Finite Group Databasebeta

← C3×#24329 #10642936

Subgroups — C3×#24329 #10642936

Order 768.

1102 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
1C2×C3×Q82C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic
3C2×C3×C4×SD16C211yesmaximal, characteristic
4C3×#2538C211yesmaximal, characteristic
5C3×#2058C212yesmaximal, complemented
6C3×#2023C212yesmaximal
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1984C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal
10C3×#1865C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24329C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C3×Q82—24no—
15C3×Q82—24no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8—22no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C3×#476—24no—
27C3×C4×SD16—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×#106C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×C4×C8C2211yescharacteristic
36C3×#396—22no—
37C3×#390C2212yes—
38C3×#389C2212yes—
39C3×#384C2211yescharacteristic
40C3×#380—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×Q82C2×C311yescharacteristic, complemented
47C2×#476C2×C311yescharacteristic
48C2×C4×SD16C2×C311yescharacteristic
49#2538C2×C311yescharacteristic
50#2058C2×C312yescomplemented
51#2023C2×C312yes—
52#1990C2×C311yescharacteristic, complemented
53#1984C2×C311yescharacteristic, complemented
54#1880C2×C312yes—
55#1865C2×C312yescomplemented
56#1803C2×C311yescharacteristic, complemented
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C2×C3×SD16Q814yescomplemented
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2312yes—
68C22×C3×C8C2311yescharacteristic
69C3×C4⋊Q8—216no—
70C3×C4⋊Q8—28no—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—24no—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8D414yes—
75C3×C22⋊Q8—44no—
76C3×C4×Q8—28no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8D414yes—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D412yes—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2311yescharacteristic
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×C42—22no—
102C2×C3×C42—21no—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C3×#106—22no—
106C3×#106—24no—
107C3×C4⋊C8—22no—
108C3×#103—22no—
109C3×#103—24no—
110C3×#102—22no—
111C3×#102—24no—
112C3×C4×C8—22no—
113C3×#95—22no—
114C22×SD16C22×C311yescharacteristic
115Q82—24no—
116Q82—24no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8C22×C312yes—
120C2×C4⋊Q8C22×C311yescharacteristic
121C2×C22⋊Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C22×C312yes—
124C2×C4×Q8C22×C311yescharacteristic
125C2×C4×Q8C22×C311yescharacteristic
126C2×C4×D4C22×C311yescharacteristic
127#476—24no—
128C4×SD16—24no—
129C2×#106C22×C311yescharacteristic
130C2×#106C22×C312yes—
131C2×C4⋊C8C22×C311yescharacteristic
132C2×#103C22×C312yes—
133C2×#103C22×C311yescharacteristic
134C2×#102C22×C312yes—
135C2×#102C22×C311yescharacteristic
136C2×C4×C8C22×C311yescharacteristic
137#396—22no—
138#390C22×C312yes—
139#389C22×C312yes—
140#384C22×C311yescharacteristic
141#380—21no—
142#340C22×C312yes—
143#340C22×C312yes—
144#340C22×C312yes—
145#336C22×C312yes—
146#336C22×C312yes—
147C24×C3—21no—
148C2×C3×Q8—28no—
149C2×C3×Q8—22no—
150C2×C3×Q8—24no—
151C2×C3×Q8—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—81no—
156C2×C3×Q8—81no—
157C2×C3×Q8—82no—
158C2×C3×Q8—82no—
159C2×C3×Q8SD1614yescomplemented
160C2×C3×Q8C2×D412yes—
161C2×C3×Q8C2×Q812yes—
162C2×C3×Q8#4112yes—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4C2×Q812yes—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C3×SD16—28no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8C2×Q812yes—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—44no—
206C3×C4⋊C4—81no—
207C3×C4⋊C4C2×D414yes—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—44no—
211C3×C22⋊C4—82no—
212C3×C42—28no—
213C3×C42—24no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218C22×Q8—22no—
219C22×Q8—21no—
220C22×Q8C3×D411yescharacteristic
221C22×Q8C23×C311yescharacteristic
222C22×D4C23×C311yescharacteristic
223C23×C4—21no—
224C2×SD16—22no—
225C2×SD16—22no—
226C2×SD16C3×Q814yescomplemented
227C22×C8C23×C311yescharacteristic
228C22×C8C23×C312yes—
229C22×C8C23×C311yescharacteristic
230C4⋊Q8—216no—
231C4⋊Q8—28no—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8C3×D414yes—
236C22⋊Q8—44no—
237C4×Q8—28no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8C3×D414yes—
244C4×D4—22no—
245C4×D4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4C3×D411yescharacteristic
254C2×C4⋊C4C3×D412yes—
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C4⋊C4C23×C312yes—
257C2×C4⋊C4C23×C312yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42C23×C312yes—
265C2×C42C23×C311yescharacteristic
266#106—22no—
267#106—24no—
