Finite Group Databasebeta

← C3×#14807 #10633414

Subgroups — C3×#14807 #10633414

Order 768.

2398 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×#399C212yesmaximal, complemented
3C2×C3×#389C211yesmaximal, characteristic, complemented
4C2×C3×#385C214yesmaximal, complemented
5C3×#1444C212yesmaximal, complemented
6C3×#1439C212yesmaximal, complemented
7C3×#1438C212yesmaximal
8C3×#1434C211yesmaximal, characteristic, complemented
9#14807C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4C2212yes—
12C2×C3×C4×D4—28no—
13C2×C3×C4×D4—24no—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2214yes—
19C22×C3×C22⋊C4C2214yes—
20C22×C3×C22⋊C4C2214yes—
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C3×#399—28no—
24C3×#389—24no—
25C3×#385—216no—
26C2×C3×#95C2212yes—
27C2×C3×#95C2214yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C22×C4×D4C2×C311yescharacteristic
32C2×#399C2×C312yescomplemented
33C2×#389C2×C311yescharacteristic, complemented
34C2×#385C2×C314yescomplemented
35#1444C2×C312yescomplemented
36#1439C2×C312yescomplemented
37#1438C2×C312yes—
38#1434C2×C311yescharacteristic, complemented
39C25×C3C2312yes—
40C22×C3×D4—24no—
41C22×C3×D4—24no—
42C22×C3×D4—22no—
43C22×C3×D4—22no—
44C22×C3×D4Q814yes—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—28no—
49C23×C3×C4—44no—
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2314yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C3×C4×D4—232no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—28no—
63C2×C3×C4⋊C4—28no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—28no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—44no—
69C2×C3×C22⋊C4—28no—
70C2×C3×C22⋊C4—28no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—28no—
74C2×C3×C22⋊C4—28no—
75C2×C3×C22⋊C4—28no—
76C2×C3×C22⋊C4—44no—
77C2×C3×C22⋊C4D418yes—
78C2×C3×C42—24no—
79C2×C3×C42D414yes—
80C3×#95—28no—
81C3×#95—24no—
82C3×#95—24no—
83C3×#95—24no—
84C3×#95D418yescomplemented
85C23×D4C22×C311yescharacteristic
86C24×C4C22×C312yes—
87C2×C4×D4—28no—
88C2×C4×D4—24no—
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C311yescharacteristic
91C22×C4⋊C4C22×C311yescharacteristic
92C22×C4⋊C4C22×C312yes—
93C22×C4⋊C4C22×C314yes—
94C22×C22⋊C4C22×C314yes—
95C22×C22⋊C4C22×C314yes—
96C22×C22⋊C4C22×C312yes—
97C22×C42C22×C311yescharacteristic
98#399—28no—
99#389—24no—
100#385—216no—
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C314yes—
105C2×#95C22×C312yes—
106C24×C3—24no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3—24no—
110C24×C3—22no—
111C24×C3—22no—
112C24×C3—24no—
113C24×C3—24no—
114C24×C3C2×D414yes—
115C24×C3C2×Q814yes—
116C24×C3#4114yes—
117C24×C3C2411yescharacteristic
118C2×C3×D4—28no—
119C2×C3×D4—28no—
120C2×C3×D4—216no—
121C2×C3×D4—216no—
122C2×C3×D4—28no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—216no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—416no—
150C22×C3×C4—416no—
151C22×C3×C4—48no—
152C22×C3×C4—48no—
153C22×C3×C4C2×D418yes—
154C22×C3×C4C2×D414yes—
155C22×C3×C4C2×D414yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D414yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×Q812yes—
161C22×C3×C4#4114yes—
162C22×C3×C4#4112yes—
163C3×C4⋊C4—28no—
164C3×C4⋊C4—28no—
165C3×C4⋊C4—416no—
166C3×C4⋊C4—44no—
167C3×C4⋊C4—48no—
168C3×C22⋊C4—232no—
169C3×C22⋊C4—216no—
170C3×C22⋊C4—416no—
171C3×C42—28no—
172C25C23×C312yes—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4C3×Q814yes—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—44no—
183C23×C4C23×C312yes—
184C23×C4C23×C314yes—
185C23×C4C23×C312yes—
186C23×C4C23×C312yes—
187C23×C4C23×C311yescharacteristic
188C23×C4C23×C311yescharacteristic
189C23×C4C23×C311yescharacteristic
190C4×D4—232no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—44no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4C3×D418yes—
211C2×C42—24no—
212C2×C42C3×D414yes—
213#95—28no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95C3×D418yescomplemented
218C23×C3—24no—
219C23×C3—24no—
220C23×C3—24no—
221C23×C3—28no—
222C23×C3—24no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—28no—
226C23×C3—28no—
227C23×C3—28no—
228C23×C3—28no—
229C23×C3—28no—
230C23×C3—42no—
231C23×C3—44no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—44no—
236C23×C3—44no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3C22⋊Q818yes—
241C23×C3C22×D411yescharacteristic
242C23×C3C22×D412yes—
243C23×C3C22×D412yes—
244C23×C3C22×Q811yescharacteristic
245C23×C3C2×#4112yes—
246C23×C3C2×#4111yescharacteristic
247C23×C3C2×#4112yes—
248C23×C3H(5,𝔽2)12yes—
249C23×C32−1+412yes—
250C3×D4—232no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—28no—
