Finite Group Databasebeta

← #26246266

Subgroups — #26246266

Order 12032.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13093998C211yesmaximal, characteristic
2#20244945C211yesmaximal, characteristic, complemented
3#20244917C211yesmaximal, characteristic, complemented
4#20244860C211yesmaximal, characteristic, complemented
5#20244814C212yesmaximal, complemented
6#20244792C211yesmaximal, characteristic, complemented
7#20244570C212yesmaximal
8C4×#13093957C212yesmaximal, complemented
9C4×#13093944C212yesmaximal, complemented
10C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
11C47×#2285C211yesmaximal, characteristic, complemented
12C2×#13093998C2211yescharacteristic
13C2×C4×D188C2211yescharacteristic
14C2×C4×C22⋊D47—21no—
15C2×#13093957C2212yes—
16C2×#13093957C2212yes—
17C2×#13093944C2212yes—
18C2×#13093944C2212yes—
19C2×C22⋊(C47⋊C8)C2212yes—
20C2×C4×C47⋊C8C2212yes—
21#15689025C2211yescharacteristic
22#15689021C2211yescharacteristic
23#15689014—21no—
24#15689002—21no—
25#15688983C2211yescharacteristic
26C4×C4⋊Dic47C2211yescharacteristic
27C4×(C2×D47)⋊C4—21no—
28#15688885C2212yes—
29#15688885C2212yes—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688878C2211yescharacteristic
33#15688878C2211yescharacteristic
34C2×#117×C47—21no—
35C22×C42×C47C2211yescharacteristic
36C47×#384C2211yescharacteristic
37C47×#384C2211yescharacteristic
38C47×#382—21no—
39C47×#380—21no—
40C47×#380C2212yes—
41C4×C4⋊C4×C47—21no—
42C4×C4⋊C4×C47C2212yes—
43C22×D188C2311yescharacteristic
44C22×C22⋊D47—21no—
45C22×C4×D47—21no—
46C22×C47⋊C8C2312yes—
47C22×C47⋊C8C2312yes—
48#13093998—22no—
49#13093998—22no—
50C4×D188—24no—
51C4×C22⋊D47—44no—
52C2×C4⋊Dic47C2311yescharacteristic
53C2×C4⋊Dic47C2311yescharacteristic
54C2×C4⋊Dic47C2311yescharacteristic
55C2×Dic47⋊C4—21no—
56C2×C22⋊Dic47—21no—
57C2×(C2×D47)⋊C4—21no—
58C2×(C2×D47)⋊C4—21no—
59C2×(C2×D47)⋊C4—21no—
60C2×(C2×D47)⋊C4—21no—
61C2×C4×Dic47—21no—
62#13093957—24no—
63#13093957—24no—
64#13093944—24no—
65#13093944—24no—
66C22⋊(C47⋊C8)—24no—
67C4×C47⋊C8—24no—
68C47⋊#95—21no—
69C47⋊#95—21no—
70C47⋊#95—21no—
71C23×C4×C47D411yescharacteristic
72C23×C4×C47D411yescharacteristic
73C23×C4×C47C2311yescharacteristic
74#117×C47—44no—
75C2×C4⋊C4×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—21no—
78C2×C4⋊C4×C47—21no—
79C2×C4⋊C4×C47C2312yes—
80C2×C4⋊C4×C47C2312yes—
81C2×C22⋊C4×C47—22no—
82C2×C42×C47—24no—
83C2×C42×C47—22no—
84C2×C42×C47D411yescharacteristic
85C2×C42×C47D411yescharacteristic
86C2×C42×C47C2311yescharacteristic
87C2×C42×C47C2311yescharacteristic
88#95×C47—22no—
89#95×C47—22no—
90C24×C47C2×D411yescharacteristic
91C23×D47—21no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—41no—
103C22×C4×C47D814yes—
