Finite Group Databasebeta

← #26234555

Subgroups — #26234555

Order 12032.

1122 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×#15689233C212yesmaximal
2C2×C4×#10625334C211yesmaximal, characteristic
3C4⋊C4×D188C211yesmaximal, characteristic, complemented
4#20244123C212yesmaximal, complemented
5#20243649C212yesmaximal, complemented
6#20243648C212yesmaximal, complemented
7#20243544C211yesmaximal, characteristic, complemented
8#20243388C211yesmaximal, characteristic, complemented
9#20243385C211yesmaximal, characteristic, complemented
10#20243257C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C47C211yesmaximal, characteristic, complemented
12C22×#10625334C2211yescharacteristic
13C2×C4×D188C2212yes—
14C2×C4×D188C2211yescharacteristic
15C2×C4⋊C4×D47—21no—
16#15689233—28no—
17C4×#10625334C418yescomplemented
18C2×#13093957C2211yescharacteristic
19C2×#13093955C2212yes—
20C2×#13093955C2212yes—
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×#13093945C2212yes—
23C2×#13093945C2211yescharacteristic
24C2×#13093944C2212yes—
25C2×#13093944C2211yescharacteristic
26C2×#13093935C2212yes—
27C2×C4×C47⋊C8C2211yescharacteristic
28#15689021C2212yes—
29#15688997—22no—
30#15688993C2211yescharacteristic
31#15688980—21no—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688878C2212yes—
35C2×C4×D4×C47C2212yes—
36C2×C4×D4×C47C2211yescharacteristic
37C22×C4⋊C4×C47—21no—
38C47×#389C2212yes—
39C47×#385—22no—
40C47×#384C2211yescharacteristic
41C47×#380—21no—
42C4×C4⋊C4×C47C2211yescharacteristic
43C22×D188C2311yescharacteristic
44C22×C4×D47—21no—
45C22×C4×D47—22no—
46C2×#10625334C2×C414yes—
47C2×#10625334D414yes—
48C2×#10625334Q814yes—
49C22×C47⋊C8C2312yes—
50C22×C47⋊C8C2311yescharacteristic
51C22×C47⋊C8C2311yescharacteristic
52C4×D188—22no—
53C4×D188—24no—
54C4×D188—42no—
55C4×D188C2×C414yescomplemented
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47C2312yes—
59C2×C4⋊Dic47C2311yescharacteristic
60C2×Dic47⋊C4—22no—
61C4⋊C4×D47—44no—
62C2×(C2×D47)⋊C4—21no—
63C2×(C2×D47)⋊C4—22no—
64#13093957C2×C414yescomplemented
65#13093955—24no—
66#13093955—42no—
67C4⋊(C47⋊C8)—22no—
68#13093945—24no—
69#13093945C2×C414yescomplemented
70#13093944—24no—
71#13093944C2×C414yescomplemented
72#13093935—24no—
73C4×C47⋊C8C2×C414yescomplemented
74C47⋊#95—22no—
75C22×D4×C47C2311yescharacteristic
76C23×C4×C47—21no—
77C23×C4×C47—22no—
78C4×D4×C47—24no—
79C4×D4×C47—22no—
80C4×D4×C47—42no—
81C4×D4×C47C2×C414yescomplemented
82C2×C4⋊C4×C47—21no—
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—44no—
86C2×C4⋊C4×C47D411yescharacteristic
87C2×C4⋊C4×C47D411yescharacteristic
88C2×C4⋊C4×C47C2312yes—
89C2×C4⋊C4×C47C2311yescharacteristic
90C2×C22⋊C4×C47—21no—
91C2×C22⋊C4×C47—22no—
92C2×C42×C47C2312yes—
93C2×C42×C47C2311yescharacteristic
94#95×C47—22no—
95#95×C47D412yes—
96C24×C47—21no—
97C23×D47—21no—
98C2×D4×C47—21no—
99C2×D4×C47—21no—
100C2×D4×C47—21no—
101C2×D4×C47C22×C412yes—
102C2×D4×C47C2×D412yes—
103C2×D4×C47C2×Q812yes—
104C22×C4×C47—21no—
105C22×C4×C47—24no—
106C22×C4×C47—22no—
107C22×C4×C47—44no—
108C22×C4×C47—44no—
109C22×C4×C47—42no—
110C22×C4×C47—42no—
111C22×C4×C47—82no—
