Finite Group Databasebeta

← #11788180

Subgroups — #11788180

Order 1152.

969 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
1C3×#5828C211yesmaximal, characteristic
2C2×C4×D3×Dic3C211yesmaximal, characteristic, complemented
3C2×#63664C211yesmaximal, characteristic, complemented
4C2×C3×#837C211yesmaximal, characteristic, complemented
5#10591069C211yesmaximal, characteristic, complemented
6#10589161C211yesmaximal, characteristic, complemented
7#10589160C211yesmaximal, characteristic, complemented
8#71711—31nomaximal
9#70382—31nomaximal
10C2×C32×#41C2211yescharacteristic
11C22×C3×C4×D3C2211yescharacteristic
12C22×D3×Dic3—21no—
13C2×C4×C3⋊Dic3C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C3×#946C412yescomplemented
16C3×#938C412yescomplemented
17C3×#937—21no—
18C3×#935—21no—
19C3×C4×#67—22no—
20C3×#926C2211yescharacteristic
21C2×C3×C4×Dic3C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C2×Dic32—21no—
24C4×D3×Dic3—22no—
25C4×D3×Dic3—22no—
26#63664—22no—
27C3×#837—22no—
28(C2×D3)⋊(C3⋊C8)—22no—
29C3×#525—31no—
30#5828D311yescharacteristic
31C22×C4×Dic3—31no—
32C2×C42×D3—31no—
33C2×C3×#104—31no—
34C2×#837—31no—
35C2×#837—31no—
36C2×#837—61no—
37C2×#837D311yescharacteristic, complemented
38#5207—31no—
39#5126—31no—
40#4555—31no—
41#4532—31no—
42#4532—31no—
43#4523—31no—
44C23×C3×D3D411yescharacteristic
45C32×#41—21no—
46C32×#41—21no—
47C32×#41—21no—
48C32×#41—21no—
49C2×C32×Q8C2×C411yescharacteristic, complemented
50C2×C32×D4—21no—
51C2×C32×D4C2×C411yescharacteristic, complemented
52C22×C32×C4—21no—
53C22×C32×C4C2311yescharacteristic
54C22×C3⋊Dic3—21no—
55C2×C3×#67—22no—
56C22×C3×Dic3—21no—
57C22×C3×Dic3D411yescharacteristic
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3—21no—
61C2×C3×C4×D3—21no—
62C2×C3×C4×D3C2×C412yes—
63C2×C3×C4×D3D412yes—
64C2×D3×Dic3—28no—
65C2×D3×Dic3—24no—
66C4×C3⋊Dic3—21no—
67C4×C3⋊Dic3D412yescomplemented
68#3748—21no—
69#3748—21no—
70C2×C3⋊(C3⋊C8)—21no—
71C3×#217—21no—
72C3×#217—22no—
73C3×(C2×D3)⋊C4—22no—
74C3×C4⋊Dic3C2×C412yescomplemented
75C3×Dic3⋊C4—22no—
76C3×C4×Dic3—21no—
77C3×C4×Dic3—21no—
78C3×C4×Dic3D412yescomplemented
79C3×#208—21no—
80C3×#208—21no—
81C2×C3×C3⋊C8—21no—
82Dic32—24no—
83#1896—91noSylow
84C2×C3×#41—21no—
85C2×C3×#41C2×D311yescharacteristic
86C2×C3×#41C2×D311yescharacteristic
87C23×C3×C4—31no—
88C23×Dic3—61no—
89C22×C4×D3—61no—
90C22×C4×D3C2×D311yescharacteristic
91C2×C3×M16—31no—
92C3×#123—61no—
93C3×C22⋊Q8—32no—
94C3×#121—32no—
95C3×#120—61no—
96C3×C4×D4—62no—
97C3×#117—31no—
98C2×C3×C42—31no—
99#946Dic312yescomplemented
100#938Dic312yescomplemented
101#937—21no—
102#935—21no—
103C4×#67—22no—
104#926C2×D311yescharacteristic
105C2×C4×Dic3—31no—
106C2×C4×Dic3—31no—
107C2×C4×Dic3—61no—
108C2×C4×Dic3—62no—
109C2×C4×Dic3—61no—
110C2×C4×Dic3—61no—
111C2×C4×Dic3—62no—
112C2×C4×Dic3C2×D311yescharacteristic
113C2×#208—31no—
114C2×#208—31no—
115C2×#208—61no—
116C2×#208C2×D311yescharacteristic
117C42×D3—62no—
118C42×D3—62no—
119C3×#104—62no—
120#837—22no—
121#837—62no—
122#837—62no—
123#837—64no—
124#830—62no—
125(C2×D3)⋊C8—62no—
126C23×C32—21no—
127C23×C32C2×D411yescharacteristic
