Finite Group Databasebeta

← C3×#29725 #10648332

Subgroups — C3×#29725 #10648332

Order 768.

1070 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#449C212yesmaximal, complemented
2C3×#2619C211yesmaximal, characteristic, complemented
3C3×#2311C211yesmaximal, characteristic, complemented
4C3×#2074C211yesmaximal, characteristic, complemented
5C3×#2073C211yesmaximal, characteristic, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2054C212yesmaximal, complemented
8C3×#2014C211yesmaximal, characteristic, complemented
9C3×#2011C211yesmaximal, characteristic, complemented
10C3×#1874C212yesmaximal, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29725C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C2×C3×#127C2211yescharacteristic
17C2×C3×#121C2212yes—
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C3×#449—28no—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×C22⋊C8C2212yes—
27C3×#394—22no—
28C3×#392—22no—
29C3×#391C2211yescharacteristic
30C3×#391C2211yescharacteristic
31C3×#390C2212yes—
32C3×#390C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#386—22no—
35C3×#382C2211yescharacteristic
36C3×#380—22no—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#336C2211yescharacteristic
40C3×#336C2211yescharacteristic
41C2×#449C2×C312yescomplemented
42#2619C2×C311yescharacteristic, complemented
43#2311C2×C311yescharacteristic, complemented
44#2074C2×C311yescharacteristic, complemented
45#2073C2×C311yescharacteristic, complemented
46#2055C2×C312yescomplemented
47#2054C2×C312yescomplemented
48#2014C2×C311yescharacteristic, complemented
49#2011C2×C311yescharacteristic, complemented
50#1874C2×C312yescomplemented
51#1798C2×C311yescharacteristic, complemented
52C2×C3×#41—22no—
53C2×C3×#41—42no—
54C2×C3×#41—42no—
55C2×C3×#41D414yescomplemented
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4C2312yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C2×C3×SD16—24no—
66C2×C3×SD16—42no—
67C2×C3×SD16—42no—
68C2×C3×D8—24no—
69C2×C3×D8—42no—
70C2×C3×D8—42no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2312yes—
73C3×#127—42no—
74C3×#121—24no—
75C3×#121—24no—
76C3×#117—42no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—42no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
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⋊C4D412yes—
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C3×C4⋊C8—42no—
92C3×#103—24no—
93C3×#102—24no—
94C3×#102—42no—
95C3×#102—42no—
96C3×C22⋊C8—24no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C22×#41C22×C311yescharacteristic
101C22×SD16C22×C312yes—
102C22×D8C22×C312yes—
103C2×#127C22×C311yescharacteristic
104C2×#121C22×C312yes—
105C2×#117C22×C311yescharacteristic
106C22×C4⋊C4C22×C311yescharacteristic
107#449—28no—
108C2×C4⋊C8C22×C312yes—
109C2×#103C22×C312yes—
110C2×#102C22×C312yes—
111C2×#102C22×C312yes—
112C2×#102C22×C312yes—
113C2×C22⋊C8C22×C312yes—
114#394—22no—
115#392—22no—
116#391C22×C311yescharacteristic
117#391C22×C311yescharacteristic
118#390C22×C311yescharacteristic
119#390C22×C312yes—
120#390C22×C311yescharacteristic
121#386—22no—
122#382C22×C311yescharacteristic
123#380—22no—
124#340C22×C312yes—
125#340C22×C312yes—
126#336C22×C311yescharacteristic
127#336C22×C311yescharacteristic
128C24×C3—21no—
129C24×C3—22no—
130C24×C3C2×D411yescharacteristic
131C3×#41—48no—
132C3×#41—48no—
133C2×C3×Q8—21no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8C2×D412yes—
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—44no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—44no—
151C2×C3×D4—82no—
152C2×C3×D4—82no—
153C2×C3×D4—82no—
154C2×C3×D4—82no—
155C2×C3×D4C2×D412yes—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—82no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C3×SD16—48no—
174C3×D8—48no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—44no—
189C3×C22⋊C4—82no—
190C3×C22⋊C4—82no—
191C3×C22⋊C4—82no—
192C3×C22⋊C4—82no—
193C3×C42—22no—
194C3×C42—22no—
195C2×#41—22no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41C3×D414yescomplemented
199C22×Q8C23×C311yescharacteristic
200C22×D4—21no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4C23×C312yes—
204C22×D4C23×C311yescharacteristic
205C23×C4—21no—
206C23×C4—21no—
207C23×C4C23×C311yescharacteristic
208C2×SD16—24no—
209C2×SD16—42no—
210C2×SD16—42no—
211C2×D8—24no—
212C2×D8—42no—
213C2×D8—42no—
214C22×C8C23×C312yes—
215C22×C8C23×C312yes—
216#127—42no—
217#121—24no—
218#121—24no—
