Finite Group Databasebeta

← #26123859

Subgroups — #26123859

Order 12032.

818 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689141C211yesmaximal, characteristic
2C2×#15689117C211yesmaximal, characteristic
3C2×#15689105C211yesmaximal, characteristic
4C2×#15688893C211yesmaximal, characteristic, complemented
5C2×#15688887C211yesmaximal, characteristic, complemented
6#20243085C211yesmaximal, characteristic, complemented
7C2×C47×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12#15689141—22no—
13#15689141—22no—
14C2×#13093952—22no—
15#15689117—22no—
16#15689117—22no—
17C2×#13093947—21no—
18C2×#13093944—21no—
19#15689105—22no—
20#15689105C414yescomplemented
21C2×#13093942—21no—
22C2×#13093940—21no—
23#15688893—22no—
24#15688887—22no—
25#15688878—22no—
26C22×M16×C47C2211yescharacteristic
27C2×#117×C47C2211yescharacteristic
28C2×#117×C47C2211yescharacteristic
29C47×H(3,ℤ/4)—22no—
30C2×#10625350—21no—
31C2×#10625350—21no—
32C2×#10625350—21no—
33C2×#10625350—21no—
34C2×#10625350—21no—
35C2×#10625350—21no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350C2×C412yes—
41C2×#10625350D412yes—
42C22×Dic94D411yescharacteristic
43C22×D188D411yescharacteristic
44C22×C22⋊D47—21no—
45C22×C4×D47—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×#10625329C2×C412yes—
57C2×#10625329D412yes—
58C22×C47⋊C8—21no—
59C22×C47⋊C8—21no—
60C22×C47⋊C8—21no—
61C22×C47⋊C8—21no—
62C22×C47⋊C8—21no—
63C22×C47⋊C8—21no—
64C22×C47⋊C8—21no—
65C22×C47⋊C8—21no—
66C22×C47⋊C8D412yes—
67#13093952—28no—
68#13093947—24no—
69#13093944—24no—
70#13093942—24no—
71#13093940—24no—
72C23×C4×C47C2311yescharacteristic
73C2×M16×C47—21no—
74C2×M16×C47—21no—
75C2×M16×C47—21no—
76C2×M16×C47—21no—
77C2×M16×C47—21no—
78C2×M16×C47C2×C412yes—
79C22×C8×C47—21no—
80#117×C47—21no—
81#117×C47—21no—
82#117×C47—21no—
83#117×C47—21no—
84#117×C47—21no—
85#117×C47—21no—
86#117×C47—21no—
87#117×C47D412yes—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—21no—
90C2×C22⋊C4×C47—21no—
91C2×C22⋊C4×C47—21no—
92C2×C42×C47—21no—
93C2×C42×C47D412yes—
94C24×C47#4111yescharacteristic
95C23×D47—21no—
96C22×C4×C47—22no—
97C22×C4×C47—41no—
98C22×C4×C47C22×C411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47#4111yescharacteristic
106C22×C4×C47#4111yescharacteristic
107C22×C4×C47#4111yescharacteristic
108C22×C4×C47#4111yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C2×C22⋊D47—22no—
113C2×C22⋊D47—22no—
114C2×C22⋊D47—41no—
115C2×C22⋊D47—41no—
116C2×C22⋊D47—41no—
117C2×C22⋊D47—41no—
118C22×Dic47—21no—
119#10625350—22no—
120#10625350—22no—
121#10625350—22no—
122#10625350—22no—
123#10625350—22no—
124#10625350—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—41no—
128#10625350—41no—
129#10625350—41no—
130#10625350—41no—
131#10625350—41no—
132#10625350—41no—
133#10625350—41no—
134#10625350—41no—
135C2×D188—21no—
136C2×D188—21no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188C22⋊C412yes—
141C2×C4×D47—22no—
142C2×C4×D47—22no—
143C2×C4×D47—41no—
144C2×C4×D47—41no—
145C2×C4×D47—41no—
146C2×C4×D47—41no—
147C2×Dic94—21no—
148C2×Dic94—21no—
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94C22⋊C412yes—
153M16×C47—22no—
154M16×C47—22no—
155M16×C47—41no—
