Finite Group Databasebeta

← C3×#24671 #10643278

Subgroups — C3×#24671 #10643278

Order 768.

786 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3039C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic, complemented
3C3×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic
4C3×#2925C211yesmaximal, characteristic, complemented
5C3×#2052C212yesmaximal, complemented
6C3×#2019C212yesmaximal
7C3×#1977C212yesmaximal, complemented
8C3×#1966C211yesmaximal, characteristic, complemented
9C3×#1950C211yesmaximal, characteristic, complemented
10C3×#1930C212yesmaximal, complemented
11C3×#1800C211yesmaximal, characteristic, complemented
12#24671C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C3×C4×#41C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C3×#462—21no—
20C3×#462—21no—
21C3×#461—21no—
22C3×#461—21no—
23C3×#461—21no—
24C3×#461—21no—
25C3×#461—22no—
26C3×#442—21no—
27C3×#440—21no—
28C3×C4×SD16—21no—
29C3×C4×D8—21no—
30C3×#433—22no—
31C3×#432—21no—
32C3×#431—21no—
33C2×C3×#108C2212yes—
34C3×#428C2211yescharacteristic
35C3×#423C2211yescharacteristic
36C3×#423C2211yescharacteristic
37C2×C3×C4⋊C8C2211yescharacteristic
38C3×#419C2212yes—
39C3×#418C2211yescharacteristic
40C3×#417C2211yescharacteristic
41C2×C3×#103C2212yes—
42C2×C3×#101C2212yes—
43C2×C3×#101C2212yes—
44C2×C3×#100C2212yes—
45C3×#408C2212yes—
46C3×#405C2212yes—
47C3×C4×M16C2211yescharacteristic
48C3×#343C2212yes—
49C3×#341C2212yes—
50#3039C2×C311yescharacteristic, complemented
51C2×#461C2×C311yescharacteristic, complemented
52C4×AGL(1,ℤ/8)C2×C311yescharacteristic
53#2925C2×C311yescharacteristic, complemented
54#2052C2×C312yescomplemented
55#2019C2×C312yes—
56#1977C2×C312yescomplemented
57#1966C2×C311yescharacteristic, complemented
58#1950C2×C311yescharacteristic, complemented
59#1930C2×C312yescomplemented
60#1800C2×C311yescharacteristic, complemented
61C2×C3×#41C2311yescharacteristic
62C22×C3×Q8C2312yes—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C3×#137—42no—
66C3×#137—42no—
67C3×AGL(1,ℤ/8)—22no—
68C3×AGL(1,ℤ/8)—22no—
69C2×C3×Q16—22no—
70C2×C3×SD16—21no—
71C2×C3×SD16—22no—
72C2×C3×SD16—24no—
73C2×C3×SD16—22no—
74C2×C3×SD16—22no—
75C2×C3×SD16—42no—
76C2×C3×D8—21no—
77C2×C3×M16C2312yes—
78C2×C3×M16C2311yescharacteristic
79C2×C3×M16C2312yes—
80C2×C3×M16C2311yescharacteristic
81C22×C3×C8C2312yes—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8—21no—
85C3×C4⋊Q8—21no—
86C3×C4×Q8D411yescharacteristic
87C3×C4×Q8D411yescharacteristic
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C3×C4×D4D411yescharacteristic
95C3×C4×D4D411yescharacteristic
96C3×#117—21no—
97C3×#117C2311yescharacteristic
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—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—22no—
116C3×#103—22no—
117C3×#103—22no—
118C3×#103—22no—
119C3×#102—21no—
120C3×#102—21no—
121C3×#102—21no—
122C3×#101—24no—
123C3×#101—42no—
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×C312yes—
132C2×AGL(1,ℤ/8)C22×C311yescharacteristic
133C22×SD16C22×C312yes—
134C2×C4⋊Q8C22×C311yescharacteristic
135C4×#41C22×C311yescharacteristic
136C2×C4×D4C22×C311yescharacteristic
137#462—21no—
138#462—21no—
139#461—21no—
140#461—21no—
141#461—21no—
142#461—21no—
143#461—22no—
144#442—21no—
145#440—21no—
146C4×SD16—21no—
147C4×D8—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×C312yes—
157#418C22×C311yescharacteristic
158#417C22×C311yescharacteristic
159C2×#103C22×C312yes—
160C2×#101C22×C312yes—
161C2×#101C22×C312yes—
162C2×#100C22×C312yes—
163#408C22×C312yes—
164#405C22×C312yes—
165C4×M16C22×C311yescharacteristic
