Finite Group Databasebeta

← C3×#37267 #10655874

Subgroups — C3×#37267 #10655874

Order 768.

1724 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
1C3×#2861C212yesmaximal, complemented
2C3×#2811C211yesmaximal, characteristic, complemented
3C3×#2804C212yesmaximal, complemented
4C3×#2733C211yesmaximal, characteristic
5C3×#2731C212yesmaximal, complemented
6C3×#2720C214yesmaximal, complemented
7C3×#2710C214yesmaximal, complemented
8C3×#2680C212yesmaximal, complemented
9C3×#2632C212yesmaximal, complemented
10C3×#2532C214yesmaximal, complemented
11C3×#2461C214yesmaximal, complemented
12C3×#2326C212yesmaximal, complemented
13C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
14#37267C311yesmaximal, characteristic, direct product factor, Sylow
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2212yes—
17C2×C3×#124C2212yes—
18C2×C3×#124C2212yes—
19C2×C3×#124C2212yes—
20C2×C3×#123C2214yes—
21C2×C3×C22⋊Q8C2214yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C3×#400C2214yes—
26C3×#399C2214yes—
27C3×#398C2214yes—
28C3×#397C2214yes—
29C3×#396C2214yes—
30C3×#395C2212yes—
31C3×#395C2212yes—
32C3×#395C2214yes—
33C3×#394C2212yes—
34C3×#394C2212yes—
35C3×#394C2214yes—
36C3×#393C2214yes—
37C3×#393C2214yes—
38C3×#392C2214yes—
39C3×#391C2214yes—
40C3×#391C2212yes—
41C3×#391C2211yescharacteristic
42C3×#391C2212yes—
43C3×#391C2211yescharacteristic
44C3×#391C2212yes—
45C3×#390C2212yes—
46C3×#390C2212yes—
47C3×#390C2212yes—
48C3×#389C2214yes—
49C3×#388C2214yes—
50C3×#388C2214yes—
51C3×#388C2214yes—
52C3×#388C2214yes—
53C3×#388C2214yes—
54C3×#388C2214yes—
55C3×#388C2214yes—
56C3×#388C2214yes—
57C3×#387C2214yes—
58C3×#386C2214yes—
59C3×#385C2214yes—
60C3×#385C2214yes—
61C3×#385C2214yes—
62C3×#384C2211yescharacteristic
63C3×#384C2212yes—
64C3×#383C2212yes—
65C3×C4×C4⋊C4C2214yes—
66C3×#376C2212yes—
67C3×C43C2211yescharacteristic
68#2861C2×C312yescomplemented
69#2811C2×C311yescharacteristic, complemented
70#2804C2×C312yescomplemented
71#2733C2×C311yescharacteristic
72#2731C2×C312yescomplemented
73#2720C2×C314yescomplemented
74#2710C2×C314yescomplemented
75#2680C2×C312yescomplemented
76#2632C2×C312yescomplemented
77#2532C2×C314yescomplemented
78#2461C2×C314yescomplemented
79#2326C2×C312yescomplemented
80C4×C4⋊Q8C2×C311yescharacteristic, complemented
81C22×C3×Q8C2312yes—
82C22×C3×Q8C2311yescharacteristic
83C22×C3×Q8C2312yes—
84C22×C3×Q8C2311yescharacteristic
85C22×C3×D4C2312yes—
86C22×C3×D4C2312yes—
87C22×C3×D4C2312yes—
88C23×C3×C4C2314yes—
89C3×C4⋊Q8—24no—
90C3×#125—44no—
91C3×#124—28no—
92C3×#124—44no—
93C3×#124—44no—
94C3×#123—48no—
95C3×C22⋊Q8—48no—
96C3×C4×Q8—42no—
97C3×C4×Q8—42no—
98C3×C4×D4—44no—
99C2×C3×C4⋊C4C2314yes—
100C2×C3×C4⋊C4C2314yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2314yes—
