Finite Group Databasebeta

← #26232486

Subgroups — #26232486

Order 12032.

1474 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689777C211yesmaximal, characteristic
2C2×D4×C47⋊C8C211yesmaximal, characteristic, complemented
3#20244348C211yesmaximal, characteristic, complemented
4#20244340C211yesmaximal, characteristic, complemented
5#20243814C212yesmaximal, complemented
6#20243705C212yesmaximal, complemented
7#20243472C212yesmaximal, complemented
8#20243385C211yesmaximal, characteristic, complemented
9C4×#13093944C211yesmaximal, characteristic, complemented
10C2×#15688885C212yesmaximal, complemented
11D4×C4⋊C4×C47C211yesmaximal, characteristic, complemented
12C22×#10625352—21no—
13C23×C47⋊C8C2212yes—
14#15689777—24no—
15#15689777C418yescomplemented
16C2×#13094024C2211yescharacteristic
17C2×#13094008—22no—
18C2×#13094004—21no—
19C2×#13093998C2212yes—
20C2×C4×D188C2211yescharacteristic
21C22×C4⋊Dic47C2212yes—
22D4×C47⋊C8—24no—
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×#13093944C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093944C2212yes—
27C2×C22⋊(C47⋊C8)C2212yes—
28C2×C4×C47⋊C8C2211yescharacteristic
29#15688885C2211yescharacteristic
30#15688885C2211yescharacteristic
31#15688885C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35#15688885C2212yes—
36C2×C4×D4×C47—21no—
37C2×C4×D4×C47C2211yescharacteristic
38C22×C4⋊C4×C47C2212yes—
39C47×#389—21no—
40C47×#385—22no—
41C47×#384C2211yescharacteristic
42C47×#380C2212yes—
43C4×C4⋊C4×C47C2211yescharacteristic
44C2×#10625352—28no—
45C2×#10625352—44no—
46C2×#10625352—44no—
47C22×Dic94—21no—
48C22×D188C2311yescharacteristic
49C22×C22⋊D47—22no—
50C23×Dic47—22no—
51C22×C4×D47—21no—
52C22×C47⋊C8—24no—
53C22×C47⋊C8—24no—
54C22×C47⋊C8—42no—
55C22×C47⋊C8C2311yescharacteristic
56C22×C47⋊C8C2312yes—
57C22×C47⋊C8C2311yescharacteristic
58#13094024—22no—
59#13094024C2×C414yescomplemented
60#13094008—216no—
61#13094004—28no—
62#13093998—24no—
63#13093998C2×C418yescomplemented
64C4×D188—22no—
65C4×D188C2×C414yescomplemented
66C2×C4⋊Dic47—24no—
67C2×C4⋊Dic47—22no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47C2×C418yescomplemented
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2312yes—
72C2×(C2×D47)⋊C4—21no—
73C2×(C2×D47)⋊C4—22no—
74C4⋊(C47⋊C8)—22no—
75#13093944—22no—
76#13093944—22no—
77#13093944—42no—
78C22⋊(C47⋊C8)—24no—
79C4×C47⋊C8—22no—
80C22×D4×C47D411yescharacteristic
81C23×C4×C47—22no—
82C23×C4×C47C2312yes—
83C4×D4×C47—44no—
84C4×D4×C47C2×C414yes—
85C4×D4×C47D414yes—
86C2×C4⋊C4×C47—24no—
87C2×C4⋊C4×C47—22no—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—21no—
90C2×C4⋊C4×C47—42no—
91C2×C4⋊C4×C47D411yescharacteristic
92C2×C4⋊C4×C47C2312yes—
93C2×C4⋊C4×C47C2311yescharacteristic
94C2×C4⋊C4×C47C2311yescharacteristic
95C2×C22⋊C4×C47—22no—
96C2×C22⋊C4×C47D412yes—
97C2×C42×C47—21no—
98C2×C42×C47C2311yescharacteristic
99#95×C47—21no—
100#95×C47—22no—
101C24×C47C2×D412yes—
102C23×D47—21no—
103C2×D4×C47C22⋊C414yes—
104C2×D4×C47D814yes—
105C2×D4×C47SD1614yes—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—44no—
114C22×C4×C47—44no—
115C22×C4×C47—82no—
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—44no—
