Finite Group Databasebeta

← #26233033

Subgroups — #26233033

Order 12032.

1172 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246265C212yesmaximal, complemented
2#20244225C212yesmaximal, complemented
3#20244225C214yesmaximal, complemented
4#20243376C211yesmaximal, characteristic, complemented
5#20243376C212yesmaximal, complemented
6C4×#13093939C211yesmaximal, characteristic, complemented
7C4×#13093939C212yesmaximal, complemented
8C2×C47×#553C211yesmaximal, characteristic
9C2×#13093939C2214yes—
10C2×#13093939C2212yes—
11C2×#13093939C2214yes—
12C2×#13093939C2212yes—
13#15689022—22no—
14#15689009—24no—
15#15688983C2212yes—
16#15688983C2214yes—
17C4⋊C4×Dic47—22no—
18C4×C22⋊Dic47C2212yes—
19C4×C22⋊Dic47C2214yes—
20#15688890C2211yescharacteristic
21#15688890C2212yes—
22#15688890C2211yescharacteristic
23C47×#553—22no—
24C47×#553—24no—
25C47×#553—22no—
26C2×#126×C47C412yescomplemented
27C2×#125×C47C411yescharacteristic, complemented
28C2×#123×C47C412yescomplemented
29C2×#121×C47C411yescharacteristic, complemented
30C2×#121×C47C2212yes—
31C2×#121×C47C2211yescharacteristic
32C2×C4×D4×C47C411yescharacteristic, complemented
33C2×C4×D4×C47C2212yes—
34C2×C4×D4×C47C2211yescharacteristic
35C2×#117×C47C411yescharacteristic, complemented
36C22×C4⋊C4×C47C2211yescharacteristic
37C2×C4⋊Dic47—22no—
38C2×C4⋊Dic47—24no—
39C2×C22⋊Dic47C2312yes—
40C2×C22⋊Dic47C2314yes—
41C2×C22⋊Dic47C2312yes—
42C2×C4×Dic47—22no—
43C2×C4×Dic47—24no—
44#13093939—24no—
45#13093939—28no—
46#13093939—44no—
47#13093939—42no—
48C47⋊#95—22no—
49C47⋊#95—24no—
50C47⋊#95—24no—
51C47⋊#95—22no—
52C22×D4×C47C2×C411yescharacteristic
53C22×D4×C47C2×C412yes—
54C22×D4×C47C2312yes—
55C22×D4×C47C2311yescharacteristic
56C23×C4×C47C2×C411yescharacteristic
57C23×C4×C47C2×C412yes—
58C23×C4×C47C2311yescharacteristic
59C23×C4×C47C2312yes—
60#126×C47—24no—
61#126×C47—24no—
62#125×C47—22no—
63#125×C47—22no—
64#123×C47—24no—
65#123×C47—24no—
66#121×C47—24no—
67#121×C47—24no—
68#121×C47—22no—
69#121×C47—22no—
70#121×C47—22no—
71#121×C47—22no—
72C4×D4×C47—24no—
73C4×D4×C47—24no—
74C4×D4×C47—22no—
75C4×D4×C47—22no—
76C4×D4×C47—22no—
77C4×D4×C47—21no—
78C4×D4×C47—21no—
79#117×C47—22no—
80#117×C47—21no—
81#117×C47—21no—
82C2×C4⋊C4×C47C2×C411yescharacteristic
83C2×C4⋊C4×C47C2×C412yes—
84C2×C4⋊C4×C47C2×C411yescharacteristic
85C2×C4⋊C4×C47C2×C412yes—
86C2×C4⋊C4×C47C2×C411yescharacteristic
87C2×C4⋊C4×C47C2×C411yescharacteristic
88C2×C4⋊C4×C47C2×C412yes—
89C2×C4⋊C4×C47D412yes—
90C2×C4⋊C4×C47D414yes—
91C2×C4⋊C4×C47D412yes—
92C2×C22⋊C4×C47C2×C412yes—
93C2×C22⋊C4×C47C2×C412yes—
94C2×C22⋊C4×C47C2×C412yes—
95C2×C22⋊C4×C47C2×C411yescharacteristic
96C2×C22⋊C4×C47C2×C412yes—
97C2×C22⋊C4×C47C2311yescharacteristic
98C2×C42×C47C2×C412yes—
99C2×C42×C47C2×C411yescharacteristic
100C24×C47C22×C411yescharacteristic
101C24×C47C22×C412yes—
102C24×C47C2411yescharacteristic
103C2×D4×C47—24no—
104C2×D4×C47—22no—
105C2×D4×C47—24no—
106C2×D4×C47—22no—
107C2×D4×C47—22no—
108C2×D4×C47—21no—
109C2×D4×C47—22no—
110C2×D4×C47—21no—
111C2×D4×C47—21no—
112C2×D4×C47—21no—
113C2×D4×C47—24no—
