Finite Group Databasebeta

← #26274340

Subgroups — #26274340

Order 12032.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20247112C211yesmaximal, characteristic
2#20244956C211yesmaximal, characteristic, complemented
3#20244889C211yesmaximal, characteristic, complemented
4#20244889C211yesmaximal, characteristic, complemented
5#20244726C211yesmaximal, characteristic, complemented
6#20244345C211yesmaximal, characteristic, complemented
7#20244156C211yesmaximal, characteristic, complemented
8#20243966C211yesmaximal, characteristic, complemented
9#20243966C211yesmaximal, characteristic, complemented
10#20243580C211yesmaximal, characteristic
11#20243573C211yesmaximal, characteristic
12#20243483C211yesmaximal, characteristic
13#20243419C211yesmaximal, characteristic
14#20243109C211yesmaximal, characteristic
15C47×#2346C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C2×#13094016C2211yescharacteristic
18C22×C4⋊Dic47C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25C2×C22⋊(C47⋊C8)C2211yescharacteristic
26#15689065—21no—
27#15689057—21no—
28#15689021C2211yescharacteristic
29#15689021C2211yescharacteristic
30#15688995—21no—
31#15688989C2211yescharacteristic
32#15688983C2211yescharacteristic
33#15688981—21no—
34#15688885C2211yescharacteristic
35#15688885C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×#117×C47C2211yescharacteristic
49C47×#389C2211yescharacteristic
50C47×#389C2211yescharacteristic
51C47×#388—21no—
52C47×#388—21no—
53C47×#382—21no—
54C47×#380C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C22⋊C4×C47C2211yescharacteristic
58C47×#376—21no—
59C23×Dic47—21no—
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329—41no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13094016—22no—
68#13094016—41no—
69C2×C4⋊Dic47—22no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—41no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×C22⋊Dic47—21no—
81#13093957—41no—
82#13093955—22no—
83#13093955—22no—
84#13093955—41no—
85C4⋊(C47⋊C8)—22no—
86C4⋊(C47⋊C8)—41no—
87C22⋊(C47⋊C8)—22no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C23×C4×C47C2311yescharacteristic
94#117×C47—22no—
95#117×C47Q814yes—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C4⋊C4×C47C2311yescharacteristic
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47D411yescharacteristic
105C2×C22⋊C4×C47D411yescharacteristic
106C2×C42×C47—21no—
107C2×C42×C47—21no—
108C2×C42×C47—21no—
109C2×C42×C47C2311yescharacteristic
110C2×C42×C47C2311yescharacteristic
111#95×C47—21no—
112#95×C47—21no—
113#95×C47—21no—
114#95×C47—21no—
115#95×C47—21no—
116#95×C47D411yescharacteristic
117#95×C47D411yescharacteristic
118C24×C47C2×D411yescharacteristic
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×Q812yes—
133C22×C4×C47#4112yes—
134C22×C4×C47C2411yescharacteristic
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—42no—
140C22×Dic47—42no—
141C22×Dic47—81no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—81no—
149C4⋊C4×C47—81no—
150C4⋊C4×C47C2×Q812yes—
151C4⋊C4×C47C2×Q812yes—
152C22⋊C4×C47—24no—
153C22⋊C4×C47—41no—
154C22⋊C4×C47—41no—
155C22⋊C4×C47—81no—
156C42×C47—24no—
157C42×C47—21no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—21no—
161C42×C47—42no—
162C42×C47—42no—
163C22⋊Dic47—42no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—42no—
172C4⋊Dic47#4112yes—
173Dic47⋊C4—42no—
174Dic47⋊C4—42no—
175Dic47⋊C4—81no—
176Dic47⋊C4—81no—
177#10625329—44no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8#4112yes—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47C22⋊Q812yes—
197C23×C47C22×D411yescharacteristic
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47C22⋊Q812yes—
229C2×C4×C47C22×Q811yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47H(5,𝔽2)11yescharacteristic
234C2×C4×C472−1+411yescharacteristic
235C2×Dic47—22no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—84no—
248C47⋊C8—22no—
249C47⋊C8—41no—
250C47⋊C8—41no—
251C47⋊C8—41no—
252#26019—471nomaximal, Sylow
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—41no—
257C22×C47—41no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47C2×C22⋊Q811yescharacteristic
261C22×C47#54611yescharacteristic
262C22×C47#54711yescharacteristic
263C22×C47#57511yescharacteristic
264C22×C47#57511yescharacteristic
265C22×C47#57711yescharacteristic
266C22×C47#57711yescharacteristic
267C4×C47—22no—
268C4×C47—42no—
269C4×C47—42no—
270C4×C47—41no—
271C4×C47—41no—
272C4×C47—41no—
273C4×C47—41no—
274C4×C47—81no—
275C4×C47—81no—
276C4×C47#53311yescharacteristic
277C4×C47#54411yescharacteristic
278C4×C47#55111yescharacteristic
279C4×C47#55211yescharacteristic
280Dic47—42no—
281Dic47—81no—
282Dic47—81no—
283Dic47—81no—
284#2442—471no—
285#2346D4711yescharacteristic
286#2101—471no—
287#2088—471no—
288#2088—471no—
289#2059—471no—
290#1989—471no—
291#1949—471no—
292#1919—471no—
293#1919—471no—
294#1853—471no—
295#1851—471no—
296#1825—471no—
297#1809—471no—
298#1736—471no—
299C2×C47—42no—
300C2×C47#243811yescharacteristic