124C2×C22⋊D47—44no—
125C22×Dic47—28no—
126C22×Dic47—21no—
127C22×Dic47—22no—
128C22×Dic47—44no—
129C22×Dic47—44no—
130#10625352—432no—
131C2×D188—21no—
132C2×D188—21no—
133C2×D188—21no—
134C2×D188—21no—
135C2×D188—21no—
136C2×D188C22×C412yes—
137C2×C4×D47—24no—
138C2×C4×D47—42no—
139C2×C4×D47—42no—
140C2×Dic94—24no—
141C2×Dic94—42no—
142C2×Dic94—42no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—22no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—44no—
149C4⋊C4×C47—42no—
150C4⋊C4×C47—81no—
151C4⋊C4×C47C22⋊C414yes—
152C4⋊C4×C47#4112yescomplemented
153C22⋊C4×C47—44no—
154C22⋊C4×C47C22⋊C418yes—
155C42×C47—42no—
156C42×C47C22×C412yes—
157C42×C47C2×D412yes—
158(C2×D47)⋊C4—24no—
159(C2×D47)⋊C4—28no—
160C4⋊Dic47—28no—
161C4⋊Dic47—24no—
162C4⋊Dic47—21no—
163C4⋊Dic47—22no—
164C4⋊Dic47C22×C412yes—
165C4⋊Dic47C22×C414yes—
166C2×C47⋊C8—21no—
167C2×C47⋊C8—21no—
168C2×C47⋊C8—21no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—21no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8—48no—
175C2×C47⋊C8#4112yescomplemented
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47C2×C22⋊C414yes—
181C23×C47C2×D814yes—
182C23×C47C2×SD1614yes—
183C23×C47C22×D411yescharacteristic
184C22×D47—22no—
185C22×D47—41no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47—41no—
189C22×D47—41no—
190C22×D47—41no—
191D4×C47#102116yescomplemented
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—21no—
195C2×C4×C47—21no—
196C2×C4×C47—24no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—22no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—42no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—41no—
209C2×C4×C47—88no—
210C2×C4×C47C2×C22⋊C412yes—
211C2×C4×C47C2×C22⋊C414yes—
212C2×C4×C47C2×C22⋊C412yes—
213C2×C4×C47C2×D812yes—
214C2×C4×C47C2×SD1612yes—
215C2×C4×C47C23×C411yescharacteristic
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C47H(5,𝔽2)11yescharacteristic
220C2×C4×C472−1+411yescharacteristic
221C22⋊D47—416no—
222C2×Dic47—24no—
223C2×Dic47—22no—
224C2×Dic47—42no—
225C2×Dic47—41no—
226C2×Dic47—42no—
227C2×Dic47—41no—
228C2×Dic47—416no—
229D188—22no—
230D188—22no—
231D188—41no—
232D188—41no—
233C4×D47—48no—
234Dic94—48no—
235C47⋊C8—22no—
236C47⋊C8—42no—
237C47⋊C8—41no—
238#22485—471nomaximal, Sylow
239C22×C47—24no—
240C22×C47—24no—
241C22×C47—24no—
242C22×C47—24no—
243C22×C47—24no—
244C22×C47—24no—
245C22×C47C2×#10218yes—
246C22×C47C22×C22⋊C411yescharacteristic
247C22×C47#54611yescharacteristic
248C22×C47#54711yescharacteristic
249C22×C47C22×D811yescharacteristic
250C22×C47C22×SD1611yescharacteristic
251C22×C47#57711yescharacteristic
252C22×C47#57811yescharacteristic
253C2×D47—42no—
254C2×D47—42no—
255C2×D47—42no—
256C2×D47—81no—
257C2×D47—81no—
258C2×D47—81no—
259C2×D47—81no—
260C4×C47—24no—
261C4×C47—22no—
262C4×C47—42no—
263C4×C47—81no—
264C4×C47—82no—
265C4×C47C2×#10214yes—
266C4×C47C4×#4111yescharacteristic
267C4×C47#52011yescharacteristic
268C4×C47#54611yescharacteristic
269C4×C47#54711yescharacteristic
270Dic47—44no—
271Dic47—42no—
272C2×#547—471no—
273C2×C8×D4—471no—
274D4×C4⋊C4D4711yescharacteristic, complemented
275#1990—471no—
276#1988—471no—
277#1893—472no—
