Finite Group Databasebeta

← #26316262

Subgroups — #26316262

Order 12032.

912 conjugacy classes of subgroups.

The subgroup lattice has 524 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#20244713C211yesmaximal, characteristic, complemented
5#20244521C211yesmaximal, characteristic, complemented
6#20243956C212yesmaximal, complemented
7#20243814C212yesmaximal
8#20243634C212yesmaximal, complemented
9#20243415C212yesmaximal, complemented
10C2×#15688878C211yesmaximal, characteristic, complemented
11C47×#2857C211yesmaximal, characteristic, complemented
12C23×C47⋊C8C2212yes—
13C2×#13093998C2211yescharacteristic
14C2×C4×D188C2211yescharacteristic
15C2×C4×C22⋊D47—21no—
16C2×#13093957C2212yes—
17C2×#13093957C2212yes—
18C2×#13093944C2212yes—
19C2×#13093944C2212yes—
20#15689025C2211yescharacteristic
21#15689021C2211yescharacteristic
22#15689014—21no—
23#15689002—21no—
24#15688983C2211yescharacteristic
25C4×C4⋊Dic47C2211yescharacteristic
26C4×(C2×D47)⋊C4—21no—
27#15688885C2212yes—
28#15688885C2212yes—
29#15688885C2212yes—
30#15688885C2212yes—
31#15688878C2211yescharacteristic
32#15688878C2212yes—
33#15688878C2211yescharacteristic
34C2×C4⋊Q8×C47C2211yescharacteristic
35C2×C22⋊Q8×C47—21no—
36C22×C42×C47C2211yescharacteristic
37C47×#391—21no—
38C47×#389—21no—
39C47×#389C2212yes—
40C47×#384C2211yescharacteristic
41C47×#380—21no—
42C47×#380C2212yes—
43C22×D188C2311yescharacteristic
44C22×C22⋊D47—21no—
45C22×C4×D47—21no—
46C22×C47⋊C8—24no—
47C22×C47⋊C8—24no—
48C22×C47⋊C8—42no—
49C22×C47⋊C8C2312yes—
50C22×C47⋊C8C2312yes—
51#13093998—22no—
52#13093998Q814yescomplemented
53C4×D188—42no—
54C4×C22⋊D47—44no—
55C2×C4⋊Dic47C2311yescharacteristic
56C2×C4⋊Dic47C2311yescharacteristic
57C2×C4⋊Dic47C2311yescharacteristic
58C2×Dic47⋊C4—21no—
59C2×C22⋊Dic47—21no—
60C2×(C2×D47)⋊C4—21no—
61C2×(C2×D47)⋊C4—21no—
62C2×(C2×D47)⋊C4—21no—
63C2×(C2×D47)⋊C4—21no—
64C2×C4×Dic47—21no—
65#13093957—24no—
66#13093957—42no—
67#13093944—24no—
68#13093944—42no—
69C47⋊#95—21no—
70C47⋊#95—21no—
71C47⋊#95—21no—
72C22×Q8×C47—21no—
73C23×C4×C47D411yescharacteristic
74C23×C4×C47D411yescharacteristic
75C23×C4×C47C2311yescharacteristic
76C4⋊Q8×C47—42no—
77C22⋊Q8×C47—44no—
78C2×C4⋊C4×C47—21no—
79C2×C4⋊C4×C47—22no—
80C2×C4⋊C4×C47—21no—
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47C2312yes—
83C2×C4⋊C4×C47C2312yes—
84C2×C22⋊C4×C47—22no—
85C2×C42×C47—42no—
86C2×C42×C47D411yescharacteristic
87C2×C42×C47D411yescharacteristic
88C2×C42×C47C2311yescharacteristic
89C2×C42×C47C2311yescharacteristic
90#95×C47—22no—
91#95×C47—22no—
92C24×C47C2×D411yescharacteristic
93C23×D47—21no—
94C2×Q8×C47—42no—
95C2×Q8×C47—42no—
96C2×Q8×C47—42no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—41no—
104C22×C4×C47—41no—
105C22×C4×C47—41no—
106C22×C4×C47D814yes—
