Finite Group Databasebeta

← C3×#25578 #10644185

Subgroups — C3×#25578 #10644185

Order 768.

1226 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#555C211yesmaximal, characteristic, complemented
2C2×C3×#489C211yesmaximal, characteristic, complemented
3C2×C3×#463C211yesmaximal, characteristic, complemented
4C3×#2540C211yesmaximal, characteristic, complemented
5C3×#2000C212yesmaximal, complemented
6C3×#1999C212yesmaximal, complemented
7C3×#1988C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1875C212yesmaximal, complemented
10C3×#1873C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25578C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×D8C2211yescharacteristic
15C3×#555—24no—
16C3×#555—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#124—22no—
19C2×C3×#124C2211yescharacteristic
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121—21no—
22C2×C3×#121C2212yes—
23C2×C3×#121C2212yes—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117—21no—
27C2×C3×#117C2212yes—
28C3×#489—24no—
29C3×#463—24no—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2211yescharacteristic
36C2×C3×C22⋊C8C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C2×C3×#97C2211yescharacteristic
39C3×#399—21no—
40C3×#392—21no—
41C3×#390—22no—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#555C2×C311yescharacteristic, complemented
47C2×#489C2×C311yescharacteristic, complemented
48C2×#463C2×C311yescharacteristic, complemented
49#2540C2×C311yescharacteristic, complemented
50#2000C2×C312yescomplemented
51#1999C2×C312yescomplemented
52#1988C2×C311yescharacteristic, complemented
53#1955C2×C311yescharacteristic, complemented
54#1875C2×C312yescomplemented
55#1873C2×C312yescomplemented
56#1762C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—22no—
60C22×C3×D4—22no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2312yes—
66C2×C3×SD16—22no—
67C2×C3×SD16—42no—
68C2×C3×D8—22no—
69C2×C3×D8—42no—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2312yes—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8D414yescomplemented
75C3×#124—216no—
76C3×#124—22no—
77C3×#124—22no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—42no—
80C3×#121—28no—
81C3×#121—24no—
82C3×#121—24no—
83C3×#121—24no—
84C3×#121—42no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4D414yescomplemented
89C3×#117—28no—
90C3×#117—24no—
91C3×#117—24no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4D412yes—
106C2×C3×C42—22no—
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C3×C4⋊C8—22no—
110C3×#103—22no—
111C3×#103—24no—
112C3×#102—22no—
113C3×#102—24no—
114C3×#102—24no—
115C3×C22⋊C8—42no—
116C3×C22⋊C8—42no—
117C3×#97—22no—
118C3×#95—21no—
119C22×SD16C22×C311yescharacteristic
120C22×D8C22×C311yescharacteristic
121#555—24no—
122#555—24no—
123C2×C4⋊Q8C22×C311yescharacteristic
124C2×#124—22no—
125C2×#124C22×C311yescharacteristic
126C2×C22⋊Q8C22×C312yes—
127C2×#121—21no—
128C2×#121C22×C312yes—
129C2×#121C22×C312yes—
130C2×C4×D4C22×C311yescharacteristic
131C2×C4×D4C22×C311yescharacteristic
132C2×#117—21no—
133C2×#117C22×C312yes—
134#489—24no—
135#463—24no—
136C2×C4⋊C8C22×C311yescharacteristic
137C2×#103C22×C312yes—
138C2×#103C22×C311yescharacteristic
139C2×#102C22×C312yes—
140C2×#102C22×C311yescharacteristic
141C2×#102C22×C312yes—
142C2×C22⋊C8C22×C312yes—
143C2×C22⋊C8C22×C312yes—
144C2×#97C22×C311yescharacteristic
145#399—21no—
146#392—21no—
147#390—22no—
148C4×C4⋊C4C22×C311yescharacteristic
149#340C22×C312yes—
150#340C22×C312yes—
151#340C22×C312yes—
152C24×C3—21no—
153C24×C3—21no—
154C24×C3C2×D412yes—
155C2×C3×Q8—24no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—81no—
161C2×C3×Q8—81no—
162C2×C3×D4—28no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—42no—
172C2×C3×D4—44no—
173C2×C3×D4—44no—
174C2×C3×D4—82no—
175C2×C3×D4—82no—
176C2×C3×D4—82no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—44no—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C3×SD16—44no—
195C3×D8—44no—
196C2×C3×C8—22no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—44no—
209C3×C4⋊C4C2×D412yes—
210C3×C4⋊C4C2×D414yes—
211C3×C4⋊C4#4114yes—
212C3×C22⋊C4—28no—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—44no—
217C3×C22⋊C4—44no—
218C3×C22⋊C4—42no—
219C3×C22⋊C4—42no—
220C3×C42—28no—
221C3×C42—81no—
222C3×C42C2×D412yes—
223C3×C42C2×D412yes—
224C22×Q8—21no—
225C22×Q8C23×C311yescharacteristic
226C22×D4—22no—
227C22×D4—22no—
228C22×D4C3×D411yescharacteristic
229C22×D4C23×C311yescharacteristic
230C22×D4C23×C311yescharacteristic
231C23×C4—21no—
232C23×C4C23×C312yes—
233C2×SD16—22no—
234C2×SD16—42no—
235C2×D8—22no—
236C2×D8—42no—
237C22×C8C23×C311yescharacteristic
238C22×C8C23×C311yescharacteristic
