Finite Group Databasebeta

← C3×#24673 #10643280

Subgroups — C3×#24673 #10643280

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3038C211yesmaximal, characteristic
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C3×C4×#137C211yesmaximal, characteristic, complemented
4C3×#2925C211yesmaximal, characteristic, complemented
5C3×#2051C212yesmaximal, complemented
6C3×#2019C212yesmaximal, complemented
7C3×#1977C212yesmaximal, complemented
8C3×#1965C211yesmaximal, characteristic, complemented
9C3×#1950C211yesmaximal, characteristic, complemented
10C3×#1931C212yesmaximal, complemented
11C3×#1801C211yesmaximal, characteristic, complemented
12#24673C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2211yescharacteristic
14C2×C3×AGL(1,ℤ/8)C2212yes—
15C22×C3×SD16C2212yes—
16C2×C3×#127C2211yescharacteristic
17C3×C4×#41C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C3×#460—21no—
20C3×#460—21no—
21C3×#460—21no—
22C3×#460—21no—
23C3×#460—22no—
24C3×#459—21no—
25C3×#459—21no—
26C3×#441—21no—
27C3×#440—21no—
28C3×C4×Q16—21no—
29C3×C4×SD16—21no—
30C3×#433—22no—
31C3×#432—21no—
32C3×#431—21no—
33C2×C3×#108C2212yes—
34C3×#428C2211yescharacteristic
35C3×#423C2211yescharacteristic
36C3×#423C2211yescharacteristic
37C2×C3×C4⋊C8C2211yescharacteristic
38C3×#419C2211yescharacteristic
39C3×#418C2212yes—
40C3×#417C2211yescharacteristic
41C2×C3×#102C2212yes—
42C2×C3×#101C2212yes—
43C2×C3×#100C2212yes—
44C2×C3×#100C2212yes—
45C3×#408C2212yes—
46C3×#405C2212yes—
47C3×C4×M16C2211yescharacteristic
48C3×#343C2212yes—
49C3×#341C2212yes—
50#3038C2×C311yescharacteristic
51C2×#460C2×C311yescharacteristic, complemented
52C4×#137C2×C311yescharacteristic, complemented
53#2925C2×C311yescharacteristic, complemented
54#2051C2×C312yescomplemented
55#2019C2×C312yescomplemented
56#1977C2×C312yescomplemented
57#1965C2×C311yescharacteristic, complemented
58#1950C2×C311yescharacteristic, complemented
59#1931C2×C312yescomplemented
60#1801C2×C311yescharacteristic, complemented
61C2×C3×#41C2311yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C3×#137—22no—
67C3×#137—22no—
68C3×AGL(1,ℤ/8)—42no—
69C3×AGL(1,ℤ/8)—42no—
70C2×C3×Q16—21no—
71C2×C3×SD16—21no—
72C2×C3×SD16—22no—
73C2×C3×SD16—24no—
74C2×C3×SD16—22no—
75C2×C3×SD16—22no—
76C2×C3×SD16—42no—
77C2×C3×D8—22no—
78C2×C3×M16C2312yes—
79C2×C3×M16C2311yescharacteristic
80C2×C3×M16C2312yes—
81C2×C3×M16C2311yescharacteristic
82C22×C3×C8C2312yes—
83C3×#127—21no—
84C3×#127—21no—
85C3×#127—21no—
86C3×#127—21no—
87C3×C4×Q8—21no—
88C3×C4×Q8—21no—
89C3×C4×Q8—21no—
90C3×C4×Q8—21no—
91C3×C4×Q8D411yescharacteristic
92C3×C4×Q8D411yescharacteristic
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4D411yescharacteristic
96C3×C4×D4D411yescharacteristic
97C3×#117—21no—
98C3×#117C2311yescharacteristic
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C42—21no—
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C3×#108—42no—
105C3×#107—21no—
106C3×#106—21no—
107C3×C4⋊C8—21no—
108C3×C4⋊C8—21no—
109C3×C4⋊C8—21no—
110C3×C4⋊C8—21no—
111C3×C4⋊C8D411yescharacteristic
112C3×C4⋊C8D411yescharacteristic
113C3×C4⋊C8D412yes—
114C3×#103—21no—
115C3×#103—21no—
116C3×#103—21no—
117C3×#102—21no—
118C3×#102—22no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#102—22no—
122C3×#101—42no—
123C3×#100—24no—
124C3×#100—42no—
125C3×C22⋊C8—22no—
126C3×C22⋊C8—22no—
127C3×#97—22no—
128C3×#97—21no—
129C3×C4×C8—22no—
130C3×C4×C8—21no—
131C2×#137C22×C311yescharacteristic
132C2×AGL(1,ℤ/8)C22×C312yes—
133C22×SD16C22×C312yes—
134C2×#127C22×C311yescharacteristic
135C4×#41C22×C311yescharacteristic
136C2×C4×Q8C22×C311yescharacteristic
137#460—21no—
138#460—21no—
139#460—21no—
140#460—21no—
141#460—22no—
142#459—21no—
143#459—21no—
144#441—21no—
145#440—21no—
146C4×Q16—21no—
147C4×SD16—21no—
148#433—22no—
149#432—21no—
150#431—21no—
151C2×#108C22×C312yes—
