Finite Group Databasebeta

← C3×#14786 #10633393

Subgroups — C3×#14786 #10633393

Order 768.

2190 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic
2C2×C3×#398C212yesmaximal, complemented
3C2×C3×#385C211yesmaximal, characteristic
4C2×C3×#385C212yesmaximal
5C3×#1445C212yesmaximal, complemented
6C3×#1444C212yesmaximal, complemented
7C3×#1441C212yesmaximal, complemented
8C3×C22⋊#95C211yesmaximal, characteristic
9C3×C22⋊#95C212yesmaximal
10#14786C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2212yes—
13C22×C3×C4⋊C4—24no—
14C22×C3×C4⋊C4—28no—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2214yes—
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2212yes—
23C3×#398—28no—
24C3×#385—24no—
25C3×#385—28no—
26C2×C3×#95C2214yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2214yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2214yes—
33C23×C4⋊C4C2×C311yescharacteristic
34C2×#398C2×C312yescomplemented
35C2×#385C2×C311yescharacteristic
36C2×#385C2×C312yes—
37#1445C2×C312yescomplemented
38#1444C2×C312yescomplemented
39#1441C2×C312yescomplemented
40C22⋊#95C2×C312yes—
41C22⋊#95C2×C311yescharacteristic
42C25×C3C2311yescharacteristic
43C23×C3×C4—24no—
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—28no—
48C23×C3×C4—24no—
49C23×C3×C4—24no—
50C23×C3×C4—44no—
51C23×C3×C4D414yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2314yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2314yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C2×C3×C4⋊C4—232no—
61C2×C3×C4⋊C4—28no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—28no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—28no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—44no—
72C2×C3×C4⋊C4D418yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4D414yes—
80C2×C3×C22⋊C4Q818yes—
81C3×#95—28no—
82C3×#95—22no—
83C3×#95—22no—
84C3×#95—24no—
85C3×#95—28no—
86C3×#95—24no—
87C3×#95—44no—
88C24×C4C22×C311yescharacteristic
89C24×C4C22×C312yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4C22×C311yescharacteristic
93C22×C4⋊C4C22×C312yes—
94C22×C4⋊C4C22×C311yescharacteristic
95C22×C4⋊C4C22×C314yes—
96C22×C4⋊C4C22×C312yes—
97C22×C22⋊C4C22×C311yescharacteristic
98C22×C22⋊C4C22×C311yescharacteristic
99C22×C22⋊C4C22×C312yes—
100#398—28no—
101#385—24no—
102#385—28no—
103C2×#95C22×C314yes—
104C2×#95C22×C312yes—
105C2×#95C22×C312yes—
106C2×#95C22×C314yes—
107C2×#95C22×C311yescharacteristic
108C2×#95C22×C311yescharacteristic
109C2×#95C22×C314yes—
110C24×C3—22no—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3C2×D412yes—
115C24×C3C2×D412yes—
116C24×C3C2×Q814yes—
117C24×C3#4112yes—
118C24×C3#4114yes—
119C24×C3C2411yescharacteristic
120C22×C3×C4—232no—
121C22×C3×C4—28no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—28no—
127C22×C3×C4—28no—
128C22×C3×C4—216no—
129C22×C3×C4—24no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—416no—
158C22×C3×C4—416no—
159C22×C3×C4—48no—
160C22×C3×C4—48no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D418yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D414yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×Q818yes—
168C22×C3×C4#4114yes—
169C3×C4⋊C4—216no—
170C3×C4⋊C4—216no—
171C3×C4⋊C4—416no—
172C3×C4⋊C4—44no—
173C3×C4⋊C4—44no—
174C3×C22⋊C4—28no—
175C3×C22⋊C4—216no—
176C3×C22⋊C4—44no—
177C25C23×C311yescharacteristic
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—44no—
186C23×C4C3×D414yes—
187C23×C4C23×C311yescharacteristic
188C23×C4C23×C311yescharacteristic
189C23×C4C23×C311yescharacteristic
190C23×C4C23×C311yescharacteristic
191C23×C4C23×C314yes—
192C23×C4C23×C314yes—
193C23×C4C23×C311yescharacteristic
194C23×C4C23×C311yescharacteristic
195C2×C4⋊C4—232no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4C3×D418yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4C3×D414yes—
215C2×C22⋊C4C3×Q818yes—
216#95—28no—
217#95—22no—
218#95—22no—
219#95—24no—
220#95—28no—
221#95—24no—
222#95—44no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—28no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—28no—
229C23×C3—24no—
230C23×C3—24no—
231C23×C3—24no—
232C23×C3—28no—
233C23×C3—24no—
234C23×C3—28no—
235C23×C3#12114yes—
236C23×C3C22⋊Q818yes—
237C23×C3#12314yes—
238C23×C3#12518yes—
239C23×C3C4⋊Q814yes—
240C23×C3C22×D411yescharacteristic
241C23×C3C22×D412yes—
242C23×C3C22×D411yescharacteristic
243C23×C3C22×Q812yes—
244C23×C3C2×#4111yescharacteristic
245C23×C3C2×#4112yes—
246C23×C3C2×#4112yes—
247C23×C3H(5,𝔽2)12yes—
248C23×C32−1+412yes—
249C2×C3×C4—232no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—28no—
