Finite Group Databasebeta

← #26225968

Subgroups — #26225968

Order 12032.

1166 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689652C211yesmaximal, characteristic
2C2×#15689174C211yesmaximal, characteristic, complemented
3#382×D47C211yesmaximal, characteristic, complemented
4#20244353C211yesmaximal, characteristic, complemented
5#20244352C211yesmaximal, characteristic, complemented
6#20244200C211yesmaximal, characteristic, complemented
7#20243702C212yesmaximal
8#20243692C212yesmaximal, complemented
9#20243401C212yesmaximal, complemented
10#20243396C212yesmaximal, complemented
11C47×#1958C211yesmaximal, characteristic, complemented
12#15689652—24no—
13#15689652C418yescomplemented
14C2×#13094017—21no—
15C2×#13094016C2211yescharacteristic
16C2×#13094006—21no—
17C2×#13094000C2211yescharacteristic
18C2×#13093995C2211yescharacteristic
19C2×Q8×Dic47C2211yescharacteristic
20C2×D4×Dic47C2211yescharacteristic
21C2×(C22⋊D47)⋊C4—21no—
22C2×#13093980—21no—
23C2×C4⋊C4×D47C2212yes—
24C2×C42×D47C2211yescharacteristic
25#15689174—24no—
26C2×Dic47⋊C8C2212yes—
27C2×#13093948C2212yes—
28C2×#13093945C2212yes—
29C2×(C2×D47)⋊C8C2212yes—
30C2×C4×C47⋊C8C2211yescharacteristic
31#15688989C2211yescharacteristic
32#15688988C2212yes—
33#15688987—21no—
34#15688987—21no—
35#95×D47—21no—
36#95×D47—21no—
37#15688886C2212yes—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×#124×C47C2211yescharacteristic
42C2×#103×C47C2212yes—
43C2×#102×C47C2212yes—
44C2×C4×C8×C47C2211yescharacteristic
45C47×#382C2211yescharacteristic
46C22×C22⋊D47—21no—
47C23×Dic47—21no—
48C22×C4×D47—22no—
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C22×C4×D47C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53#13094017—28no—
54#13094016—22no—
55#13094016C2×C414yescomplemented
56#13094006—28no—
57#13094000—22no—
58#13094000C2×C414yes—
59#13093995—22no—
60#13093995C2×C414yes—
61Q8×Dic47—22no—
62Q8×Dic47C2×C414yescomplemented
63D4×Dic47—22no—
64D4×Dic47C2×C414yescomplemented
65(C22⋊D47)⋊C4—28no—
66#13093980—28no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2312yes—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—22no—
75C4⋊C4×D47—42no—
76C4⋊C4×D47—42no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47D411yescharacteristic
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84C42×D47C2×C414yes—
85C42×D47D414yes—
86Dic47⋊C8—24no—
87#13093948—42no—
88#13093945—42no—
89(C2×D47)⋊C8—24no—
90C4×C47⋊C8—22no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—22no—
94C47⋊#95—22no—
95C22×Q8×C47C2311yescharacteristic
96C22×D4×C47C2311yescharacteristic
97C22×C8×C47C2312yes—
98#124×C47—22no—
99#124×C47C2×C414yescomplemented
100C2×C4⋊C4×C47C2312yes—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—21no—
103C2×C42×C47C2311yescharacteristic
104#103×C47—42no—
105#102×C47—42no—
106C4×C8×C47—22no—
107#95×C47—21no—
108#95×C47—21no—
109C24×C47—21no—
