Finite Group Databasebeta

← C3×#25640 #10644247

Subgroups — C3×#25640 #10644247

Order 768.

1102 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×C3×#497C211yesmaximal, characteristic
3C2×C3×#464C211yesmaximal, characteristic
4C3×#2592C211yesmaximal, characteristic
5C3×#2016C212yesmaximal, complemented
6C3×#2013C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1981C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal
10C3×#1878C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25640C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C22×C3×SD16C2211yescharacteristic
17C3×#557—24no—
18C3×#557—24no—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#125—22no—
22C2×C3×#125—21no—
23C2×C3×#124C2211yescharacteristic
24C2×C3×C4×Q8—21no—
25C2×C3×C4×Q8C2212yes—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C3×#497—24no—
29C3×#464—24no—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2211yescharacteristic
36C2×C3×#97C2211yescharacteristic
37C3×#397—21no—
38C3×#393—21no—
39C3×#391C2212yes—
40C3×#390C2212yes—
41C3×#388—22no—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#557C2×C311yescharacteristic, complemented
47C2×#497C2×C311yescharacteristic
48C2×#464C2×C311yescharacteristic
49#2592C2×C311yescharacteristic
50#2016C2×C312yescomplemented
51#2013C2×C312yes—
52#1985C2×C311yescharacteristic, complemented
53#1981C2×C311yescharacteristic, complemented
54#1880C2×C312yes—
55#1878C2×C312yescomplemented
56#1763C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×C3×Q8C2312yes—
61C22×C3×D4C2311yescharacteristic
62C2×C3×Q16—22no—
63C2×C3×Q16—24no—
64C2×C3×Q16—24no—
65C2×C3×Q16—42no—
66C2×C3×Q16—42no—
67C2×C3×SD16—22no—
68C2×C3×SD16—24no—
69C2×C3×SD16—24no—
70C2×C3×SD16—42no—
71C2×C3×SD16—42no—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2312yes—
75C3×C4⋊Q8—24no—
76C3×C4⋊Q8—24no—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8D414yes—
79C3×#125—216no—
80C3×#125—28no—
81C3×#124—22no—
82C3×#124—22no—
83C3×C4×Q8—28no—
84C3×C4×Q8—24no—
85C3×C4×Q8—24no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8D414yes—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4D412yes—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C42—22no—
105C2×C3×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42C2312yes—
108C2×C3×C42C2311yescharacteristic
109C3×C4⋊C8—22no—
110C3×#103—24no—
111C3×#103—22no—
112C3×#103—42no—
113C3×#103—42no—
114C3×#102—22no—
115C3×#97—22no—
116C3×#95—21no—
117C3×#95—22no—
118C3×#95—22no—
119C22×Q16C22×C312yes—
120C22×Q16C22×C311yescharacteristic
121C22×SD16C22×C312yes—
122C22×SD16C22×C311yescharacteristic
123#557—24no—
124#557—24no—
125C2×C4⋊Q8C22×C312yes—
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#125—22no—
128C2×#125—21no—
129C2×#124C22×C311yescharacteristic
130C2×C4×Q8—21no—
131C2×C4×Q8C22×C312yes—
132C2×C4×Q8C22×C311yescharacteristic
133C2×C4×Q8C22×C311yescharacteristic
134#497—24no—
135#464—24no—
136C2×C4⋊C8C22×C311yescharacteristic
137C2×#103C22×C311yescharacteristic
138C2×#103C22×C312yes—
139C2×#103C22×C312yes—
140C2×#103C22×C312yes—
141C2×#102C22×C311yescharacteristic
142C2×#97C22×C311yescharacteristic
143#397—21no—
144#393—21no—
145#391C22×C312yes—
146#390C22×C312yes—
147#388—22no—
148C4×C4⋊C4C22×C311yescharacteristic
149#340C22×C312yes—
150#336C22×C312yes—
151#336C22×C312yes—
152C24×C3—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8C2×D414yes—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2411yescharacteristic
183C3×Q16—48no—
184C3×Q16—44no—
185C3×SD16—48no—
186C3×SD16—44no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8#4114yes—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—28no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C2×D412yes—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—42no—
208C3×C22⋊C4—44no—
209C3×C22⋊C4—82no—
210C3×C42—28no—
211C3×C42—24no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—81no—
215C3×C42C2×D412yes—
216C3×C42C2×D412yes—
217C22×Q8C3×D411yescharacteristic
218C22×Q8C23×C311yescharacteristic
219C22×Q8C23×C311yescharacteristic
220C22×Q8C23×C312yes—
221C22×D4C23×C311yescharacteristic
222C2×Q16—22no—
223C2×Q16—24no—
224C2×Q16—24no—
225C2×Q16—42no—
226C2×Q16—42no—
