Finite Group Databasebeta

← C3×#39366 #10657973

Subgroups — C3×#39366 #10657973

Order 768.

1994 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
1C2×C3×#544C212yesmaximal, complemented
2C3×#2862C211yesmaximal, characteristic, complemented
3C3×#2713C214yesmaximal, complemented
4C3×#2692C214yesmaximal
5C3×#2628C212yesmaximal, complemented
6C3×#2556C214yesmaximal, complemented
7C3×#2550C212yesmaximal, complemented
8C3×#2492C214yesmaximal, complemented
9C3×#2442C212yesmaximal, complemented
10C3×#2439C212yesmaximal, complemented
11C3×#2352C212yesmaximal
12C3×C4×#125C211yesmaximal, characteristic, complemented
13C3×#2293C211yesmaximal, characteristic
14#39366C311yesmaximal, characteristic, direct product factor, Sylow
15C3×#544—216no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2214yes—
19C2×C3×#125C2212yes—
20C2×C3×#125C2214yes—
21C2×C3×#125C2211yescharacteristic
22C2×C3×C22⋊Q8C2214yes—
23C2×C3×C22⋊Q8C2214yes—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117C2212yes—
26C2×C3×#117C2212yes—
27C22×C3×C4⋊C4C2212yes—
28C3×#400C2214yes—
29C3×#400C2214yes—
30C3×#399C2214yes—
31C3×#398C2214yes—
32C3×#398C2214yes—
33C3×#396C2214yes—
34C3×#395C2214yes—
35C3×#395C2214yes—
36C3×#393C2214yes—
37C3×#393C2214yes—
38C3×#391C2214yes—
39C3×#391C2214yes—
40C3×#391C2214yes—
41C3×#391C2214yes—
42C3×#390C2212yes—
43C3×#389C2212yes—
44C3×#389C2212yes—
45C3×#389C2212yes—
46C3×#389C2212yes—
47C3×#389C2214yes—
48C3×#388C2212yes—
49C3×#388C2214yes—
50C3×#388C2212yes—
51C3×#387C2214yes—
52C3×#387C2214yes—
53C3×#387C2214yes—
54C3×#386C2214yes—
55C3×#385C2214yes—
56C3×#382C2211yescharacteristic
57C3×#382C2211yescharacteristic
58C3×#382C2211yescharacteristic
59C3×#382C2212yes—
60C3×#380C2212yes—
61C3×#380C2212yes—
62C3×#380C2214yes—
63C3×C4×C4⋊C4C2214yes—
64C3×C4×C4⋊C4C2211yescharacteristic
65C3×C4×C4⋊C4C2212yes—
66C3×C4×C22⋊C4C2212yes—
67C3×C4×C22⋊C4C2214yes—
68C3×C43C2211yescharacteristic
69C2×#544C2×C312yescomplemented
70#2862C2×C311yescharacteristic, complemented
71#2713C2×C314yescomplemented
72#2692C2×C314yes—
73#2628C2×C312yescomplemented
74#2556C2×C314yescomplemented
75#2550C2×C312yescomplemented
76#2492C2×C314yescomplemented
77#2442C2×C312yescomplemented
78#2439C2×C312yescomplemented
79#2352C2×C312yes—
80C4×#125C2×C311yescharacteristic, complemented
81#2293C2×C311yescharacteristic
82C22×C3×Q8C2314yes—
83C22×C3×Q8C2314yes—
84C22×C3×D4C2311yescharacteristic
85C23×C3×C4C2312yes—
86C23×C3×C4C2314yes—
87C3×C4⋊Q8—216no—
88C3×C4⋊Q8—28no—
89C3×C4⋊Q8—28no—
90C3×#125—216no—
91C3×#125—28no—
92C3×#125D418yescomplemented
93C3×C22⋊Q8—216no—
94C3×C22⋊Q8—216no—
95C3×C4×D4Q818yescomplemented
96C3×#117—28no—
97C3×#117—28no—
98C2×C3×C4⋊C4—28no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2314yes—