128C22×C3×D3—21no—
129C22×C3×D3—21no—
130C22×C3×D3—24no—
131C22×C3×D3C22⋊C412yes—
132C32×Q8—21no—
133C32×Q8—21no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—41no—
137C32×D4—41no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4—41no—
141C2×C32×C4C22×C411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4#4111yescharacteristic
146C2×C32×C4#4111yescharacteristic
147C2×C3⋊Dic3—24no—
148C2×C3⋊Dic3—22no—
149C3×#67—44no—
150C2×C3×Dic3—24no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3C22⋊C412yes—
156C3×C4×D3—22no—
157C3×C4×D3—22no—
158C3×C4×D3—22no—
159C3×C4×D3—22no—
160D3×Dic3—216no—
161C3⋊(C3⋊C8)—41no—
162C3×C3⋊C8—41no—
163#525—31no—
164C22×C42—91no—
165C2×#104—31no—
166C2×#104—91no—
167#407—91no—
168#339—91no—
169#339—91no—
170C24×C3—31no—
171C23×D3#6711yescharacteristic
172C3×#41—22no—
173C3×#41—22no—
174C3×#41—21no—
175C3×#41—21no—
176C3×#41—21no—
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C3×#41—21no—
181C3×#41—21no—
182C3×#41—41no—
183C3×#41—41no—
184C2×C3×Q8—21no—
185C2×C3×Q8C4×D311yescharacteristic, complemented
186C2×C3×Q8C2×Dic311yescharacteristic, complemented
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—41no—
191C2×C3×D4—62no—
192C2×C3×D4C4×D311yescharacteristic, complemented
193C2×C3×D4C2×Dic311yescharacteristic, complemented
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—31no—
198C22×C3×C4—32no—
199C22×C3×C4—32no—
200C22×C3×C4—41no—
201C22×C3×C4—61no—
202C22×C3×C4—61no—
203C22×C3×C4—61no—
204C22×C3×C4—61no—
205C22×C3×C4—61no—
206C22×C3×C4C22×D311yescharacteristic
207C22×C3×C4C22×D311yescharacteristic
208C2×#67—22no—
209C22×Dic3—21no—
210C22×Dic3—64no—
211C22×Dic3—68no—
212C22×Dic3—61no—
213C22×Dic3—61no—
214C22×Dic3—121no—
215C22×Dic3#6711yescharacteristic
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—64no—
221C2×C4×D3—68no—
222C2×C4×D3C2×Dic312yes—
223C2×C4×D3#6712yes—
224C3×M16—61no—
225C3×M16—61no—
226C2×C3×C8—61no—
227C3×C4⋊C4—32no—
228C3×C4⋊C4—62no—
229C3×C22⋊C4—62no—
230C3×C22⋊C4—62no—
231C3×C22⋊C4—61no—
232C3×C42—32no—
233C3×C42—61no—
234C3×C42—61no—
235#217—21no—
236#217—22no—
237(C2×D3)⋊C4—22no—
238C4⋊Dic3C2×Dic312yescomplemented
239Dic3⋊C4—22no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—32no—
243C4×Dic3—32no—
244C4×Dic3—62no—
245C4×Dic3—64no—
246C4×Dic3—64no—
247C4×Dic3—61no—
248C4×Dic3—61no—
249C4×Dic3—121no—
250C4×Dic3#6712yescomplemented
251#208—21no—
252#208—21no—
253#208—62no—
254#208—61no—
255#208—61no—
256#208—61no—
257#208—61no—
258#208—121no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8—61no—
262C2×C3⋊C8—121no—
263C22×C32—22no—
264C22×C32—41no—
265C22×C32—42no—
266C22×C32C2×C22⋊C411yescharacteristic
267C22×C32#12011yescharacteristic
268C22×C32#12311yescharacteristic
269C2×C3×D3—24no—
270C2×C3×D3—24no—
271C2×C3×D3—22no—
272C2×C3×D3—22no—
273C2×C3×D3#10414yescomplemented
274C32×C4—41no—
275C32×C4C4×D411yescharacteristic
276C32×C4C4×D411yescharacteristic
277C32×C4#12111yescharacteristic
278C32×C4#12111yescharacteristic
279C3⋊Dic3—24no—
280C3×Dic3—24no—
281C3×Dic3—22no—
282C3×Dic3—42no—
283C2×#41D3211yescharacteristic