255C2×C3×C4—28no—
256C2×C3×C4—28no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—28no—
260C2×C3×C4—24no—
261C2×C3×C4—28no—
262C2×C3×C4—28no—
263C2×C3×C4—24no—
264C2×C3×C4—28no—
265C2×C3×C4—28no—
266C2×C3×C4—432no—
267C2×C3×C4—41no—
268C2×C3×C4—44no—
269C2×C3×C4—41no—
270C2×C3×C4—44no—
271C2×C3×C4—41no—
272C2×C3×C4—44no—
273C2×C3×C4—41no—
274C2×C3×C4—44no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4—42no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—44no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—44no—
289C2×C3×C4#12014yes—
290C2×C3×C4#12314yes—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24C2×C3×D412yes—
296C24C2×C3×D412yes—
297C24C2×C3×Q814yes—
298C24C3×#4112yes—
299C24C3×#4114yes—
300C24C24×C311yescharacteristic
301C22×C4—232no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—216no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—416no—
339C22×C4—416no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4C2×C3×D412yes—
343C22×C4C2×C3×D418yes—
344C22×C4C2×C3×D412yes—
345C22×C4C2×C3×D412yes—
346C22×C4C2×C3×D414yes—
347C22×C4C2×C3×D412yes—
348C22×C4C2×C3×Q818yes—
349C22×C4C3×#4114yes—
350C4⋊C4—216no—
351C4⋊C4—216no—
352C4⋊C4—416no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C22⋊C4—28no—
356C22⋊C4—216no—
357C22⋊C4—44no—
358C22×C3—24no—
359C22×C3—24no—
360C22×C3—24no—
361C22×C3—24no—
362C22×C3—216no—
363C22×C3—216no—
364C22×C3—24no—
365C22×C3—24no—
366C22×C3#39818yescomplemented
367C22×C3C2×#12011yescharacteristic
368C22×C3C2×#12111yescharacteristic
369C22×C3C2×C22⋊Q812yes—
370C22×C3C2×#12311yescharacteristic
371C22×C3C2×#12311yescharacteristic
372C22×C3#52514yes—
373C22×C3C2×#12512yes—
374C22×C3C2×C4⋊Q811yescharacteristic
375C22×C3#53511yescharacteristic
376C22×C3#53611yescharacteristic
377C22×C3#53914yes—
378C22×C3D4212yes—
379C22×C3#54612yes—
380C22×C3#54712yes—
381C22×C3D4×Q814yes—
382C22×C3#55212yes—
383C22×C3#55414yes—
384C3×C4—24no—
385C3×C4—24no—
386C3×C4—42no—
387C3×C4—48no—
388C3×C4—48no—
389C3×C4—42no—
390C3×C4—81no—
391C3×C4—81no—
392C23—22no—
393C23—22no—
394C23—28no—
395C23—22no—
396C23—22no—
397C23—28no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—28no—
402C23—24no—
403C23—28no—
404C23C3×#12114yes—
405C23C3×C22⋊Q818yes—
406C23C3×#12314yes—
407C23C3×#12518yes—
408C23C3×C4⋊Q814yes—
409C23C22×C3×D411yescharacteristic
410C23C22×C3×D412yes—
411C23C22×C3×D411yescharacteristic
412C23C22×C3×Q812yes—
413C23C2×C3×#4111yescharacteristic
414C23C2×C3×#4112yes—
415C23C2×C3×#4112yes—
416C23C3×H(5,𝔽2)12yes—
417C23C3×2−1+412yes—
418C2×C4—232no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—432no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4C3×#12014yes—
459C2×C4C3×#12314yes—
460C2×C3—28no—
461C2×C3C2×#39811yescharacteristic
462C2×C3#240811yescharacteristic
463C2×C3#243411yescharacteristic
464C2×C3#244914yes—
465C2×C3#247314yes—
466C2×C3#248611yescharacteristic
467C2×C3#250211yescharacteristic
468C2×C3#262311yescharacteristic
469C2×C3#284911yescharacteristic
470C22—24no—
471C22—24no—
472C22—24no—
473C22—216no—
474C22—216no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22C3×#39818yescomplemented
479C22C2×C3×#12011yescharacteristic
480C22C2×C3×#12111yescharacteristic
481C22C2×C3×C22⋊Q812yes—
482C22C2×C3×#12311yescharacteristic
483C22C2×C3×#12311yescharacteristic
484C22C3×#52514yes—
485C22C2×C3×#12512yes—
486C22C2×C3×C4⋊Q811yescharacteristic
487C22C3×#53511yescharacteristic
488C22C3×#53611yescharacteristic
489C22C3×#53914yes—
490C22C3×D4212yes—
491C22C3×#54612yes—
492C22C3×#54712yes—
493C22C3×D4×Q814yes—
494C22C3×#55212yes—
495C22C3×#55414yes—
496C4—24no—
497C4—24no—
498C4—48no—
499C4—42no—
500C4—48no—
501C4—42no—
502C4—81no—
503C4—81no—
504C3#1478611yescharacteristic, direct product factor, Sylow
505C2—28no—
506C2C2×C3×#39811yescharacteristic
507C2C3×#240811yescharacteristic
508C2C3×#243411yescharacteristic
509C2C3×#244914yes—
510C2C3×#247314yes—
511C2C3×#248611yescharacteristic
512C2C3×#250211yescharacteristic
513C2C3×#262311yescharacteristic
514C2C3×#284911yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#398C2semidirect
5. C3×#1445C2semidirect
6. C3×#1444C2semidirect
7. C3×#1441C2semidirect
10. #14786C3direct (N × H)
34. C2×#398C2×C3semidirect
37. #1445C2×C3semidirect
38. #1444C2×C3semidirect
39. #1441C2×C3semidirect
366. C22×C3#398semidirect
478. C22C3×#398semidirect
504. C3#14786direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C2×C3×#3982
C2×C3×#3852
C3×#14452
C3×#14442
C3×#14412
C3×C22⋊#952
#147863