Finite Group Databasebeta

← #26272419

Subgroups — #26272419

Order 12032.

1076 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689833C211yesmaximal, characteristic
2C2×#15689529C211yesmaximal, characteristic, complemented
3C2×#15689410C211yesmaximal, characteristic
4#20244726C212yesmaximal, complemented
5#20244721C212yesmaximal, complemented
6#20244345C211yesmaximal, characteristic, complemented
7#20244329C211yesmaximal, characteristic, complemented
8#20243648C212yesmaximal
9#20243634C212yesmaximal
10#20243216C211yesmaximal, characteristic
11C47×#2546C211yesmaximal, characteristic, complemented
12#15689833—24no—
13#15689833—24no—
14C2×#13094024C2211yescharacteristic
15C2×#13094016C2211yescharacteristic
16C2×#13094016C2211yescharacteristic
17C2×#13094002—21no—
18C2×#13094002—21no—
19C2×#13094000—21no—
20C2×C4×D188C2211yescharacteristic
21C2×C4×D188C2211yescharacteristic
22C2×C4×D188C2211yescharacteristic
23C2×C4⋊C4×D47—21no—
24#15689529—24no—
25#15689410—24no—
26C2×#13093957C2212yes—
27C2×#13093955C2212yes—
28C2×#13093955C2212yes—
29C2×C4⋊(C47⋊C8)C2211yescharacteristic
30C2×#13093944C2212yes—
31C2×#13093944C2212yes—
32C2×#13093935C2211yescharacteristic
33#15688885C2212yes—
34#15688885C2212yes—
35#15688885C2212yes—
36#15688878C2212yes—
37#15688878C2212yes—
38C2×C4⋊Q8×C47—21no—
39C2×C4⋊Q8×C47C2211yescharacteristic
40C2×C4×Q8×C47C2211yescharacteristic
41C47×#398—21no—
42C47×#398—21no—
43C47×#391—21no—
44C47×#389C2212yes—
45C47×#389C2212yes—
46C4×C4⋊C4×C47C2211yescharacteristic
47C22×Dic94—21no—
48C22×D188C2311yescharacteristic
49C22×C4×D47—21no—
50C22×C4×D47—21no—
51C22×C4×D47—21no—
52C22×C47⋊C8C2312yes—
53C22×C47⋊C8C2312yes—
54#13094024—22no—
55#13094024—22no—
56#13094016—22no—
57#13094016—22no—
58#13094016—22no—
59#13094016D414yes—
60#13094002—28no—
61#13094002—28no—
62#13094000—28no—
63C4×D188—22no—
64C4×D188—22no—
65C4×D188—22no—
66C4×D188—22no—
67C4×D188—22no—
68C4×D188D414yes—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C4⋊C4×D47—28no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81#13093957—24no—
82#13093955—24no—
83#13093955—24no—
84C4⋊(C47⋊C8)—22no—
85#13093944—24no—
86#13093944Q818yescomplemented
87#13093935—22no—
88C22×Q8×C47—21no—
89C22×Q8×C47D411yescharacteristic
90C4⋊Q8×C47—22no—
91C4⋊Q8×C47—22no—
92C4⋊Q8×C47—44no—
93C4×Q8×C47—22no—
94C4×Q8×C47—22no—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47D411yescharacteristic
101C2×C4⋊C4×C47C2312yes—
102C2×C4⋊C4×C47C2312yes—
103C2×C42×C47—21no—
104C2×C42×C47C2311yescharacteristic
105C2×C42×C47C2311yescharacteristic
106C2×C42×C47C2311yescharacteristic
107#95×C47—21no—
108#95×C47—22no—
109#95×C47—22no—
110C23×D47—21no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47—41no—
114C2×Q8×C47—42no—
115C2×Q8×C47—42no—
116C2×Q8×C47—42no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2411yescharacteristic
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47—21no—
129C2×D188—21no—
130C2×D188—22no—
131C2×D188C2×D412yes—
132C2×D188C2×Q814yes—
133C2×C4×D47—24no—
134C2×C4×D47—24no—
135C2×C4×D47—24no—
136C2×C4×D47—42no—
137C2×C4×D47—42no—
138C2×C4×D47—82no—
139C2×C4×D47—82no—
140C2×Dic94—24no—
141C2×Dic94—82no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—22no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—44no—
149C4⋊C4×C47—42no—
150C4⋊C4×C47—44no—
151C4⋊C4×C47C2×Q814yes—
152C42×C47—21no—
153C42×C47—21no—
154C42×C47—21no—
155C42×C47—21no—
156C42×C47—42no—
157C42×C47C2×D412yes—
158C42×C47C2×D412yes—
159(C2×D47)⋊C4—24no—
160(C2×D47)⋊C4—24no—
161(C2×D47)⋊C4—24no—
162C4⋊Dic47—24no—
