Finite Group Databasebeta

← C3×#24360 #10642967

Subgroups — C3×#24360 #10642967

Order 768.

1382 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Q16C211yesmaximal, characteristic
2C2×C3×#462C212yesmaximal, complemented
3C2×C3×#451C211yesmaximal, characteristic
4C3×#2434C211yesmaximal, characteristic
5C3×Q8×C22⋊C4C211yesmaximal, characteristic
6C3×#2056C212yesmaximal, complemented
7C3×#2016C212yesmaximal, complemented
8C3×#1892C211yesmaximal, characteristic
9C3×#1878C212yesmaximal, complemented
10C3×#1850C211yesmaximal, characteristic
11C3×#1787C211yesmaximal, characteristic
12#24360C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×Q16—24no—
16C22×C3×Q16—28no—
17C22×C3×Q16C2212yes—
18C22×C3×Q16C2211yescharacteristic
19C22×C3×Q16C2212yes—
20C22×C3×Q16C2211yescharacteristic
21C23×C3×C8C2211yescharacteristic
22C2×C3×C4⋊Q8C2212yes—
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×C4×Q8C2212yes—
25C3×#462—28no—
26C3×#451—24no—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#395—22no—
35C3×#391—22no—
36C3×#391C2212yes—
37C3×#391C2212yes—
38C3×#385—21no—
39C3×#385—21no—
40C3×#380—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C22⋊C4—21no—
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×#340C2212yes—
45C3×#336C2212yes—
46C23×Q16C2×C311yescharacteristic
47C2×#462C2×C312yescomplemented
48C2×#451C2×C311yescharacteristic
49#2434C2×C311yescharacteristic
50Q8×C22⋊C4C2×C311yescharacteristic
51#2056C2×C312yescomplemented
52#2016C2×C312yescomplemented
53#1892C2×C311yescharacteristic
54#1878C2×C312yescomplemented
55#1850C2×C311yescharacteristic
56#1787C2×C311yescharacteristic
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×Q8C2311yescharacteristic
69C22×C3×Q8C2312yes—
70C22×C3×Q8C2311yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C2×C3×Q16—232no—
75C2×C3×Q16—24no—
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×Q16—24no—
79C2×C3×Q16—22no—
80C2×C3×Q16—22no—
81C2×C3×Q16—24no—
82C2×C3×Q16—24no—
83C2×C3×Q16—42no—
84C2×C3×Q16D418yes—
85C22×C3×C8—22no—
86C22×C3×C8—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—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—24no—
113C3×#103—42no—
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×C311yescharacteristic
127C22×Q16C22×C312yes—
128C22×Q16C22×C311yescharacteristic
129C23×C8C22×C311yescharacteristic
130C2×C4⋊Q8C22×C312yes—
131C2×C22⋊Q8C22×C311yescharacteristic
132C2×C4×Q8C22×C312yes—
133#462—28no—
134#451—24no—
135C2×C4⋊C8C22×C312yes—
136C2×#103C22×C311yescharacteristic
137C2×#103C22×C312yes—
138C2×#103C22×C311yescharacteristic
139C2×#103C22×C312yes—
140C2×#103C22×C312yes—
141C2×C22⋊C8C22×C311yescharacteristic
142#395—22no—
143#391—22no—
144#391C22×C312yes—
145#391C22×C312yes—
146#385—21no—
147#385—21no—
148#380—21no—
149#380C22×C311yescharacteristic
150C4×C22⋊C4—21no—
151C4×C22⋊C4C22×C311yescharacteristic
152#340C22×C312yes—
153#336C22×C312yes—
154C24×C3C2×D411yescharacteristic
155C2×C3×Q8—24no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—28no—
161C2×C3×Q8—28no—
162C2×C3×Q8—22no—
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—22no—
172C2×C3×Q8—44no—
173C2×C3×Q8—44no—
174C2×C3×Q8—44no—
175C2×C3×Q8—48no—
176C2×C3×Q8—48no—
177C2×C3×Q8—44no—
178C2×C3×Q8—42no—
179C2×C3×Q8C2×D414yes—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8C2×D412yes—
182C2×C3×Q8C2×D414yes—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—24no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
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—216no—
203C3×Q16—44no—
204C3×Q16—44no—
205C2×C3×C8—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—22no—
214C3×C4⋊C4—21no—
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—22no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—42no—
224C3×C22⋊C4Q1614yescomplemented
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—44no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8—24no—
232C22×Q8—24no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—42no—
236C22×Q8—42no—
237C22×Q8C3×D414yes—
238C22×Q8C23×C311yescharacteristic
239C22×Q8C23×C312yes—
240C22×Q8C23×C311yescharacteristic
241C22×Q8C23×C311yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C23×C311yescharacteristic
245C2×Q16—232no—
246C2×Q16—24no—
247C2×Q16—24no—
248C2×Q16—24no—
249C2×Q16—24no—
250C2×Q16—22no—
251C2×Q16—22no—
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16—42no—
255C2×Q16C3×D418yes—
256C22×C8—22no—
257C22×C8—24no—
258C22×C8C23×C311yescharacteristic
