Finite Group Databasebeta

← C3×#23946 #10642553

Subgroups — C3×#23946 #10642553

Order 768.

1222 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic
2C2×C3×#451C211yesmaximal, characteristic, complemented
3C2×C3×#448C211yesmaximal, characteristic
4C3×#2310C211yesmaximal, characteristic, complemented
5C3×#2024C212yesmaximal, complemented
6C3×#2015C212yesmaximal, complemented
7C3×#1878C212yesmaximal, complemented
8C3×#1877C212yesmaximal, complemented
9C3×#1875C211yesmaximal, characteristic
10C3×#1873C211yesmaximal, characteristic, complemented
11C3×#1818C211yesmaximal, characteristic, complemented
12#23946C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C3×#542—24no—
17C3×#542—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#126—21no—
20C2×C3×#124—21no—
21C2×C3×#124C2211yescharacteristic
22C2×C3×#123—21no—
23C2×C3×C22⋊Q8—21no—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×#121C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×#117C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C3×#451—24no—
31C3×#448—24no—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2211yescharacteristic
38C3×#391—21no—
39C3×#391—21no—
40C3×#391C2212yes—
41C3×#391C2212yes—
42C3×#381—21no—
43C3×#381—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#542C2×C311yescharacteristic
49C2×#451C2×C311yescharacteristic, complemented
50C2×#448C2×C311yescharacteristic
51#2310C2×C311yescharacteristic, complemented
52#2024C2×C312yescomplemented
53#2015C2×C312yescomplemented
54#1878C2×C312yescomplemented
55#1877C2×C312yescomplemented
56#1875C2×C311yescharacteristic
57#1873C2×C311yescharacteristic, complemented
58#1818C2×C311yescharacteristic, complemented
59C22×C3×Q8—24no—
60C22×C3×Q8—21no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8—42no—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×Q8C2312yes—
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C2×C3×Q16—24no—
73C2×C3×Q16—24no—
74C2×C3×Q16—42no—
75C2×C3×SD16—24no—
76C2×C3×SD16—24no—
77C2×C3×SD16—42no—
78C22×C3×C8C2312yes—
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—22no—
82C3×#126—28no—
83C3×#124—28no—
84C3×#124—22no—
85C3×#124—22no—
86C3×#123—28no—
87C3×C22⋊Q8—28no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8D414yes—
90C3×#121—22no—
91C3×#121D414yes—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×#117—22no—
97C3×#117—22no—
98C3×#117—42no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4D411yescharacteristic
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C3×#103—24no—
120C3×#103—24no—
121C3×#103—42no—
122C3×#102—42no—
123C3×C22⋊C8—22no—
124C3×C22⋊C8—22no—
125C3×#95—22no—
126C3×#95—22no—
127C3×#95—22no—
128C3×#95—22no—
129C23×Q8C22×C311yescharacteristic
130C22×Q16C22×C312yes—
131C22×SD16C22×C312yes—
132#542—24no—
133#542—24no—
134C2×C4⋊Q8C22×C311yescharacteristic
135C2×#126—21no—
136C2×#124—21no—
137C2×#124C22×C311yescharacteristic
138C2×#123—21no—
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×#121C22×C311yescharacteristic
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4C22×C311yescharacteristic
144C2×#117C22×C311yescharacteristic
145C2×#117C22×C311yescharacteristic
146#451—24no—
147#448—24no—
148C2×#103C22×C312yes—
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#102C22×C312yes—
152C2×C22⋊C8C22×C311yescharacteristic
153C2×C22⋊C8C22×C311yescharacteristic
154#391—21no—
155#391—21no—
156#391C22×C312yes—
157#391C22×C312yes—
158#381—21no—
159#381—21no—
160#340C22×C312yes—
161#340C22×C312yes—
162#336C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—24no—
167C2×C3×Q8—24no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—48no—
177C2×C3×Q8—48no—
178C2×C3×Q8—42no—
179C2×C3×Q8—42no—
180C2×C3×Q8—82no—
181C2×C3×Q8C2×D412yes—
182C2×C3×Q8#4114yes—
183C2×C3×D4—24no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—82no—
188C2×C3×D4C2×D412yes—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—24no—
197C22×C3×C4—22no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4—82no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2411yescharacteristic
209C3×Q16—48no—
210C3×SD16—48no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4C2×D412yes—
227C3×C4⋊C4C2×D412yes—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—42no—
236C3×C42—24no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—42no—
242C22×Q8—24no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—42no—
248C22×Q8C23×C311yescharacteristic
