Finite Group Databasebeta

← #26228320

Subgroups — #26228320

Order 12032.

848 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689400C211yesmaximal, characteristic
2C4⋊Q8×Dic47C211yesmaximal, characteristic, complemented
3#20245849C211yesmaximal, characteristic
4#20244146C211yesmaximal, characteristic, complemented
5#20244120C212yesmaximal, complemented
6#20244072C212yesmaximal
7#20244006C212yesmaximal
8#20243957C212yesmaximal, complemented
9#20243389C211yesmaximal, characteristic
10C4⋊C8×Dic47C211yesmaximal, characteristic, complemented
11C47×#1988C211yesmaximal, characteristic, complemented
12C2×Q8×Dic47—21no—
13C2×C4×D188C2211yescharacteristic
14C2×#13093981—21no—
15#15689400C418yescomplemented
16C2×#13093957C2212yes—
17C2×#13093957C2212yes—
18C2×C4⋊(C47⋊C8)C2211yescharacteristic
19C2×#13093946C2212yes—
20C2×#13093944C2212yes—
21C2×C8×Dic47C2212yes—
22#15689030—21no—
23#15689024C2211yescharacteristic
24#15689024C2211yescharacteristic
25#15689022—21no—
26#15689022C2212yes—
27#15689019C2211yescharacteristic
28#15689019C2211yescharacteristic
29#15689012—21no—
30#15689012—21no—
31#15688993C2211yescharacteristic
32C4⋊C4×Dic47—21no—
33C4⋊C4×Dic47C2212yes—
34C4×(C2×D47)⋊C4—21no—
35C42×Dic47C2211yescharacteristic
36#15688886C2212yes—
37#15688885C2212yes—
38#15688877C2212yes—
39C2×C4⋊Q8×C47C2211yescharacteristic
40C2×C4⋊C8×C47C2211yescharacteristic
41C4×C4⋊C4×C47C2211yescharacteristic
42C47×#340C2212yes—
43C47×#340C2212yes—
44C22×D188C2311yescharacteristic
45C22×C4×D47—21no—
46C22×C47⋊C8C2312yes—
47Q8×Dic47—44no—
48C4×D188—22no—
49C4×D188C2×C414yes—
50#13093981—44no—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47C2312yes—
55C2×C4⋊Dic47C2311yescharacteristic
56C2×Dic47⋊C4—21no—
57C2×Dic47⋊C4—21no—
58C2×(C2×D47)⋊C4—21no—
59C2×(C2×D47)⋊C4—21no—
60C2×(C2×D47)⋊C4—21no—
61C2×(C2×D47)⋊C4—21no—
62C2×(C2×D47)⋊C4—21no—
63C2×C4×Dic47—22no—
64C2×C4×Dic47D411yescharacteristic
65C2×C4×Dic47D411yescharacteristic
66C2×C4×Dic47D411yescharacteristic
67C2×C4×Dic47D411yescharacteristic
68C2×C4×Dic47C2311yescharacteristic
69C2×C4×Dic47C2311yescharacteristic
70#13093957—42no—
71#13093957C2×C418yes—
72C4⋊(C47⋊C8)C2×C414yes—
73#13093946—42no—
74#13093944C2×C418yes—
75C8×Dic47—42no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C47⋊#95—22no—
79C47⋊#95—22no—
80C22×Q8×C47—21no—
81C22×C8×C47C2312yes—
82C4⋊Q8×C47—22no—
83C4⋊Q8×C47C2×C414yes—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47C2311yescharacteristic
87C2×C4⋊C4×C47C2312yes—
88C2×C42×C47—21no—
89C2×C42×C47C2311yescharacteristic
90C4⋊C8×C47—22no—
91#95×C47—21no—
92C23×D47—21no—
93C2×Q8×C47—24no—
94C2×Q8×C47—42no—
95C2×Q8×C47—42no—
96C22×C4×C47—22no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47C2411yescharacteristic
102C22×Dic47—21no—
103C22×Dic47—22no—
104C22×Dic47C2×D411yescharacteristic
