Finite Group Databasebeta

← C3×#24672 #10643279

Subgroups — C3×#24672 #10643279

Order 768.

754 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3039C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C3×C4×#137C211yesmaximal, characteristic
4C3×#2925C211yesmaximal, characteristic
5C3×#2052C212yesmaximal
6C3×#2020C212yesmaximal
7C3×#1979C212yesmaximal, complemented
8C3×#1966C211yesmaximal, characteristic, complemented
9C3×#1950C211yesmaximal, characteristic, complemented
10C3×#1930C212yesmaximal, complemented
11C3×#1801C211yesmaximal, characteristic, complemented
12#24672C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2211yescharacteristic
14C2×C3×#137C2212yes—
15C22×C3×Q16C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C3×C4×#41C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C3×#462—21no—
20C3×#462—21no—
21C3×#462—21no—
22C3×#462—21no—
23C3×#462—22no—
24C3×#461—21no—
25C3×#461—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×#419C2212yes—
40C3×#417C2211yescharacteristic
41C2×C3×#103C2212yes—
42C2×C3×#101C2212yes—
43C2×C3×#101C2212yes—
44C2×C3×#101C2212yes—
45C3×#408C2212yes—
46C3×#405C2212yes—
47C3×C4×M16C2211yescharacteristic
48C3×#343C2212yes—
49C3×#341C2212yes—
50#3039C2×C311yescharacteristic
51C2×#462C2×C311yescharacteristic, complemented
52C4×#137C2×C311yescharacteristic
53#2925C2×C311yescharacteristic
54#2052C2×C312yes—
55#2020C2×C312yes—
56#1979C2×C312yescomplemented
57#1966C2×C311yescharacteristic, complemented
58#1950C2×C311yescharacteristic, complemented
59#1930C2×C312yescomplemented
60#1801C2×C311yescharacteristic, complemented
61C2×C3×#41C2311yescharacteristic
62C22×C3×Q8C2312yes—
63C22×C3×Q8C2311yescharacteristic
64C3×#137—22no—
65C3×#137—22no—
66C3×#137—42no—
67C3×#137—42no—
68C2×C3×Q16—21no—
69C2×C3×Q16—22no—
70C2×C3×Q16—24no—
71C2×C3×Q16—22no—
72C2×C3×Q16—22no—
73C2×C3×Q16—42no—
74C2×C3×SD16—21no—
75C2×C3×SD16—22no—
76C2×C3×M16C2312yes—
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2312yes—
79C2×C3×M16C2311yescharacteristic
80C22×C3×C8C2312yes—
81C3×C4⋊Q8—21no—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8—21no—
85C3×C4×Q8—21no—
86C3×C4×Q8—21no—
87C3×C4×Q8—21no—
88C3×C4×Q8—21no—
89C3×C4×Q8D411yescharacteristic
90C3×C4×Q8D411yescharacteristic
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C3×C4×D4D411yescharacteristic
94C3×C4×D4D411yescharacteristic
95C3×#117—21no—
96C3×#117C2311yescharacteristic
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
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—22no—
116C3×#103—22no—
117C3×#103—21no—
118C3×#103—21no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#102—21no—
122C3×#101—24no—
123C3×#101—42no—
124C3×#101—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×#137C22×C312yes—
133C22×Q16C22×C312yes—
134C2×C4⋊Q8C22×C311yescharacteristic
135C4×#41C22×C311yescharacteristic
136C2×C4×Q8C22×C311yescharacteristic
137#462—21no—
138#462—21no—
139#462—21no—
140#462—21no—
141#462—22no—
142#461—21no—
143#461—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#419C22×C312yes—
158#417C22×C311yescharacteristic
159C2×#103C22×C312yes—
160C2×#101C22×C312yes—
161C2×#101C22×C312yes—
162C2×#101C22×C312yes—
163#408C22×C312yes—
164#405C22×C312yes—
165C4×M16C22×C311yescharacteristic
166#343C22×C312yes—
167#341C22×C312yes—