152#428C22×C311yescharacteristic
153#423C22×C311yescharacteristic
154#423C22×C311yescharacteristic
155C2×C4⋊C8C22×C311yescharacteristic
156#419C22×C311yescharacteristic
157#418C22×C312yes—
158#417C22×C311yescharacteristic
159C2×#102C22×C312yes—
160C2×#101C22×C312yes—
161C2×#100C22×C312yes—
162C2×#100C22×C312yes—
163#408C22×C312yes—
164#405C22×C312yes—
165C4×M16C22×C311yescharacteristic
166#343C22×C312yes—
167#341C22×C312yes—
168C24×C3—22no—
169C3×#41—21no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8C2×D411yescharacteristic
179C2×C3×Q8#4112yes—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—41no—
188C2×C3×D4—41no—
189C2×C3×D4—41no—
190C2×C3×D4—41no—
191C2×C3×D4—82no—
192C2×C3×D4—82no—
193C2×C3×D4C2×D411yescharacteristic
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4#4111yescharacteristic
200C22×C3×C4#4111yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C3×Q16—24no—
203C3×SD16—24no—
204C3×SD16—44no—
205C3×SD16—44no—
206C3×SD16—44no—
207C3×D8—44no—
208C3×M16—21no—
209C3×M16—22no—
210C3×M16—21no—
211C3×M16—22no—
212C3×M16—22no—
213C3×M16—21no—
214C3×M16—22no—
215C3×M16#4112yes—
216C2×C3×C8—21no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—21no—
221C2×C3×C8—22no—
222C2×C3×C8C2×D412yes—
223C2×C3×C8C2×D412yes—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—41no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4C2×D411yescharacteristic
231C3×C4⋊C4C2×D411yescharacteristic
232C3×C22⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—41no—
239C3×C42—41no—
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42C2×D411yescharacteristic
244C2×#41C23×C311yescharacteristic
245C22×Q8C23×C311yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C23×C312yes—
249#137—22no—
250#137—22no—
251AGL(1,ℤ/8)—42no—
252AGL(1,ℤ/8)—42no—
253C2×Q16—21no—
254C2×SD16—21no—
255C2×SD16—22no—
256C2×SD16—24no—
257C2×SD16—22no—
258C2×SD16—22no—
259C2×SD16—42no—
260C2×D8—22no—
261C2×M16C23×C312yes—
262C2×M16C23×C312yes—
263C2×M16C23×C311yescharacteristic
264C2×M16C23×C311yescharacteristic
265C22×C8C23×C312yes—
266#127—21no—
267#127—21no—
268#127—21no—
269#127—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8C3×D411yescharacteristic
275C4×Q8C3×D411yescharacteristic
276C4×D4—21no—
277C4×D4—21no—
278C4×D4C3×D411yescharacteristic
279C4×D4C3×D411yescharacteristic
280#117—21no—
281#117C23×C311yescharacteristic
282C2×C4⋊C4—21no—
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42C23×C311yescharacteristic
287#108—42no—
288#107—21no—
289#106—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8C3×D411yescharacteristic
295C4⋊C8C3×D411yescharacteristic
296C4⋊C8C3×D412yes—
297#103—21no—
298#103—21no—
299#103—21no—
300#102—21no—
301#102—22no—
302#102—22no—
303#102—22no—
304#102—22no—
305#101—42no—
306#100—24no—
307#100—42no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310#97—22no—
311#97—21no—
312C4×C8—22no—
313C4×C8—21no—
314C23×C3—21no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—82no—
319C23×C3—82no—
320C23×C3C2×#4111yescharacteristic
321C3×Q8—22no—
322C3×Q8—21no—
323C3×Q8—21no—
324C3×Q8—22no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×D4—21no—
328C3×D4—21no—
329C3×D4—41no—
330C3×D4—41no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—82no—
336C3×D4—82no—
337C3×D4—82no—
338C3×D4—82no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—81no—
352C2×C3×C4—81no—
353C2×C3×C4#12111yescharacteristic
354C2×C3×C4#12111yescharacteristic
355C2×C3×C4#12111yescharacteristic
356C2×C3×C4#12111yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C3×C8—22no—
364C3×C8—22no—
365C3×C8—22no—
366C3×C8—41no—
367C3×C8—42no—
368C3×C8—42no—
369C24—22no—
370#41—21no—
371#41—21no—
372#41—21no—
373#41—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8C2×C3×D411yescharacteristic
