Finite Group Databasebeta

← #26118623

Subgroups — #26118623

Order 12032.

628 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689410C211yesmaximal, characteristic
2#20248306C211yesmaximal, characteristic
3#20244191C211yesmaximal, characteristic, complemented
4C2×Q8⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
5C2×#15688850C211yesmaximal, characteristic
6#20241272C211yesmaximal, characteristic, complemented
7C2×C47×#477C211yesmaximal, characteristic, complemented
8C2×#13094000—21no—
9C2×C4×D188C2211yescharacteristic
10#15689410—22no—
11#15689410—22no—
12C2×#13093955—21no—
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093948—22no—
15C2×#13093944—21no—
16C2×C4×C47⋊C8C2211yescharacteristic
17#15689045—22no—
18#15689025—21no—
19#15689021—21no—
20#15688993C2211yescharacteristic
21#15688980—21no—
22Q8⋊(C47⋊C8)—22no—
23#15688850—22no—
24C2×C4⋊Q8×C47C2211yescharacteristic
25C2×C4×Q8×C47C2211yescharacteristic
26C47×#477—22no—
27C47×#477—22no—
28C2×#107×C47—21no—
29C2×C4⋊C8×C47C2211yescharacteristic
30C2×#103×C47—21no—
31C22×D188D411yescharacteristic
32C22×C4×D47—21no—
33C22×C47⋊C8—21no—
34C22×C47⋊C8—21no—
35#13094000—42no—
36#13094000—42no—
37C4×D188—21no—
38C4×D188—21no—
39C4×D188—21no—
40C4×D188—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47D411yescharacteristic
44C2×C4⋊Dic47D412yes—
45C2×Dic47⋊C4—22no—
46C2×(C2×D47)⋊C4—22no—
47C2×(C2×D47)⋊C4—21no—
48C2×(C2×D47)⋊C4—22no—
49#13093955—24no—
50C4⋊(C47⋊C8)—21no—
51C4⋊(C47⋊C8)D412yes—
52#13093948—44no—
53#13093944—24no—
54C4×C47⋊C8—21no—
55C4×C47⋊C8—21no—
56C47⋊#95—22no—
57C22×Q8×C47—21no—
58C22×Q8×C47D411yescharacteristic
59C22×C8×C47—21no—
60C4⋊Q8×C47—21no—
61C4⋊Q8×C47—21no—
62C4⋊Q8×C47—21no—
63C4⋊Q8×C47D412yes—
64C4×Q8×C47—21no—
65C4×Q8×C47—21no—
66C4×Q8×C47—21no—
67C4×Q8×C47—21no—
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47D411yescharacteristic
72C2×C42×C47—21no—
73C2×C42×C47C2311yescharacteristic
74#107×C47—24no—
75C4⋊C8×C47—21no—
76C4⋊C8×C47D412yes—
77#103×C47—24no—
78C23×D47—21no—
79C2×Q8×C47—21no—
80C2×Q8×C47—21no—
81C2×Q8×C47—21no—
82C2×Q8×C47—22no—
83C2×Q8×C47—21no—
84C2×Q8×C47—21no—
85C2×Q8×C47—42no—
86C2×Q8×C47—42no—
87C2×Q8×C47—41no—
88C2×Q8×C47SD1612yes—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—41no—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D411yescharacteristic
95C22×Dic47—21no—
96C22×Dic47—22no—
97C2×D188—21no—
98C2×D188—21no—
99C2×D188—21no—
100C2×D188—21no—
101C2×D188—21no—
102C2×D188Q1612yes—
103C2×C4×D47—41no—
104C2×C4×D47—41no—
105C2×C4×D47—41no—
106C2×C4×D47—41no—
107C2×C4×D47—41no—
108C2×C4×D47—41no—
109C2×C8×C47—22no—
110C2×C8×C47—41no—
111C2×C8×C47—41no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—21no—
114C4⋊C4×C47—22no—
115C4⋊C4×C47—21no—
116C4⋊C4×C47—41no—
117C4⋊C4×C47—41no—
118C4⋊C4×C47—41no—
119C4⋊C4×C47—41no—
120C42×C47—41no—
121C42×C47—41no—
122C42×C47C2×D411yescharacteristic
123C42×C47C2×D411yescharacteristic
124C42×C47C2×D411yescharacteristic
125C42×C47#4111yescharacteristic
126(C2×D47)⋊C4—44no—
127(C2×D47)⋊C4—41no—
128(C2×D47)⋊C4—41no—
129(C2×D47)⋊C4—82no—
130C4⋊Dic47—21no—
131C4⋊Dic47—21no—
132C4⋊Dic47—41no—
133C4⋊Dic47—41no—
134C4⋊Dic47—41no—
135C4⋊Dic47—41no—
136C4⋊Dic47—42no—
137Dic47⋊C4—44no—
138C2×C47⋊C8—22no—
139C2×C47⋊C8—22no—
140C2×C47⋊C8—41no—
141C2×C47⋊C8—41no—
142C2×C47⋊C8—41no—
143C2×C47⋊C8—41no—
144C23×C47#12011yescharacteristic
145C22×D47—22no—
146C22×D47—41no—
147C22×D47—41no—
148C22×D47—41no—
149C22×D47—41no—
150C22×D47—41no—
151C22×D47—41no—
152Q8×C47—22no—
153Q8×C47—22no—
154Q8×C47—44no—
155Q8×C47—41no—
156Q8×C47—41no—
