Finite Group Databasebeta

← C3×#14652 #10633259

Subgroups — C3×#14652 #10633259

Order 768.

1226 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#397C211yesmaximal, characteristic
2C3×#1442C212yesmaximal, complemented
3C3×#1442C212yesmaximal, complemented
4C3×#1441C212yesmaximal, complemented
5C3×#1440C211yesmaximal, characteristic
6C3×#1439C212yesmaximal
7C3×#1439C212yesmaximal
8C3×#1437C211yesmaximal, characteristic
9C3×#1436C212yesmaximal, complemented
10#14652C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C3×#397C418yescomplemented
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×#397C2×C311yescharacteristic
34#1442C2×C312yescomplemented
35#1442C2×C312yescomplemented
36#1441C2×C312yescomplemented
37#1440C2×C311yescharacteristic
38#1439C2×C312yes—
39#1439C2×C312yes—
40#1437C2×C311yescharacteristic
41#1436C2×C312yescomplemented
42C25×C3C2311yescharacteristic
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C2×C3×C4⋊C4—24no—
52C2×C3×C4⋊C4—41no—
53C2×C3×C4⋊C4—41no—
54C2×C3×C4⋊C4—42no—
55C2×C3×C4⋊C4—42no—
56C2×C3×C4⋊C4—42no—
57C2×C3×C4⋊C4—42no—
58C2×C3×C4⋊C4—42no—
59C2×C3×C4⋊C4C2×C414yes—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—42no—
62C2×C3×C22⋊C4—41no—
63C2×C3×C22⋊C4—41no—
64C2×C3×C22⋊C4—42no—
65C2×C3×C22⋊C4—42no—
66C2×C3×C22⋊C4—42no—
67C2×C3×C22⋊C4—41no—
68C2×C3×C22⋊C4—42no—
69C2×C3×C22⋊C4—42no—
70C2×C3×C22⋊C4—41no—
71C2×C3×C22⋊C4C2×C414yes—
72C2×C3×C22⋊C4C2×C418yes—
73C2×C3×C42—42no—
74C2×C3×C42—41no—
75C3×#95—24no—
76C3×#95—42no—
77C3×#95—42no—
78C3×#95—42no—
79C3×#95—42no—
80C3×#95—42no—
81C3×#95—41no—
82C3×#95—42no—
83C3×#95—42no—
84C3×#95—42no—
85C3×#95C2×C418yes—
86C3×#95C2×C414yes—
87C22×C4⋊C4C22×C312yes—
88C22×C4⋊C4C22×C312yes—
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C22⋊C4C22×C312yes—
91C22×C22⋊C4C22×C311yescharacteristic
92C22×C22⋊C4C22×C311yescharacteristic
93C22×C22⋊C4C22×C311yescharacteristic
94C22×C22⋊C4C22×C312yes—
95C22×C42C22×C311yescharacteristic
96#397C3×C418yescomplemented
97C2×#95C22×C312yes—
98C2×#95C22×C312yes—
99C2×#95C22×C312yes—
100C2×#95C22×C312yes—
101C2×#95C22×C312yes—
102C2×#95C22×C311yescharacteristic
103C2×#95C22×C312yes—
104C2×#95C22×C312yes—
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C311yescharacteristic
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—21no—
112C24×C3—21no—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3C22×C412yes—
118C24×C3C2411yescharacteristic
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C414yes—
168C22×C3×C4C22×C414yes—
169C22×C3×C4C22×C412yes—
170C22×C3×C4#4114yes—
171C3×C4⋊C4—44no—
172C3×C4⋊C4—48no—
173C3×C4⋊C4—48no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—48no—
176C3×C22⋊C4—48no—
177C3×C22⋊C4—44no—
178C3×C22⋊C4—44no—
179C3×C42—44no—
180C25C23×C311yescharacteristic
181C23×C4C23×C312yes—
182C23×C4C23×C312yes—
183C23×C4C23×C312yes—
184C23×C4C23×C312yes—
185C23×C4C23×C311yescharacteristic
186C23×C4C23×C312yes—
187C23×C4C23×C311yescharacteristic
188C23×C4C23×C312yes—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4C2×C3×C414yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4C2×C3×C414yes—
210C2×C22⋊C4C2×C3×C418yes—
211C2×C42—42no—
212C2×C42—41no—
213#95—24no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—41no—
220#95—42no—
221#95—42no—
222#95—42no—
223#95C2×C3×C418yes—
224#95C2×C3×C414yes—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—24no—
229C23×C3—24no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—41no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3C23×C411yescharacteristic
