Finite Group Databasebeta

← C3×#32956 #10651563

Subgroups — C3×#32956 #10651563

Order 768.

1764 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×#476C212yesmaximal, complemented
3C3×#2857C211yesmaximal, characteristic, complemented
4C3×#1864C214yesmaximal, complemented
5C3×#1853C212yesmaximal, complemented
6C3×#1802C212yesmaximal, complemented
7C3×#1797C211yesmaximal, characteristic, complemented
8C3×#1786C212yesmaximal
9#32956C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4—21no—
12C2×C3×C4⋊Q8C2211yescharacteristic, complemented
13C2×C3×C22⋊Q8—21no—
14C2×C3×C4×D4—28no—
15C2×C3×C4×D4—21no—
16C2×C3×C4×D4—21no—
17C2×C3×C4×D4C2212yes—
18C2×C3×C4×D4C2212yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4—21no—
21C22×C3×C42C2211yescharacteristic
22C3×#476—28no—
23C2×C3×#106C2214yescomplemented
24C2×C3×C4⋊C8C2212yescomplemented
25C2×C3×#102C2214yes—
26C2×C3×#102C2214yes—
27C2×C3×C22⋊C8C2212yes—
28C3×#391—21no—
29C3×#389—21no—
30C3×#389C2212yescomplemented
31C3×#384C2211yescharacteristic, complemented
32C3×#380—21no—
33C3×#380C2212yes—
34C3×#340C2214yescomplemented
35C3×#336C2212yescomplemented
36C22×C4×D4C2×C311yescharacteristic
37C2×#476C2×C312yescomplemented
38#2857C2×C311yescharacteristic, complemented
39#1864C2×C314yescomplemented
40#1853C2×C312yescomplemented
41#1802C2×C312yescomplemented
42#1797C2×C311yescharacteristic, complemented
43#1786C2×C312yes—
44C25×C3—21no—
45C22×C3×Q8—21no—
46C22×C3×D4—24no—
47C22×C3×D4—24no—
48C22×C3×D4—21no—
49C22×C3×D4—22no—
50C22×C3×D4Q814yescomplemented
51C22×C3×D4C2312yes—
52C23×C3×C4—28no—
53C23×C3×C4—21no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—44no—
57C23×C3×C4—44no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×C3×C8C2314yes—
62C3×C4⋊Q8—24no—
63C3×C22⋊Q8—44no—
64C3×C4×D4—28no—
65C3×C4×D4—28no—
66C3×C4×D4—24no—
67C3×C4×D4—24no—
68C3×C4×D4—24no—
69C3×C4×D4D418yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—28no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—24no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C3×#106—28no—
89C3×C4⋊C8—24no—
90C3×#102—28no—
91C3×#102—28no—
92C3×C22⋊C8—24no—
93C3×#95—22no—
94C3×#95—22no—
95C23×D4C22×C311yescharacteristic
96C24×C4—21no—
97C2×C4⋊Q8C22×C311yescharacteristic, complemented
98C2×C22⋊Q8—21no—
99C2×C4×D4—28no—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4C22×C312yes—
103C2×C4×D4C22×C312yes—
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C42C22×C311yescharacteristic
107#476—28no—
108C2×#106C22×C314yescomplemented
109C2×C4⋊C8C22×C312yescomplemented
110C2×#102C22×C314yes—
111C2×#102C22×C314yes—
112C2×C22⋊C8C22×C312yes—
113#391—21no—
114#389—21no—
115#389C22×C312yescomplemented
116#384C22×C311yescharacteristic, complemented
117#380—21no—
118#380C22×C312yes—
119#340C22×C314yescomplemented
120#336C22×C312yescomplemented
121C24×C3—24no—
122C24×C3—24no—
123C24×C3—24no—
124C24×C3—22no—
125C24×C3—42no—
126C24×C3—42no—
127C24×C3—42no—
128C24×C3—42no—
129C24×C3C2×D411yescharacteristic
130C2×C3×Q8—42no—
131C2×C3×Q8—42no—
132C2×C3×Q8—42no—
133C2×C3×D4—24no—
134C2×C3×D4—24no—
135C2×C3×D4—216no—
136C2×C3×D4—216no—
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4C2×D414yes—
142C2×C3×D4C2×Q814yes—
143C2×C3×D4#4114yes—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—28no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—28no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—41no—
157C22×C3×C4—44no—
