Finite Group Databasebeta

← C3×#25733 #10644340

Subgroups — C3×#25733 #10644340

Order 768.

1566 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3C2×C3×#492C211yesmaximal, characteristic, complemented
4C2×C3×#449C212yesmaximal, complemented
5C2×C3×#448C212yesmaximal, complemented
6C3×#2619C211yesmaximal, characteristic, complemented
7C3×#2013C212yesmaximal, complemented
8C3×#2012C212yesmaximal, complemented
9C3×#1960C211yesmaximal, characteristic, complemented
10C3×#1958C211yesmaximal, characteristic, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25733C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C22×C3×SD16C2212yes—
17C22×C3×SD16C2212yes—
18C22×C3×D8C2212yes—
19C3×#536—24no—
20C3×#536—24no—
21C2×C3×C4⋊Q8C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×#127C2211yescharacteristic
24C2×C3×#124C2211yescharacteristic
25C2×C3×#124C2211yescharacteristic
26C2×C3×#123—22no—
27C2×C3×C22⋊Q8—22no—
28C2×C3×#121C2212yes—
29C2×C3×#117C2211yescharacteristic
30C3×#496—24no—
31C3×#492—24no—
32C3×#449—28no—
33C3×#448—28no—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#103C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C2×C3×#97C2211yescharacteristic
40C2×C3×C4×C8C2211yescharacteristic
41C3×#394—22no—
42C3×#392—22no—
43C3×#390C2212yes—
44C3×#382C2211yescharacteristic
45C3×#336C2212yes—
46C2×#536C2×C311yescharacteristic, complemented
47C2×#496C2×C311yescharacteristic, complemented
48C2×#492C2×C311yescharacteristic, complemented
49C2×#449C2×C312yescomplemented
50C2×#448C2×C312yescomplemented
51#2619C2×C311yescharacteristic, complemented
52#2013C2×C312yescomplemented
53#2012C2×C312yescomplemented
54#1960C2×C311yescharacteristic, complemented
55#1958C2×C311yescharacteristic, complemented
56#1750C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—24no—
60C2×C3×#41—42no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—21no—
64C22×C3×Q8—24no—
65C22×C3×Q8—42no—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—21no—
70C22×C3×D4—22no—
71C22×C3×D4—22no—
72C22×C3×D4C2312yes—
73C22×C3×D4C2311yescharacteristic
74C23×C3×C4—21no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×SD16—42no—
84C2×C3×SD16—42no—
85C2×C3×SD16—42no—
86C2×C3×D8—24no—
87C2×C3×D8—24no—
88C2×C3×D8—42no—
89C22×C3×C8C2312yes—
90C22×C3×C8C2312yes—
91C3×C4⋊Q8—22no—
92C3×C4⋊Q8—22no—
93C3×C4⋊Q8—22no—
94C3×C4⋊Q8D414yescomplemented
95C3×#127—22no—
96C3×#127—22no—
97C3×#124—22no—
98C3×#124—22no—
99C3×#124—22no—
100C3×#124D414yescomplemented
101C3×#123—216no—
102C3×C22⋊Q8—216no—
103C3×#121—24no—
104C3×#121—24no—
105C3×#117—22no—
106C3×#117—22no—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4D412yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—24no—
115C2×C3×C22⋊C4D412yes—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×#103—24no—
119C3×#103—24no—
120C3×#103—24no—
121C3×#102—24no—
122C3×C22⋊C8D418yescomplemented
123C3×#97—22no—
124C3×C4×C8—22no—
125C3×#95—22no—
126C22×#41C22×C311yescharacteristic
127C23×Q8C22×C311yescharacteristic
128C22×SD16C22×C312yes—
129C22×SD16C22×C312yes—
130C22×SD16C22×C312yes—
131C22×D8C22×C312yes—
132#536—24no—
133#536—24no—
134C2×C4⋊Q8C22×C311yescharacteristic
135C2×C4⋊Q8C22×C311yescharacteristic
136C2×#127C22×C311yescharacteristic
137C2×#124C22×C311yescharacteristic
138C2×#124C22×C311yescharacteristic
139C2×#123—22no—
140C2×C22⋊Q8—22no—
141C2×#121C22×C312yes—
