Finite Group Databasebeta

← C3×#24189 #10642796

Subgroups — C3×#24189 #10642796

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C2×C3×C4×SD16C211yesmaximal, characteristic
4C3×#2582C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal
6C3×#2015C212yesmaximal
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1865C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic
12#24189C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2211yescharacteristic
15C3×#556—24no—
16C3×#556—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#126—22no—
19C2×C3×#124C2212yes—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117—21no—
26C3×#462—24no—
27C3×C4×SD16—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2212yes—
36C2×C3×C4×C8C2211yescharacteristic
37C3×#398—22no—
38C3×#391—21no—
39C3×#391C2212yes—
40C3×#389—21no—
41C3×#389C2212yes—
42C3×#384C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#556C2×C311yescharacteristic
47C2×#462C2×C311yescharacteristic, complemented
48C2×C4×SD16C2×C311yescharacteristic
49#2582C2×C311yescharacteristic, complemented
50#2042C2×C312yes—
51#2015C2×C312yes—
52#1984C2×C311yescharacteristic, complemented
53#1956C2×C311yescharacteristic, complemented
54#1878C2×C312yescomplemented
55#1865C2×C312yescomplemented
56#1802C2×C311yescharacteristic
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C2×C3×Q16—24no—
63C2×C3×Q16—24no—
64C2×C3×Q16—42no—
65C2×C3×SD16—22no—
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8D414yes—
73C3×#126—216no—
74C3×#124—24no—
75C3×#124—24no—
76C3×C22⋊Q8—28no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—22no—
84C3×C4×D4D414yes—
85C3×#117—28no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
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×C4⋊C8—22no—
107C3×#103—24no—
108C3×#103—24no—
109C3×#103—22no—
110C3×#103—42no—
111C3×#102—22no—
112C3×#102—42no—
113C3×C4×C8—22no—
114C3×#95—22no—
115C3×#95—22no—
116C3×#95—22no—
117C22×Q16C22×C312yes—
118C22×SD16C22×C311yescharacteristic
119#556—24no—
120#556—24no—
121C2×C4⋊Q8C22×C311yescharacteristic
122C2×#126—22no—
123C2×#124C22×C312yes—
124C2×C22⋊Q8—21no—
125C2×C4×Q8C22×C311yescharacteristic
126C2×C4×Q8C22×C312yes—
127C2×C4×Q8C22×C311yescharacteristic
128C2×C4×D4C22×C311yescharacteristic
129C2×#117—21no—
130#462—24no—
131C4×SD16—24no—
132C2×#106C22×C311yescharacteristic
133C2×C4⋊C8C22×C311yescharacteristic
134C2×#103C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C311yescharacteristic
137C2×#103C22×C312yes—
138C2×#102C22×C311yescharacteristic
139C2×#102C22×C312yes—
140C2×C4×C8C22×C311yescharacteristic
141#398—22no—
142#391—21no—
143#391C22×C312yes—
144#389—21no—
145#389C22×C312yes—
146#384C22×C311yescharacteristic
147#340C22×C312yes—
148#336C22×C312yes—
149#336C22×C312yes—
150C24×C3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8C2×D414yes—
165C2×C3×Q8#4112yes—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4C2×D412yes—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C3×Q16—48no—
186C3×SD16—28no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8#4112yes—
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—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4—81no—
209C3×C4⋊C4—82no—
210C3×C4⋊C4C2×D412yes—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—28no—
213C3×C22⋊C4—24no—
214C3×C42—28no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42—42no—
218C3×C42C2×D412yes—
219C3×C42C2×D412yes—
220C22×Q8C3×D411yescharacteristic
221C22×Q8C23×C312yes—
222C22×Q8C23×C311yescharacteristic
223C22×D4C23×C311yescharacteristic
224C23×C4—21no—
225C2×Q16—24no—
226C2×Q16—24no—
227C2×Q16—42no—
228C2×SD16—22no—
229C2×SD16—22no—
230C2×SD16—22no—
231C22×C8C23×C312yes—
232C22×C8C23×C311yescharacteristic
233C22×C8C23×C311yescharacteristic
234C4⋊Q8—22no—
235C4⋊Q8C3×D414yes—
236#126—216no—
237#124—24no—
238#124—24no—
239C22⋊Q8—28no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×D4—22no—
247C4×D4C3×D414yes—
248#117—28no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C3×D411yescharacteristic
257C2×C4⋊C4C3×D412yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C312yes—
260C2×C4⋊C4C23×C311yescharacteristic
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42C23×C312yes—
267C2×C42C23×C311yescharacteristic