156M16×C47—41no—
157C2×C8×C47—22no—
158C2×C8×C47—41no—
159C2×C8×C47—41no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—41no—
162C4⋊C4×C47—41no—
163C4⋊C4×C47—41no—
164C22⋊C4×C47—22no—
165C22⋊C4×C47—41no—
166C22⋊C4×C47—41no—
167C22⋊C4×C47—41no—
168C42×C47—22no—
169C42×C47—22no—
170C42×C47—22no—
171C42×C47—41no—
172#10625329—22no—
173#10625329—22no—
174#10625329—22no—
175#10625329—22no—
176#10625329—41no—
177#10625329—41no—
178#10625329—41no—
179#10625329—41no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—41no—
186C2×C47⋊C8—41no—
187C2×C47⋊C8—41no—
188C2×C47⋊C8—41no—
189C2×C47⋊C8—41no—
190C2×C47⋊C8—41no—
191C2×C47⋊C8—41no—
192C2×C47⋊C8—41no—
193C23×C47—42no—
194C23×C47#11711yescharacteristic
195C23×C47#12111yescharacteristic
196C23×C47#12311yescharacteristic
197C23×C47#12311yescharacteristic
198C23×C47#12311yescharacteristic
199C23×C47#12411yescharacteristic
200C23×C47#12611yescharacteristic
201C22×D47—22no—
202C22×D47—41no—
203C22×D47—41no—
204C22×D47—82no—
205C2×C4×C47—44no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—82no—
209C2×C4×C47C2×C22⋊C411yescharacteristic
210C2×C4×C47#11711yescharacteristic
211C2×C4×C47C4×D411yescharacteristic
212C2×C4×C47C4×D411yescharacteristic
213C2×C4×C47C4×D411yescharacteristic
214C2×C4×C47C4×D411yescharacteristic
215C2×C4×C47#12011yescharacteristic
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#12111yescharacteristic
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#12311yescharacteristic
231C2×C4×C47#12411yescharacteristic
232C2×C4×C47#12411yescharacteristic
233C2×C4×C47#12411yescharacteristic
234C2×C4×C47#12611yescharacteristic
235C2×C4×C47#12611yescharacteristic
236C2×C4×C47#12611yescharacteristic
237C22⋊D47—42no—
238C22⋊D47—42no—
239C22⋊D47—42no—
240C22⋊D47—42no—
241C2×Dic47—22no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244D188—22no—
245D188—22no—
246D188—41no—
247D188—41no—
248C4×D47—42no—
249C4×D47—42no—
250C4×D47—42no—
251C4×D47—42no—
252Dic94—22no—
253Dic94—22no—
254Dic94—41no—
255Dic94—41no—
256C8×C47—42no—
257C47⋊C8—42no—
258C47⋊C8—42no—
259C47⋊C8—42no—
260#11150—471nomaximal, Sylow
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—22no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47#38111yescharacteristic
269C22×C47#38811yescharacteristic
270C22×C47#38811yescharacteristic
271C22×C47#39411yescharacteristic
272C22×C47#39411yescharacteristic
273C22×C47#39711yescharacteristic
274C22×C47#39711yescharacteristic
275C2×D47—42no—
276C2×D47—82no—
277C2×D47—82no—
278C2×D47—82no—
279C4×C47—82no—
280C4×C47—82no—
281C4×C47#38611yescharacteristic
282C4×C47#38611yescharacteristic
283C4×C47#38811yescharacteristic
284C4×C47#38811yescharacteristic
285C4×C47#39411yescharacteristic
286C4×C47#39411yescharacteristic
287C4×C47#39711yescharacteristic
288C4×C47#39711yescharacteristic
289Dic47—42no—
290C2×#421—471no—
291C2×#416—471no—
292C2×#413—471no—
293C2×#344—471no—
294C2×#341—471no—
295C2×H(3,ℤ/4)D4711yescharacteristic, complemented
296#1729—471no—
297C2×C47—21no—
298C2×C47—21no—
299C2×C47—21no—
300C2×C47—21no—
301C2×C47—21no—
302C2×C47—21no—
303C2×C47#144011yescharacteristic
304C2×C47#208012yes—
305D47—82no—
306C22×#41—471no—