112C22×C4×C47C2×D412yes—
113C22×C4×C47C2×D412yes—
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2411yescharacteristic
117C22×Dic47—21no—
118C22×Dic47—22no—
119C2×D188—21no—
120C2×D188—21no—
121C2×D188—21no—
122C2×D188C22×C412yes—
123C2×D188C2×D412yes—
124C2×D188C2×Q812yes—
125C2×C4×D47—24no—
126C2×C4×D47—42no—
127C2×C4×D47—42no—
128C2×C4×D47—44no—
129C2×C4×D47—44no—
130C2×C4×D47—82no—
131C4⋊C4×C47—22no—
132C4⋊C4×C47—22no—
133C4⋊C4×C47—21no—
134C4⋊C4×C47—22no—
135C4⋊C4×C47—22no—
136C4⋊C4×C47—44no—
137C4⋊C4×C47—42no—
138C4⋊C4×C47—81no—
139C4⋊C4×C47C22×C412yes—
140C22⋊C4×C47—24no—
141C22⋊C4×C47—44no—
142C42×C47—22no—
143C42×C47—22no—
144C42×C47C22×C412yes—
145C42×C47#4112yes—
146#10625334C4⋊C4116yescomplemented
147(C2×D47)⋊C4—24no—
148(C2×D47)⋊C4—44no—
149C4⋊Dic47—21no—
150C4⋊Dic47—22no—
151C4⋊Dic47—22no—
152C4⋊Dic47—42no—
153C4⋊Dic47—81no—
154C4⋊Dic47C22×C412yes—
155Dic47⋊C4—44no—
156C2×C47⋊C8—21no—
157C2×C47⋊C8—21no—
158C2×C47⋊C8—22no—
159C2×C47⋊C8—22no—
160C2×C47⋊C8—22no—
161C2×C47⋊C8C22×C412yes—
162C2×C47⋊C8C22×C412yes—
163C2×C47⋊C8C2×D412yes—
164C2×C47⋊C8C2×Q812yes—
165C23×C47—22no—
166C23×C47—22no—
167C23×C47—22no—
168C23×C47—41no—
169C23×C47—41no—
170C23×C47—41no—
171C23×C47—41no—
172C23×C47C22×D411yescharacteristic
173C22×D47—22no—
174C22×D47—22no—
175C22×D47—22no—
176C22×D47—41no—
177C22×D47—41no—
178C22×D47—41no—
179C22×D47—41no—
180D4×C47—22no—
181D4×C47—22no—
182D4×C47—22no—
183D4×C47C2×C4⋊C414yescomplemented
184C2×C4×C47—21no—
185C2×C4×C47—21no—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—22no—
189C2×C4×C47—22no—
190C2×C4×C47—22no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—41no—
194C2×C4×C47—41no—
195C2×C4×C47—48no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—88no—
200C2×C4×C47C4×D412yes—
201C2×C4×C47C4×D412yes—
202C2×C4×C47AGL(1,ℤ/8)12yes—
203C2×C4×C47AGL(1,ℤ/8)12yes—
204C2×C4×C47C23×C411yescharacteristic
205C2×C4×C47C22×D411yescharacteristic
206C2×C4×C47C22×Q811yescharacteristic
207C2×C4×C47C2×#4111yescharacteristic
208C2×C4×C47H(5,𝔽2)11yescharacteristic
209C2×C4×C472−1+411yescharacteristic
210C2×Dic47—22no—
211C2×Dic47—41no—
212C2×Dic47—41no—
213C2×Dic47—42no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216D188—22no—
217D188—22no—
218D188—22no—
219D188C2×C4⋊C414yescomplemented
220C4×D47—48no—
221C4×D47—88no—
222C47⋊C8—22no—
223C47⋊C8—42no—
224C47⋊C8C2×C4⋊C414yescomplemented
225#22670—471nomaximal, Sylow
226C22×C47—24no—
227C22×C47—42no—
228C22×C47—42no—
229C22×C47—42no—
230C22×C47—42no—
231C22×C47—42no—
232C22×C47—42no—
233C22×C47C2×C4×D411yescharacteristic
234C22×C47D4211yescharacteristic
235C22×C47D4×Q811yescharacteristic
236C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C47#57711yescharacteristic
239C22×C47#57711yescharacteristic
240C2×D47—24no—
241C2×D47—42no—
242C2×D47—42no—
243C2×D47—42no—
244C2×D47—42no—
245C2×D47—42no—
246C2×D47—42no—
247C4×C47—22no—
248C4×C47—42no—
249C4×C47—42no—
250C4×C47—42no—
251C4×C47—82no—
252C4×C47#44214yescomplemented
253C4×C47C22×C4⋊C411yescharacteristic
254C4×C47#52011yescharacteristic
255C4×C47#54711yescharacteristic
256C4×C47#55411yescharacteristic
257Dic47—42no—
