Finite Group Databasebeta

← #26225989

Subgroups — #26225989

Order 12032.

1106 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689675C211yesmaximal, characteristic
2C2×#15689173C211yesmaximal, characteristic
3#20245791C211yesmaximal, characteristic
4#20244347C212yesmaximal, complemented
5#20244188C212yesmaximal, complemented
6#20243701C211yesmaximal, characteristic
7#20243700C211yesmaximal, characteristic
8#20243698C211yesmaximal, characteristic
9#20243402C212yesmaximal, complemented
10#20243401C212yesmaximal, complemented
11C47×#1873C211yesmaximal, characteristic
12#15689675—24no—
13#15689675C418yescomplemented
14C2×#13094023C2211yescharacteristic
15C2×#13094022C2211yescharacteristic
16C2×#13094014—21no—
17C2×#13094003C2211yescharacteristic
18C2×#13094001—21no—
19C2×Q8×Dic47C2211yescharacteristic
20C2×#13093989—21no—
21C2×#13093989C2211yescharacteristic
22C2×#13093983C2211yescharacteristic
23C2×#13093980—21no—
24C22×C4×Dic47C2211yescharacteristic
25#15689173—24no—
26C2×Dic47⋊C8C2212yes—
27C2×#13093949C2212yes—
28C2×#13093948C2212yes—
29C2×#13093947C2212yes—
30C2×C22⋊(C47⋊C8)C2211yescharacteristic
31C2×C8×Dic47C2212yes—
32#15688992C2212yes—
33#15688988C2212yes—
34#15688981—21no—
35#15688981—21no—
36C4×Dic47⋊C4—21no—
37C4×Dic47⋊C4—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C22⋊Q8×C47C2211yescharacteristic
42C2×#117×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C2×C22⋊C8×C47C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94—21no—
47C22×Dic94C2311yescharacteristic
48C23×Dic47D411yescharacteristic
49C23×Dic47D411yescharacteristic
50C22×C47⋊C8C2312yes—
51#13094023—22no—
52#13094023C2×C414yes—
53#13094022—22no—
54#13094022C2×C414yes—
55#13094014—28no—
56#13094003—22no—
57#13094003C2×C414yes—
58#13094001—28no—
59Q8×Dic47—22no—
60Q8×Dic47C2×C414yes—
61#13093989—28no—
62#13093989—22no—
63#13093989C2×C414yes—
64#13093983—42no—
65#13093980—28no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47C2312yes—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—22no—
75C2×Dic47⋊C4—22no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47C2×C414yes—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D414yes—
85C2×C4×Dic47C2311yescharacteristic
86C2×C4×Dic47C2311yescharacteristic
87Dic47⋊C8—24no—
88#13093949—42no—
89#13093948—42no—
90#13093947—42no—
91C22⋊(C47⋊C8)—22no—
92C8×Dic47—24no—
93C47⋊#95—22no—
94C47⋊#95—22no—
95C22×Q8×C47C2311yescharacteristic
96C23×C4×C47C2311yescharacteristic
97C22×C8×C47C2312yes—
98C22⋊Q8×C47—22no—
99C22⋊Q8×C47C2×C414yes—
100#117×C47—42no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47C2312yes—
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C42×C47—21no—
107#103×C47—42no—
108C22⋊C8×C47—22no—
109C24×C47C2×D411yescharacteristic
110C2×Q8×C47—21no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47C22×C412yes—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D412yes—
122C22×C4×C47C2411yescharacteristic
123C22×Dic47—24no—
124C22×Dic47—24no—
125C22×Dic47—21no—
126C22×Dic47—21no—
127C22×Dic47—22no—
128C22×Dic47C22⋊C414yes—
129C22×Dic47C22⋊C414yes—
130C22×Dic47C2×D411yescharacteristic
131C22×Dic47C2×D411yescharacteristic
132C22×Dic47C2×D411yescharacteristic
133C22×Dic47C2×D411yescharacteristic
134C2×Dic94—22no—
135C2×Dic94—22no—
136C2×Dic94—21no—
137C2×Dic94—24no—
138C2×Dic94—44no—
139C2×Dic94C22×C412yes—
140C2×C8×C47—22no—
141C2×C8×C47—22no—
142C2×C8×C47—22no—
143C4⋊C4×C47—24no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—22no—
147C4⋊C4×C47C22×C412yes—
148C22⋊C4×C47—24no—