307C22×M16—471no—
308C22×M16—471no—
309C22×M16C2×D4711yescharacteristic
310C23×C8—471no—
311C2×#117C2×D4711yescharacteristic
312C2×#117C2×D4711yescharacteristic
313#421—942no—
314#421—942no—
315C2×#104—942no—
316#416—942no—
317#416—942no—
318C2×#103—941no—
319C2×#102—941no—
320#413—474no—
321#413—942no—
322C2×#101—941no—
323C2×#100—941no—
324#344—942no—
325#341—942no—
326H(3,ℤ/4)—22no—
327#336—942no—
328C47#1115011yescharacteristic, complemented, Sylow
329C2×#41—472no—
330C2×#41—472no—
331C2×#41—941no—
332C2×#41—941no—
333C2×#41—941no—
334C2×#41—941no—
335C2×#41—941no—
336C2×#41—941no—
337C2×#41—941no—
338C2×#41—941no—
339C2×#41—941no—
340C2×#41—941no—
341C22×Q8—471no—
342C22×D4—471no—
343C22×D4—941no—
344C23×C4—941no—
345C23×C4C22×D4711yescharacteristic
346C2×M16—21no—
347C2×M16—21no—
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—472no—
352C2×M16—472no—
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×M16C4×D4712yes—
364C22×C8—21no—
365C22×C8—472no—
366C22×C8—941no—
367C22×C8—941no—
368C22×C8—941no—
369C22×C8—941no—
370C22×C8—941no—
371C22×C8—941no—
372C22×C8—941no—
373C22×C8—941no—
374#117—21no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117—21no—
379#117—21no—
380#117—21no—
381#117C22⋊D4712yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42D18812yes—
388#104—948no—
389#103—944no—
390#102—944no—
391#101—944no—
392#100—944no—
393C24—941no—
394C24#1062535211yescharacteristic
395#41—942no—
396#41—942no—
397#41—942no—
398#41—942no—
399#41—942no—
400#41—942no—
401#41—942no—
402#41—942no—
403#41—1881no—
404#41—1881no—
405#41—1881no—
406#41—1881no—
407#41—1881no—
408#41—1881no—
409#41—1881no—
410#41—1881no—
411C2×Q8—472no—
412C2×Q8—941no—
413C2×Q8—941no—
414C2×Q8—941no—
415C2×Q8—941no—
416C2×Q8—941no—
417C2×D4—472no—
418C2×D4—942no—
419C2×D4—942no—
420C2×D4—941no—
421C2×D4—941no—
422C2×D4—941no—
423C2×D4—941no—
424C2×D4—941no—
425C2×D4—1881no—
426C2×D4—1881no—
427C2×D4—1881no—
428C2×D4—1881no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—941no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—1881no—
435C22×C4—1881no—
436C22×C4—1881no—
437C22×C4—1881no—
438C22×C4C2×C4×D4711yescharacteristic
439C22×C4C2×D18811yescharacteristic
440C22×C4#1062535011yescharacteristic
441C22×C4#1062535011yescharacteristic
442C22×C4#1062535011yescharacteristic
443C22×C4#1062535011yescharacteristic
444C22×C4D4×D4711yescharacteristic
445C22×C4D4×D4711yescharacteristic
446C22×C4D4×D4711yescharacteristic
447C22×C4D4×D4711yescharacteristic
448C22×C4#1062535211yescharacteristic
449C22×C4#1062535411yescharacteristic
450C22×C4#1062535411yescharacteristic
451C22×C4C2×C22⋊D4711yescharacteristic
452M16—22no—
453M16—22no—
454M16—41no—
455M16—41no—
456M16—942no—
457M16—942no—
458M16—942no—
459M16—942no—
460M16—1881no—
461M16—1881no—
462M16—1881no—
463M16—1881no—
464C2×C8—22no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—942no—
468C2×C8—942no—
469C2×C8—942no—
470C2×C8—942no—
471C2×C8—942no—
472C2×C8—1881no—
473C2×C8—1881no—
474C2×C8—1881no—
475C2×C8—1881no—
476C2×C8—1881no—
477C2×C8—1881no—
478C2×C8—1881no—
479C2×C8—1881no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C22⋊C4—22no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—41no—