219#117—42no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C3×D412yes—
223C2×C4⋊C4C23×C311yescharacteristic
224C2×C4⋊C4C23×C311yescharacteristic
225C2×C4⋊C4C23×C312yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4C3×D412yes—
232C2×C42C23×C311yescharacteristic
233C2×C42C23×C311yescharacteristic
234C4⋊C8—42no—
235#103—24no—
236#102—24no—
237#102—42no—
238#102—42no—
239C22⋊C8—24no—
240#95—22no—
241#95—22no—
242#95—22no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3#12112yes—
260C23×C3C22×D411yescharacteristic
261C3×Q8—24no—
262C3×Q8—42no—
263C3×D4—24no—
264C3×D4—48no—
265C3×D4—42no—
266C3×D4—42no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—42no—
270C3×D4—88no—
271C3×D4—88no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—48no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—84no—
294C2×C3×C4#12112yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4H(5,𝔽2)12yes—
298C2×C3×C4H(5,𝔽2)12yes—
299C3×C8—42no—
300C3×C8—42no—
301C24—21no—
302C24—22no—
303C24C2×C3×D411yescharacteristic
304#41—48no—
305#41—48no—
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8C2×C3×D412yes—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—21no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—44no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—44no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C2×D4C2×C3×D412yes—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—82no—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D412yes—
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D412yes—
344C22×C4C3×#4112yes—
345C22×C4C24×C311yescharacteristic
346SD16—48no—
347D8—48no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—44no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C42—22no—
367C42—22no—
368C22×C3—22no—
369C22×C3—24no—
370C22×C3—42no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—42no—
374C22×C3—44no—
375C22×C3—82no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3C2×#12111yescharacteristic
386C22×C3#53411yescharacteristic
387C22×C3#53511yescharacteristic
388C22×C3#57612yes—
389C22×C3#57712yes—
390C3×C4—22no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—81no—
395C3×C4—81no—
396C3×C4—82no—
397C3×C4#53412yes—
398C3×C4#53812yes—
399C23—22no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23C3×#12112yes—
416C23C22×C3×D411yescharacteristic
417Q8—24no—
418Q8—42no—
419D4—24no—
420D4—48no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—88no—
427D4—88no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—48no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—84no—
450C2×C4C3×#12112yes—
451C2×C4C22×C3×D411yescharacteristic
452C2×C4C2×C3×#4111yescharacteristic
453C2×C4C3×H(5,𝔽2)12yes—
454C2×C4C3×H(5,𝔽2)12yes—
455C8—42no—
456C8—42no—
457C2×C3—42no—
458C2×C3—82no—
459C2×C3—162no—
460C2×C3#261711yescharacteristic
461C2×C3#304512yes—
462C2×C3#319812yes—
463C2×C3#322011yescharacteristic
464C2×C3#322111yescharacteristic
465C22—22no—
466C22—24no—
467C22—42no—
468C22—41no—
469C22—41no—
470C22—42no—
471C22—44no—
472C22—82no—
473C22—81no—
474C22—81no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22C2×C3×#12111yescharacteristic
483C22C3×#53411yescharacteristic
484C22C3×#53511yescharacteristic
485C22C3×#57612yes—
486C22C3×#57712yes—
487C4—22no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—81no—
492C4—81no—
493C4—82no—
494C4C3×#53412yes—
495C4C3×#53812yes—
496C3#2972511yescharacteristic, direct product factor, Sylow
497C2—42no—
498C2—82no—
499C2—162no—
500C2C3×#261711yescharacteristic
501C2C3×#304512yes—
502C2C3×#319812yes—
503C2C3×#322011yescharacteristic
504C2C3×#322111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#449C2semidirect
2. C3×#2619C2semidirect
3. C3×#2311C2semidirect
4. C3×#2074C2semidirect
5. C3×#2073C2semidirect
6. C3×#2055C2semidirect
7. C3×#2054C2semidirect
8. C3×#2014C2semidirect
9. C3×#2011C2semidirect
10. C3×#1874C2semidirect
11. C3×#1798C2semidirect
12. #29725C3direct (N × H)
41. C2×#449C2×C3semidirect
42. #2619C2×C3semidirect
43. #2311C2×C3semidirect
44. #2074C2×C3semidirect
45. #2073C2×C3semidirect
46. #2055C2×C3semidirect
47. #2054C2×C3semidirect
48. #2014C2×C3semidirect
49. #2011C2×C3semidirect
50. #1874C2×C3semidirect
51. #1798C2×C3semidirect
55. C2×C3×#41D4semidirect
198. C2×#41C3×D4semidirect
496. C3#29725direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4492
C3×#26192
C3×#23112
C3×#20742
C3×#20732
C3×#20552
C3×#20542
C3×#20142
C3×#20112
C3×#18742
C3×#17982
#297253