Finite Group Databasebeta

← #26110856

Subgroups — #26110856

Order 12032.

672 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625334C211yesmaximal, characteristic
2C2×C8×D188C211yesmaximal, characteristic, complemented
3C2×D4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
4C2×#15688850C211yesmaximal, characteristic, complemented
5#20241222C211yesmaximal, characteristic, complemented
6C47⋊#1273C211yesmaximal, characteristic, complemented
7C47×#1954C211yesmaximal, characteristic, complemented
8C22×#10625334C411yescharacteristic
9C22×C8×D47—21no—
10C2×C4×D188C2211yescharacteristic
11C4×#10625334—22no—
12C4×#10625334—22no—
13C8×D188—22no—
14C8×D188—22no—
15C2×#13093957C411yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×#13093945C411yescharacteristic
18C2×#13093944C411yescharacteristic
19C2×(C2×D47)⋊C8—21no—
20C2×C4×C47⋊C8C2211yescharacteristic
21C2×C4×C47⋊C8C2211yescharacteristic
22D4⋊(C47⋊C8)—22no—
23#15688850—22no—
24C2×C4×D4×C47C2211yescharacteristic
25C2×C4⋊C8×C47C2211yescharacteristic
26C2×C22⋊C8×C47—21no—
27C2×C22⋊C8×C47—21no—
28C2×C4×C8×C47C2211yescharacteristic
29C22×D188C2×C411yescharacteristic
30C22×C4×D47—21no—
31C2×#10625334—22no—
32C2×#10625334—22no—
33C2×#10625334C814yescomplemented
34C22×C47⋊C8—21no—
35C22×C47⋊C8—21no—
36C22×C47⋊C8C2×C411yescharacteristic
37C22×C47⋊C8C2×C411yescharacteristic
38C2×C8×D47—24no—
39C2×C8×D47—42no—
40C2×C8×D47—42no—
41C4×D188—21no—
42C4×D188—21no—
43C4×D188D412yes—
44C4×D188D412yes—
45C2×C4⋊Dic47C2×C411yescharacteristic
46C2×(C2×D47)⋊C4—21no—
47#13093957—22no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)D412yes—
50#13093945—22no—
51#13093944—22no—
52(C2×D47)⋊C8—24no—
53C4×C47⋊C8—21no—
54C4×C47⋊C8—21no—
55C4×C47⋊C8—21no—
56C4×C47⋊C8—21no—
57C22×D4×C47C2×C411yescharacteristic
58C23×C4×C47—21no—
59C22×C8×C47—21no—
60C22×C8×C47D411yescharacteristic
61C22×C8×C47D411yescharacteristic
62C4×D4×C47—21no—
63C4×D4×C47—21no—
64C4×D4×C47—21no—
65C4×D4×C47—21no—
66C2×C4⋊C4×C47C2×C411yescharacteristic
67C2×C22⋊C4×C47—21no—
68C2×C42×C47C2311yescharacteristic
69C4⋊C8×C47—21no—
70C4⋊C8×C47—21no—
71C22⋊C8×C47—42no—
72C22⋊C8×C47—42no—
73C4×C8×C47—21no—
74C4×C8×C47—21no—
75C24×C47—21no—
76C23×D47—21no—
77C2×D4×C47—21no—
78C2×D4×C47—21no—
79C2×D4×C47—21no—
80C2×D4×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47C2×C812yescomplemented
83C22×C4×C47—21no—
84C22×C4×C47—41no—
85C22×C4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47—41no—
89C22×C4×C47—41no—
90C22×C4×C47C22×C411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47#4111yescharacteristic
93C22×Dic47—21no—
94C2×D188—21no—
95C2×D188—21no—
96C2×D188—21no—
97C2×D188C22⋊C412yes—
98C2×D188C2×C812yes—
99C2×D188M1612yes—
100C2×C4×D47—22no—
101C2×C4×D47—22no—
102C2×C4×D47—41no—
103C2×C4×D47—41no—
104C2×C4×D47—41no—
105C2×C4×D47—41no—
106C2×C8×C47—21no—
107C2×C8×C47—21no—