380C2×Q8C3×#4112yes—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4C2×C3×D411yescharacteristic
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4C3×#4111yescharacteristic
401C22×C4C3×#4111yescharacteristic
402C22×C4C24×C311yescharacteristic
403Q16—24no—
404SD16—24no—
405SD16—44no—
406SD16—44no—
407SD16—44no—
408D8—44no—
409M16—21no—
410M16—22no—
411M16—21no—
412M16—22no—
413M16—22no—
414M16—21no—
415M16—22no—
416M16C3×#4112yes—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8C2×C3×D412yes—
424C2×C8C2×C3×D412yes—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4C2×C3×D411yescharacteristic
432C4⋊C4C2×C3×D411yescharacteristic
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—41no—
440C42—41no—
441C42C2×C3×D411yescharacteristic
442C42C2×C3×D411yescharacteristic
443C42C2×C3×D411yescharacteristic
444C42C2×C3×D411yescharacteristic
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—41no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3—162no—
454C22×C3—162no—
455C22×C3C2×#12111yescharacteristic
456C22×C3#55311yescharacteristic
457C22×C3#55511yescharacteristic
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—41no—
462C3×C4—42no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—81no—
466C3×C4#47112yes—
467C3×C4#52511yescharacteristic
468C3×C4#53511yescharacteristic
469C3×C4#54611yescharacteristic
470C3×C4#55011yescharacteristic
471C23—21no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—82no—
476C23—82no—
477C23C2×C3×#4111yescharacteristic
478Q8—22no—
479Q8—21no—
480Q8—21no—
481Q8—22no—
482Q8—41no—
483Q8—41no—
484D4—21no—
485D4—21no—
486D4—41no—
487D4—41no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—82no—
493D4—82no—
494D4—82no—
495D4—82no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4C3×#12111yescharacteristic
511C2×C4C3×#12111yescharacteristic
512C2×C4C3×#12111yescharacteristic
513C2×C4C3×#12111yescharacteristic
514C2×C4C22×C3×D411yescharacteristic
515C2×C4C22×C3×D411yescharacteristic
516C2×C4C2×C3×#4111yescharacteristic
517C2×C4C2×C3×#4111yescharacteristic
518C2×C4C3×H(5,𝔽2)11yescharacteristic
519C2×C4C3×H(5,𝔽2)11yescharacteristic
520C8—22no—
521C8—22no—
522C8—22no—
523C8—42no—
524C8—41no—
525C8—42no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—81no—
529C2×C3—162no—
530C2×C3#241411yescharacteristic
531C2×C3C2×#47111yescharacteristic
532C2×C3#306811yescharacteristic
533C22—21no—
534C22—21no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—162no—
542C22—162no—
543C22C2×C3×#12111yescharacteristic
544C22C3×#55311yescharacteristic
545C22C3×#55511yescharacteristic
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—42no—
551C4—41no—
552C4—41no—
553C4—81no—
554C4C3×#47112yes—
555C4C3×#52511yescharacteristic
556C4C3×#53511yescharacteristic
557C4C3×#54611yescharacteristic
558C4C3×#55011yescharacteristic
559C3#2467311yescharacteristic, direct product factor, Sylow
560C2—21no—
561C2—21no—
562C2—81no—
563C2—162no—
564C2C3×#241411yescharacteristic
565C2C2×C3×#47111yescharacteristic
566C2C3×#306811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3038C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#460C2semidirect
3. C3×C4×#137C2semidirect
4. C3×#2925C2semidirect
5. C3×#2051C2semidirect
6. C3×#2019C2semidirect
7. C3×#1977C2semidirect
8. C3×#1965C2semidirect
9. C3×#1950C2semidirect
10. C3×#1931C2semidirect
11. C3×#1801C2semidirect
12. #24673C3direct (N × H)
51. C2×#460C2×C3semidirect
52. C4×#137C2×C3semidirect
53. #2925C2×C3semidirect
54. #2051C2×C3semidirect
55. #2019C2×C3semidirect
56. #1977C2×C3semidirect
57. #1965C2×C3semidirect
58. #1950C2×C3semidirect
59. #1931C2×C3semidirect
60. #1801C2×C3semidirect
559. C3#24673direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30382
C2×C3×#4602
C3×C4×#1372
C3×#29252
C3×#20512
C3×#20192
C3×#19772
C3×#19652
C3×#19502
C3×#19312
C3×#18012
#246733