Finite Group Databasebeta

← #26123892

Subgroups — #26123892

Order 12032.

1018 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093940C211yesmaximal, characteristic
2#20245968C211yesmaximal, characteristic
3#20245968C211yesmaximal, characteristic
4#20243099C211yesmaximal, characteristic, complemented
5#20243098C211yesmaximal, characteristic, complemented
6C2×#15688879C211yesmaximal, characteristic, complemented
7C2×C47×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C23×D188C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C2×#13093983C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12C2×#13093940—22no—
13C2×#13093940—22no—
14C2×#13093940—22no—
15#15689025—22no—
16#15689025C414yescomplemented
17#15689024—22no—
18#15689024—22no—
19#15688978—22no—
20#15688978—22no—
21#15688884—22no—
22#15688883—22no—
23#15688879—22no—
24C22×M16×C47C2211yescharacteristic
25C2×#117×C47C2211yescharacteristic
26C2×#117×C47C2211yescharacteristic
27C47×H(3,ℤ/4)—22no—
28C24×D47D412yes—
29C22×D188—21no—
30C22×D188—21no—
31C22×D188—21no—
32C22×D188—21no—
33C22×D188—21no—
34C22×D188—21no—
35C22×D188—21no—
36C22×D188—21no—
37C22×D188—21no—
38C22×D188—21no—
39C22×D188—21no—
40C22×D188—21no—
41C22×D188C2×C412yes—
42C22×D188D412yes—
43C2×#10625329—21no—
44C2×#10625329—21no—
45C2×#10625329—21no—
46C2×#10625329—21no—
47C2×#10625329—21no—
48C2×#10625329D412yes—
49C22×C47⋊C8—21no—
50#13093983—21no—
51#13093983—21no—
52#13093983—21no—
53#13093983—21no—
54#13093983—21no—
55#13093983—21no—
56#13093983—21no—
57#13093983—21no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47C2×C412yes—
60C2×C22⋊Dic47—21no—
61C2×C22⋊Dic47—21no—
62C2×(C2×D47)⋊C4—28no—
63C2×(C2×D47)⋊C4—28no—
64C2×C4×Dic47—21no—
65C2×C4×Dic47D412yes—
66#13093940—28no—
67C23×C4×C47C2311yescharacteristic
68C2×M16×C47—21no—
69C2×M16×C47—21no—
70C2×M16×C47—21no—
71C2×M16×C47—21no—
72C2×M16×C47—21no—
73C2×M16×C47D412yes—
74C22×C8×C47—21no—
75#117×C47—21no—
76#117×C47—21no—
77#117×C47—21no—
78#117×C47—21no—
79#117×C47—21no—
80#117×C47—21no—
81#117×C47—21no—
82#117×C47—21no—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47—21no—
85C2×C22⋊C4×C47—21no—
86C2×C22⋊C4×C47—21no—
87C2×C42×C47C2×C412yes—
88C2×C42×C47D412yes—
89C24×C47C2×D411yescharacteristic
90C23×D47—22no—
91C23×D47—22no—
92C23×D47—22no—
93C23×D47—22no—
94C23×D47—22no—
95C23×D47—22no—
96C23×D47—84no—
97C23×D47C22⋊C414yes—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47C22×C411yescharacteristic
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47#4111yescharacteristic
107C22×C4×C47#4111yescharacteristic
108C22×C4×C47#4111yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4111yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×Dic47—22no—
115C22×Dic47—22no—
116C2×D188—22no—
117C2×D188—22no—
118C2×D188—22no—
119C2×D188—22no—
120C2×D188—22no—
121C2×D188—22no—
122C2×D188—22no—
123C2×D188—22no—
124C2×D188—22no—
125C2×D188—22no—
126C2×D188—22no—
127C2×D188—22no—
128C2×D188—41no—
129C2×D188—41no—
130C2×D188—41no—
131C2×D188—41no—
132C2×D188—41no—
133C2×D188—41no—
134C2×D188—41no—
135C2×D188—41no—
136C2×D188—41no—
137C2×D188—41no—
138C2×D188—41no—
139C2×D188—41no—
140C2×D188—41no—
141C2×D188—41no—
142C2×D188—41no—
143C2×D188—41no—
144M16×C47—22no—
145M16×C47—22no—
146M16×C47—41no—
147M16×C47—41no—
148C2×C8×C47—22no—
149C2×C8×C47—41no—
150C2×C8×C47—41no—
151C4⋊C4×C47—41no—
152C4⋊C4×C47—41no—