105C22×Dic47C2×D411yescharacteristic
106C22×Dic47C2×D411yescharacteristic
107C22×Dic47C2×D411yescharacteristic
108C2×D188—21no—
109C2×D188—21no—
110C2×D188—21no—
111C2×D188—21no—
112C2×D188—21no—
113C2×D188C22×C412yes—
114C2×C4×D47—24no—
115C2×C4×D47—42no—
116C2×C4×D47—42no—
117C2×C8×C47—22no—
118C2×C8×C47—22no—
119C2×C8×C47—22no—
120C4⋊C4×C47—24no—
121C4⋊C4×C47—22no—
122C4⋊C4×C47—21no—
123C4⋊C4×C47—21no—
124C4⋊C4×C47—21no—
125C4⋊C4×C47—21no—
126C4⋊C4×C47C22×C414yes—
127C42×C47—42no—
128C42×C47C22×C412yes—
129C42×C47#4112yes—
130(C2×D47)⋊C4—24no—
131(C2×D47)⋊C4—42no—
132(C2×D47)⋊C4—42no—
133(C2×D47)⋊C4—42no—
134(C2×D47)⋊C4—42no—
135C4⋊Dic47—21no—
136C4⋊Dic47—22no—
137C4⋊Dic47—22no—
138C4⋊Dic47—42no—
139C4⋊Dic47—42no—
140C4⋊Dic47—44no—
141C4⋊Dic47C22×C412yes—
142Dic47⋊C4—42no—
143Dic47⋊C4—42no—
144C4×Dic47—22no—
145C4×Dic47—21no—
146C4×Dic47—21no—
147C4×Dic47—21no—
148C4×Dic47—21no—
149C4×Dic47—22no—
150C4×Dic47—41no—
151C4×Dic47—44no—
152C4×Dic47D814yes—
153C2×C47⋊C8—22no—
154C2×C47⋊C8—22no—
155C2×C47⋊C8C22×C414yes—
156C23×C47C22×D411yescharacteristic
157C22×D47—22no—
158C22×D47—41no—
159C22×D47—41no—
160C22×D47—41no—
161C22×D47—41no—
162C22×D47—41no—
163C22×D47—41no—
164Q8×C47—48no—
165C2×C4×C47—21no—
166C2×C4×C47—21no—
167C2×C4×C47—24no—
168C2×C4×C47—42no—
169C2×C4×C47—42no—
170C2×C4×C47—41no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47—41no—
176C2×C4×C47C4×D412yes—
177C2×C4×C47C4×D412yes—
178C2×C4×C47C2×D812yes—
179C2×C4×C47#13512yes—
180C2×C4×C47C23×C411yescharacteristic
181C2×C4×C47C2×#4111yescharacteristic
182C2×C4×C47H(5,𝔽2)11yescharacteristic
183C2×C4×C472−1+411yescharacteristic
184C2×C4×C472−1+411yescharacteristic
185C2×C4×C472−1+411yescharacteristic
186C2×Dic47—21no—
187C2×Dic47—21no—
188C2×Dic47—21no—
189C2×Dic47—21no—
190C2×Dic47—21no—
191C2×Dic47—21no—
192C2×Dic47—22no—
193C2×Dic47—21no—
194C2×Dic47—21no—
195C2×Dic47—21no—
196C2×Dic47—41no—
197C2×Dic47—41no—
198C2×Dic47—42no—
199C2×Dic47—42no—
200C2×Dic47—42no—
201C2×Dic47C2×D812yes—
202C2×Dic47C2×D812yes—
203C2×Dic47#13712yes—
204D188—22no—
205D188—22no—
206D188—41no—
207D188—41no—
208C4×D47—48no—
209C8×C47—42no—
210C47⋊C8—24no—
211#22110—471nomaximal, Sylow
212C22×C47C2×C4×D411yescharacteristic
213C22×C47#53611yescharacteristic
214C22×C47#53711yescharacteristic
215C22×C47C22×D811yescharacteristic
216C22×C47C2×#13511yescharacteristic
217C22×C47C2×#13711yescharacteristic
218C22×C47#57511yescharacteristic
219C2×D47—42no—
220C2×D47—42no—
221C2×D47—42no—
222C2×D47—81no—
223C2×D47—81no—
224C2×D47—81no—
225C2×D47—81no—
226C4×C47—22no—
227C4×C47—44no—
228C4×C47—81no—
229C4×C47—81no—
230C4×C47C4×D814yes—
231C4×C47#51911yescharacteristic
232C4×C47#52011yescharacteristic
233C4×C47#54111yescharacteristic
234C4×C47#54211yescharacteristic
235Dic47—22no—
236Dic47—22no—
