Finite Group Databasebeta

← C3×#32960 #10651567

Subgroups — C3×#32960 #10651567

Order 768.

1288 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic
2C2×C3×#479C212yesmaximal, complemented
3C3×#2851C211yesmaximal, characteristic
4C3×#1865C212yesmaximal, complemented
5C3×#1863C212yesmaximal, complemented
6C3×#1853C211yesmaximal, characteristic
7C3×#1852C211yesmaximal, characteristic
8C3×#1805C212yesmaximal, complemented
9C3×#1797C211yesmaximal, characteristic
10C3×#1787C211yesmaximal, characteristic
11C3×#1787C211yesmaximal, characteristic
12#32960C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×C3×C4×Q8—28no—
16C2×C3×C4×Q8—21no—
17C2×C3×C4×Q8—21no—
18C2×C3×C4×Q8C2212yes—
19C2×C3×C4×Q8C2212yes—
20C22×C3×C4⋊C4—21no—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C42—21no—
23C22×C3×C42C2211yescharacteristic
24C3×#479—28no—
25C2×C3×#107C2212yes—
26C2×C3×#106C2212yes—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#387—21no—
37C3×#387—21no—
38C3×#385—21no—
39C3×#385—21no—
40C3×#382C2211yescharacteristic
41C3×#380C2211yescharacteristic
42C3×#380C2211yescharacteristic
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#336C2212yes—
46C22×C4×Q8C2×C311yescharacteristic
47C2×#479C2×C312yescomplemented
48#2851C2×C311yescharacteristic
49#1865C2×C312yescomplemented
50#1863C2×C312yescomplemented
51#1853C2×C311yescharacteristic
52#1852C2×C311yescharacteristic
53#1805C2×C312yescomplemented
54#1797C2×C311yescharacteristic
55#1787C2×C311yescharacteristic
56#1787C2×C311yescharacteristic
57C22×C3×Q8—24no—
58C22×C3×Q8—24no—
59C22×C3×Q8—21no—
60C22×C3×Q8—22no—
61C22×C3×Q8Q814yes—
62C22×C3×Q8C2312yes—
63C23×C3×C4—21no—
64C23×C3×C4—21no—
65C23×C3×C4D411yescharacteristic
66C23×C3×C4D411yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×#125—24no—
71C3×C4×Q8—28no—
72C3×C4×Q8—28no—
73C3×C4×Q8—24no—
74C3×C4×Q8—24no—
75C3×C4×Q8—24no—
76C3×C4×Q8D418yes—
77C2×C3×C4⋊C4—28no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—28no—
94C2×C3×C42—22no—
95C2×C3×C42—22no—
96C2×C3×C42—24no—
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C3×#107—24no—
102C3×#106—24no—
103C3×C4⋊C8—24no—
104C3×#103—24no—
105C3×#103—24no—
106C3×#103—24no—
107C3×#103—24no—
108C3×C22⋊C8—22no—
109C3×C22⋊C8—22no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C23×Q8C22×C311yescharacteristic
117C2×#125C22×C311yescharacteristic
118C2×C4×Q8—28no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8C22×C312yes—
122C2×C4×Q8C22×C312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C22×C311yescharacteristic
125C22×C42—21no—
126C22×C42C22×C311yescharacteristic
127#479—28no—
128C2×#107C22×C312yes—
129C2×#106C22×C312yes—
130C2×C4⋊C8C22×C312yes—
131C2×#103C22×C312yes—
132C2×#103C22×C312yes—
133C2×#103C22×C312yes—
134C2×#103C22×C312yes—
135C2×C22⋊C8C22×C311yescharacteristic
136C2×C22⋊C8C22×C311yescharacteristic
137#389C22×C311yescharacteristic
138#389C22×C311yescharacteristic
139#387—21no—
140#387—21no—
141#385—21no—
142#385—21no—
143#382C22×C311yescharacteristic
144#380C22×C311yescharacteristic
145#380C22×C311yescharacteristic
146#340C22×C312yes—
147#340C22×C312yes—
148#336C22×C312yes—
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—24no—
151C2×C3×Q8—216no—
152C2×C3×Q8—24no—
153C2×C3×Q8—216no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8C2×D414yes—
159C2×C3×Q8C2×Q814yes—
160C2×C3×Q8#4114yes—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—41no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×Q812yes—
187C22×C3×C4#4112yes—
188C22×C3×C4C2411yescharacteristic
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C3×C4⋊C4—28no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4C2×D414yes—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—42no—
210C3×C42—28no—
211C3×C42—28no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42C2×D414yes—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8C3×Q814yes—
221C22×Q8C23×C312yes—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C3×D411yescharacteristic
225C23×C4C3×D411yescharacteristic
226C23×C4C23×C311yescharacteristic
227C22×C8C23×C312yes—
228C22×C8C23×C312yes—
229#125—24no—
230C4×Q8—28no—
231C4×Q8—28no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8C3×D418yes—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C23×C311yescharacteristic
246C2×C4⋊C4C23×C311yescharacteristic
247C2×C4⋊C4C23×C311yescharacteristic
248C2×C4⋊C4C23×C311yescharacteristic