114C2×D4×C47—42no—
115C2×D4×C47—82no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—42no—
121C22×C4×C47—42no—
122C22×C4×C47—82no—
123C22×C4×C47—82no—
124C22×C4×C47C22×C412yes—
125C22×C4×C47C22×C411yescharacteristic
126C22×C4×C47C22×C411yescharacteristic
127C22×C4×C47C22×C411yescharacteristic
128C22×C4×C47C22×C411yescharacteristic
129C22×C4×C47C22×C412yes—
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C22×C412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D414yes—
135C22×C4×C47C2×D414yes—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—24no—
139C4⋊C4×C47—21no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—24no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—24no—
144C4⋊C4×C47C22⋊C412yes—
145C4⋊C4×C47C22⋊C411yescharacteristic
146C4⋊C4×C47C22⋊C411yescharacteristic
147C4⋊C4×C47C22⋊C412yes—
148C4⋊C4×C47C22⋊C412yes—
149C4⋊C4×C47C22⋊C414yes—
150C4⋊C4×C47C22⋊C414yes—
151C22⋊C4×C47—22no—
152C22⋊C4×C47—22no—
153C22⋊C4×C47—22no—
154C22⋊C4×C47—24no—
155C22⋊C4×C47—22no—
156C22⋊C4×C47—22no—
157C22⋊C4×C47—22no—
158C22⋊C4×C47—24no—
159C42×C47—22no—
160C42×C47—22no—
161C42×C47—21no—
162C42×C47—21no—
163C22⋊Dic47—22no—
164C22⋊Dic47—22no—
165C22⋊Dic47—24no—
166C22⋊Dic47—22no—
167C22⋊Dic47—24no—
168C22⋊Dic47#4114yes—
169C4⋊Dic47—44no—
170C4⋊Dic47—84no—
171C4×Dic47—44no—
172C4×Dic47—84no—
173C23×C47—21no—
174C23×C47—22no—
175C23×C47—21no—
176C23×C47—22no—
177C23×C47—21no—
178C23×C47—22no—
179C23×C47—41no—
180C23×C47—41no—
181C23×C47—41no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—41no—
185C23×C47C23×C411yescharacteristic
186C23×C47C22×D412yes—
187C23×C47C2×#4112yes—
188C23×C47H(5,𝔽2)11yescharacteristic
189C23×C472−1+411yescharacteristic
190D4×C47—42no—
191D4×C47—44no—
192D4×C47—42no—
193D4×C47—44no—
194D4×C47—42no—
195D4×C47—42no—
196D4×C47—84no—
197C2×C4×C47—24no—
198C2×C4×C47—22no—
199C2×C4×C47—21no—
200C2×C4×C47—22no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—22no—
204C2×C4×C47—21no—
205C2×C4×C47—22no—
206C2×C4×C47—21no—
207C2×C4×C47—44no—
208C2×C4×C47—44no—
209C2×C4×C47—41no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—82no—
218C2×C4×C47—82no—
219C2×C4×C47C2×C22⋊C412yes—
220C2×C4×C47C2×C22⋊C412yes—
221C2×C4×C47C2×C22⋊C411yescharacteristic
222C2×C4×C47C2×C22⋊C412yes—
223C2×C4×C47C2×C22⋊C414yes—
224C2×C4×C47C2×C22⋊C411yescharacteristic
225C2×Dic47—24no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—44no—
230C2×Dic47—44no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—44no—
234#22535—471nomaximal, Sylow
235C22×C47—22no—
236C22×C47—21no—
237C22×C47—22no—
238C22×C47—21no—
239C22×C47—42no—
240C22×C47—41no—
241C22×C47—41no—
242C22×C47—41no—
243C22×C47—42no—
244C22×C47—41no—
245C22×C47—42no—
246C22×C47—42no—
247C22×C47—41no—
248C22×C47—81no—
249C22×C47—82no—
250C22×C47—81no—
251C22×C47—82no—
252C22×C47C22×C22⋊C411yescharacteristic
253C22×C47C4×#4111yescharacteristic
254C22×C47#52011yescharacteristic
255C22×C47#54812yes—
256C22×C47#55012yes—
257C4×C47—21no—
258C4×C47—22no—
259C4×C47—22no—
260C4×C47—22no—
261C4×C47—24no—
262C4×C47—42no—