249C22×Q8C23×C312yes—
250C22×D4—21no—
251C22×D4C23×C311yescharacteristic
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C23×C311yescharacteristic
255C2×Q16—24no—
256C2×Q16—24no—
257C2×Q16—42no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—42no—
261C22×C8C23×C312yes—
262C22×C8C23×C312yes—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265#126—28no—
266#124—28no—
267#124—22no—
268#124—22no—
269#123—28no—
270C22⋊Q8—28no—
271C22⋊Q8—22no—
272C22⋊Q8C3×D414yes—
273#121—22no—
274#121C3×D414yes—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×D4—22no—
278C4×D4—22no—
279#117—22no—
280#117—22no—
281#117—42no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4C3×D411yescharacteristic
286C2×C4⋊C4C3×D411yescharacteristic
287C2×C4⋊C4C23×C312yes—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4C3×D411yescharacteristic
297C2×C22⋊C4C3×D411yescharacteristic
298C2×C42—21no—
299C2×C42—21no—
300C2×C42C23×C311yescharacteristic
301C2×C42C23×C311yescharacteristic
302#103—24no—
303#103—24no—
304#103—42no—
305#102—42no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3#12012yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—24no—
324C3×Q8—24no—
325C3×Q8—24no—
326C3×Q8—48no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—84no—
331C3×D4—24no—
332C3×D4—42no—
333C3×D4—84no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—48no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—84no—
361C2×C3×C4#12012yes—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4112yes—
365C2×C3×C4H(5,𝔽2)12yes—
366C3×C8—42no—
367C3×C8—42no—
368C24—21no—
369C24C2×C3×D411yescharacteristic
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—82no—
385C2×Q8C2×C3×D412yes—
386C2×Q8C3×#4114yes—
387C2×D4—24no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—82no—
392C2×D4C2×C3×D412yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—82no—
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D412yes—
411C22×C4C2×C3×D412yes—
412C22×C4C24×C311yescharacteristic
413Q16—48no—
414SD16—48no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4C2×C3×D412yes—
431C4⋊C4C2×C3×D412yes—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—24no—
439C22⋊C4—42no—
440C42—24no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—42no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#52511yescharacteristic
458C22×C3#53511yescharacteristic
459C22×C3#57712yes—
460C22×C3#57812yes—
461C3×C4—22no—
462C3×C4—44no—
463C3×C4—42no—
464C3×C4—41no—
465C3×C4—41no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—82no—
471C3×C4#54612yes—
472C3×C4#54612yes—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23C3×#12012yes—
483C23C22×C3×D411yescharacteristic
484Q8—24no—
485Q8—24no—
486Q8—24no—
487Q8—48no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—84no—
492D4—24no—
493D4—42no—
494D4—84no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—84no—
522C2×C4C3×#12012yes—
523C2×C4C22×C3×D411yescharacteristic
524C2×C4C22×C3×D411yescharacteristic
525C2×C4C2×C3×#4112yes—
526C2×C4C3×H(5,𝔽2)12yes—
527C8—42no—
528C8—42no—
529C2×C3—42no—
530C2×C3—82no—
531C2×C3#240411yescharacteristic
532C2×C3#300911yescharacteristic
533C2×C3#301111yescharacteristic
534C2×C3#312112yes—
535C2×C3#322512yes—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22C2×C3×#12011yescharacteristic
547C22C3×#52511yescharacteristic
548C22C3×#53511yescharacteristic
549C22C3×#57712yes—
550C22C3×#57812yes—
551C4—22no—
552C4—44no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—82no—
561C4C3×#54612yes—
562C4C3×#54612yes—
563C3#2394611yescharacteristic, direct product factor, Sylow
564C2—42no—
565C2—82no—
566C2C3×#240411yescharacteristic
567C2C3×#300911yescharacteristic
568C2C3×#301111yescharacteristic
569C2C3×#312112yes—
570C2C3×#322512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#451C2semidirect
4. C3×#2310C2semidirect
5. C3×#2024C2semidirect
6. C3×#2015C2semidirect
7. C3×#1878C2semidirect
8. C3×#1877C2semidirect
10. C3×#1873C2semidirect
11. C3×#1818C2semidirect
12. #23946C3direct (N × H)
49. C2×#451C2×C3semidirect
51. #2310C2×C3semidirect
52. #2024C2×C3semidirect
53. #2015C2×C3semidirect
54. #1878C2×C3semidirect
55. #1877C2×C3semidirect
57. #1873C2×C3semidirect
58. #1818C2×C3semidirect
563. C3#23946direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5422
C2×C3×#4512
C2×C3×#4482
C3×#23102
C3×#20242
C3×#20152
C3×#18782
C3×#18772
C3×#18752
C3×#18732
C3×#18182
#239463