158C22×C3×C4—432no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—44no—
162C22×C3×C4—44no—
163C22×C3×C4—44no—
164C22×C3×C4SD1614yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×Q812yes—
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C2×C3×C8—24no—
174C2×C3×C8—24no—
175C2×C3×C8—24no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—44no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4C2×D414yes—
185C3×C22⋊C4—28no—
186C3×C22⋊C4—28no—
187C3×C22⋊C4—44no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42C2×D414yes—
192C25—21no—
193C22×Q8—21no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—21no—
197C22×D4—22no—
198C22×D4C3×Q814yescomplemented
199C22×D4C23×C312yes—
200C23×C4—28no—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C3×D411yescharacteristic
207C23×C4C3×D411yescharacteristic
208C23×C4C23×C311yescharacteristic
209C22×C8C23×C314yes—
210C4⋊Q8—24no—
211C22⋊Q8—44no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4C3×D418yes—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C23×C312yes—
225C2×C4⋊C4C23×C312yes—
226C2×C4⋊C4C23×C312yes—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C42—24no—
232C2×C42C3×D411yescharacteristic
233C2×C42C3×D411yescharacteristic
234C2×C42C23×C311yescharacteristic
235C2×C42C23×C311yescharacteristic
236#106—28no—
237C4⋊C8—24no—
238#102—28no—
239#102—28no—
240C22⋊C8—24no—
241#95—22no—
242#95—22no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—216no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—48no—
252C23×C3—48no—
253C23×C3—48no—
254C23×C3—48no—
255C23×C3—48no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—48no—
261C23×C3C22⋊Q812yes—
262C23×C3C2×SD1612yes—
263C23×C3AGL(1,ℤ/8)12yes—
264C23×C3C22×D411yescharacteristic
265C3×Q8—84no—
266C3×D4—216no—
267C3×D4—24no—
268C3×D4—24no—
269C3×D4—24no—
270C3×D4C22⋊Q818yescomplemented
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—28no—
280C2×C3×C4—21no—
281C2×C3×C4—24no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—44no—
285C2×C3×C4—416no—
286C2×C3×C4—48no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—416no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4#12012yes—
297C2×C3×C4#12012yes—
298C2×C3×C4C22⋊Q812yes—
299C2×C3×C4C2×SD1612yes—
300C2×C3×C4C2×SD1612yes—
301C2×C3×C4AGL(1,ℤ/8)12yes—
302C2×C3×C4C22×D412yes—
303C2×C3×C4C22×Q811yescharacteristic
304C2×C3×C4C2×#4112yes—
305C2×C3×C4C2×#4111yescharacteristic
306C3×C8—44no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C2×C3×D411yescharacteristic
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—216no—
322C2×D4—216no—
323C2×D4—24no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4C2×C3×D414yes—
328C2×D4C2×C3×Q814yes—
329C2×D4C3×#4114yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—41no—
343C22×C4—44no—
344C22×C4—432no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4C3×SD1614yes—
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×Q812yes—
357C22×C4C3×#4112yes—
358C22×C4C24×C311yescharacteristic
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4C2×C3×D414yes—
371C22⋊C4—28no—
372C22⋊C4—28no—
373C22⋊C4—44no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42C2×C3×D414yes—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—28no—
385C22×C3—432no—
386C22×C3—44no—
387C22×C3—44no—
388C22×C3—44no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3#47614yescomplemented
393C22×C3C2×#12011yescharacteristic
394C22×C3C2×C22⋊Q811yescharacteristic
395C22×C3#52511yescharacteristic