258Dic47—82no—
259C2×#442—472no—
260C2×C4×D8—471no—
261D4×C4⋊C4—471no—
262D4×C4⋊C4D4711yescharacteristic, complemented
263#1943—472no—
264#1864—472no—
265#1864—472no—
266#1844—471no—
267#1802—471no—
268#1802—471no—
269#1770—471no—
270C2×C47—44no—
271C2×C47D4×C4⋊C411yescharacteristic
272C2×C47C2×#44211yescharacteristic
273C2×C47#295211yescharacteristic
274C2×C47#327211yescharacteristic
275C2×C47#327211yescharacteristic
276C2×C47#338411yescharacteristic
277C2×C47#338411yescharacteristic
278D47—44no—
279C22×D8—471no—
280C2×C4×D4—471no—
281C2×C4×D4—472no—
282C2×C4×D4C2×D4712yes—
283C2×C4×D4C2×D4711yescharacteristic
284C22×C4⋊C4—21no—
285C22×C4⋊C4—941no—
286#442—948no—
287C4×D8—478no—
288C2×#107—471no—
289C2×#106—472no—
290C2×#106—472no—
291C2×C4⋊C8—471no—
292C2×#102—471no—
293C2×#102—471no—
294C2×#102—472no—
295C2×#102—472no—
296C2×#97—472no—
297C2×C4×C8—471no—
298#389—472no—
299#389C2×D4712yes—
300#385—22no—
301#385—942no—
302#384—471no—
303#384C2×D4711yescharacteristic
304#380—21no—
305#380—941no—
306C4×C4⋊C4C2×D4711yescharacteristic
307#340—472no—
308#340—472no—
309#336—472no—
310C47#2267011yescharacteristic, complemented, Sylow
311C22×D4—471no—
312C22×D4C22×D4711yescharacteristic
313C23×C4—21no—
314C23×C4—22no—
315C23×C4—941no—
316C23×C4—942no—
317C2×D8—474no—
318C2×D8—474no—
319C2×D8—474no—
320C22×C8—471no—
321C22×C8—471no—
322C22×C8—472no—
323C4×D4—24no—
324C4×D4—22no—
325C4×D4—42no—
326C4×D4—474no—
327C4×D4—944no—
328C4×D4—942no—
329C4×D4—1882no—
330C4×D4C4×D4714yescomplemented
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—471no—
336C2×C4⋊C4—472no—
337C2×C4⋊C4—941no—
338C2×C4⋊C4—942no—
339C2×C4⋊C4—941no—
340C2×C4⋊C4—1884no—
341C2×C4⋊C4C22⋊D4711yescharacteristic
342C2×C4⋊C4C22⋊D4711yescharacteristic
343C2×C4⋊C4C22×D4712yes—
344C2×C4⋊C4C22×D4711yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—941no—
348C2×C22⋊C4—942no—
349C2×C42C22×D4712yes—
350C2×C42C22×D4711yescharacteristic
351#107—474no—
352#106—944no—
353#106—1882no—
354C4⋊C8—942no—
355#102—474no—
356#102—474no—
357#102—944no—
358#102—944no—
359#97—944no—
360C4×C8—474no—
361#95—22no—
362#95—942no—
363#95C22⋊D4712yes—
364C24—21no—
365C24—941no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—472no—
370C2×D4—472no—
371C2×D4—472no—
372C2×D4—941no—
373C2×D4—941no—
374C2×D4—941no—
375C2×D4C2×C4×D4712yes—
376C2×D4D4×D4712yes—
377C2×D4Q8×D4712yes—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—82no—
386C22×C4—941no—
387C22×C4—944no—
388C22×C4—942no—
389C22×C4—3762no—
390C22×C4—1884no—
391C22×C4—1884no—
392C22×C4—1882no—
393C22×C4—1882no—
394C22×C4C2×C22⋊D4712yes—
395C22×C4C2×C22⋊D4712yes—
396C22×C4C2×C22⋊D4711yescharacteristic
397C22×C4C2×C22⋊D4711yescharacteristic
398C22×C4C23×D4711yescharacteristic
399D8—4716no—
400C2×C8—472no—
401C2×C8—472no—
402C2×C8—472no—
403C2×C8—472no—
404C2×C8—941no—
405C2×C8—941no—
406C2×C8—942no—
407C2×C8—942no—
408C2×C8—942no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—472no—
418C4⋊C4—941no—
419C4⋊C4—942no—
420C4⋊C4—942no—
421C4⋊C4—3761no—
422C4⋊C4—1884no—
423C4⋊C4—1882no—
424C4⋊C4C2×C4×D4712yes—
425C22⋊C4—24no—
426C22⋊C4—44no—
427C22⋊C4—944no—
