Finite Group Databasebeta

← C3×#25638 #10644245

Subgroups — C3×#25638 #10644245

Order 768.

1134 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic, complemented
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#463C211yesmaximal, characteristic
4C3×#2592C211yesmaximal, characteristic, complemented
5C3×#2015C212yesmaximal, complemented
6C3×#2012C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1981C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1878C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25638C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C22×C3×SD16C2211yescharacteristic
16C22×C3×D8C2211yescharacteristic
17C3×#556—24no—
18C3×#556—24no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#126—22no—
21C2×C3×#124C2211yescharacteristic
22C2×C3×#124C2212yes—
23C2×C3×C22⋊Q8—21no—
24C2×C3×C4×Q8C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×#117—21no—
28C3×#497—24no—
29C3×#463—24no—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2211yescharacteristic
36C2×C3×#97C2211yescharacteristic
37C3×#397—21no—
38C3×#393—21no—
39C3×#391C2212yes—
40C3×#390C2212yes—
41C3×#388—22no—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#556C2×C311yescharacteristic, complemented
47C2×#497C2×C311yescharacteristic, complemented
48C2×#463C2×C311yescharacteristic
49#2592C2×C311yescharacteristic, complemented
50#2015C2×C312yescomplemented
51#2012C2×C312yes—
52#1985C2×C311yescharacteristic, complemented
53#1981C2×C311yescharacteristic, complemented
54#1880C2×C312yescomplemented
55#1878C2×C312yescomplemented
56#1762C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2312yes—
60C22×C3×D4C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C2×C3×Q16—24no—
64C2×C3×Q16—24no—
65C2×C3×Q16—42no—
66C2×C3×SD16—24no—
67C2×C3×SD16—22no—
68C2×C3×SD16—24no—
69C2×C3×SD16—42no—
70C2×C3×SD16—42no—
71C2×C3×D8—22no—
72C2×C3×D8—42no—
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2312yes—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8D414yes—
78C3×#126—216no—
79C3×#124—24no—
80C3×#124—24no—
81C3×#124—22no—
82C3×#124—22no—
83C3×C22⋊Q8—28no—
84C3×C4×Q8—24no—
85C3×C4×Q8—24no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×D4—22no—
89C3×C4×D4D414yes—
90C3×#117—28no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C42—22no—
106C2×C3×C42—21no—
107C2×C3×C42C2312yes—
108C2×C3×C42C2311yescharacteristic
109C3×C4⋊C8—22no—
110C3×#103—22no—
111C3×#103—42no—
112C3×#103—42no—
113C3×#102—24no—
114C3×#102—22no—
115C3×#97—22no—
116C3×#95—21no—
117C3×#95—22no—
118C3×#95—22no—
119C22×Q16C22×C312yes—
120C22×SD16C22×C312yes—
121C22×SD16C22×C311yescharacteristic
122C22×D8C22×C311yescharacteristic
123#556—24no—
124#556—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×#126—22no—
127C2×#124C22×C311yescharacteristic
128C2×#124C22×C312yes—
129C2×C22⋊Q8—21no—
130C2×C4×Q8C22×C312yes—
131C2×C4×Q8C22×C311yescharacteristic
132C2×C4×D4C22×C311yescharacteristic
133C2×#117—21no—
134#497—24no—
135#463—24no—
136C2×C4⋊C8C22×C311yescharacteristic
137C2×#103C22×C311yescharacteristic
138C2×#103C22×C312yes—
139C2×#103C22×C312yes—
140C2×#102C22×C311yescharacteristic
141C2×#102C22×C312yes—
142C2×#97C22×C311yescharacteristic
143#397—21no—
144#393—21no—
145#391C22×C312yes—
146#390C22×C312yes—
147#388—22no—
148C4×C4⋊C4C22×C311yescharacteristic
149#340C22×C312yes—
150#336C22×C312yes—
151#336C22×C312yes—
152C24×C3—21no—
153C24×C3—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—42no—
165C2×C3×Q8C2×D414yes—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—44no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C3×Q16—48no—
186C3×SD16—48no—
187C3×SD16—44no—
188C3×D8—44no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8#4114yes—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4C2×D412yes—
205C3×C22⋊C4—28no—
206C3×C22⋊C4—28no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—44no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—82no—
212C3×C42—28no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—81no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218C22×Q8C3×D411yescharacteristic
219C22×Q8C23×C311yescharacteristic
220C22×Q8C23×C312yes—
221C22×D4C23×C311yescharacteristic
222C22×D4C23×C311yescharacteristic
223C23×C4—21no—
224C2×Q16—24no—
225C2×Q16—24no—
226C2×Q16—42no—
227C2×SD16—24no—
228C2×SD16—22no—
229C2×SD16—24no—
230C2×SD16—42no—
231C2×SD16—42no—
232C2×D8—22no—
233C2×D8—42no—
234C22×C8C23×C311yescharacteristic
235C22×C8C23×C311yescharacteristic
