Finite Group Databasebeta

← C3×#29743 #10648350

Subgroups — C3×#29743 #10648350

Order 768.

1090 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#470C211yesmaximal, characteristic, complemented
2C2×C3×#407C211yesmaximal, characteristic
3C3×#2848C212yesmaximal
4C3×#2077C212yesmaximal, complemented
5C3×#2036C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#1894C212yesmaximal, complemented
9C3×#1809C211yesmaximal, characteristic, complemented
10#29743C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×Q16C2211yescharacteristic, complemented
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×C22⋊Q8C2212yes—
18C3×#470—24no—
19C2×C3×#107C2211yescharacteristic, complemented
20C2×C3×#106C2211yescharacteristic, complemented
21C2×C3×#106C2211yescharacteristic, complemented
22C2×C3×#103C2212yescomplemented
23C2×C3×#103C2212yescomplemented
24C2×C3×#103C2212yescomplemented
25C3×#407—24no—
26C2×C3×C22⋊C8C2212yes—
27C2×C3×C22⋊C8C2212yes—
28C3×#385—22no—
29C3×#385—22no—
30C3×#385—22no—
31C3×#380C2212yes—
32C3×#380C2212yes—
33C3×#380C2212yes—
34C3×#379—22no—
35C3×#340C2212yescomplemented
36C3×#340C2212yescomplemented
37C3×#340C2212yescomplemented
38C2×#470C2×C311yescharacteristic, complemented
39C2×#407C2×C311yescharacteristic
40#2848C2×C312yes—
41#2077C2×C312yescomplemented
42#2036C2×C312yescomplemented
43#2035C2×C312yescomplemented
44#2001C2×C312yescomplemented
45#1894C2×C312yescomplemented
46#1809C2×C311yescharacteristic, complemented
47C25×C3D411yescharacteristic
48C22×C3×Q8C2312yes—
49C23×C3×C4—22no—
50C23×C3×C4—22no—
51C23×C3×C4—41no—
52C23×C3×C4—41no—
53C23×C3×C4—41no—
54C23×C3×C4—41no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D412yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C2×C3×Q16—22no—
60C2×C3×Q16—41no—
61C2×C3×Q16—41no—
62C2×C3×M16—22no—
63C2×C3×M16—41no—
64C2×C3×M16—41no—
65C2×C3×M16—41no—
66C2×C3×M16—41no—
67C2×C3×M16D414yes—
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—42no—
74C3×C22⋊Q8—42no—
75C3×C22⋊Q8—42no—
76C3×C22⋊Q8—42no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C3×#107—41no—
90C3×#106—22no—
91C3×#106—41no—
92C3×#103—24no—
93C3×#103—42no—
94C3×#103—42no—
95C3×C22⋊C8—24no—
96C3×C22⋊C8—24no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C24×C4C22×C311yescharacteristic
101C22×Q16C22×C311yescharacteristic, complemented
102C22×M16C22×C311yescharacteristic
103C22×M16C22×C311yescharacteristic
104C2×C22⋊Q8C22×C312yes—
105C2×C22⋊Q8C22×C312yes—
106C2×C22⋊Q8C22×C312yes—
107#470—24no—
108C2×#107C22×C311yescharacteristic, complemented
109C2×#106C22×C311yescharacteristic, complemented
110C2×#106C22×C311yescharacteristic, complemented
111C2×#103C22×C312yescomplemented
112C2×#103C22×C312yescomplemented
113C2×#103C22×C312yescomplemented
114#407—24no—
115C2×C22⋊C8C22×C312yes—
116C2×C22⋊C8C22×C312yes—
117#385—22no—
118#385—22no—
119#385—22no—
120#380C22×C312yes—
121#380C22×C312yes—
122#380C22×C312yes—
123#379—22no—
124#340C22×C312yescomplemented
125#340C22×C312yescomplemented
126#340C22×C312yescomplemented
127C24×C3—22no—
128C24×C3—24no—
129C24×C3—42no—
130C24×C3—41no—
131C24×C3—41no—
132C24×C3C2×D411yescharacteristic
133C24×C3C2×D412yes—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—41no—
159C22×C3×C4—41no—
160C22×C3×C4—44no—
161C22×C3×C4—44no—
162C22×C3×C4—44no—
163C22×C3×C4—44no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4#4112yes—
168C22×C3×C4C2411yescharacteristic
169C3×Q16—44no—
170C3×M16—44no—
171C3×M16—44no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8C2×D412yes—
180C2×C3×C8C2×D412yes—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—44no—
188C3×C4⋊C4—44no—
189C3×C4⋊C4—44no—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—44no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C25C3×D411yescharacteristic
197C22×Q8C23×C312yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—41no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4—41no—
204C23×C4C3×D411yescharacteristic
205C23×C4C3×D412yes—
206C23×C4C23×C312yes—
207C23×C4C23×C311yescharacteristic
208C2×Q16—22no—
209C2×Q16—41no—
210C2×Q16—41no—
211C2×M16—22no—
212C2×M16—41no—
213C2×M16—41no—
214C2×M16—41no—
215C2×M16—41no—
216C2×M16C3×D414yes—
217C22×C8C23×C312yes—
218C22×C8C23×C311yescharacteristic
219C22×C8C23×C311yescharacteristic
220C22⋊Q8—24no—
221C22⋊Q8—24no—
222C22⋊Q8—42no—
223C22⋊Q8—42no—
224C22⋊Q8—42no—
225C22⋊Q8—42no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4C23×C312yes—
230C2×C4⋊C4C23×C312yes—
231C2×C4⋊C4C23×C312yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238#107—41no—
239#106—22no—
240#106—41no—
