Finite Group Databasebeta

← #26231748

Subgroups — #26231748

Order 12032.

1542 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689794C211yesmaximal, characteristic
2C2×#15689120C212yesmaximal, complemented
3#20246337C211yesmaximal, characteristic, complemented
4#20244314C211yesmaximal, characteristic, complemented
5#20244194C211yesmaximal, characteristic, complemented
6#20243816C212yesmaximal, complemented
7#20243708C212yesmaximal, complemented
8#20243394C211yesmaximal, characteristic, complemented
9#20243329C212yesmaximal, complemented
10C4×#13093949C211yesmaximal, characteristic, complemented
11C2×C8×D4×C47C211yesmaximal, characteristic, complemented
12C22×#10625350C2212yes—
13#15689794—24no—
14#15689794C418yescomplemented
15C2×#13094021C2211yescharacteristic
16C2×#13094020—21no—
17C2×#13093998C2212yes—
18C2×#13093997—22no—
19C2×C4×Dic94C2211yescharacteristic
20C2×#13093986—21no—
21C2×#13093983C2212yes—
22#15689120—28no—
23C2×#13093949C2211yescharacteristic
24C2×#13093949C2212yes—
25C2×#13093949C2211yescharacteristic
26C2×#13093946C2212yes—
27C2×#13093946C2212yes—
28C2×#13093946C2212yes—
29C2×#13093946C2212yes—
30#15689026—21no—
31#15689016—22no—
32#15688993C2211yescharacteristic
33#15688983C2212yes—
34C4⋊C4×Dic47—21no—
35C4×C4⋊Dic47C2211yescharacteristic
36#15688886C2211yescharacteristic
37#15688886C2211yescharacteristic
38#15688886C2212yes—
39C23×C8×C47C2212yes—
40C2×C4×D4×C47C2211yescharacteristic
41C8×D4×C47—24no—
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×C22⋊C8×C47C2212yes—
44C2×C4×C8×C47C2211yescharacteristic
45C2×#10625350—24no—
46C2×#10625350—24no—
47C2×#10625350—24no—
48C2×#10625350—42no—
49C2×#10625350—42no—
50C2×#10625350C2×C418yes—
51C22×Dic94C2311yescharacteristic
52C22×D188—21no—
53C22×D188—21no—
54C22×D188C2312yes—
55C22×C22⋊D47—22no—
56C22×C22⋊D47—22no—
57C22×C4×D47—22no—
58#13094021—22no—
59#13094021C2×C414yes—
60#13094020—28no—
61#13093998—24no—
62#13093998C2×C418yes—
63#13093997—216no—
64C4×Dic94—22no—
65C4×Dic94C2×C414yes—
66#13093986—28no—
67#13093983—24no—
68#13093983—42no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2312yes—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—22no—
77C2×C22⋊Dic47—22no—
78C2×(C2×D47)⋊C4—22no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—22no—
82#13093949—24no—
83#13093949—22no—
84#13093949D414yescomplemented
85#13093946—24no—
86#13093946—24no—
87#13093946—42no—
88#13093946—42no—
89C47⋊#95—21no—
90C47⋊#95—22no—
91C22×D4×C47D411yescharacteristic
92C23×C4×C47C2312yes—
93C22×C8×C47—24no—
94C22×C8×C47—24no—
95C22×C8×C47—42no—
96C22×C8×C47C2312yes—
97C22×C8×C47C2311yescharacteristic
98C22×C8×C47C2311yescharacteristic
99C4×D4×C47C2×C414yes—
100C4×D4×C47D414yes—
101C2×C4⋊C4×C47D411yescharacteristic
102C2×C22⋊C4×C47D412yes—
103C2×C42×C47C2311yescharacteristic
104C4⋊C8×C47—22no—
105C22⋊C8×C47—24no—
106C4×C8×C47—22no—
107C24×C47C2×D412yes—
108C23×D47—22no—
109C2×D4×C47—22no—
110C2×D4×C47—22no—
111C2×D4×C47C22⋊C414yes—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47C22×C414yes—
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2×D414yes—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2411yescharacteristic