108C2×C8×C47—21no—
109C2×C8×C47—21no—
110C2×C8×C47—21no—
111C2×C8×C47—41no—
112C2×C8×C47—41no—
113C2×C8×C47—41no—
114C2×C8×C47D812yescomplemented
115C4⋊C4×C47—21no—
116C4⋊C4×C47—21no—
117C22⋊C4×C47—41no—
118C22⋊C4×C47—41no—
119C42×C47C2×D411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47#4111yescharacteristic
123#10625334—28no—
124(C2×D47)⋊C4—22no—
125(C2×D47)⋊C4—22no—
126C4⋊Dic47—21no—
127C4⋊Dic47C22⋊C412yes—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—21no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8—21no—
134C2×C47⋊C8—21no—
135C2×C47⋊C8—41no—
136C2×C47⋊C8—41no—
137C2×C47⋊C8—41no—
138C2×C47⋊C8—41no—
139C2×C47⋊C8C2×C812yescomplemented
140C8×D47—48no—
141C23×C47—22no—
142C23×C47—41no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47—41no—
146C23×C47—41no—
147C23×C47—41no—
148C23×C47C4×D411yescharacteristic
149C22×D47—22no—
150C22×D47—22no—
151C22×D47—22no—
152C22×D47—41no—
153C22×D47—41no—
154C22×D47—41no—
155C22×D47—41no—
156D4×C47—22no—
157D4×C47—22no—
158D4×C47—41no—
159D4×C47—41no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—81no—
166C2×C4×C47—81no—
167C2×C4×C47—81no—
168C2×C4×C47—81no—
169C2×C4×C47C2×C22⋊C411yescharacteristic
170C2×C4×C47C4×D411yescharacteristic
171C2×C4×C47#12011yescharacteristic
172C2×C4×C47#12111yescharacteristic
173C2×C4×C47#12111yescharacteristic
174C2×C4×C47#12311yescharacteristic
175C2×C4×C47C22×C811yescharacteristic
176C2×C4×C47C2×M1611yescharacteristic
177C2×C4×C47#13111yescharacteristic
178C2×C4×C47#13111yescharacteristic
179C2×C4×C47C2×D811yescharacteristic
180C2×C4×C47#13511yescharacteristic
181C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
183C2×Dic47—22no—
184C2×Dic47—41no—
185C2×Dic47—41no—
186D188—22no—
187D188—22no—
188D188—22no—
189D188C22⋊C814yescomplemented
190C4×D47—42no—
191C4×D47—42no—
192C4×D47—42no—
193C4×D47—42no—
194C8×C47—22no—
195C8×C47—41no—
196C8×C47—81no—
197C47⋊C8—22no—
198C47⋊C8—42no—
199C47⋊C8—41no—
200C47⋊C8—42no—
201#8947—471nomaximal, Sylow
202C22×C47—42no—
203C22×C47—42no—
204C22×C47—42no—
205C22×C47—81no—
206C22×C47—81no—
207C22×C47—81no—
208C22×C47—81no—
209C22×C47#38611yescharacteristic
210C22×C47C8×D411yescharacteristic
211C22×C47#43511yescharacteristic
212C22×C47C4×D811yescharacteristic
213C22×C47#44211yescharacteristic
214C22×C47#44311yescharacteristic
215C22×C47#44411yescharacteristic
216C2×D47—24no—
217C2×D47—42no—
218C2×D47—42no—
219C2×D47—42no—
220C2×D47—42no—
221C2×D47—42no—
222C2×D47—42no—
223C4×C47—21no—
224C4×C47—21no—
225C4×C47—81no—
226C4×C47C2×C22⋊C811yescharacteristic
227C4×C47#40811yescharacteristic
228C4×C47C8×D411yescharacteristic
229C4×C47#43511yescharacteristic
230C4×C47#44711yescharacteristic
231C4×C47#44911yescharacteristic
232C4×C47#45911yescharacteristic
233C4×C47#46311yescharacteristic
234Dic47—42no—
235C2×C4×D8—471no—
236C2×C8×D4—471no—
237#1954D4711yescharacteristic, complemented
