Finite Group Databasebeta

← C3×#25789 #10644396

Subgroups — C3×#25789 #10644396

Order 768.

1500 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Q16C211yesmaximal, characteristic, complemented
2C2×C3×#497C212yesmaximal, complemented
3C2×C3×#465C211yesmaximal, characteristic
4C2×C3×#451C211yesmaximal, characteristic, complemented
5C2×C3×#448C211yesmaximal, characteristic
6C3×#2436C211yesmaximal, characteristic
7C3×#2434C211yesmaximal, characteristic, complemented
8C3×#2016C212yesmaximal, complemented
9C3×#2015C212yesmaximal, complemented
10C3×#1971C212yesmaximal, complemented
11C3×#1752C211yesmaximal, characteristic, complemented
12#25789C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×Q16—28no—
16C22×C3×Q16—24no—
17C22×C3×Q16C2212yes—
18C22×C3×Q16C2212yes—
19C22×C3×Q16C2212yes—
20C22×C3×SD16C2211yescharacteristic
21C22×C3×SD16C2212yes—
22C23×C3×C8C2211yescharacteristic
23C2×C3×C4⋊Q8C2212yes—
24C2×C3×#124C2212yes—
25C2×C3×C22⋊Q8C2211yescharacteristic
26C2×C3×#121C2211yescharacteristic
27C3×#497—28no—
28C3×#465—24no—
29C3×#451—24no—
30C3×#448—24no—
31C2×C3×#106C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#102C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×#97C2212yes—
38C3×#395—22no—
39C3×#393—22no—
40C3×#391C2212yes—
41C3×#391C2212yes—
42C3×#386—22no—
43C3×#385—22no—
44C3×C4×C22⋊C4C2211yescharacteristic
45C3×#336C2212yes—
46C23×Q16C2×C311yescharacteristic, complemented
47C2×#497C2×C312yescomplemented
48C2×#465C2×C311yescharacteristic
49C2×#451C2×C311yescharacteristic, complemented
50C2×#448C2×C311yescharacteristic
51#2436C2×C311yescharacteristic
52#2434C2×C311yescharacteristic, complemented
53#2016C2×C312yescomplemented
54#2015C2×C312yescomplemented
55#1971C2×C312yescomplemented
56#1752C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8—24no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8—22no—
65C22×C3×Q8—42no—
66C22×C3×Q8—42no—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2312yes—
69C22×C3×Q8C2312yes—
70C22×C3×D4—21no—
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—21no—
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C2×C3×Q16—232no—
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×Q16—24no—
79C2×C3×Q16—24no—
80C2×C3×Q16—24no—
81C2×C3×Q16—24no—
82C2×C3×Q16—24no—
83C2×C3×Q16—42no—
84C2×C3×Q16D418yescomplemented
85C2×C3×SD16—24no—
86C2×C3×SD16—24no—
87C2×C3×SD16—22no—
88C2×C3×SD16—24no—
89C2×C3×SD16—42no—
90C22×C3×C8—24no—
91C22×C3×C8D414yes—
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2312yes—
94C22×C3×C8C2311yescharacteristic
95C3×C4⋊Q8—24no—
96C3×C4⋊Q8—42no—
97C3×#124—24no—
98C3×#124—42no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8—22no—
101C3×#121—22no—
102C3×#121—22no—
103C2×C3×C4⋊C4—24no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—24no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4D412yes—
111C2×C3×C42C2312yes—
112C3×#106—22no—
113C3×#103—24no—
114C3×#103—24no—
115C3×#103—22no—
116C3×#102—22no—
117C3×C22⋊C8D414yes—
118C3×#97—24no—
119C3×#95—24no—
120C3×#95—24no—
121C3×#95D412yes—
122C23×Q8C22×C311yescharacteristic
123C23×Q8C22×C311yescharacteristic
124C22×Q16—28no—
125C22×Q16—24no—
126C22×Q16C22×C312yes—
127C22×Q16C22×C312yes—
128C22×Q16C22×C312yes—
129C22×SD16C22×C311yescharacteristic
130C22×SD16C22×C312yes—
131C23×C8C22×C311yescharacteristic
132C2×C4⋊Q8C22×C312yes—
133C2×#124C22×C312yes—
134C2×C22⋊Q8C22×C311yescharacteristic