396C22×C3C22×SD1611yescharacteristic
397C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
398C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
399C22×C3#57511yescharacteristic
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—44no—
404C3×C4—41no—
405C3×C4—82no—
406C3×C4—82no—
407C3×C4#45014yes—
408C3×C4C2×C22⋊Q812yes—
409C3×C4#52512yes—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—216no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—48no—
419C23—48no—
420C23—48no—
421C23—48no—
422C23—48no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—48no—
428C23C3×C22⋊Q812yes—
429C23C2×C3×SD1612yes—
430C23C3×AGL(1,ℤ/8)12yes—
431C23C22×C3×D411yescharacteristic
432Q8—84no—
433D4—216no—
434D4—24no—
435D4—24no—
436D4—24no—
437D4C3×C22⋊Q818yescomplemented
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—44no—
452C2×C4—416no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—416no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4C3×#12012yes—
464C2×C4C3×#12012yes—
465C2×C4C3×C22⋊Q812yes—
466C2×C4C2×C3×SD1612yes—
467C2×C4C2×C3×SD1612yes—
468C2×C4C3×AGL(1,ℤ/8)12yes—
469C2×C4C22×C3×D412yes—
470C2×C4C22×C3×Q811yescharacteristic
471C2×C4C2×C3×#4112yes—
472C2×C4C2×C3×#4111yescharacteristic
473C8—44no—
474C2×C3—24no—
475C2×C3—48no—
476C2×C3#284811yescharacteristic
477C2×C3C2×#45011yescharacteristic
478C2×C3#300611yescharacteristic
479C2×C3C2×#47611yescharacteristic
480C2×C3#307711yescharacteristic
481C2×C3#310111yescharacteristic
482C2×C3#310911yescharacteristic
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—28no—
490C22—432no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22C3×#47614yescomplemented
498C22C2×C3×#12011yescharacteristic
499C22C2×C3×C22⋊Q811yescharacteristic
500C22C3×#52511yescharacteristic
501C22C22×C3×SD1611yescharacteristic
502C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
503C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
504C22C3×#57511yescharacteristic
505C4—22no—
506C4—22no—
507C4—22no—
508C4—44no—
509C4—41no—
510C4—82no—
511C4—82no—
512C4C3×#45014yes—
513C4C2×C3×C22⋊Q812yes—
514C4C3×#52512yes—
515C3#3295611yescharacteristic, direct product factor, Sylow
516C2—24no—
517C2—48no—
518C2C3×#284811yescharacteristic
519C2C2×C3×#45011yescharacteristic
520C2C3×#300611yescharacteristic
521C2C2×C3×#47611yescharacteristic
522C2C3×#307711yescharacteristic
523C2C3×#310111yescharacteristic
524C2C3×#310911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#476C2semidirect
3. C3×#2857C2semidirect
4. C3×#1864C2semidirect
5. C3×#1853C2semidirect
6. C3×#1802C2semidirect
7. C3×#1797C2semidirect
9. #32956C3direct (N × H)
12. C2×C3×C4⋊Q8C22semidirect
23. C2×C3×#106C22semidirect
24. C2×C3×C4⋊C8C22semidirect
30. C3×#389C22semidirect
31. C3×#384C22semidirect
34. C3×#340C22semidirect
35. C3×#336C22semidirect
37. C2×#476C2×C3semidirect
38. #2857C2×C3semidirect
39. #1864C2×C3semidirect
40. #1853C2×C3semidirect
41. #1802C2×C3semidirect
42. #1797C2×C3semidirect
50. C22×C3×D4Q8semidirect
97. C2×C4⋊Q8C22×C3semidirect
108. C2×#106C22×C3semidirect
109. C2×C4⋊C8C22×C3semidirect
115. #389C22×C3semidirect
116. #384C22×C3semidirect
119. #340C22×C3semidirect
120. #336C22×C3semidirect
198. C22×D4C3×Q8semidirect
270. C3×D4C22⋊Q8semidirect
392. C22×C3#476semidirect
437. D4C3×C22⋊Q8semidirect
497. C22C3×#476semidirect
515. C3#32956direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×C3×#4762
C3×#28572
C3×#18642
C3×#18532
C3×#18022
C3×#17972
C3×#17862
#329563