Finite Group Databasebeta

← #26123789

Subgroups — #26123789

Order 12032.

950 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
1C22×#13093940C211yesmaximal, characteristic
2C22×#13093940C211yesmaximal, characteristic
3C22×#13093940C211yesmaximal, characteristic
4#20243115C211yesmaximal, characteristic, complemented
5#20243114C211yesmaximal, characteristic, complemented
6C2×C47⋊#338C211yesmaximal, characteristic, complemented
7C2×C47×#430C211yesmaximal, characteristic, complemented
8C23×D188C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C2×#13093940—22no—
13C2×#13093940—22no—
14C2×#13093940—22no—
15C2×#13093940—22no—
16C2×#13093940—22no—
17C2×#13093940—22no—
18C2×#13093940—22no—
19C2×#13093940—22no—
20C2×#13093940C414yes—
21#15688888—22no—
22#15688887—22no—
23C47⋊#338—22no—
24C22×M16×C47C2211yescharacteristic
25C22×M16×C47C2211yescharacteristic
26C22×M16×C47C2211yescharacteristic
27C47×#430—22no—
28C47×#430—22no—
29C2×#108×C47—22no—
30C24×D47Q812yes—
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×D188—21no—
42C22×D188—21no—
43C22×D188C2×C412yes—
44C22×D188D412yes—
45C2×#10625329—21no—
46C2×#10625329—21no—
47C2×#10625329—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329—21no—
58C2×#10625329—21no—
59C2×#10625329—21no—
60C2×#10625329C2×C412yes—
61C2×#10625329D412yes—
62C2×#10625329D412yes—
63C22×C47⋊C8—21no—
64C22×C47⋊C8—21no—
65C22×C47⋊C8—21no—
66#13093940—28no—
67#13093940—28no—
68#13093940—28no—
69C23×C4×C47C2311yescharacteristic
70C2×M16×C47—21no—
71C2×M16×C47—21no—
72C2×M16×C47—21no—
73C2×M16×C47—21no—
74C2×M16×C47—21no—
75C2×M16×C47—21no—
76C2×M16×C47—21no—
77C2×M16×C47—21no—
78C2×M16×C47—21no—
79C2×M16×C47—21no—
80C2×M16×C47—21no—
81C2×M16×C47—21no—
82C2×M16×C47—21no—
83C2×M16×C47C2×C412yes—
84C2×M16×C47D412yes—
85C2×M16×C47D412yes—
86C2×M16×C47D412yes—
87C2×M16×C47D412yes—
88C22×C8×C47—21no—
89C22×C8×C47—21no—
90C22×C8×C47—21no—
91#108×C47—28no—
92C24×C47C2×Q811yescharacteristic
93C23×D47—22no—
94C23×D47—22no—
95C23×D47—22no—
96C23×D47—22no—
97C23×D47—22no—
98C23×D47—22no—
99C23×D47—84no—
100C23×D47C4⋊C414yes—
101C22×C4×C47C22×C411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4111yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×C4×C47#4111yescharacteristic
115C2×D188—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—41no—
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—
143M16×C47—22no—
144M16×C47—22no—
145M16×C47—22no—
146M16×C47—22no—
147M16×C47—22no—
148M16×C47—22no—
149M16×C47—22no—
150M16×C47—41no—
151M16×C47—41no—
152M16×C47—41no—
153M16×C47—41no—
154M16×C47C22⋊C414yes—
155C2×C8×C47—22no—
156C2×C8×C47—22no—
157C2×C8×C47—22no—
158C2×C8×C47—22no—
159C2×C8×C47—22no—
160C2×C8×C47—41no—
161C2×C8×C47—41no—
162C2×C8×C47—41no—
163C2×C8×C47—41no—
164#10625329—22no—
165#10625329—22no—
166#10625329—22no—
167#10625329—22no—
168#10625329—22no—
169#10625329—22no—
170#10625329—41no—
171#10625329—41no—
172#10625329—41no—
173#10625329—41no—
174#10625329—41no—