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C22⋊C4C2314yes—
113C2×C3×C22⋊C4C2314yes—
114C2×C3×C22⋊C4C2314yes—
115C2×C3×C22⋊C4C2314yes—
116C2×C3×C22⋊C4C2314yes—
117C2×C3×C22⋊C4C2314yes—
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C22⋊C4C2314yes—
120C2×C3×C22⋊C4C2314yes—
121C2×C3×C22⋊C4C2314yes—
122C2×C3×C22⋊C4C2314yes—
123C2×C3×C42C2312yes—
124C2×C3×C42C2311yescharacteristic
125C2×C3×C42C2311yescharacteristic
126C2×C3×C42C2312yes—
127C2×C3×C42C2312yes—
128C2×C3×C42C2312yes—
129C2×C3×C42C2312yes—
130C2×C3×C42C2311yescharacteristic
131C2×C3×C42C2312yes—
132C3×#95C2314yes—
133C3×#95C2314yes—
134C3×#95C2314yes—
135C3×#95C2314yes—
136C3×#95C2314yes—
137C3×#95C2312yes—
138C3×#95C2312yes—
139C3×#95C2314yes—
140C3×#95C2314yes—
141C3×#95C2314yes—
142C3×#95C2314yes—
143C3×#95C2314yes—
144C2×C4⋊Q8C22×C311yescharacteristic
145C2×#125C22×C312yes—
146C2×#124C22×C312yes—
147C2×#124C22×C312yes—
148C2×#124C22×C312yes—
149C2×#123C22×C314yes—
150C2×C22⋊Q8C22×C314yes—
151C2×C4×Q8C22×C311yescharacteristic
152C2×C4×Q8C22×C311yescharacteristic
153C2×C4×D4C22×C312yes—
154#400C22×C314yes—
155#399C22×C314yes—
156#398C22×C314yes—
157#397C22×C314yes—
158#396C22×C314yes—
159#395C22×C312yes—
160#395C22×C312yes—
161#395C22×C314yes—
162#394C22×C312yes—
163#394C22×C312yes—
164#394C22×C314yes—
165#393C22×C314yes—
166#393C22×C314yes—
167#392C22×C314yes—
168#391C22×C311yescharacteristic
169#391C22×C311yescharacteristic
170#391C22×C312yes—
171#391C22×C312yes—
172#391C22×C312yes—
173#391C22×C314yes—
174#390C22×C312yes—
175#390C22×C312yes—
176#390C22×C312yes—
177#389C22×C314yes—
178#388C22×C314yes—
179#388C22×C314yes—
180#388C22×C314yes—
181#388C22×C314yes—
182#388C22×C314yes—
183#388C22×C314yes—
184#388C22×C314yes—
185#388C22×C314yes—
186#387C22×C314yes—
187#386C22×C314yes—
188#385C22×C314yes—
189#385C22×C314yes—
190#385C22×C314yes—
191#384C22×C311yescharacteristic
192#384C22×C312yes—
193#383C22×C312yes—
194C4×C4⋊C4C22×C314yes—
195#376C22×C312yes—
196C43C22×C311yescharacteristic
197C24×C3C2414yes—
198C2×C3×Q8—22no—
199C2×C3×Q8—24no—
200C2×C3×Q8—41no—
201C2×C3×Q8—41no—
202C2×C3×Q8—41no—
203C2×C3×Q8—41no—
204C2×C3×Q8—44no—
205C2×C3×Q8—42no—
206C2×C3×Q8—42no—
207C2×C3×Q8—42no—
208C2×C3×Q8#4114yes—
209C2×C3×D4—24no—
210C2×C3×D4—24no—
211C2×C3×D4—42no—
212C2×C3×D4—42no—
213C2×C3×D4—44no—
214C2×C3×D4—42no—
215C2×C3×D4—42no—
216C2×C3×D4—42no—
217C22×C3×C4—44no—
218C22×C3×C4—44no—
219C22×C3×C4—44no—
220C22×C3×C4C2414yes—
221C22×C3×C4C2414yes—
222C22×C3×C4C2412yes—
223C22×C3×C4C2412yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2414yes—
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2412yes—
231C22×C3×C4C2412yes—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—28no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—41no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—44no—