166#343C22×C312yes—
167#341C22×C312yes—
168C24×C3—21no—
169C3×#41—21no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8C2×D411yescharacteristic
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4C2×D411yescharacteristic
187C2×C3×D4#4112yes—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—41no—
194C22×C3×C4—41no—
195C22×C3×C4—82no—
196C22×C3×C4#4111yescharacteristic
197C22×C3×C4#4111yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C3×Q16—44no—
200C3×SD16—24no—
201C3×SD16—44no—
202C3×SD16—44no—
203C3×SD16—44no—
204C3×D8—24no—
205C3×M16—21no—
206C3×M16—22no—
207C3×M16—22no—
208C3×M16—22no—
209C3×M16—21no—
210C3×M16—21no—
211C3×M16—22no—
212C3×M16#4112yes—
213C2×C3×C8—21no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—21no—
218C2×C3×C8—22no—
219C2×C3×C8C2×D412yes—
220C2×C3×C8C2×D412yes—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—41no—
226C3×C4⋊C4—41no—
227C3×C4⋊C4—41no—
228C3×C4⋊C4—41no—
229C3×C4⋊C4C2×D411yescharacteristic
230C3×C4⋊C4C2×D411yescharacteristic
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—41no—
235C3×C22⋊C4—41no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C2×#41C23×C311yescharacteristic
244C22×Q8C23×C312yes—
245C22×D4C23×C311yescharacteristic
246C23×C4—21no—
247#137—42no—
248#137—42no—
249AGL(1,ℤ/8)—22no—
250AGL(1,ℤ/8)—22no—
251C2×Q16—22no—
252C2×SD16—21no—
253C2×SD16—22no—
254C2×SD16—24no—
255C2×SD16—22no—
256C2×SD16—22no—
257C2×SD16—42no—
258C2×D8—21no—
259C2×M16C23×C312yes—
260C2×M16C23×C312yes—
261C2×M16C23×C311yescharacteristic
262C2×M16C23×C311yescharacteristic
263C22×C8C23×C312yes—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4×Q8C3×D411yescharacteristic
269C4×Q8C3×D411yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4C3×D411yescharacteristic
277C4×D4C3×D411yescharacteristic
278#117—21no—
279#117C23×C311yescharacteristic
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C22⋊C4—21no—
284C2×C42—21no—
285C2×C42C23×C311yescharacteristic
286#108—42no—
287#107—21no—
288#106—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8C3×D411yescharacteristic
294C4⋊C8C3×D411yescharacteristic
295C4⋊C8C3×D412yes—
296#103—21no—
297#103—22no—
298#103—22no—
299#103—22no—
300#103—22no—
301#102—21no—
302#102—21no—
303#102—21no—
304#101—24no—
305#101—42no—
306#100—42no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309#97—22no—
310#97—21no—
311C4×C8—22no—
312C4×C8—21no—
313C23×C3—21no—
314C23×C3—22no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—81no—
318C23×C3—81no—
319C23×C3C2×#4111yescharacteristic
320C3×Q8—21no—
321C3×Q8—21no—
322C3×Q8—42no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—42no—
326C3×D4—21no—
327C3×D4—22no—
328C3×D4—21no—
329C3×D4—22no—
330C3×D4—41no—
331C3×D4—41no—
332C3×D4—41no—
333C3×D4—41no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—81no—
347C2×C3×C4—82no—
348C2×C3×C4—82no—
349C2×C3×C4—81no—
350C2×C3×C4#12111yescharacteristic
351C2×C3×C4#12111yescharacteristic
352C2×C3×C4#12111yescharacteristic
353C2×C3×C4#12111yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4H(5,𝔽2)11yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C3×C8—22no—
361C3×C8—22no—
362C3×C8—22no—
363C3×C8—41no—
364C3×C8—42no—
365C3×C8—42no—
366C24—21no—
367#41—21no—
368#41—21no—
369#41—21no—
370#41—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8C2×C3×D411yescharacteristic
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4C2×C3×D411yescharacteristic
385C2×D4C3×#4112yes—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—82no—
394C22×C4C3×#4111yescharacteristic
395C22×C4C3×#4111yescharacteristic