122C2×C22⋊D47—28no—
123C2×C22⋊D47—28no—
124C2×C22⋊D47—44no—
125C2×C22⋊D47—44no—
126C2×C22⋊D47—82no—
127C2×C22⋊D47—82no—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—21no—
131C22×Dic47—21no—
132C22×Dic47—22no—
133#10625350—216no—
134#10625350—48no—
135#10625350—48no—
136#10625350—48no—
137C2×D188—22no—
138C2×D188—22no—
139C2×D188—22no—
140C2×D188—22no—
141C2×D188—24no—
142C2×D188—22no—
143C2×D188—24no—
144C2×D188—42no—
145C2×D188—42no—
146C2×D188—81no—
147C2×D188—81no—
148C2×D188C22×C414yes—
149C2×C4×D47—28no—
150C2×C4×D47—44no—
151C2×C4×D47—44no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94C22×C412yes—
156C2×Dic94C2×D412yes—
157C2×Dic94#4112yes—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—22no—
165C2×C8×C47—21no—
166C2×C8×C47—48no—
167C2×C8×C47C2×D412yes—
168C4⋊C4×C47C22⋊C414yes—
169C22⋊C4×C47C22⋊C418yes—
170C42×C47C22×C412yes—
171C42×C47C2×D412yes—
172C22⋊Dic47—44no—
173C22⋊Dic47—82no—
174(C2×D47)⋊C4—28no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—22no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—22no—
181C4⋊Dic47—42no—
182C4⋊Dic47—81no—
183C4⋊Dic47C22×C412yes—
184C4⋊Dic47C2×D412yes—
185Dic47⋊C4—24no—
186C4×Dic47—24no—
187C4×Dic47—44no—
188C4×Dic47—42no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47C2×C22⋊C414yes—
196C23×C47C22×D411yescharacteristic
197C22×D47—24no—
198C22×D47—42no—
199C22×D47—42no—
200C22×D47—42no—
201C22×D47—42no—
202C22×D47—42no—
203C22×D47—42no—
204D4×C47—28no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—24no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—24no—
213C2×C4×C47C2×C22⋊C412yes—
214C2×C4×C47C2×C22⋊C414yes—
215C2×C4×C47C2×C22⋊C412yes—
216C2×C4×C47#13512yes—
217C2×C4×C47#13512yes—
218C2×C4×C47C23×C411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47H(5,𝔽2)11yescharacteristic
223C2×C4×C47H(5,𝔽2)11yescharacteristic
224C22⋊D47—416no—
225C22⋊D47—88no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—42no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—41no—
241D188—24no—
242D188—24no—
243D188—42no—
244D188—42no—
245D188—48no—
246D188—84no—
247C4×D47—416no—
248Dic94—22no—
249Dic94—22no—
250Dic94—22no—
251Dic94C4×D414yes—
252C8×C47—22no—
253C8×C47—42no—
254C8×C47—41no—
255#22418—471nomaximal, Sylow
256C22×C47—24no—
257C22×C47—24no—
258C22×C47—24no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47C22×C22⋊C411yescharacteristic
264C22×C47D4211yescharacteristic
265C22×C47#54611yescharacteristic
266C22×C47C2×#13511yescharacteristic
267C22×C47C2×#13511yescharacteristic
268C22×C47#57611yescharacteristic
269C22×C47#57711yescharacteristic
270C2×D47—44no—
271C2×D47—44no—
272C2×D47—44no—
273C2×D47—82no—
274C2×D47—82no—
275C2×D47—82no—
276C2×D47—82no—
277C4×C47—24no—
278C4×C47—22no—
279C4×C47#41614yescomplemented
280C4×C47C2×C4×D411yescharacteristic
281C4×C47#51811yescharacteristic
282C4×C47D4211yescharacteristic
283C4×C47#54611yescharacteristic
284Dic47—24no—
285Dic47—42no—
286Dic47—42no—
287Dic47—82no—
288Dic47—81no—
289C2×#550—471no—
290C2×C8×D4D4711yescharacteristic, complemented
291C2×#416—472no—
292#2354—471no—
293#1982—471no—
294#1957—471no—
295#1893—472no—
296#1875—472no—
297#1803—471no—
298#1788—472no—