237Dic47—22no—
238Dic47—42no—
239Dic47—41no—
240Dic47—82no—
241C2×#474—471no—
242C4×C4⋊Q8—471no—
243#2293—471no—
244#1988D4711yescharacteristic, complemented
245#1947—471no—
246#1942—472no—
247#1934—472no—
248#1925—472no—
249#1918—472no—
250#1802—471no—
251C4×C4⋊C8—471no—
252C2×C47#232311yescharacteristic
253C2×C47C2×C4×D811yescharacteristic
254C2×C47C4×#13711yescharacteristic
255C2×C47#322611yescharacteristic
256C2×C47#323311yescharacteristic
257C2×C47#324511yescharacteristic
258C2×C47#324811yescharacteristic
259D47—82no—
260C2×C4⋊Q8C2×D4711yescharacteristic
261C2×C4×Q8—941no—
262C2×C4×D4—471no—
263C2×#117—941no—
264#474—478no—
265C2×#107—472no—
266C2×#107—472no—
267C2×C4⋊C8—471no—
268C2×C4⋊C8C2×D4711yescharacteristic
269C2×#102—472no—
270C2×#102—472no—
271C2×C4×C8—472no—
272#391—941no—
273#390—471no—
274#390—471no—
275#389—472no—
276#389—471no—
277#389—471no—
278#389—941no—
279#388—941no—
280#388—941no—
281#384—471no—
282C4×C4⋊C4—472no—
283C4×C4⋊C4—941no—
284C4×C4⋊C4C2×D4711yescharacteristic
285C4×C22⋊C4—941no—
286C43—471no—
287#340—472no—
288#340—472no—
289#340C2×D4712yes—
290#340C2×D4712yes—
291#336—472no—
292C47#2211011yescharacteristic, complemented, Sylow
293C22×Q8—21no—
294C22×D4—471no—
295C23×C4—941no—
296C22×C8—472no—
297C22×C8C22×D4712yes—
298C4⋊Q8—22no—
299C4⋊Q8C4×D4714yes—
300C4×Q8—1884no—
301C4×D4—474no—
302C4×D4—942no—
303#117—1884no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—471no—
306C2×C4⋊C4—472no—
307C2×C4⋊C4—941no—
308C2×C4⋊C4—941no—
309C2×C4⋊C4—942no—
310C2×C4⋊C4—941no—
311C2×C4⋊C4—941no—
312C2×C4⋊C4C22×D4711yescharacteristic
313C2×C4⋊C4C22×D4711yescharacteristic
314C2×C4⋊C4C22×D4712yes—
315C2×C22⋊C4—941no—
316C2×C22⋊C4—941no—
317C2×C22⋊C4—941no—
318C2×C22⋊C4—941no—
319C2×C22⋊C4—941no—
320C2×C42—21no—
321C2×C42—471no—
322C2×C42—471no—
323C2×C42—471no—
324C2×C42—471no—
325C2×C42—471no—
326C2×C42—471no—
327C2×C42—942no—
328C2×C42C22×D4711yescharacteristic
329#107—478no—
330#107—1882no—
331C4⋊C8—22no—
332C4⋊C8—474no—
333#102—478no—
334#102—1882no—
335C4×C8—1882no—
336#95—21no—
337#95—941no—
338#95—941no—
339#95—942no—
340#95—942no—
341C24—941no—
342C2×Q8—24no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×D4—472no—
346C2×D4—941no—
347C2×D4—941no—
348C2×D4—941no—
349C2×D4—941no—
350C2×D4—941no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—471no—
355C22×C4—471no—
356C22×C4—471no—
357C22×C4—471no—
358C22×C4—941no—
359C22×C4—944no—
360C22×C4—942no—
361C22×C4—1882no—
362C22×C4—1882no—
363C22×C4D4×D4711yescharacteristic
364C22×C4D4×D4711yescharacteristic
365C22×C4C23×D4711yescharacteristic
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—474no—
370C2×C8—942no—
371C2×C8—942no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—472no—
379C4⋊C4—941no—
380C4⋊C4—942no—
381C4⋊C4—942no—
382C4⋊C4—1882no—