241#103—24no—
242#103—42no—
243#103—42no—
244C22⋊C8—24no—
245C22⋊C8—24no—
246#95—22no—
247#95—22no—
248#95—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—44no—
260C23×C3—48no—
261C23×C3—44no—
262C23×C3—48no—
263C23×C3—44no—
264C23×C3#12112yes—
265C23×C3#13714yes—
266C23×C3C22×D411yescharacteristic
267C3×Q8—42no—
268C3×Q8—42no—
269C3×Q8—42no—
270C3×Q8—42no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—48no—
281C2×C3×C4—44no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
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—42no—
297C2×C3×C4—42no—
298C2×C3×C4#12112yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4H(5,𝔽2)11yescharacteristic
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C2×C3×C4H(5,𝔽2)11yescharacteristic
304C2×C3×C4H(5,𝔽2)11yescharacteristic
305C3×C8—22no—
306C3×C8—22no—
307C3×C8—42no—
308C24—22no—
309C24—24no—
310C24—41no—
311C24—41no—
312C24—42no—
313C24C2×C3×D411yescharacteristic
314C24C2×C3×D412yes—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4C2×C3×D412yes—
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×D412yes—
348C22×C4C3×#4112yes—
349C22×C4C24×C311yescharacteristic
350Q16—44no—
351M16—44no—
352M16—44no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8C2×C3×D412yes—
361C2×C8C2×C3×D412yes—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22×C3—22no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—416no—
383C22×C3—42no—
384C22×C3—41no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—41no—
389C22×C3—42no—
390C22×C3C2×#12111yescharacteristic
391C22×C3#53411yescharacteristic
392C22×C3#53411yescharacteristic
393C22×C3C2×#13712yes—
394C22×C3#57512yes—
395C3×C4—22no—
396C3×C4—42no—
397C3×C4—82no—
398C3×C4—82no—
399C3×C4—82no—
400C3×C4—82no—
401C3×C4#53411yescharacteristic
402C3×C4#53511yescharacteristic
403C3×C4#53811yescharacteristic
404C3×C4#53811yescharacteristic
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—21no—
414C23—21no—
415C23—44no—
416C23—48no—
417C23—44no—
418C23—48no—
419C23—44no—
420C23C3×#12112yes—
421C23C3×#13714yes—
422C23C22×C3×D411yescharacteristic
423Q8—42no—
424Q8—42no—
425Q8—42no—
426Q8—42no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—48no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4C3×#12112yes—
455C2×C4C22×C3×D411yescharacteristic
456C2×C4C2×C3×#4111yescharacteristic
457C2×C4C3×H(5,𝔽2)11yescharacteristic
458C2×C4C3×H(5,𝔽2)11yescharacteristic
459C2×C4C3×H(5,𝔽2)11yescharacteristic
460C2×C4C3×H(5,𝔽2)11yescharacteristic
461C8—22no—
462C8—22no—
463C8—42no—
464C2×C3—42no—
465C2×C3—44no—
466C2×C3#261711yescharacteristic
467C2×C3#305612yes—
468C2×C3#319511yescharacteristic
469C2×C3#319612yes—
470C2×C3#319711yescharacteristic
471C22—22no—
472C22—24no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—416no—
477C22—42no—
478C22—41no—
479C22—42no—
480C22—42no—
481C22—41no—
482C22—42no—
483C22—42no—
484C22C2×C3×#12111yescharacteristic
485C22C3×#53411yescharacteristic
486C22C3×#53411yescharacteristic
487C22C2×C3×#13712yes—
488C22C3×#57512yes—
489C4—22no—
490C4—42no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4C3×#53411yescharacteristic
496C4C3×#53511yescharacteristic
497C4C3×#53811yescharacteristic
498C4C3×#53811yescharacteristic
499C3#2974311yescharacteristic, direct product factor, Sylow
500C2—42no—
501C2—44no—
502C2C3×#261711yescharacteristic
503C2C3×#305612yes—
504C2C3×#319511yescharacteristic
505C2C3×#319612yes—
506C2C3×#319711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#470C2semidirect
4. C3×#2077C2semidirect
5. C3×#2036C2semidirect
6. C3×#2035C2semidirect
7. C3×#2001C2semidirect
8. C3×#1894C2semidirect
9. C3×#1809C2semidirect
10. #29743C3direct (N × H)
12. C22×C3×Q16C22semidirect
19. C2×C3×#107C22semidirect
20. C2×C3×#106C22semidirect
21. C2×C3×#106C22semidirect
22. C2×C3×#103C22semidirect
23. C2×C3×#103C22semidirect
24. C2×C3×#103C22semidirect
35. C3×#340C22semidirect
36. C3×#340C22semidirect
37. C3×#340C22semidirect
38. C2×#470C2×C3semidirect
41. #2077C2×C3semidirect
42. #2036C2×C3semidirect
43. #2035C2×C3semidirect
44. #2001C2×C3semidirect
45. #1894C2×C3semidirect
46. #1809C2×C3semidirect
101. C22×Q16C22×C3semidirect
108. C2×#107C22×C3semidirect
109. C2×#106C22×C3semidirect
110. C2×#106C22×C3semidirect
111. C2×#103C22×C3semidirect
112. C2×#103C22×C3semidirect
113. C2×#103C22×C3semidirect
124. #340C22×C3semidirect
125. #340C22×C3semidirect
126. #340C22×C3semidirect
499. C3#29743direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4702
C2×C3×#4072
C3×#28482
C3×#20772
C3×#20362
C3×#20352
C3×#20012
C3×#18942
C3×#18092
#297433