Finite Group Databasebeta

← #26226217

Subgroups — #26226217

Order 12032.

1086 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689677C211yesmaximal, characteristic
2C2×#15689145C211yesmaximal, characteristic
3#20245781C211yesmaximal, characteristic
4#20244334C212yesmaximal, complemented
5#20244309C212yesmaximal, complemented
6#20243820C212yesmaximal
7#20243818C212yesmaximal
8#20243404C211yesmaximal, characteristic, complemented
9#20243403C211yesmaximal, characteristic, complemented
10#20243400C211yesmaximal, characteristic, complemented
11C47×#1805C211yesmaximal, characteristic, complemented
12C22×#10625324C2212yes—
13#15689677—24no—
14#15689677C418yescomplemented
15C2×#13094023C2211yescharacteristic
16C2×#13094022C2211yescharacteristic
17C2×#13094014—21no—
18C2×#13094004C2211yescharacteristic
19C2×#13094002—21no—
20C2×#13094000C2211yescharacteristic
21C2×C4×Dic94—21no—
22C2×C4×Dic94C2211yescharacteristic
23C2×#13093981—21no—
24C2×C42×D47C2211yescharacteristic
25#15689145—24no—
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×Dic47⋊C8C2212yes—
28C2×#13093949C2212yes—
29C2×#13093948C2212yes—
30C2×#13093947C2212yes—
31#15688990—21no—
32#15688989C2211yescharacteristic
33#15688985—21no—
34#15688980C2212yes—
35C4⋊C4×Dic47C2212yes—
36#15688970—21no—
37#15688968—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C47×#382C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94—21no—
47C22×Dic94C2311yescharacteristic
48C22×C4×D47D411yescharacteristic
49C22×C4×D47D411yescharacteristic
50C22×C4×D47C2311yescharacteristic
51C2×#10625324—24no—
52C2×#10625324—42no—
53C2×#10625324—42no—
54C22×C47⋊C8C2312yes—
55#13094023—22no—
56#13094023C2×C414yes—
57#13094022—22no—
58#13094022C2×C414yes—
59#13094014—28no—
60#13094004—22no—
61#13094004C2×C414yes—
62#13094002—28no—
63#13094000—22no—
64#13094000C2×C414yes—
65C4×Dic94—22no—
66C4×Dic94—28no—
67C4×Dic94C2×C414yes—
68#13093981—28no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47C2312yes—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—22no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C42×D47C2×C414yes—
86C42×D47D414yes—
87C4⋊(C47⋊C8)—22no—
88Dic47⋊C8—24no—
89#13093949—42no—
90#13093948—42no—
91#13093947—42no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—22no—
95C47⋊#95—22no—
96C22×Q8×C47C2311yescharacteristic
97C22×C8×C47C2312yes—
98C4×Q8×C47—22no—
99C4×Q8×C47C2×C414yes—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2312yes—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C42×C47—21no—
104C2×C42×C47C2311yescharacteristic
105C4⋊C8×C47—22no—
106#103×C47—42no—
107#95×C47—21no—
108#95×C47—21no—
109C23×D47C2×D411yescharacteristic
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47C22×C412yes—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2411yescharacteristic
122C22×Dic47—21no—
