Finite Group Databasebeta

← #26270447

Subgroups — #26270447

Order 12032.

1028 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689482C211yesmaximal, characteristic
2C2×#15689408C211yesmaximal, characteristic, complemented
3#20248593C211yesmaximal, characteristic, complemented
4#20244631C212yesmaximal, complemented
5#20244624C212yesmaximal
6#20244329C211yesmaximal, characteristic, complemented
7#20244321C211yesmaximal, characteristic
8#20243656C212yesmaximal, complemented
9#20243649C212yesmaximal
10#20243218C211yesmaximal, characteristic, complemented
11C2×C47×#556C211yesmaximal, characteristic
12C2×#13094024C2211yescharacteristic
13#15689482—24no—
14#15689408—24no—
15C2×#13093955C2212yes—
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×#13093948C2212yes—
18C2×#13093948C2212yes—
19C2×#13093948C2212yes—
20C2×#13093948C2212yes—
21C2×#13093945C2212yes—
22C2×#13093945C2212yes—
23C2×#13093935C2211yescharacteristic
24#15689043—21no—
25#15689043—21no—
26#15689030—21no—
27#15689021C2212yes—
28#15689021C2212yes—
29#15688992—21no—
30C4×C4⋊Dic47C2211yescharacteristic
31#15688885C2212yes—
32#15688878C2212yes—
33#15688878C2212yes—
34C47×#556—24no—
35C47×#556—24no—
36C2×C4⋊Q8×C47C2211yescharacteristic
37C2×#126×C47—21no—
38C2×#126×C47—21no—
39C2×#124×C47C2211yescharacteristic
40C2×#124×C47C2211yescharacteristic
41C2×C22⋊Q8×C47—21no—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4×Q8×C47C2211yescharacteristic
44C2×C4×Q8×C47C2211yescharacteristic
45C2×C4×D4×C47C2211yescharacteristic
46C2×#117×C47—21no—
47C22×Dic94—21no—
48C22×C47⋊C8C2312yes—
49C22×C47⋊C8C2312yes—
50#13094024—22no—
51#13094024—22no—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47C2312yes—
54C2×C4⋊Dic47C2312yes—
55C2×Dic47⋊C4—22no—
56C2×Dic47⋊C4—22no—
57C2×C4×Dic47—21no—
58#13093955—24no—
59C4⋊(C47⋊C8)—22no—
60#13093948—24no—
61#13093948—24no—
62#13093948—24no—
63#13093948—24no—
64#13093945—24no—
65#13093945—24no—
66#13093935—22no—
67C47⋊#95—21no—
68C47⋊#95—22no—
69C47⋊#95—22no—
70C22×Q8×C47D411yescharacteristic
71C22×Q8×C47C2311yescharacteristic
72C22×Q8×C47C2311yescharacteristic
73C22×D4×C47C2311yescharacteristic
74C23×C4×C47—21no—
75C4⋊Q8×C47—22no—
76C4⋊Q8×C47—22no—
77#126×C47—28no—
78#126×C47—28no—
79#124×C47—22no—
80#124×C47—22no—
81#124×C47—22no—
82#124×C47D414yes—
83C22⋊Q8×C47—28no—
84C4×Q8×C47—22no—
85C4×Q8×C47—22no—
86C4×Q8×C47—22no—
87C4×Q8×C47—22no—
88C4×Q8×C47—22no—
89C4×Q8×C47D414yes—
90C4×D4×C47—22no—
91C4×D4×C47—22no—
92#117×C47—28no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47D411yescharacteristic
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C42×C47—21no—
107C2×C42×C47—21no—
108C2×C42×C47C2311yescharacteristic
109C2×C42×C47C2311yescharacteristic
110C2×C42×C47C2311yescharacteristic
111C24×C47—21no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47—22no—
117C2×Q8×C47—22no—
118C2×Q8×C47—22no—
119C2×Q8×C47—22no—
120C2×Q8×C47—41no—
121C2×Q8×C47C2×D412yes—
122C2×Q8×C47C2×D412yes—
123C2×D4×C47—21no—
124C2×D4×C47—22no—
125C2×D4×C47—22no—
126C2×D4×C47C2×D412yes—
127C22×C4×C47—24no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—22no—
141C22×Dic47—22no—
142C2×Dic94—42no—
143C2×Dic94—82no—
144C4⋊C4×C47—24no—
145C4⋊C4×C47—24no—
146C4⋊C4×C47—24no—
147C4⋊C4×C47—24no—
148C4⋊C4×C47—22no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47C2×D412yes—
153C22⋊C4×C47—24no—
154C22⋊C4×C47—24no—
155C22⋊C4×C47—24no—