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2411yescharacteristic
110C2×C22⋊D47—24no—
111C2×C22⋊D47—42no—
112C2×C22⋊D47—42no—
113C22×Dic47—21no—
114C22×Dic47—21no—
115C22×Dic47—21no—
116C2×D188—22no—
117C2×D188—22no—
118C2×D188—21no—
119C2×D188#4112yes—
120C2×C4×D47—42no—
121C2×C4×D47—42no—
122C2×C4×D47—42no—
123C4⋊C4×C47—22no—
124C4⋊C4×C47—22no—
125C4⋊C4×C47—22no—
126C4⋊C4×C47—22no—
127C4⋊C4×C47—42no—
128C4⋊C4×C47—42no—
129C4⋊C4×C47—42no—
130C4⋊C4×C47—42no—
131C22⋊C4×C47—44no—
132C42×C47—22no—
133C42×C47—22no—
134C42×C47—22no—
135C42×C47—44no—
136C42×C47#4114yes—
137C22⋊Dic47—42no—
138(C2×D47)⋊C4—24no—
139(C2×D47)⋊C4—42no—
140(C2×D47)⋊C4—42no—
141(C2×D47)⋊C4—42no—
142C4⋊Dic47—21no—
143C4⋊Dic47—22no—
144C4⋊Dic47—22no—
145C4⋊Dic47#4112yes—
146Dic47⋊C4—42no—
147C4×Dic47—42no—
148C2×C47⋊C8—22no—
149C2×C47⋊C8—22no—
150C2×C47⋊C8—22no—
151C2×C47⋊C8—22no—
152C2×C47⋊C8—22no—
153C2×C47⋊C8#4114yes—
154C23×C47—21no—
155C23×C47—21no—
156C23×C47—21no—
157C23×C47—21no—
158C23×C47—21no—
159C23×C47—21no—
160C23×C47C2×D812yes—
161C23×C47C22×D411yescharacteristic
162C22×D47—22no—
163C22×D47—41no—
164C22×D47—41no—
165C22×D47—41no—
166C22×D47—41no—
167C22×D47—41no—
168C22×D47—41no—
169C2×C4×C47—21no—
170C2×C4×C47—21no—
171C2×C4×C47—21no—
172C2×C4×C47—21no—
173C2×C4×C47—21no—
174C2×C4×C47—21no—
175C2×C4×C47—28no—
176C2×C4×C47—21no—
177C2×C4×C47—21no—
178C2×C4×C47—21no—
179C2×C4×C47—44no—
180C2×C4×C47—42no—
181C2×C4×C47—42no—
182C2×C4×C47—42no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47—42no—
186C2×C4×C47—42no—
187C2×C4×C47—42no—
188C2×C4×C47#12312yes—
189C2×C4×C47#12312yes—
190C2×C4×C47C2×D812yes—
191C2×C4×C47C2×D812yes—
192C2×C4×C47#13512yes—
193C2×C4×C47#13712yes—
194C2×C4×C47C2×#4111yescharacteristic
195C2×C4×C47C2×#4111yescharacteristic
196C2×C4×C47C2×#4112yes—
197C2×C4×C47C2×#4112yes—
198C22⋊D47—48no—
199C2×Dic47—22no—
200C2×Dic47—41no—
201C2×Dic47—41no—
202C2×Dic47—41no—
203C2×Dic47—41no—
204C2×Dic47—41no—
205C2×Dic47—41no—
206C2×Dic47—41no—
207C2×Dic47—41no—
208D188—24no—
209D188—42no—
210C4×D47—84no—
211C47⋊C8—24no—
212C47⋊C8—42no—
213#23754—471nomaximal, Sylow
214C22×C47—22no—
215C22×C47—22no—
216C22×C47—22no—
217C22×C47—41no—
218C22×C47—41no—
219C22×C47—41no—
220C22×C47—41no—
221C22×C47C2×#12311yescharacteristic
222C22×C47#52511yescharacteristic
223C22×C47#53211yescharacteristic
224C22×C47C22×D811yescharacteristic
225C22×C47C2×#13511yescharacteristic
226C22×C47C2×#13711yescharacteristic
227C22×C47#57511yescharacteristic
228C2×D47—42no—
229C2×D47—42no—
230C2×D47—42no—
231C2×D47—81no—
232C2×D47—81no—
233C2×D47—81no—
234C2×D47—81no—
235C4×C47—22no—
236C4×C47—22no—
237C4×C47—22no—
238C4×C47—41no—
239C4×C47—82no—
240C4×C47—82no—
241C4×C47#48014yes—
242C4×C47#52912yes—
243C4×C47#52912yes—
244Dic47—42no—