268#106—22no—
269C4⋊C8—22no—
270#103—24no—
271#103—24no—
272#103—22no—
273#103—42no—
274#102—22no—
275#102—42no—
276C4×C8—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3C22×D411yescharacteristic
288C3×Q8—22no—
289C3×Q8—22no—
290C3×Q8—24no—
291C3×Q8—24no—
292C3×Q8—42no—
293C3×Q8—42no—
294C3×Q8—41no—
295C3×Q8—41no—
296C3×Q8—44no—
297C3×D4—22no—
298C3×D4—22no—
299C3×D4—41no—
300C3×D4—41no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—48no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4#12012yes—
326C2×C3×C4#12012yes—
327C2×C3×C4#13512yes—
328C2×C3×C4#13712yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334C2×C3×C42−1+411yescharacteristic
335C3×C8—22no—
336C3×C8—41no—
337C3×C8—42no—
338C24—21no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8C2×C3×D414yes—
353C2×Q8C3×#4112yes—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4C2×C3×D412yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4C2×C3×D412yes—
369C22×C4C2×C3×D412yes—
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C24×C311yescharacteristic
373Q16—48no—
374SD16—28no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8C3×#4112yes—
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—81no—
397C4⋊C4—82no—
398C4⋊C4C2×C3×D412yes—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—24no—
402C42—28no—
403C42—22no—
404C42—22no—
405C42—42no—
406C42C2×C3×D412yes—
407C42C2×C3×D412yes—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3C2×#12011yescharacteristic
416C22×C3#54811yescharacteristic
417C22×C3#55011yescharacteristic
418C22×C3C2×#13511yescharacteristic
419C22×C3C2×#13711yescharacteristic
420C22×C3#57711yescharacteristic
421C22×C3#57811yescharacteristic
422C3×C4—22no—
423C3×C4—44no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—82no—
429C3×C4—81no—
430C3×C4#45214yes—
431C3×C4#52511yescharacteristic
432C3×C4#53711yescharacteristic
433C3×C4#54611yescharacteristic
434C3×C4#54711yescharacteristic
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C22×C3×D411yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—24no—
446Q8—24no—
447Q8—44no—
448Q8—42no—
449Q8—42no—
450Q8—41no—
451Q8—41no—
452D4—22no—
453D4—22no—
454D4—41no—
455D4—41no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4C3×#12012yes—
481C2×C4C3×#12012yes—
482C2×C4C3×#13512yes—
483C2×C4C3×#13712yes—
484C2×C4C22×C3×D411yescharacteristic
485C2×C4C22×C3×D411yescharacteristic
486C2×C4C2×C3×#4111yescharacteristic
487C2×C4C2×C3×#4111yescharacteristic
488C2×C4C3×H(5,𝔽2)11yescharacteristic
489C2×C4C3×2−1+411yescharacteristic
490C8—22no—
491C8—41no—
492C8—42no—
493C2×C3—82no—
494C2×C3#240711yescharacteristic
495C2×C3C2×#45211yescharacteristic
496C2×C3#301311yescharacteristic
497C2×C3#336411yescharacteristic
498C2×C3#336911yescharacteristic
499C2×C3#339411yescharacteristic
500C2×C3#339811yescharacteristic
501C22—42no—
502C22—42no—
503C22—42no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22C2×C3×#12011yescharacteristic
509C22C3×#54811yescharacteristic
510C22C3×#55011yescharacteristic
511C22C2×C3×#13511yescharacteristic
512C22C2×C3×#13711yescharacteristic
513C22C3×#57711yescharacteristic
514C22C3×#57811yescharacteristic
515C4—22no—
516C4—44no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—82no—
522C4—81no—
523C4C3×#45214yes—
524C4C3×#52511yescharacteristic
525C4C3×#53711yescharacteristic
526C4C3×#54611yescharacteristic
527C4C3×#54711yescharacteristic
528C3#2418911yescharacteristic, direct product factor, Sylow
529C2—82no—
530C2C3×#240711yescharacteristic
531C2C2×C3×#45211yescharacteristic
532C2C3×#301311yescharacteristic
533C2C3×#336411yescharacteristic
534C2C3×#336911yescharacteristic
535C2C3×#339411yescharacteristic
536C2C3×#339811yescharacteristic
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×#462C2semidirect
4. C3×#2582C2semidirect
7. C3×#1984C2semidirect
8. C3×#1956C2semidirect
9. C3×#1878C2semidirect
10. C3×#1865C2semidirect
12. #24189C3direct (N × H)
47. C2×#462C2×C3semidirect
49. #2582C2×C3semidirect
52. #1984C2×C3semidirect
53. #1956C2×C3semidirect
54. #1878C2×C3semidirect
55. #1865C2×C3semidirect
528. C3#24189direct (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×#4622
C2×C3×C4×SD162
C3×#25822
C3×#20422
C3×#20152
C3×#19842
C3×#19562
C3×#18782
C3×#18652
C3×#18022
#241893