278#1874—472no—
279#1823—472no—
280#1802—471no—
281C4×#102—471no—
282C2×#340—472no—
283C2×C47—28no—
284C2×C47#233911yescharacteristic
285C2×C47C22×#10211yescharacteristic
286C2×C47#289811yescharacteristic
287C2×C47#329411yescharacteristic
288C2×C47#329811yescharacteristic
289C2×C47#333411yescharacteristic
290C2×C47#333511yescharacteristic
291D47—82no—
292C22×#41—941no—
293C23×C8—472no—
294#547—478no—
295#547—944no—
296C2×C4⋊Q8—471no—
297C2×#123—942no—
298C2×C22⋊Q8—941no—
299C2×#121—472no—
300C2×C4×D4—21no—
301C2×C4×D4—471no—
302C2×C4×D4C2×D4711yescharacteristic
303C22×C4⋊C4—472no—
304C22×C4⋊C4C2×D4712yes—
305C8×D4—944no—
306C2×C4⋊C8—471no—
307C2×#102—471no—
308C2×#102—471no—
309C2×#102—472no—
310C2×C22⋊C8—472no—
311C2×C4×C8—471no—
312#389—21no—
313#385—22no—
314#384C2×D4711yescharacteristic
315#380C2×D4712yes—
316C4×C4⋊C4C2×D4711yescharacteristic
317#340—472no—
318#340—471no—
319#340—472no—
320#340—472no—
321#340—472no—
322#340—472no—
323#340—471no—
324C47#2248511yescharacteristic, complemented, Sylow
325C2×#41—948no—
326C2×#41—1884no—
327C2×#41—1884no—
328C22×Q8—941no—
329C22×D4—471no—
330C22×D4—942no—
331C22×D4C22⋊D4711yescharacteristic
332C23×C4—22no—
333C23×C4—942no—
334C23×C4—941no—
335C23×C4C22×D4712yes—
336C22×C8—471no—
337C22×C8—472no—
338C22×C8—471no—
339C22×C8—944no—
340C22×C8—944no—
341C22×C8—1882no—
342C4⋊Q8—474no—
343C4⋊Q8—942no—
344#123—9416no—
345C22⋊Q8—948no—
346#121—478no—
347#121—944no—
348C4×D4—44no—
349C4×D4—474no—
350C4×D4—942no—
351C4×D4C4×D4714yes—
352C4×D4D18814yes—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—478no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—472no—
361C2×C4⋊C4—944no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4C22⋊D4711yescharacteristic
365C2×C4⋊C4C22×D4712yes—
366C2×C4⋊C4C22×D4711yescharacteristic
367C2×C4⋊C4C22×D4711yescharacteristic
368C2×C22⋊C4—22no—
369C2×C22⋊C4—941no—
370C2×C22⋊C4—942no—
371C2×C22⋊C4C22⋊D4712yes—
372C2×C42—21no—
373C2×C42C22×D4711yescharacteristic
374C4⋊C8—942no—
375#102—942no—
376#102—942no—
377#102—1882no—
378C22⋊C8—944no—
379C4×C8—942no—
380#95—21no—
381#95—22no—
382C24—941no—
383C24C2×C22⋊D4712yes—
384#41—18832no—
385C2×Q8—944no—
386C2×Q8—1882no—
387C2×Q8—1882no—
388C2×D4—472no—
389C2×D4—948no—
390C2×D4—941no—
391C2×D4—941no—
392C2×D4—941no—
393C2×D4—941no—
394C2×D4—941no—
395C2×D4—1884no—
396C2×D4—1884no—
397C2×D4(C2×D47)⋊C414yes—
398C2×D4#1062533414yes—
399C2×D4#1062533614yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—82no—
410C22×C4—948no—
411C22×C4—941no—
412C22×C4—942no—
413C22×C4—944no—
414C22×C4—1884no—
415C22×C4—1884no—
416C22×C4—1882no—
417C22×C4—1882no—
418C22×C4C2×C4×D4714yes—
419C22×C4C2×D18814yes—
420C22×C4C2×C22⋊D4711yescharacteristic
421C22×C4C2×C22⋊D4712yes—
422C22×C4C2×C22⋊D4711yescharacteristic
423C22×C4C23×D4711yescharacteristic
424C2×C8—472no—
425C2×C8—941no—
426C2×C8—941no—
427C2×C8—941no—
428C2×C8—942no—
429C2×C8—942no—
430C2×C8—941no—
431C2×C8—942no—
432C2×C8—941no—
433C2×C8—1888no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—81no—
442C4⋊C4—472no—
443C4⋊C4—474no—
444C4⋊C4—948no—
445C4⋊C4—944no—
446C4⋊C4—941no—
447C4⋊C4—942no—
448C4⋊C4(C2×D47)⋊C414yes—