163C4⋊Dic47—21no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47C2×D412yes—
167C4⋊Dic47C2×D412yes—
168C4⋊Dic47C2×D412yes—
169Dic47⋊C4—24no—
170Dic47⋊C4—24no—
171Dic47⋊C4—24no—
172Dic47⋊C4—24no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—24no—
177C2×C47⋊C8C2×Q814yes—
178C23×C47C22×D411yescharacteristic
179C22×D47—22no—
180C22×D47—24no—
181C22×D47—41no—
182C22×D47—41no—
183C22×D47—42no—
184Q8×C47—44no—
185Q8×C47—84no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—21no—
189C2×C4×C47—21no—
190C2×C4×C47—22no—
191C2×C4×C47—21no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—24no—
195C2×C4×C47—21no—
196C2×C4×C47—21no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47#12012yes—
205C2×C4×C47#12012yes—
206C2×C4×C47AGL(1,ℤ/8)12yes—
207C2×C4×C47C22×D411yescharacteristic
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47C22×Q812yes—
210C2×C4×C472−1+412yes—
211C2×Dic47—22no—
212C2×Dic47—22no—
213C2×Dic47—22no—
214C2×Dic47—41no—
215C2×Dic47—41no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218D188—22no—
219D188—24no—
220D188C4⋊Q814yescomplemented
221C4×D47—48no—
222C4×D47—84no—
223C4×D47—84no—
224Dic94—84no—
225C47⋊C8—24no—
226C47⋊C8—42no—
227#25851—471nomaximal, Sylow
228C22×C47C2×#12011yescharacteristic
229C22×C47D4×Q812yes—
230C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
231C22×C47#57511yescharacteristic
232C22×C47#57712yes—
233C2×D47—24no—
234C2×D47—44no—
235C2×D47—44no—
236C2×D47—42no—
237C2×D47—42no—
238C4×C47—22no—
239C4×C47—22no—
240C4×C47—22no—
241C4×C47—44no—
242C4×C47—41no—
243C4×C47—41no—
244C4×C47—41no—
245C4×C47—82no—
246C4×C47C2×C4⋊Q811yescharacteristic
247C4×C47#53611yescharacteristic
248C4×C47D4×Q812yes—
249Dic47—42no—
250Dic47—42no—
251Dic47—42no—
252C2×#554—471no—
253C2×#501—471no—
254C2×#476—471no—
255#2546D4711yescharacteristic, complemented
256#2059—472no—
257#2058—472no—
258#1989—471no—
259#1985—471no—
260#1864—472no—
261#1862—472no—
262#1762—471no—
263C2×C47#243411yescharacteristic
264C2×C47#300611yescharacteristic
265C2×C47#300911yescharacteristic
266C2×C47#338412yes—
267C2×C47#338812yes—
268D47—44no—
269#554—944no—
270#554—944no—
271C2×C4⋊Q8—21no—
272C2×C4⋊Q8—471no—
273C2×C4⋊Q8C2×D4711yescharacteristic
274C2×#125—471no—
275C2×#125—471no—
276C2×C22⋊Q8—941no—
277C2×C22⋊Q8—941no—
278C2×C22⋊Q8—941no—
279C2×C4×Q8C2×D4711yescharacteristic
280C2×C4×D4—471no—
281C2×C4×D4—471no—
282C2×C4×D4—471no—
283C22×C4⋊C4—941no—
284#501—944no—
285#476—944no—
286C2×#107—472no—
287C2×#106—472no—
288C2×#106—472no—
289C2×C4⋊C8—471no—
290C2×#102—472no—
291C2×#102—472no—
292C2×#97—471no—
293#398—21no—
294#398—21no—
295#391—21no—
296#389C2×D4712yes—
297#389C2×D4712yes—
298C4×C4⋊C4C2×D4711yescharacteristic
299#340—472no—
300#340—472no—
301#340—472no—
302#336—472no—
303#336—472no—
304C47#2585111yescharacteristic, complemented, Sylow
305C22×Q8—21no—
306C22×Q8—941no—
307C22×Q8C22⋊D4711yescharacteristic
308C22×D4—471no—
309C23×C4—941no—
310C23×C4—941no—
311C23×C4—941no—
312C22×C8—472no—
313C22×C8—472no—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C4⋊Q8—44no—
317C4⋊Q8—942no—
318C4⋊Q8—942no—
319#125—474no—
320#125—942no—
321#125—942no—
322#125—942no—
323C22⋊Q8—948no—
324C22⋊Q8—948no—
325C22⋊Q8—948no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×D4—474no—
329C4×D4—942no—
330C4×D4—942no—
331C4×D4—942no—
332C4×D4—942no—
333C4×D4—942no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—471no—
338C2×C4⋊C4—471no—
339C2×C4⋊C4—471no—
340C2×C4⋊C4—948no—
341C2×C4⋊C4—941no—
342C2×C4⋊C4—941no—
343C2×C4⋊C4—941no—
344C2×C4⋊C4—941no—
345C2×C4⋊C4—941no—
346C2×C4⋊C4C22⋊D4711yescharacteristic