239C22×C8C23×C312yes—
240C4⋊Q8—22no—
241C4⋊Q8C3×D414yescomplemented
242#124—216no—
243#124—22no—
244#124—22no—
245C22⋊Q8—24no—
246C22⋊Q8—42no—
247#121—28no—
248#121—24no—
249#121—24no—
250#121—24no—
251#121—42no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4C3×D414yescomplemented
256#117—28no—
257#117—24no—
258#117—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4C3×D411yescharacteristic
262C2×C4⋊C4C23×C311yescharacteristic
263C2×C4⋊C4C23×C312yes—
264C2×C4⋊C4C23×C312yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4C3×D412yes—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42C23×C311yescharacteristic
276C4⋊C8—22no—
277#103—22no—
278#103—24no—
279#102—22no—
280#102—24no—
281#102—24no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284#97—22no—
285#95—21no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3C22×D411yescharacteristic
304C3×Q8—41no—
305C3×Q8—41no—
306C3×Q8—42no—
307C3×Q8—84no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—44no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—88no—
316C3×D4—88no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—48no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4#12012yes—
339C2×C3×C4#12012yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4112yes—
343C2×C3×C4H(5,𝔽2)12yes—
344C3×C8—41no—
345C3×C8—41no—
346C3×C8—42no—
347C24—21no—
348C24—21no—
349C24C2×C3×D412yes—
350C2×Q8—24no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×D4—28no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4C2×C3×D412yes—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—44no—
385C22×C4C2×C3×D412yes—
386C22×C4C2×C3×D411yescharacteristic
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C24×C311yescharacteristic
389SD16—44no—
390D8—44no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—44no—
404C4⋊C4C2×C3×D412yes—
405C4⋊C4C2×C3×D414yes—
406C4⋊C4C3×#4114yes—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C42—28no—
416C42—81no—
417C42C2×C3×D412yes—
418C42C2×C3×D412yes—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3C2×#12011yescharacteristic
437C22×C3#54612yes—
438C22×C3#57512yes—
439C22×C3#57611yescharacteristic
440C22×C3#57711yescharacteristic
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—44no—
446C3×C4—44no—
447C3×C4—81no—
448C3×C4#53211yescharacteristic
449C3×C4#53511yescharacteristic
450C3×C4#55012yes—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23C22×C3×D411yescharacteristic
469Q8—41no—
470Q8—41no—
471Q8—42no—
472Q8—84no—
473D4—22no—
474D4—22no—
475D4—44no—
476D4—41no—
477D4—41no—
478D4—42no—
479D4—42no—
480D4—88no—
481D4—88no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—48no—
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—41no—
502C2×C4—42no—
503C2×C4C3×#12012yes—
504C2×C4C3×#12012yes—
505C2×C4C22×C3×D411yescharacteristic
506C2×C4C22×C3×D411yescharacteristic
507C2×C4C2×C3×#4112yes—
508C2×C4C3×H(5,𝔽2)12yes—
509C8—41no—
510C8—41no—
511C8—42no—
512C2×C3—82no—
513C2×C3—82no—
514C2×C3—161no—
515C2×C3#243111yescharacteristic
516C2×C3#300511yescharacteristic
517C2×C3#301211yescharacteristic
518C2×C3#331912yes—
519C2×C3#332112yes—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—82no—
536C22—82no—
537C22C2×C3×#12011yescharacteristic
538C22C3×#54612yes—
539C22C3×#57512yes—
540C22C3×#57611yescharacteristic
541C22C3×#57711yescharacteristic
542C4—22no—
543C4—22no—
544C4—42no—
545C4—42no—
546C4—44no—
547C4—44no—
548C4—81no—
549C4C3×#53211yescharacteristic
550C4C3×#53511yescharacteristic
551C4C3×#55012yes—
552C3#2557811yescharacteristic, direct product factor, Sylow
553C2—82no—
554C2—82no—
555C2—161no—
556C2C3×#243111yescharacteristic
557C2C3×#300511yescharacteristic
558C2C3×#301211yescharacteristic
559C2C3×#331912yes—
560C2C3×#332112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#555C2semidirect
2. C2×C3×#489C2semidirect
3. C2×C3×#463C2semidirect
4. C3×#2540C2semidirect
5. C3×#2000C2semidirect
6. C3×#1999C2semidirect
7. C3×#1988C2semidirect
8. C3×#1955C2semidirect
9. C3×#1875C2semidirect
10. C3×#1873C2semidirect
11. C3×#1762C2semidirect
12. #25578C3direct (N × H)
46. C2×#555C2×C3semidirect
47. C2×#489C2×C3semidirect
48. C2×#463C2×C3semidirect
49. #2540C2×C3semidirect
50. #2000C2×C3semidirect
51. #1999C2×C3semidirect
52. #1988C2×C3semidirect
53. #1955C2×C3semidirect
54. #1875C2×C3semidirect
55. #1873C2×C3semidirect
56. #1762C2×C3semidirect
74. C3×C4⋊Q8D4semidirect
88. C3×C4×D4D4semidirect
241. C4⋊Q8C3×D4semidirect
255. C4×D4C3×D4semidirect
552. C3#25578direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5552
C2×C3×#4892
C2×C3×#4632
C3×#25402
C3×#20002
C3×#19992
C3×#19882
C3×#19552
C3×#18752
C3×#18732
C3×#17622
#255783