428C22⋊C4—1884no—
429C42—22no—
430C42—22no—
431C42C2×C4×D4712yes—
432C42#1062535012yes—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—942no—
441C23—942no—
442C23—942no—
443C23—1881no—
444C23—1881no—
445C23—1881no—
446C23—1881no—
447C23C22×C22⋊D4711yescharacteristic
448D4—22no—
449D4—22no—
450D4—22no—
451D4—474no—
452D4—942no—
453D4—942no—
454D4—942no—
455D4C4⋊C4×D4714yescomplemented
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—48no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—88no—
472C2×C4—942no—
473C2×C4—3768no—
474C2×C4—1882no—
475C2×C4—1882no—
476C2×C4—1882no—
477C2×C4—1881no—
478C2×C4—1881no—
479C2×C4—1888no—
480C2×C4C4×C22⋊D4712yes—
481C2×C4C4×C22⋊D4712yes—
482C2×C4#1309404912yes—
483C2×C4#1309405012yes—
484C2×C4C22×C4×D4711yescharacteristic
485C2×C4C2×D4×D4711yescharacteristic
486C2×C4C2×Q8×D4711yescharacteristic
487C2×C4C2×#1062535011yescharacteristic
488C2×C4#1309406911yescharacteristic
489C2×C4#1309407211yescharacteristic
490C8—474no—
491C8—942no—
492C8—1882no—
493C22—24no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—944no—
501C22—1882no—
502C22—1882no—
503C22—1882no—
504C22—1882no—
505C22—1882no—
506C22—1882no—
507C22C2×C4×C22⋊D4711yescharacteristic
508C22D4×C22⋊D4711yescharacteristic
509C22Q8×C22⋊D4711yescharacteristic
510C22C2×#1309404911yescharacteristic
511C22C2×#1309405011yescharacteristic
512C22#1568993111yescharacteristic
513C22#1568993411yescharacteristic
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—82no—
519C4—3762no—
520C4—1882no—
521C4#1568923314yescomplemented
522C4C2×C4⋊C4×D4711yescharacteristic
523C4#1568960411yescharacteristic
524C4#1568977411yescharacteristic
525C4#1568982811yescharacteristic
526C2—44no—
527C2—1884no—
528C2C4⋊C4×C22⋊D4711yescharacteristic
529C2C2×#1568923311yescharacteristic
530C2#2025226911yescharacteristic
531C2#2025498411yescharacteristic
532C2#2025498811yescharacteristic
533C2#2025628311yescharacteristic
534C2#2025628711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingD4×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC22×#10625334C22
cosocleC22×C4×C47C24
hypercenterD4×C4⋊C4D47
hypocenterC47#22670
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×D188C2semidirect
4. #20244123C2semidirect
5. #20243649C2semidirect
6. #20243648C2semidirect
7. #20243544C2semidirect
8. #20243388C2semidirect
9. #20243385C2semidirect
10. #20243257C2semidirect
11. D4×C4⋊C4×C47C2semidirect
17. C4×#10625334C4semidirect
55. C4×D188C2×C4semidirect
64. #13093957C2×C4semidirect
69. #13093945C2×C4semidirect
71. #13093944C2×C4semidirect
73. C4×C47⋊C8C2×C4semidirect
81. C4×D4×C47C2×C4semidirect
146. #10625334C4⋊C4semidirect
183. D4×C47C2×C4⋊C4semidirect
219. D188C2×C4⋊C4semidirect
224. C47⋊C8C2×C4⋊C4semidirect
252. C4×C47#442semidirect
262. D4×C4⋊C4D47semidirect
310. C47#22670semidirect
330. C4×D4C4×D47semidirect
455. D4C4⋊C4×D47semidirect
521. C4#15689233semidirect

Maximal subgroups

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

subgroupindex
C2×#156892332
C2×C4×#106253342
C4⋊C4×D1882
#202441232
#202436492
#202436482
#202435442
#202433882
#202433852
#202432572
D4×C4⋊C4×C472
#2267047