249C2×C4⋊C4C23×C312yes—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C42—28no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42C3×D411yescharacteristic
257C2×C42C3×D411yescharacteristic
258C2×C42C23×C311yescharacteristic
259C2×C42C23×C311yescharacteristic
260#107—24no—
261#106—24no—
262C4⋊C8—24no—
263#103—24no—
264#103—24no—
265#103—24no—
266#103—24no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3C22⋊Q812yes—
280C23×C3#13512yes—
281C23×C3#13712yes—
282C23×C3C22×D411yescharacteristic
283C3×Q8—216no—
284C3×Q8—24no—
285C3×Q8—24no—
286C3×Q8—24no—
287C3×Q8C22⋊Q818yes—
288C2×C3×C4—28no—
289C2×C3×C4—216no—
290C2×C3×C4—21no—
291C2×C3×C4—24no—
292C2×C3×C4—21no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—21no—
304C2×C3×C4—24no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—44no—
308C2×C3×C4—48no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4#12012yes—
324C2×C3×C4#12012yes—
325C2×C3×C4C22⋊Q812yes—
326C2×C3×C4#13712yes—
327C2×C3×C4C22×D412yes—
328C2×C3×C4C22×Q811yescharacteristic
329C2×C3×C4C2×#4112yes—
330C2×C3×C4C2×#4111yescharacteristic
331C3×C8—42no—
332C3×C8—42no—
333C24C2×C3×D411yescharacteristic
334C2×Q8—24no—
335C2×Q8—216no—
336C2×Q8—24no—
337C2×Q8—216no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8C2×C3×D414yes—
343C2×Q8C2×C3×Q814yes—
344C2×Q8C3×#4114yes—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×Q812yes—
371C22×C4C3×#4112yes—
372C22×C4C24×C311yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C4⋊C4—28no—
380C4⋊C4—24no—
381C4⋊C4—28no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4C2×C3×D414yes—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C42—28no—
395C42—28no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42C2×C3×D414yes—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3#47914yescomplemented
407C22×C3C2×#12011yescharacteristic
408C22×C3C2×C22⋊Q811yescharacteristic
409C22×C3#52511yescharacteristic
410C22×C3C2×#13511yescharacteristic
411C22×C3C2×#13711yescharacteristic
412C22×C3C2×#13711yescharacteristic
413C22×C3#57511yescharacteristic
414C3×C4—28no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—41no—
420C3×C4—44no—
421C3×C4—81no—
422C3×C4—81no—
423C3×C4—81no—
424C3×C4—81no—
425C3×C4C2×C22⋊Q812yes—
426C3×C4#52512yes—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23C3×C22⋊Q812yes—
432C23C3×#13512yes—
433C23C3×#13712yes—
434C23C22×C3×D411yescharacteristic
435Q8—216no—
436Q8—24no—
437Q8—24no—
438Q8—24no—
439Q8C3×C22⋊Q818yes—
440C2×C4—216no—
441C2×C4—28no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—44no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4C3×#12012yes—
476C2×C4C3×#12012yes—
477C2×C4C3×C22⋊Q812yes—
478C2×C4C3×#13712yes—
479C2×C4C22×C3×D412yes—
480C2×C4C22×C3×Q811yescharacteristic
481C2×C4C2×C3×#4112yes—
482C2×C4C2×C3×#4111yescharacteristic
483C8—42no—
484C8—42no—
485C2×C3—24no—
486C2×C3#284811yescharacteristic
487C2×C3#300211yescharacteristic
488C2×C3#300411yescharacteristic
489C2×C3C2×#47911yescharacteristic
490C2×C3#307811yescharacteristic
491C2×C3#310811yescharacteristic
492C2×C3#311711yescharacteristic
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22C3×#47914yescomplemented
500C22C2×C3×#12011yescharacteristic
501C22C2×C3×C22⋊Q811yescharacteristic
502C22C3×#52511yescharacteristic
503C22C2×C3×#13511yescharacteristic
504C22C2×C3×#13711yescharacteristic
505C22C2×C3×#13711yescharacteristic
506C22C3×#57511yescharacteristic
507C4—22no—
508C4—22no—
509C4—22no—
510C4—28no—
511C4—22no—
512C4—41no—
513C4—44no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4C2×C3×C22⋊Q812yes—
519C4C3×#52512yes—
520C3#3296011yescharacteristic, direct product factor, Sylow
521C2—24no—
522C2C3×#284811yescharacteristic
523C2C3×#300211yescharacteristic
524C2C3×#300411yescharacteristic
525C2C2×C3×#47911yescharacteristic
526C2C3×#307811yescharacteristic
527C2C3×#310811yescharacteristic
528C2C3×#311711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#479C2semidirect
4. C3×#1865C2semidirect
5. C3×#1863C2semidirect
8. C3×#1805C2semidirect
12. #32960C3direct (N × H)
47. C2×#479C2×C3semidirect
49. #1865C2×C3semidirect
50. #1863C2×C3semidirect
53. #1805C2×C3semidirect
406. C22×C3#479semidirect
499. C22C3×#479semidirect
520. C3#32960direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×C3×#4792
C3×#28512
C3×#18652
C3×#18632
C3×#18532
C3×#18522
C3×#18052
C3×#17972
C3×#17872
#329603