175#10625329—41no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—41no—
180C2×C47⋊C8—41no—
181C2×C47⋊C8—41no—
182C2×C47⋊C8—41no—
183C2×C47⋊C8—41no—
184C2×C47⋊C8—41no—
185C23×C47—42no—
186C23×C47C2×C4⋊C411yescharacteristic
187C23×C47C22⋊Q811yescharacteristic
188C23×C47C22⋊Q811yescharacteristic
189C23×C47C22⋊Q811yescharacteristic
190C23×C47C22⋊Q811yescharacteristic
191C23×C47C22⋊Q811yescharacteristic
192C23×C47C22⋊Q811yescharacteristic
193C22×D47—24no—
194C22×D47—24no—
195C22×D47—24no—
196C22×D47—42no—
197C22×D47—42no—
198C22×D47—42no—
199C22×D47—42no—
200C22×D47—84no—
201C22×D47—84no—
202C22×D47—84no—
203C22×D47—84no—
204C22×D47—84no—
205C22×D47—84no—
206C22×D47—84no—
207C2×C4×C47C2×C22⋊C411yescharacteristic
208C2×C4×C47#11711yescharacteristic
209C2×C4×C47C4×D411yescharacteristic
210C2×C4×C47C4×D411yescharacteristic
211C2×C4×C47C4×D411yescharacteristic
212C2×C4×C47C4×D411yescharacteristic
213C2×C4×C47#12011yescharacteristic
214C2×C4×C47#12011yescharacteristic
215C2×C4×C47#12011yescharacteristic
216C2×C4×C47#12111yescharacteristic
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#12111yescharacteristic
224C2×C4×C47#12311yescharacteristic
225C2×C4×C47#12311yescharacteristic
226C2×C4×C47#12311yescharacteristic
227C2×C4×C47#12311yescharacteristic
228C2×C4×C47#12311yescharacteristic
229C2×C4×C47#12411yescharacteristic
230C2×C4×C47#12411yescharacteristic
231C2×C4×C47#12411yescharacteristic
232C2×C4×C47#12611yescharacteristic
233C2×C4×C47#12611yescharacteristic
234C2×C4×C47#12711yescharacteristic
235D188—42no—
236D188—42no—
237D188—42no—
238D188—42no—
239D188—42no—
240D188—42no—
241D188—42no—
242D188—42no—
243C8×C47—24no—
244C8×C47—42no—
245C8×C47—42no—
246C47⋊C8—42no—
247C47⋊C8—42no—
248C47⋊C8—42no—
249#11139—471nomaximal, Sylow
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—41no—
254C22×C47—41no—
255C22×C47—41no—
256C22×C47—41no—
257C22×C47#38011yescharacteristic
258C22×C47#38511yescharacteristic
259C22×C47#38511yescharacteristic
260C22×C47#39311yescharacteristic
261C22×C47#39611yescharacteristic
262C22×C47#39611yescharacteristic
263C22×C47#39911yescharacteristic
264C2×D47—44no—
265C2×D47—84no—
266C2×D47—84no—
267C2×D47—84no—
268C2×D47—84no—
269C2×D47—84no—
270C2×D47—84no—
271C2×D47—84no—
272C4×C47#38611yescharacteristic
273C4×C47#38811yescharacteristic
274C4×C47#38811yescharacteristic
275C4×C47#39011yescharacteristic
276C4×C47#39211yescharacteristic
277C4×C47#39411yescharacteristic
278C4×C47#39411yescharacteristic
279C4×C47#39711yescharacteristic
280C2×#430D4711yescharacteristic, complemented
281C22×#100—471no—
282C22×#100—471no—
283C22×#100—471no—
284#1738—471no—
285#1738—471no—
286C2×#338—471no—
287C2×C47—21no—
288C2×C47—21no—
289C2×C47—21no—
290C2×C47—21no—
291C2×C47—21no—
292C2×C47—21no—
293C2×C47#143911yescharacteristic
294C2×C47#206912yes—
295D47—84no—
296C23×D4—471no—
297C22×M16—471no—
298C22×M16—471no—
299C22×M16—471no—
300C22×M16C2×D4711yescharacteristic
301C22×M16C2×D4711yescharacteristic