110C23×D47C2×D411yescharacteristic
111C2×Q8×C47—21no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47C22×C412yes—
115C2×D4×C47—21no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47C22×C412yes—
119C22×C4×C47—21no—
120C22×C4×C47—22no—
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2411yescharacteristic
124C2×C22⋊D47—24no—
125C2×C22⋊D47—44no—
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47—24no—
129C22×Dic47—22no—
130C22×Dic47—44no—
131C22×Dic47C2×D411yescharacteristic
132C22×Dic47C2×D411yescharacteristic
133C22×Dic47C2×D411yescharacteristic
134C2×C4×D47—24no—
135C2×C4×D47—22no—
136C2×C4×D47—22no—
137C2×C4×D47—42no—
138C2×C4×D47—44no—
139C2×C4×D47—44no—
140C2×C4×D47—44no—
141C2×C4×D47C22⋊C414yes—
142C2×C4×D47C22⋊C414yes—
143C2×C4×D47C22×C412yes—
144C2×C4×D47C2×D412yes—
145C2×C8×C47—22no—
146C2×C8×C47—22no—
147C2×C8×C47—22no—
148C4⋊C4×C47—22no—
149C4⋊C4×C47—22no—
150C22⋊C4×C47—24no—
151C22⋊C4×C47—24no—
152C42×C47C22×C412yes—
153C42×C47C2×D412yes—
154C22⋊Dic47—24no—
155C22⋊Dic47—24no—
156(C2×D47)⋊C4—24no—
157C4⋊Dic47—21no—
158C4⋊Dic47—21no—
159C4⋊Dic47—24no—
160C4⋊Dic47—22no—
161C4⋊Dic47—22no—
162C4⋊Dic47C22×C412yes—
163C4⋊Dic47C22×C412yes—
164Dic47⋊C4—24no—
165Dic47⋊C4—24no—
166Dic47⋊C4—24no—
167Dic47⋊C4—44no—
168C4×Dic47—24no—
169C4×Dic47C22⋊C414yes—
170C4×Dic47C22⋊C414yes—
171C4×Dic47C22×C412yes—
172C4×Dic47C2×D412yes—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C23×C47—22no—
177C23×C47—41no—
178C23×C47—41no—
179C23×C47—42no—
180C23×C47—42no—
181C23×C47C22×D411yescharacteristic
182C22×D47—21no—
183C22×D47—21no—
184C22×D47—22no—
185C22×D47C2×C22⋊C412yes—
186C22×D47#13514yes—
187Q8×C47—24no—
188Q8×C47—42no—
189D4×C47—24no—
190D4×C47—42no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—41no—
195C2×C4×C47—41no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47C2×C22⋊C412yes—
200C2×C4×C47C2×C22⋊C412yes—
201C2×C4×C47C23×C411yescharacteristic
202C2×C4×C47C22×D411yescharacteristic
203C2×C4×C47H(5,𝔽2)12yes—
204C2×C4×C472−1+412yes—
205C22⋊D47—48no—
206C2×Dic47—21no—
207C2×Dic47—21no—
208C2×Dic47—22no—
209C2×Dic47—22no—
210C2×Dic47—22no—
211C2×Dic47—22no—
212C2×Dic47—48no—
213C2×Dic47—42no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218C2×Dic47C2×C22⋊C412yes—
219C2×Dic47C2×C22⋊C412yes—
220C2×Dic47C2×C22⋊C412yes—
221C4×D47—28no—
222C4×D47—24no—
223C4×D47—24no—
224C4×D47—44no—
225C4×D47—88no—
226C8×C47—42no—
227C47⋊C8—42no—
228#21861—471nomaximal, Sylow
229C22×C47—42no—
230C22×C47—42no—
231C22×C47—42no—
232C22×C47—82no—
233C22×C47—82no—
234C22×C47C22×C22⋊C411yescharacteristic
235C22×C47#53511yescharacteristic
236C22×C47#53611yescharacteristic
237C22×C47C2×#13512yes—
238C22×C47#57512yes—
239C2×D47—22no—
240C2×D47—22no—
241C2×D47—24no—
242C2×D47—24no—
243C2×D47#41614yes—
244C4×C47—22no—
245C4×C47—22no—
246C4×C47—42no—
247C4×C47—82no—
248C4×C47#52012yes—
249C4×C47#53712yes—
250Dic47—22no—
251Dic47—22no—