123C22×Dic47—21no—
124C22×Dic47—22no—
125C22×Dic47C2×D411yescharacteristic
126C22×Dic47C2×D411yescharacteristic
127C22×Dic47C2×D411yescharacteristic
128C2×C4×D47—21no—
129C2×C4×D47—21no—
130C2×C4×D47—21no—
131C2×C4×D47—21no—
132C2×C4×D47—24no—
133C2×C4×D47—42no—
134C2×C4×D47C22⋊C414yes—
135C2×C4×D47C22⋊C414yes—
136C2×C4×D47C22×C412yes—
137C2×C4×D47C2×D412yes—
138C2×Dic94—21no—
139C2×Dic94—21no—
140C2×Dic94—21no—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—24no—
144C2×Dic94—44no—
145C2×Dic94C22×C412yes—
146C2×C8×C47—22no—
147C2×C8×C47—22no—
148C2×C8×C47—22no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47C22×C412yes—
154C42×C47—24no—
155C42×C47C22×C412yes—
156C42×C47C2×D412yes—
157(C2×D47)⋊C4—24no—
158(C2×D47)⋊C4—24no—
159(C2×D47)⋊C4—44no—
160C4⋊Dic47—21no—
161C4⋊Dic47—24no—
162C4⋊Dic47—22no—
163C4⋊Dic47—22no—
164C4⋊Dic47C22×C412yes—
165Dic47⋊C4—24no—
166Dic47⋊C4—24no—
167Dic47⋊C4—24no—
168Dic47⋊C4—24no—
169Dic47⋊C4—24no—
170C4×Dic47—24no—
171C4×Dic47—44no—
172C4×Dic47C22⋊C414yes—
173C4×Dic47C22⋊C414yes—
174C4×Dic47C22×C412yes—
175C4×Dic47C2×D412yes—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179#10625324—48no—
180C23×C47C22×D411yescharacteristic
181C22×D47—21no—
182C22×D47—21no—
183C22×D47—22no—
184C22×D47—22no—
185C22×D47C2×C22⋊C412yes—
186Q8×C47—22no—
187Q8×C47—22no—
188Q8×C47—41no—
189Q8×C47—41no—
190C2×C4×C47—22no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—41no—
195C2×C4×C47—41no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47C2×C22⋊C412yes—
201C2×C4×C47C2×C22⋊C412yes—
202C2×C4×C47C23×C411yescharacteristic
203C2×C4×C47C22×D411yescharacteristic
204C2×C4×C47H(5,𝔽2)11yescharacteristic
205C2×C4×C47H(5,𝔽2)11yescharacteristic
206C2×C4×C472−1+411yescharacteristic
207C2×C4×C472−1+411yescharacteristic
208C2×Dic47—21no—
209C2×Dic47—21no—
210C2×Dic47—22no—
211C2×Dic47—22no—
212C2×Dic47—22no—
213C2×Dic47—41no—
214C2×Dic47—41no—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47C2×C22⋊C412yes—
220C2×Dic47C2×C22⋊C412yes—
221C2×Dic47C2×C22⋊C412yes—
222C2×Dic47#13714yes—
223C4×D47—22no—
224C4×D47—22no—
225C4×D47—22no—
226C4×D47—22no—
227C4×D47—28no—
228C4×D47—44no—
229Dic94—22no—
230Dic94—22no—
231Dic94—48no—
232Dic94—41no—
233Dic94—41no—
234C8×C47—42no—
235C47⋊C8—42no—
236#21892—471nomaximal, Sylow
237C22×C47C22×C22⋊C411yescharacteristic
238C22×C47#53711yescharacteristic
239C22×C47#53711yescharacteristic
240C22×C47C2×#13712yes—
241C22×C47#57512yes—
242C2×D47—22no—
243C2×D47—22no—
244C2×D47—22no—
245C2×D47—42no—
246C2×D47—42no—
247C4×C47—22no—
248C4×C47—22no—
249C4×C47—42no—
250C4×C47—42no—
251C4×C47—82no—
252C4×C47#52011yescharacteristic
253C4×C47#52011yescharacteristic
254C4×C47#53511yescharacteristic
255C4×C47#53611yescharacteristic
256Dic47—22no—
257Dic47—22no—
258Dic47—42no—
259Dic47—42no—
260Dic47—82no—
