Finite Group Databasebeta

← C3×#25613 #10644220

Subgroups — C3×#25613 #10644220

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C3×#2621C211yesmaximal, characteristic, complemented
5C3×#2001C212yesmaximal, complemented
6C3×#2000C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1958C211yesmaximal, characteristic, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1877C212yesmaximal, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25613C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C3×#542—24no—
16C3×#542—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
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×C22⋊Q8C2212yes—
26C2×C3×#121C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C3×#497—24no—
31C3×#487—24no—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C2×C3×#97C2211yescharacteristic
38C2×C3×C4×C8C2211yescharacteristic
39C3×#398—21no—
40C3×#398—21no—
41C3×#397—21no—
42C3×#397—21no—
43C3×#391C2212yes—
44C3×#382C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#542C2×C311yescharacteristic
49C2×#497C2×C311yescharacteristic, complemented
50C2×#487C2×C311yescharacteristic, complemented
51#2621C2×C311yescharacteristic, complemented
52#2001C2×C312yescomplemented
53#2000C2×C312yes—
54#1993C2×C311yescharacteristic, complemented
55#1958C2×C311yescharacteristic, complemented
56#1878C2×C312yescomplemented
57#1877C2×C312yescomplemented
58#1750C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C2×C3×Q16—24no—
67C2×C3×Q16—24no—
68C2×C3×Q16—42no—
69C2×C3×SD16—24no—
70C2×C3×SD16—24no—
71C2×C3×SD16—42no—
72C22×C3×C8C2312yes—
73C22×C3×C8C2312yes—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8D414yes—
78C3×#126—28no—
79C3×#124—28no—
80C3×#124—22no—
81C3×#124D414yes—
82C3×#123—28no—
83C3×C22⋊Q8—28no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—42no—
87C3×C22⋊Q8—42no—
88C3×#121—22no—
89C3×#121—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×#117—22no—
95C3×#117—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
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—22no—
113C2×C3×C22⋊C4D411yescharacteristic
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C42—21no—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×#103—24no—
119C3×#103—24no—
120C3×#103—42no—
121C3×#102—42no—
122C3×C22⋊C8—24no—
123C3×#97—22no—
124C3×C4×C8—22no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—22no—
128C3×#95—22no—
129C22×Q16C22×C312yes—
130C22×SD16C22×C312yes—
131#542—24no—
132#542—24no—
133C2×C4⋊Q8C22×C311yescharacteristic
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×C22⋊Q8C22×C312yes—
142C2×#121C22×C311yescharacteristic
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×D4C22×C311yescharacteristic
145C2×#117C22×C311yescharacteristic
146#497—24no—
147#487—24no—
148C2×#103C22×C312yes—
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#102C22×C312yes—
152C2×C22⋊C8C22×C312yes—
153C2×#97C22×C311yescharacteristic
154C2×C4×C8C22×C311yescharacteristic
155#398—21no—
156#398—21no—
157#397—21no—
158#397—21no—
159#391C22×C312yes—
160#382C22×C311yescharacteristic
161#340C22×C312yes—
162#340C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—24no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
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—82no—
177C2×C3×Q8C2×D412yes—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—82no—
183C2×C3×D4C2×D412yes—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—82no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C3×Q16—48no—
202C3×SD16—48no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8#4114yes—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—44no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4C2×D412yes—
222C3×C4⋊C4C2×D412yes—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—44no—
231C3×C42—24no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42C2×D412yes—
235C3×C42C2×D412yes—
236C22×Q8—21no—
237C22×Q8C23×C312yes—
238C22×Q8C23×C311yescharacteristic
239C22×D4—21no—
240C22×D4C23×C311yescharacteristic
241C23×C4—21no—
242C23×C4C23×C311yescharacteristic
243C2×Q16—24no—
244C2×Q16—24no—
245C2×Q16—42no—
246C2×SD16—24no—
247C2×SD16—24no—
248C2×SD16—42no—
249C22×C8C23×C312yes—
250C22×C8C23×C312yes—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8C3×D414yes—
255#126—28no—
256#124—28no—
257#124—22no—
258#124C3×D414yes—