156C22⋊C4×C47—24no—
157C22⋊C4×C47—24no—
158C42×C47—24no—
159C42×C47—24no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—21no—
163C42×C47—21no—
164C42×C47C2×D412yes—
165C42×C47C2×D412yes—
166C4⋊Dic47—22no—
167C4⋊Dic47—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47—42no—
170C4⋊Dic47#4114yes—
171Dic47⋊C4—82no—
172Dic47⋊C4—82no—
173C4×Dic47—42no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C23×C47—22no—
181C23×C47—42no—
182C23×C47—41no—
183C23×C47—41no—
184C23×C47—42no—
185C23×C47C22×D411yescharacteristic
186Q8×C47—22no—
187Q8×C47—22no—
188Q8×C47—24no—
189Q8×C47—44no—
190Q8×C47—41no—
191Q8×C47—41no—
192Q8×C47—42no—
193D4×C47—24no—
194D4×C47—41no—
195D4×C47—41no—
196C2×C4×C47—21no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—22no—
201C2×C4×C47—21no—
202C2×C4×C47—22no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—48no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47#12012yes—
217C2×C4×C47#12012yes—
218C2×C4×C47#13712yes—
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C2×#4112yes—
222C2×C4×C472−1+412yes—
223C2×Dic47—24no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229Dic94—84no—
230C47⋊C8—42no—
231C47⋊C8—42no—
232#25662—471nomaximal, Sylow
233C22×C47—42no—
234C22×C47—42no—
235C22×C47—42no—
236C22×C47—82no—
237C22×C47—82no—
238C22×C47C2×#12011yescharacteristic
239C22×C47#54812yes—
240C22×C47C2×#13711yescharacteristic
241C22×C47#57511yescharacteristic
242C22×C47#57712yes—
243C4×C47—22no—
244C4×C47—22no—
245C4×C47—22no—
246C4×C47—42no—
247C4×C47—42no—
248C4×C47—41no—
249C4×C47—42no—
250C4×C47—41no—
251C4×C47—41no—
252C4×C47#53211yescharacteristic
253C4×C47#53611yescharacteristic
254C4×C47#54712yes—
255Dic47—44no—
256Dic47—82no—
257C2×#556D4711yescharacteristic
258C2×#489—471no—
259C2×#475—471no—
260#2567—471no—
261#2043—472no—
262#2042—472no—
263#1985—471no—
264#1983—471no—
265#1865—472no—
266#1864—472no—
267#1763—471no—
268C2×C47—82no—
269C2×C47#243211yescharacteristic
270C2×C47#300711yescharacteristic
271C2×C47#301011yescharacteristic
272C2×C47#336912yes—
273C2×C47#337612yes—
274#556—24no—
275#556—24no—
276C2×C4⋊Q8—471no—
277C2×C4⋊Q8C2×D4711yescharacteristic
278C2×#126—21no—
279C2×#126—21no—
280C2×#124C2×D4711yescharacteristic
281C2×#124C2×D4711yescharacteristic
282C2×C22⋊Q8—21no—
283C2×C4×Q8C2×D4711yescharacteristic
284C2×C4×Q8C2×D4711yescharacteristic
285C2×C4×Q8C2×D4711yescharacteristic
286C2×C4×D4C2×D4711yescharacteristic
287C2×#117—21no—
288#489—944no—
289#475—944no—
290C2×#106—472no—
291C2×C4⋊C8—471no—
292C2×#103—472no—
293C2×#103—472no—
294C2×#103—472no—
295C2×#103—472no—
296C2×#102—472no—
297C2×#102—472no—
298C2×#97—471no—
299#395—941no—
300#395—941no—
301#391—941no—
302#389—472no—
303#389—472no—
304#384—941no—
305C4×C4⋊C4—471no—
306#340—472no—
307#336—472no—
308#336—472no—
309C47#2566211yescharacteristic, complemented, Sylow
310C22×Q8—941no—
311C22×Q8C22⋊D4711yescharacteristic
312C22×Q8C22×D4711yescharacteristic
313C22×Q8C22×D4711yescharacteristic
314C22×D4C22×D4711yescharacteristic
315C23×C4—21no—
316C22×C8—472no—
317C22×C8—472no—
318C4⋊Q8—22no—
319C4⋊Q8—22no—
320C4⋊Q8—942no—
321C4⋊Q8—942no—
322#126—28no—
323#126—28no—
324#124—22no—
325#124—22no—
326#124—22no—
327#124C22⋊D4714yes—
328C22⋊Q8—28no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8C22⋊D4714yes—
335C4×D4—22no—
336C4×D4—22no—
337#117—28no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—472no—
