Finite Group Databasebeta

← C3×#15722 #10634329

Subgroups — C3×#15722 #10634329

Order 768.

1386 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#397C211yesmaximal, characteristic
2C2×C3×#385C211yesmaximal, characteristic
3C3×#1443C212yesmaximal, complemented
4C3×#1442C212yesmaximal, complemented
5C3×#1441C212yesmaximal, complemented
6C3×#1440C212yesmaximal
7C3×#1437C211yesmaximal, characteristic
8C3×#1434C212yesmaximal, complemented
9C3×C22⋊#95C212yesmaximal
10#15722C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C3×#397—24no—
20C3×#385—24no—
21C2×C3×#95C2212yes—
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×#397C2×C311yescharacteristic
35C2×#385C2×C311yescharacteristic
36#1443C2×C312yescomplemented
37#1442C2×C312yescomplemented
38#1441C2×C312yescomplemented
39#1440C2×C312yes—
40#1437C2×C311yescharacteristic
41#1434C2×C312yescomplemented
42C22⋊#95C2×C312yes—
43C25×C3C2311yescharacteristic
44C23×C3×C4—22no—
45C23×C3×C4—24no—
46C23×C3×C4—41no—
47C23×C3×C4—41no—
48C23×C3×C4—42no—
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—42no—
61C2×C3×C4⋊C4—41no—
62C2×C3×C4⋊C4—41no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—24no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—41no—
71C2×C3×C22⋊C4—41no—
72C2×C3×C22⋊C4—42no—
73C2×C3×C22⋊C4D414yes—
74C2×C3×C42—24no—
75C2×C3×C42—41no—
76C3×#95—24no—
77C3×#95—22no—
78C3×#95—24no—
79C3×#95—24no—
80C3×#95—24no—
81C3×#95—24no—
82C3×#95—24no—
83C3×#95—42no—
84C3×#95—42no—
85C3×#95—41no—
86C3×#95—42no—
87C3×#95—42no—
88C3×#95—42no—
89C24×C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C312yes—
91C22×C4⋊C4C22×C311yescharacteristic
92C22×C22⋊C4C22×C312yes—
93C22×C22⋊C4C22×C311yescharacteristic
94C22×C22⋊C4C22×C312yes—
95C22×C22⋊C4C22×C311yescharacteristic
96C22×C42C22×C311yescharacteristic
97#397—24no—
98#385—24no—
99C2×#95C22×C312yes—
100C2×#95C22×C311yescharacteristic
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C312yes—
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C311yescharacteristic
111C2×#95C22×C312yes—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3C2×D412yes—
120C24×C3#4114yes—
121C24×C3C2411yescharacteristic
122C22×C3×C4—22no—
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—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
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×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—44no—
170C22×C3×C4—48no—
171C22×C3×C4—48no—
172C22×C3×C4—44no—
173C22×C3×C4—44no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4#4114yes—
177C22×C3×C4#4114yes—
178C22×C3×C4#4114yes—
179C3×C4⋊C4—48no—
180C3×C4⋊C4—44no—
181C3×C22⋊C4—28no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—48no—
184C3×C22⋊C4—48no—
185C3×C42—44no—
186C25C23×C311yescharacteristic
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—41no—
190C23×C4—41no—
191C23×C4—42no—
192C23×C4C23×C312yes—
193C23×C4C23×C312yes—
194C23×C4C23×C312yes—
195C23×C4C23×C311yescharacteristic
196C23×C4C23×C311yescharacteristic
197C23×C4C23×C312yes—
198C23×C4C23×C312yes—
199C23×C4C23×C312yes—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4C3×D414yes—
217C2×C42—24no—
218C2×C42—41no—
219#95—24no—
220#95—22no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—42no—
227#95—42no—
228#95—41no—
229#95—42no—
230#95—42no—
231#95—42no—
232C23×C3—22no—
233C23×C3—24no—
234C23×C3—22no—
235C23×C3—24no—
236C23×C3—24no—
237C23×C3—24no—
238C23×C3—24no—
239C23×C3—24no—
240C23×C3—24no—
241C23×C3—24no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3#12414yes—
253C23×C3C22×D411yescharacteristic
254C23×C3C2×#4112yes—
255C23×C3C2×#4112yes—
256C23×C3C2×#4112yes—
257C23×C3C2×#4112yes—
258C23×C3H(5,𝔽2)11yescharacteristic
259C23×C3H(5,𝔽2)11yescharacteristic
260C23×C3H(5,𝔽2)12yes—