245Dic47—81no—
246Dic47—81no—
247C4×#121—471no—
248#2285D4711yescharacteristic, complemented
249#2099—471no—
250#2093—471no—
251#2083—471no—
252#2075—472no—
253#2071—471no—
254#2033—472no—
255C4×#107—472no—
256C4×#102—472no—
257C4×C22⋊C8—471no—
258C2×C47—42no—
259C2×C47#239511yescharacteristic
260C2×C47C2×#48011yescharacteristic
261C2×C47#309111yescharacteristic
262C2×C47#310011yescharacteristic
263C2×C47#310711yescharacteristic
264C2×C47#316911yescharacteristic
265C2×C47#317211yescharacteristic
266D47—82no—
267C2×#121—471no—
268C2×C4×D4—471no—
269C2×C4×D4—941no—
270C2×#117—21no—
271C22×C42C2×D4711yescharacteristic
272C2×#107—472no—
273C2×#107—472no—
274C2×#102—472no—
275C2×#102—472no—
276C2×C22⋊C8—472no—
277C2×C4×C8—472no—
278#390—471no—
279#389—471no—
280#388—941no—
281#386—941no—
282#384C2×D4711yescharacteristic
283#384C2×D4711yescharacteristic
284#382—21no—
285#380—21no—
286#380—471no—
287#380C2×D4712yes—
288C4×C4⋊C4—21no—
289C4×C4⋊C4—471no—
290C4×C4⋊C4C2×D4712yes—
291C4×C22⋊C4—941no—
292#340—472no—
293#340—472no—
294#340—472no—
295#340—472no—
296#336—471no—
297#336—471no—
298C47#2375411yescharacteristic, complemented, Sylow
299C22×D4—471no—
300C22×D4—941no—
301C23×C4—941no—
302C23×C4C22⋊D4711yescharacteristic
303C23×C4C22⋊D4711yescharacteristic
304C23×C4C22×D4711yescharacteristic
305C22×C8—472no—
306C22×C8—472no—
307#121—942no—
308#121—942no—
309C4×D4—944no—
310C4×D4—1884no—
311#117—44no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—471no—
317C2×C4⋊C4—471no—
318C2×C4⋊C4—471no—
319C2×C4⋊C4—941no—
320C2×C4⋊C4C22×D4712yes—
321C2×C4⋊C4C22×D4712yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—941no—
324C2×C22⋊C4—941no—
325C2×C22⋊C4—941no—
326C2×C22⋊C4—941no—
327C2×C22⋊C4—941no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—941no—
331C2×C42C22⋊D4711yescharacteristic
332C2×C42C22⋊D4711yescharacteristic
333C2×C42C22×D4711yescharacteristic
334C2×C42C22×D4711yescharacteristic
335#107—944no—
336#107—944no—
337#102—944no—
338#102—944no—
339C22⋊C8—944no—
340C4×C8—944no—
341#95—22no—
342#95—22no—
343#95—941no—
344#95—941no—
345#95—941no—
346C24—941no—
347C24C2×C22⋊D4711yescharacteristic
348C2×D4—472no—
349C2×D4—944no—
350C2×D4—942no—
351C2×D4—941no—
352C2×D4—942no—
353C2×D4—1882no—
354C2×D4—1882no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—41no—
366C22×C4—941no—
367C22×C4—941no—
368C22×C4—941no—
369C22×C4—1882no—
370C22×C4—1882no—
371C22×C4—1882no—
372C22×C4#1062533414yes—
373C22×C4C2×C22⋊D4711yescharacteristic
374C22×C4C2×C22⋊D4711yescharacteristic
375C22×C4C2×C22⋊D4711yescharacteristic
376C22×C4C2×C22⋊D4711yescharacteristic
377C22×C4C2×C22⋊D4711yescharacteristic
378C22×C4C23×D4711yescharacteristic
379C2×C8—474no—
380C2×C8—942no—
381C2×C8—942no—
382C2×C8—942no—