142C2×#117C22×C311yescharacteristic
143#496—24no—
144#492—24no—
145#449—28no—
146#448—28no—
147C2×#103C22×C312yes—
148C2×#103C22×C312yes—
149C2×#103C22×C312yes—
150C2×#102C22×C312yes—
151C2×C22⋊C8C22×C312yes—
152C2×#97C22×C311yescharacteristic
153C2×C4×C8C22×C311yescharacteristic
154#394—22no—
155#392—22no—
156#390C22×C312yes—
157#382C22×C311yescharacteristic
158#336C22×C312yes—
159C24×C3—21no—
160C24×C3—22no—
161C24×C3C2×D411yescharacteristic
162C3×#41—48no—
163C3×#41—48no—
164C2×C3×Q8—24no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—48no—
175C2×C3×Q8—48no—
176C2×C3×Q8—42no—
177C2×C3×Q8—42no—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8C2×D412yes—
181C2×C3×D4—24no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—42no—
192C2×C3×D4—42no—
193C2×C3×D4—44no—
194C2×C3×D4—44no—
195C2×C3×D4—84no—
196C2×C3×D4—84no—
197C2×C3×D4C2×D412yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—22no—
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D414yes—
215C22×C3×C4C2411yescharacteristic
216C3×SD16—48no—
217C3×SD16—48no—
218C3×SD16—48no—
219C3×D8—48no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8C2×D414yes—
225C2×C3×C8C2×D414yes—
226C3×C4⋊C4—28no—
227C3×C4⋊C4—28no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—28no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—44no—
235C3×C22⋊C4—84no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—24no—
243C2×#41—42no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—21no—
247C22×Q8—24no—
248C22×Q8—42no—
249C22×Q8C23×C311yescharacteristic
250C22×Q8C23×C311yescharacteristic
251C22×Q8C23×C311yescharacteristic
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4C23×C312yes—
256C22×D4C23×C311yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C23×C311yescharacteristic
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C2×SD16—42no—
268C2×SD16—42no—
269C2×D8—24no—
270C2×D8—24no—
271C2×D8—42no—
272C22×C8C23×C312yes—
273C22×C8C23×C312yes—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8C3×D414yescomplemented
278#127—22no—
279#127—22no—
280#124—22no—
281#124—22no—
282#124—22no—
283#124C3×D414yescomplemented
284#123—216no—
285C22⋊Q8—216no—
286#121—24no—
287#121—24no—
288#117—22no—
289#117—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C3×D412yes—
293C2×C4⋊C4C23×C312yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4C3×D412yes—
299C2×C42C23×C311yescharacteristic
300C2×C42C23×C311yescharacteristic
301#103—24no—
302#103—24no—
303#103—24no—
304#102—24no—
305C22⋊C8C3×D418yescomplemented
306#97—22no—
307C4×C8—22no—
308#95—22no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—44no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3#12712yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—24no—
324C3×Q8—24no—
325C3×Q8—24no—
326C3×Q8—48no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×D4—24no—
331C3×D4—48no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—42no—
337C3×D4—88no—
338C3×D4—88no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—48no—
350C2×C3×C4—48no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—44no—
359C2×C3×C4#12014yes—
360C2×C3×C4#12712yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D412yes—
364C2×C3×C4H(5,𝔽2)12yes—
365C3×C8—24no—
366C3×C8—24no—
367C24—21no—
368C24—22no—
369C24C2×C3×D411yescharacteristic
370#41—48no—
371#41—48no—
372C2×Q8—24no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8C2×C3×D412yes—