261C23×C32−1+412yes—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—416no—
276C2×C3×C4—42no—
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—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
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—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24C2×C3×D412yes—
323C24C3×#4114yes—
324C24C24×C311yescharacteristic
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
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×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4C2×C3×D412yes—
378C22×C4C2×C3×D412yes—
379C22×C4C3×#4114yes—
380C22×C4C3×#4114yes—
381C22×C4C3×#4114yes—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C22⋊C4—28no—
385C22⋊C4—44no—
386C22⋊C4—48no—
387C22⋊C4—48no—
388C42—44no—
389C22×C3—24no—
390C22×C3—28no—
391C22×C3—44no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3#52511yescharacteristic
399C22×C3C2×#12411yescharacteristic
400C22×C3#52912yes—
401C22×C3#52912yes—
402C22×C3#53411yescharacteristic
403C22×C3#53812yes—
404C22×C3#53912yes—
405C22×C3#54212yes—
406C22×C3#54212yes—
407C22×C3#54612yes—
408C22×C3#54712yes—
409C22×C3#55111yescharacteristic
410C22×C3#55112yes—
411C22×C3#55111yescharacteristic
412C22×C3#55212yes—
413C22×C3#55212yes—
414C22×C3#55312yes—
415C22×C3#55312yes—
416C22×C3#55312yes—
417C22×C3#56011yescharacteristic
418C22×C3#56311yescharacteristic
419C3×C4—44no—
420C3×C4—42no—
421C3×C4—44no—
422C3×C4—42no—
423C3×C4—44no—
424C3×C4—82no—
425C3×C4—82no—
426C3×C4—82no—
427C23—22no—
428C23—24no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23C3×#12414yes—
448C23C22×C3×D411yescharacteristic
449C23C2×C3×#4112yes—
450C23C2×C3×#4112yes—
451C23C2×C3×#4112yes—
452C23C2×C3×#4112yes—
453C23C3×H(5,𝔽2)11yescharacteristic
454C23C3×H(5,𝔽2)12yes—
455C23C3×H(5,𝔽2)11yescharacteristic
456C23C3×2−1+412yes—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—416no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
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—41no—
486C2×C4—41no—
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×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C3—44no—
511C2×C3#246812yes—
512C2×C3#248812yes—
513C2×C3#253012yes—
514C2×C3#257612yes—
515C2×C3#259111yescharacteristic
516C2×C3#269711yescharacteristic
517C2×C3#271711yescharacteristic
518C2×C3#275712yes—
519C2×C3#277012yes—
520C22—28no—
521C22—24no—
522C22—42no—
523C22—42no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22C3×#52511yescharacteristic
530C22C2×C3×#12411yescharacteristic
531C22C3×#52912yes—
532C22C3×#52912yes—
533C22C3×#53411yescharacteristic
534C22C3×#53812yes—
535C22C3×#53912yes—
536C22C3×#54212yes—
537C22C3×#54212yes—
538C22C3×#54612yes—
539C22C3×#54712yes—
540C22C3×#55111yescharacteristic
541C22C3×#55111yescharacteristic
542C22C3×#55112yes—
543C22C3×#55212yes—
544C22C3×#55212yes—
545C22C3×#55312yes—
546C22C3×#55312yes—
547C22C3×#55312yes—
548C22C3×#56011yescharacteristic
549C22C3×#56311yescharacteristic
550C4—44no—
551C4—42no—
552C4—42no—
553C4—44no—
554C4—44no—
555C4—82no—
556C4—82no—
557C4—82no—
558C3#1572211yescharacteristic, direct product factor, Sylow
559C2—44no—
560C2C3×#246812yes—
561C2C3×#248812yes—
562C2C3×#253012yes—
563C2C3×#257612yes—
564C2C3×#259111yescharacteristic
565C2C3×#269711yescharacteristic
566C2C3×#271711yescharacteristic
567C2C3×#275712yes—
568C2C3×#277012yes—
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
3. C3×#1443C2semidirect
4. C3×#1442C2semidirect
5. C3×#1441C2semidirect
8. C3×#1434C2semidirect
10. #15722C3direct (N × H)
36. #1443C2×C3semidirect
37. #1442C2×C3semidirect
38. #1441C2×C3semidirect
41. #1434C2×C3semidirect
558. C3#15722direct (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
C2×C3×#3852
C3×#14432
C3×#14422
C3×#14412
C3×#14402
C3×#14372
C3×#14342
C3×C22⋊#952
#157223