Finite Group Databasebeta

← #26232831

Subgroups — #26232831

Order 12032.

1114 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689830C211yesmaximal, characteristic
2#20246263C212yesmaximal, complemented
3#20244373C212yesmaximal, complemented
4#20244371C214yesmaximal, complemented
5#20243376C211yesmaximal, characteristic
6#20243374C212yesmaximal
7#99×Dic47C212yesmaximal
8C4×#99×C47C211yesmaximal, characteristic
9#15689830—22no—
10#15689830—24no—
11#15689830—22no—
12C2×#13094024C411yescharacteristic, complemented
13C2×#13094015C412yescomplemented
14C2×#13094003C411yescharacteristic, complemented
15C2×#13094003C2211yescharacteristic
16C2×#13094001C412yescomplemented
17C2×#13094001C2212yes—
18C2×C4×Dic94C411yescharacteristic, complemented
19C2×D4×Dic47C2212yes—
20C22×C4⋊Dic47C411yescharacteristic, complemented
21C22×C4⋊Dic47C2211yescharacteristic
22C2×#13093939C2212yes—
23#15689021—22no—
24#15689006—24no—
25#15688983C2212yes—
26#15688981C2214yes—
27C4×C4⋊Dic47—22no—
28C22⋊C4×Dic47C2214yes—
29#15688890C2212yes—
30#15688889C2214yes—
31#15688889C2214yes—
32C2×C4×D4×C47C2211yescharacteristic
33C2×#99×C47C2211yescharacteristic
34C2×#99×C47C2211yescharacteristic
35C4×C22⋊C4×C47C2212yes—
36C47×#342C2212yes—
37C22×Dic94C2×C411yescharacteristic
38C22×Dic94C2×C412yes—
39C23×Dic47C2×C412yes—
40C23×Dic47C2312yes—
41#13094024—22no—
42#13094024—22no—
43#13094015—24no—
44#13094015—24no—
45#13094003—22no—
46#13094003—22no—
47#13094003—22no—
48#13094003—22no—
49#13094001—24no—
50#13094001—24no—
51#13094001—24no—
52#13094001—24no—
53C4×Dic94—22no—
54C4×Dic94—21no—
55C4×Dic94—21no—
56D4×Dic47—24no—
57D4×Dic47—24no—
58C2×C4⋊Dic47—22no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47—22no—
62C2×C4⋊Dic47C2×C412yes—
63C2×C4⋊Dic47C2×C411yescharacteristic
64C2×C4⋊Dic47C2×C412yes—
65C2×C4⋊Dic47C2×C411yescharacteristic
66C2×C4⋊Dic47C2×C412yes—
67C2×C4⋊Dic47C2×C411yescharacteristic
68C2×C4⋊Dic47D412yes—
69C2×C4⋊Dic47D414yes—
70C2×C4⋊Dic47D412yes—
71C2×Dic47⋊C4—24no—
72C2×Dic47⋊C4C2×C412yes—
73C2×Dic47⋊C4C2×C412yes—
74C2×Dic47⋊C4C2×C412yes—
75C2×C22⋊Dic47C2×C412yes—
76C2×C22⋊Dic47C2314yes—
77C2×C22⋊Dic47C2312yes—
78C2×C4×Dic47—24no—
79C2×C4×Dic47C2×C412yes—
80#13093939—24no—
81C47⋊#95—22no—
82C47⋊#95—24no—
83C47⋊#95—24no—
84C22×D4×C47C2311yescharacteristic
85C23×C4×C47C2×C411yescharacteristic
86C23×C4×C47C2311yescharacteristic
87C4×D4×C47—22no—
88C4×D4×C47—22no—
89C2×C4⋊C4×C47C2×C411yescharacteristic
90C2×C22⋊C4×C47C2×C411yescharacteristic
91C2×C22⋊C4×C47C2312yes—
92C2×C22⋊C4×C47C2311yescharacteristic
93C2×C22⋊C4×C47C2312yes—
94C2×C42×C47—22no—
95C2×C42×C47C2×C411yescharacteristic
96#99×C47—22no—
97#99×C47Q814yescomplemented
98#95×C47—22no—
99C24×C47C22×C411yescharacteristic
100C24×C47C2411yescharacteristic
101C2×D4×C47—22no—
102C2×D4×C47—22no—
103C2×D4×C47C2×Q812yes—
104C2×D4×C47#4112yes—