261Dic47#41914yes—
262C2×#533—471no—
263C2×#423—471no—
264#2283—471no—
265#1987—472no—
266#1981—472no—
267#1894—472no—
268#1894—472no—
269#1805—471no—
270#1805—471no—
271#1805—471no—
272#1805D4711yescharacteristic, complemented
273C2×C47#231111yescharacteristic
274C2×C47C2×#41911yescharacteristic
275C2×C47#289611yescharacteristic
276C2×C47#325212yes—
277C2×C47#325212yes—
278D47—42no—
279C22×M16—472no—
280#533—478no—
281#533—944no—
282C2×C4⋊Q8—471no—
283C2×C4⋊Q8—471no—
284C2×#125—941no—
285C2×C22⋊Q8—471no—
286C2×C22⋊Q8—471no—
287C2×C22⋊Q8—941no—
288C2×C4×Q8—471no—
289C2×C4×Q8—941no—
290C2×C4×Q8C2×D4711yescharacteristic
291C2×#117—941no—
292C22×C42—471no—
293#423—944no—
294C2×C4⋊C8—472no—
295C2×C4⋊C8—471no—
296C2×C4⋊C8C2×D4711yescharacteristic
297C2×#103—472no—
298C2×#103—472no—
299C2×#103—472no—
300C2×#103C2×D4712yes—
301#383—941no—
302#382—471no—
303#382C2×D4711yescharacteristic
304#381—941no—
305#380—472no—
306C4×C4⋊C4—472no—
307C4×C22⋊C4—941no—
308#376—941no—
309#340—472no—
310#340—472no—
311#340—472no—
312#340C2×D4712yes—
313C47#2189211yescharacteristic, complemented, Sylow
314C22×Q8—471no—
315C22×Q8—941no—
316C22×Q8C22×D4711yescharacteristic
317C23×C4—471no—
318C23×C4—471no—
319C23×C4—471no—
320C2×M16—944no—
321C2×M16—1882no—
322C2×M16—1882no—
323C22×C8—472no—
324C22×C8C22×D4712yes—
325C4⋊Q8—474no—
326C4⋊Q8—474no—
327C4⋊Q8—942no—
328C4⋊Q8—942no—
329#125—948no—
330C22⋊Q8—474no—
331C22⋊Q8—474no—
332C22⋊Q8—942no—
333C22⋊Q8—948no—
334C22⋊Q8—942no—
335C4×Q8—22no—
336C4×Q8—474no—
337C4×Q8—948no—
338C4×Q8—942no—
339C4×Q8C4×D4714yes—
340#117—948no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—471no—
343C2×C4⋊C4—472no—
344C2×C4⋊C4—941no—
345C2×C4⋊C4—941no—
346C2×C4⋊C4—941no—
347C2×C4⋊C4—941no—
348C2×C4⋊C4—941no—
349C2×C4⋊C4—941no—
350C2×C4⋊C4C22×D4712yes—
351C2×C4⋊C4C22×D4711yescharacteristic
352C2×C22⋊C4—941no—
353C2×C22⋊C4—941no—
354C2×C22⋊C4—942no—
355C2×C42—21no—
356C2×C42—471no—
357C2×C42—471no—
358C2×C42—471no—
359C2×C42—474no—
360C2×C42—474no—
361C2×C42—941no—
362C2×C42—942no—
363C2×C42C22×D4711yescharacteristic
364C4⋊C8—22no—
365C4⋊C8—944no—
366C4⋊C8—942no—
367#103—42no—
368#103—1882no—
369#103—1882no—
370#103—1882no—
371#95—21no—
372#95—21no—
373#95—942no—
374#95—942no—
375#95—941no—
376#95—941no—
377C24—471no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—472no—
384C2×Q8—944no—
385C2×Q8—941no—
386C2×Q8—941no—
387C2×Q8—941no—
388C2×Q8—941no—
389C2×Q8—941no—
390C2×Q8—1884no—
391C2×Q8C2×C4×D4712yes—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—471no—
396C22×C4—471no—
397C22×C4—471no—
398C22×C4—474no—
399C22×C4—472no—
400C22×C4—472no—
401C22×C4—474no—
402C22×C4—944no—
403C22×C4—941no—
404C22×C4—941no—
405C22×C4—941no—
406C22×C4—941no—
407C22×C4—941no—
408C22×C4—941no—
409C22×C4—942no—
410C22×C4—1882no—
411C22×C4D4×D4711yescharacteristic