168C3×#41—21no—
169C3×#41—21no—
170C3×#41—21no—
171C3×#41—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—21no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—21no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8C2×D411yescharacteristic
184C2×C3×Q8#4112yes—
185C2×C3×D4—21no—
186C2×C3×D4C2×D411yescharacteristic
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4#4111yescharacteristic
194C22×C3×C4#4111yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C3×Q16—24no—
197C3×Q16—44no—
198C3×Q16—44no—
199C3×Q16—44no—
200C3×SD16—24no—
201C3×SD16—44no—
202C3×M16—21no—
203C3×M16—22no—
204C3×M16—21no—
205C3×M16—22no—
206C3×M16—22no—
207C3×M16—21no—
208C3×M16—22no—
209C3×M16#4112yes—
210C2×C3×C8—21no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—21no—
215C2×C3×C8—22no—
216C2×C3×C8C2×D412yes—
217C2×C3×C8C2×D412yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—41no—
223C3×C4⋊C4—41no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—41no—
226C3×C4⋊C4—41no—
227C3×C4⋊C4—41no—
228C3×C4⋊C4C2×D411yescharacteristic
229C3×C4⋊C4C2×D411yescharacteristic
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—41no—
237C3×C42—41no—
238C3×C42C2×D411yescharacteristic
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C2×#41C23×C311yescharacteristic
243C22×Q8C23×C312yes—
244C22×Q8C23×C311yescharacteristic
245#137—22no—
246#137—22no—
247#137—42no—
248#137—42no—
249C2×Q16—21no—
250C2×Q16—22no—
251C2×Q16—24no—
252C2×Q16—22no—
253C2×Q16—22no—
254C2×Q16—42no—
255C2×SD16—21no—
256C2×SD16—22no—
257C2×M16C23×C312yes—
258C2×M16C23×C312yes—
259C2×M16C23×C311yescharacteristic
260C2×M16C23×C311yescharacteristic
261C22×C8C23×C312yes—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8C3×D411yescharacteristic
271C4×Q8C3×D411yescharacteristic
272C4×D4—21no—
273C4×D4—21no—
274C4×D4C3×D411yescharacteristic
275C4×D4C3×D411yescharacteristic
276#117—21no—
277#117C23×C311yescharacteristic
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C23×C311yescharacteristic
285#108—42no—
286#107—21no—
287#106—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8C3×D411yescharacteristic
293C4⋊C8C3×D411yescharacteristic
294C4⋊C8C3×D412yes—
295#103—21no—
296#103—22no—
297#103—22no—
298#103—21no—
299#103—21no—
300#103—22no—
301#103—22no—
302#102—21no—
303#101—24no—
304#101—42no—
305#101—42no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308#97—22no—
309#97—21no—
310C4×C8—22no—
311C4×C8—21no—
312C23×C3—21no—
313C23×C3C2×#4111yescharacteristic
314C3×Q8—22no—
315C3×Q8—21no—
316C3×Q8—21no—
317C3×Q8—22no—
318C3×Q8—41no—
319C3×Q8—41no—
320C3×Q8—42no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×Q8—42no—
324C3×D4—21no—
325C3×D4—21no—
326C3×D4—41no—
327C3×D4—41no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—81no—
344C2×C3×C4—81no—
345C2×C3×C4#12111yescharacteristic
346C2×C3×C4#12111yescharacteristic
347C2×C3×C4#12111yescharacteristic
348C2×C3×C4#12111yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4H(5,𝔽2)11yescharacteristic
354C2×C3×C4H(5,𝔽2)11yescharacteristic
355C3×C8—22no—
356C3×C8—22no—
357C3×C8—22no—
358C3×C8—41no—
359C3×C8—42no—
360C3×C8—42no—
361#41—21no—
362#41—21no—
363#41—21no—
364#41—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8C2×C3×D411yescharacteristic
377C2×Q8C3×#4112yes—
378C2×D4—21no—
379C2×D4C2×C3×D411yescharacteristic
