Finite Group Databasebeta

← #13178758

Subgroups — #13178758

Order 1728.

1630 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C4×#1113C211yesmaximal, characteristic
3C3×#64155C212yesmaximal, complemented
4#11723700C212yesmaximal
5#11723689C211yesmaximal, characteristic, complemented
6#11723688C214yesmaximal, complemented
7#11723682C212yesmaximal, complemented
8#11723674C212yesmaximal, complemented
9#10595429—31nomaximal
10D3×#946—32nomaximal
11C22×#1113C2211yescharacteristic
12C2×C33×Q8C2211yescharacteristic, complemented
13C3×Q8×C3⋊D3—24no—
14C2×C3×C4×C3⋊D3C2212yes—
15C2×C3×C4×C3⋊D3C2212yes—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic, complemented
18C2×C3×#634C2212yescomplemented
19C2×#6294C2211yescharacteristic
20C2×#6294C2212yes—
21C2×#6294C2212yes—
22C4×#1113—24no—
23C3×#3752C2214yes—
24C3×C4⋊(C3⋊Dic3)C2212yescomplemented
25C3×#3750C2214yescomplemented
26#88815C2214yescomplemented
27#88815C2212yescomplemented
28#88814C2214yes—
29C2×Q8×C3⋊D3D311yescharacteristic, complemented
30C2×C3×D3×Q8—32no—
31C2×C3×D3×Q8—61no—
32C2×C4×D32—32no—
33C2×C4×D32—31no—
34#64155D312yescomplemented
35C3×#946—32no—
36C3×#946—32no—
37C3×#946—62no—
38#63921—32no—
39D3×(C2×D3)⋊C4—34no—
40#63900—31no—
41#63896—34no—
42D3×C4⋊Dic3—32no—
43#63888—34no—
44#63887—34no—
45#63883—32no—
46D3×Dic3⋊C4—34no—
47#63871—32no—
48#63857—32no—
49#63856—32no—
50C22×C3×C3⋊D3C2312yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C2×#1113—22no—
53C2×#1113—22no—
54C2×#1113Q814yescomplemented
55C33×Q8—22no—
56C2×C33×C4C2312yes—
57C2×C33×C4C2311yescharacteristic
58C2×C3×C3⋊Dic3C2312yes—
59C2×C3×C3⋊Dic3C2314yes—
60C2×C3×C3⋊Dic3C2312yes—
61C2×C3×C3⋊Dic3C2311yescharacteristic
62C3×C4×C3⋊D3—24no—
63C3×C4×C3⋊D3—24no—
64C3×C4×C3⋊D3D414yes—
65C3×#634—22no—
66C3×#634—24no—
67#6294—22no—
68#6294—24no—
69#6294—24no—
70C2×#946—91no—
71D3×C22⋊Q8—92no—
72C22×D32—32no—
73C22×D32—31no—
74C2×C32×Q8—21no—
75C2×C32×Q8—22no—
76C2×C32×Q8—41no—
77C2×C32×Q8C2×D311yescharacteristic, complemented
78C2×C32×Q8C2×D312yescomplemented
79Q8×C3⋊D3—24no—
80C2×C4×C3⋊D3C2×D312yes—
81C2×C4×C3⋊D3C2×D312yes—
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×#634C2×D311yescharacteristic, complemented
84C2×#634C2×D312yescomplemented
85C3×D3×Q8—68no—
86C3×D3×Q8—68no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—34no—
91C2×C3×C4×D3—61no—
92C2×C3×C4×D3—62no—
93C2×C3×C4×D3—62no—
94C2×C3×Dic6—32no—
95C2×C3×Dic6—34no—
96C2×C3×Dic6—61no—
97C2×C3×Dic6—62no—
98C2×#624—31no—
99C2×#624—32no—