153C4⋊C4×C47—41no—
154C4⋊C4×C47—41no—
155C22⋊C4×C47—41no—
156C22⋊C4×C47—41no—
157C22⋊C4×C47—41no—
158C22⋊C4×C47—41no—
159C42×C47—22no—
160C42×C47—22no—
161C42×C47—22no—
162C42×C47—22no—
163C22⋊Dic47—41no—
164C22⋊Dic47—41no—
165C22⋊Dic47—41no—
166C22⋊Dic47—41no—
167(C2×D47)⋊C4—88no—
168(C2×D47)⋊C4—88no—
169C4⋊Dic47—22no—
170C4⋊Dic47—22no—
171C4⋊Dic47—41no—
172C4⋊Dic47—41no—
173C4×Dic47—22no—
174C4×Dic47—22no—
175C4×Dic47—41no—
176C4×Dic47—41no—
177#10625329—22no—
178#10625329—22no—
179#10625329—41no—
180#10625329—41no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—41no—
183C2×C47⋊C8—41no—
184C23×C47—42no—
185C23×C47C2×C22⋊C411yescharacteristic
186C23×C47#12011yescharacteristic
187C23×C47#12111yescharacteristic
188C23×C47#12111yescharacteristic
189C23×C47#12311yescharacteristic
190C23×C47#12311yescharacteristic
191C23×C47#12311yescharacteristic
192C22×D47—24no—
193C22×D47—24no—
194C22×D47—24no—
195C22×D47—48no—
196C22×D47—48no—
197C22×D47—42no—
198C22×D47—42no—
199C22×D47—42no—
200C22×D47—42no—
201C22×D47—84no—
202C22×D47—84no—
203C22×D47—84no—
204C22×D47—84no—
205C22×D47—84no—
206C2×C4×C47—44no—
207C2×C4×C47—44no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47#11711yescharacteristic
211C2×C4×C47#11711yescharacteristic
212C2×C4×C47C4×D411yescharacteristic
213C2×C4×C47C4×D411yescharacteristic
214C2×C4×C47C4×D411yescharacteristic
215C2×C4×C47C4×D411yescharacteristic
216C2×C4×C47#12011yescharacteristic
217C2×C4×C47#12111yescharacteristic
218C2×C4×C47#12111yescharacteristic
219C2×C4×C47#12111yescharacteristic
220C2×C4×C47#12111yescharacteristic
221C2×C4×C47#12111yescharacteristic
222C2×C4×C47#12111yescharacteristic
223C2×C4×C47#12311yescharacteristic
224C2×C4×C47#12311yescharacteristic
225C2×C4×C47#12311yescharacteristic
226C2×C4×C47#12311yescharacteristic
227C2×C4×C47#12311yescharacteristic
228C2×C4×C47#12311yescharacteristic
229C2×C4×C47#12311yescharacteristic
230C2×C4×C47#12411yescharacteristic
231C2×C4×C47#12411yescharacteristic
232C2×C4×C47#12411yescharacteristic
233C2×C4×C47#12411yescharacteristic
234C2×C4×C47#12611yescharacteristic
235C2×C4×C47#12611yescharacteristic
236C2×C4×C47#12611yescharacteristic
237C2×C4×C47#12611yescharacteristic
238C2×Dic47—44no—
239C2×Dic47—44no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242D188—42no—
243D188—42no—
244D188—42no—
245D188—42no—
246D188—42no—
247D188—42no—
248D188—42no—
249D188—42no—
250C8×C47—42no—
251C47⋊C8—42no—
252#11155—471nomaximal, Sylow
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—41no—
257C22×C47—41no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47#38111yescharacteristic
261C22×C47#38611yescharacteristic
262C22×C47#38611yescharacteristic
263C22×C47#39411yescharacteristic
264C22×C47#39411yescharacteristic
265C22×C47#39711yescharacteristic
266C22×C47#39711yescharacteristic
267C2×D47—48no—
268C2×D47—48no—
269C2×D47—44no—
270C2×D47—84no—
271C2×D47—84no—
272C2×D47—84no—
273C2×D47—84no—
274C2×D47—84no—
275C4×C47—82no—
276C4×C47—82no—
277C4×C47#38811yescharacteristic
278C4×C47#38811yescharacteristic
279C4×C47#38811yescharacteristic
280C4×C47#38811yescharacteristic
281C4×C47#39411yescharacteristic
282C4×C47#39411yescharacteristic
283C4×C47#39711yescharacteristic
284C4×C47#39711yescharacteristic
285Dic47—82no—
286Dic47—82no—
287C22×#100—471no—
288#2309—471no—
289#2309—471no—
290#1733—471no—
291#1733—471no—