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4C3×#4111yescharacteristic
387C22×C4C3×#4111yescharacteristic
388C22×C4C24×C311yescharacteristic
389Q16—24no—
390Q16—44no—
391Q16—44no—
392Q16—44no—
393SD16—24no—
394SD16—44no—
395M16—21no—
396M16—22no—
397M16—21no—
398M16—22no—
399M16—22no—
400M16—21no—
401M16—22no—
402M16C3×#4112yes—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8C2×C3×D412yes—
410C2×C8C2×C3×D412yes—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4C2×C3×D411yescharacteristic
422C4⋊C4C2×C3×D411yescharacteristic
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—41no—
430C42—41no—
431C42C2×C3×D411yescharacteristic
432C42C2×C3×D411yescharacteristic
433C42C2×C3×D411yescharacteristic
434C42C2×C3×D411yescharacteristic
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3C2×#12111yescharacteristic
441C22×C3#55311yescharacteristic
442C22×C3#55511yescharacteristic
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—41no—
447C3×C4—42no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—82no—
451C3×C4—81no—
452C3×C4#47312yes—
453C3×C4#52511yescharacteristic
454C3×C4#53511yescharacteristic
455C3×C4#54611yescharacteristic
456C3×C4#55011yescharacteristic
457C23—21no—
458C23C2×C3×#4111yescharacteristic
459Q8—22no—
460Q8—21no—
461Q8—21no—
462Q8—22no—
463Q8—41no—
464Q8—41no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469D4—21no—
470D4—21no—
471D4—41no—
472D4—41no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4C3×#12111yescharacteristic
491C2×C4C3×#12111yescharacteristic
492C2×C4C3×#12111yescharacteristic
493C2×C4C3×#12111yescharacteristic
494C2×C4C22×C3×D411yescharacteristic
495C2×C4C22×C3×D411yescharacteristic
496C2×C4C2×C3×#4111yescharacteristic
497C2×C4C2×C3×#4111yescharacteristic
498C2×C4C3×H(5,𝔽2)11yescharacteristic
499C2×C4C3×H(5,𝔽2)11yescharacteristic
500C8—22no—
501C8—22no—
502C8—22no—
503C8—42no—
504C8—41no—
505C8—42no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—81no—
509C2×C3#241411yescharacteristic
510C2×C3C2×#47311yescharacteristic
511C2×C3#306911yescharacteristic
512C22—21no—
513C22—21no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22C2×C3×#12111yescharacteristic
518C22C3×#55311yescharacteristic
519C22C3×#55511yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—41no—
524C4—42no—
525C4—41no—
526C4—41no—
527C4—82no—
528C4—81no—
529C4C3×#47312yes—
530C4C3×#52511yescharacteristic
531C4C3×#53511yescharacteristic
532C4C3×#54611yescharacteristic
533C4C3×#55011yescharacteristic
534C3#2467211yescharacteristic, direct product factor, Sylow
535C2—21no—
536C2—21no—
537C2—81no—
538C2C3×#241411yescharacteristic
539C2C2×C3×#47311yescharacteristic
540C2C3×#306911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#41C23
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×#462C2semidirect
7. C3×#1979C2semidirect
8. C3×#1966C2semidirect
9. C3×#1950C2semidirect
10. C3×#1930C2semidirect
11. C3×#1801C2semidirect
12. #24672C3direct (N × H)
51. C2×#462C2×C3semidirect
56. #1979C2×C3semidirect
57. #1966C2×C3semidirect
58. #1950C2×C3semidirect
59. #1930C2×C3semidirect
60. #1801C2×C3semidirect
534. C3#24672direct (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×#4622
C3×C4×#1372
C3×#29252
C3×#20522
C3×#20202
C3×#19792
C3×#19662
C3×#19502
C3×#19302
C3×#18012
#246723