263C4×C47—41no—
264C4×C47—41no—
265C4×C47—41no—
266C4×C47#41012yes—
267Dic47—44no—
268Dic47—82no—
269Dic47—84no—
270C2×#553D4711yescharacteristic
271#2345—472no—
272#1964—474no—
273#1964—472no—
274#1799—472no—
275#1799—471no—
276C4×#99—472no—
277C4×#99—471no—
278C2×C47—22no—
279C2×C47—22no—
280C2×C47—21no—
281C2×C47—21no—
282C2×C47—81no—
283C2×C47—82no—
284C2×C47#234111yescharacteristic
285C2×C47C2×#41011yescharacteristic
286C2×C47#288311yescharacteristic
287#553—22no—
288#553—24no—
289#553—22no—
290C2×#126Dic4712yescomplemented
291C2×#125Dic4711yescharacteristic, complemented
292C2×#123Dic4712yescomplemented
293C2×#121Dic4711yescharacteristic, complemented
294C2×#121C2×D4712yes—
295C2×#121C2×D4711yescharacteristic
296C2×C4×D4Dic4711yescharacteristic, complemented
297C2×C4×D4C2×D4712yes—
298C2×C4×D4C2×D4711yescharacteristic
299C2×#117Dic4711yescharacteristic, complemented
300C22×C4⋊C4C2×D4711yescharacteristic
301C2×#99—474no—
302C2×#99—472no—
303C2×#99—472no—
304C2×#99—474no—
305#389—942no—
306#387—944no—
307#380—474no—
308#380—472no—
309C4×C4⋊C4—942no—
310C4×C22⋊C4—472no—
311C4×C22⋊C4—474no—
312#342—472no—
313#342—471no—
314#342—471no—
315C47#2253511yescharacteristic, complemented, Sylow
316C22×D4C2×Dic4711yescharacteristic
317C22×D4C2×Dic4712yes—
318C22×D4C22×D4712yes—
319C22×D4C22×D4711yescharacteristic
320C23×C4C2×Dic4711yescharacteristic
321C23×C4C2×Dic4712yes—
322C23×C4C22×D4711yescharacteristic
323C23×C4C22×D4712yes—
324#126—24no—
325#126—24no—
326#125—22no—
327#125—22no—
328#123—24no—
329#123—24no—
330#121—24no—
331#121—24no—
332#121—22no—
333#121—22no—
334#121—22no—
335#121—22no—
336C4×D4—24no—
337C4×D4—24no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—21no—
342C4×D4—21no—
343#117—22no—
344#117—21no—
345#117—21no—
346C2×C4⋊C4—944no—
347C2×C4⋊C4—942no—
348C2×C4⋊C4C2×Dic4711yescharacteristic
349C2×C4⋊C4C2×Dic4712yes—
350C2×C4⋊C4C2×Dic4711yescharacteristic
351C2×C4⋊C4C2×Dic4712yes—
352C2×C4⋊C4C2×Dic4711yescharacteristic
353C2×C4⋊C4C2×Dic4711yescharacteristic
354C2×C4⋊C4C2×Dic4712yes—
355C2×C4⋊C4C22⋊D4712yes—
356C2×C4⋊C4C22⋊D4714yes—
357C2×C4⋊C4C22⋊D4712yes—
358C2×C22⋊C4—472no—
359C2×C22⋊C4—474no—
360C2×C22⋊C4—472no—
361C2×C22⋊C4C2×Dic4712yes—
362C2×C22⋊C4C2×Dic4712yes—
363C2×C22⋊C4C2×Dic4712yes—
364C2×C22⋊C4C2×Dic4711yescharacteristic
365C2×C22⋊C4C2×Dic4712yes—
366C2×C22⋊C4C22×D4711yescharacteristic
367C2×C42—944no—
368C2×C42—942no—
369C2×C42C2×Dic4712yes—
370C2×C42C2×Dic4711yescharacteristic
371#99—948no—
372#99—944no—
373#99—1882no—
374#99—1884no—
375#95—942no—
376#95—944no—
377#95—944no—
378#95—942no—
379C24C22×Dic4711yescharacteristic
380C24C22×Dic4712yes—
381C24C23×D4711yescharacteristic
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—24no—
393C2×D4—42no—
394C2×D4—82no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—942no—
404C22×C4—944no—
405C22×C4—942no—
406C22×C4C22×Dic4712yes—
407C22×C4C22×Dic4711yescharacteristic
408C22×C4C22×Dic4711yescharacteristic
409C22×C4C22×Dic4711yescharacteristic
410C22×C4C22×Dic4711yescharacteristic
411C22×C4C22×Dic4712yes—
