Finite Group Databasebeta

← C3×#25580 #10644187

Subgroups — C3×#25580 #10644187

Order 768.

1162 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic, complemented
2C2×C3×#489C211yesmaximal, characteristic, complemented
3C2×C3×#464C211yesmaximal, characteristic, complemented
4C3×#2540C211yesmaximal, characteristic
5C3×#2001C212yesmaximal, complemented
6C3×#2000C212yesmaximal, complemented
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1875C212yesmaximal, complemented
10C3×#1873C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25580C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic
15C3×#556—24no—
16C3×#556—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#126—22no—
19C2×C3×#124—21no—
20C2×C3×#124C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×#121C2212yes—
24C2×C3×C4×Q8—21no—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×#117C2212yes—
28C3×#489—24no—
29C3×#464—24no—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#103C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×#102C2211yescharacteristic
36C2×C3×C22⋊C8C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C2×C3×#97C2211yescharacteristic
39C3×#399—21no—
40C3×#392—21no—
41C3×#390—22no—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#556C2×C311yescharacteristic, complemented
47C2×#489C2×C311yescharacteristic, complemented
48C2×#464C2×C311yescharacteristic, complemented
49#2540C2×C311yescharacteristic
50#2001C2×C312yescomplemented
51#2000C2×C312yescomplemented
52#1987C2×C311yescharacteristic, complemented
53#1955C2×C311yescharacteristic, complemented
54#1875C2×C312yescomplemented
55#1873C2×C312yescomplemented
56#1763C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—22no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4C2312yes—
64C2×C3×Q16—22no—
65C2×C3×Q16—42no—
66C2×C3×SD16—22no—
67C2×C3×SD16—42no—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8D414yescomplemented
73C3×#126—216no—
74C3×#124—28no—
75C3×#124—22no—
76C3×#124—22no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—42no—
81C3×#121—24no—
82C3×#121—42no—
83C3×C4×Q8—28no—
84C3×C4×Q8—22no—
85C3×C4×Q8D414yescomplemented
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×#117—24no—
89C3×#117—24no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
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⋊C4D412yes—
105C2×C3×C42—22no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C3×C4⋊C8—22no—
110C3×#103—22no—
111C3×#103—24no—
112C3×#103—24no—
113C3×#102—22no—
114C3×#102—24no—
115C3×C22⋊C8—42no—
116C3×C22⋊C8—42no—
117C3×#97—22no—
118C3×#95—21no—
119C22×Q16C22×C311yescharacteristic
120C22×SD16C22×C311yescharacteristic
121#556—24no—
122#556—24no—
123C2×C4⋊Q8C22×C311yescharacteristic
124C2×#126—22no—
125C2×#124—21no—
126C2×#124C22×C311yescharacteristic
127C2×C22⋊Q8C22×C312yes—
128C2×C22⋊Q8C22×C312yes—
129C2×#121C22×C312yes—
130C2×C4×Q8—21no—
131C2×C4×Q8C22×C311yescharacteristic
132C2×C4×D4C22×C311yescharacteristic
133C2×#117C22×C312yes—
134#489—24no—
135#464—24no—
136C2×C4⋊C8C22×C311yescharacteristic
137C2×#103C22×C312yes—
138C2×#103C22×C311yescharacteristic
139C2×#103C22×C312yes—
140C2×#102C22×C312yes—
141C2×#102C22×C311yescharacteristic
142C2×C22⋊C8C22×C312yes—
143C2×C22⋊C8C22×C312yes—
144C2×#97C22×C311yescharacteristic
145#399—21no—
146#392—21no—
147#390—22no—
148C4×C4⋊C4C22×C311yescharacteristic
149#340C22×C312yes—
150#340C22×C312yes—
151#340C22×C312yes—
152C24×C3—21no—
153C24×C3C2×D412yes—
154C2×C3×Q8—24no—
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—44no—
163C2×C3×Q8C2×D412yes—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4—44no—
171C2×C3×D4—44no—
172C2×C3×D4—82no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C3×Q16—44no—
189C3×SD16—44no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4C2×D412yes—
206C3×C4⋊C4C2×D414yes—
207C3×C4⋊C4#4114yes—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—44no—
212C3×C22⋊C4—44no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C42—28no—
216C3×C42—24no—
217C3×C42—81no—
218C3×C42C2×D412yes—
219C3×C42C2×D412yes—
220C22×Q8—21no—
221C22×Q8C23×C311yescharacteristic
222C22×Q8C23×C311yescharacteristic
223C22×D4—22no—
224C22×D4C3×D411yescharacteristic
225C22×D4C23×C311yescharacteristic
226C23×C4C23×C312yes—
227C2×Q16—22no—
228C2×Q16—42no—
229C2×SD16—22no—
230C2×SD16—42no—
231C22×C8C23×C311yescharacteristic
232C22×C8C23×C311yescharacteristic
233C22×C8C23×C312yes—
234C4⋊Q8—22no—
235C4⋊Q8C3×D414yescomplemented
236#126—216no—
237#124—28no—
238#124—22no—
239#124—22no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—24no—
