Finite Group Databasebeta

← #26186134

Subgroups — #26186134

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252506C211yesmaximal, characteristic, complemented
2#20252505C211yesmaximal, characteristic, complemented
3C4×#13094044C211yesmaximal, characteristic
4#20242299C212yesmaximal, complemented
5#20242294C212yesmaximal, complemented
6#20242281C212yesmaximal, complemented
7#20242275C212yesmaximal, complemented
8C2×#15688873C211yesmaximal, characteristic, complemented
9#20241776C211yesmaximal, characteristic, complemented
10#20241672C211yesmaximal, characteristic, complemented
11C47×#1477C211yesmaximal, characteristic, complemented
12C2×#13094044C411yescharacteristic
13C4×#10625350C2211yescharacteristic
14#15689254C2212yes—
15#15689250C2212yes—
16C4×#10625337C411yescharacteristic
17C4×#10625337C2211yescharacteristic
18C4×#10625336C411yescharacteristic
19C4×#10625336C2211yescharacteristic
20C4×#10625335C411yescharacteristic
21C4×#10625335C2211yescharacteristic
22C4×#10625334C411yescharacteristic
23C4×#10625334C2211yescharacteristic
24#15689201C2212yes—
25#15689200C2212yes—
26#15689195—22no—
27#15689194—22no—
28#15689180—21no—
29#15689179—21no—
30#15689176C2211yescharacteristic
31#15689159C411yescharacteristic
32#15689147—21no—
33#15689145—21no—
34C2×C4⋊(C47⋊C8)C2211yescharacteristic
35#15689119C411yescharacteristic
36#15689117C411yescharacteristic
37C2×C4×C47⋊C8C2211yescharacteristic
38#15688873C2212yes—
39#15688873C2212yes—
40#15688855C2212yes—
41#15688852C2212yes—
42#15688845C2212yes—
43C8×C47⋊C8C2212yes—
44C4×#41×C47C2211yescharacteristic
45C47×#431C2211yescharacteristic
46C2×C4⋊C8×C47C2211yescharacteristic
47C47×Q8⋊C8C2212yes—
48C47×D4⋊C8C2212yes—
49C2×#10625350C2×C411yescharacteristic
50#13094044—24no—
51C2×#10625337C2×C411yescharacteristic
52C2×#10625335C2×C411yescharacteristic
53C2×#10625336C2×C411yescharacteristic
54C2×#10625334C2×C411yescharacteristic
55C2×#10625329—22no—
56C2×#10625324—22no—
57C22×C47⋊C8—21no—
58C22×C47⋊C8C2×C411yescharacteristic
59C22×C47⋊C8C2×C411yescharacteristic
60C4×Dic94C2×C411yescharacteristic
61C4×Dic94C2311yescharacteristic
62C4×D188C2×C411yescharacteristic
63C4×D188C2311yescharacteristic
64C4×C22⋊D47—21no—
65C4×C22⋊D47—21no—
66#13093983C2×C411yescharacteristic
67#13093981—21no—
68C42×D47—21no—
69#13093957C2×C411yescharacteristic
70#13093957C2×C411yescharacteristic
71#13093955C2×C411yescharacteristic
72#13093955C2×C411yescharacteristic
73C4⋊(C47⋊C8)—21no—
74C4⋊(C47⋊C8)—22no—
75C4⋊(C47⋊C8)D411yescharacteristic
76C4⋊(C47⋊C8)D411yescharacteristic
77C4⋊(C47⋊C8)C2312yes—
78Dic47⋊C8—22no—
79#13093948C2×C411yescharacteristic
80#13093948C2×C411yescharacteristic
81#13093947C2×C411yescharacteristic
82#13093947C2×C411yescharacteristic
83#13093945C2×C411yescharacteristic
84#13093945C2×C411yescharacteristic
85#13093944C2×C411yescharacteristic
86#13093944C2×C411yescharacteristic
87C22⋊(C47⋊C8)—22no—
88C22⋊(C47⋊C8)D412yes—
89(C2×D47)⋊C8—22no—
90#13093935—22no—
91#13093934—22no—
92C4×C47⋊C8C2×C411yescharacteristic
93C4×C47⋊C8C2×C411yescharacteristic