301C2×C47#308011yescharacteristic
302C2×C47#308411yescharacteristic
303C2×C47#332111yescharacteristic
304C2×C47#332311yescharacteristic
305C2×C47#334811yescharacteristic
306C2×C47#335111yescharacteristic
307C22×M16—471no—
308C2×#125—471no—
309C2×#117C2×D4711yescharacteristic
310C22×C4⋊C4—471no—
311C2×#107—471no—
312C2×#106—471no—
313C2×#106—471no—
314C2×#106—471no—
315C2×C4⋊C8—471no—
316C2×C4⋊C8—471no—
317C2×C22⋊C8—471no—
318#400—941no—
319#398—941no—
320#389—471no—
321#389—471no—
322#389C2×D4711yescharacteristic
323#389C2×D4711yescharacteristic
324#388—21no—
325#388—21no—
326#385—941no—
327#382—21no—
328#382—471no—
329#380—471no—
330#380—941no—
331#380C2×D4711yescharacteristic
332C4×C4⋊C4C2×D4711yescharacteristic
333C4×C4⋊C4C2×D4711yescharacteristic
334C4×C22⋊C4C2×D4711yescharacteristic
335#376—21no—
336#340—471no—
337#340—471no—
338#340—471no—
339#340—471no—
340#340—471no—
341#340—471no—
342#340—471no—
343#340—471no—
344#340—471no—
345#340—471no—
346#340—471no—
347#340—471no—
348#336—471no—
349#336—471no—
350C47#2601911yescharacteristic, complemented, Sylow
351C23×C4—941no—
352C23×C4C22×D4711yescharacteristic
353C2×M16—942no—
354C2×M16—942no—
355C2×M16—1881no—
356C22×C8—471no—
357C22×C8—471no—
358C22×C8—471no—
359C22×C8—471no—
360#125—942no—
361#125—1881no—
362#117—22no—
363#117Dic9414yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—1881no—
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—941no—
384C2×C22⋊C4C22⋊D4711yescharacteristic
385C2×C22⋊C4C22⋊D4711yescharacteristic
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42C22×D4711yescharacteristic
390C2×C42C22×D4711yescharacteristic
391#107—1881no—
392#106—942no—
393#106—942no—
394#106—1881no—
395C4⋊C8—942no—
396C4⋊C8—1881no—
397C22⋊C8—942no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—941no—
404#95—941no—
405#95—941no—
406#95—941no—
407#95—941no—
408#95C22⋊D4711yescharacteristic
409#95C22⋊D4711yescharacteristic
410C24C2×C22⋊D4711yescharacteristic
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4—941no—
423C22×C4—3761no—
424C22×C4—1882no—
425C22×C4—1882no—
426C22×C4C2×Dic9412yes—
427C22×C4#1062535012yes—
428C22×C4C2×C22⋊D4711yescharacteristic
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C2×C22⋊D4711yescharacteristic
432C22×C4C2×C22⋊D4711yescharacteristic
433C22×C4C23×D4711yescharacteristic
434M16—1884no—
435C2×C8—472no—
436C2×C8—941no—
437C2×C8—941no—
438C2×C8—941no—
439C2×C8—941no—
440C2×C8—941no—
441C2×C8—941no—
442C2×C8—941no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—472no—
456C4⋊C4—941no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—941no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—3761no—
464C4⋊C4—3761no—
465C4⋊C4—1882no—
466C4⋊C4—1882no—
467C4⋊C4—1882no—
468C4⋊C4C2×Dic9412yes—
469C4⋊C4C2×Dic9412yes—
470C22⋊C4—24no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—81no—
474C22⋊C4—1882no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—42no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1309400312yes—
489C23C22×C22⋊D4711yescharacteristic
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—942no—
521C2×C4—3764no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4#1309400312yes—
534C2×C4C22×Dic9411yescharacteristic
535C2×C4#41×D4711yescharacteristic
536C2×C4#41×D4711yescharacteristic
537C2×C4C2×#1062535011yescharacteristic
538C2×C4#1309407011yescharacteristic
539C2×C4#1309407111yescharacteristic
540C8—942no—
541C8—1881no—
542C8—1881no—
543C8—1881no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1309400311yescharacteristic
552C22#1568976111yescharacteristic
553C22#1568977211yescharacteristic
554C22#1568992111yescharacteristic
555C22#1568992111yescharacteristic
556C22#1568993111yescharacteristic
557C22#1568993411yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—3761no—
568C4—3761no—
569C4—3761no—
570C4—1882no—
571C4#1568967611yescharacteristic
572C4#1568975111yescharacteristic
573C4#1568979711yescharacteristic
574C4#1568980611yescharacteristic
575C2—42no—
576C2#2024705811yescharacteristic
577C2#2025321011yescharacteristic
578C2#2025325011yescharacteristic
579C2#2025556811yescharacteristic
580C2#2025559811yescharacteristic
581C2#2025590311yescharacteristic
582C2#2025594811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2346C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2346D47
hypocenterC47#26019
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2346

Complements

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

NH ≅ G/Ntype
2. #20244956C2semidirect
3. #20244889C2semidirect
4. #20244889C2semidirect
5. #20244726C2semidirect
6. #20244345C2semidirect
7. #20244156C2semidirect
8. #20243966C2semidirect
9. #20243966C2semidirect
350. C47#26019semidirect

Maximal subgroups

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

subgroupindex
#202471122
#202449562
#202448892
#202447262
#202443452
#202441562
#202439662
#202435802
#202435732
#202434832
#202434192
#202431092
C47×#23462
#2601947