Finite Group Databasebeta

← #26108195

Subgroups — #26108195

Order 12032.

942 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244341C211yesmaximal, characteristic, complemented
2#20243408C211yesmaximal, characteristic, complemented
3#20242065C211yesmaximal, characteristic, complemented
4#20242060C211yesmaximal, characteristic, complemented
5#20241883C211yesmaximal, characteristic, complemented
6#20241775C211yesmaximal, characteristic, complemented
7C47×#3499C211yesmaximal, characteristic
8#15689176C2211yescharacteristic
9#15689147C2211yescharacteristic
10C2×C4⋊(C47⋊C8)C2211yescharacteristic
11C2×#13093951—21no—
12C4×C4⋊Dic47C2211yescharacteristic
13#15688885—22no—
14#15688882—21no—
15#15688869—22no—
16#15688869—21no—
17#15688867—21no—
18#15688857—22no—
19Q8⋊(C47⋊C8)—21no—
20D4⋊(C47⋊C8)—21no—
21C2×2−1+4×C47—21no—
22D4×Q8×C47—21no—
23D4×Q8×C47C412yescomplemented
24C47×#547—22no—
25C47×#547—21no—
26C47×#547—22no—
27C47×#547C412yescomplemented
28C47×#536—22no—
29C47×#536—21no—
30C47×#532—21no—
31C47×#532C2211yescharacteristic
32C2×C4⋊Q8×C47—21no—
33C2×C4⋊Q8×C47C2211yescharacteristic
34C4×#41×C47C2211yescharacteristic
35C2×#10625329—21no—
36C22×C47⋊C8—21no—
37C2×C4⋊Dic47—21no—
38C2×C4×Dic47—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44#13093951—42no—
45C22⋊(C47⋊C8)—21no—
46C4×C47⋊C8—21no—
47C47⋊#95—21no—
482−1+4×C47—42no—
492−1+4×C47—42no—
502−1+4×C47—42no—
512−1+4×C47—42no—
52C2×#41×C47—22no—
53C2×#41×C47—24no—
54C2×#41×C47—22no—
55C2×#41×C47—21no—
56C2×#41×C47D411yescharacteristic
57C22×Q8×C47—22no—
58C22×Q8×C47—22no—
59C22×Q8×C47—21no—
60C4⋊Q8×C47—22no—
61C4⋊Q8×C47—22no—
62C4⋊Q8×C47—22no—
63C4⋊Q8×C47—21no—
64C4⋊Q8×C47—21no—
65C4⋊Q8×C47C2×C412yes—
66C4⋊Q8×C47D411yescharacteristic
67C4⋊Q8×C47D412yes—
68#127×C47D411yescharacteristic
69#124×C47—22no—
70#124×C47—21no—
71#123×C47—24no—
72#123×C47—24no—
73#123×C47—24no—
74C22⋊Q8×C47—24no—
75C22⋊Q8×C47—22no—
76C22⋊Q8×C47—24no—
77C22⋊Q8×C47—22no—
78#121×C47—22no—
79#121×C47—22no—
80#121×C47—21no—
81#121×C47D412yes—
82C4×Q8×C47C2×C411yescharacteristic
83C4×Q8×C47D411yescharacteristic
84C4×D4×C47—22no—
85C4×D4×C47—22no—
86C4×D4×C47—21no—
87C4×D4×C47—21no—
88C4×D4×C47—21no—
89C4×D4×C47C2×C411yescharacteristic
90C4×D4×C47C2×C412yes—
91C4×D4×C47D411yescharacteristic
92#117×C47—21no—
93#117×C47D411yescharacteristic
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47D412yes—
98C2×C42×C47—21no—
99C2×C42×C47C2311yescharacteristic
100#41×C47—21no—
101#41×C47—21no—
102#41×C47—44no—
103#41×C47—44no—
104#41×C47—48no—
105#41×C47—48no—
106#41×C47—44no—
107#41×C47—44no—
108#41×C47—41no—
109#41×C47—41no—
110#41×C47—41no—
111#41×C47—41no—
112#41×C47D812yes—
113#41×C47SD1612yes—
114C2×Q8×C47—24no—
115C2×Q8×C47—22no—
116C2×Q8×C47—22no—
117C2×Q8×C47—21no—
118C2×Q8×C47—44no—
119C2×Q8×C47—44no—
120C2×Q8×C47—44no—
121C2×Q8×C47—44no—
122C2×Q8×C47—41no—
123C2×Q8×C47—41no—
124C2×Q8×C47—41no—
125C2×Q8×C47—41no—
126C2×Q8×C47C22⋊C411yescharacteristic
127C2×D4×C47—21no—
128C2×D4×C47—24no—
129C2×D4×C47—22no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47C22⋊C411yescharacteristic
133C2×D4×C47C22⋊C412yes—
134C22×C4×C47—21no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—22no—
138C22×C4×C47—21no—
139C22×C4×C47—22no—
140C22×C4×C47—21no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×Dic47—41no—
145C4⋊C4×C47—24no—
146C4⋊C4×C47—24no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47C22⋊C412yes—
154C4⋊C4×C47C22⋊C414yes—
155C4⋊C4×C47C22⋊C411yescharacteristic
156C4⋊C4×C47C22⋊C411yescharacteristic
157C22⋊C4×C47—24no—
158C22⋊C4×C47—24no—
159C22⋊C4×C47—21no—
160C22⋊C4×C47—21no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47—21no—
163C22⋊C4×C47—22no—
164C42×C47—21no—
165C42×C47—21no—
166C42×C47—21no—