299C4×#103—471no—
300C2×C47—44no—
301C2×C47D4×C22⋊C411yescharacteristic
302C2×C47C2×#41611yescharacteristic
303C2×C47#289711yescharacteristic
304C2×C47#328411yescharacteristic
305C2×C47#328711yescharacteristic
306C2×C47#330411yescharacteristic
307C2×C47#330511yescharacteristic
308D47—84no—
309C22×#41—472no—
310C23×C8C2×D4712yes—
311#550—478no—
312#550—944no—
313C2×#127—471no—
314C2×#127—941no—
315C2×#121—472no—
316C2×#121—942no—
317C2×C4×Q8—471no—
318C2×C4×D4—941no—
319C2×C4×D4C2×D4711yescharacteristic
320C2×#117—472no—
321C8×D4—24no—
322C2×C4⋊C8C2×D4711yescharacteristic
323#416—948no—
324C2×#103—471no—
325C2×#103—472no—
326C2×#103—471no—
327C2×#102—472no—
328C2×#102—472no—
329C2×#102—472no—
330C2×#102—472no—
331C2×C22⋊C8C2×D4712yes—
332C2×C4×C8C2×D4711yescharacteristic
333#390—941no—
334#388—942no—
335#384—471no—
336#380—472no—
337C4×C4⋊C4—471no—
338C4×C4⋊C4—941no—
339#340—472no—
340#340—471no—
341#340—471no—
342C47#2241811yescharacteristic, complemented, Sylow
343C2×#41—478no—
344C2×#41—944no—
345C2×#41—944no—
346C2×#41—944no—
347C2×#41—1882no—
348C2×#41—1882no—
349C22×Q8—471no—
350C22×D4—472no—
351C22×D4—942no—
352C22×D4—942no—
353C22×D4—941no—
354C22×D4—941no—
355C22×D4D18811yescharacteristic
356C23×C4—942no—
357C23×C4C22×D4712yes—
358C22×C8—24no—
359C22×C8—24no—
360C22×C8—42no—
361C22×C8C22×D4712yes—
362C22×C8C22×D4711yescharacteristic
363C22×C8C22×D4711yescharacteristic
364#127—474no—
365#127—948no—
366#127—942no—
367#121—478no—
368#121—944no—
369#121—9416no—
370C4×Q8—474no—
371C4×Q8—942no—
372C4×D4—948no—
373C4×D4C4×D4714yes—
374C4×D4C22⋊D4714yes—
375#117—944no—
376#117—1882no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—472no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4D18811yescharacteristic
385C2×C22⋊C4—942no—
386C2×C22⋊C4—942no—
387C2×C22⋊C4—942no—
388C2×C22⋊C4D18812yes—
389C2×C42—941no—
390C2×C42—942no—
391C2×C42—941no—
392C2×C42C22×D4711yescharacteristic
393C4⋊C8—22no—
394#103—474no—
395#103—942no—
396#103—944no—
397#102—944no—
398#102—944no—
399#102—1882no—
400#102—1882no—
401C22⋊C8—24no—
402C4×C8—22no—
403#95—941no—
404#95—942no—
405C24—942no—
406C24C2×D18812yes—
407#41—9416no—
408#41—1888no—
409#41—1888no—
410#41—1888no—
411C2×Q8—472no—
412C2×Q8—472no—
413C2×Q8—472no—
414C2×Q8—941no—
415C2×Q8—941no—
416C2×Q8—941no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—474no—
420C2×D4—948no—
421C2×D4—948no—
422C2×D4—944no—
423C2×D4—944no—
424C2×D4—942no—
425C2×D4—942no—
426C2×D4—942no—
427C2×D4—942no—
428C2×D4—942no—
429C2×D4—3761no—
430C2×D4—3761no—
431C2×D4—3762no—
432C2×D4—3762no—
433C2×D4—1884no—
434C2×D4—1884no—
435C2×D4—1882no—
436C2×D4—1882no—
437C2×D4(C2×D47)⋊C414yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—948no—
446C22×C4—941no—
447C22×C4—942no—
448C22×C4—1884no—
449C22×C4—1884no—
450C22×C4C2×C4×D4714yes—
451C22×C4C2×D18812yes—
452C22×C4C2×D18811yescharacteristic
453C22×C4C2×D18811yescharacteristic
454C22×C4C2×C22⋊D4714yes—
455C22×C4C23×D4711yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8—21no—
460C2×C8—22no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—21no—
464C2×C8—48no—
465C2×C8D4×D4712yes—
466C4⋊C4—472no—
467C4⋊C4—472no—