396C22×C4C24×C311yescharacteristic
397Q16—44no—
398SD16—24no—
399SD16—44no—
400SD16—44no—
401SD16—44no—
402D8—24no—
403M16—21no—
404M16—22no—
405M16—22no—
406M16—22no—
407M16—21no—
408M16—21no—
409M16—22no—
410M16C3×#4112yes—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8C2×C3×D412yes—
418C2×C8C2×C3×D412yes—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4C2×C3×D411yescharacteristic
428C4⋊C4C2×C3×D411yescharacteristic
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—21no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42C2×C3×D411yescharacteristic
438C42C2×C3×D411yescharacteristic
439C42C2×C3×D411yescharacteristic
440C42C2×C3×D411yescharacteristic
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—41no—
444C22×C3—42no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3C2×#12111yescharacteristic
452C22×C3#55311yescharacteristic
453C22×C3#55511yescharacteristic
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—41no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—82no—
461C3×C4—81no—
462C3×C4#47112yes—
463C3×C4#52511yescharacteristic
464C3×C4#53511yescharacteristic
465C3×C4#54611yescharacteristic
466C3×C4#55011yescharacteristic
467C23—21no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—81no—
472C23—81no—
473C23C2×C3×#4111yescharacteristic
474Q8—21no—
475Q8—21no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480D4—21no—
481D4—22no—
482D4—21no—
483D4—22no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—41no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4C3×#12111yescharacteristic
505C2×C4C3×#12111yescharacteristic
506C2×C4C3×#12111yescharacteristic
507C2×C4C3×#12111yescharacteristic
508C2×C4C22×C3×D411yescharacteristic
509C2×C4C22×C3×D411yescharacteristic
510C2×C4C2×C3×#4111yescharacteristic
511C2×C4C2×C3×#4111yescharacteristic
512C2×C4C3×H(5,𝔽2)11yescharacteristic
513C2×C4C3×H(5,𝔽2)11yescharacteristic
514C8—22no—
515C8—22no—
516C8—22no—
517C8—42no—
518C8—41no—
519C8—42no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—81no—
523C2×C3—82no—
524C2×C3#241411yescharacteristic
525C2×C3C2×#47111yescharacteristic
526C2×C3#306911yescharacteristic
527C22—21no—
528C22—21no—
529C22—41no—
530C22—42no—
531C22—41no—
532C22—41no—
533C22—82no—
534C22—82no—
535C22—81no—
536C22—81no—
537C22C2×C3×#12111yescharacteristic
538C22C3×#55311yescharacteristic
539C22C3×#55511yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—82no—
547C4—81no—
548C4C3×#47112yes—
549C4C3×#52511yescharacteristic
550C4C3×#53511yescharacteristic
551C4C3×#54611yescharacteristic
552C4C3×#55011yescharacteristic
553C3#2467111yescharacteristic, direct product factor, Sylow
554C2—21no—
555C2—21no—
556C2—81no—
557C2—82no—
558C2C3×#241411yescharacteristic
559C2C2×C3×#47111yescharacteristic
560C2C3×#306911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×AGL(1,ℤ/8)C22
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
1. C3×#3039C2semidirect
2. C2×C3×#461C2semidirect
4. C3×#2925C2semidirect
5. C3×#2052C2semidirect
7. C3×#1977C2semidirect
8. C3×#1966C2semidirect
9. C3×#1950C2semidirect
10. C3×#1930C2semidirect
11. C3×#1800C2semidirect
12. #24671C3direct (N × H)
50. #3039C2×C3semidirect
51. C2×#461C2×C3semidirect
53. #2925C2×C3semidirect
54. #2052C2×C3semidirect
56. #1977C2×C3semidirect
57. #1966C2×C3semidirect
58. #1950C2×C3semidirect
59. #1930C2×C3semidirect
60. #1800C2×C3semidirect
553. C3#24671direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30392
C2×C3×#4612
C3×C4×AGL(1,ℤ/8)2
C3×#29252
C3×#20522
C3×#20192
C3×#19772
C3×#19662
C3×#19502
C3×#19302
C3×#18002
#246713