Finite Group Databasebeta

← C3×#24331 #10642938

Subgroups — C3×#24331 #10642938

Order 768.

1078 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3C2×C3×C4×SD16C211yesmaximal, characteristic
4C3×#2582C211yesmaximal, characteristic, complemented
5C3×#2059C212yesmaximal, complemented
6C3×#2026C212yesmaximal, complemented
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1984C211yesmaximal, characteristic, complemented
9C3×#1877C212yesmaximal
10C3×#1862C212yesmaximal
11C3×#1802C211yesmaximal, characteristic
12#24331C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C3×#554—24no—
15C3×#554—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C3×#477—24no—
26C3×C4×SD16—24no—
27C2×C3×#107C2212yes—
28C2×C3×#106C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×C4×C8C2211yescharacteristic
35C3×#398—22no—
36C3×#391—21no—
37C3×#391C2212yes—
38C3×#389—21no—
39C3×#389C2212yes—
40C3×#384C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#554C2×C311yescharacteristic
47C2×#477C2×C311yescharacteristic, complemented
48C2×C4×SD16C2×C311yescharacteristic
49#2582C2×C311yescharacteristic, complemented
50#2059C2×C312yescomplemented
51#2026C2×C312yescomplemented
52#1990C2×C311yescharacteristic, complemented
53#1984C2×C311yescharacteristic, complemented
54#1877C2×C312yes—
55#1862C2×C312yes—
56#1802C2×C311yescharacteristic
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C2×C3×SD16Q814yescomplemented
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2312yes—
68C22×C3×C8C2311yescharacteristic
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yes—
71C3×#125—24no—
72C3×#125—24no—
73C3×C22⋊Q8—28no—
74C3×C22⋊Q8—216no—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×C4×D4—22no—
82C3×C4×D4D414yes—
83C2×C3×C4⋊C4—28no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D412yes—
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42—21no—
100C2×C3×C42C2312yes—
101C2×C3×C42C2311yescharacteristic
102C3×#107—24no—
103C3×#106—22no—
104C3×C4⋊C8—22no—
105C3×#103—22no—
106C3×#103—24no—
107C3×#102—22no—
108C3×#102—24no—
109C3×C4×C8—22no—
110C3×#95—22no—
111C3×#95—22no—
112C3×#95—22no—
113C22×SD16C22×C311yescharacteristic
114#554—24no—
115#554—24no—
116C2×C4⋊Q8C22×C311yescharacteristic
117C2×#125C22×C312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—22no—
120C2×C4×Q8C22×C311yescharacteristic
121C2×C4×Q8C22×C311yescharacteristic
122C2×C4×D4C22×C312yes—
123C2×C4×D4C22×C311yescharacteristic
124C22×C4⋊C4—21no—
125#477—24no—
126C4×SD16—24no—
127C2×#107C22×C312yes—
128C2×#106C22×C311yescharacteristic
129C2×C4⋊C8C22×C311yescharacteristic
130C2×#103C22×C312yes—
131C2×#103C22×C311yescharacteristic
132C2×#102C22×C312yes—
133C2×#102C22×C311yescharacteristic
134C2×C4×C8C22×C311yescharacteristic
135#398—22no—
136#391—21no—
137#391C22×C312yes—
138#389—21no—
139#389C22×C312yes—
140#384C22×C311yescharacteristic
141#340C22×C312yes—
142#340C22×C312yes—
143#340C22×C312yes—
144#336C22×C312yes—
145#336C22×C312yes—
146C24×C3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—81no—
157C2×C3×Q8—81no—
158C2×C3×Q8C2×Q812yes—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4C2×D412yes—
163C2×C3×D4C2×Q812yes—
164C2×C3×D4#4112yes—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—28no—
169C22×C3×C4—24no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—82no—
175C22×C3×C4—82no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C3×SD16—28no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8C2×Q812yes—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—81no—
205C3×C4⋊C4—82no—
206C3×C4⋊C4C2×D412yes—
207C3×C4⋊C4C2×D414yes—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—24no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—42no—
213C3×C42C2×D412yes—
214C3×C42C2×D412yes—
215C22×Q8—21no—
216C22×Q8C3×D411yescharacteristic
217C22×Q8C23×C311yescharacteristic
218C22×D4C23×C311yescharacteristic
219C23×C4—21no—
220C23×C4—22no—
221C2×SD16—22no—
222C2×SD16—22no—
223C2×SD16C3×Q814yescomplemented
224C22×C8C23×C311yescharacteristic
225C22×C8C23×C312yes—
226C22×C8C23×C311yescharacteristic
227C4⋊Q8—22no—
228C4⋊Q8C3×D414yes—
229#125—24no—
230#125—24no—
231C22⋊Q8—28no—
232C22⋊Q8—216no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—22no—
240C4×D4C3×D414yes—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4C3×D411yescharacteristic
250C2×C4⋊C4C3×D412yes—
251C2×C4⋊C4C23×C311yescharacteristic
252C2×C4⋊C4C23×C312yes—
253C2×C4⋊C4C23×C311yescharacteristic