100C2×#624—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—32no—
104C2×D3×Dic3—34no—
105C4×D32—68no—
106C4×D32—64no—
107#3752C2×D314yes—
108C4⋊(C3⋊Dic3)C2×D312yescomplemented
109#3750C2×D314yescomplemented
110C3×(C2×D3)⋊C4—34no—
111C3×(C2×D3)⋊C4—34no—
112C3×(C2×D3)⋊C4—64no—
113C3×C4⋊Dic3—32no—
114C3×C4⋊Dic3—32no—
115C3×C4⋊Dic3—62no—
116C3×Dic3⋊C4—34no—
117C3×Dic3⋊C4—34no—
118C3×Dic3⋊C4—64no—
119#3724—34no—
120#3724—32no—
121Dic3⋊Dic3—34no—
122Dic3⋊Dic3—34no—
123Dic3⋊Dic3—34no—
124#3722—34no—
125(C2×D3)⋊Dic3—34no—
126(C2×D3)⋊Dic3—34no—
127C22×C33C2411yescharacteristic
128C2×C3×C3⋊D3—22no—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3C2×D412yes—
131C2×C3×C3⋊D3C2×Q812yes—
132C2×C3×C3⋊D3C2×Q814yescomplemented
133C2×C3×C3⋊D3#4112yes—
134#1113—28no—
135C33×C4—22no—
136C33×C4C2×D412yes—
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—24no—
140C3×C3⋊Dic3C2×D412yes—
141C22×C3×Q8—91no—
142C2×D3×Q8—32no—
143C2×D3×Q8—61no—
144C22×C4×D3—91no—
145C22×C4×D3—92no—
146C3×C22⋊Q8—92no—
147#946—32no—
148#946—32no—
149#946—62no—
150#946—94no—
151#946—92no—
152C2×(C2×D3)⋊C4—92no—
153C2×C4⋊Dic3—91no—
154#924—92no—
155C2×Dic3⋊C4—92no—
156#897—92no—
157#896—94no—
158D3×C4⋊C4—94no—
159D3×C4⋊C4—92no—
160D3×C22⋊C4—94no—
161#878—94no—
162C22×C3⋊D3C22×D312yes—
163C22×C3⋊D3C22×D311yescharacteristic
164C22×C3×D3—32no—
165C22×C3×D3—32no—
166C22×C3×D3—32no—
167C22×C3×D3—61no—
168C22×C3×D3—62no—
169C2×D32—34no—
170C2×D32—38no—
171C2×D32—64no—
172C2×D32—64no—
173C2×D32—62no—
174C2×D32—62no—
175C32×Q8—22no—
176C32×Q8—24no—
177C32×Q8—24no—
178C32×Q8—44no—
179C32×Q8—44no—
180C2×C32×C4—22no—
181C2×C32×C4—22no—
182C2×C32×C4—21no—
183C2×C32×C4—24no—
184C2×C32×C4—41no—
185C2×C32×C4—42no—
186C2×C32×C4C22×D312yes—
187C2×C32×C4C22×D314yes—
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D314yes—
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C4×C3⋊D3—24no—
195C4×C3⋊D3—24no—
196C4×C3⋊D3#6714yes—
197#634—22no—
198#634—24no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—64no—
209C3×C4×D3—38no—
210C3×C4×D3—64no—
211C3×C4×D3—68no—
212C3×C4×D3—68no—
213C3×C4×D3—64no—
214C3×C4×D3—64no—
215C3×C4×D3—124no—
216C3×Dic6—68no—
217C3×Dic6—68no—
218C3×Dic6—64no—
219C3×Dic6—64no—
220#624—64no—
221#624—64no—
222#624—62no—
223D3×Dic3—64no—
224D3×Dic3—64no—
225D3×Dic3—68no—
226D3×Dic3—64no—
227C2×C22⋊Q8—271noSylow
228C2×C33C22×D411yescharacteristic
229C2×C33C22×Q811yescharacteristic
230C2×C33C2×#4111yescharacteristic
231C3×C3⋊D3—24no—