167C42×C47C22×C411yescharacteristic
168C42×C47C2×D411yescharacteristic
169C42×C47C2×D411yescharacteristic
170C42×C47C2×D411yescharacteristic
171C4⋊Dic47—22no—
172C4⋊Dic47—41no—
173C4×Dic47—41no—
174C4×Dic47—41no—
175#10625329—41no—
176#10625329—41no—
177C2×C47⋊C8—42no—
178C2×C47⋊C8—41no—
179C2×C47⋊C8—41no—
180C2×C47⋊C8—41no—
181C23×C47—21no—
182C23×C47—22no—
183C23×C47#12011yescharacteristic
184Q8×C47—21no—
185Q8×C47—48no—
186Q8×C47—42no—
187Q8×C47—42no—
188Q8×C47—41no—
189Q8×C47—42no—
190Q8×C47—41no—
191Q8×C47—42no—
192Q8×C47#10212yes—
193D4×C47—22no—
194D4×C47—22no—
195D4×C47—21no—
196D4×C47—42no—
197D4×C47—48no—
198D4×C47—42no—
199D4×C47—42no—
200D4×C47—41no—
201D4×C47—41no—
202D4×C47—41no—
203D4×C47—41no—
204D4×C47#10212yes—
205C2×C4×C47—21no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—24no—
213C2×C4×C47—22no—
214C2×C4×C47—41no—
215C2×C4×C47—42no—
216C2×C4×C47—48no—
217C2×C4×C47—48no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47C2×C22⋊C411yescharacteristic
225C2×C4×C47C2×C22⋊C411yescharacteristic
226C2×C4×C47C2×C22⋊C411yescharacteristic
227C2×C4×C47#12011yescharacteristic
228C2×C4×C47#12011yescharacteristic
229C2×C4×C47#12011yescharacteristic
230C2×C4×C47C2×D811yescharacteristic
231C2×C4×C47C2×SD1611yescharacteristic
232C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
233C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
234C2×Dic47—42no—
235C2×Dic47—81no—
236C2×Dic47—81no—
237C47⋊C8—81no—
238C47⋊C8—81no—
239C47⋊C8—81no—
240#8421—471nomaximal, Sylow
241C22×C47—21no—
242C22×C47—21no—
243C22×C47—22no—
244C22×C47—41no—
245C22×C47—41no—
246C22×C47—42no—
247C22×C47—42no—
248C22×C47—42no—
249C22×C47#37911yescharacteristic
250C22×C47#44711yescharacteristic
251C22×C47#45011yescharacteristic
252C4×C47—21no—
253C4×C47—21no—
254C4×C47—21no—
255C4×C47—22no—
256C4×C47—41no—
257C4×C47—41no—
258C4×C47—44no—
259C4×C47—42no—
260C4×C47C2×#10211yescharacteristic
261C4×C47#41811yescharacteristic
262C4×C47#44711yescharacteristic
263C4×C47#45011yescharacteristic
264C4×C47#45411yescharacteristic
265C4×C47#45611yescharacteristic
266Dic47—82no—
267#3499D4711yescharacteristic
268#1988—471no—
269#1806—471no—
270#1547—471no—
271#1546—471no—
272#1502—471no—
273#1477—471no—
274C2×C47—21no—
275C2×C47—21no—
276C2×C47—42no—
277C2×C47—82no—
278C2×C47#178611yescharacteristic
279C2×C47#179311yescharacteristic
280C2×C47#179611yescharacteristic
281C2×2−1+4—21no—
282D4×Q8—21no—
283D4×Q8Dic4712yescomplemented
284#547—22no—
285#547—21no—
286#547—22no—
287#547Dic4712yescomplemented
288#536—22no—
289#536—21no—
290#532—21no—
291#532C2×D4711yescharacteristic
292C2×C4⋊Q8—21no—
293C2×C4⋊Q8C2×D4711yescharacteristic
294C4×#41C2×D4711yescharacteristic
295#431—471no—
296#423—471no—
297C2×C4⋊C8—471no—
298C2×#104—941no—
299C4×C4⋊C4—471no—
300#340—942no—
301#339—941no—
302#332—942no—
303#332—941no—
304#331—941no—
305#327—942no—
306Q8⋊C8—941no—
307D4⋊C8—941no—
308C47#842111yescharacteristic, complemented, Sylow
3092−1+4—42no—
3102−1+4—42no—
3112−1+4—42no—
3122−1+4—42no—
313C2×#41—22no—
314C2×#41—24no—
315C2×#41—22no—
316C2×#41—21no—
317C2×#41C22⋊D4711yescharacteristic
318C22×Q8—22no—
319C22×Q8—22no—
320C22×Q8—21no—
321C2×M16—941no—
322C22×C8—941no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—22no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8C2×Dic4712yes—
329C4⋊Q8C22⋊D4711yescharacteristic
330C4⋊Q8C22⋊D4712yes—
331#127C22⋊D4711yescharacteristic
332#124—22no—
333#124—21no—
334#123—24no—
335#123—24no—
336#123—24no—
337C22⋊Q8—24no—
338C22⋊Q8—22no—
339C22⋊Q8—24no—
340C22⋊Q8—22no—
341#121—22no—
342#121—22no—
343#121—21no—
344#121C22⋊D4712yes—
345C4×Q8C2×Dic4711yescharacteristic
346C4×Q8C22⋊D4711yescharacteristic