468C4⋊C4—944no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—942no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—942no—
475C4⋊C4—3761no—
476C4⋊C4—1882no—
477C4⋊C4(C2×D47)⋊C414yes—
478C22⋊C4—948no—
479C22⋊C4—3762no—
480C22⋊C4—1884no—
481C22⋊C4(C2×D47)⋊C418yes—
482C42—944no—
483C42—1884no—
484C42—1882no—
485C42C2×C4×D4712yes—
486C42C2×C22⋊D4712yes—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—944no—
494C23—1882no—
495C23—1882no—
496C23—1882no—
497C23—1882no—
498C23—1882no—
499C23—1882no—
500C23C2×(C2×D47)⋊C414yes—
501C23C22×D18811yescharacteristic
502Q8—474no—
503Q8—942no—
504Q8—942no—
505Q8—942no—
506D4—28no—
507D4—944no—
508D4—944no—
509D4—3764no—
510D4—3768no—
511D4—18816no—
512D4—1882no—
513D4—1882no—
514D4—1888no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—18816no—
529C2×C4—1882no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1882no—
533C2×C4—1881no—
534C2×C4—1882no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4C2×(C2×D47)⋊C412yes—
540C2×C4C2×(C2×D47)⋊C414yes—
541C2×C4C2×(C2×D47)⋊C412yes—
542C2×C4#1309404612yes—
543C2×C4#1309404612yes—
544C2×C4C22×C4×D4711yescharacteristic
545C2×C4C2×D4×D4711yescharacteristic
546C2×C4C22×C22⋊D4711yescharacteristic
547C2×C4#41×D4711yescharacteristic
548C2×C4#1309406911yescharacteristic
549C2×C4#1309407011yescharacteristic
550C8—22no—
551C8—42no—
552C8—41no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—3762no—
561C22—3762no—
562C22—3762no—
563C22—3762no—
564C22—1884no—
565C22—1884no—
566C22—1884no—
567C22C22×(C2×D47)⋊C411yescharacteristic
568C22D4×D18811yescharacteristic
569C22#1568976811yescharacteristic
570C22C2×#1309404611yescharacteristic
571C22C2×#1309404611yescharacteristic
572C22#1568992811yescharacteristic
573C22#1568993511yescharacteristic
574C4—22no—
575C4—24no—
576C4—944no—
577C4—3761no—
578C4—3762no—
579C4—1882no—
580C4—1882no—
581C4#1568912014yescomplemented
582C4C4×D4×D4711yescharacteristic
583C4#1568959511yescharacteristic
584C4D4×C22⋊D4711yescharacteristic
585C4#1568976111yescharacteristic
586C2—44no—
587C2—3764no—
588C2D4×(C2×D47)⋊C411yescharacteristic
589C2C2×#1568912011yescharacteristic
590C2#2025191511yescharacteristic
591C2#2025512011yescharacteristic
592C2#2025514811yescharacteristic
593C2#2025532811yescharacteristic
594C2#2025533911yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C8×D4×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689794C2
cosocleC22×C4×C47C24
hypercenterC2×C8×D4D47
hypocenterC47#22418
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
2. C2×#15689120C2semidirect
3. #20246337C2semidirect
4. #20244314C2semidirect
5. #20244194C2semidirect
6. #20243816C2semidirect
7. #20243708C2semidirect
8. #20243394C2semidirect
9. #20243329C2semidirect
10. C4×#13093949C2semidirect
11. C2×C8×D4×C47C2semidirect
14. #15689794C4semidirect
84. #13093949D4semidirect
279. C4×C47#416semidirect
290. C2×C8×D4D47semidirect
342. C47#22418semidirect
581. C4#15689120semidirect

Maximal subgroups

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

subgroupindex
C2×#156897942
C2×#156891202
#202463372
#202443142
#202441942
#202438162
#202437082
#202433942
#202433292
C4×#130939492
C2×C8×D4×C472
#2241847