232C3×C3⋊D3C22⋊Q814yescomplemented
233C23×D3—92no—
234C23×D3—91no—
235C2×C3×Q8—21no—
236C2×C3×Q8—22no—
237C2×C3×Q8—41no—
238C2×C3×Q8—92no—
239C2×C3×Q8—91no—
240C2×C3×Q8—184no—
241C2×C3×Q8D3211yescharacteristic, complemented
242C2×C3×Q8D3212yescomplemented
243C22×C3×C4—91no—
244C22×C3×C4—92no—
245C22×C3×C4—92no—
246C22×Dic3—92no—
247C22×Dic3—91no—
248C22×Dic3—92no—
249D3×Q8—68no—
250D3×Q8—68no—
251C2×C4×D3—32no—
252C2×C4×D3—34no—
253C2×C4×D3—32no—
254C2×C4×D3—32no—
255C2×C4×D3—61no—
256C2×C4×D3—62no—
257C2×C4×D3—62no—
258C2×C4×D3—94no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—92no—
262C2×C4×D3—188no—
263C2×C4×D3—184no—
264C2×Dic6—34no—
265C2×Dic6—32no—
266C2×Dic6—61no—
267C2×Dic6—62no—
268C3×C4⋊C4—94no—
269C3×C4⋊C4—92no—
270C3×C22⋊C4—94no—
271#217—94no—
272(C2×D3)⋊C4—34no—
273(C2×D3)⋊C4—34no—
274(C2×D3)⋊C4—64no—
275(C2×D3)⋊C4—94no—
276(C2×D3)⋊C4—94no—
277C4⋊Dic3—32no—
278C4⋊Dic3—32no—
279C4⋊Dic3—62no—
280C4⋊Dic3—94no—
281C4⋊Dic3—92no—
282Dic3⋊C4—34no—
283Dic3⋊C4—34no—
284Dic3⋊C4—64no—
285Dic3⋊C4—94no—
286Dic3⋊C4—94no—
287Dic3⋊C4—94no—
288C22×C32—22no—
289C22×C32—21no—
290C22×C32—41no—
291C22×C32C23×D312yes—
292C22×C32C23×D311yescharacteristic
293C2×C3⋊D3—22no—
294C2×C3⋊D3—22no—
295C2×C3⋊D3D3×Q812yes—
296C2×C3⋊D3D3×Q814yescomplemented
297C2×C3⋊D3#23912yes—
298C2×C3⋊D3C2×#6712yes—
299C2×C3×D3—34no—
300C2×C3×D3—34no—
301C2×C3×D3—34no—
302C2×C3×D3—34no—
303C2×C3×D3—34no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—64no—
308C2×C3×D3—62no—
309C2×C3×D3—62no—
310C2×C3×D3—62no—
311C2×C3×D3—62no—
312C2×C3×D3—122no—
313C2×C3×D3—122no—
314D32—616no—
315D32—68no—
316C32×C4—22no—
317C32×C4—24no—
318C32×C4—24no—
319C32×C4—24no—
320C32×C4—22no—
321C32×C4—42no—
322C32×C4—44no—
323C32×C4—44no—
324C32×C4D3×D414yes—
325C32×C4C2×#6712yes—
326C3⋊Dic3—22no—
327C3⋊Dic3—22no—
328C3⋊Dic3—24no—
329C3⋊Dic3C2×#6712yes—
330C3×Dic3—34no—
331C3×Dic3—62no—
332C3×Dic3—64no—
333C3×Dic3—62no—
334C3×Dic3—64no—
335C3×Dic3—64no—
336C3×Dic3—64no—
337C3×Dic3—62no—
338C3×Dic3—122no—
339C3×Dic3—124no—
340C22×Q8—91no—
341C23×C4—271no—
342C22⋊Q8—92no—
343C22⋊Q8—274no—
344C22⋊Q8—272no—
345C2×C4⋊C4—272no—
346C2×C4⋊C4—271no—
347C2×C22⋊C4—272no—
348C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
349C23×C3—91no—
350C23×C3—92no—
351C22×D3—32no—
352C22×D3—32no—
353C22×D3—32no—
354C22×D3—61no—
355C22×D3—62no—
356C22×D3—98no—
357C22×D3—94no—
358C22×D3—184no—
359C22×D3—184no—
360C22×D3—182no—
361C22×D3—182no—
362C3×Q8—24no—
363C3×Q8—24no—
364C3×Q8—22no—
365C3×Q8—44no—
