Finite Group Databasebeta

← C3×#24374 #10642981

Subgroups — C3×#24374 #10642981

Order 768.

1396 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#137C211yesmaximal, characteristic
2C2×C3×#462C212yesmaximal, complemented
3C2×C3×#452C211yesmaximal, characteristic
4C3×#2434C211yesmaximal, characteristic, complemented
5C3×#2341C211yesmaximal, characteristic
6C3×#2055C212yesmaximal, complemented
7C3×#2013C212yesmaximal, complemented
8C3×#1894C211yesmaximal, characteristic, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1851C211yesmaximal, characteristic, complemented
11C3×#1787C211yesmaximal, characteristic, complemented
12#24374C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C2×C3×#137—24no—
16C2×C3×#137—28no—
17C22×C3×Q16C2211yescharacteristic
18C22×C3×Q16C2212yes—
19C22×C3×SD16C2212yes—
20C22×C3×SD16C2211yescharacteristic
21C22×C3×M16C2211yescharacteristic
22C2×C3×C4⋊Q8C2212yes—
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×C4×Q8C2212yes—
25C3×#462—28no—
26C3×#452—24no—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#395—21no—
35C3×#395—21no—
36C3×#391C2212yes—
37C3×#390—22no—
38C3×#390C2212yes—
39C3×#385—22no—
40C3×#380—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C22⋊C4—21no—
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×#340C2212yes—
45C3×#336C2212yes—
46C22×#137C2×C311yescharacteristic
47C2×#462C2×C312yescomplemented
48C2×#452C2×C311yescharacteristic
49#2434C2×C311yescharacteristic, complemented
50#2341C2×C311yescharacteristic
51#2055C2×C312yescomplemented
52#2013C2×C312yescomplemented
53#1894C2×C311yescharacteristic, complemented
54#1878C2×C312yescomplemented
55#1851C2×C311yescharacteristic, complemented
56#1787C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—24no—
59C2×C3×#41—42no—
60C2×C3×#41D414yes—
61C22×C3×Q8—21no—
62C22×C3×Q8—24no—
63C22×C3×Q8—22no—
64C22×C3×Q8—22no—
65C22×C3×Q8—42no—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×Q8C2312yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—21no—
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C3×#137—232no—
75C2×C3×Q16—24no—
76C2×C3×Q16—24no—
77C2×C3×Q16—22no—
78C2×C3×Q16—24no—
79C2×C3×Q16D418yes—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16—22no—
83C2×C3×SD16—24no—
84C2×C3×SD16—42no—
85C2×C3×M16—22no—
86C2×C3×M16—24no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2312yes—
90C3×C4⋊Q8—24no—
91C3×C4⋊Q8—42no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—22no—
94C3×C4×Q8—24no—
95C3×C4×Q8—42no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4D412yes—
107C2×C3×C42—22no—
108C2×C3×C42C2312yes—
109C3×C4⋊C8—24no—
110C3×#103—22no—
111C3×#103—24no—
112C3×#103—24no—
113C3×#103—42no—
114C3×#102—22no—
115C3×C22⋊C8—22no—
116C3×#95—21no—
117C3×#95—22no—
118C3×#95—22no—
119C3×#95D411yescharacteristic
120C3×#95D411yescharacteristic
121C22×#41C22×C311yescharacteristic
122C23×Q8C22×C311yescharacteristic
123C2×#137—24no—
124C2×#137—28no—
125C22×Q16C22×C311yescharacteristic
126C22×Q16C22×C312yes—
127C22×SD16C22×C312yes—
128C22×SD16C22×C311yescharacteristic
129C22×M16C22×C311yescharacteristic
130C2×C4⋊Q8C22×C312yes—
131C2×C22⋊Q8C22×C311yescharacteristic
132C2×C4×Q8C22×C312yes—
133#462—28no—
134#452—24no—
135C2×C4⋊C8C22×C312yes—
136C2×#103C22×C312yes—
137C2×#103C22×C311yescharacteristic
138C2×#103C22×C312yes—
139C2×#103C22×C312yes—
140C2×#102C22×C311yescharacteristic
141C2×C22⋊C8C22×C311yescharacteristic
142#395—21no—
143#395—21no—
144#391C22×C312yes—
145#390—22no—
146#390C22×C312yes—
147#385—22no—
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
156C3×#41—28no—
157C3×#41—44no—
158C3×#41—48no—
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—28no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—44no—
172C2×C3×Q8—48no—
173C2×C3×Q8—44no—
174C2×C3×Q8—42no—
175C2×C3×Q8C2×D414yes—
176C2×C3×Q8C2×D412yes—
177C2×C3×Q8C2×D414yes—
178C2×C3×D4—24no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—44no—
183C2×C3×D4C2×D412yes—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—24no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—44no—
193C22×C3×C4—44no—
194C22×C3×C4—42no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4#4112yes—
200C22×C3×C4C2411yescharacteristic
201C3×Q16—216no—
202C3×Q16—44no—
203C3×SD16—216no—
204C3×SD16—44no—
205C3×M16—28no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8C2×D414yes—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—44no—
218C3×C4⋊C4—44no—
219C3×C4⋊C4—44no—
220C3×C4⋊C4—42no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—82no—
225C3×C22⋊C4—81no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—44no—
229C2×#41—22no—
230C2×#41—24no—
231C2×#41—42no—
232C2×#41C3×D414yes—
233C22×Q8—21no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—42no—
238C22×Q8C23×C311yescharacteristic
239C22×Q8C23×C312yes—
240C22×Q8C23×C311yescharacteristic
241C22×D4—21no—
242C22×D4C23×C311yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C23×C311yescharacteristic