284C23×C4—31no—
285C23×C4—181no—
286C2×M16—31no—
287C2×M16—91no—
288#123—61no—
289C22⋊Q8—32no—
290#121—32no—
291#120—61no—
292C4×D4—62no—
293#117—31no—
294C2×C42—31no—
295C2×C42—91no—
296C2×C42—182no—
297C2×C42—182no—
298C2×C42—181no—
299#104—62no—
300#104—182no—
301C22⋊C8—182no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—32no—
306C23×C3—41no—
307C23×C3—64no—
308C23×C3—61no—
309C23×C3—61no—
310C23×C3D3×D411yescharacteristic
311C23×C3C2×#6711yescharacteristic
312C22×D3—21no—
313C22×D3—21no—
314C22×D3—24no—
315C22×D3#21712yes—
316C3×Q8—22no—
317C3×Q8—21no—
318C3×Q8—21no—
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—41no—
322C3×D4—22no—
323C3×D4—21no—
324C3×D4—21no—
325C3×D4—21no—
326C3×D4—21no—
327C3×D4—41no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—41no—
331C3×D4—41no—
332C3×D4—41no—
333C3×D4—41no—
334C3×D4—124no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—32no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—64no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4C2×C4×D311yescharacteristic
361C2×C3×C4#23511yescharacteristic
362C2×C3×C4D3×D411yescharacteristic
363C2×C3×C4D3×D411yescharacteristic
364C2×C3×C4D3×D411yescharacteristic
365C2×C3×C4D3×D411yescharacteristic
366C2×C3×C4#23711yescharacteristic
367C2×C3×C4#23711yescharacteristic
368C2×C3×C4#23711yescharacteristic
369C2×C3×C4C22×Dic311yescharacteristic
370C2×C3×C4C2×#6711yescharacteristic
371C2×C3×C4C2×#6711yescharacteristic
372#67—44no—
373C2×Dic3—24no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—616no—
379C2×Dic3—64no—
380C2×Dic3—64no—
381C2×Dic3—62no—
382C2×Dic3—62no—
383C2×Dic3—122no—
384C2×Dic3—124no—
385C2×Dic3#21712yes—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—22no—
389C4×D3—22no—
390C4×D3—616no—
391C3×C8—121no—
392C3⋊C8—41no—
393C3⋊C8—122no—
394C3⋊C8—121no—
395C3⋊C8—121no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—81no—
399C2×C32#38611yescharacteristic
400C2×C32C2×#10412yes—
401C3×D3—24no—
402C24—31no—
403#41—21no—
404#41—21no—
405#41—21no—
406#41—21no—
407C2×Q8D3×Dic311yescharacteristic, complemented
408C2×D4—21no—
409C2×D4—62no—
410C2×D4D3×Dic311yescharacteristic, complemented
411C22×C4—21no—
412C22×C4—31no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—184no—
421C22×C4—188no—
422C22×C4—181no—
423C22×C4C2×D3211yescharacteristic
424M16—61no—
425M16—61no—
426M16—181no—
427M16—181no—
428C2×C8—61no—
429C2×C8—181no—
430C4⋊C4—32no—
431C4⋊C4—62no—
432C22⋊C4—62no—
433C22⋊C4—62no—
434C22⋊C4—61no—
435C42—32no—
436C42—61no—
437C42—61no—
438C42—92no—
439C42—184no—
440C42—181no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—34no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—64no—
454C22×C3—64no—
455C22×C3—62no—
456C22×C3—62no—
457C22×C3—82no—
458C22×C3D3×C22⋊C411yescharacteristic
459C22×C3#88311yescharacteristic
460C22×C3#93511yescharacteristic
461C22×C3#93711yescharacteristic
462C22×C3#93711yescharacteristic
463C22×C3C2×#21711yescharacteristic
464C2×D3—24no—
465C2×D3—24no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3#83714yescomplemented
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—64no—