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2×Q812yes—
113C22×C4×C47#4112yes—
114C22×C4×C47C2411yescharacteristic
115C2×C22⋊D47—24no—
116C2×C22⋊D47—42no—
117C2×C22⋊D47—42no—
118C22×Dic47—21no—
119C22×Dic47—21no—
120C22×Dic47—21no—
121C2×D188—22no—
122C2×D188—22no—
123C2×D188—21no—
124C2×D188C2×Q812yes—
125C2×C4×D47—42no—
126C2×C4×D47—42no—
127C2×C4×D47—42no—
128C4⋊C4×C47—22no—
129C4⋊C4×C47—22no—
130C4⋊C4×C47—22no—
131C4⋊C4×C47—22no—
132C4⋊C4×C47—42no—
133C4⋊C4×C47—42no—
134C4⋊C4×C47—44no—
135C4⋊C4×C47—42no—
136C22⋊C4×C47—44no—
137C42×C47—22no—
138C42×C47—22no—
139C42×C47—41no—
140C42×C47—41no—
141C22⋊Dic47—42no—
142(C2×D47)⋊C4—24no—
143(C2×D47)⋊C4—42no—
144(C2×D47)⋊C4—42no—
145(C2×D47)⋊C4—42no—
146C4⋊Dic47—21no—
147C4⋊Dic47—22no—
148C4⋊Dic47—22no—
149C4⋊Dic47C2×Q812yes—
150Dic47⋊C4—42no—
151C4×Dic47—42no—
152C2×C47⋊C8—22no—
153C2×C47⋊C8—22no—
154C2×C47⋊C8—22no—
155C2×C47⋊C8—22no—
156C2×C47⋊C8—22no—
157C2×C47⋊C8—48no—
158C2×C47⋊C8#4114yes—
159C23×C47—21no—
160C23×C47—21no—
161C23×C47—21no—
162C23×C47—21no—
163C23×C47C22⋊Q812yes—
164C23×C47C2×D812yes—
165C23×C47#13712yes—
166C23×C47C22×D411yescharacteristic
167C22×D47—22no—
168C22×D47—41no—
169C22×D47—41no—
170C22×D47—41no—
171C22×D47—41no—
172C22×D47—41no—
173C22×D47—41no—
174Q8×C47—84no—
175C2×C4×C47—21no—
176C2×C4×C47—21no—
177C2×C4×C47—21no—
178C2×C4×C47—21no—
179C2×C4×C47—21no—
180C2×C4×C47—21no—
181C2×C4×C47—21no—
182C2×C4×C47—21no—
183C2×C4×C47—24no—
184C2×C4×C47—24no—
185C2×C4×C47—21no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—44no—
189C2×C4×C47—44no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47C22⋊Q812yes—
197C2×C4×C47C2×D812yes—
198C2×C4×C47C2×D812yes—
199C2×C4×C47#13512yes—
200C2×C4×C47C22×Q811yescharacteristic
201C2×C4×C47C2×#4112yes—
202C2×C4×C47C2×#4111yescharacteristic
203C2×C4×C47H(5,𝔽2)12yes—
204C22⋊D47—48no—
205C2×Dic47—22no—
206C2×Dic47—41no—
207C2×Dic47—41no—
208C2×Dic47—41no—
209C2×Dic47—41no—
210C2×Dic47—41no—
211C2×Dic47—41no—
212C2×Dic47—41no—
213C2×Dic47—41no—
214D188—24no—
215D188—42no—
216C4×D47—84no—
217C47⋊C8—24no—
218C47⋊C8—42no—
219#29157—471nomaximal, Sylow
220C22×C47—22no—
221C22×C47—22no—
222C22×C47—22no—
223C22×C47—22no—
224C22×C47—22no—
225C22×C47—22no—
226C22×C47#47414yes—
227C22×C47C2×C22⋊Q811yescharacteristic
228C22×C47#53211yescharacteristic
229C22×C47#53411yescharacteristic
230C22×C47C22×D811yescharacteristic
231C22×C47C2×#13511yescharacteristic
232C22×C47C2×#13711yescharacteristic
233C22×C47#57511yescharacteristic
234C2×D47—42no—
235C2×D47—42no—
236C2×D47—42no—
237C2×D47—81no—
238C2×D47—81no—
239C2×D47—81no—
240C2×D47—81no—
241C4×C47—22no—
242C4×C47—22no—
243C4×C47—22no—
244C4×C47—22no—
245C4×C47—41no—
246C4×C47—82no—
247C4×C47—82no—
248C4×C47#55112yes—
249C4×C47#55412yes—
250Dic47—42no—
