Finite Group Databasebeta

← C3×#32920 #10651527

Subgroups — C3×#32920 #10651527

Order 768.

1590 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#3038C212yesmaximal, complemented
3C2×C3×#453C212yesmaximal, complemented
4C3×#2541C211yesmaximal, characteristic, complemented
5C3×#1887C212yesmaximal, complemented
6C3×#1883C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1800C211yesmaximal, characteristic, complemented
9#32920C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C2×C3×AGL(1,ℤ/8)C2214yes—
12C2×C3×#127C2212yes—
13C2×C3×#127C2211yescharacteristic, complemented
14C2×C3×#121—22no—
15C3×#520—22no—
16C3×#520C2212yes—
17C3×#518—22no—
18C3×#518—22no—
19C3×#518C2211yescharacteristic
20C3×C4×#41—22no—
21C3×C4×#41C2212yes—
22C2×C3×C4×D4—22no—
23C2×C3×C4×D4—22no—
24C2×C3×C4×D4C2211yescharacteristic
25C3×#460—22no—
26C3×#460—22no—
27C3×#459—22no—
28C3×#459—22no—
29C3×#453—28no—
30C3×#423C2212yescomplemented
31C2×C3×#104C2214yes—
32C3×#418C2214yescomplemented
33C2×C3×#100C2212yes—
34C3×#392—22no—
35C3×#390C2212yescomplemented
36C3×C4×C4⋊C4C2211yescharacteristic, complemented
37C3×#343C2212yescomplemented
38C3×#339C2214yescomplemented
39C4×H(5,𝔽2)C2×C311yescharacteristic, complemented
40#3038C2×C312yescomplemented
41C2×#453C2×C312yescomplemented
42#2541C2×C311yescharacteristic, complemented
43#1887C2×C312yescomplemented
44#1883C2×C314yescomplemented
45#1806C2×C312yescomplemented
46#1800C2×C311yescharacteristic, complemented
47C3×H(5,𝔽2)—24no—
48C3×H(5,𝔽2)—22no—
49C3×H(5,𝔽2)D414yescomplemented
50C2×C3×#41—22no—
51C2×C3×#41C2312yes—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4—24no—
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2312yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—22no—
60C23×C3×C4D412yes—
61C3×AGL(1,ℤ/8)—28no—
62C3×AGL(1,ℤ/8)—28no—
63C2×C3×SD16—24no—
64C2×C3×D8—24no—
65C2×C3×M16C2314yes—
66C3×#127—24no—
67C3×#127—24no—
68C3×#127—22no—
69C3×#127—22no—
70C3×#121—48no—
71C3×C4×Q8—22no—
72C3×C4×Q8D412yescomplemented
73C3×C4×Q8D412yescomplemented
74C3×C4×D4—24no—
75C3×C4×D4—24no—
76C3×C4×D4—24no—
77C3×C4×D4—24no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4—24no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4D412yescomplemented
87C3×C4×D4D412yescomplemented
88C3×#117—24no—
89C3×#117—22no—
90C3×#117—22no—
91C3×#117C2312yes—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—24no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—24no—
100C2×C3×C22⋊C4D412yes—
101C2×C3×C42—22no—
102C2×C3×C42—22no—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C3×C4⋊C8—22no—
106C3×C4⋊C8—22no—
107C3×#104—28no—
108C3×#102—24no—
109C3×#102—24no—
110C3×#100—24no—
111C3×#95—21no—
112C2×H(5,𝔽2)C22×C311yescharacteristic
113C2×AGL(1,ℤ/8)C22×C314yes—
114C2×#127C22×C312yes—
115C2×#127C22×C311yescharacteristic, complemented
116C2×#121—22no—
117#520—22no—
118#520C22×C312yes—
119#518—22no—
120#518—22no—
121#518C22×C311yescharacteristic
122C4×#41—22no—
123C4×#41C22×C312yes—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4C22×C311yescharacteristic
127#460—22no—
128#460—22no—
129#459—22no—
130#459—22no—
131#453—28no—
132#423C22×C312yescomplemented
133C2×#104C22×C314yes—
134#418C22×C314yescomplemented
135C2×#100C22×C312yes—
136#392—22no—
137#390C22×C312yescomplemented
138C4×C4⋊C4C22×C311yescharacteristic, complemented
139#343C22×C312yescomplemented
140#339C22×C314yescomplemented
141C24×C3—22no—
142C24×C3—22no—
143C24×C3C2×D412yes—
144C3×#41—28no—
145C3×#41—28no—
146C3×#41—22no—
147C3×#41—22no—
148C3×#41C2×D414yes—
149C3×#41#4114yes—
150C2×C3×Q8C2×D412yes—
151C2×C3×D4—24no—