366C3×Q8—44no—
367C3×Q8—184no—
368C3×Q8—182no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—24no—
373C2×C3×C4—42no—
374C2×C3×C4—41no—
375C2×C3×C4—91no—
376C2×C3×C4—94no—
377C2×C3×C4—92no—
378C2×C3×C4—92no—
379C2×C3×C4—94no—
380C2×C3×C4—184no—
381C2×C3×C4—184no—
382C2×C3×C4C2×D3212yes—
383C2×C3×C4C2×D3214yes—
384C2×C3×C4C2×D3212yes—
385C2×C3×C4C2×D3211yescharacteristic
386C2×Dic3—32no—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—34no—
390C2×Dic3—34no—
391C2×Dic3—34no—
392C2×Dic3—61no—
393C2×Dic3—62no—
394C2×Dic3—62no—
395C2×Dic3—64no—
396C2×Dic3—184no—
397C2×Dic3—182no—
398C2×Dic3—184no—
399C4×D3—38no—
400C4×D3—64no—
401C4×D3—68no—
402C4×D3—68no—
403C4×D3—64no—
404C4×D3—64no—
405C4×D3—124no—
406C4×D3—184no—
407C4×D3—184no—
408C4×D3—188no—
409C4×D3—184no—
410Dic6—68no—
411Dic6—68no—
412Dic6—64no—
413Dic6—64no—
414C2×C32—22no—
415C2×C32—21no—
416C2×C32—22no—
417C2×C32—21no—
418C2×C32—22no—
419C2×C32—21no—
420C2×C32—41no—
421C2×C32—41no—
422C2×C32—41no—
423C2×C32C2×D3×D412yes—
424C2×C32C2×D3×Q811yescharacteristic
425C2×C32C2×D3×Q812yes—
426C2×C32C2×#23911yescharacteristic
427C2×C32D3×#4112yes—
428C2×C32C22×#6711yescharacteristic
429C3⋊D3—24no—
430C3⋊D3#94614yescomplemented
431C3×D3—38no—
432C3×D3—64no—
433C3×D3—64no—
434C3×D3—64no—
435C3×D3—124no—
436C24—271no—
437C2×Q8—92no—
438C2×Q8—91no—
439C2×Q8—184no—
440C2×Q8#111311yescharacteristic, complemented
441C22×C4—91no—
442C22×C4—92no—
443C22×C4—92no—
444C22×C4—272no—
445C22×C4—271no—
446C22×C4—272no—
447C22×C4—544no—
448C4⋊C4—94no—
449C4⋊C4—92no—
450C4⋊C4—272no—
451C4⋊C4—274no—
452C22⋊C4—94no—
453C22⋊C4—274no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—41no—
457C22×C3—92no—
458C22×C3—92no—
459C22×C3—92no—
460C22×C3—94no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3C22×D3212yes—
464C22×C3C22×D3211yescharacteristic
465C2×D3—34no—
466C2×D3—34no—
467C2×D3—34no—
468C2×D3—34no—
469C2×D3—34no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—64no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—122no—
479C2×D3—122no—
480C2×D3—1816no—
481C2×D3—188no—
482C3×C4—24no—
483C3×C4—24no—
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—24no—
487C3×C4—44no—
488C3×C4—44no—
489C3×C4—42no—
490C3×C4—92no—
491C3×C4—184no—
492C3×C4—182no—
493C3×C4—182no—
494C3×C4D3×#6714yes—
495C3×C4#381112yes—
496Dic3—34no—
497Dic3—62no—
498Dic3—64no—
499Dic3—64no—
500Dic3—64no—
501Dic3—64no—
502Dic3—62no—
503Dic3—62no—
504Dic3—122no—
505Dic3—124no—
506C32—22no—
507C32—21no—