157Q8×C47—82no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—22no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—42no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—81no—
168C2×C4×C47—81no—
169C2×C4×C47#12011yescharacteristic
170C2×C4×C47#12011yescharacteristic
171C2×C4×C47#12111yescharacteristic
172C2×C4×C47#12111yescharacteristic
173C2×C4×C47#12111yescharacteristic
174C2×C4×C47#12711yescharacteristic
175C2×C4×C47C2×SD1611yescharacteristic
176C2×C4×C47C2×Q1611yescharacteristic
177C2×C4×C47#13511yescharacteristic
178C2×C4×C47#13511yescharacteristic
179C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C47#13711yescharacteristic
181C2×C4×C47#13711yescharacteristic
182C2×C4×C47#13711yescharacteristic
183C2×Dic47—41no—
184C2×Dic47—41no—
185C2×Dic47—42no—
186C2×Dic47—42no—
187C2×Dic47—41no—
188C2×Dic47—42no—
189D188—22no—
190D188—22no—
191D188—41no—
192D188—41no—
193C4×D47—81no—
194C4×D47—81no—
195C4×D47—81no—
196C4×D47—81no—
197C8×C47—42no—
198C47⋊C8—42no—
199C47⋊C8—42no—
200#10236—471nomaximal, Sylow
201C22×C47#39211yescharacteristic
202C22×C47#45011yescharacteristic
203C22×C47#45111yescharacteristic
204C22×C47#45211yescharacteristic
205C22×C47#45211yescharacteristic
206C22×C47#45311yescharacteristic
207C22×C47#45411yescharacteristic
208C2×D47—42no—
209C2×D47—42no—
210C2×D47—42no—
211C2×D47—81no—
212C2×D47—81no—
213C2×D47—81no—
214C2×D47—81no—
215C4×C47—21no—
216C4×C47—21no—
217C4×C47—42no—
218C4×C47—81no—
219C4×C47—82no—
220C4×C47#44811yescharacteristic
221C4×C47#44911yescharacteristic
222C4×C47#45111yescharacteristic
223C4×C47#45211yescharacteristic
224C4×C47#46511yescharacteristic
225C4×C47#46711yescharacteristic
226C4×C47#46811yescharacteristic
227C4×C47#47011yescharacteristic
228Dic47—82no—
229Dic47—81no—
230C2×#477D4711yescharacteristic, complemented
231C2×#476—471no—
232#2538—471no—
233#1956—471no—
234C2×Q8⋊C8—471no—
235C2×D4⋊C8—471no—
236#1296—471no—
237C2×C47#163111yescharacteristic
238C2×C47#163211yescharacteristic
239C2×C47#163711yescharacteristic
240C2×C47#163811yescharacteristic
241C2×C47#200011yescharacteristic
242C2×C47#200111yescharacteristic
243C2×C47#200611yescharacteristic
244D47—82no—
245C2×C4⋊Q8C2×D4711yescharacteristic
246C2×C22⋊Q8—941no—
247C2×C4×Q8C2×D4711yescharacteristic
248C2×C4×D4—471no—
249#477—22no—
250#477—22no—
251#476—942no—
252#476—942no—
253C2×#107—21no—
254C2×#106—941no—
255C2×C4⋊C8—471no—
256C2×C4⋊C8C2×D4711yescharacteristic
257C2×#103—21no—
258C2×#103—942no—
259C2×#102—941no—
260C2×C4×C8—471no—
261#396—942no—
262#390—941no—
263#389—941no—
264#384—471no—
265#380—941no—
266Q8⋊C8—942no—
267D4⋊C8—942no—
268C47#1023611yescharacteristic, complemented, Sylow
269C22×Q8—21no—
270C22×Q8C22⋊D4711yescharacteristic
271C22×D4—471no—
272C23×C4—941no—
273C22×C8—21no—
274C22×C8—941no—
275C22×C8—941no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8C22⋊D4712yes—
280C22⋊Q8—1882no—
281C22⋊Q8—1882no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×D4—941no—
287C4×D4—941no—
288C4×D4—941no—
289C4×D4—941no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—471no—
294C2×C4⋊C4—472no—
295C2×C4⋊C4—942no—
296C2×C4⋊C4—941no—
297C2×C4⋊C4—941no—
298C2×C4⋊C4C22⋊D4711yescharacteristic
299C2×C22⋊C4—942no—
300C2×C22⋊C4—942no—
301C2×C22⋊C4—941no—
302C2×C42—21no—
303C2×C42C22×D4711yescharacteristic
304#107—24no—
305#106—944no—
306C4⋊C8—21no—
307C4⋊C8—472no—
308C4⋊C8—941no—
309C4⋊C8C22⋊D4712yes—
310#103—24no—
311#103—1884no—
312#102—944no—
313C4×C8—941no—
314C4×C8—941no—
315#95—942no—
316C24—941no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—22no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×Q8#1062533612yes—