259#123—28no—
260C22⋊Q8—28no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265#121—22no—
266#121—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×D4—22no—
270C4×D4—22no—
271#117—22no—
272#117—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C3×D411yescharacteristic
280C2×C4⋊C4C3×D411yescharacteristic
281C2×C4⋊C4C23×C312yes—
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4C3×D411yescharacteristic
291C2×C22⋊C4C3×D411yescharacteristic
292C2×C42—21no—
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#103—24no—
296#103—24no—
297#103—42no—
298#102—42no—
299C22⋊C8—24no—
300#97—22no—
301C4×C8—22no—
302#95—21no—
303#95—21no—
304#95—22no—
305#95—22no—
306C23×C3—21no—
307C23×C3—22no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C22×D411yescharacteristic
317C3×Q8—24no—
318C3×Q8—42no—
319C3×Q8—42no—
320C3×Q8—42no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×Q8—84no—
324C3×D4—24no—
325C3×D4—42no—
326C3×D4—84no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—84no—
352C2×C3×C4#12012yes—
353C2×C3×C4#12012yes—
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C2×#4112yes—
357C2×C3×C4H(5,𝔽2)12yes—
358C3×C8—24no—
359C3×C8—42no—
360C24—21no—
361C24C2×C3×D411yescharacteristic
362C2×Q8—24no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—82no—
373C2×Q8C2×C3×D412yes—
374C2×D4—24no—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—82no—
379C2×D4C2×C3×D412yes—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—82no—
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C24×C311yescharacteristic
397Q16—48no—
398SD16—48no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8C3×#4114yes—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4C2×C3×D412yes—
418C4⋊C4C2×C3×D412yes—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—44no—
427C42—24no—
428C42—21no—
429C42—21no—
430C42C2×C3×D412yes—
431C42C2×C3×D412yes—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—42no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3C2×#12011yescharacteristic
443C22×C3#53211yescharacteristic
444C22×C3#53511yescharacteristic
445C22×C3#57712yes—
446C22×C3#57812yes—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—41no—
455C3×C4—82no—
456C3×C4—82no—
457C3×C4#54612yes—
458C3×C4#55012yes—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—22no—
463C23—21no—
464C23—21no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23C22×C3×D411yescharacteristic
470Q8—24no—
471Q8—42no—
472Q8—42no—
473Q8—42no—
474Q8—42no—
475Q8—42no—
476Q8—84no—
477D4—24no—
478D4—42no—
479D4—84no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
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—84no—
505C2×C4C3×#12012yes—
506C2×C4C3×#12012yes—
507C2×C4C22×C3×D411yescharacteristic
508C2×C4C22×C3×D411yescharacteristic
509C2×C4C2×C3×#4112yes—
510C2×C4C3×H(5,𝔽2)12yes—
511C8—24no—
512C8—42no—
513C2×C3—81no—
514C2×C3—82no—
515C2×C3#243111yescharacteristic
516C2×C3#300911yescharacteristic
517C2×C3#301111yescharacteristic
518C2×C3#317812yes—
519C2×C3#322512yes—
520C22—42no—
521C22—41no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—42no—
528C22—82no—
529C22—82no—
530C22C2×C3×#12011yescharacteristic
531C22C3×#53211yescharacteristic
532C22C3×#53511yescharacteristic
533C22C3×#57712yes—
534C22C3×#57812yes—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—42no—
541C4—42no—
542C4—41no—
543C4—82no—
544C4—82no—
545C4C3×#54612yes—
546C4C3×#55012yes—
547C3#2561311yescharacteristic, direct product factor, Sylow
548C2—81no—
549C2—82no—
550C2C3×#243111yescharacteristic
551C2C3×#300911yescharacteristic
552C2C3×#301111yescharacteristic
553C2C3×#317812yes—
554C2C3×#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×#497C2semidirect
3. C2×C3×#487C2semidirect
4. C3×#2621C2semidirect
5. C3×#2001C2semidirect
7. C3×#1993C2semidirect
8. C3×#1958C2semidirect
9. C3×#1878C2semidirect
10. C3×#1877C2semidirect
11. C3×#1750C2semidirect
12. #25613C3direct (N × H)
49. C2×#497C2×C3semidirect
50. C2×#487C2×C3semidirect
51. #2621C2×C3semidirect
52. #2001C2×C3semidirect
54. #1993C2×C3semidirect
55. #1958C2×C3semidirect
56. #1878C2×C3semidirect
57. #1877C2×C3semidirect
58. #1750C2×C3semidirect
547. C3#25613direct (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×#4972
C2×C3×#4872
C3×#26212
C3×#20012
C3×#20002
C3×#19932
C3×#19582
C3×#18782
C3×#18772
C3×#17502
#256133