243C22⋊Q8—42no—
244#121—24no—
245#121—42no—
246C4×Q8—28no—
247C4×Q8—22no—
248C4×Q8C3×D414yescomplemented
249C4×D4—22no—
250C4×D4—22no—
251#117—24no—
252#117—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C3×D411yescharacteristic
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C312yes—
260C2×C4⋊C4C23×C312yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4C3×D412yes—
268C2×C42—22no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42C23×C311yescharacteristic
272C4⋊C8—22no—
273#103—22no—
274#103—24no—
275#103—24no—
276#102—22no—
277#102—24no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280#97—22no—
281#95—21no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—41no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3C22×D411yescharacteristic
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—48no—
298C3×Q8—42no—
299C3×Q8—41no—
300C3×Q8—41no—
301C3×Q8—42no—
302C3×D4—44no—
303C3×D4—41no—
304C3×D4—41no—
305C3×D4—42no—
306C3×D4—88no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—44no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4#12012yes—
330C2×C3×C4#12012yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4112yes—
334C2×C3×C4H(5,𝔽2)12yes—
335C3×C8—41no—
336C3×C8—41no—
337C3×C8—42no—
338C24—21no—
339C24C2×C3×D412yes—
340C2×Q8—24no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—44no—
349C2×Q8C2×C3×D412yes—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—82no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4C2×C3×D412yes—
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C2×C3×D411yescharacteristic
373C22×C4C24×C311yescharacteristic
374Q16—44no—
375SD16—44no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C4⋊C4—28no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—44no—
391C4⋊C4C2×C3×D412yes—
392C4⋊C4C2×C3×D414yes—
393C4⋊C4C3×#4114yes—
394C22⋊C4—28no—
395C22⋊C4—28no—
396C22⋊C4—24no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C42—28no—
402C42—24no—
403C42—81no—
404C42C2×C3×D412yes—
405C42C2×C3×D412yes—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3—81no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3C2×#12011yescharacteristic
419C22×C3#54612yes—
420C22×C3#57512yes—
421C22×C3#57711yescharacteristic
422C22×C3#57811yescharacteristic
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—44no—
428C3×C4—42no—
429C3×C4—44no—
430C3×C4—81no—
431C3×C4#53211yescharacteristic
432C3×C4#53511yescharacteristic
433C3×C4#55012yes—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23C22×C3×D411yescharacteristic
447Q8—22no—
448Q8—22no—
449Q8—48no—
450Q8—42no—
451Q8—41no—
452Q8—41no—
453Q8—42no—
454D4—44no—
455D4—41no—
456D4—41no—
457D4—42no—
458D4—88no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4C3×#12012yes—
482C2×C4C3×#12012yes—
483C2×C4C22×C3×D411yescharacteristic
484C2×C4C22×C3×D411yescharacteristic
485C2×C4C2×C3×#4112yes—
486C2×C4C3×H(5,𝔽2)12yes—
487C8—41no—
488C8—41no—
489C8—42no—
490C2×C3—82no—
491C2×C3—161no—
492C2×C3#243111yescharacteristic
493C2×C3#300511yescharacteristic
494C2×C3#301311yescharacteristic
495C2×C3#331112yes—
496C2×C3#331312yes—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—82no—
508C22—82no—
509C22C2×C3×#12011yescharacteristic
510C22C3×#54612yes—
511C22C3×#57512yes—
512C22C3×#57711yescharacteristic
513C22C3×#57811yescharacteristic
514C4—22no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—44no—
519C4—42no—
520C4—44no—
521C4—81no—
522C4C3×#53211yescharacteristic
523C4C3×#53511yescharacteristic
524C4C3×#55012yes—
525C3#2558011yescharacteristic, direct product factor, Sylow
526C2—82no—
527C2—161no—
528C2C3×#243111yescharacteristic
529C2C3×#300511yescharacteristic
530C2C3×#301311yescharacteristic
531C2C3×#331112yes—
532C2C3×#331312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#556C2semidirect
2. C2×C3×#489C2semidirect
3. C2×C3×#464C2semidirect
5. C3×#2001C2semidirect
6. C3×#2000C2semidirect
7. C3×#1987C2semidirect
8. C3×#1955C2semidirect
9. C3×#1875C2semidirect
10. C3×#1873C2semidirect
11. C3×#1763C2semidirect
12. #25580C3direct (N × H)
46. C2×#556C2×C3semidirect
47. C2×#489C2×C3semidirect
48. C2×#464C2×C3semidirect
50. #2001C2×C3semidirect
51. #2000C2×C3semidirect
52. #1987C2×C3semidirect
53. #1955C2×C3semidirect
54. #1875C2×C3semidirect
55. #1873C2×C3semidirect
56. #1763C2×C3semidirect
72. C3×C4⋊Q8D4semidirect
85. C3×C4×Q8D4semidirect
235. C4⋊Q8C3×D4semidirect
248. C4×Q8C3×D4semidirect
525. C3#25580direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4892
C2×C3×#4642
C3×#25402
C3×#20012
C3×#20002
C3×#19872
C3×#19552
C3×#18752
C3×#18732
C3×#17632
#255803