48no—
429C23—44no—
430C23—88no—
431C23—88no—
432C23C3×#12012yes—
433C23C3×#13512yes—
434C23C3×AGL(1,ℤ/8)14yes—
435C23C3×#13712yes—
436C23C22×C3×D411yescharacteristic
437Q8—42no—
438Q8—42no—
439D4—24no—
440D4—24no—
441D4—48no—
442D4—432no—
443D4—44no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—48no—
452C2×C4—44no—
453C2×C4—48no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4C3×#12012yes—
465C2×C4C22×C3×D411yescharacteristic
466C2×C4C22×C3×D411yescharacteristic
467C2×C4C3×H(5,𝔽2)14yes—
468C8—42no—
469C8—42no—
470C2×C3—24no—
471C2×C3—44no—
472C2×C3—84no—
473C2×C3#261311yescharacteristic
474C2×C3C2×#45211yescharacteristic
475C2×C3#300311yescharacteristic
476C2×C3#318812yes—
477C2×C3#319312yes—
478C22—24no—
479C22—24no—
480C22—22no—
481C22—22no—
482C22—24no—
483C22—24no—
484C22—24no—
485C22—416no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—44no—
491C22—44no—
492C22—44no—
493C22—816no—
494C22—84no—
495C22—84no—
496C22C3×#45214yes—
497C22C2×C3×#12011yescharacteristic
498C22C3×#53412yes—
499C22C2×C3×#13511yescharacteristic
500C22C2×C3×AGL(1,ℤ/8)12yes—
501C22C2×C3×#13711yescharacteristic
502C4—22no—
503C4—44no—
504C4—42no—
505C4—82no—
506C4—81no—
507C4—81no—
508C4C3×#53412yes—
509C4C3×#53512yes—
510C3#2959811yescharacteristic, direct product factor, Sylow
511C2—24no—
512C2—44no—
513C2—84no—
514C2C3×#261311yescharacteristic
515C2C2×C3×#45211yescharacteristic
516C2C3×#300311yescharacteristic
517C2C3×#318812yes—
518C2C3×#319312yes—
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×#484C2semidirect
3. C2×C3×#450C2semidirect
4. C3×#2848C2semidirect
5. C3×#2033C2semidirect
6. C3×#2000C2semidirect
7. C3×#1824C2semidirect
8. C3×#1786C2semidirect
9. C3×#1779C2semidirect
10. #29598C3direct (N × H)
14. C22×C3×SD16C22semidirect
16. C2×C3×C22⋊Q8C22semidirect
25. C2×C3×#107C22semidirect
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×#484C2×C3semidirect
40. C2×#450C2×C3semidirect
41. #2848C2×C3semidirect
42. #2033C2×C3semidirect
43. #2000C2×C3semidirect
44. #1824C2×C3semidirect
45. #1786C2×C3semidirect
46. #1779C2×C3semidirect
102. C22×SD16C22×C3semidirect
104. C2×C22⋊Q8C22×C3semidirect
113. C2×#107C22×C3semidirect
114. C2×#103C22×C3semidirect
115. C2×#102C22×C3semidirect
116. C2×#102C22×C3semidirect
119. C2×C22⋊C8C22×C3semidirect
123. #380C22×C3semidirect
125. #340C22×C3semidirect
510. C3#29598direct (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×#4842
C2×C3×#4502
C3×#28482
C3×#20332
C3×#20002
C3×#18242
C3×#17862
C3×#17792
#295983