412C22×C4D4×D4711yescharacteristic
413C22×C4C23×D4711yescharacteristic
414M16—1888no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—942no—
419C2×C8—942no—
420C2×C8—942no—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—472no—
426C4⋊C4—944no—
427C4⋊C4—944no—
428C4⋊C4—944no—
429C4⋊C4—944no—
430C4⋊C4—944no—
431C4⋊C4—944no—
432C4⋊C4—941no—
433C4⋊C4—942no—
434C4⋊C4—942no—
435C4⋊C4C2×C4×D4712yes—
436C22⋊C4—944no—
437C22⋊C4—944no—
438C22⋊C4—1884no—
439C42—24no—
440C42—474no—
441C42—474no—
442C42—472no—
443C42—472no—
444C42—944no—
445C42—1884no—
446C42C2×C4×D4712yes—
447C42D4×D4712yes—
448C23—472no—
449C23—942no—
450C23—942no—
451C23—941no—
452C23—941no—
453C23C2×D4×D4711yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—41no—
457Q8—41no—
458Q8—942no—
459Q8—942no—
460Q8—1888no—
461Q8—1881no—
462Q8—1881no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—472no—
474C2×C4—474no—
475C2×C4—472no—
476C2×C4—472no—
477C2×C4—948no—
478C2×C4—942no—
479C2×C4—941no—
480C2×C4—941no—
481C2×C4—942no—
482C2×C4—942no—
483C2×C4—942no—
484C2×C4—942no—
485C2×C4—942no—
486C2×C4—942no—
487C2×C4—1884no—
488C2×C4—1882no—
489C2×C4—1882no—
490C2×C4—1882no—
491C2×C4—1882no—
492C2×C4—1881no—
493C2×C4—1881no—
494C2×C4C22⋊C4×D4712yes—
495C2×C4C22⋊C4×D4712yes—
496C2×C4C22×C4×D4711yescharacteristic
497C2×C4C2×D4×D4711yescharacteristic
498C2×C4#1309406911yescharacteristic
499C2×C4#1309407011yescharacteristic
500C2×C4#1309407111yescharacteristic
501C2×C4#1309407211yescharacteristic
502C8—42no—
503C8—1882no—
504C22—942no—
505C22—942no—
506C22—942no—
507C22—1882no—
508C22—1882no—
509C22C2×C22⋊C4×D4711yescharacteristic
510C22#1568970211yescharacteristic
511C22#1568970311yescharacteristic
512C22#137×D4712yes—
513C22#1568992012yes—
514C4—22no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—82no—
519C4—474no—
520C4—942no—
521C4—942no—
522C4—3762no—
523C4—1882no—
524C4—1882no—
525C4#1568960311yescharacteristic
526C4#1568960411yescharacteristic
527C4#1568968911yescharacteristic
528C4#1568969611yescharacteristic
529C2—1882no—
530C2#2024597711yescharacteristic
531C2#419×D4711yescharacteristic
532C2#2025190311yescharacteristic
533C2#2025475812yes—
534C2#2025476012yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1805C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1805D47
hypocenterC47#21892
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #20244334C2semidirect
5. #20244309C2semidirect
8. #20243404C2semidirect
9. #20243403C2semidirect
10. #20243400C2semidirect
11. C47×#1805C2semidirect
14. #15689677C4semidirect
272. #1805D47semidirect
313. C47#21892semidirect

Maximal subgroups

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

subgroupindex
C2×#156896772
C2×#156891452
#202457812
#202443342
#202443092
#202438202
#202438182
#202434042
#202434032
#202434002
C47×#18052
#2189247