254C2×C4⋊C4C23×C312yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C42—21no—
258C2×C42C23×C312yes—
259C2×C42C23×C311yescharacteristic
260#107—24no—
261#106—22no—
262C4⋊C8—22no—
263#103—22no—
264#103—24no—
265#102—22no—
266#102—24no—
267C4×C8—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3C22×D411yescharacteristic
279C3×Q8—22no—
280C3×Q8—22no—
281C3×Q8—41no—
282C3×Q8—41no—
283C3×Q8—44no—
284C3×Q8—84no—
285C3×D4—22no—
286C3×D4—22no—
287C3×D4—22no—
288C3×D4C22⋊Q814yes—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—48no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—88no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4#13512yes—
317C2×C3×C4#13712yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×Q811yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C42−1+411yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C3×C8—22no—
325C3×C8—41no—
326C3×C8—42no—
327C24—21no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—81no—
338C2×Q8—81no—
339C2×Q8C2×C3×Q812yes—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4C2×C3×D412yes—
344C2×D4C2×C3×Q812yes—
345C2×D4C3×#4112yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4C2×C3×D412yes—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C24×C311yescharacteristic
362SD16—28no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8C2×C3×Q812yes—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—81no—
386C4⋊C4—82no—
387C4⋊C4C2×C3×D412yes—
388C4⋊C4C2×C3×D414yes—
389C22⋊C4—28no—
390C22⋊C4—24no—
391C42—22no—
392C42—22no—
393C42—42no—
394C42C2×C3×D412yes—
395C42C2×C3×D412yes—
396C22×C3—24no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54811yescharacteristic
405C22×C3D4×Q811yescharacteristic
406C22×C3C2×#13511yescharacteristic
407C22×C3C2×#13711yescharacteristic
408C22×C3#57611yescharacteristic
409C22×C3#57711yescharacteristic
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—44no—
416C3×C4—82no—
417C3×C4—81no—
418C3×C4#45214yes—
419C3×C4C2×C22⋊Q811yescharacteristic
420C3×C4#53611yescharacteristic
421C3×C4#54811yescharacteristic
422C3×C4D4×Q811yescharacteristic
423C23—22no—
424C23—22no—
425C23—22no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23C22×C3×D411yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—44no—
434Q8—41no—
435Q8—41no—
436Q8—84no—
437D4—22no—
438D4—22no—
439D4—22no—
440D4C3×C22⋊Q814yes—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—88no—
466C2×C4C3×#12012yes—
467C2×C4C3×#12012yes—
468C2×C4C3×#13512yes—
469C2×C4C3×#13712yes—
470C2×C4C22×C3×D411yescharacteristic
471C2×C4C22×C3×D411yescharacteristic
472C2×C4C22×C3×Q811yescharacteristic
473C2×C4C2×C3×#4111yescharacteristic
474C2×C4C3×2−1+411yescharacteristic
475C2×C4C3×2−1+411yescharacteristic
476C8—22no—
477C8—41no—
478C8—42no—
479C2×C3—44no—
480C2×C3#240911yescharacteristic
481C2×C3C2×#45211yescharacteristic
482C2×C3#301211yescharacteristic
483C2×C3#336111yescharacteristic
484C2×C3#337211yescharacteristic
485C2×C3#338111yescharacteristic
486C2×C3#338511yescharacteristic
487C22—24no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22C2×C3×#12011yescharacteristic
495C22C3×#54811yescharacteristic
496C22C3×D4×Q811yescharacteristic
497C22C2×C3×#13511yescharacteristic
498C22C2×C3×#13711yescharacteristic
499C22C3×#57611yescharacteristic
500C22C3×#57711yescharacteristic
501C4—22no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—44no—
507C4—81no—
508C4—82no—
509C4C3×#45214yes—
510C4C2×C3×C22⋊Q811yescharacteristic
511C4C3×#53611yescharacteristic
512C4C3×#54811yescharacteristic
513C4C3×D4×Q811yescharacteristic
514C3#2433111yescharacteristic, direct product factor, Sylow
515C2—44no—
516C2C3×#240911yescharacteristic
517C2C2×C3×#45211yescharacteristic
518C2C3×#301211yescharacteristic
519C2C3×#336111yescharacteristic
520C2C3×#337211yescharacteristic
521C2C3×#338111yescharacteristic
522C2C3×#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
2. C2×C3×#477C2semidirect
4. C3×#2582C2semidirect
5. C3×#2059C2semidirect
6. C3×#2026C2semidirect
7. C3×#1990C2semidirect
8. C3×#1984C2semidirect
12. #24331C3direct (N × H)
47. C2×#477C2×C3semidirect
49. #2582C2×C3semidirect
50. #2059C2×C3semidirect
51. #2026C2×C3semidirect
52. #1990C2×C3semidirect
53. #1984C2×C3semidirect
65. C2×C3×SD16Q8semidirect
223. C2×SD16C3×Q8semidirect
514. C3#24331direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×C3×#4772
C2×C3×C4×SD162
C3×#25822
C3×#20592
C3×#20262
C3×#19902
C3×#19842
C3×#18772
C3×#18622
C3×#18022
#243313