94C4×C47⋊C8C2311yescharacteristic
95C4×C47⋊C8C2311yescharacteristic
96C4×C47⋊C8C2312yes—
97C2×#41×C47C2×C411yescharacteristic
98C22×C8×C47—21no—
99C4×Q8×C47C2×C411yescharacteristic
100C4×Q8×C47C2311yescharacteristic
101C4×D4×C47—21no—
102C4×D4×C47—21no—
103C4×D4×C47C2×C411yescharacteristic
104C4×D4×C47C2311yescharacteristic
105#117×C47—21no—
106#117×C47C2×C411yescharacteristic
107C2×C42×C47—21no—
108C2×C42×C47C2311yescharacteristic
109C4⋊C8×C47—21no—
110C4⋊C8×C47—21no—
111C4⋊C8×C47C2312yes—
112C22⋊C8×C47—21no—
113C4×C8×C47C2312yes—
114#41×C47—22no—
115#41×C47—22no—
116C2×Q8×C47C22×C411yescharacteristic
117C2×D4×C47—21no—
118C2×D4×C47C22×C411yescharacteristic
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47C22×C411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47#4111yescharacteristic
125C2×C22⋊D47—21no—
126#10625350—22no—
127#10625350—22no—
128C2×D188C22×C411yescharacteristic
129C2×C4×D47—21no—
130C2×C4×D47—21no—
131C2×C4×D47—21no—
132C2×Dic94C22×C411yescharacteristic
133C2×C8×C47—22no—
134C2×C8×C47—22no—
135C2×C8×C47—81no—
136C4⋊C4×C47—21no—
137C4⋊C4×C47—21no—
138C4⋊C4×C47C22×C411yescharacteristic
139C4⋊C4×C47C22×C411yescharacteristic
140C22⋊C4×C47—21no—
141C22⋊C4×C47—21no—
142C22⋊C4×C47—21no—
143C42×C47—21no—
144C42×C47—21no—
145C42×C47—21no—
146C42×C47C22×C411yescharacteristic
147C42×C47C2×D411yescharacteristic
148C42×C47#4111yescharacteristic
149C42×C47C2411yescharacteristic
150C22⋊Dic47—21no—
151#10625337—22no—
152#10625336—22no—
153#10625335—22no—
154#10625334—22no—
155(C2×D47)⋊C4—21no—
156(C2×D47)⋊C4—21no—
157C4⋊Dic47C22×C411yescharacteristic
158C4⋊Dic47C22×C411yescharacteristic
159Dic47⋊C4—21no—
160Dic47⋊C4—21no—
161C4×Dic47—21no—
162C4×Dic47—21no—
163C4×Dic47—21no—
164#10625329—82no—
165C2×C47⋊C8—22no—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—81no—
168C2×C47⋊C8M1614yescomplemented
169C2×C47⋊C8C22×C411yescharacteristic
170C2×C47⋊C8C22×C411yescharacteristic
171C2×C47⋊C8C22×C411yescharacteristic
172C2×C47⋊C8C22×C411yescharacteristic
173C2×C47⋊C8C2×D412yes—
174C2×C47⋊C8C2×D412yes—
175#10625324—82no—
176C23×C47—21no—
177C23×C47C4×D411yescharacteristic
178C22×D47—21no—
179Q8×C47—21no—
180Q8×C47—21no—
181D4×C47—21no—
182D4×C47—21no—
183D4×C47—42no—
184C2×C4×C47—21no—
185C2×C4×C47—21no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—21no—
189C2×C4×C47—41no—
190C2×C4×C47—41no—
191C2×C4×C47—41no—
192C2×C4×C47—42no—
193C2×C4×C47—81no—
194C2×C4×C47C4×D411yescharacteristic
195C2×C4×C47C4×D411yescharacteristic
196C2×C4×C47C4×D411yescharacteristic
197C2×C4×C47C2×M1612yes—
198C2×C4×C47#13112yes—
199C2×C4×C47C23×C411yescharacteristic
200C2×C4×C47C22×D411yescharacteristic
201C2×C4×C47C2×#4111yescharacteristic
202C22⋊D47—42no—
203C2×Dic47—21no—
204C2×Dic47—21no—
205C2×Dic47—21no—
206D188—21no—
207D188—21no—
208C4×D47—41no—
209C4×D47—41no—
210C4×D47—42no—
211C4×D47—81no—
212Dic94—21no—
213Dic94—21no—
214C8×C47—42no—
215C8×C47—42no—
216C47⋊C8—21no—
217C47⋊C8—21no—
218C47⋊C8—22no—
219C47⋊C8—22no—
220C47⋊C8—42no—
221C47⋊C8C2×M1612yes—
222C47⋊C8C2×M1612yes—