477C3×C4—62no—
478C3×C4—122no—
479C3×C4#67⋊C411yescharacteristic
480C3×C4#88411yescharacteristic
481C3×C4C4×#6711yescharacteristic
482C3×C4D4×Dic311yescharacteristic
483C3×C4D4×Dic311yescharacteristic
484C3×C4#93811yescharacteristic
485C3×C4#93911yescharacteristic
486C3×C4#93911yescharacteristic
487Dic3—24no—
488Dic3—22no—
489Dic3—42no—
490Dic3—64no—
491Dic3—64no—
492Dic3—124no—
493C32#189611yescharacteristic, complemented, Sylow
494C23—21no—
495C23—32no—
496C23—64no—
497C23—61no—
498C23—61no—
499C23D3×#6711yescharacteristic
500Q8—21no—
501Q8—21no—
502D4—21no—
503D4—21no—
504D4—41no—
505D4—41no—
506D4—124no—
507C2×C4—21no—
508C2×C4—32no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—64no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—1816no—
521C2×C4—184no—
522C2×C4—182no—
523C2×C4C2×D3×Dic311yescharacteristic
524C2×C4#380411yescharacteristic
525C2×C4#380511yescharacteristic
526C2×C4D3×#6711yescharacteristic
527C2×C4D3×#6711yescharacteristic
528C2×C4#381111yescharacteristic
529C8—121no—
530C8—361no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—22no—
536C2×C3—21no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—64no—
540C2×C3—82no—
541C2×C3—81no—
542C2×C3—81no—
543C2×C3D3×#10412yes—
544C2×C3#495611yescharacteristic
545C2×C3#522011yescharacteristic
546C2×C3C2×#83712yes—
547D3—24no—
548C22—22no—
549C22—34no—
550C22—42no—
551C22—41no—
552C22—64no—
553C22—64no—
554C22—62no—
555C22—62no—
556C22D3×#21711yescharacteristic
557C22#6396911yescharacteristic
558C22#6398311yescharacteristic
559C4—41no—
560C4—64no—
561C4—62no—
562C4—122no—
563C4—184no—
564C4#6397011yescharacteristic
565C4#6397111yescharacteristic
566C4Dic3×#6711yescharacteristic
567C4#67⋊Dic311yescharacteristic
568C3—21no—
569C3#7038211yescharacteristic, complemented
570C3#7171111yescharacteristic, complemented
571C2—21no—
572C2—21no—
573C2—64no—
574C2—81no—
575C2D3×#83712yes—
576C2#1059172111yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×#41C22
socleC22×C32C2×C22⋊C4
prime radicalC3×#5828C2
cosocleC22×C32×C4C23
hypercenterC2×#41D32
hypocenterC32#1896
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×C4×D3×Dic3C2semidirect
3. C2×#63664C2semidirect
4. C2×C3×#837C2semidirect
5. #10591069C2semidirect
6. #10589161C2semidirect
7. #10589160C2semidirect
15. C3×#946C4semidirect
16. C3×#938C4semidirect
37. C2×#837D3semidirect
49. C2×C32×Q8C2×C4semidirect
51. C2×C32×D4C2×C4semidirect
67. C4×C3⋊Dic3D4semidirect
74. C3×C4⋊Dic3C2×C4semidirect
78. C3×C4×Dic3D4semidirect
99. #946Dic3semidirect
100. #938Dic3semidirect
185. C2×C3×Q8C4×D3semidirect
186. C2×C3×Q8C2×Dic3semidirect
192. C2×C3×D4C4×D3semidirect
193. C2×C3×D4C2×Dic3semidirect
238. C4⋊Dic3C2×Dic3semidirect
250. C4×Dic3#67semidirect
273. C2×C3×D3#104semidirect
407. C2×Q8D3×Dic3semidirect
410. C2×D4D3×Dic3semidirect
468. C2×D3#837semidirect
493. C32#1896semidirect
569. C3#70382semidirect
570. C3#71711semidirect

Maximal subgroups

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

subgroupindex
C3×#58282
C2×C4×D3×Dic32
C2×#636642
C2×C3×#8372
#105910692
#105891612
#105891602
#717113
#703823