149C22⋊C4×C47—42no—
150C42×C47—42no—
151C22⋊Dic47—24no—
152C22⋊Dic47—42no—
153C4⋊Dic47—21no—
154C4⋊Dic47—24no—
155C4⋊Dic47—22no—
156C4⋊Dic47—22no—
157C4⋊Dic47C22×C412yes—
158Dic47⋊C4—24no—
159Dic47⋊C4—24no—
160Dic47⋊C4—24no—
161Dic47⋊C4—24no—
162Dic47⋊C4—24no—
163Dic47⋊C4—44no—
164Dic47⋊C4—44no—
165C4×Dic47—24no—
166C4×Dic47—24no—
167C4×Dic47—42no—
168C4×Dic47C22⋊C414yes—
169C4×Dic47C22⋊C414yes—
170C4×Dic47C22×C412yes—
171C4×Dic47C22×C412yes—
172C4×Dic47C2×D412yes—
173C4×Dic47C2×D412yes—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47C2×C22⋊C412yes—
182C23×C47C22×D411yescharacteristic
183Q8×C47—24no—
184Q8×C47—42no—
185C2×C4×C47—21no—
186C2×C4×C47—21no—
187C2×C4×C47—24no—
188C2×C4×C47—24no—
189C2×C4×C47—22no—
190C2×C4×C47—22no—
191C2×C4×C47—41no—
192C2×C4×C47—41no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47C2×C22⋊C412yes—
198C2×C4×C47C23×C411yescharacteristic
199C2×C4×C47C22×D411yescharacteristic
200C2×C4×C47H(5,𝔽2)12yes—
201C2×C4×C472−1+412yes—
202C2×Dic47—22no—
203C2×Dic47—22no—
204C2×Dic47—28no—
205C2×Dic47—28no—
206C2×Dic47—22no—
207C2×Dic47—22no—
208C2×Dic47—41no—
209C2×Dic47—41no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—42no—
214C2×Dic47C2×C22⋊C412yes—
215C2×Dic47C2×C22⋊C412yes—
216C2×Dic47C2×C22⋊C412yes—
217C2×Dic47C2×C22⋊C412yes—
218C2×Dic47#13514yes—
219C2×Dic47#13714yes—
220Dic94—24no—
221Dic94—42no—
222Dic94—48no—
223C8×C47—42no—
224C47⋊C8—42no—
225#21864—471nomaximal, Sylow
226C22×C47—22no—
227C22×C47—22no—
228C22×C47—22no—
229C22×C47—42no—
230C22×C47—42no—
231C22×C47C22×C22⋊C411yescharacteristic
232C22×C47#53511yescharacteristic
233C22×C47#53611yescharacteristic
234C22×C47C2×#13512yes—
235C22×C47C2×#13712yes—
236C4×C47—22no—
237C4×C47—42no—
238C4×C47—42no—
239C4×C47—82no—
240C4×C47#52012yes—
241C4×C47#53712yes—
242Dic47—22no—
243Dic47—22no—
244Dic47—42no—
245Dic47—42no—
246Dic47—82no—
247Dic47#41614yes—
248Dic47#41914yes—
249C2×#533—471no—
250C2×#431—471no—
251#2285—471no—
252#1990—472no—
253#1956—472no—
254#1873—471no—
255#1873—471no—
256#1873—471no—
257#1873D4711yescharacteristic
258#1805—472no—
259#1805—472no—
260C2×C47—42no—
261C2×C47#231111yescharacteristic
262C2×C47C2×#41611yescharacteristic
263C2×C47C2×#41911yescharacteristic
264C2×C47#321012yes—
265C2×C47#323112yes—
266#533—478no—
267#533—944no—
268C2×C4⋊Q8—471no—
269C2×C4⋊Q8—471no—
270C2×#125—941no—
271C2×C22⋊Q8—471no—
272C2×C22⋊Q8—941no—
273C2×C22⋊Q8C2×D4711yescharacteristic
274C2×C4×Q8—471no—
275C2×C4×Q8—471no—
276C2×C4×Q8—941no—
277C2×#117—471no—
278C2×#117—941no—
279C2×#117C2×D4711yescharacteristic
280C22×C42—471no—
281#431—944no—
282C2×C4⋊C8—472no—
283C2×#103—472no—
284C2×#103—472no—
285C2×#103—472no—
286C2×#103C2×D4712yes—
287C2×C22⋊C8—471no—
288C2×C22⋊C8C2×D4711yescharacteristic
289C2×C4×C8—472no—
290#384—472no—
291#382—472no—
292#380—941no—
293#380—941no—
294C4×C4⋊C4—941no—
295C4×C4⋊C4—941no—
296#340—472no—
297#340—472no—
298#340—472no—
299#340C2×D4712yes—
300C47#2186411yescharacteristic, complemented, Sylow
301C22×Q8—471no—
302C22×Q8—941no—
303C22×Q8C22×D4711yescharacteristic
304C23×C4—471no—
305C23×C4—471no—
306C23×C4C22×D4711yescharacteristic
307C22×C8—472no—
308C22×C8C22×D4712yes—
309C4⋊Q8—474no—
310C4⋊Q8—474no—
311C4⋊Q8—942no—
312C4⋊Q8—942no—
313#125—948no—
314C22⋊Q8—22no—
315C22⋊Q8—474no—
316C22⋊Q8—948no—
317C22⋊Q8—942no—
318C22⋊Q8C4×D4714yes—
319C4×Q8—474no—
320C4×Q8—474no—
321C4×Q8—942no—
322C4×Q8—948no—
323C4×Q8—942no—
324#117—42no—
325#117—948no—
326#117—1882no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—471no—