238C2×D4⋊C8—471no—
239C2×D4⋊C8—471no—
240#1277—471no—
241#1273—471no—
242C2×C47—82no—
243C2×C47C8×D811yescharacteristic
244C2×C47#158011yescharacteristic
245C2×C47#158111yescharacteristic
246C2×C47#158611yescharacteristic
247C2×C47#186911yescharacteristic
248C2×C47#187611yescharacteristic
249C2×C47#187911yescharacteristic
250D47—44no—
251C22×D8—471no—
252C23×C8—941no—
253C2×C4×D4—471no—
254C2×C4×D4C2×D4711yescharacteristic
255C4×D8—942no—
256C4×D8—942no—
257C8×D4—942no—
258C8×D4—942no—
259C2×#107—471no—
260C2×C4⋊C8—471no—
261C2×C4⋊C8C2×D4711yescharacteristic
262C2×#102—471no—
263C2×#102—471no—
264C2×C22⋊C8—21no—
265C2×C22⋊C8—21no—
266C2×C22⋊C8—941no—
267C2×C4×C8—471no—
268C2×C4×C8—471no—
269C2×C4×C8C2×D4711yescharacteristic
270D4⋊C8—942no—
271D4⋊C8—942no—
272C47#894711yescharacteristic, complemented, Sylow
273C22×D4—471no—
274C22×D4C4×D4711yescharacteristic
275C23×C4—21no—
276C23×C4—941no—
277C2×D8—474no—
278C2×D8—942no—
279C2×D8—942no—
280C22×C8—21no—
281C22×C8—471no—
282C22×C8—471no—
283C22×C8—944no—
284C22×C8—941no—
285C22×C8—941no—
286C22×C8—1882no—
287C22×C8—1882no—
288C22×C8C22⋊D4711yescharacteristic
289C22×C8C22⋊D4711yescharacteristic
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—472no—
295C4×D4—472no—
296C4×D4—941no—
297C4×D4—941no—
298C2×C4⋊C4—471no—
299C2×C4⋊C4C4×D4711yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—941no—
302C2×C42C22×D4711yescharacteristic
303#107—942no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—472no—
307C4⋊C8—941no—
308#102—942no—
309#102—942no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312C22⋊C8—944no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—941no—
316C4×C8—941no—
317C4×C8—941no—
318C4×C8—941no—
319C24—21no—
320C24—941no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—472no—
327C2×D4—472no—
328C2×D4—472no—
329C2×D4—941no—
330C2×D4—941no—
331C2×D4—941no—
332C2×D4C8×D4712yescomplemented
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—941no—
341C22×C4—942no—
342C22×C4—942no—
343C22×C4—1881no—
344C22×C4—1881no—
345C22×C4—1881no—
346C22×C4—1881no—
347C22×C4C2×C4×D4711yescharacteristic
348C22×C4#1062535011yescharacteristic
349C22×C4C2×C22⋊D4711yescharacteristic
350D8—948no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—472no—
360C2×C8—941no—
361C2×C8—941no—
362C2×C8—942no—
363C2×C8—941no—
364C2×C8—942no—
365C2×C8—941no—
366C2×C8—941no—
367C2×C8—1888no—
368C2×C8—1881no—
369C2×C8—1881no—
370C2×C8—1881no—
371C2×C8—1881no—
372C2×C8#1062533412yescomplemented
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—472no—
376C4⋊C4—941no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—942no—
380C22⋊C4—942no—
381C42D4×D4711yescharacteristic
382C42D4×D4711yescharacteristic
383C42D4×D4711yescharacteristic
384C42#1062535211yescharacteristic
385C23—22no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—942no—