223#17899—471nomaximal, Sylow
224C22×C47—21no—
225C22×C47—21no—
226C22×C47—41no—
227C22×C47—41no—
228C22×C47—41no—
229C22×C47#43612yes—
230C22×C47C2×C4×D411yescharacteristic
231C2×D47—41no—
232C2×D47—41no—
233C2×D47—41no—
234C4×C47—21no—
235C4×C47—21no—
236C4×C47—21no—
237C4×C47—41no—
238C4×C47—41no—
239C4×C47—41no—
240C4×C47#43612yes—
241C4×C47C22×M1611yescharacteristic
242C4×C47C2×#13111yescharacteristic
243C4×C47#57711yescharacteristic
244C4×C47#57711yescharacteristic
245Dic47—41no—
246Dic47—41no—
247Dic47—41no—
248#2989—471no—
249#2989—471no—
250C4×#135—471no—
251#1594—472no—
252#1593—472no—
253#1590—472no—
254#1590—472no—
255C2×#334—471no—
256#1477—471no—
257#1477D4711yescharacteristic, complemented
258#1450—471no—
259C2×C47—22no—
260C2×C47—81no—
261C2×C47C2×#43611yescharacteristic
262C2×C47#295211yescharacteristic
263C2×C47#295711yescharacteristic
264D47—81no—
265C2×#135—471no—
266C4×#41—471no—
267C4×#41C2×D4711yescharacteristic
268#446—472no—
269#446—472no—
270C4×Q16—471no—
271C4×Q16—471no—
272C4×SD16—471no—
273C4×SD16—471no—
274C4×SD16—471no—
275C4×SD16—471no—
276C4×D8—471no—
277C4×D8—471no—
278#436—472no—
279#436—472no—
280#435—942no—
281#435—942no—
282#432—941no—
283#432—941no—
284#431—471no—
285#431C2×D4711yescharacteristic
286#427—471no—
287#423—941no—
288#423—941no—
289C2×C4⋊C8—471no—
290C2×C4⋊C8C2×D4711yescharacteristic
291#416—471no—
292#416—471no—
293C2×C4×C8—471no—
294#334—472no—
295#334—472no—
296Q8⋊C8—472no—
297Q8⋊C8C2×D4712yes—
298D4⋊C8—472no—
299D4⋊C8C2×D4712yes—
300#322—472no—
301C82—472no—
302C47#1789911yescharacteristic, complemented, Sylow
303C2×#41—471no—
304C2×#41C4×D4711yescharacteristic
305#135—944no—
306C2×Q16—471no—
307C2×SD16—471no—
308C2×SD16—471no—
309C2×D8—471no—
310C2×M16—942no—
311C2×M16—942no—
312C22×C8—21no—
313C22×C8—471no—
314C22×C8—471no—
315C22×C8—941no—
316C4×Q8—471no—
317C4×Q8—471no—
318C4×Q8C4×D4711yescharacteristic
319C4×Q8C22×D4711yescharacteristic
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—471no—
323C4×D4—471no—
324C4×D4—941no—
325C4×D4—941no—
326C4×D4C4×D4711yescharacteristic
327C4×D4C22×D4711yescharacteristic
328#117—21no—
329#117—471no—
330#117—941no—
331#117C4×D4711yescharacteristic
332C2×C42—21no—
333C2×C42—941no—
334C2×C42C22×D4711yescharacteristic
335#107—471no—
336#107—471no—
337#106—471no—
338#106—471no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—471no—
342C4⋊C8—471no—
343C4⋊C8—472no—
344C4⋊C8—942no—
345C4⋊C8—942no—
346C4⋊C8—941no—
347C4⋊C8C22×D4712yes—
348#103—471no—
349#103—471no—
350#103—471no—
351#103—471no—
352#102—471no—
353#102—471no—
354#102—471no—
355#102—471no—
356C22⋊C8—21no—
357C22⋊C8—472no—
358C22⋊C8—942no—
359C22⋊C8—942no—
360#97—942no—
361#97—942no—
362C4×C8—471no—
363C4×C8—471no—
364C4×C8—471no—
365C4×C8—472no—
366C4×C8—471no—
367C4×C8C22×D4712yes—
368#41—22no—
369#41—22no—
370#41—942no—
371#41—942no—
372C2×Q8—471no—
373C2×Q8C2×C4×D4711yescharacteristic
374C2×D4—21no—
375C2×D4—471no—
376C2×D4—941no—
377C2×D4C2×C4×D4711yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—941no—
