Finite Group Databasebeta

← C3×#23941 #10642548

Subgroups — C3×#23941 #10642548

Order 768.

1334 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×#540C211yesmaximal, characteristic, complemented
2C2×C3×#449C212yesmaximal, complemented
3C3×#2311C211yesmaximal, characteristic, complemented
4C3×#2026C211yesmaximal, characteristic, complemented
5C3×#2025C211yesmaximal, characteristic, complemented
6C3×#2014C211yesmaximal, characteristic, complemented
7C3×#2011C211yesmaximal, characteristic, complemented
8C3×#1879C212yesmaximal, complemented
9C3×#1877C212yesmaximal, complemented
10C3×#1874C212yesmaximal, complemented
11C3×#1818C211yesmaximal, characteristic, complemented
12#23941C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C3×#540—28no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×#123—22no—
20C2×C3×#121—22no—
21C2×C3×#121C2212yes—
22C2×C3×C4×D4C2212yes—
23C2×C3×#117C2211yescharacteristic
24C22×C3×C4⋊C4C2211yescharacteristic
25C3×#449—28no—
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2212yes—
31C3×#391C2211yescharacteristic
32C3×#391C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#386—22no—
36C3×#380—22no—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#336C2211yescharacteristic
40C3×#336C2211yescharacteristic
41C3×#336C2212yes—
42C2×#540C2×C311yescharacteristic, complemented
43C2×#449C2×C312yescomplemented
44#2311C2×C311yescharacteristic, complemented
45#2026C2×C311yescharacteristic, complemented
46#2025C2×C311yescharacteristic, complemented
47#2014C2×C311yescharacteristic, complemented
48#2011C2×C311yescharacteristic, complemented
49#1879C2×C312yescomplemented
50#1877C2×C312yescomplemented
51#1874C2×C312yescomplemented
52#1818C2×C311yescharacteristic, complemented
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—24no—
56C2×C3×#41—42no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4—22no—
60C22×C3×D4—22no—
61C22×C3×D4C2312yes—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4—22no—
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C2×C3×SD16—24no—
68C2×C3×SD16—24no—
69C2×C3×SD16—42no—
70C2×C3×D8—24no—
71C2×C3×D8—24no—
72C2×C3×D8—42no—
73C22×C3×C8C2312yes—
74C22×C3×C8C2312yes—
75C3×#127—24no—
76C3×#125—24no—
77C3×#123—216no—
78C3×#121—216no—
79C3×#121—24no—
80C3×#121D418yes—
81C3×C4×D4—24no—
82C3×C4×D4—24no—
83C3×#117—24no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—42no—
87C2×C3×C4⋊C4D412yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4D412yes—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C3×#103—24no—
99C3×#102—24no—
100C3×#102—42no—
101C3×#102—42no—
102C3×C22⋊C8—24no—
103C3×#95—22no—
104C3×#95—22no—
105C22×#41C22×C311yescharacteristic
106C22×SD16C22×C312yes—
107C22×D8C22×C312yes—
108#540—28no—
109C2×#127C22×C311yescharacteristic
110C2×#125C22×C311yescharacteristic
111C2×#123—22no—
112C2×#121—22no—
113C2×#121C22×C312yes—
114C2×C4×D4C22×C312yes—
115C2×#117C22×C311yescharacteristic
116C22×C4⋊C4C22×C311yescharacteristic
117#449—28no—
118C2×#103C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121C2×#102C22×C312yes—
122C2×C22⋊C8C22×C312yes—
123#391C22×C311yescharacteristic
124#391C22×C311yescharacteristic
125#390C22×C311yescharacteristic
126#390C22×C311yescharacteristic
127#386—22no—
128#380—22no—
129#340C22×C312yes—
130#340C22×C312yes—
131#336C22×C312yes—
132#336C22×C311yescharacteristic
133#336C22×C311yescharacteristic
134C24×C3—21no—
135C24×C3—22no—
136C24×C3C2×D411yescharacteristic
137C3×#41—48no—
138C3×#41—48no—
139C2×C3×Q8—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8#4112yes—
143C2×C3×D4—24no—
144C2×C3×D4—28no—
145C2×C3×D4—28no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4—82no—
157C2×C3×D4—82no—
158C2×C3×D4—82no—
159C2×C3×D4—82no—
160C2×C3×D4C2×D414yes—
161C2×C3×D4#4112yes—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—28no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—82no—
175C22×C3×C4—82no—
176C22×C3×C4—82no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2411yescharacteristic
182C3×SD16—48no—
183C3×D8—48no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4C2×D414yes—
197C3×C22⋊C4—28no—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—28no—
200C3×C22⋊C4—82no—
201C3×C22⋊C4—82no—
202C3×C42—22no—
203C3×C42—22no—
204C2×#41—22no—
205C2×#41—22no—
206C2×#41—24no—
207C2×#41—42no—
208C22×Q8C23×C311yescharacteristic
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4C23×C312yes—
213C22×D4C23×C311yescharacteristic
214C23×C4—21no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4C23×C311yescharacteristic
218C2×SD16—24no—
219C2×SD16—24no—
220C2×SD16—42no—
221C2×D8—24no—
222C2×D8—24no—
223C2×D8—42no—
224C22×C8C23×C312yes—