135C2×#121C22×C311yescharacteristic
136#497—28no—
137#465—24no—
138#451—24no—
139#448—24no—
140C2×#106C22×C311yescharacteristic
141C2×#103C22×C312yes—
142C2×#103C22×C312yes—
143C2×#103C22×C311yescharacteristic
144C2×#102C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×#97C22×C312yes—
147#395—22no—
148#393—22no—
149#391C22×C312yes—
150#391C22×C312yes—
151#386—22no—
152#385—22no—
153C4×C22⋊C4C22×C311yescharacteristic
154#336C22×C312yes—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—28no—
160C2×C3×Q8—28no—
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×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—44no—
174C2×C3×Q8—44no—
175C2×C3×Q8—48no—
176C2×C3×Q8—48no—
177C2×C3×Q8—44no—
178C2×C3×Q8—44no—
179C2×C3×Q8—42no—
180C2×C3×Q8—42no—
181C2×C3×Q8C2×D414yes—
182C2×C3×Q8C2×D414yes—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—42no—
188C2×C3×D4—44no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—44no—
197C22×C3×C4—44no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4C2×D414yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2411yescharacteristic
205C3×Q16—216no—
206C3×Q16—216no—
207C3×Q16—48no—
208C3×SD16—48no—
209C3×SD16—44no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—28no—
214C2×C3×C8C2×D414yes—
215C2×C3×C8C2×D412yes—
216C2×C3×C8C2×D414yes—
217C2×C3×C8C2×D412yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—48no—
223C3×C4⋊C4—42no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—48no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C42—22no—
229C3×C42—22no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8—24no—
233C22×Q8—24no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8C23×C311yescharacteristic
241C22×Q8C23×C312yes—
242C22×Q8C23×C312yes—
243C22×D4—21no—
244C22×D4C23×C311yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C23×C311yescharacteristic
248C2×Q16—232no—
249C2×Q16—24no—
250C2×Q16—24no—
251C2×Q16—24no—
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16—24no—
255C2×Q16—24no—
256C2×Q16—42no—
257C2×Q16C3×D418yescomplemented
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—22no—
261C2×SD16—24no—
262C2×SD16—42no—
263C22×C8—24no—
264C22×C8C3×D414yes—
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C312yes—
267C22×C8C23×C311yescharacteristic
268C4⋊Q8—24no—
269C4⋊Q8—42no—
270#124—24no—
271#124—42no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274#121—22no—
275#121—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4C3×D412yes—
284C2×C42C23×C312yes—
285#106—22no—
286#103—24no—
287#103—24no—
288#103—22no—
289#102—22no—
290C22⋊C8C3×D414yes—
291#97—24no—
292#95—24no—
293#95—24no—
294#95C3×D412yes—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3#12712yes—
302C23×C3#13714yes—
303C23×C3C22×D411yescharacteristic
304C3×Q8—24no—
305C3×Q8—24no—
306C3×Q8—24no—
307C3×Q8—24no—
308C3×Q8—48no—
309C3×Q8—48no—
310C3×Q8—42no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×Q8—42no—
314C3×Q8—41no—
315C3×Q8—41no—
316C3×Q8—42no—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—42no—
320C3×D4—84no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—48no—
331C2×C3×C4—48no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4#12712yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D412yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4H(5,𝔽2)12yes—
347C3×C8—22no—
348C3×C8—24no—
349C3×C8#12014yes—
350C24—21no—
351C24C2×C3×D411yescharacteristic