252Dic47—22no—
253Dic47—42no—
254Dic47—42no—
255Dic47—82no—
256C2×#529—471no—
257C2×#431—471no—
258C2×#382—471no—
259#1991—471no—
260#1991—471no—
261#1958—471no—
262#1958D4711yescharacteristic, complemented
263#1874—472no—
264#1872—472no—
265#1805—472no—
266#1804—472no—
267C2×C47—82no—
268C2×C47#231111yescharacteristic
269C2×C47C2×#41611yescharacteristic
270C2×C47#289611yescharacteristic
271C2×C47#321112yes—
272C2×C47#323212yes—
273D47—24no—
274#529—478no—
275#529—944no—
276C2×#126—941no—
277C2×#125—471no—
278C2×#124C2×D4711yescharacteristic
279C2×#123—941no—
280C2×C22⋊Q8—471no—
281C2×#121—471no—
282C2×C4×Q8—471no—
283C2×C4×D4—471no—
284C2×C4×D4—941no—
285C2×#117—941no—
286C22×C4⋊C4—472no—
287C22×C42—471no—
288#431—944no—
289C2×C4⋊C8—472no—
290C2×#103—472no—
291C2×#103C2×D4712yes—
292C2×#102—472no—
293C2×#102C2×D4712yes—
294C2×C22⋊C8—472no—
295C2×C4×C8—471no—
296C2×C4×C8C2×D4711yescharacteristic
297#382—472no—
298#382—471no—
299#382—941no—
300#382—941no—
301#382C2×D4711yescharacteristic
302C2×#95—941no—
303C2×#95—941no—
304#340—472no—
305#340—472no—
306#340—472no—
307#340—472no—
308C47#2186111yescharacteristic, complemented, Sylow
309C22×Q8C22×D4711yescharacteristic
310C22×D4—941no—
311C22×D4C22×D4711yescharacteristic
312C23×C4—471no—
313C23×C4—471no—
314C23×C4—471no—
315C23×C4—942no—
316C23×C4—941no—
317C22×C8—472no—
318C22×C8C22×D4712yes—
319#126—948no—
320#125—474no—
321#125—942no—
322#124—22no—
323#124C4×D4714yescomplemented
324#123—948no—
325C22⋊Q8—474no—
326C22⋊Q8—942no—
327#121—474no—
328#121—942no—
329C4×Q8—474no—
330C4×Q8—942no—
331C4×D4—474no—
332C4×D4—942no—
333C4×D4—948no—
334#117—948no—
335C2×C4⋊C4—471no—
336C2×C4⋊C4—471no—
337C2×C4⋊C4—472no—
338C2×C4⋊C4—941no—
339C2×C4⋊C4—941no—
340C2×C4⋊C4—941no—
341C2×C4⋊C4—941no—
342C2×C4⋊C4—942no—
343C2×C4⋊C4—1882no—
344C2×C4⋊C4—1882no—
345C2×C4⋊C4C22×D4712yes—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—941no—
349C2×C22⋊C4—941no—
350C2×C22⋊C4—941no—
351C2×C42—471no—
352C2×C42—471no—
353C2×C42—471no—
354C2×C42—474no—
355C2×C42—474no—
356C2×C42—941no—
357C2×C42C22×D4711yescharacteristic
358C4⋊C8—944no—
359#103—42no—
360#103—1882no—
361#102—42no—
362#102—1882no—
363C22⋊C8—944no—
364C4×C8—22no—
365C4×C8—942no—
366#95—21no—
367#95—21no—
368#95—942no—
369#95—942no—
370#95—941no—
371#95—941no—
372C24—21no—
373C24—471no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8C2×C4×D4712yes—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—944no—
382C2×D4—1884no—
383C2×D4C2×C4×D4712yes—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—471no—
387C22×C4—471no—
388C22×C4—471no—
389C22×C4—474no—
390C22×C4—472no—
391C22×C4—472no—
392C22×C4—474no—
393C22×C4—944no—
394C22×C4—944no—
395C22×C4—942no—
396C22×C4—942no—
397C22×C4—941no—
398C22×C4—941no—
399C22×C4—942no—
400C22×C4—1882no—
401C22×C4—1884no—
402C22×C4—1884no—
403C22×C4—1884no—
404C22×C4—1884no—