292C2×H(3,ℤ/4)—471no—
293C2×H(3,ℤ/4)D4711yescharacteristic, complemented
294C2×C47—21no—
295C2×C47—21no—
296C2×C47—21no—
297C2×C47—21no—
298C2×C47—21no—
299C2×C47—21no—
300C2×C47#144011yescharacteristic
301C2×C47#191112yes—
302D47—84no—
303C23×D4—471no—
304C22×M16—471no—
305C22×M16C2×D4711yescharacteristic
306C2×#117—471no—
307C2×#117—471no—
308C2×#117C2×D4711yescharacteristic
309C2×#117C2×D4711yescharacteristic
310C2×#100—942no—
311C2×#100—942no—
312C2×#100—942no—
313#390—474no—
314#390—942no—
315#390—942no—
316#390—942no—
317#379—942no—
318#379—942no—
319#339—942no—
320#339—942no—
321H(3,ℤ/4)—22no—
322H(3,ℤ/4)—942no—
323C47#1115511yescharacteristic, complemented, Sylow
324C25—472no—
325C22×D4—472no—
326C22×D4—472no—
327C22×D4—941no—
328C22×D4—941no—
329C22×D4—941no—
330C22×D4—941no—
331C22×D4—941no—
332C22×D4—941no—
333C22×D4—941no—
334C22×D4—941no—
335C22×D4—941no—
336C22×D4—941no—
337C22×D4—941no—
338C22×D4—941no—
339C23×C4C22×D4711yescharacteristic
340C2×M16—21no—
341C2×M16—21no—
342C2×M16—21no—
343C2×M16—21no—
344C2×M16—21no—
345C2×M16—472no—
346C2×M16—941no—
347C2×M16—941no—
348C2×M16—941no—
349C2×M16—941no—
350C2×M16—941no—
351C2×M16C22⋊D4712yes—
352C22×C8—21no—
353C22×C8—941no—
354#117—21no—
355#117—21no—
356#117—21no—
357#117—21no—
358#117—21no—
359#117—21no—
360#117—21no—
361#117—21no—
362#117—941no—
363#117—941no—
364#117—941no—
365#117—941no—
366#117—941no—
367#117—941no—
368#117—941no—
369#117—941no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—472no—
373C2×C4⋊C4—941no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—948no—
377C2×C22⋊C4—948no—
378C2×C22⋊C4—941no—
379C2×C22⋊C4—941no—
380C2×C42—472no—
381C2×C42—941no—
382C2×C42C4×D4712yes—
383C2×C42D18812yes—
384#100—948no—
385C24—474no—
386C24—942no—
387C24—942no—
388C24—942no—
389C24—942no—
390C24—942no—
391C24—942no—
392C24—3764no—
393C24D4×D4711yescharacteristic
394C2×D4—942no—
395C2×D4—942no—
396C2×D4—942no—
397C2×D4—942no—
398C2×D4—942no—
399C2×D4—942no—
400C2×D4—942no—
401C2×D4—942no—
402C2×D4—942no—
403C2×D4—942no—
404C2×D4—942no—
405C2×D4—942no—
406C2×D4—1881no—
407C2×D4—1881no—
408C2×D4—1881no—
409C2×D4—1881no—
410C2×D4—1881no—
411C2×D4—1881no—
412C2×D4—1881no—
413C2×D4—1881no—
414C2×D4—1881no—
415C2×D4—1881no—
416C2×D4—1881no—
417C2×D4—1881no—
418C2×D4—1881no—
419C2×D4—1881no—
420C2×D4—1881no—
421C2×D4—1881no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4C2×C4×D4711yescharacteristic
427C22×C4C2×D18811yescharacteristic
428C22×C4#1062535011yescharacteristic
429C22×C4#1062535011yescharacteristic
430C22×C4#1062535011yescharacteristic
431C22×C4#1062535011yescharacteristic
432C22×C4D4×D4711yescharacteristic
433C22×C4D4×D4711yescharacteristic
434C22×C4D4×D4711yescharacteristic
435C22×C4#1062535211yescharacteristic
436C22×C4#1062535211yescharacteristic
437C22×C4#1062535411yescharacteristic
438C22×C4#1062535411yescharacteristic
439C22×C4C2×C22⋊D4711yescharacteristic
440M16—22no—
441M16—22no—
442M16—41no—
443M16—41no—
444M16—942no—
445M16—942no—
446M16—1881no—
447M16—1881no—
448C2×C8—22no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—942no—
452C2×C8—1881no—
453C2×C8—1881no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—942no—
459C4⋊C4—942no—
460C4⋊C4—1881no—
461C4⋊C4—1881no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—3768no—
467C22⋊C4—3768no—
468C22⋊C4—1881no—
469C22⋊C4—1881no—
470C22⋊C4—1881no—