225C22×C8C23×C312yes—
226#127—24no—
227#125—24no—
228#123—216no—
229#121—216no—
230#121—24no—
231#121C3×D418yes—
232C4×D4—24no—
233C4×D4—24no—
234#117—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C3×D412yes—
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C311yescharacteristic
241C2×C4⋊C4C23×C312yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4C3×D412yes—
247C2×C42C23×C311yescharacteristic
248C2×C42C23×C311yescharacteristic
249#103—24no—
250#102—24no—
251#102—42no—
252#102—42no—
253C22⋊C8—24no—
254#95—22no—
255#95—22no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3—41no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3#12012yes—
273C23×C3C22×D411yescharacteristic
274C3×Q8—24no—
275C3×Q8—42no—
276C3×D4—24no—
277C3×D4—24no—
278C3×D4—24no—
279C3×D4—48no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—88no—
284C3×D4—88no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—48no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—88no—
307C2×C3×C4—84no—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D412yes—
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4H(5,𝔽2)12yes—
313C3×C8—42no—
314C3×C8—42no—
315C24—21no—
316C24—22no—
317C24C2×C3×D411yescharacteristic
318#41—48no—
319#41—48no—
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8C3×#4112yes—
324C2×D4—24no—
325C2×D4—28no—
326C2×D4—28no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4C2×C3×D414yes—
342C2×D4C3×#4112yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D414yes—
362C22×C4C24×C311yescharacteristic
363SD16—48no—
364D8—48no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C4⋊C4—28no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4C2×C3×D414yes—
378C22⋊C4—28no—
379C22⋊C4—24no—
380C22⋊C4—28no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C42—22no—
384C42—22no—
385C22×C3—22no—
386C22×C3—24no—
387C22×C3—42no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—44no—
391C22×C3—42no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—81no—
398C22×C3—82no—
399C22×C3—81no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3C2×#12011yescharacteristic
403C22×C3#52511yescharacteristic
404C22×C3#53511yescharacteristic
405C22×C3#57612yes—
406C22×C3#57712yes—
407C3×C4—22no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—44no—
412C3×C4—81no—
413C3×C4—81no—
414C3×C4#54612yes—
415C3×C4#54612yes—
416C23—22no—
417C23—22no—
418C23—21no—
419C23—21no—
420C23—24no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—42no—
430C23—41no—
431C23—42no—
432C23C3×#12012yes—
433C23C22×C3×D411yescharacteristic
434Q8—24no—
435Q8—42no—
436D4—24no—
437D4—24no—
438D4—24no—
439D4—48no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—88no—
444D4—88no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—88no—
467C2×C4—84no—
468C2×C4C3×#12012yes—
469C2×C4C22×C3×D412yes—
470C2×C4C2×C3×#4111yescharacteristic
471C2×C4C2×C3×#4111yescharacteristic
472C2×C4C3×H(5,𝔽2)12yes—
473C8—42no—
474C8—42no—
475C2×C3—42no—
476C2×C3—82no—
477C2×C3—84no—
478C2×C3#240411yescharacteristic
479C2×C3#301212yes—
480C2×C3#311912yes—
481C2×C3#322011yescharacteristic
482C2×C3#322111yescharacteristic
483C22—22no—
484C22—24no—
485C22—41no—
486C22—41no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—44no—
493C22—81no—
494C22—82no—
495C22—82no—
496C22—81no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22C2×C3×#12011yescharacteristic
501C22C3×#52511yescharacteristic
502C22C3×#53511yescharacteristic
503C22C3×#57612yes—
504C22C3×#57712yes—
505C4—22no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—44no—
510C4—81no—
511C4—81no—
512C4C3×#54612yes—
513C4C3×#54612yes—
514C3#2394111yescharacteristic, direct product factor, Sylow
515C2—42no—
516C2—82no—
517C2—84no—
518C2C3×#240411yescharacteristic
519C2C3×#301212yes—
520C2C3×#311912yes—
521C2C3×#322011yescharacteristic
522C2C3×#322111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#540C2semidirect
2. C2×C3×#449C2semidirect
3. C3×#2311C2semidirect
4. C3×#2026C2semidirect
5. C3×#2025C2semidirect
6. C3×#2014C2semidirect
7. C3×#2011C2semidirect
8. C3×#1879C2semidirect
9. C3×#1877C2semidirect
10. C3×#1874C2semidirect
11. C3×#1818C2semidirect
12. #23941C3direct (N × H)
42. C2×#540C2×C3semidirect
43. C2×#449C2×C3semidirect
44. #2311C2×C3semidirect
45. #2026C2×C3semidirect
46. #2025C2×C3semidirect
47. #2014C2×C3semidirect
48. #2011C2×C3semidirect
49. #1879C2×C3semidirect
50. #1877C2×C3semidirect
51. #1874C2×C3semidirect
52. #1818C2×C3semidirect
514. C3#23941direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×C3×#4492
C3×#23112
C3×#20262
C3×#20252
C3×#20142
C3×#20112
C3×#18792
C3×#18772
C3×#18742
C3×#18182
#239413