Finite Group Databasebeta

← C3×#14843 #10633450

Subgroups — C3×#14843 #10633450

Order 768.

2046 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic
2C2×C3×#400C212yesmaximal, complemented
3C2×C3×#385C211yesmaximal, characteristic
4C3×#1445C212yesmaximal, complemented
5C3×#1442C212yesmaximal, complemented
6C3×#1441C212yesmaximal, complemented
7C3×C22⋊#95C211yesmaximal, characteristic
8C3×C22⋊#95C212yesmaximal
9C3×C22⋊#95C212yesmaximal
10#14843C311yesmaximal, 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⋊C4C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2211yescharacteristic
22C3×#400—28no—
23C3×#385—24no—
24C2×C3×#95C2214yes—
25C2×C3×#95C2214yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2214yes—
31C2×C3×#95C2214yes—
32C23×C4⋊C4C2×C311yescharacteristic
33C2×#400C2×C312yescomplemented
34C2×#385C2×C311yescharacteristic
35#1445C2×C312yescomplemented
36#1442C2×C312yescomplemented
37#1441C2×C312yescomplemented
38C22⋊#95C2×C311yescharacteristic
39C22⋊#95C2×C312yes—
40C22⋊#95C2×C312yes—
41C25×C3C2311yescharacteristic
42C23×C3×C4—24no—
43C23×C3×C4—24no—
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4—28no—
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—44no—
50C23×C3×C4D414yes—
51C23×C3×C4C2314yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2314yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C2×C3×C4⋊C4—232no—
60C2×C3×C4⋊C4—28no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4D418yes—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4Q814yes—
77C3×#95—22no—
78C3×#95—22no—
79C3×#95—28no—
80C3×#95—24no—
81C3×#95—28no—
82C3×#95—24no—
83C3×#95—44no—
84C3×#95—44no—
85C24×C4C22×C311yescharacteristic
86C24×C4C22×C312yes—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C312yes—
91C22×C4⋊C4C22×C311yescharacteristic
92C22×C4⋊C4C22×C312yes—
93C22×C22⋊C4C22×C312yes—
94C22×C22⋊C4C22×C311yescharacteristic
95C22×C22⋊C4C22×C311yescharacteristic
96#400—28no—
97#385—24no—
98C2×#95C22×C314yes—
99C2×#95C22×C314yes—
100C2×#95C22×C312yes—
101C2×#95C22×C312yes—
102C2×#95C22×C311yescharacteristic
103C2×#95C22×C311yescharacteristic
104C2×#95C22×C314yes—
105C2×#95C22×C314yes—
106C24×C3—22no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3—22no—
110C24×C3C2×D412yes—
111C24×C3C2×Q812yes—
112C24×C3#4112yes—
113C24×C3#4114yes—
114C24×C3#4114yes—
115C24×C3C2411yescharacteristic
116C22×C3×C4—232no—
117C22×C3×C4—28no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—28no—
123C22×C3×C4—28no—
124C22×C3×C4—216no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—22no—
139C22×C3×C4—24no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—416no—
155C22×C3×C4—416no—
156C22×C3×C4—48no—
157C22×C3×C4—48no—
158C22×C3×C4C2×D418yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D414yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×Q812yes—
163C22×C3×C4C2×Q812yes—
164C22×C3×C4#4114yes—
165C3×C4⋊C4—216no—
166C3×C4⋊C4—216no—
167C3×C4⋊C4—44no—
168C3×C4⋊C4—44no—
169C3×C22⋊C4—28no—
170C3×C22⋊C4—44no—
171C3×C22⋊C4—48no—
172C25C23×C311yescharacteristic
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—28no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—44no—
181C23×C4C3×D414yes—
182C23×C4C23×C311yescharacteristic
183C23×C4C23×C314yes—
184C23×C4C23×C311yescharacteristic
185C23×C4C23×C314yes—
186C23×C4C23×C311yescharacteristic
187C23×C4C23×C311yescharacteristic
188C23×C4C23×C311yescharacteristic
189C23×C4C23×C311yescharacteristic
190C2×C4⋊C4—232no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C3×D418yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4C3×Q814yes—
208#95—22no—
209#95—22no—
210#95—28no—
211#95—24no—
212#95—28no—
213#95—24no—
214#95—44no—
215#95—44no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—28no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—28no—
222C23×C3—28no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—28no—
226C23×C3—24no—
227C23×C3—24no—
228C23×C3C22⋊Q814yes—
229C23×C3#12314yes—
230C23×C3#12414yes—
231C23×C3#12518yes—
232C23×C3#12618yes—
233C23×C3C22×D411yescharacteristic
234C23×C3C22×D412yes—
235C23×C3C22×Q811yescharacteristic
236C23×C3C2×#4111yescharacteristic
237C23×C3C2×#4112yes—
238C23×C3C2×#4112yes—
239C23×C3C2×#4112yes—
240C23×C3H(5,𝔽2)12yes—
241C23×C32−1+412yes—
242C2×C3×C4—232no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—28no—
248C2×C3×C4—28no—