302C22×M16C2×D4711yescharacteristic
303#430—22no—
304#430—22no—
305C2×#108—22no—
306C2×#100—474no—
307C2×#100—942no—
308C2×#100—942no—
309C2×#100—942no—
310C2×#100—942no—
311C2×#100—942no—
312C2×#100—942no—
313C2×#100—942no—
314C2×#100—942no—
315#341—942no—
316#341—942no—
317#338—942no—
318C47#1113911yescharacteristic, complemented, Sylow
319C25—472no—
320C22×D4—472no—
321C22×D4—472no—
322C22×D4—941no—
323C22×D4—941no—
324C22×D4—941no—
325C22×D4—941no—
326C22×D4—941no—
327C22×D4—941no—
328C22×D4—941no—
329C22×D4—941no—
330C22×D4—941no—
331C22×D4—941no—
332C22×D4—941no—
333C22×D4—941no—
334C23×C4C22×D4711yescharacteristic
335C2×M16—21no—
336C2×M16—21no—
337C2×M16—21no—
338C2×M16—21no—
339C2×M16—21no—
340C2×M16—21no—
341C2×M16—21no—
342C2×M16—21no—
343C2×M16—21no—
344C2×M16—21no—
345C2×M16—21no—
346C2×M16—21no—
347C2×M16—21no—
348C2×M16—472no—
349C2×M16—472no—
350C2×M16—472no—
351C2×M16—941no—
352C2×M16—941no—
353C2×M16—941no—
354C2×M16—941no—
355C2×M16—941no—
356C2×M16—941no—
357C2×M16—941no—
358C2×M16—941no—
359C2×M16—941no—
360C2×M16—941no—
361C2×M16—941no—
362C2×M16—941no—
363C2×M16—941no—
364C2×M16—941no—
365C2×M16—941no—
366C2×M16C4×D4712yes—
367C2×M16D18812yes—
368C2×M16C22⋊D4712yes—
369C2×M16C22⋊D4712yes—
370C2×M16C22⋊D4712yes—
371C22×C8—21no—
372C22×C8—21no—
373C22×C8—21no—
374C22×C8—941no—
375C22×C8—941no—
376C22×C8—941no—
377#108—28no—
378#100—948no—
379#100—948no—
380#100—948no—
381C24—474no—
382C24—942no—
383C24—942no—
384C24—942no—
385C24—942no—
386C24—942no—
387C24—942no—
388C24—3764no—
389C24Q8×D4711yescharacteristic
390C2×D4—942no—
391C2×D4—942no—
392C2×D4—942no—
393C2×D4—942no—
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—1881no—
403C2×D4—1881no—
404C2×D4—1881no—
405C2×D4—1881no—
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—
418C22×C4C2×C4×D4711yescharacteristic
419C22×C4C2×D18811yescharacteristic
420C22×C4#1062535011yescharacteristic
421C22×C4#1062535011yescharacteristic
422C22×C4D4×D4711yescharacteristic
423C22×C4D4×D4711yescharacteristic
424C22×C4D4×D4711yescharacteristic
425C22×C4#1062535211yescharacteristic
426C22×C4#1062535411yescharacteristic
427C22×C4#1062535411yescharacteristic
428C22×C4#1062535411yescharacteristic
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C2×C22⋊D4711yescharacteristic
432M16—22no—
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—22no—
438M16—22no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—942no—
444M16—942no—
445M16—942no—
446M16—942no—
447M16—942no—
448M16—942no—
449M16—1881no—
450M16—1881no—
451M16—1881no—
452M16—1881no—
453M16—1881no—
454M16—1881no—
455M16(C2×D47)⋊C414yes—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—942no—
466C2×C8—942no—
467C2×C8—942no—
468C2×C8—1881no—
469C2×C8—1881no—
470C2×C8—1881no—
471C2×C8—1881no—
472C2×C8—1881no—
473C2×C8—1881no—
474C23—42no—
475C23—944no—
476C23—944no—
477C23—944no—
478C23—3764no—
479C23—3764no—
480C23—3764no—
481C23—3764no—
482C23—3764no—