251Dic47—81no—
252Dic47—81no—
253#2857D4711yescharacteristic, complemented
254C4×#121—471no—
255#2099—471no—
256#2093—471no—
257#2057—471no—
258#2023—471no—
259#1918—472no—
260#1893—472no—
261#1862—472no—
262#1808—472no—
263C2×#336—471no—
264C2×C47—24no—
265C2×C47#256411yescharacteristic
266C2×C47C2×#47411yescharacteristic
267C2×C47#307611yescharacteristic
268C2×C47#316911yescharacteristic
269C2×C47#318111yescharacteristic
270C2×C47#318611yescharacteristic
271C2×C47#319011yescharacteristic
272D47—82no—
273C23×C8—472no—
274C2×C4⋊Q8C2×D4711yescharacteristic
275C2×C22⋊Q8—21no—
276C2×#121—471no—
277C2×C4×D4—471no—
278C2×C4×D4—941no—
279C22×C42C2×D4711yescharacteristic
280C2×#107—472no—
281C2×#107—472no—
282C2×#102—472no—
283C2×#102—472no—
284#391—21no—
285#390—471no—
286#389—21no—
287#389—471no—
288#389C2×D4712yes—
289#388—941no—
290#386—941no—
291#384C2×D4711yescharacteristic
292#380—21no—
293#380—471no—
294#380C2×D4712yes—
295C4×C4⋊C4—471no—
296C4×C22⋊C4—941no—
297#340—472no—
298#340—472no—
299#340—472no—
300#340—472no—
301#336—471no—
302#336—471no—
303#336—472no—
304C47#2915711yescharacteristic, complemented, Sylow
305C22×Q8—21no—
306C22×D4—471no—
307C22×D4—941no—
308C23×C4—941no—
309C23×C4C22⋊D4711yescharacteristic
310C23×C4C22⋊D4711yescharacteristic
311C23×C4C22×D4711yescharacteristic
312C22×C8—472no—
313C22×C8—472no—
314C22×C8—944no—
315C22×C8—944no—
316C22×C8—1882no—
317C4⋊Q8—42no—
318C22⋊Q8—44no—
319#121—474no—
320#121—942no—
321C4×D4—1882no—
322C4×D4—1884no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—471no—
328C2×C4⋊C4—471no—
329C2×C4⋊C4—471no—
330C2×C4⋊C4—941no—
331C2×C4⋊C4C22×D4712yes—
332C2×C4⋊C4C22×D4712yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—941no—
335C2×C22⋊C4—941no—
336C2×C22⋊C4—941no—
337C2×C22⋊C4—941no—
338C2×C22⋊C4—941no—
339C2×C42—42no—
340C2×C42—941no—
341C2×C42C22⋊D4711yescharacteristic
342C2×C42C22⋊D4711yescharacteristic
343C2×C42C22×D4711yescharacteristic
344C2×C42C22×D4711yescharacteristic
345#107—944no—
346#107—1882no—
347#102—944no—
348#102—1882no—
349#95—22no—
350#95—22no—
351#95—941no—
352#95—941no—
353#95—941no—
354C24—941no—
355C24C2×C22⋊D4711yescharacteristic
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×D4—472no—
360C2×D4—944no—
361C2×D4—942no—
362C2×D4—941no—
363C2×D4—942no—
364C2×D4—1882no—
365C2×D4—1882no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—941no—
376C22×C4—941no—
377C22×C4—941no—
378C22×C4—1882no—
379C22×C4—1882no—
380C22×C4—1882no—
381C22×C4#1062533414yes—
382C22×C4Q8×D4712yes—
383C22×C4#1062535412yes—
384C22×C4C2×C22⋊D4711yescharacteristic
385C22×C4C2×C22⋊D4711yescharacteristic
386C22×C4C2×C22⋊D4711yescharacteristic
387C22×C4C2×C22⋊D4711yescharacteristic
388C22×C4C2×C22⋊D4711yescharacteristic
389C22×C4C23×D4711yescharacteristic