449C4⋊C4#1062535212yescomplemented
450C22⋊C4—44no—
451C22⋊C4—944no—
452C22⋊C4—948no—
453C22⋊C4(C2×D47)⋊C418yes—
454C42—42no—
455C42C2×C4×D4712yes—
456C42C2×D18812yes—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—942no—
462C23—1881no—
463C23—1881no—
464C23—1881no—
465C23—1881no—
466C23—1881no—
467C23—1881no—
468C23C2×(C2×D47)⋊C414yes—
469C23C2×#1062533414yes—
470C23C2×#1062533614yes—
471C23C22×C22⋊D4711yescharacteristic
472Q8—1888no—
473D4—942no—
474D4—942no—
475D4—18816no—
476D4—1881no—
477D4—1881no—
478D4#13093944116yescomplemented
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—88no—
497C2×C4—942no—
498C2×C4—944no—
499C2×C4—18816no—
500C2×C4—1881no—
501C2×C4—1882no—
502C2×C4—1881no—
503C2×C4—1882no—
504C2×C4—1888no—
505C2×C4C2×(C2×D47)⋊C412yes—
506C2×C4C2×(C2×D47)⋊C414yes—
507C2×C4C2×(C2×D47)⋊C412yes—
508C2×C4C2×#1062533412yes—
509C2×C4C2×#1062533612yes—
510C2×C4C22×C4×D4711yescharacteristic
511C2×C4C22×D18811yescharacteristic
512C2×C4#41×D4711yescharacteristic
513C2×C4C2×#1062535211yescharacteristic
514C2×C4#1309406911yescharacteristic
515C2×C4#1309407111yescharacteristic
516C8—942no—
517C8—1882no—
518C8—1881no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—3761no—
526C22—3761no—
527C22—3761no—
528C22—3761no—
529C22—1882no—
530C22—1882no—
531C22—1882no—
532C22C2×#1309394418yes—
533C22C22×(C2×D47)⋊C411yescharacteristic
534C22#1568976711yescharacteristic
535C22#1568977211yescharacteristic
536C22C22×#1062533411yescharacteristic
537C22C22×#1062533611yescharacteristic
538C22#1568993411yescharacteristic
539C22#1568993811yescharacteristic
540C4—22no—
541C4—24no—
542C4—42no—
543C4—82no—
544C4—81no—
545C4—1882no—
546C4—1884no—
547C4C2×#1309394414yes—
548C4C4×#1062535211yescharacteristic
549C4#1568960211yescharacteristic
550C4#1568976811yescharacteristic
551C4#1568977711yescharacteristic
552C2—28no—
553C2—3762no—
554C2#2024621311yescharacteristic
555C2C22×#1309394411yescharacteristic
556C2#2025191811yescharacteristic
557C2#2025520111yescharacteristic
558C2#2025525311yescharacteristic
559C2#2025573111yescharacteristic
560C2#2025573911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingD4×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689777C2
cosocleC22×C4×C47C24
hypercenterD4×C4⋊C4D47
hypocenterC47#22485
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×D4×C47⋊C8C2semidirect
3. #20244348C2semidirect
4. #20244340C2semidirect
5. #20243814C2semidirect
6. #20243705C2semidirect
7. #20243472C2semidirect
8. #20243385C2semidirect
9. C4×#13093944C2semidirect
10. C2×#15688885C2semidirect
11. D4×C4⋊C4×C47C2semidirect
15. #15689777C4semidirect
59. #13094024C2×C4semidirect
63. #13093998C2×C4semidirect
65. C4×D188C2×C4semidirect
69. C2×C4⋊Dic47C2×C4semidirect
152. C4⋊C4×C47#41semidirect
175. C2×C47⋊C8#41semidirect
191. D4×C47#102semidirect
274. D4×C4⋊C4D47semidirect
324. C47#22485semidirect
449. C4⋊C4#10625352semidirect
478. D4#13093944semidirect

Maximal subgroups

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

subgroupindex
C2×#156897772
C2×D4×C47⋊C82
#202443482
#202443402
#202438142
#202437052
#202434722
#202433852
C4×#130939442
C2×#156888852
D4×C4⋊C4×C472
#2248547