246#137—232no—
247C2×Q16—24no—
248C2×Q16—24no—
249C2×Q16—22no—
250C2×Q16—24no—
251C2×Q16C3×D418yes—
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—22no—
255C2×SD16—24no—
256C2×SD16—42no—
257C2×M16—22no—
258C2×M16—24no—
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C312yes—
262C4⋊Q8—24no—
263C4⋊Q8—42no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C4×Q8—24no—
267C4×Q8—42no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4C23×C312yes—
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C3×D412yes—
279C2×C42—22no—
280C2×C42C23×C312yes—
281C4⋊C8—24no—
282#103—22no—
283#103—24no—
284#103—24no—
285#103—42no—
286#102—22no—
287C22⋊C8—22no—
288#95—21no—
289#95—22no—
290#95—22no—
291#95C3×D411yescharacteristic
292#95C3×D411yescharacteristic
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3#12112yes—
303C23×C3C22×D411yescharacteristic
304C3×Q8—22no—
305C3×Q8—24no—
306C3×Q8—24no—
307C3×Q8—24no—
308C3×Q8—48no—
309C3×Q8—42no—
310C3×Q8—42no—
311C3×Q8—41no—
312C3×Q8—41no—
313C3×Q8—42no—
314C3×Q8#12014yes—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—48no—
318C3×D4—42no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—48no—
332C2×C3×C4—48no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4#12112yes—
344C2×C3×C4#13714yescomplemented
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D412yes—
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4H(5,𝔽2)12yes—
349C3×C8—24no—
350C3×C8—41no—
351C3×C8—41no—
352C24—21no—
353C24C2×C3×D411yescharacteristic
354#41—28no—
355#41—44no—
356#41—48no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—28no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—44no—
370C2×Q8—48no—
371C2×Q8—44no—
372C2×Q8—42no—
373C2×Q8C2×C3×D414yes—
374C2×Q8C2×C3×D412yes—
375C2×Q8C2×C3×D414yes—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—44no—
381C2×D4C2×C3×D412yes—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4C2×C3×D412yes—
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D412yes—
397C22×C4C3×#4112yes—
398C22×C4C24×C311yescharacteristic
399Q16—216no—
400Q16—44no—
401SD16—216no—
402SD16—44no—
403M16—28no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8C2×C3×D414yes—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C22⋊C4—24no—
420C22⋊C4—42no—
421C22⋊C4—44no—
422C22⋊C4—82no—
423C22⋊C4—81no—
424C42—22no—
425C42—22no—
426C42—44no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C2×#12111yescharacteristic
438C22×C3D4212yes—
439C22×C3C2×#13712yes—
440C22×C3#57711yescharacteristic
441C22×C3#57811yescharacteristic
442C3×C4—22no—
443C3×C4—24no—
444C3×C4—24no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—81no—
448C3×C4—82no—
449C3×C4—81no—
450C3×C4C2×#12011yescharacteristic
451C3×C4#53411yescharacteristic
452C3×C4#54612yes—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23C3×#12112yes—
463C23C22×C3×D411yescharacteristic
464Q8—22no—
465Q8—24no—
466Q8—24no—
467Q8—24no—
468Q8—48no—
469Q8—42no—
470Q8—42no—
471Q8—41no—
472Q8—41no—
473Q8—42no—
474Q8C3×#12014yes—
475D4—22no—
476D4—22no—
477D4—48no—
478D4—42no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4C3×#12112yes—
504C2×C4C3×#13714yescomplemented
505C2×C4C22×C3×D411yescharacteristic
506C2×C4C22×C3×D412yes—
507C2×C4C2×C3×#4111yescharacteristic
508C2×C4C3×H(5,𝔽2)12yes—
509C8—24no—
510C8—41no—
511C8—41no—
512C2×C3—42no—
513C2×C3—82no—
514C2×C3#241011yescharacteristic
515C2×C3#303911yescharacteristic
516C2×C3#304611yescharacteristic
517C2×C3#327012yes—
518C2×C3#327812yes—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—82no—
528C22—82no—
529C22C2×C3×#12111yescharacteristic
530C22C3×D4212yes—
531C22C2×C3×#13712yes—
532C22C3×#57711yescharacteristic
533C22C3×#57811yescharacteristic
534C4—22no—
535C4—24no—
536C4—24no—
537C4—44no—
538C4—42no—
539C4—81no—
540C4—82no—
541C4—81no—
542C4C2×C3×#12011yescharacteristic
543C4C3×#53411yescharacteristic
544C4C3×#54612yes—
545C3#2437411yescharacteristic, direct product factor, Sylow
546C2—42no—
547C2—82no—
548C2C3×#241011yescharacteristic
549C2C3×#303911yescharacteristic
550C2C3×#304611yescharacteristic
551C2C3×#327012yes—
552C2C3×#327812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
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
2. C2×C3×#462C2semidirect
4. C3×#2434C2semidirect
6. C3×#2055C2semidirect
7. C3×#2013C2semidirect
8. C3×#1894C2semidirect
9. C3×#1878C2semidirect
10. C3×#1851C2semidirect
11. C3×#1787C2semidirect
12. #24374C3direct (N × H)
47. C2×#462C2×C3semidirect
49. #2434C2×C3semidirect
51. #2055C2×C3semidirect
52. #2013C2×C3semidirect
53. #1894C2×C3semidirect
54. #1878C2×C3semidirect
55. #1851C2×C3semidirect
56. #1787C2×C3semidirect
344. C2×C3×C4#137semidirect
504. C2×C4C3×#137semidirect
545. C3#24374direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1372
C2×C3×#4622
C2×C3×#4522
C3×#24342
C3×#23412
C3×#20552
C3×#20132
C3×#18942
C3×#18782
C3×#18512
C3×#17872
#243743