329C2×C4⋊C4—472no—
330C2×C4⋊C4—941no—
331C2×C4⋊C4—941no—
332C2×C4⋊C4—941no—
333C2×C4⋊C4—941no—
334C2×C4⋊C4—941no—
335C2×C4⋊C4—941no—
336C2×C4⋊C4—942no—
337C2×C4⋊C4—942no—
338C2×C4⋊C4C22×D4712yes—
339C2×C4⋊C4C22×D4711yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—941no—
343C2×C22⋊C4—941no—
344C2×C42—21no—
345C2×C42—474no—
346C2×C42—471no—
347C2×C42—471no—
348C2×C42—474no—
349C2×C42—471no—
350C2×C42—471no—
351C2×C42—941no—
352C2×C42—941no—
353C2×C42—941no—
354C4⋊C8—944no—
355#103—42no—
356#103—1882no—
357#103—1882no—
358#103—1882no—
359C22⋊C8—22no—
360C22⋊C8—942no—
361C4×C8—944no—
362#95—942no—
363#95—942no—
364C24D4×D4711yescharacteristic
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—472no—
369C2×Q8—944no—
370C2×Q8—941no—
371C2×Q8—942no—
372C2×Q8—942no—
373C2×Q8—1884no—
374C2×Q8C2×C4×D4712yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—471no—
381C22×C4—471no—
382C22×C4—474no—
383C22×C4—474no—
384C22×C4—471no—
385C22×C4—471no—
386C22×C4—944no—
387C22×C4—944no—
388C22×C4—941no—
389C22×C4—941no—
390C22×C4—942no—
391C22×C4C2×C4×D4712yes—
392C22×C4D4×D4711yescharacteristic
393C22×C4D4×D4712yes—
394C22×C4C23×D4711yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—942no—
399C2×C8—942no—
400C2×C8—942no—
401C4⋊C4—24no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—472no—
406C4⋊C4—944no—
407C4⋊C4—944no—
408C4⋊C4—944no—
409C4⋊C4—944no—
410C4⋊C4—944no—
411C4⋊C4—944no—
412C4⋊C4—941no—
413C4⋊C4—942no—
414C4⋊C4—942no—
415C4⋊C4—1884no—
416C4⋊C4—1884no—
417C4⋊C4C2×C4×D4712yes—
418C22⋊C4—24no—
419C22⋊C4—42no—
420C22⋊C4—944no—
421C22⋊C4—1882no—
422C42—42no—
423C42—474no—
424C42—474no—
425C42—472no—
426C42—472no—
427C42—472no—
428C42—472no—
429C42—944no—
430C42—944no—
431C42—1882no—
432C23—22no—
433C23—22no—
434C23—21no—
435C23—21no—
436C23C22⋊C4×D4712yes—
437C23C2×D4×D4711yescharacteristic
438Q8—24no—
439Q8—42no—
440Q8—944no—
441Q8—1888no—
442Q8—1882no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—472no—
456C2×C4—472no—
457C2×C4—474no—
458C2×C4—474no—
459C2×C4—472no—
460C2×C4—472no—
461C2×C4—948no—
462C2×C4—948no—
463C2×C4—942no—
464C2×C4—942no—
465C2×C4—942no—
466C2×C4—942no—
467C2×C4—1882no—
468C2×C4—1882no—
469C2×C4—1882no—
470C2×C4—1882no—
471C2×C4—1881no—
472C2×C4—1881no—
473C2×C4C22⋊C4×D4712yes—
474C2×C4C22×C4×D4711yescharacteristic
475C2×C4C2×D4×D4711yescharacteristic
476C2×C4#1309406912yes—
477C2×C4#1309407112yes—
478C8—42no—
479C8—1882no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22C2×C22⋊C4×D4711yescharacteristic
486C22#1568968811yescharacteristic
487C22#1568969511yescharacteristic
488C22#135×D4712yes—
489C22#137×D4712yes—
490C4—22no—
491C4—42no—
492C4—42no—
493C4—82no—
494C4—474no—
495C4—474no—
496C4—942no—
497C4—942no—
498C4—3762no—
499C4—1882no—
500C4—1882no—
501C4#1568960212yes—
502C4#1568970012yes—
503C2—42no—
504C2#2024597511yescharacteristic
505C2#416×D4711yescharacteristic
506C2#419×D4711yescharacteristic
507C2#2025434212yes—
508C2#2025454512yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1873C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1873D47
hypocenterC47#21864
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
4. #20244347C2semidirect
5. #20244188C2semidirect
9. #20243402C2semidirect
10. #20243401C2semidirect
13. #15689675C4semidirect
300. C47#21864semidirect

Maximal subgroups

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

subgroupindex
C2×#156896752
C2×#156891732
#202457912
#202443472
#202441882
#202437012
#202437002
#202436982
#202434022
#202434012
C47×#18732
#2186447