508C32—41no—
509C32D3×C22⋊Q812yescomplemented
510C32C2×#94611yescharacteristic, complemented
511C23—91no—
512C23—92no—
513C23—274no—
514C23—542no—
515C23—542no—
516Q8—22no—
517Q8—184no—
518Q8—182no—
519C2×C4—91no—
520C2×C4—94no—
521C2×C4—94no—
522C2×C4—92no—
523C2×C4—92no—
524C2×C4—184no—
525C2×C4—184no—
526C2×C4—544no—
527C2×C4—542no—
528C2×C4—544no—
529C2×C4C2×#111312yes—
530C2×C4C2×#111311yescharacteristic
531C2×C3—21no—
532C2×C3—22no—
533C2×C3—22no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—22no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—41no—
540C2×C3—94no—
541C2×C3—184no—
542C2×C3D32×Q812yes—
543C2×C3D32×Q811yescharacteristic
544C2×C3D3×#23912yes—
545C2×C3#6431911yescharacteristic
546C2×C3C2×D3×#6712yes—
547C2×C3C2×#381111yescharacteristic
548D3—38no—
549D3—64no—
550D3—64no—
551D3—64no—
552D3—124no—
553C22—92no—
554C22—92no—
555C22—92no—
556C22—94no—
557C22—182no—
558C22—182no—
559C22—548no—
560C22C22×#111311yescharacteristic
561C4—22no—
562C4—92no—
563C4—184no—
564C4—182no—
565C4—182no—
566C4#8904912yes—
567C3—21no—
568C3—22no—
569C3—41no—
570C3D3×#94612yescomplemented
571C3#1059542911yescharacteristic, complemented
572C2—94no—
573C2—184no—
574C2Q8×#111311yescharacteristic
575C2#1172462611yescharacteristic
576C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC22×#1113C22
hypercenterC2×Q8#1113
hypocenterC33C2×C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×C3⋊D3C2semidirect
3. C3×#64155C2semidirect
5. #11723689C2semidirect
6. #11723688C2semidirect
7. #11723682C2semidirect
8. #11723674C2semidirect
12. C2×C33×Q8C22semidirect
17. C2×C3×#634C22semidirect
18. C2×C3×#634C22semidirect
24. C3×C4⋊(C3⋊Dic3)C22semidirect
25. C3×#3750C22semidirect
26. #88815C22semidirect
27. #88815C22semidirect
29. C2×Q8×C3⋊D3D3semidirect
34. #64155D3semidirect
54. C2×#1113Q8semidirect
77. C2×C32×Q8C2×D3semidirect
78. C2×C32×Q8C2×D3semidirect
83. C2×#634C2×D3semidirect
84. C2×#634C2×D3semidirect
108. C4⋊(C3⋊Dic3)C2×D3semidirect
109. #3750C2×D3semidirect
132. C2×C3×C3⋊D3C2×Q8semidirect
232. C3×C3⋊D3C22⋊Q8semidirect
241. C2×C3×Q8D32semidirect
242. C2×C3×Q8D32semidirect
296. C2×C3⋊D3D3×Q8semidirect
348. C33C2×C22⋊Q8semidirect
430. C3⋊D3#946semidirect
440. C2×Q8#1113semidirect
509. C32D3×C22⋊Q8semidirect
510. C32C2×#946semidirect
570. C3D3×#946semidirect
571. C3#10595429semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C4×#11132
C3×#641552
#117237002
#117236892
#117236882
#117236822
#117236742
#105954293
D3×#9463