247C23×C3C2×#4112yes—
248C23×C3H(5,𝔽2)11yescharacteristic
249C23×C3H(5,𝔽2)12yes—
250C23×C3H(5,𝔽2)11yescharacteristic
251C23×C3H(5,𝔽2)12yes—
252C23×C3H(5,𝔽2)12yes—
253C23×C32−1+412yes—
254C23×C32−1+412yes—
255C2×C3×C4—24no—
256C2×C3×C4—22no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24C22×C3×C412yes—
316C24C24×C311yescharacteristic
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4C22×C3×C414yes—
365C22×C4C22×C3×C412yes—
366C22×C4C22×C3×C414yes—
367C22×C4C22×C3×C412yes—
368C22×C4C3×#4114yes—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C22⋊C4—44no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C42—44no—
378C22×C3—42no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—82no—
386C22×C3—82no—
387C22×C3C4×#4111yescharacteristic
388C22×C3#51811yescharacteristic
389C22×C3#51811yescharacteristic
390C22×C3#52012yes—
391C22×C3#52012yes—
392C22×C3#53812yes—
393C22×C3#53812yes—
394C22×C3#53912yes—
395C22×C3#54012yes—
396C22×C3#54112yes—
397C22×C3#54212yes—
398C22×C3#56012yes—
399C22×C3#56011yescharacteristic
400C22×C3H(3,𝔽4)11yescharacteristic
401C22×C3#56212yes—
402C22×C3#56212yes—
403C22×C3#56212yes—
404C22×C3#56212yes—
405C22×C3#56311yescharacteristic
406C22×C3#56312yes—
407C22×C3#56311yescharacteristic
408C3×C4—44no—
409C3×C4—44no—
410C3×C4—42no—
411C3×C4—44no—
412C3×C4—42no—
413C3×C4—82no—
414C3×C4—82no—
415C3×C4—82no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—41no—
422C23—42no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23C23×C3×C411yescharacteristic
438C23C2×C3×#4112yes—
439C23C3×H(5,𝔽2)12yes—
440C23C3×H(5,𝔽2)12yes—
441C23C3×H(5,𝔽2)12yes—
442C23C3×H(5,𝔽2)11yescharacteristic
443C23C3×H(5,𝔽2)11yescharacteristic
444C23C3×2−1+412yes—
445C23C3×2−1+412yes—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C3—82no—
499C2×C3#233311yescharacteristic
500C2×C3#233412yes—
501C2×C3#237612yes—
502C2×C3#237611yescharacteristic
503C2×C3#237911yescharacteristic
504C2×C3#282412yes—
505C2×C3#282912yes—
506C2×C3#283112yes—
507C2×C3#283612yes—
508C22—44no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—44no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22C3×C4×#4111yescharacteristic
518C22C3×#51811yescharacteristic
519C22C3×#51811yescharacteristic
520C22C3×#52012yes—
521C22C3×#52012yes—
522C22C3×#53812yes—
523C22C3×#53812yes—
524C22C3×#53912yes—
525C22C3×#54012yes—
526C22C3×#54112yes—
527C22C3×#54212yes—
528C22C3×#56012yes—
529C22C3×#56011yescharacteristic
530C22C3×H(3,𝔽4)11yescharacteristic
531C22C3×#56212yes—
532C22C3×#56212yes—
533C22C3×#56212yes—
534C22C3×#56212yes—
535C22C3×#56312yes—
536C22C3×#56311yescharacteristic
537C22C3×#56311yescharacteristic
538C4—42no—
539C4—44no—
540C4—42no—
541C4—44no—
542C4—44no—
543C4—82no—
544C4—82no—
545C4—82no—
546C3#1465211yescharacteristic, direct product factor, Sylow
547C2—82no—
548C2C3×#233311yescharacteristic
549C2C3×#233412yes—
550C2C3×#237612yes—
551C2C3×#237611yescharacteristic
552C2C3×#237911yescharacteristic
553C2C3×#282412yes—
554C2C3×#282912yes—
555C2C3×#283112yes—
556C2C3×#283612yes—
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#1442C2semidirect
3. C3×#1442C2semidirect
4. C3×#1441C2semidirect
9. C3×#1436C2semidirect
10. #14652C3direct (N × H)
20. C3×#397C4semidirect
34. #1442C2×C3semidirect
35. #1442C2×C3semidirect
36. #1441C2×C3semidirect
41. #1436C2×C3semidirect
96. #397C3×C4semidirect
546. C3#14652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3972
C3×#14422
C3×#14412
C3×#14402
C3×#14392
C3×#14372
C3×#14362
#146523