236C22×C8C23×C312yes—
237C4⋊Q8—22no—
238C4⋊Q8C3×D414yes—
239#126—216no—
240#124—24no—
241#124—24no—
242#124—22no—
243#124—22no—
244C22⋊Q8—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×D4—22no—
250C4×D4C3×D414yes—
251#117—28no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4C3×D411yescharacteristic
256C2×C4⋊C4C3×D412yes—
257C2×C4⋊C4C23×C312yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42C23×C312yes—
269C2×C42C23×C311yescharacteristic
270C4⋊C8—22no—
271#103—22no—
272#103—42no—
273#103—42no—
274#102—24no—
275#102—22no—
276#97—22no—
277#95—21no—
278#95—22no—
279#95—22no—
280C23×C3—22no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3C22×D411yescharacteristic
291C3×Q8—24no—
292C3×Q8—24no—
293C3×Q8—44no—
294C3×Q8—42no—
295C3×Q8—42no—
296C3×Q8—41no—
297C3×Q8—41no—
298C3×Q8—42no—
299C3×D4—22no—
300C3×D4—22no—
301C3×D4—41no—
302C3×D4—41no—
303C3×D4—42no—
304C3×D4—42no—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—48no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4#12012yes—
326C2×C3×C4#12012yes—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4112yes—
330C2×C3×C4H(5,𝔽2)12yes—
331C3×C8—24no—
332C3×C8—41no—
333C3×C8—41no—
334C24—21no—
335C24—21no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—42no—
347C2×Q8C2×C3×D414yes—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4C2×C3×D412yes—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—44no—
362C22×C4C2×C3×D412yes—
363C22×C4C2×C3×D412yes—
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C24×C311yescharacteristic
367Q16—48no—
368SD16—48no—
369SD16—44no—
370D8—44no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8C3×#4114yes—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4C2×C3×D412yes—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—24no—
390C22⋊C4—44no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—82no—
394C42—28no—
395C42—22no—
396C42—22no—
397C42—81no—
398C42C2×C3×D412yes—
399C42C2×C3×D412yes—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3C2×#12011yescharacteristic
411C22×C3#55012yes—
412C22×C3#57512yes—
413C22×C3#57711yescharacteristic
414C22×C3#57811yescharacteristic
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—44no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—82no—
422C3×C4—81no—
423C3×C4#53211yescharacteristic
424C3×C4#53511yescharacteristic
425C3×C4#54612yes—
426C23—22no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—42no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23C22×C3×D411yescharacteristic
437Q8—24no—
438Q8—24no—
439Q8—44no—
440Q8—42no—
441Q8—42no—
442Q8—41no—
443Q8—41no—
444Q8—42no—
445D4—22no—
446D4—22no—
447D4—41no—
448D4—41no—
449D4—42no—
450D4—42no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—48no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4C3×#12012yes—
472C2×C4C3×#12012yes—
473C2×C4C22×C3×D411yescharacteristic
474C2×C4C22×C3×D411yescharacteristic
475C2×C4C2×C3×#4112yes—
476C2×C4C3×H(5,𝔽2)12yes—
477C8—24no—
478C8—41no—
479C8—41no—
480C2×C3—82no—
481C2×C3—161no—
482C2×C3#243111yescharacteristic
483C2×C3#300511yescharacteristic
484C2×C3#301311yescharacteristic
485C2×C3#340112yes—
486C2×C3#340212yes—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22C2×C3×#12011yescharacteristic
498C22C3×#55012yes—
499C22C3×#57512yes—
500C22C3×#57711yescharacteristic
501C22C3×#57811yescharacteristic
502C4—22no—
503C4—22no—
504C4—44no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—81no—
510C4C3×#53211yescharacteristic
511C4C3×#53511yescharacteristic
512C4C3×#54612yes—
513C3#2563811yescharacteristic, direct product factor, Sylow
514C2—82no—
515C2—161no—
516C2C3×#243111yescharacteristic
517C2C3×#300511yescharacteristic
518C2C3×#301311yescharacteristic
519C2C3×#340112yes—
520C2C3×#340212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#556C2semidirect
2. C2×C3×#497C2semidirect
4. C3×#2592C2semidirect
5. C3×#2015C2semidirect
7. C3×#1985C2semidirect
8. C3×#1981C2semidirect
9. C3×#1880C2semidirect
10. C3×#1878C2semidirect
11. C3×#1762C2semidirect
12. #25638C3direct (N × H)
46. C2×#556C2×C3semidirect
47. C2×#497C2×C3semidirect
49. #2592C2×C3semidirect
50. #2015C2×C3semidirect
52. #1985C2×C3semidirect
53. #1981C2×C3semidirect
54. #1880C2×C3semidirect
55. #1878C2×C3semidirect
56. #1762C2×C3semidirect
513. C3#25638direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4972
C2×C3×#4632
C3×#25922
C3×#20152
C3×#20122
C3×#19852
C3×#19812
C3×#18802
C3×#18782
C3×#17622
#256383