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C4⋊C4C2314yes—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C22⋊C4C2314yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C22⋊C4C2314yes—
121C2×C3×C22⋊C4C2312yes—
122C2×C3×C22⋊C4C2314yes—
123C2×C3×C22⋊C4C2312yes—
124C2×C3×C42C2311yescharacteristic
125C2×C3×C42C2312yes—
126C2×C3×C42C2314yes—
127C2×C3×C42C2312yes—
128C2×C3×C42C2312yes—
129C2×C3×C42C2314yes—
130C2×C3×C42C2311yescharacteristic
131C2×C3×C42C2311yescharacteristic
132C3×#95C2314yes—
133C3×#95C2314yes—
134C3×#95C2312yes—
135C3×#95C2312yes—
136C3×#95C2314yes—
137C3×#95C2314yes—
138C3×#95C2314yes—
139C3×#95C2311yescharacteristic
140C3×#95C2311yescharacteristic
141C3×#95C2314yes—
142C3×#95C2314yes—
143C3×#95C2314yes—
144C3×#95C2314yes—
145C3×#95C2314yes—
146C3×#95C2314yes—
147#544—216no—
148C2×C4⋊Q8C22×C312yes—
149C2×C4⋊Q8C22×C312yes—
150C2×C4⋊Q8C22×C314yes—
151C2×#125C22×C312yes—
152C2×#125C22×C314yes—
153C2×#125C22×C311yescharacteristic
154C2×C22⋊Q8C22×C314yes—
155C2×C22⋊Q8C22×C314yes—
156C2×C4×D4C22×C311yescharacteristic
157C2×#117C22×C312yes—
158C2×#117C22×C312yes—
159C22×C4⋊C4C22×C312yes—
160#400C22×C314yes—
161#400C22×C314yes—
162#399C22×C314yes—
163#398C22×C314yes—
164#398C22×C314yes—
165#396C22×C314yes—
166#395C22×C314yes—
167#395C22×C314yes—
168#393C22×C314yes—
169#393C22×C314yes—
170#391C22×C314yes—
171#391C22×C314yes—
172#391C22×C314yes—
173#391C22×C314yes—
174#390C22×C312yes—
175#389C22×C312yes—
176#389C22×C312yes—
177#389C22×C312yes—
178#389C22×C312yes—
179#389C22×C314yes—
180#388C22×C314yes—
181#388C22×C312yes—
182#388C22×C312yes—
183#387C22×C314yes—
184#387C22×C314yes—
185#387C22×C314yes—
186#386C22×C314yes—
187#385C22×C314yes—
188#382C22×C311yescharacteristic
189#382C22×C312yes—
190#382C22×C311yescharacteristic
191#382C22×C311yescharacteristic
192#380C22×C312yes—
193#380C22×C312yes—
194#380C22×C314yes—
195C4×C4⋊C4C22×C312yes—
196C4×C4⋊C4C22×C314yes—
197C4×C4⋊C4C22×C311yescharacteristic
198C4×C22⋊C4C22×C314yes—
199C4×C22⋊C4C22×C312yes—
200C43C22×C311yescharacteristic
201C24×C3C2412yes—
202C2×C3×Q8—28no—
203C2×C3×Q8—28no—
204C2×C3×Q8—44no—
205C2×C3×Q8—44no—
206C2×C3×Q8—44no—
207C2×C3×Q8—44no—
208C2×C3×D4—42no—
209C2×C3×D4C2×Q814yes—
210C22×C3×C4—24no—
211C22×C3×C4—24no—
212C22×C3×C4—28no—
213C22×C3×C4—44no—
214C22×C3×C4—44no—
215C22×C3×C4C2×Q818yes—
216C22×C3×C4C2412yes—
217C22×C3×C4C2414yes—
218C22×C3×C4C2414yes—
219C22×C3×C4C2412yes—
220C22×C3×C4C2414yes—
221C22×C3×C4C2412yes—
222C22×C3×C4C2412yes—
223C22×C3×C4C2414yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2412yes—
227C22×C3×C4C2411yescharacteristic
228C3×C4⋊C4—28no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—28no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—28no—
236C3×C4⋊C4—28no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—44no—
240C3×C4⋊C4—44no—
241C3×C4⋊C4—44no—
242C3×C4⋊C4C2×D418yes—
243C3×C4⋊C4C2×D4116yes—
