Finite Group Databasebeta

← C3×#24361 #10642968

Subgroups — C3×#24361 #10642968

Order 768.

1408 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Q16C211yesmaximal, characteristic, complemented
2C2×C3×#464C212yesmaximal, complemented
3C2×C3×#448C211yesmaximal, characteristic
4C3×#2436C211yesmaximal, characteristic
5C3×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
6C3×#2056C212yesmaximal, complemented
7C3×#2015C212yesmaximal, complemented
8C3×#1893C211yesmaximal, characteristic
9C3×#1878C212yesmaximal, complemented
10C3×#1850C211yesmaximal, characteristic, complemented
11C3×#1787C211yesmaximal, characteristic, complemented
12#24361C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×Q16—24no—
16C22×C3×Q16—28no—
17C22×C3×Q16C2212yes—
18C22×C3×Q16C2212yes—
19C22×C3×SD16C2211yescharacteristic
20C22×C3×SD16C2211yescharacteristic
21C23×C3×C8C2211yescharacteristic
22C2×C3×#124C2212yes—
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×Q8C2212yes—
25C3×#464—28no—
26C3×#448—24no—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#393—22no—
35C3×#391—22no—
36C3×#391C2212yes—
37C3×#391C2212yes—
38C3×#386—21no—
39C3×#386—21no—
40C3×#380—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C22⋊C4—21no—
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×#340C2212yes—
45C3×#336C2212yes—
46C23×Q16C2×C311yescharacteristic, complemented
47C2×#464C2×C312yescomplemented
48C2×#448C2×C311yescharacteristic
49#2436C2×C311yescharacteristic
50Q8×C22⋊C4C2×C311yescharacteristic, complemented
51#2056C2×C312yescomplemented
52#2015C2×C312yescomplemented
53#1893C2×C311yescharacteristic
54#1878C2×C312yescomplemented
55#1850C2×C311yescharacteristic, complemented
56#1787C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—24no—
61C22×C3×Q8—24no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8—42no—
65C22×C3×Q8—42no—
66C22×C3×Q8D414yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2312yes—
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4—21no—
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—21no—
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C2×C3×Q16—232no—
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×Q16—24no—
79C2×C3×Q16—24no—
80C2×C3×Q16—24no—
81C2×C3×Q16D418yescomplemented
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—24no—
85C2×C3×SD16—42no—
86C22×C3×C8—22no—
87C22×C3×C8—24no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2312yes—
91C3×#124—24no—
92C3×#124—42no—
93C3×#121—22no—
94C3×#121—22no—
95C3×C4×Q8—24no—
96C3×C4×Q8—42no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C42—22no—
107C2×C3×C42C2312yes—
108C3×C4⋊C8—24no—
109C3×#103—22no—
110C3×#103—22no—
111C3×#103—24no—
112C3×#103—42no—
113C3×#102—24no—
114C3×C22⋊C8—22no—
115C3×#95—22no—
116C3×#95—22no—
117C3×#95—21no—
118C3×#95—22no—
119C3×#95—22no—
120C3×#95D412yes—
121C23×Q8C22×C311yescharacteristic
122C23×Q8C22×C311yescharacteristic
123C22×Q16—24no—
124C22×Q16—28no—
125C22×Q16C22×C312yes—
126C22×Q16C22×C312yes—
127C22×SD16C22×C311yescharacteristic
128C22×SD16C22×C311yescharacteristic
129C23×C8C22×C311yescharacteristic
130C2×#124C22×C312yes—
131C2×#121C22×C311yescharacteristic
132C2×C4×Q8C22×C312yes—
133#464—28no—
134#448—24no—
135C2×C4⋊C8C22×C312yes—
136C2×#103C22×C311yescharacteristic
137C2×#103C22×C312yes—
138C2×#103C22×C311yescharacteristic
139C2×#103C22×C312yes—
140C2×#102C22×C312yes—
141C2×C22⋊C8C22×C311yescharacteristic
142#393—22no—
143#391—22no—
144#391C22×C312yes—
145#391C22×C312yes—
146#386—21no—
147#386—21no—
148#380—21no—
149#380C22×C311yescharacteristic
150C4×C22⋊C4—21no—
151C4×C22⋊C4C22×C311yescharacteristic
152#340C22×C312yes—
153#336C22×C312yes—
154C24×C3—21no—
155C24×C3C2×D411yescharacteristic
156C2×C3×Q8—24no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—28no—
162C2×C3×Q8—28no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—44no—
170C2×C3×Q8—44no—
171C2×C3×Q8—44no—
172C2×C3×Q8—48no—
173C2×C3×Q8—48no—
174C2×C3×Q8—44no—
175C2×C3×Q8—42no—
176C2×C3×Q8C2×D414yes—
177C2×C3×Q8C2×D412yes—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8C2×D414yes—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—21no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—44no—
196C22×C3×C4—44no—
197C22×C3×C4—42no—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D412yes—
202C22×C3×C4#4112yes—
203C22×C3×C4C2411yescharacteristic
204C3×Q16—216no—
205C3×Q16—216no—
206C3×SD16—44no—
207C3×SD16—44no—
208C2×C3×C8—28no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8C2×D414yes—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—44no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—44no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—44no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—24no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—42no—
238C22×Q8—42no—
239C22×Q8C3×D414yes—
240C22×Q8C23×C312yes—
241C22×Q8C23×C311yescharacteristic
242C22×Q8C23×C311yescharacteristic
243C22×D4—21no—
244C22×D4C23×C311yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C23×C311yescharacteristic