152C2×C3×D4—28no—
153C2×C3×D4—28no—
154C2×C3×D4—28no—
155C2×C3×D4—22no—
156C2×C3×D4—28no—
157C2×C3×D4—22no—
158C2×C3×D4—24no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—24no—
163C2×C3×D4—44no—
164C2×C3×D4—48no—
165C2×C3×D4—48no—
166C2×C3×D4—42no—
167C2×C3×D4—41no—
168C2×C3×D4—41no—
169C2×C3×D4—82no—
170C2×C3×D4—84no—
171C2×C3×D4C2×D412yes—
172C2×C3×D4C2×D412yes—
173C2×C3×D4#4112yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—44no—
185C22×C3×C4—84no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C3×SD16—48no—
191C3×D8—48no—
192C3×M16—24no—
193C3×M16—24no—
194C2×C3×C8—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4C2×D412yes—
201C3×C4⋊C4C2×D412yes—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—48no—
209C3×C42—24no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42C2×D412yes—
216C3×C42C2×D412yes—
217H(5,𝔽2)—24no—
218H(5,𝔽2)—22no—
219H(5,𝔽2)C3×D414yescomplemented
220C2×#41—22no—
221C2×#41C23×C312yes—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4C23×C312yes—
228C22×D4C23×C311yescharacteristic
229C23×C4—22no—
230C23×C4C3×D412yes—
231AGL(1,ℤ/8)—28no—
232AGL(1,ℤ/8)—28no—
233C2×SD16—24no—
234C2×D8—24no—
235C2×M16C23×C314yes—
236#127—24no—
237#127—24no—
238#127—22no—
239#127—22no—
240#121—48no—
241C4×Q8—22no—
242C4×Q8C3×D412yescomplemented
243C4×Q8C3×D412yescomplemented
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—24no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4C3×D412yescomplemented
257C4×D4C3×D412yescomplemented
258#117—24no—
259#117—22no—
260#117—22no—
261#117C23×C312yes—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C23×C311yescharacteristic
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4C3×D412yes—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C23×C312yes—
274C2×C42C23×C311yescharacteristic
275C4⋊C8—22no—
276C4⋊C8—22no—
277#104—28no—
278#102—24no—
279#102—24no—
280#100—24no—
281#95—21no—
282C23×C3—28no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—44no—
289C23×C3—44no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—82no—
294C23×C3—82no—
295C23×C3#12114yes—
296C23×C3C22×D411yescharacteristic
297C3×Q8—22no—
298C3×Q8#12114yescomplemented
299C3×D4—216no—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—24no—
303C3×D4—22no—
304C3×D4—24no—
305C3×D4—44no—
306C3×D4—44no—
307C3×D4—82no—
308C3×D4—84no—
309C3×D4—816no—
310C3×D4—82no—
311C3×D4#12114yescomplemented
312C2×C3×C4—28no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—44no—
325C2×C3×C4—44no—
326C2×C3×C4—42no—
327C2×C3×C4—82no—
328C2×C3×C4—82no—
329C2×C3×C4—82no—
330C2×C3×C4—84no—
331C2×C3×C4—82no—
332C2×C3×C4#12012yes—
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D412yes—
337C2×C3×C4C2×#4112yes—
338C2×C3×C4C2×#4111yescharacteristic
339C3×C8—44no—
340C24—22no—
341C24—22no—
342C24C2×C3×D412yes—
343#41—28no—
344#41—28no—
345#41—22no—
346#41—22no—
347#41C2×C3×D414yes—
348#41C3×#4114yes—
349C2×Q8C2×C3×D412yes—
350C2×D4—24no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—22no—
355C2×D4—28no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—82no—
369C2×D4—84no—
370C2×D4C2×C3×D412yes—
371C2×D4C2×C3×D412yes—
372C2×D4C3×#4112yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—44no—
384C22×C4—84no—
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D412yes—
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C24×C311yescharacteristic
389SD16—48no—
390D8—48no—
391M16—24no—
392M16—24no—
393C2×C8—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4C2×C3×D412yes—
400C4⋊C4C2×C3×D412yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—24no—