105C22×C4×C47—22no—
106C22×C4×C47—21no—
107C22×C4×C47—21no—
108C22×C4×C47—22no—
109C22×C4×C47—42no—
110C22×C4×C47—42no—
111C22×C4×C47C22×C411yescharacteristic
112C22×C4×C47C22×C411yescharacteristic
113C22×C4×C47C22×C411yescharacteristic
114C22×C4×C47C22×C411yescharacteristic
115C22×C4×C47C22×C411yescharacteristic
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D412yes—
118C22×Dic47—22no—
119C22×Dic47—22no—
120C22×Dic47—24no—
121C22×Dic47—82no—
122C22×Dic47—82no—
123C22×Dic47C22×C412yes—
124C22×Dic47C22×C412yes—
125C22×Dic47C22×C412yes—
126C22×Dic47C22×C412yes—
127C22×Dic47C2×D414yes—
128C22×Dic47C2×D414yes—
129C2×Dic94—21no—
130C2×Dic94—21no—
131C2×Dic94—24no—
132C2×Dic94—42no—
133C2×Dic94—82no—
134C4⋊C4×C47—22no—
135C22⋊C4×C47—22no—
136C22⋊C4×C47—22no—
137C22⋊C4×C47—22no—
138C22⋊C4×C47—22no—
139C22⋊C4×C47—22no—
140C22⋊C4×C47C2×Q814yes—
141C42×C47—21no—
142C42×C47—21no—
143C42×C47—44no—
144C22⋊Dic47—24no—
145C22⋊Dic47—24no—
146C22⋊Dic47—24no—
147C22⋊Dic47—24no—
148C4⋊Dic47—21no—
149C4⋊Dic47—21no—
150C4⋊Dic47—22no—
151C4⋊Dic47—22no—
152C4⋊Dic47—24no—
153C4⋊Dic47—44no—
154C4⋊Dic47C22⋊C412yes—
155C4⋊Dic47C22⋊C411yescharacteristic
156C4⋊Dic47C22⋊C411yescharacteristic
157C4⋊Dic47C22⋊C414yes—
158C4⋊Dic47C22⋊C414yes—
159C4⋊Dic47C22⋊C412yes—
160C4⋊Dic47C22⋊C412yes—
161Dic47⋊C4—24no—
162Dic47⋊C4—22no—
163Dic47⋊C4—22no—
164Dic47⋊C4—24no—
165Dic47⋊C4—84no—
166C4×Dic47—22no—
167C4×Dic47—22no—
168C4×Dic47—84no—
169C23×C47—21no—
170C23×C47—21no—
171C23×C47—41no—
172C23×C47—41no—
173C23×C47—41no—
174C23×C47—41no—
175C23×C47C4×Q812yes—
176C23×C47C23×C411yescharacteristic
177C23×C47C22×D412yes—
178C23×C47C22×Q811yescharacteristic
179C23×C47C2×#4111yescharacteristic
180C23×C472−1+412yes—
181D4×C47—24no—
182D4×C47—24no—
183C2×C4×C47—21no—
184C2×C4×C47—21no—
185C2×C4×C47—24no—
186C2×C4×C47—22no—
187C2×C4×C47—21no—
188C2×C4×C47—21no—
189C2×C4×C47—24no—
190C2×C4×C47—41no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47—41no—
197C2×C4×C47—41no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47C2×C22⋊C412yes—
202C2×C4×C47C2×C22⋊C411yescharacteristic
203C2×C4×C47C2×C22⋊C411yescharacteristic
204C2×C4×C47C4×Q812yes—
205C2×Dic47—22no—
206C2×Dic47—22no—
207C2×Dic47—44no—
208C2×Dic47—44no—
209C2×Dic47—42no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—44no—
214C2×Dic47—44no—
215C2×Dic47—44no—
216C2×Dic47—82no—
217C2×Dic47—82no—
218C2×Dic47C2×C22⋊C412yes—
219C2×Dic47C2×C22⋊C412yes—
220C2×Dic47C2×C22⋊C414yes—
221Dic94—42no—
222Dic94—42no—
223Dic94—84no—
224#22513—471nomaximal, Sylow
225C22×C47—22no—
226C22×C47—21no—
227C22×C47—22no—
228C22×C47—21no—
229C22×C47—22no—
230C22×C47—41no—
231C22×C47—41no—
232C22×C47—42no—
233C22×C47—42no—
234C22×C47—42no—
235C22×C47—41no—
236C22×C47—41no—
237C22×C47C22×C22⋊C411yescharacteristic
238C22×C47C2×C4×Q811yescharacteristic
239C22×C47#51911yescharacteristic