387C2×Q8C2×C3×D412yes—
388C2×Q8C2×C3×D412yes—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—84no—
404C2×D4—84no—
405C2×D4C2×C3×D412yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4C2×C3×D412yes—
420C22×C4C2×C3×D412yes—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D414yes—
423C22×C4C24×C311yescharacteristic
424SD16—48no—
425SD16—48no—
426SD16—48no—
427D8—48no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8C2×C3×D414yes—
433C2×C8C2×C3×D414yes—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C22⋊C4—28no—
440C22⋊C4—28no—
441C22⋊C4—24no—
442C22⋊C4—44no—
443C22⋊C4—84no—
444C42—21no—
445C42—21no—
446C42C2×C3×D412yes—
447C42C2×C3×D412yes—
448C22×C3—24no—
449C22×C3—22no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—44no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3—84no—
460C22×C3—84no—
461C22×C3C2×#12011yescharacteristic
462C22×C3C2×#12711yescharacteristic
463C22×C3#53511yescharacteristic
464C22×C3#57714yes—
465C3×C4—22no—
466C3×C4—24no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—82no—
472C3×C4D4212yes—
473C3×C4D4212yes—
474C23—24no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—44no—
479C23—41no—
480C23—41no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23C3×#12712yes—
487C23C22×C3×D411yescharacteristic
488Q8—24no—
489Q8—24no—
490Q8—24no—
491Q8—48no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—88no—
503D4—88no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4C3×#12014yes—
525C2×C4C3×#12712yes—
526C2×C4C22×C3×D411yescharacteristic
527C2×C4C22×C3×D411yescharacteristic
528C2×C4C22×C3×D412yes—
529C2×C4C3×H(5,𝔽2)12yes—
530C8—24no—
531C8—24no—
532C2×C3—42no—
533C2×C3—82no—
534C2×C3—162no—
535C2×C3#243311yescharacteristic
536C2×C3#301012yes—
537C2×C3#315412yes—
538C2×C3#322112yes—
539C22—24no—
540C22—22no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—44no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—84no—
551C22—84no—
552C22C2×C3×#12011yescharacteristic
553C22C2×C3×#12711yescharacteristic
554C22C3×#53511yescharacteristic
555C22C3×#57714yes—
556C4—22no—
557C4—24no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—82no—
563C4C3×D4212yes—
564C4C3×D4212yes—
565C3#2573311yescharacteristic, direct product factor, Sylow
566C2—42no—
567C2—82no—
568C2—162no—
569C2C3×#243311yescharacteristic
570C2C3×#301012yes—
571C2C3×#315412yes—
572C2C3×#322112yes—
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×#536C2semidirect
2. C2×C3×#496C2semidirect
3. C2×C3×#492C2semidirect
4. C2×C3×#449C2semidirect
5. C2×C3×#448C2semidirect
6. C3×#2619C2semidirect
7. C3×#2013C2semidirect
8. C3×#2012C2semidirect
9. C3×#1960C2semidirect
10. C3×#1958C2semidirect
11. C3×#1750C2semidirect
12. #25733C3direct (N × H)
46. C2×#536C2×C3semidirect
47. C2×#496C2×C3semidirect
48. C2×#492C2×C3semidirect
49. C2×#449C2×C3semidirect
50. C2×#448C2×C3semidirect
51. #2619C2×C3semidirect
52. #2013C2×C3semidirect
53. #2012C2×C3semidirect
54. #1960C2×C3semidirect
55. #1958C2×C3semidirect
56. #1750C2×C3semidirect
94. C3×C4⋊Q8D4semidirect
100. C3×#124D4semidirect
122. C3×C22⋊C8D4semidirect
277. C4⋊Q8C3×D4semidirect
283. #124C3×D4semidirect
305. C22⋊C8C3×D4semidirect
565. C3#25733direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#4962
C2×C3×#4922
C2×C3×#4492
C2×C3×#4482
C3×#26192
C3×#20132
C3×#20122
C3×#19602
C3×#19582
C3×#17502
#257333