407C22⋊C4—48no—
408C42—24no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42C2×C3×D412yes—
415C42C2×C3×D412yes—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—44no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—82no—
426C22×C3—84no—
427C22×C3—84no—
428C22×C3—88no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3C2×#12011yescharacteristic
433C22×C3C2×#12111yescharacteristic
434C22×C3#52511yescharacteristic
435C3×C4—24no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—44no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4#45312yescomplemented
443C3×C4C2×#12112yes—
444C3×C4#52512yes—
445C23—28no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—82no—
457C23—82no—
458C23C3×#12114yes—
459C23C22×C3×D411yescharacteristic
460Q8—22no—
461Q8C3×#12114yescomplemented
462D4—216no—
463D4—24no—
464D4—24no—
465D4—24no—
466D4—22no—
467D4—24no—
468D4—44no—
469D4—44no—
470D4—82no—
471D4—84no—
472D4—816no—
473D4—82no—
474D4C3×#12114yescomplemented
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4C3×#12012yes—
496C2×C4C3×#12011yescharacteristic
497C2×C4C3×#12011yescharacteristic
498C2×C4C22×C3×D411yescharacteristic
499C2×C4C22×C3×D412yes—
500C2×C4C2×C3×#4112yes—
501C2×C4C2×C3×#4111yescharacteristic
502C8—44no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—44no—
506C2×C3—44no—
507C2×C3—162no—
508C2×C3#284711yescharacteristic
509C2×C3C2×#45311yescharacteristic
510C2×C3#302111yescharacteristic
511C22—21no—
512C22—21no—
513C22—24no—
514C22—24no—
515C22—44no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—82no—
521C22—84no—
522C22—84no—
523C22—88no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22C2×C3×#12011yescharacteristic
528C22C2×C3×#12111yescharacteristic
529C22C3×#52511yescharacteristic
530C4—21no—
531C4—22no—
532C4—24no—
533C4—44no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4C3×#45312yescomplemented
538C4C2×C3×#12112yes—
539C4C3×#52512yes—
540C3#3292011yescharacteristic, direct product factor, Sylow
541C2—21no—
542C2—21no—
543C2—44no—
544C2—44no—
545C2—162no—
546C2C3×#284711yescharacteristic
547C2C2×C3×#45311yescharacteristic
548C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
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×C4×H(5,𝔽2)C2semidirect
2. C3×#3038C2semidirect
3. C2×C3×#453C2semidirect
4. C3×#2541C2semidirect
5. C3×#1887C2semidirect
6. C3×#1883C2semidirect
7. C3×#1806C2semidirect
8. C3×#1800C2semidirect
9. #32920C3direct (N × H)
13. C2×C3×#127C22semidirect
30. C3×#423C22semidirect
32. C3×#418C22semidirect
35. C3×#390C22semidirect
36. C3×C4×C4⋊C4C22semidirect
37. C3×#343C22semidirect
38. C3×#339C22semidirect
39. C4×H(5,𝔽2)C2×C3semidirect
40. #3038C2×C3semidirect
41. C2×#453C2×C3semidirect
42. #2541C2×C3semidirect
43. #1887C2×C3semidirect
44. #1883C2×C3semidirect
45. #1806C2×C3semidirect
46. #1800C2×C3semidirect
49. C3×H(5,𝔽2)D4semidirect
72. C3×C4×Q8D4semidirect
73. C3×C4×Q8D4semidirect
86. C3×C4×D4D4semidirect
87. C3×C4×D4D4semidirect
115. C2×#127C22×C3semidirect
132. #423C22×C3semidirect
134. #418C22×C3semidirect
137. #390C22×C3semidirect
138. C4×C4⋊C4C22×C3semidirect
139. #343C22×C3semidirect
140. #339C22×C3semidirect
219. H(5,𝔽2)C3×D4semidirect
242. C4×Q8C3×D4semidirect
243. C4×Q8C3×D4semidirect
256. C4×D4C3×D4semidirect
257. C4×D4C3×D4semidirect
298. C3×Q8#121semidirect
311. C3×D4#121semidirect
442. C3×C4#453semidirect
461. Q8C3×#121semidirect
474. D4C3×#121semidirect
537. C4C3×#453semidirect
540. C3#32920direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C3×#30382
C2×C3×#4532
C3×#25412
C3×#18872
C3×#18832
C3×#18062
C3×#18002
#329203