483C23—3764no—
484C23—3764no—
485C23—1882no—
486C23—1882no—
487C23—1882no—
488C23—1882no—
489C23C4⋊C4×D4711yescharacteristic
490C23#1309400011yescharacteristic
491C23#1309400011yescharacteristic
492C23#1309400011yescharacteristic
493C23#1309400211yescharacteristic
494C23#1309400211yescharacteristic
495C23#1309400411yescharacteristic
496D4—1882no—
497D4—1882no—
498D4—1882no—
499D4—1882no—
500D4—1882no—
501D4—1882no—
502D4—1882no—
503D4—1882no—
504C2×C4C2×(C2×D47)⋊C411yescharacteristic
505C2×C4#1309398211yescharacteristic
506C2×C4C4×C22⋊D4711yescharacteristic
507C2×C4C4×C22⋊D4711yescharacteristic
508C2×C4#1309398611yescharacteristic
509C2×C4#1309398611yescharacteristic
510C2×C4#1309399211yescharacteristic
511C2×C4#1309399211yescharacteristic
512C2×C4#1309399311yescharacteristic
513C2×C4#1309399511yescharacteristic
514C2×C4#1309399611yescharacteristic
515C2×C4#1309399611yescharacteristic
516C2×C4#1309399811yescharacteristic
517C2×C4#1309399811yescharacteristic
518C2×C4#1309399911yescharacteristic
519C2×C4#1309399911yescharacteristic
520C2×C4#1309399911yescharacteristic
521C2×C4#1309400711yescharacteristic
522C2×C4#1309400911yescharacteristic
523C2×C4#1309400911yescharacteristic
524C2×C4#1309400911yescharacteristic
525C2×C4#1309400911yescharacteristic
526C2×C4#1309401311yescharacteristic
527C2×C4#1309401311yescharacteristic
528C2×C4#1309401311yescharacteristic
529C2×C4#1309401911yescharacteristic
530C2×C4#1309401911yescharacteristic
531C2×C4#1309402011yescharacteristic
532C8—24no—
533C8—42no—
534C8—42no—
535C8—1882no—
536C8—1882no—
537C8—1882no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—3764no—
546C22—3764no—
547C22—3764no—
548C22—3764no—
549C22—3764no—
550C22—3764no—
551C22—3764no—
552C22—1884no—
553C22#1568898011yescharacteristic
554C22#1568899411yescharacteristic
555C22#1568899411yescharacteristic
556C22#1568903411yescharacteristic
557C22#1568904511yescharacteristic
558C22#1568904511yescharacteristic
559C22#1568905811yescharacteristic
560C4#1568900211yescharacteristic
561C4#1568901311yescharacteristic
562C4#1568901311yescharacteristic
563C4#1568902411yescharacteristic
564C4#1568903111yescharacteristic
565C4#1568904011yescharacteristic
566C4#1568904011yescharacteristic
567C4#1568904911yescharacteristic
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—21no—
573C2—21no—
574C2—3764no—
575C2#2024163011yescharacteristic
576C2#2024478512yes—
subgroupG/H
centerC22#15689045
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#430C2
socleC22×C47#396
prime radicalC23×D188C22
cosocleC23×C4×C47C23
hypercenterC2×#430D47
hypocenterC47#11139
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
4. #20243115C2semidirect
5. #20243114C2semidirect
6. C2×C47⋊#338C2semidirect
7. C2×C47×#430C2semidirect
280. C2×#430D47semidirect
318. C47#11139semidirect

Maximal subgroups

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

subgroupindex
C22×#130939402
#202431152
#202431142
C2×C47⋊#3382
C2×C47×#4302
#1113947