248C2×Q16—232no—
249C2×Q16—24no—
250C2×Q16—24no—
251C2×Q16—24no—
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16C3×D418yescomplemented
255C2×SD16—22no—
256C2×SD16—22no—
257C2×SD16—24no—
258C2×SD16—42no—
259C22×C8—22no—
260C22×C8—24no—
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C22×C8C23×C312yes—
264#124—24no—
265#124—42no—
266#121—22no—
267#121—22no—
268C4×Q8—24no—
269C4×Q8—42no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C3×D411yescharacteristic
278C2×C22⋊C4C3×D411yescharacteristic
279C2×C42—22no—
280C2×C42C23×C312yes—
281C4⋊C8—24no—
282#103—22no—
283#103—22no—
284#103—24no—
285#103—42no—
286#102—24no—
287C22⋊C8—22no—
288#95—22no—
289#95—22no—
290#95—21no—
291#95—22no—
292#95—22no—
293#95C3×D412yes—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—42no—
301C23×C3#12112yes—
302C23×C3#13512yes—
303C23×C3#13712yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—22no—
306C3×Q8—24no—
307C3×Q8—22no—
308C3×Q8—24no—
309C3×Q8—22no—
310C3×Q8—24no—
311C3×Q8—48no—
312C3×Q8—48no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—42no—
316C3×Q8#12014yes—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—42no—
320C3×D4—84no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—48no—
336C2×C3×C4—48no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4#12112yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D412yes—
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)12yes—
351C3×C8—24no—
352C3×C8—41no—
353C3×C8—41no—
354C24—21no—
355C24C2×C3×D411yescharacteristic
356C2×Q8—24no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×Q8C2×C3×D414yes—
377C2×Q8C2×C3×D412yes—
378C2×Q8C2×C3×D412yes—
379C2×Q8C2×C3×D414yes—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4C2×C3×D412yes—
399C22×C4C2×C3×D412yes—
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D412yes—
402C22×C4C3×#4112yes—
403C22×C4C24×C311yescharacteristic
404Q16—216no—
405Q16—216no—
406SD16—44no—
407SD16—44no—
408C2×C8—28no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8C2×C3×D414yes—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—44no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C42—22no—
428C42—22no—
429C42—44no—
430C22×C3—22no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—81no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3#46414yescomplemented
440C22×C3C2×#12111yescharacteristic
441C22×C3D4212yes—
442C22×C3C2×#13511yescharacteristic
443C22×C3C2×#13711yescharacteristic
444C22×C3#57812yes—
445C3×C4—22no—
446C3×C4—24no—
447C3×C4—42no—
448C3×C4—44no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—82no—
452C3×C4—81no—
453C3×C4C2×#12011yescharacteristic
454C3×C4#53411yescharacteristic
455C3×C4#54612yes—
456C23—22no—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—41no—
462C23—42no—
463C23C3×#12112yes—
464C23C3×#13512yes—
465C23C3×#13712yes—
466C23C22×C3×D411yescharacteristic
467Q8—22no—
468Q8—24no—
469Q8—22no—
470Q8—24no—
471Q8—22no—
472Q8—24no—
473Q8—48no—
474Q8—48no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8C3×#12014yes—
479D4—41no—
480D4—41no—
481D4—42no—
482D4—84no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4C3×#12112yes—
509C2×C4C22×C3×D411yescharacteristic
510C2×C4C22×C3×D412yes—
511C2×C4C2×C3×#4111yescharacteristic
512C2×C4C3×H(5,𝔽2)12yes—
513C8—24no—
514C8—41no—
515C8—41no—
516C2×C3—24no—
517C2×C3—161no—
518C2×C3#241011yescharacteristic
519C2×C3C2×#46411yescharacteristic
520C2×C3#304411yescharacteristic
521C2×C3#327012yes—
522C2×C3#328512yes—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—22no—
527C22—81no—
528C22—82no—
529C22—82no—
530C22—81no—
531C22—81no—
532C22C3×#46414yescomplemented
533C22C2×C3×#12111yescharacteristic
534C22C3×D4212yes—
535C22C2×C3×#13511yescharacteristic
536C22C2×C3×#13711yescharacteristic
537C22C3×#57812yes—
538C4—22no—
539C4—24no—
540C4—42no—
541C4—44no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—81no—
546C4C2×C3×#12011yescharacteristic
547C4C3×#53411yescharacteristic
548C4C3×#54612yes—
549C3#2436111yescharacteristic, direct product factor, Sylow
550C2—24no—
551C2—161no—
552C2C3×#241011yescharacteristic
553C2C2×C3×#46411yescharacteristic
554C2C3×#304411yescharacteristic
555C2C3×#327012yes—
556C2C3×#328512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C3×Q16C2semidirect
2. C2×C3×#464C2semidirect
5. C3×Q8×C22⋊C4C2semidirect
6. C3×#2056C2semidirect
7. C3×#2015C2semidirect
9. C3×#1878C2semidirect
10. C3×#1850C2semidirect
11. C3×#1787C2semidirect
12. #24361C3direct (N × H)
46. C23×Q16C2×C3semidirect
47. C2×#464C2×C3semidirect
50. Q8×C22⋊C4C2×C3semidirect
51. #2056C2×C3semidirect
52. #2015C2×C3semidirect
54. #1878C2×C3semidirect
55. #1850C2×C3semidirect
56. #1787C2×C3semidirect
81. C2×C3×Q16D4semidirect
254. C2×Q16C3×D4semidirect
439. C22×C3#464semidirect
532. C22C3×#464semidirect
549. C3#24361direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×Q162
C2×C3×#4642
C2×C3×#4482
C3×#24362
C3×Q8×C22⋊C42
C3×#20562
C3×#20152
C3×#18932
C3×#18782
C3×#18502
C3×#17872
#243613