412C22×C4C22×Dic4711yescharacteristic
413C22×C4C22×Dic4712yes—
414C22×C4C2×C22⋊D4712yes—
415C22×C4C2×C22⋊D4712yes—
416C22×C4C2×C22⋊D4714yes—
417C22×C4C2×C22⋊D4714yes—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—3764no—
424C4⋊C4—1884no—
425C4⋊C4C22⋊Dic4712yes—
426C4⋊C4C22⋊Dic4711yescharacteristic
427C4⋊C4C22⋊Dic4711yescharacteristic
428C4⋊C4C22⋊Dic4712yes—
429C4⋊C4C22⋊Dic4712yes—
430C4⋊C4C22⋊Dic4714yes—
431C4⋊C4C22⋊Dic4714yes—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—24no—
440C22⋊C4—474no—
441C22⋊C4—942no—
442C22⋊C4—944no—
443C22⋊C4—942no—
444C22⋊C4—944no—
445C22⋊C4—942no—
446C42—22no—
447C42—22no—
448C42—21no—
449C42—21no—
450C42—3764no—
451C42—1884no—
452C23—21no—
453C23—22no—
454C23—21no—
455C23—22no—
456C23—21no—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23C23×Dic4711yescharacteristic
465C23C22×C22⋊D4712yes—
466C23#41×D4712yes—
467C23#1309407011yescharacteristic
468C23#1309407111yescharacteristic
469D4—42no—
470D4—44no—
471D4—42no—
472D4—44no—
473D4—42no—
474D4—42no—
475D4—84no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—944no—
499C2×C4—1884no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1884no—
503C2×C4—1884no—
504C2×C4—1882no—
505C2×C4—1882no—
506C2×C4—1882no—
507C2×C4C2×C22⋊Dic4712yes—
508C2×C4C2×C22⋊Dic4712yes—
509C2×C4C2×C22⋊Dic4711yescharacteristic
510C2×C4C2×C22⋊Dic4712yes—
511C2×C4C2×C22⋊Dic4714yes—
512C2×C4C2×C22⋊Dic4711yescharacteristic
513C22—21no—
514C22—22no—
515C22—21no—
516C22—22no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—42no—
523C22—41no—
524C22—41no—
525C22—42no—
526C22—81no—
527C22—82no—
528C22—81no—
529C22—82no—
530C22C22×C22⋊Dic4711yescharacteristic
531C22#41×Dic4711yescharacteristic
532C22#1568960611yescharacteristic
533C22#1568978012yes—
534C22#1568979212yes—
535C4—21no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—24no—
540C4—41no—
541C4—41no—
542C4—42no—
543C4—41no—
544C4—3764no—
545C4—3762no—
546C4—1884no—
547C4#1568909712yes—
548C2—22no—
549C2—22no—
550C2—21no—
551C2—21no—
552C2—81no—
553C2—82no—
554C2#2024622911yescharacteristic
555C2C2×#1568909711yescharacteristic
556C2#2025184911yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
Fitting, prime radicalC2×C47×#553C2
socleC22×C47C22×C22⋊C4
cosocleC24×C47C24
hypercenterC2×#553D47
hypocenterC47#22535
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. #20246265C2semidirect
2. #20244225C2semidirect
3. #20244225C2semidirect
4. #20243376C2semidirect
5. #20243376C2semidirect
6. C4×#13093939C2semidirect
7. C4×#13093939C2semidirect
26. C2×#126×C47C4semidirect
27. C2×#125×C47C4semidirect
28. C2×#123×C47C4semidirect
29. C2×#121×C47C4semidirect
32. C2×C4×D4×C47C4semidirect
35. C2×#117×C47C4semidirect
290. C2×#126Dic47semidirect
291. C2×#125Dic47semidirect
292. C2×#123Dic47semidirect
293. C2×#121Dic47semidirect
296. C2×C4×D4Dic47semidirect
299. C2×#117Dic47semidirect
315. C47#22535semidirect

Maximal subgroups

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

subgroupindex
#202462652
#202442252
#202433762
C4×#130939392
C2×C47×#5532
#2253547