244C3×C4⋊C4C2×Q814yes—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—28no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—44no—
251C3×C22⋊C4—44no—
252C3×C22⋊C4C2×Q818yes—
253C3×C42—28no—
254C3×C42—24no—
255C3×C42—22no—
256C3×C42—44no—
257C3×C42—42no—
258C3×C42—42no—
259C3×C42C2×D414yes—
260C3×C42C2×Q814yes—
261C22×Q8C23×C314yes—
262C22×Q8C23×C314yes—
263C22×D4C23×C311yescharacteristic
264C23×C4C23×C312yes—
265C23×C4C23×C314yes—
266C4⋊Q8—216no—
267C4⋊Q8—28no—
268C4⋊Q8—28no—
269#125—216no—
270#125—28no—
271#125C3×D418yescomplemented
272C22⋊Q8—216no—
273C22⋊Q8—216no—
274C4×D4C3×Q818yescomplemented
275#117—28no—
276#117—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4C23×C312yes—
279C2×C4⋊C4C23×C314yes—
280C2×C4⋊C4C23×C312yes—
281C2×C4⋊C4C23×C312yes—
282C2×C4⋊C4C23×C312yes—
283C2×C4⋊C4C23×C314yes—
284C2×C4⋊C4C23×C314yes—
285C2×C4⋊C4C23×C314yes—
286C2×C4⋊C4C23×C314yes—
287C2×C4⋊C4C23×C312yes—
288C2×C4⋊C4C23×C314yes—
289C2×C4⋊C4C23×C312yes—
290C2×C4⋊C4C23×C314yes—
291C2×C4⋊C4C23×C314yes—
292C2×C4⋊C4C23×C312yes—
293C2×C4⋊C4C23×C314yes—
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C22⋊C4C23×C314yes—
296C2×C22⋊C4C23×C312yes—
297C2×C22⋊C4C23×C312yes—
298C2×C22⋊C4C23×C312yes—
299C2×C22⋊C4C23×C314yes—
300C2×C22⋊C4C23×C312yes—
301C2×C22⋊C4C23×C314yes—
302C2×C22⋊C4C23×C312yes—
303C2×C42C23×C312yes—
304C2×C42C23×C314yes—
305C2×C42C23×C311yescharacteristic
306C2×C42C23×C312yes—
307C2×C42C23×C312yes—
308C2×C42C23×C314yes—
309C2×C42C23×C311yescharacteristic
310C2×C42C23×C311yescharacteristic
311#95C23×C314yes—
312#95C23×C314yes—
313#95C23×C312yes—
314#95C23×C312yes—
315#95C23×C314yes—
316#95C23×C314yes—
317#95C23×C314yes—
318#95C23×C311yescharacteristic
319#95C23×C311yescharacteristic
320#95C23×C314yes—
321#95C23×C314yes—
322#95C23×C314yes—
323#95C23×C314yes—
324#95C23×C314yes—
325#95C23×C314yes—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3C22×Q814yes—
333C23×C3C2511yescharacteristic
334C3×Q8—88no—
335C3×Q8—88no—
336C3×D4—44no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—216no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—88no—
365C2×C3×C4C22×D418yes—
366C2×C3×C4C22×D414yes—
367C2×C3×C4C22×D412yes—
368C2×C3×C4C22×Q812yes—
369C2×C3×C4C22×Q812yes—
370C2×C3×C4C22×Q814yes—
371C2×C3×C4C22×Q812yes—
372C2×C3×C42−1+412yes—
373C24C24×C312yes—
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×D4—42no—
381C2×D4C2×C3×Q814yes—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4C2×C3×Q818yes—
388C22×C4C24×C312yes—
389C22×C4C24×C314yes—
390C22×C4C24×C314yes—
391C22×C4C24×C312yes—
392C22×C4C24×C314yes—
393C22×C4C24×C312yes—
394C22×C4C24×C312yes—
395C22×C4C24×C314yes—
396C22×C4C24×C311yescharacteristic
397C22×C4C24×C311yescharacteristic
398C22×C4C24×C312yes—
399C22×C4C24×C311yescharacteristic
400C4⋊C4—28no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—28no—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4C2×C3×D418yes—