383C4⋊C4—1882no—
384C4⋊C4—1882no—
385C4⋊C4—1884no—
386C4⋊C4—1882no—
387C4⋊C4C2×C4×D4714yes—
388C22⋊C4—944no—
389C22⋊C4—1882no—
390C22⋊C4—1882no—
391C22⋊C4—1882no—
392C22⋊C4—1882no—
393C42—42no—
394C42—474no—
395C42—942no—
396C42—942no—
397C42—941no—
398C42—941no—
399C42—941no—
400C42—941no—
401C42—1881no—
402C42—1884no—
403C42C2×C4×D4712yes—
404C42#1062535212yes—
405C23—942no—
406C23—1881no—
407C23—1881no—
408C23—1881no—
409C23—1881no—
410C23—1881no—
411C23—1881no—
412C23C2×D4×D4711yescharacteristic
413Q8—48no—
414D4—942no—
415D4—942no—
416D4—1881no—
417D4—1881no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—472no—
430C2×C4—472no—
431C2×C4—472no—
432C2×C4—941no—
433C2×C4—941no—
434C2×C4—941no—
435C2×C4—941no—
436C2×C4—941no—
437C2×C4—941no—
438C2×C4—942no—
439C2×C4—941no—
440C2×C4—941no—
441C2×C4—941no—
442C2×C4—1882no—
443C2×C4—1882no—
444C2×C4—1882no—
445C2×C4—1881no—
446C2×C4—1881no—
447C2×C4—1888no—
448C2×C4(C22⋊D47)⋊C412yes—
449C2×C4(C22⋊D47)⋊C412yes—
450C2×C4D8×D4712yes—
451C2×C4#1309404512yes—
452C2×C4C22×C4×D4711yescharacteristic
453C2×C4C2×#1062535211yescharacteristic
454C2×C4#1309407011yescharacteristic
455C2×C4#1309407111yescharacteristic
456C2×C4#1309407211yescharacteristic
457C2×C4#1309407211yescharacteristic
458C8—42no—
459C8—944no—
460C22—3761no—
461C22—3761no—
462C22—3761no—
463C22—3761no—
464C22—1882no—
465C22—1882no—
466C22—1882no—
467C22C2×(C22⋊D47)⋊C411yescharacteristic
468C22#1568969611yescharacteristic
469C22#1568970311yescharacteristic
470C22C2×D8×D4711yescharacteristic
471C22C2×#1309404511yescharacteristic
472C22#137×D4711yescharacteristic
473C22#1568992211yescharacteristic
474C4—22no—
475C4—44no—
476C4—81no—
477C4—81no—
478C4—942no—
479C4—942no—
480C4—942no—
481C4—3762no—
482C4—1881no—
483C4—1882no—
484C4#1568920314yes—
485C4#1568959911yescharacteristic
486C4#1568960511yescharacteristic
487C4#1568973411yescharacteristic
488C4#1568974211yescharacteristic
489C2—3762no—
490C2#2024607311yescharacteristic
491C2C2×#1568920311yescharacteristic
492C2#2025220311yescharacteristic
493C2#2025450811yescharacteristic
494C2#2025457111yescharacteristic
495C2#2025469711yescharacteristic
496C2#2025472811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1988C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1988D47
hypocenterC47#22110
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×Dic47C2semidirect
4. #20244146C2semidirect
5. #20244120C2semidirect
8. #20243957C2semidirect
10. C4⋊C8×Dic47C2semidirect
11. C47×#1988C2semidirect
15. #15689400C4semidirect
244. #1988D47semidirect
292. C47#22110semidirect

Maximal subgroups

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

subgroupindex
C2×#156894002
C4⋊Q8×Dic472
#202458492
#202441462
#202441202
#202440722
#202440062
#202439572
#202433892
C4⋊C8×Dic472
C47×#19882
#2211047