383C2×C8—942no—
384C2×C8—942no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—472no—
394C4⋊C4—941no—
395C4⋊C4—942no—
396C4⋊C4—942no—
397C4⋊C4—1882no—
398C22⋊C4—44no—
399C22⋊C4—944no—
400C22⋊C4—1882no—
401C22⋊C4—1882no—
402C22⋊C4—1882no—
403C22⋊C4—1882no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—44no—
408C42—1882no—
409C42#1062535014yes—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—942no—
417C23—1881no—
418C23—1881no—
419C23—1881no—
420C23—1881no—
421C23—1881no—
422C23—1881no—
423C23C2×#1062533412yes—
424C23C22×C22⋊D4711yescharacteristic
425D4—944no—
426D4—1888no—
427D4—1882no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—28no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—942no—
448C2×C4—3764no—
449C2×C4—1881no—
450C2×C4—1881no—
451C2×C4—1881no—
452C2×C4—1881no—
453C2×C4—1881no—
454C2×C4—1881no—
455C2×C4—1881no—
456C2×C4—1881no—
457C2×C4#1309400712yes—
458C2×C4#1309400712yes—
459C2×C4C2×#1062533412yes—
460C2×C4C2×#1062533412yes—
461C2×C4#1309404412yes—
462C2×C4#1309405512yes—
463C2×C4#41×D4711yescharacteristic
464C2×C4#41×D4711yescharacteristic
465C2×C4#41×D4712yes—
466C2×C4C2×#1062535012yes—
467C8—944no—
468C8—1882no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—3761no—
477C22—3761no—
478C22—3761no—
479C22—3761no—
480C22—1882no—
481C22—1882no—
482C22—1882no—
483C22C2×#1309400711yescharacteristic
484C22#1568963011yescharacteristic
485C22#1568966911yescharacteristic
486C22C22×#1062533411yescharacteristic
487C22C2×#1309404411yescharacteristic
488C22C2×#1309405511yescharacteristic
489C22#1568992111yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—41no—
494C4—82no—
495C4—82no—
496C4—3761no—
497C4—3761no—
498C4—1882no—
499C4#1568943614yes—
500C4#1568965112yes—
501C4#1568965112yes—
502C2—42no—
503C2—3762no—
504C2#2024664511yescharacteristic
505C2C2×#1568943611yescharacteristic
506C2#2025330111yescharacteristic
507C2#2025338411yescharacteristic
508C2#2025345311yescharacteristic
509C2#2025396311yescharacteristic
510C2#2025398811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2285C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenter#2285D47
hypocenterC47#23754
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #20244945C2semidirect
3. #20244917C2semidirect
4. #20244860C2semidirect
5. #20244814C2semidirect
6. #20244792C2semidirect
8. C4×#13093957C2semidirect
9. C4×#13093944C2semidirect
10. C4×C22⋊(C47⋊C8)C2semidirect
11. C47×#2285C2semidirect
248. #2285D47semidirect
298. C47#23754semidirect

Maximal subgroups

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

subgroupindex
C4×#130939982
#202449452
#202449172
#202448602
#202448142
#202447922
#202445702
C4×#130939572
C4×#130939442
C4×C22⋊(C47⋊C8)2
C47×#22852
#2375447