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—28no—
355C2×Q8—28no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8C2×C3×D414yes—
377C2×Q8C2×C3×D414yes—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—42no—
383C2×D4—44no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4C2×C3×D414yes—
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D412yes—
398C22×C4C2×C3×D412yes—
399C22×C4C24×C311yescharacteristic
400Q16—216no—
401Q16—216no—
402Q16—48no—
403SD16—48no—
404SD16—44no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—28no—
409C2×C8C2×C3×D414yes—
410C2×C8C2×C3×D412yes—
411C2×C8C2×C3×D414yes—
412C2×C8C2×C3×D412yes—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—48no—
418C4⋊C4—42no—
419C22⋊C4—24no—
420C22⋊C4—48no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C42—22no—
424C42—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—28no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—84no—
432C22×C3#49714yescomplemented
433C22×C3C2×#12711yescharacteristic
434C22×C3D4212yes—
435C22×C3C2×#13712yes—
436C22×C3#57812yes—
437C3×C4—22no—
438C3×C4—44no—
439C3×C4—44no—
440C3×C4—44no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4—81no—
444C3×C4C2×#12011yescharacteristic
445C3×C4#53511yescharacteristic
446C3×C4D4212yes—
447C23—22no—
448C23—22no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—42no—
453C23C3×#12712yes—
454C23C3×#13714yes—
455C23C22×C3×D411yescharacteristic
456Q8—24no—
457Q8—24no—
458Q8—24no—
459Q8—24no—
460Q8—48no—
461Q8—48no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465Q8—42no—
466Q8—41no—
467Q8—41no—
468Q8—42no—
469D4—41no—
470D4—41no—
471D4—42no—
472D4—84no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4C3×#12712yes—
495C2×C4C22×C3×D411yescharacteristic
496C2×C4C22×C3×D412yes—
497C2×C4C22×C3×D411yescharacteristic
498C2×C4C3×H(5,𝔽2)12yes—
499C8—22no—
500C8—24no—
501C8C3×#12014yes—
502C2×C3—24no—
503C2×C3—161no—
504C2×C3#243311yescharacteristic
505C2×C3C2×#49711yescharacteristic
506C2×C3#315611yescharacteristic
507C2×C3#327014yes—
508C22—22no—
509C22—22no—
510C22—28no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—84no—
515C22C3×#49714yescomplemented
516C22C2×C3×#12711yescharacteristic
517C22C3×D4212yes—
518C22C2×C3×#13712yes—
519C22C3×#57812yes—
520C4—22no—
521C4—44no—
522C4—44no—
523C4—44no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4C2×C3×#12011yescharacteristic
528C4C3×#53511yescharacteristic
529C4C3×D4212yes—
530C3#2578911yescharacteristic, direct product factor, Sylow
531C2—24no—
532C2—161no—
533C2C3×#243311yescharacteristic
534C2C2×C3×#49711yescharacteristic
535C2C3×#315611yescharacteristic
536C2C3×#327014yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C3×Q16C2semidirect
2. C2×C3×#497C2semidirect
4. C2×C3×#451C2semidirect
7. C3×#2434C2semidirect
8. C3×#2016C2semidirect
9. C3×#2015C2semidirect
10. C3×#1971C2semidirect
11. C3×#1752C2semidirect
12. #25789C3direct (N × H)
46. C23×Q16C2×C3semidirect
47. C2×#497C2×C3semidirect
49. C2×#451C2×C3semidirect
52. #2434C2×C3semidirect
53. #2016C2×C3semidirect
54. #2015C2×C3semidirect
55. #1971C2×C3semidirect
56. #1752C2×C3semidirect
84. C2×C3×Q16D4semidirect
257. C2×Q16C3×D4semidirect
432. C22×C3#497semidirect
515. C22C3×#497semidirect
530. C3#25789direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×Q162
C2×C3×#4972
C2×C3×#4652
C2×C3×#4512
C2×C3×#4482
C3×#24362
C3×#24342
C3×#20162
C3×#20152
C3×#19712
C3×#17522
#257893