241C3×C4⋊C4—44no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—44no—
245C3×C22⋊C4—28no—
246C3×C22⋊C4—28no—
247C3×C22⋊C4—28no—
248C3×C22⋊C4—28no—
249C3×C22⋊C4—28no—
250C3×C22⋊C4—28no—
251C3×C22⋊C4—44no—
252C3×C22⋊C4—44no—
253C3×C22⋊C4—44no—
254C3×C22⋊C4—44no—
255C3×C22⋊C4—44no—
256C3×C42—24no—
257C3×C42—22no—
258C3×C42—24no—
259C3×C42—22no—
260C3×C42—24no—
261C3×C42—24no—
262C3×C42—24no—
263C3×C42—41no—
264C3×C42—42no—
265C22×Q8C23×C311yescharacteristic
266C22×Q8C23×C312yes—
267C22×Q8C23×C312yes—
268C22×Q8C23×C311yescharacteristic
269C22×D4C23×C312yes—
270C22×D4C23×C312yes—
271C22×D4C23×C312yes—
272C23×C4C23×C314yes—
273C4⋊Q8—24no—
274#125—44no—
275#124—28no—
276#124—44no—
277#124—44no—
278#123—48no—
279C22⋊Q8—48no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×D4—44no—
283C2×C4⋊C4C23×C314yes—
284C2×C4⋊C4C23×C314yes—
285C2×C4⋊C4C23×C314yes—
286C2×C4⋊C4C23×C312yes—
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C312yes—
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C312yes—
291C2×C4⋊C4C23×C314yes—
292C2×C4⋊C4C23×C314yes—
293C2×C4⋊C4C23×C314yes—
294C2×C4⋊C4C23×C312yes—
295C2×C4⋊C4C23×C312yes—
296C2×C22⋊C4C23×C314yes—
297C2×C22⋊C4C23×C314yes—
298C2×C22⋊C4C23×C314yes—
299C2×C22⋊C4C23×C314yes—
300C2×C22⋊C4C23×C314yes—
301C2×C22⋊C4C23×C314yes—
302C2×C22⋊C4C23×C314yes—
303C2×C22⋊C4C23×C314yes—
304C2×C22⋊C4C23×C314yes—
305C2×C22⋊C4C23×C314yes—
306C2×C22⋊C4C23×C314yes—
307C2×C42C23×C312yes—
308C2×C42C23×C311yescharacteristic
309C2×C42C23×C312yes—
310C2×C42C23×C312yes—
311C2×C42C23×C312yes—
312C2×C42C23×C311yescharacteristic
313C2×C42C23×C312yes—
314C2×C42C23×C311yescharacteristic
315C2×C42C23×C312yes—
316#95C23×C314yes—
317#95C23×C314yes—
318#95C23×C314yes—
319#95C23×C314yes—
320#95C23×C314yes—
321#95C23×C312yes—
322#95C23×C312yes—
323#95C23×C314yes—
324#95C23×C314yes—
325#95C23×C314yes—
326#95C23×C314yes—
327#95C23×C314yes—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3—24no—
331C23×C3—24no—
332C23×C3—24no—
333C23×C3—24no—
334C23×C3—24no—
335C23×C3C2511yescharacteristic
336C3×Q8—44no—
337C3×Q8—44no—
338C3×Q8—84no—
339C3×Q8—84no—
340C3×D4—48no—
341C3×D4—84no—
342C3×D4—84no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—88no—
371C2×C3×C4C2×#4112yes—
372C2×C3×C4C2×#4114yes—
373C2×C3×C4H(5,𝔽2)12yes—
374C2×C3×C4H(5,𝔽2)14yes—
375C2×C3×C42−1+414yes—
376C24C24×C314yes—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8C3×#4114yes—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4C24×C314yes—
400C22×C4C24×C314yes—
401C22×C4C24×C312yes—
402C22×C4C24×C312yes—
403C22×C4C24×C312yes—
404C22×C4C24×C314yes—
405C22×C4C24×C311yescharacteristic
406C22×C4C24×C311yescharacteristic
407C22×C4C24×C312yes—