415C4⋊C4C2×C3×D4116yes—
416C4⋊C4C2×C3×Q814yes—
417C22⋊C4—24no—
418C22⋊C4—28no—
419C22⋊C4—24no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4C2×C3×Q818yes—
425C42—28no—
426C42—24no—
427C42—22no—
428C42—44no—
429C42—42no—
430C42—42no—
431C42C2×C3×D414yes—
432C42C2×C3×Q814yes—
433C22×C3—24no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3C23×D411yescharacteristic
441C22×C3C23×Q811yescharacteristic
442C22×C3C2×2−1+411yescharacteristic
443C22×C3#58512yes—
444C22×C3#58512yes—
445C3×C4—22no—
446C3×C4—24no—
447C3×C4—28no—
448C3×C4—24no—
449C3×C4—22no—
450C3×C4—42no—
451C3×C4—44no—
452C3×C4—44no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—44no—
456C3×C4D4×Q814yes—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23C22×C3×Q814yes—
464C23C25×C311yescharacteristic
465Q8—88no—
466Q8—88no—
467D4—44no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—216no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—88no—
496C2×C4C22×C3×D418yes—
497C2×C4C22×C3×D414yes—
498C2×C4C22×C3×D412yes—
499C2×C4C22×C3×Q812yes—
500C2×C4C22×C3×Q812yes—
501C2×C4C22×C3×Q814yes—
502C2×C4C22×C3×Q812yes—
503C2×C4C3×2−1+412yes—
504C2×C3—44no—
505C2×C3#345811yescharacteristic
506C2×C3C2×D4×Q811yescharacteristic
507C2×C3#350012yes—
508C2×C3#350411yescharacteristic
509C2×C3#356512yes—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C23×C3×D411yescharacteristic
518C22C23×C3×Q811yescharacteristic
519C22C2×C3×2−1+411yescharacteristic
520C22C3×#58512yes—
521C22C3×#58512yes—
522C4—22no—
523C4—24no—
524C4—28no—
525C4—24no—
526C4—22no—
527C4—44no—
528C4—44no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—44no—
533C4C3×D4×Q814yes—
534C3#3936611yescharacteristic, direct product factor, Sylow
535C2—44no—
536C2C3×#345811yescharacteristic
537C2C2×C3×D4×Q811yescharacteristic
538C2C3×#350012yes—
539C2C3×#350411yescharacteristic
540C2C3×#356512yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC22×C3×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#544C2semidirect
2. C3×#2862C2semidirect
3. C3×#2713C2semidirect
5. C3×#2628C2semidirect
6. C3×#2556C2semidirect
7. C3×#2550C2semidirect
8. C3×#2492C2semidirect
9. C3×#2442C2semidirect
10. C3×#2439C2semidirect
12. C3×C4×#125C2semidirect
14. #39366C3direct (N × H)
69. C2×#544C2×C3semidirect
70. #2862C2×C3semidirect
71. #2713C2×C3semidirect
73. #2628C2×C3semidirect
74. #2556C2×C3semidirect
75. #2550C2×C3semidirect
76. #2492C2×C3semidirect
77. #2442C2×C3semidirect
78. #2439C2×C3semidirect
80. C4×#125C2×C3semidirect
92. C3×#125D4semidirect
95. C3×C4×D4Q8semidirect
271. #125C3×D4semidirect
274. C4×D4C3×Q8semidirect
534. C3#39366direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C3×#28622
C3×#27132
C3×#26922
C3×#26282
C3×#25562
C3×#25502
C3×#24922
C3×#24422
C3×#24392
C3×#23522
C3×C4×#1252
C3×#22932
#393663