260C2×C3×C4—28no—
261C2×C3×C4—24no—
262C2×C3×C4—28no—
263C2×C3×C4—28no—
264C2×C3×C4—24no—
265C2×C3×C4—28no—
266C2×C3×C4—28no—
267C2×C3×C4—28no—
268C2×C3×C4—432no—
269C2×C3×C4—42no—
270C2×C3×C4—44no—
271C2×C3×C4—44no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4#12014yes—
285C2×C3×C4C22⋊Q814yes—
286C2×C3×C4#12714yes—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—22no—
292C24—22no—
293C24—24no—
294C24—24no—
295C24C2×C3×D414yes—
296C24C2×C3×Q814yes—
297C24C3×#4114yes—
298C24C24×C311yescharacteristic
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—216no—
302C2×D4—216no—
303C2×D4—28no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—216no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—416no—
331C22×C4—416no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4C2×C3×D418yes—
335C22×C4C2×C3×D414yes—
336C22×C4C2×C3×D414yes—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D412yes—
339C22×C4C2×C3×D414yes—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×Q812yes—
342C22×C4C3×#4114yes—
343C22×C4C3×#4112yes—
344C4⋊C4—28no—
345C4⋊C4—28no—
346C4⋊C4—416no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C22⋊C4—232no—
350C22⋊C4—216no—
351C22⋊C4—416no—
352C42—28no—
353C22×C3—28no—
354C22×C3—216no—
355C22×C3—44no—
356C22×C3—44no—
357C22×C3—44no—
358C22×C3—44no—
359C22×C3—44no—
360C22×C3—48no—
361C22×C3—48no—
362C22×C3—48no—
363C22×C3—44no—
364C22×C3C2×#12011yescharacteristic
365C22×C3C2×C22⋊Q811yescharacteristic
366C22×C3C2×C22⋊Q812yes—
367C22×C3#52511yescharacteristic
368C22×C3#52511yescharacteristic
369C22×C3#52512yes—
370C22×C3#52514yes—
371C22×C3C2×#12711yescharacteristic
372C22×C3#53511yescharacteristic
373C22×C3#53611yescharacteristic
374C22×C3#53914yes—
375C22×C3D4214yes—
376C22×C3#54714yes—
377C22×C3#54712yes—
378C22×C3D4×Q812yes—
379C22×C3#55412yes—
380C22×C3#55912yes—
381C3×C4—24no—
382C3×C4—24no—
383C3×C4—24no—
384C3×C4—44no—
385C3×C4—44no—
386C3×C4—48no—
387C3×C4—82no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—28no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—28no—
396C23—28no—
397C23—28no—
398C23—28no—
399C23—28no—
400C23—44no—
401C23—42no—
402C23—44no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23C3×C22⋊Q818yes—
411C23C22×C3×D411yescharacteristic
412C23C22×C3×D412yes—
413C23C22×C3×D412yes—
414C23C22×C3×Q811yescharacteristic
415C23C2×C3×#4111yescharacteristic
416C23C2×C3×#4112yes—
417C23C2×C3×#4112yes—
418C23C3×H(5,𝔽2)12yes—
419C23C3×2−1+412yes—
420D4—232no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—24no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—432no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4C3×#12014yes—
455C2×C4C3×C22⋊Q814yes—
456C2×C4C3×#12714yes—
457C2×C3—48no—
458C2×C3#239711yescharacteristic
459C2×C3#240611yescharacteristic
460C2×C3#243311yescharacteristic
461C2×C3#244914yes—
462C2×C3#245514yes—
463C2×C3#254211yescharacteristic
464C2×C3#258011yescharacteristic
465C2×C3#263311yescharacteristic
466C2×C3#284811yescharacteristic
467C22—216no—
468C22—28no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—44no—
473C22—48no—
474C22—48no—
475C22—48no—
476C22—44no—
477C22—44no—
478C22C2×C3×#12011yescharacteristic
479C22C2×C3×C22⋊Q812yes—
480C22C2×C3×C22⋊Q811yescharacteristic
481C22C3×#52511yescharacteristic
482C22C3×#52511yescharacteristic
483C22C3×#52514yes—
484C22C3×#52512yes—
485C22C2×C3×#12711yescharacteristic
486C22C3×#53511yescharacteristic
487C22C3×#53611yescharacteristic
488C22C3×#53914yes—
489C22C3×D4214yes—
490C22C3×#54714yes—
491C22C3×#54712yes—
492C22C3×D4×Q812yes—
493C22C3×#55412yes—
494C22C3×#55912yes—
495C4—24no—
496C4—24no—
497C4—24no—
498C4—44no—
499C4—44no—
500C4—48no—
501C4—82no—
502C3#1480711yescharacteristic, direct product factor, Sylow
503C2—48no—
504C2C3×#239711yescharacteristic
505C2C3×#240611yescharacteristic
506C2C3×#243311yescharacteristic
507C2C3×#244914yes—
508C2C3×#245514yes—
509C2C3×#254211yescharacteristic
510C2C3×#258011yescharacteristic
511C2C3×#263311yescharacteristic
512C2C3×#284811yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#399C2semidirect
3. C2×C3×#389C2semidirect
4. C2×C3×#385C2semidirect
5. C3×#1444C2semidirect
6. C3×#1439C2semidirect
8. C3×#1434C2semidirect
9. #14807C3direct (N × H)
32. C2×#399C2×C3semidirect
33. C2×#389C2×C3semidirect
34. C2×#385C2×C3semidirect
35. #1444C2×C3semidirect
36. #1439C2×C3semidirect
38. #1434C2×C3semidirect
84. C3×#95D4semidirect
217. #95C3×D4semidirect
502. C3#14807direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×C3×#3992
C2×C3×#3892
C2×C3×#3852
C3×#14442
C3×#14392
C3×#14382
C3×#14342
#148073