240C22×C47#54812yes—
241C22×C47D4×Q812yes—
242C4×C47—22no—
243C4×C47—21no—
244C4×C47—22no—
245C4×C47—44no—
246C4×C47—41no—
247C4×C47—41no—
248C4×C47—82no—
249C4×C47#41012yes—
250Dic47—24no—
251Dic47—22no—
252Dic47—22no—
253Dic47—42no—
254Dic47—42no—
255Dic47—84no—
256C2×#554—471no—
257#2345—472no—
258#1996—472no—
259#1996—474no—
260#1799—472no—
261#1799—471no—
262C4×#99—472no—
263C4×#99D4711yescharacteristic
264C2×C47—22no—
265C2×C47—22no—
266C2×C47—21no—
267C2×C47—21no—
268C2×C47—42no—
269C2×C47Q8×C22⋊C411yescharacteristic
270C2×C47C2×#41011yescharacteristic
271C2×C47#288311yescharacteristic
272#554—942no—
273#554—944no—
274#554—942no—
275C2×C4⋊Q8—471no—
276C2×#125—472no—
277C2×C22⋊Q8—471no—
278C2×C22⋊Q8—471no—
279C2×C22⋊Q8—472no—
280C2×C22⋊Q8—472no—
281C2×C4×Q8—471no—
282C2×C4×D4—472no—
283C2×C4×D4C2×D4711yescharacteristic
284C22×C4⋊C4—471no—
285C22×C4⋊C4—471no—
286C2×#99—472no—
287C2×#99C2×D4711yescharacteristic
288C2×#99C2×D4711yescharacteristic
289#389—942no—
290#387—944no—
291#380—472no—
292#380—474no—
293C4×C4⋊C4—942no—
294C4×C22⋊C4—474no—
295C4×C22⋊C4C2×D4712yes—
296#342—474no—
297#342—472no—
298#342—474no—
299#342C2×D4712yes—
300C47#2251311yescharacteristic, complemented, Sylow
301C22×Q8—472no—
302C22×Q8—471no—
303C22×D4C22×D4711yescharacteristic
304C23×C4—472no—
305C23×C4—472no—
306C23×C4C4×D4711yescharacteristic
307C23×C4C22×D4711yescharacteristic
308C4⋊Q8—942no—
309C4⋊Q8—942no—
310#125—944no—
311#125—944no—
312C22⋊Q8—944no—
313C22⋊Q8—944no—
314C22⋊Q8—944no—
315C22⋊Q8—944no—
316C22⋊Q8—942no—
317C22⋊Q8—942no—
318C22⋊Q8—942no—
319C22⋊Q8—942no—
320C4×Q8—942no—
321C4×Q8—941no—
322C4×Q8—941no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—944no—
326C4×D4—944no—
327C2×C4⋊C4—471no—
328C2×C4⋊C4—472no—
329C2×C4⋊C4—471no—
330C2×C4⋊C4—472no—
331C2×C4⋊C4—474no—
332C2×C4⋊C4—472no—
333C2×C4⋊C4—472no—
334C2×C4⋊C4—471no—
335C2×C4⋊C4—472no—
336C2×C4⋊C4—472no—
337C2×C4⋊C4—472no—
338C2×C4⋊C4—472no—
339C2×C4⋊C4—942no—
340C2×C4⋊C4—941no—
341C2×C4⋊C4—941no—
342C2×C4⋊C4—944no—
343C2×C4⋊C4—942no—
344C2×C4⋊C4C4×D4711yescharacteristic
345C2×C22⋊C4—472no—
346C2×C22⋊C4—472no—
347C2×C22⋊C4—474no—
348C2×C22⋊C4C4×D4711yescharacteristic
349C2×C22⋊C4C22×D4712yes—
350C2×C22⋊C4C22×D4711yescharacteristic
351C2×C22⋊C4C22×D4712yes—
352C2×C42—22no—
353C2×C42—472no—
354C2×C42—944no—
355C2×C42C4×D4711yescharacteristic
356#99—22no—
357#99—944no—
358#99Dic9414yescomplemented
359#95—22no—
360#95—944no—
361#95—944no—
362#95—942no—
363C24C2×C4×D4711yescharacteristic
364C24C23×D4711yescharacteristic
365C2×Q8—944no—
366C2×Q8—941no—
367C2×Q8—941no—
368C2×Q8—3762no—
369C2×Q8—1882no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4C2×Dic9412yes—
373C2×D4#1062535012yes—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—472no—
381C22×C4—472no—
382C22×C4—474no—
383C22×C4—474no—
384C22×C4—472no—