343C2×C4⋊C4—472no—
344C2×C4⋊C4—942no—
345C2×C4⋊C4—942no—
346C2×C4⋊C4—941no—
347C2×C4⋊C4C22⋊D4711yescharacteristic
348C2×C4⋊C4C22⋊D4711yescharacteristic
349C2×C4⋊C4C22⋊D4711yescharacteristic
350C2×C4⋊C4C22×D4711yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—941no—
359C2×C42C22×D4711yescharacteristic
360C2×C42C22×D4711yescharacteristic
361C2×C42C22×D4711yescharacteristic
362#106—944no—
363C4⋊C8—942no—
364#103—944no—
365#103—944no—
366#103—944no—
367#103—944no—
368#102—944no—
369#102—944no—
370#97—942no—
371#95—941no—
372#95—942no—
373#95—942no—
374C24—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—3762no—
385C2×Q8—1882no—
386C2×Q8C2×C22⋊D4712yes—
387C2×Q8C2×C22⋊D4712yes—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4C2×C22⋊D4712yes—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—942no—
399C22×C4—942no—
400C22×C4C2×C22⋊D4711yescharacteristic
401C22×C4C2×C22⋊D4711yescharacteristic
402C22×C4C2×C22⋊D4711yescharacteristic
403C22×C4C2×C22⋊D4711yescharacteristic
404C22×C4C2×C22⋊D4711yescharacteristic
405C22×C4C2×C22⋊D4711yescharacteristic
406C22×C4C23×D4711yescharacteristic
407C2×C8—942no—
408C2×C8—942no—
409C2×C8—942no—
410C2×C8—942no—
411C2×C8—942no—
412C2×C8—942no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—474no—
422C4⋊C4—942no—
423C4⋊C4—942no—
424C4⋊C4—942no—
425C4⋊C4—3762no—
426C4⋊C4—3762no—
427C4⋊C4—1882no—
428C4⋊C4C2×C22⋊D4712yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C42—24no—
435C42—24no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—1882no—
441C42C2×C22⋊D4712yes—
442C42C2×C22⋊D4712yes—
443C23—22no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23C22×C22⋊D4711yescharacteristic
449Q8—24no—
450Q8—22no—
451Q8—22no—
452Q8—44no—
453Q8—42no—
454Q8—41no—
455Q8—41no—
456Q8—3764no—
457D4—24no—
458D4—41no—
459D4—41no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—944no—
481C2×C4—1882no—
482C2×C4—1882no—
483C2×C4—1882no—
484C2×C4—1882no—
485C2×C4—1882no—
486C2×C4#1309399312yes—
487C2×C4#1309399312yes—
488C2×C4#1309405412yes—
489C2×C4C22×C22⋊D4711yescharacteristic
490C2×C4C22×C22⋊D4711yescharacteristic
491C2×C4#41×D4712yes—
492C2×C4#1309407112yes—
493C8—1882no—
494C8—1882no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—82no—
499C22—82no—
500C22C2×#1309399311yescharacteristic
501C22#1568978012yes—
502C22C2×#1309405411yescharacteristic
503C22#1568992111yescharacteristic
504C22#1568993112yes—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—42no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—3762no—
515C4—1884no—
516C4#1568966911yescharacteristic
517C4#1568969411yescharacteristic
518C4#1568977212yes—
519C2—82no—
520C2#2024701211yescharacteristic
521C2#2025264811yescharacteristic
522C2#2025267411yescharacteristic
523C2#2025612412yes—
524C2#2025620912yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#556C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenterC2×#556D47
hypocenterC47#25662
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#15689408C2semidirect
3. #20248593C2semidirect
4. #20244631C2semidirect
6. #20244329C2semidirect
8. #20243656C2semidirect
10. #20243218C2semidirect
309. C47#25662semidirect

Maximal subgroups

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

subgroupindex
C2×#156894822
C2×#156894082
#202485932
#202446312
#202446242
#202443292
#202443212
#202436562
#202436492
#202432182
C2×C47×#5562
#2566247