408C22×C4C24×C311yescharacteristic
409C22×C4C24×C312yes—
410C22×C4C24×C312yes—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—28no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C42—24no—
436C42—22no—
437C42—24no—
438C42—22no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—41no—
443C42—42no—
444C22×C3—44no—
445C22×C3—44no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3C22×#4111yescharacteristic
452C22×C3C2×H(5,𝔽2)11yescharacteristic
453C22×C3C2×H(5,𝔽2)12yes—
454C22×C3C2×2−1+412yes—
455C22×C3#58511yescharacteristic
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—24no—
459C3×C4—22no—
460C3×C4—24no—
461C3×C4—42no—
462C3×C4—44no—
463C3×C4—41no—
464C3×C4—42no—
465C3×C4—44no—
466C3×C4—42no—
467C3×C4—44no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23C25×C311yescharacteristic
476Q8—44no—
477Q8—44no—
478Q8—84no—
479Q8—84no—
480D4—48no—
481D4—84no—
482D4—84no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—88no—
511C2×C4C2×C3×#4112yes—
512C2×C4C2×C3×#4114yes—
513C2×C4C3×H(5,𝔽2)12yes—
514C2×C4C3×H(5,𝔽2)14yes—
515C2×C4C3×2−1+414yes—
516C2×C3—84no—
517C2×C3#345911yescharacteristic
518C2×C3#352212yes—
519C2×C3#354411yescharacteristic
520C2×C3#354911yescharacteristic
521C2×C3#356112yes—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22C22×C3×#4111yescharacteristic
530C22C2×C3×H(5,𝔽2)11yescharacteristic
531C22C2×C3×H(5,𝔽2)12yes—
532C22C2×C3×2−1+412yes—
533C22C3×#58511yescharacteristic
534C4—24no—
535C4—22no—
536C4—24no—
537C4—22no—
538C4—24no—
539C4—42no—
540C4—44no—
541C4—42no—
542C4—44no—
543C4—42no—
544C4—41no—
545C4—44no—
546C3#3726711yescharacteristic, direct product factor, Sylow
547C2—84no—
548C2C3×#345911yescharacteristic
549C2C3×#352212yes—
550C2C3×#354411yescharacteristic
551C2C3×#354911yescharacteristic
552C2C3×#356112yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC3×#2733C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2861C2semidirect
2. C3×#2811C2semidirect
3. C3×#2804C2semidirect
5. C3×#2731C2semidirect
6. C3×#2720C2semidirect
7. C3×#2710C2semidirect
8. C3×#2680C2semidirect
9. C3×#2632C2semidirect
10. C3×#2532C2semidirect
11. C3×#2461C2semidirect
12. C3×#2326C2semidirect
13. C3×C4×C4⋊Q8C2semidirect
14. #37267C3direct (N × H)
68. #2861C2×C3semidirect
69. #2811C2×C3semidirect
70. #2804C2×C3semidirect
72. #2731C2×C3semidirect
73. #2720C2×C3semidirect
74. #2710C2×C3semidirect
75. #2680C2×C3semidirect
76. #2632C2×C3semidirect
77. #2532C2×C3semidirect
78. #2461C2×C3semidirect
79. #2326C2×C3semidirect
80. C4×C4⋊Q8C2×C3semidirect
546. C3#37267direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28612
C3×#28112
C3×#28042
C3×#27332
C3×#27312
C3×#27202
C3×#27102
C3×#26802
C3×#26322
C3×#25322
C3×#24612
C3×#23262
C3×C4×C4⋊Q82
#372673