382C22×C4—941no—
383C22×C4—941no—
384C22×C4C2×C4×D4711yescharacteristic
385C22×C4D4×D4711yescharacteristic
386C22×C4#1062535211yescharacteristic
387Q16—942no—
388SD16—942no—
389SD16—942no—
390D8—942no—
391M16—3762no—
392M16—3762no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—81no—
396C2×C8—474no—
397C2×C8—471no—
398C2×C8—471no—
399C2×C8—471no—
400C2×C8—471no—
401C2×C8—472no—
402C2×C8—472no—
403C2×C8—942no—
404C2×C8—942no—
405C2×C8—3761no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—471no—
409C4⋊C4—471no—
410C4⋊C4—941no—
411C4⋊C4—941no—
412C4⋊C4C2×C4×D4711yescharacteristic
413C4⋊C4C2×C4×D4711yescharacteristic
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—941no—
418C22⋊C4—941no—
419C22⋊C4—941no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—941no—
424C42—941no—
425C42—941no—
426C42C2×C4×D4711yescharacteristic
427C42D4×D4711yescharacteristic
428C42#1062535211yescharacteristic
429C42C23×D4711yescharacteristic
430C23—21no—
431C23—941no—
432C23(C22⋊D47)⋊C411yescharacteristic
433Q8—21no—
434Q8—21no—
435Q8—941no—
436Q8—941no—
437D4—21no—
438D4—21no—
439D4—42no—
440D4—941no—
441D4—941no—
442D4—1882no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—81no—
453C2×C4—941no—
454C2×C4—941no—
455C2×C4—941no—
456C2×C4—3761no—
457C2×C4—1881no—
458C2×C4—1881no—
459C2×C4—1882no—
460C2×C4(C22⋊D47)⋊C411yescharacteristic
461C2×C4(C22⋊D47)⋊C411yescharacteristic
462C2×C4(C22⋊D47)⋊C411yescharacteristic
463C2×C4M16×D4712yes—
464C2×C4#1309403012yes—
465C2×C4C22×C4×D4711yescharacteristic
466C2×C4C2×D4×D4711yescharacteristic
467C2×C4C2×#1062535211yescharacteristic
468C8—42no—
469C8—42no—
470C8—472no—
471C8—472no—
472C8—941no—
473C8—941no—
474C8—942no—
475C8—942no—
476C8—1882no—
477C22—21no—
478C22—21no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—1881no—
483C22—1881no—
484C22—1881no—
485C22#1568919812yes—
486C22C2×(C22⋊D47)⋊C411yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—1881no—
494C4—1881no—
495C4—1881no—
496C4#1568919812yes—
497C4C2×M16×D4711yescharacteristic
498C4C2×#1309403011yescharacteristic
499C4#1568993211yescharacteristic
500C4#1568993311yescharacteristic
501C2—22no—
502C2—81no—
503C2—3761no—
504C2C2×#1568919811yescharacteristic
505C2#2025226711yescharacteristic
506C2#2025230111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1477C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094044C4
cosocleC42×C47C24
hypercenter#1477D47
hypocenterC47#17899
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
1. #20252506C2semidirect
2. #20252505C2semidirect
4. #20242299C2semidirect
5. #20242294C2semidirect
6. #20242281C2semidirect
7. #20242275C2semidirect
8. C2×#15688873C2semidirect
9. #20241776C2semidirect
10. #20241672C2semidirect
11. C47×#1477C2semidirect
168. C2×C47⋊C8M16semidirect
257. #1477D47semidirect
302. C47#17899semidirect

Maximal subgroups

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

subgroupindex
#202525062
#202525052
C4×#130940442
#202422992
#202422942
#202422812
#202422752
C2×#156888732
#202417762
#202416722
C47×#14772
#1789947