327C2×D4—472no—
328C2×D4—941no—
329C2×D4—941no—
330C2×D4—941no—
331C2×D4—941no—
332C2×D4—941no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—41no—
336C22×C4—941no—
337C22×C4—942no—
338C22×C4—1881no—
339C22×C4—1881no—
340C22×C4—1881no—
341C22×C4—1881no—
342C22×C4—1881no—
343C22×C4—1881no—
344C22×C4D4×D4711yescharacteristic
345C22×C4D4×D4711yescharacteristic
346C22×C4C2×C22⋊D4711yescharacteristic
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—942no—
351C2×C8—942no—
352C2×C8—1881no—
353C2×C8—1881no—
354C2×C8—1881no—
355C2×C8—1881no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—941no—
365C4⋊C4—941no—
366C4⋊C4—1882no—
367C4⋊C4—1884no—
368C4⋊C4—1881no—
369C4⋊C4—1881no—
370C4⋊C4—1881no—
371C4⋊C4—1881no—
372C22⋊C4—3762no—
373C22⋊C4—1884no—
374C22⋊C4—1881no—
375C22⋊C4—1881no—
376C42—41no—
377C42—41no—
378C42D4×D4711yescharacteristic
379C42#1062535211yescharacteristic
380C42C2×C22⋊D4711yescharacteristic
381C42C2×C22⋊D4711yescharacteristic
382C23—942no—
383C23—1881no—
384C23—1881no—
385C23—1881no—
386C23—1881no—
387C23—1881no—
388C23—1881no—
389C23#1309399211yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—44no—
393Q8—41no—
394Q8—41no—
395Q8—82no—
396D4—942no—
397D4—942no—
398D4—1881no—
399D4—1881no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—3761no—
412C2×C4—3761no—
413C2×C4—3761no—
414C2×C4—3761no—
415C2×C4—1882no—
416C2×C4—1881no—
417C2×C4—1881no—
418C2×C4—1882no—
419C2×C4—1881no—
420C2×C4—1882no—
421C2×C4#1309399211yescharacteristic
422C2×C4#1309399311yescharacteristic
423C2×C4#1309399511yescharacteristic
424C2×C4#1309399611yescharacteristic
425C2×C4#1309399611yescharacteristic
426C2×C4#1309402011yescharacteristic
427C2×C4C2×#1062533611yescharacteristic
428C2×C4Q16×D4711yescharacteristic
429C2×C4#1309404411yescharacteristic
430C2×C4#1309404711yescharacteristic
431C2×C4#1309404911yescharacteristic
432C2×C4#1309405511yescharacteristic
433C2×C4#1309405611yescharacteristic
434C2×C4#1309405611yescharacteristic
435C8—42no—
436C8—1882no—
437C8—1882no—
438C22—3761no—
439C22—3761no—
440C22—3761no—
441C22—3761no—
442C22—1882no—
443C22—1882no—
444C22—1882no—
445C22#1568903111yescharacteristic
446C22#1568927311yescharacteristic
447C22#1568927711yescharacteristic
448C22#1568928311yescharacteristic
449C22#1568928411yescharacteristic
450C22#1568928911yescharacteristic
451C22#1568929411yescharacteristic
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—82no—
457C4—3762no—
458C4—3761no—
459C4#1568926211yescharacteristic
460C4#1568926811yescharacteristic
461C4#1568927711yescharacteristic
462C4#1568928411yescharacteristic
463C4#1568935411yescharacteristic
464C4#1568936111yescharacteristic
465C4#1568937111yescharacteristic
466C4#1568937811yescharacteristic
467C2—3762no—
468C2#2024251311yescharacteristic
469C2#2024252111yescharacteristic
470C2#2024255111yescharacteristic
471C2#2024255911yescharacteristic
472C2#2024438911yescharacteristic
473C2#2024439511yescharacteristic
474C2#2024442811yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC2×C47×#477C2
socleC23×C47#120
prime radicalC22×D188D4
hypercenterC2×#477D47
hypocenterC47#10236
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #20244191C2semidirect
4. C2×Q8⋊(C47⋊C8)C2semidirect
6. #20241272C2semidirect
7. C2×C47×#477C2semidirect
230. C2×#477D47semidirect
268. C47#10236semidirect

Maximal subgroups

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

subgroupindex
C2×#156894102
#202483062
#202441912
C2×Q8⋊(C47⋊C8)2
C2×#156888502
#202412722
C2×C47×#4772
#1023647