Finite Group Databasebeta

← C3×#25639 #10644246

Subgroups — C3×#25639 #10644246

Order 768.

1198 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic
3C2×C3×#464C211yesmaximal, characteristic, complemented
4C3×#2592C211yesmaximal, characteristic, complemented
5C3×#2015C212yesmaximal, complemented
6C3×#2012C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1982C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1877C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25639C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×SD16C2211yescharacteristic
16C22×C3×D8C2212yes—
17C3×#559—24no—
18C3×#559—24no—
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124—21no—
21C2×C3×#124C2211yescharacteristic
22C2×C3×#124C2212yes—
23C2×C3×#123—22no—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4—21no—
27C2×C3×C4×D4C2212yes—
28C3×#496—24no—
29C3×#464—24no—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
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×#559C2×C311yescharacteristic, complemented
47C2×#496C2×C311yescharacteristic
48C2×#464C2×C311yescharacteristic, complemented
49#2592C2×C311yescharacteristic, complemented
50#2015C2×C312yescomplemented
51#2012C2×C312yescomplemented
52#1985C2×C311yescharacteristic, complemented
53#1982C2×C311yescharacteristic, complemented
54#1879C2×C312yescomplemented
55#1877C2×C312yescomplemented
56#1763C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C22×C3×D4C2312yes—
64C23×C3×C4—22no—
65C2×C3×Q16—22no—
66C2×C3×Q16—42no—
67C2×C3×SD16—22no—
68C2×C3×SD16—24no—
69C2×C3×SD16—24no—
70C2×C3×SD16—42no—
71C2×C3×SD16—42no—
72C2×C3×D8—24no—
73C2×C3×D8—24no—
74C2×C3×D8—42no—
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2312yes—
78C3×#127—22no—
79C3×#127D414yes—
80C3×#124—28no—
81C3×#124—24no—
82C3×#124—24no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#123—216no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8D414yes—
90C3×C4×D4—28no—
91C3×C4×D4—24no—
92C3×C4×D4—24no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—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×#102—22no—
113C3×#102—42no—
114C3×#102—42no—
115C3×#97—22no—
116C3×#95—21no—
117C3×#95—22no—
118C3×#95—22no—
119C22×Q16C22×C311yescharacteristic
120C22×SD16C22×C312yes—
121C22×SD16C22×C311yescharacteristic
122C22×D8C22×C312yes—
123#559—24no—
124#559—24no—
125C2×#127C22×C311yescharacteristic
126C2×#124—21no—
127C2×#124C22×C311yescharacteristic
128C2×#124C22×C312yes—
129C2×#123—22no—
130C2×C4×Q8C22×C311yescharacteristic
131C2×C4×Q8C22×C311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4C22×C312yes—
134#496—24no—
135#464—24no—
136C2×C4⋊C8C22×C311yescharacteristic
137C2×#103C22×C311yescharacteristic
138C2×#103C22×C312yes—
139C2×#102C22×C311yescharacteristic
140C2×#102C22×C312yes—
141C2×#102C22×C312yes—
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—22no—
153C24×C3—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—42no—
163C2×C3×Q8C2×D412yes—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—44no—
176C2×C3×D4—81no—
177C2×C3×D4—81no—
178C2×C3×D4C2×D414yes—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—28no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—82no—
185C22×C3×C4—82no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C3×Q16—44no—
192C3×SD16—48no—
193C3×SD16—44no—
194C3×D8—48no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8#4114yes—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—44no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—82no—
216C3×C42—24no—
217C3×C42—22no—
218C3×C42—22no—
219C3×C42—81no—
220C3×C42C2×D412yes—
221C3×C42C2×D412yes—
222C22×Q8C3×D411yescharacteristic
223C22×Q8C23×C311yescharacteristic
224C22×Q8C23×C311yescharacteristic
225C22×D4—21no—
226C22×D4—21no—
227C22×D4C23×C311yescharacteristic
228C22×D4C23×C312yes—
229C23×C4—22no—
230C2×Q16—22no—
231C2×Q16—42no—
232C2×SD16—22no—
233C2×SD16—24no—
234C2×SD16—24no—
235C2×SD16—42no—
236C2×SD16—42no—
237C2×D8—24no—
238C2×D8—24no—
239C2×D8—42no—
240C22×C8C23×C311yescharacteristic
241C22×C8C23×C311yescharacteristic
242C22×C8C23×C312yes—
243#127—22no—
244#127C3×D414yes—
245#124—28no—
246#124—24no—
247#124—24no—
248#124—22no—
249#124—22no—
250#123—216no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8C3×D414yes—
255C4×D4—28no—
256C4×D4—24no—