393C23—942no—
394C23—942no—
395C23—1881no—
396C23—1881no—
397C23—1881no—
398C23—1881no—
399C23C4×C22⋊D4711yescharacteristic
400D4—22no—
401D4—22no—
402D4—41no—
403D4—41no—
404D4—474no—
405D4—942no—
406D4—942no—
407D4—942no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—942no—
418C2×C4—1881no—
419C2×C4—1881no—
420C2×C4—1882no—
421C2×C4—1882no—
422C2×C4—1882no—
423C2×C4—1882no—
424C2×C4C22⋊C4×D4711yescharacteristic
425C2×C4(C22⋊D47)⋊C411yescharacteristic
426C2×C4#1309399211yescharacteristic
427C2×C4#1309399511yescharacteristic
428C2×C4#1309399711yescharacteristic
429C2×C4#1309400611yescharacteristic
430C2×C4C2×C8×D4711yescharacteristic
431C2×C4M16×D4711yescharacteristic
432C2×C4#1309403011yescharacteristic
433C2×C4#1309403211yescharacteristic
434C2×C4C2×#1062533411yescharacteristic
435C2×C4#1309404411yescharacteristic
436C2×C4#1309404911yescharacteristic
437C2×C4#1309405011yescharacteristic
438C8—22no—
439C8—41no—
440C8—81no—
441C8—942no—
442C8—1882no—
443C8—1882no—
444C8—1881no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—944no—
453C22—1882no—
454C22—1882no—
455C22—1882no—
456C22—1882no—
457C22—1882no—
458C22—1882no—
459C22#1568899911yescharacteristic
460C22C8×C22⋊D4711yescharacteristic
461C22#1568919211yescharacteristic
462C22C4×#1062533411yescharacteristic
463C22#1568923311yescharacteristic
464C22#1568923811yescharacteristic
465C22#1568924311yescharacteristic
466C4—21no—
467C4—21no—
468C4—81no—
469C4—1882no—
470C4C22⋊C8×D4711yescharacteristic
471C4#1568908811yescharacteristic
472C4(C22⋊D47)⋊C811yescharacteristic
473C4#1568919711yescharacteristic
474C4#1568925611yescharacteristic
475C4#1568926611yescharacteristic
476C4#1568931511yescharacteristic
477C4#1568933511yescharacteristic
478C2—82no—
479C2—1884no—
480C2C8×#1062533411yescharacteristic
481C2#2024221011yescharacteristic
482C2#2024221511yescharacteristic
483C2#2024225011yescharacteristic
484C2#2024367711yescharacteristic
485C2#2024371811yescharacteristic
486C2#2024373911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1954C2
socleC23×C47C4×D4
prime radicalC22×#10625334C4
cosocleC2×C42×C47C23
hypercenter#1954D47
hypocenterC47#8947
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1954

Complements

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

NH ≅ G/Ntype
2. C2×C8×D188C2semidirect
3. C2×D4⋊(C47⋊C8)C2semidirect
4. C2×#15688850C2semidirect
5. #20241222C2semidirect
6. C47⋊#1273C2semidirect
7. C47×#1954C2semidirect
33. C2×#10625334C8semidirect
82. C2×D4×C47C2×C8semidirect
114. C2×C8×C47D8semidirect
139. C2×C47⋊C8C2×C8semidirect
189. D188C22⋊C8semidirect
237. #1954D47semidirect
272. C47#8947semidirect
332. C2×D4C8×D47semidirect
372. C2×C8#10625334semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253342
C2×C8×D1882
C2×D4⋊(C47⋊C8)2
C2×#156888502
#202412222
C47⋊#12732
C47×#19542
#894747