405C22×C4D4×D4711yescharacteristic
406C22×C4D4×D4711yescharacteristic
407C22×C4C23×D4711yescharacteristic
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—942no—
412C2×C8—942no—
413C2×C8—942no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—472no—
417C4⋊C4—472no—
418C4⋊C4—944no—
419C4⋊C4—944no—
420C4⋊C4—944no—
421C4⋊C4—944no—
422C4⋊C4—941no—
423C4⋊C4—941no—
424C4⋊C4—942no—
425C4⋊C4—942no—
426C4⋊C4—1884no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—944no—
430C22⋊C4—944no—
431C22⋊C4—944no—
432C42—474no—
433C42—474no—
434C42—472no—
435C42—472no—
436C42—944no—
437C42C2×C4×D4712yes—
438C42D4×D4712yes—
439C23—22no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—472no—
445C23—474no—
446C23—942no—
447C23—941no—
448C23—941no—
449C23C2×D4×D4711yescharacteristic
450Q8—24no—
451Q8—42no—
452D4—24no—
453D4—42no—
454D4—1888no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—472no—
464C2×C4—472no—
465C2×C4—472no—
466C2×C4—948no—
467C2×C4—942no—
468C2×C4—942no—
469C2×C4—941no—
470C2×C4—941no—
471C2×C4—944no—
472C2×C4—944no—
473C2×C4—942no—
474C2×C4—942no—
475C2×C4—3768no—
476C2×C4—1884no—
477C2×C4—1888no—
478C2×C4—1882no—
479C2×C4—1882no—
480C2×C4—1882no—
481C2×C4—1882no—
482C2×C4—1882no—
483C2×C4C22⋊C4×D4712yes—
484C2×C4C22⋊C4×D4712yes—
485C2×C4C22×C4×D4711yescharacteristic
486C2×C4C2×D4×D4711yescharacteristic
487C2×C4#1309406912yes—
488C2×C4#1309407112yes—
489C8—42no—
490C8—1882no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—82no—
495C22—82no—
496C22—474no—
497C22—944no—
498C22—944no—
499C22—942no—
500C22—942no—
501C22C2×C22⋊C4×D4711yescharacteristic
502C22#1568968611yescharacteristic
503C22#1568969311yescharacteristic
504C22#135×D4712yes—
505C22#1568992012yes—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—82no—
510C4—942no—
511C4—942no—
512C4—942no—
513C4—3762no—
514C4—1882no—
515C4—1882no—
516C4#1568960212yes—
517C4#1568970212yes—
518C2—82no—
519C2—944no—
520C2#2024597811yescharacteristic
521C2#416×D4711yescharacteristic
522C2#2025190311yescharacteristic
523C2#2025434812yes—
524C2#2025455112yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1958C2
socleC23×C47C22×D4
prime radicalC2×#13093995C22
cosocleC22×C4×C47C24
hypercenter#1958D47
hypocenterC47#21861
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#15689174C2semidirect
3. #382×D47C2semidirect
4. #20244353C2semidirect
5. #20244352C2semidirect
6. #20244200C2semidirect
8. #20243692C2semidirect
9. #20243401C2semidirect
10. #20243396C2semidirect
11. C47×#1958C2semidirect
13. #15689652C4semidirect
55. #13094016C2×C4semidirect
62. Q8×Dic47C2×C4semidirect
64. D4×Dic47C2×C4semidirect
99. #124×C47C2×C4semidirect
262. #1958D47semidirect
308. C47#21861semidirect
323. #124C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896522
C2×#156891742
#382×D472
#202443532
#202443522
#202442002
#202437022
#202436922
#202434012
#202433962
C47×#19582
#2186147