390C2×C8—474no—
391C2×C8—942no—
392C2×C8—942no—
393C2×C8—942no—
394C2×C8—942no—
395C2×C8—942no—
396C2×C8—1888no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—472no—
406C4⋊C4—941no—
407C4⋊C4—942no—
408C4⋊C4—942no—
409C4⋊C4—1882no—
410C22⋊C4—44no—
411C22⋊C4—944no—
412C22⋊C4—1882no—
413C22⋊C4—1882no—
414C22⋊C4—1882no—
415C22⋊C4—1882no—
416C42—22no—
417C42—22no—
418C42—41no—
419C42—41no—
420C42—1882no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—942no—
426C23—1881no—
427C23—1881no—
428C23—1881no—
429C23—1881no—
430C23—1881no—
431C23—1881no—
432C23#1309400012yes—
433C23C2×#1062533412yes—
434C23#1309405512yes—
435C23C22×C22⋊D4711yescharacteristic
436Q8—84no—
437D4—944no—
438D4—1888no—
439D4—1882no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—942no—
462C2×C4—3764no—
463C2×C4—1881no—
464C2×C4—1881no—
465C2×C4—1881no—
466C2×C4—1881no—
467C2×C4—1881no—
468C2×C4—1881no—
469C2×C4—1881no—
470C2×C4—1881no—
471C2×C4#1309400012yes—
472C2×C4C2×#1062533412yes—
473C2×C4C2×#1062533412yes—
474C2×C4#1309404412yes—
475C2×C4C2×Q8×D4711yescharacteristic
476C2×C4#41×D4712yes—
477C2×C4C2×#1062535411yescharacteristic
478C2×C4#1309406912yes—
479C8—944no—
480C8—1882no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—3761no—
488C22—3761no—
489C22—3761no—
490C22—3761no—
491C22—1882no—
492C22—1882no—
493C22—1882no—
494C22#1568940014yes—
495C22C2×#1309400011yescharacteristic
496C22#1568966911yescharacteristic
497C22#1568968211yescharacteristic
498C22C22×#1062533411yescharacteristic
499C22C2×#1309404411yescharacteristic
500C22C2×#1309405511yescharacteristic
501C22#1568992111yescharacteristic
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—41no—
507C4—82no—
508C4—82no—
509C4—3761no—
510C4—3761no—
511C4—1882no—
512C4#1568980012yes—
513C4#1568982712yes—
514C2—24no—
515C2—3762no—
516C2#2024857011yescharacteristic
517C2C2×#1568940011yescharacteristic
518C2#2025317011yescharacteristic
519C2#2025396311yescharacteristic
520C2#2025409111yescharacteristic
521C2#2025413611yescharacteristic
522C2#2025416911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2857C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenter#2857D47
hypocenterC47#29157
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. #20244945C2semidirect
3. #20244917C2semidirect
4. #20244713C2semidirect
5. #20244521C2semidirect
6. #20243956C2semidirect
8. #20243634C2semidirect
9. #20243415C2semidirect
10. C2×#15688878C2semidirect
11. C47×#2857C2semidirect
52. #13093998Q8semidirect
253. #2857D47semidirect
304. C47#29157semidirect

Maximal subgroups

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

subgroupindex
C4×#130939982
#202449452
#202449172
#202447132
#202445212
#202439562
#202438142
#202436342
#202434152
C2×#156888782
C47×#28572
#2915747