492C23—42no—
493C23—942no—
494C23—3762no—
495C23—1881no—
496C23—1881no—
497C23#1309398211yescharacteristic
498C23#1309399611yescharacteristic
499C23#1309400611yescharacteristic
500C23#1309400611yescharacteristic
501C23#1309400811yescharacteristic
502C23#1309401011yescharacteristic
503C23#1309401911yescharacteristic
504Q8—942no—
505Q8—942no—
506Q8—1881no—
507Q8—1881no—
508D4—942no—
509D4—942no—
510D4—1882no—
511D4—1882no—
512D4—1882no—
513D4—1882no—
514D4—1881no—
515D4—1881no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—82no—
520C2×C4—942no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4C22⋊C4×D4711yescharacteristic
528C2×C4#1309398111yescharacteristic
529C2×C4(C22⋊D47)⋊C411yescharacteristic
530C2×C4C4×C22⋊D4711yescharacteristic
531C2×C4#1309398611yescharacteristic
532C2×C4C4×D18811yescharacteristic
533C2×C4#1309399211yescharacteristic
534C2×C4#1309399411yescharacteristic
535C2×C4#1309399511yescharacteristic
536C2×C4#1309399711yescharacteristic
537C2×C4#1309399711yescharacteristic
538C2×C4#1309399711yescharacteristic
539C2×C4#1309399811yescharacteristic
540C2×C4#1309399911yescharacteristic
541C2×C4#1309399911yescharacteristic
542C2×C4#1309400611yescharacteristic
543C2×C4#1309400711yescharacteristic
544C2×C4#1309400911yescharacteristic
545C2×C4#1309400911yescharacteristic
546C2×C4#1309400911yescharacteristic
547C2×C4#1309400911yescharacteristic
548C2×C4#1309400911yescharacteristic
549C2×C4#1309401011yescharacteristic
550C2×C4#1309401211yescharacteristic
551C2×C4#1309401311yescharacteristic
552C2×C4#1309401811yescharacteristic
553C2×C4#1309401911yescharacteristic
554C2×C4#1309401911yescharacteristic
555C8—42no—
556C8—1882no—
557C8—1882no—
558C8—1882no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—3762no—
567C22—3762no—
568C22—3762no—
569C22—1882no—
570C22#1568898511yescharacteristic
571C22#1568901311yescharacteristic
572C22#1568901511yescharacteristic
573C22#1568903811yescharacteristic
574C22#1568903911yescharacteristic
575C22#1568905011yescharacteristic
576C22#1568905211yescharacteristic
577C4—82no—
578C4—82no—
579C4—1882no—
580C4#1568899911yescharacteristic
581C4#1568900111yescharacteristic
582C4#1568901211yescharacteristic
583C4#1568901411yescharacteristic
584C4#1568904011yescharacteristic
585C4#1568904111yescharacteristic
586C4#1568904911yescharacteristic
587C4#1568905211yescharacteristic
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—3762no—
595C2#2024163611yescharacteristic
596C2#2024484012yes—
subgroupG/H
centerC2×C4#13094012
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×H(3,ℤ/4)C2
socleC22×C47#397
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenterC2×H(3,ℤ/4)D47
hypocenterC47#11150
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. C2×#15688893C2semidirect
5. C2×#15688887C2semidirect
6. #20243085C2semidirect
7. C2×C47×H(3,ℤ/4)C2semidirect
20. #15689105C4semidirect
295. C2×H(3,ℤ/4)D47semidirect
328. C47#11150semidirect

Maximal subgroups

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

subgroupindex
C2×#156891412
C2×#156891172
C2×#156891052
C2×#156888932
C2×#156888872
#202430852
C2×C47×H(3,ℤ/4)2
#1115047