268C4⋊C8—22no—
269#103—22no—
270#103—24no—
271#102—22no—
272#102—24no—
273C4×C8—22no—
274#95—22no—
275C23×C3—22no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3C22×D411yescharacteristic
283C3×Q8—22no—
284C3×Q8—22no—
285C3×Q8—22no—
286C3×Q8—28no—
287C3×Q8—84no—
288C3×Q8—88no—
289C3×Q8C22⋊Q814yescomplemented
290C3×D4—22no—
291C3×D4—22no—
292C3×D4—41no—
293C3×D4—41no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—84no—
317C2×C3×C4#12012yes—
318C2×C3×C4#12012yes—
319C2×C3×C4C2×SD1614yes—
320C2×C3×C4C2×SD1612yes—
321C2×C3×C4#13712yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×Q811yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C42−1+411yescharacteristic
327C2×C3×C42−1+411yescharacteristic
328C3×C8—22no—
329C3×C8—41no—
330C3×C8—42no—
331C24—21no—
332C2×Q8—28no—
333C2×Q8—22no—
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—81no—
340C2×Q8—81no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8C3×SD1614yescomplemented
344C2×Q8C2×C3×D412yes—
345C2×Q8C2×C3×Q812yes—
346C2×Q8C3×#4112yes—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4C2×C3×Q812yes—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4C2×C3×D412yes—
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D412yes—
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C24×C311yescharacteristic
366SD16—28no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8C2×C3×Q812yes—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—81no—
391C4⋊C4C2×C3×D414yes—
392C4⋊C4C2×C3×D412yes—
393C22⋊C4—42no—
394C22⋊C4—44no—
395C22⋊C4—82no—
396C42—28no—
397C42—24no—
398C42—22no—
399C42—22no—
400C42C2×C3×D412yes—
401C42C2×C3×D412yes—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3C2×#12011yescharacteristic
410C22×C3#54811yescharacteristic
411C22×C3D4×Q811yescharacteristic
412C22×C3C22×SD1611yescharacteristic
413C22×C3C2×#13711yescharacteristic
414C22×C3#57611yescharacteristic
415C22×C3#57711yescharacteristic
416C3×C4—24no—
417C3×C4—24no—
418C3×C4—22no—
419C3×C4—42no—
420C3×C4—44no—
421C3×C4—42no—
422C3×C4—82no—
423C3×C4—81no—
424C3×C4#44814yes—
425C3×C4C2×C22⋊Q811yescharacteristic
426C3×C4#53611yescharacteristic
427C3×C4#54811yescharacteristic
428C3×C4D4×Q811yescharacteristic
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23C22×C3×D411yescharacteristic
437Q8—28no—
438Q8—22no—
439Q8—22no—
440Q8—22no—
441Q8—84no—
442Q8—88no—
443Q8C3×C22⋊Q814yescomplemented
444D4—22no—
445D4—22no—
446D4—41no—
447D4—41no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—84no—
471C2×C4C3×#12012yes—
472C2×C4C3×#12012yes—
473C2×C4C2×C3×SD1614yes—
474C2×C4C2×C3×SD1612yes—
475C2×C4C3×#13712yes—
476C2×C4C22×C3×D411yescharacteristic
477C2×C4C22×C3×D411yescharacteristic
478C2×C4C22×C3×Q811yescharacteristic
479C2×C4C2×C3×#4111yescharacteristic
480C2×C4C3×2−1+411yescharacteristic
481C2×C4C3×2−1+411yescharacteristic
482C8—22no—
483C8—41no—
484C8—42no—
485C2×C3—82no—
486C2×C3#240911yescharacteristic
487C2×C3C2×#44811yescharacteristic
488C2×C3#301211yescharacteristic
489C2×C3#336211yescharacteristic
490C2×C3#336911yescharacteristic
491C2×C3Q8×SD1611yescharacteristic
492C2×C3#338511yescharacteristic
493C22—42no—
494C22—42no—
495C22—42no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22C2×C3×#12011yescharacteristic
501C22C3×#54811yescharacteristic
502C22C3×D4×Q811yescharacteristic
503C22C22×C3×SD1611yescharacteristic
504C22C2×C3×#13711yescharacteristic
505C22C3×#57611yescharacteristic
506C22C3×#57711yescharacteristic
507C4—24no—
508C4—24no—
509C4—22no—
510C4—42no—
511C4—44no—
512C4—42no—
513C4—81no—
514C4—82no—
515C4C3×#44814yes—
516C4C2×C3×C22⋊Q811yescharacteristic
517C4C3×#53611yescharacteristic
518C4C3×#54811yescharacteristic
519C4C3×D4×Q811yescharacteristic
520C3#2432911yescharacteristic, direct product factor, Sylow
521C2—82no—
522C2C3×#240911yescharacteristic
523C2C2×C3×#44811yescharacteristic
524C2C3×#301211yescharacteristic
525C2C3×#336211yescharacteristic
526C2C3×#336911yescharacteristic
527C2C3×Q8×SD1611yescharacteristic
528C2C3×#338511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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×Q82C2semidirect
5. C3×#2058C2semidirect
7. C3×#1990C2semidirect
8. C3×#1984C2semidirect
10. C3×#1865C2semidirect
11. C3×#1803C2semidirect
12. #24329C3direct (N × H)
46. C2×Q82C2×C3semidirect
50. #2058C2×C3semidirect
52. #1990C2×C3semidirect
53. #1984C2×C3semidirect
55. #1865C2×C3semidirect
56. #1803C2×C3semidirect
65. C2×C3×SD16Q8semidirect
159. C2×C3×Q8SD16semidirect
226. C2×SD16C3×Q8semidirect
289. C3×Q8C22⋊Q8semidirect
343. C2×Q8C3×SD16semidirect
443. Q8C3×C22⋊Q8semidirect
520. C3#24329direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×C3×#4762
C2×C3×C4×SD162
C3×#25382
C3×#20582
C3×#20232
C3×#19902
C3×#19842
C3×#18802
C3×#18652
C3×#18032
#243293