249C2×C3×C4—28no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—28no—
253C2×C3×C4—28no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—432no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—44no—
260C2×C3×C4—41no—
261C2×C3×C4—44no—
262C2×C3×C4—41no—
263C2×C3×C4—44no—
264C2×C3×C4—44no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—44no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—44no—
281C2×C3×C4—44no—
282C2×C3×C4#12014yes—
283C2×C3×C4#12314yes—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24C2×C3×D412yes—
289C24C2×C3×Q812yes—
290C24C3×#4112yes—
291C24C3×#4114yes—
292C24C3×#4114yes—
293C24C24×C311yescharacteristic
294C22×C4—232no—
295C22×C4—28no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—216no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—416no—
333C22×C4—416no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4C2×C3×D418yes—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D414yes—
339C22×C4C2×C3×D412yes—
340C22×C4C2×C3×Q812yes—
341C22×C4C2×C3×Q812yes—
342C22×C4C3×#4114yes—
343C4⋊C4—216no—
344C4⋊C4—216no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C22⋊C4—28no—
348C22⋊C4—44no—
349C22⋊C4—48no—
350C22×C3—24no—
351C22×C3—24no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—216no—
355C22×C3—216no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3#40018yescomplemented
359C22×C3C2×#12011yescharacteristic
360C22×C3C2×C22⋊Q811yescharacteristic
361C22×C3C2×#12311yescharacteristic
362C22×C3C2×#12311yescharacteristic
363C22×C3#52514yes—
364C22×C3C2×#12411yescharacteristic
365C22×C3C2×#12512yes—
366C22×C3C2×#12612yes—
367C22×C3#53511yescharacteristic
368C22×C3#53611yescharacteristic
369C22×C3#53914yes—
370C22×C3#54614yes—
371C22×C3#54812yes—
372C22×C3D4×Q812yes—
373C22×C3#55214yes—
374C22×C3#55312yes—
375C22×C3#55412yes—
376C3×C4—24no—
377C3×C4—24no—
378C3×C4—42no—
379C3×C4—48no—
380C3×C4—42no—
381C3×C4—81no—
382C3×C4—81no—
383C3×C4—84no—
384C23—22no—
385C23—22no—
386C23—28no—
387C23—22no—
388C23—22no—
389C23—28no—
390C23—28no—
391C23—24no—
392C23—24no—
393C23—28no—
394C23—24no—
395C23—24no—
396C23C3×C22⋊Q814yes—
397C23C3×#12314yes—
398C23C3×#12414yes—
399C23C3×#12518yes—
400C23C3×#12618yes—
401C23C22×C3×D411yescharacteristic
402C23C22×C3×D412yes—
403C23C22×C3×Q811yescharacteristic
404C23C2×C3×#4111yescharacteristic
405C23C2×C3×#4112yes—
406C23C2×C3×#4112yes—
407C23C2×C3×#4112yes—
408C23C3×H(5,𝔽2)12yes—
409C23C3×2−1+412yes—
410C2×C4—232no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—432no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4C3×#12014yes—
451C2×C4C3×#12314yes—
452C2×C3—28no—
453C2×C3C2×#40011yescharacteristic
454C2×C3#240911yescharacteristic
455C2×C3#243511yescharacteristic
456C2×C3#247314yes—
457C2×C3#248114yes—
458C2×C3#248711yescharacteristic
459C2×C3#250311yescharacteristic
460C2×C3#262311yescharacteristic
461C2×C3#284911yescharacteristic
462C22—24no—
463C22—24no—
464C22—24no—
465C22—216no—
466C22—216no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22C3×#40018yescomplemented
471C22C2×C3×#12011yescharacteristic
472C22C2×C3×C22⋊Q811yescharacteristic
473C22C2×C3×#12311yescharacteristic
474C22C2×C3×#12311yescharacteristic
475C22C3×#52514yes—
476C22C2×C3×#12411yescharacteristic
477C22C2×C3×#12512yes—
478C22C2×C3×#12612yes—
479C22C3×#53511yescharacteristic
480C22C3×#53611yescharacteristic
481C22C3×#53914yes—
482C22C3×#54614yes—
483C22C3×#54812yes—
484C22C3×D4×Q812yes—
485C22C3×#55214yes—
486C22C3×#55312yes—
487C22C3×#55412yes—
488C4—24no—
489C4—24no—
490C4—42no—
491C4—48no—
492C4—42no—
493C4—81no—
494C4—84no—
495C4—81no—
496C3#1484311yescharacteristic, direct product factor, Sylow
497C2—28no—
498C2C2×C3×#40011yescharacteristic
499C2C3×#240911yescharacteristic
500C2C3×#243511yescharacteristic
501C2C3×#247314yes—
502C2C3×#248114yes—
503C2C3×#248711yescharacteristic
504C2C3×#250311yescharacteristic
505C2C3×#262311yescharacteristic
506C2C3×#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×#400C2semidirect
4. C3×#1445C2semidirect
5. C3×#1442C2semidirect
6. C3×#1441C2semidirect
10. #14843C3direct (N × H)
33. C2×#400C2×C3semidirect
35. #1445C2×C3semidirect
36. #1442C2×C3semidirect
37. #1441C2×C3semidirect
358. C22×C3#400semidirect
470. C22C3×#400semidirect
496. C3#14843direct (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×#4002
C2×C3×#3852
C3×#14452
C3×#14422
C3×#14412
C3×C22⋊#952
#148433