385C22×C4—472no—
386C22×C4—942no—
387C22×C4—942no—
388C22×C4—944no—
389C22×C4—3762no—
390C22×C4—3762no—
391C22×C4C2×C4×D4711yescharacteristic
392C22×C4C2×C4×D4711yescharacteristic
393C22×C4C2×C4×D4711yescharacteristic
394C22×C4C2×C4×D4711yescharacteristic
395C22×C4C2×C4×D4711yescharacteristic
396C22×C4D4×D4712yes—
397C22×C4D4×D4712yes—
398C4⋊C4—22no—
399C4⋊C4—472no—
400C4⋊C4—471no—
401C4⋊C4—471no—
402C4⋊C4—474no—
403C4⋊C4—474no—
404C4⋊C4—472no—
405C4⋊C4—472no—
406C4⋊C4—942no—
407C4⋊C4—942no—
408C4⋊C4—944no—
409C4⋊C4—942no—
410C4⋊C4—942no—
411C4⋊C4—944no—
412C4⋊C4—944no—
413C4⋊C4—941no—
414C4⋊C4—941no—
415C4⋊C4—3764no—
416C4⋊C4—1884no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—944no—
423C22⋊C4—944no—
424C22⋊C4—944no—
425C22⋊C4—944no—
426C22⋊C4C2×Dic9414yes—
427C42—21no—
428C42—21no—
429C42—44no—
430C42—942no—
431C42—942no—
432C42—3764no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23C4×Dic9412yes—
440C23C22×C4×D4711yescharacteristic
441C23C2×D4×D4712yes—
442C23C22×Dic9411yescharacteristic
443C23C2×#1062535011yescharacteristic
444C23#1309407112yes—
445Q8—3764no—
446Q8—1882no—
447Q8—1882no—
448D4—24no—
449D4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—472no—
469C2×C4—472no—
470C2×C4—474no—
471C2×C4—942no—
472C2×C4—942no—
473C2×C4—3762no—
474C2×C4—3762no—
475C2×C4—1884no—
476C2×C4—1884no—
477C2×C4—1882no—
478C2×C4—1882no—
479C2×C4—1884no—
480C2×C4—1884no—
481C2×C4—1884no—
482C2×C4—1882no—
483C2×C4—1882no—
484C2×C4C22⋊C4×D4711yescharacteristic
485C2×C4C22⋊C4×D4712yes—
486C2×C4C22⋊C4×D4711yescharacteristic
487C2×C4C4×Dic9412yes—
488C22—22no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22C2×C22⋊C4×D4711yescharacteristic
501C22C2×C4×Dic9411yescharacteristic
502C22#1568960011yescharacteristic
503C22#1568978212yes—
504C22D4×Dic9412yes—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—44no—
509C4—41no—
510C4—41no—
511C4—82no—
512C4—942no—
513C4—942no—
514C4—944no—
515C4—3764no—
516C4—1882no—
517C4—1882no—
518C4#1568909512yes—
519C2—22no—
520C2—22no—
521C2—21no—
522C2—21no—
523C2—42no—
524C2C22⋊C4×Dic9411yescharacteristic
525C2C2×#1568909511yescharacteristic
526C2#2025184711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC4×#99×C47C2
socleC22×C47C22×C22⋊C4
prime radicalC22×D4×C47C23
cosocleC24×C47C24
hypercenterC4×#99D47
hypocenterC47#22513
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #20246263C2semidirect
3. #20244373C2semidirect
4. #20244371C2semidirect
12. C2×#13094024C4semidirect
13. C2×#13094015C4semidirect
14. C2×#13094003C4semidirect
16. C2×#13094001C4semidirect
18. C2×C4×Dic94C4semidirect
20. C22×C4⋊Dic47C4semidirect
97. #99×C47Q8semidirect
300. C47#22513semidirect
358. #99Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156898302
#202462632
#202443732
#202443712
#202433762
#202433742
#99×Dic472
C4×#99×C472
#2251347