471C22⋊C4—1881no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—942no—
477C42—942no—
478C42—1881no—
479C42—1881no—
480C23—42no—
481C23—944no—
482C23—944no—
483C23—944no—
484C23—3764no—
485C23—3764no—
486C23—3764no—
487C23—3764no—
488C23—3764no—
489C23—1888no—
490C23—1888no—
491C23—1882no—
492C23—1882no—
493C23—1882no—
494C23—1882no—
495C23C22⋊C4×D4711yescharacteristic
496C23#1309399411yescharacteristic
497C23#1309399511yescharacteristic
498C23#1309399711yescharacteristic
499C23#1309400611yescharacteristic
500C23#1309400911yescharacteristic
501C23#1309400911yescharacteristic
502D4—1882no—
503D4—1882no—
504D4—1882no—
505D4—1882no—
506D4—1882no—
507D4—1882no—
508D4—1882no—
509D4—1882no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—1884no—
515C2×C4—1884no—
516C2×C4—1882no—
517C2×C4—1882no—
518C2×C4#1309398111yescharacteristic
519C2×C4#1309398211yescharacteristic
520C2×C4(C22⋊D47)⋊C411yescharacteristic
521C2×C4C4×C22⋊D4711yescharacteristic
522C2×C4#1309398611yescharacteristic
523C2×C4C4×D18811yescharacteristic
524C2×C4#1309399211yescharacteristic
525C2×C4#1309399611yescharacteristic
526C2×C4#1309399711yescharacteristic
527C2×C4#1309399711yescharacteristic
528C2×C4#1309399811yescharacteristic
529C2×C4#1309399911yescharacteristic
530C2×C4#1309399911yescharacteristic
531C2×C4#1309400611yescharacteristic
532C2×C4#1309400611yescharacteristic
533C2×C4#1309400711yescharacteristic
534C2×C4#1309400811yescharacteristic
535C2×C4#1309400911yescharacteristic
536C2×C4#1309400911yescharacteristic
537C2×C4#1309400911yescharacteristic
538C2×C4#1309401011yescharacteristic
539C2×C4#1309401011yescharacteristic
540C2×C4#1309401211yescharacteristic
541C2×C4#1309401311yescharacteristic
542C2×C4#1309401811yescharacteristic
543C2×C4#1309401911yescharacteristic
544C2×C4#1309401911yescharacteristic
545C2×C4#1309401911yescharacteristic
546C8—42no—
547C8—1882no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—3764no—
556C22—3764no—
557C22—3764no—
558C22—3764no—
559C22—3764no—
560C22—1888no—
561C22—1888no—
562C22—1884no—
563C22#1568898511yescharacteristic
564C22#1568899911yescharacteristic
565C22#1568900111yescharacteristic
566C22#1568903911yescharacteristic
567C22#1568904111yescharacteristic
568C22#1568904911yescharacteristic
569C22#1568905211yescharacteristic
570C4—82no—
571C4—82no—
572C4—3762no—
573C4—3762no—
574C4#1568901211yescharacteristic
575C4#1568901311yescharacteristic
576C4#1568901411yescharacteristic
577C4#1568901511yescharacteristic
578C4#1568903811yescharacteristic
579C4#1568904011yescharacteristic
580C4#1568905011yescharacteristic
581C4#1568905211yescharacteristic
582C2—21no—
583C2—21no—
584C2—21no—
585C2—21no—
586C2—21no—
587C2—21no—
588C2—3764no—
589C2#2024163611yescharacteristic
590C2#2024391912yes—
subgroupG/H
centerC22#15689001
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×H(3,ℤ/4)C2
socleC22×C47#386
prime radicalC23×D188C22
cosocleC23×C4×C47C23
hypercenterC2×H(3,ℤ/4)D47
hypocenterC47#11155
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. #20243099C2semidirect
5. #20243098C2semidirect
6. C2×#15688879C2semidirect
7. C2×C47×H(3,ℤ/4)C2semidirect
16. #15689025C4semidirect
293. C2×H(3,ℤ/4)D47semidirect
323. C47#11155semidirect

Maximal subgroups

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

subgroupindex
C22×#130939402
#202459682
#202430992
#202430982
C2×#156888792
C2×C47×H(3,ℤ/4)2
#1115547