257C4×D4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4C3×D411yescharacteristic
260C2×C4⋊C4C3×D412yes—
261C2×C4⋊C4C23×C312yes—
262C2×C4⋊C4C23×C311yescharacteristic
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42C23×C312yes—
273C2×C42C23×C311yescharacteristic
274C4⋊C8—22no—
275#103—24no—
276#103—22no—
277#102—22no—
278#102—42no—
279#102—42no—
280#97—22no—
281#95—21no—
282#95—22no—
283#95—22no—
284C23×C3—24no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3C22×D411yescharacteristic
297C3×Q8—22no—
298C3×Q8—22no—
299C3×Q8—44no—
300C3×Q8—42no—
301C3×Q8—41no—
302C3×Q8—41no—
303C3×Q8—42no—
304C3×D4—24no—
305C3×D4—24no—
306C3×D4—48no—
307C3×D4—42no—
308C3×D4—41no—
309C3×D4—42no—
310C3×D4—41no—
311C3×D4—42no—
312C3×D4—84no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—88no—
333C2×C3×C4#12012yes—
334C2×C3×C4#12012yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4112yes—
338C2×C3×C4H(5,𝔽2)12yes—
339C3×C8—24no—
340C3×C8—41no—
341C3×C8—41no—
342C24—22no—
343C24—21no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—42no—
353C2×Q8C2×C3×D412yes—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—44no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4C2×C3×D414yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—28no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4C2×C3×D412yes—
377C22×C4C2×C3×D412yes—
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C24×C311yescharacteristic
381Q16—44no—
382SD16—48no—
383SD16—44no—
384D8—48no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8C3×#4114yes—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4C2×C3×D412yes—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—28no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—82no—
406C42—24no—
407C42—22no—
408C42—22no—
409C42—81no—
410C42C2×C3×D412yes—
411C42C2×C3×D412yes—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3—81no—
424C22×C3C2×#12011yescharacteristic
425C22×C3#55012yes—
426C22×C3#57512yes—
427C22×C3#57611yescharacteristic
428C22×C3#57711yescharacteristic
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—82no—
436C3×C4—81no—
437C3×C4#53211yescharacteristic
438C3×C4#53511yescharacteristic
439C3×C4#54612yes—
440C23—24no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23C22×C3×D411yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—44no—
456Q8—42no—
457Q8—41no—
458Q8—41no—
459Q8—42no—
460D4—24no—
461D4—24no—
462D4—48no—
463D4—42no—
464D4—41no—
465D4—42no—
466D4—41no—
467D4—42no—
468D4—84no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—88no—
489C2×C4C3×#12012yes—
490C2×C4C3×#12012yes—
491C2×C4C22×C3×D411yescharacteristic
492C2×C4C22×C3×D411yescharacteristic
493C2×C4C2×C3×#4112yes—
494C2×C4C3×H(5,𝔽2)12yes—
495C8—24no—
496C8—41no—
497C8—41no—
498C2×C3—84no—
499C2×C3—161no—
500C2×C3#243111yescharacteristic
501C2×C3#300511yescharacteristic
502C2×C3#301211yescharacteristic
503C2×C3#340112yes—
504C2×C3#340312yes—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22C2×C3×#12011yescharacteristic
518C22C3×#55012yes—
519C22C3×#57512yes—
520C22C3×#57611yescharacteristic
521C22C3×#57711yescharacteristic
522C4—22no—
523C4—22no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—82no—
529C4—81no—
530C4C3×#53211yescharacteristic
531C4C3×#53511yescharacteristic
532C4C3×#54612yes—
533C3#2563911yescharacteristic, direct product factor, Sylow
534C2—84no—
535C2—161no—
536C2C3×#243111yescharacteristic
537C2C3×#300511yescharacteristic
538C2C3×#301211yescharacteristic
539C2C3×#340112yes—
540C2C3×#340312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#496C2
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×#559C2semidirect
3. C2×C3×#464C2semidirect
4. C3×#2592C2semidirect
5. C3×#2015C2semidirect
6. C3×#2012C2semidirect
7. C3×#1985C2semidirect
8. C3×#1982C2semidirect
9. C3×#1879C2semidirect
10. C3×#1877C2semidirect
11. C3×#1763C2semidirect
12. #25639C3direct (N × H)
46. C2×#559C2×C3semidirect
48. C2×#464C2×C3semidirect
49. #2592C2×C3semidirect
50. #2015C2×C3semidirect
51. #2012C2×C3semidirect
52. #1985C2×C3semidirect
53. #1982C2×C3semidirect
54. #1879C2×C3semidirect
55. #1877C2×C3semidirect
56. #1763C2×C3semidirect
533. C3#25639direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4962
C2×C3×#4642
C3×#25922
C3×#20152
C3×#20122
C3×#19852
C3×#19822
C3×#18792
C3×#18772
C3×#17632
#256393