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C312yes—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C4×Q8—24no—
266C4×Q8—42no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4C3×D411yescharacteristic
276C2×C22⋊C4C3×D411yescharacteristic
277C2×C42—22no—
278C2×C42C23×C312yes—
279C4⋊C8—24no—
280#103—22no—
281#103—22no—
282#103—24no—
283#103—24no—
284#103—42no—
285C22⋊C8—22no—
286#95—22no—
287#95—22no—
288#95—21no—
289#95—22no—
290#95—22no—
291#95C3×D412yes—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3#12112yes—
295C23×C3C2×Q1612yes—
296C23×C3#13712yes—
297C23×C3C22×D411yescharacteristic
298C3×Q8—22no—
299C3×Q8—24no—
300C3×Q8—22no—
301C3×Q8—24no—
302C3×Q8—22no—
303C3×Q8—24no—
304C3×Q8—48no—
305C3×Q8—48no—
306C3×Q8—42no—
307C3×Q8—42no—
308C3×Q8—42no—
309C3×Q8—41no—
310C3×Q8—41no—
311C3×Q8—42no—
312C3×Q8#12014yes—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—48no—
327C2×C3×C4—48no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4#12112yes—
339C2×C3×C4C2×Q1614yescomplemented
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D412yes—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4H(5,𝔽2)12yes—
344C3×C8—24no—
345C3×C8—41no—
346C3×C8—41no—
347C24C2×C3×D411yescharacteristic
348C2×Q8—24no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—28no—
354C2×Q8—28no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—44no—
371C2×Q8—42no—
372C2×Q8C2×C3×D414yes—
373C2×Q8C2×C3×D412yes—
374C2×Q8C2×C3×D412yes—
375C2×Q8C2×C3×D414yes—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4C2×C3×D412yes—
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D412yes—
391C22×C4C2×C3×D412yes—
392C22×C4C3×#4112yes—
393C22×C4C24×C311yescharacteristic
394Q16—216no—
395Q16—216no—
396Q16—44no—
397Q16—44no—
398C2×C8—28no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8C2×C3×D414yes—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C22⋊C4—22no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4C3×Q1614yescomplemented
418C42—22no—
419C42—22no—
420C42—44no—
421C22×C3—22no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3#46214yescomplemented
426C22×C3C2×#12111yescharacteristic
427C22×C3D4212yes—
428C22×C3C22×Q1611yescharacteristic
429C22×C3C2×#13711yescharacteristic
430C22×C3#57812yes—
431C3×C4—22no—
432C3×C4—24no—
433C3×C4—24no—
434C3×C4—42no—
435C3×C4—44no—
436C3×C4—42no—
437C3×C4—81no—
438C3×C4—82no—
439C3×C4—81no—
440C3×C4C2×#12011yescharacteristic
441C3×C4#53411yescharacteristic
442C3×C4#54612yes—
443C23—22no—
444C23—22no—
445C23C3×#12112yes—
446C23C2×C3×Q1612yes—
447C23C3×#13712yes—
448C23C22×C3×D411yescharacteristic
449Q8—22no—
450Q8—24no—
451Q8—22no—
452Q8—24no—
453Q8—22no—
454Q8—24no—
455Q8—48no—
456Q8—48no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460Q8—41no—
461Q8—41no—
462Q8—42no—
463Q8C3×#12014yes—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4C3×#12112yes—
490C2×C4C2×C3×Q1614yescomplemented
491C2×C4C22×C3×D411yescharacteristic
492C2×C4C22×C3×D412yes—
493C2×C4C2×C3×#4111yescharacteristic
494C2×C4C3×H(5,𝔽2)12yes—
495C8—24no—
496C8—41no—
497C8—41no—
498C2×C3—24no—
499C2×C3#241011yescharacteristic
500C2×C3C2×#46211yescharacteristic
501C2×C3#304411yescharacteristic
502C2×C3#327012yes—
503C2×C3D4×Q1612yes—
504C22—22no—
505C22—24no—
506C22—24no—
507C22—22no—
508C22C3×#46214yescomplemented
509C22C2×C3×#12111yescharacteristic
510C22C3×D4212yes—
511C22C22×C3×Q1611yescharacteristic
512C22C2×C3×#13711yescharacteristic
513C22C3×#57812yes—
514C4—22no—
515C4—24no—
516C4—24no—
517C4—42no—
518C4—44no—
519C4—42no—
520C4—81no—
521C4—82no—
522C4—81no—
523C4C2×C3×#12011yescharacteristic
524C4C3×#53411yescharacteristic
525C4C3×#54612yes—
526C3#2436011yescharacteristic, direct product factor, Sylow
527C2—24no—
528C2C3×#241011yescharacteristic
529C2C2×C3×#46211yescharacteristic
530C2C3×#304411yescharacteristic
531C2C3×#327012yes—
532C2C3×D4×Q1612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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
6. C3×#2056C2semidirect
7. C3×#2016C2semidirect
9. C3×#1878C2semidirect
12. #24360C3direct (N × H)
47. C2×#462C2×C3semidirect
51. #2056C2×C3semidirect
52. #2016C2×C3semidirect
54. #1878C2×C3semidirect
224. C3×C22⋊C4Q16semidirect
339. C2×C3×C4C2×Q16semidirect
417. C22⋊C4C3×Q16semidirect
425. C22×C3#462semidirect
490. C2×C4C2×C3×Q16semidirect
508. C22C3×#462semidirect
526. C3#24360direct (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×#4622
C2×C3×#4512
C3×#24342
C3×Q8×C22⋊C42
C3×#20562
C3×#20162
C3×#18922
C3×#18782
C3×#18502
C3×#17872
#243603