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4C2×Dic4711yescharacteristic
353C4×D4C2×Dic4712yes—
354C4×D4C22⋊D4711yescharacteristic
355#117—21no—
356#117C22⋊D4711yescharacteristic
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4C22⋊D4712yes—
362C2×C42—21no—
363C2×C42—941no—
364C2×C42C22×D4711yescharacteristic
365C4⋊C8—941no—
366C4⋊C8—941no—
367C4⋊C8—941no—
368C4⋊C8—941no—
369C4⋊C8—941no—
370#104—1882no—
371C22⋊C8—941no—
372C4×C8—941no—
373#95—941no—
374#41—21no—
375#41—21no—
376#41—44no—
377#41—44no—
378#41—48no—
379#41—48no—
380#41—44no—
381#41—44no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41#1062533412yes—
387#41#1062533512yes—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8C22⋊Dic4711yescharacteristic
401C2×D4—21no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C22⋊Dic4711yescharacteristic
407C2×D4C22⋊Dic4712yes—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—1881no—
416C22×C4C2×C22⋊D4711yescharacteristic
417C22×C4C2×C22⋊D4711yescharacteristic
418C22×C4C2×C22⋊D4711yescharacteristic
419M16—1881no—
420M16—1881no—
421C2×C8—1882no—
422C2×C8—1881no—
423C2×C8—1881no—
424C2×C8—1881no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—942no—
434C4⋊C4—1881no—
435C4⋊C4C22⋊Dic4712yes—
436C4⋊C4C22⋊Dic4714yes—
437C4⋊C4C22⋊Dic4711yescharacteristic
438C4⋊C4C22⋊Dic4711yescharacteristic
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—21no—
445C22⋊C4—22no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—1881no—
450C42—1881no—
451C42C22×Dic4711yescharacteristic
452C42C2×C22⋊D4711yescharacteristic
453C42C2×C22⋊D4711yescharacteristic
454C42C2×C22⋊D4711yescharacteristic
455C23—21no—
456C23—22no—
457C23#1309399311yescharacteristic
458Q8—21no—
459Q8—48no—
460Q8—41no—
461Q8—42no—
462Q8—41no—
463Q8—42no—
464Q8—42no—
465Q8—42no—
466Q8#1309394512yes—
467D4—22no—
468D4—22no—
469D4—21no—
470D4—42no—
471D4—48no—
472D4—42no—
473D4—42no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4#1309394512yes—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—3761no—
499C2×C4—3761no—
500C2×C4—1882no—
501C2×C4C2×C22⋊Dic4711yescharacteristic
502C2×C4C2×C22⋊Dic4711yescharacteristic
503C2×C4C2×C22⋊Dic4711yescharacteristic
504C2×C4#1309399311yescharacteristic
505C2×C4#1309399311yescharacteristic
506C2×C4#1309399311yescharacteristic
507C2×C4C2×#1062533411yescharacteristic
508C2×C4C2×#1062533511yescharacteristic
509C2×C4#1309405011yescharacteristic
510C2×C4#1309405011yescharacteristic
511C8—3761no—
512C8—3761no—
513C8—3761no—
514C22—21no—
515C22—21no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22#1568897911yescharacteristic
523C22#1568925711yescharacteristic
524C22#1568927411yescharacteristic
525C4—22no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—44no—
530C4—42no—
531C4—41no—
532C4—41no—
533C4—3762no—
534C4C2×#1309394511yescharacteristic
535C4#1568912811yescharacteristic
536C4#1568925711yescharacteristic
537C4#1568927411yescharacteristic
538C4#1568929611yescharacteristic
539C4#1568930611yescharacteristic
540C2—21no—
541C2—21no—
542C2—42no—
543C2—82no—
544C2#2024331911yescharacteristic
545C2#2024334711yescharacteristic
546C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3499C2
socleC22×C47#379
cosocleC2×C42×C47C23
hypercenter#3499D47
hypocenterC47#8421
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #20244341C2semidirect
2. #20243408C2semidirect
3. #20242065C2semidirect
4. #20242060C2semidirect
5. #20241883C2semidirect
6. #20241775C2semidirect
23. D4×Q8×C47C4semidirect
27. C47×#547C4semidirect
283. D4×Q8Dic47semidirect
287. #547Dic47semidirect
308. C47#8421semidirect

Maximal subgroups

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

subgroupindex
#202443412
#202434082
#202420652
#202420602
#202418832
#202417752
C47×#34992
#842147