347C2×C4⋊C4C22⋊D4711yescharacteristic
348C2×C4⋊C4C22⋊D4711yescharacteristic
349C2×C4⋊C4C22×D4712yes—
350C2×C4⋊C4C22×D4712yes—
351C2×C22⋊C4—941no—
352C2×C22⋊C4—941no—
353C2×C22⋊C4—941no—
354C2×C42—21no—
355C2×C42C22×D4711yescharacteristic
356C2×C42C22×D4711yescharacteristic
357C2×C42C22×D4711yescharacteristic
358#107—944no—
359#106—944no—
360#106—944no—
361C4⋊C8—942no—
362#102—478no—
363#102—944no—
364#97—942no—
365#95—21no—
366#95—22no—
367#95—22no—
368C24—941no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—41no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—944no—
376C2×Q8—3762no—
377C2×D4—472no—
378C2×D4—474no—
379C2×D4—941no—
380C2×D4—942no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—944no—
384C22×C4—944no—
385C22×C4—941no—
386C22×C4—941no—
387C22×C4—941no—
388C22×C4—944no—
389C22×C4—3762no—
390C22×C4—3762no—
391C22×C4—1882no—
392C22×C4—1882no—
393C22×C4C2×C22⋊D4711yescharacteristic
394C22×C4C2×C22⋊D4711yescharacteristic
395C22×C4C2×C22⋊D4711yescharacteristic
396C22×C4C2×C22⋊D4711yescharacteristic
397C22×C4C2×C22⋊D4711yescharacteristic
398C22×C4C2×C22⋊D4711yescharacteristic
399C22×C4C23×D4711yescharacteristic
400C2×C8—474no—
401C2×C8—942no—
402C2×C8—942no—
403C2×C8—942no—
404C2×C8—944no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—472no—
415C4⋊C4—472no—
416C4⋊C4—472no—
417C4⋊C4—944no—
418C4⋊C4—944no—
419C4⋊C4—944no—
420C4⋊C4—944no—
421C4⋊C4—944no—
422C4⋊C4—941no—
423C4⋊C4—941no—
424C4⋊C4—941no—
425C4⋊C4Q8×D4714yes—
426C22⋊C4—944no—
427C22⋊C4—944no—
428C22⋊C4—944no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—42no—
434C42D4×D4712yes—
435C42D4×D4712yes—
436C23—944no—
437C23—942no—
438C23—1881no—
439C23—1881no—
440C23—1882no—
441C23C22×C22⋊D4711yescharacteristic
442Q8—44no—
443Q8—84no—
444Q8—3764no—
445D4—474no—
446D4—944no—
447D4—942no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—942no—
467C2×C4—942no—
468C2×C4—942no—
469C2×C4—3764no—
470C2×C4—3764no—
471C2×C4—1888no—
472C2×C4—1881no—
473C2×C4—1881no—
474C2×C4—1882no—
475C2×C4—1882no—
476C2×C4#1309399212yes—
477C2×C4#1309399212yes—
478C2×C4#1309405012yes—
479C2×C4C2×D4×D4711yescharacteristic
480C2×C4C2×D4×D4711yescharacteristic
481C2×C4C2×Q8×D4712yes—
482C2×C4#1309407212yes—
483C8—944no—
484C8—1882no—
485C22—944no—
486C22—1884no—
487C22—1884no—
488C22—1882no—
489C22—1882no—
490C22C2×#1309399211yescharacteristic
491C22Q8×C22⋊D4712yes—
492C22C2×#1309405011yescharacteristic
493C22#1568992111yescharacteristic
494C22#1568993112yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—44no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—82no—
503C4—1882no—
504C4—1882no—
505C4—1882no—
506C4C4⋊Q8×D4711yescharacteristic
507C4#1568969811yescharacteristic
508C4#1568978812yes—
509C2—1884no—
510C2#2024702911yescharacteristic
511C2#2025263811yescharacteristic
512C2#2025266311yescharacteristic
513C2#2025628312yes—
514C2#2025632712yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2546C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenter#2546D47
hypocenterC47#25851
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
2. C2×#15689529C2semidirect
4. #20244726C2semidirect
5. #20244721C2semidirect
6. #20244345C2semidirect
7. #20244329C2semidirect
11. C47×#2546C2semidirect
86. #13093944Q8semidirect
220. D188C4⋊Q8semidirect
255. #2546D47semidirect
304. C47#25851semidirect

Maximal subgroups

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

subgroupindex
C2×#156898332
C2×#156895292
C2×#156894102
#202447262
#202447212
#202443452
#202443292
#202436482
#202436342
#202432162
C47×#25462
#2585147