227C2×SD16—22no—
228C2×SD16—24no—
229C2×SD16—24no—
230C2×SD16—42no—
231C2×SD16—42no—
232C22×C8C23×C311yescharacteristic
233C22×C8C23×C311yescharacteristic
234C22×C8C23×C312yes—
235C4⋊Q8—24no—
236C4⋊Q8—24no—
237C4⋊Q8—22no—
238C4⋊Q8C3×D414yes—
239#125—216no—
240#125—28no—
241#124—22no—
242#124—22no—
243C4×Q8—28no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8C3×D414yes—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C3×D412yes—
257C2×C4⋊C4C3×D411yescharacteristic
258C2×C4⋊C4C23×C312yes—
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42C23×C312yes—
268C2×C42C23×C311yescharacteristic
269C4⋊C8—22no—
270#103—24no—
271#103—22no—
272#103—42no—
273#103—42no—
274#102—22no—
275#97—22no—
276#95—21no—
277#95—22no—
278#95—22no—
279C23×C3—41no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3C22×D411yescharacteristic
285C3×Q8—28no—
286C3×Q8—22no—
287C3×Q8—22no—
288C3×Q8—24no—
289C3×Q8—24no—
290C3×Q8—42no—
291C3×Q8—42no—
292C3×Q8—42no—
293C3×Q8—41no—
294C3×Q8—41no—
295C3×Q8—42no—
296C3×D4—41no—
297C3×D4—41no—
298C3×D4—42no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4#12012yes—
321C2×C3×C4#12012yes—
322C2×C3×C4#13714yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4112yes—
326C2×C3×C4H(5,𝔽2)12yes—
327C3×C8—24no—
328C3×C8—41no—
329C3×C8—41no—
330C24—21no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8C2×C3×D414yes—
346C2×Q8C2×C3×D412yes—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4C2×C3×D412yes—
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×D412yes—
360C22×C4C24×C311yescharacteristic
361Q16—48no—
362Q16—44no—
363SD16—48no—
364SD16—44no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8C3×#4114yes—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4C2×C3×D412yes—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—82no—
388C42—28no—
389C42—24no—
390C42—22no—
391C42—22no—
392C42—81no—
393C42C2×C3×D412yes—
394C42C2×C3×D412yes—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3C2×#12011yescharacteristic
401C22×C3#55012yes—
402C22×C3C2×#13712yes—
403C22×C3#57711yescharacteristic
404C22×C3#57811yescharacteristic
405C3×C4—22no—
406C3×C4—24no—
407C3×C4—22no—
408C3×C4—42no—
409C3×C4—44no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—82no—
413C3×C4—81no—
414C3×C4#53211yescharacteristic
415C3×C4#53511yescharacteristic
416C3×C4#54612yes—
417C23—42no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23C22×C3×D411yescharacteristic
423Q8—28no—
424Q8—22no—
425Q8—22no—
426Q8—24no—
427Q8—24no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—41no—
432Q8—41no—
433Q8—42no—
434D4—41no—
435D4—41no—
436D4—42no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4C3×#12012yes—
459C2×C4C3×#12012yes—
460C2×C4C3×#13714yes—
461C2×C4C22×C3×D411yescharacteristic
462C2×C4C22×C3×D411yescharacteristic
463C2×C4C2×C3×#4112yes—
464C2×C4C3×H(5,𝔽2)12yes—
465C8—24no—
466C8—41no—
467C8—41no—
468C2×C3—161no—
469C2×C3#243111yescharacteristic
470C2×C3#300411yescharacteristic
471C2×C3#301311yescharacteristic
472C2×C3#339812yes—
473C2×C3#340012yes—
474C22—82no—
475C22—82no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22C2×C3×#12011yescharacteristic
480C22C3×#55012yes—
481C22C2×C3×#13712yes—
482C22C3×#57711yescharacteristic
483C22C3×#57811yescharacteristic
484C4—22no—
485C4—24no—
486C4—22no—
487C4—42no—
488C4—44no—
489C4—42no—
490C4—42no—
491C4—82no—
492C4—81no—
493C4C3×#53211yescharacteristic
494C4C3×#53511yescharacteristic
495C4C3×#54612yes—
496C3#2564011yescharacteristic, direct product factor, Sylow
497C2—161no—
498C2C3×#243111yescharacteristic
499C2C3×#300411yescharacteristic
500C2C3×#301311yescharacteristic
501C2C3×#339812yes—
502C2C3×#340012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
5. C3×#2016C2semidirect
7. C3×#1985C2semidirect
8. C3×#1981C2semidirect
10. C3×#1878C2semidirect
11. C3×#1763C2semidirect
12. #25640C3direct (N × H)
46. C2×#557C2×C3semidirect
50. #2016C2×C3semidirect
52. #1985C2×C3semidirect
53. #1981C2×C3semidirect
55. #1878C2×C3semidirect
56. #1763C2×C3semidirect
496. C3#25640direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×C3×#4972
C2×C3×#4642
C3×#25922
C3×#20162
C3×#20132
C3×#19852
C3×#19812
C3×#18802
C3×#18782
C3×#17632
#256403