Finite Group Databasebeta

← #26124168

Subgroups — #26124168

Order 12032.

1194 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688983C211yesmaximal, characteristic
2C2×#15688890C212yesmaximal
3C2×#15688890C212yesmaximal
4#20241611C211yesmaximal, characteristic
5C2×C47×#385C211yesmaximal, characteristic
6C22×C4⋊Dic47—21no—
7C22×C22⋊Dic47C2212yes—
8C22×C22⋊Dic47C2212yes—
9#15688983—24no—
10C2×C47⋊#95—21no—
11C2×C47⋊#95—22no—
12#15688890—24no—
13#15688890—24no—
14C24×C4×C47C2211yescharacteristic
15C22×C4⋊C4×C47C412yescomplemented
16C22×C22⋊C4×C47C2211yescharacteristic
17C22×C22⋊C4×C47C2211yescharacteristic
18C47×#385—21no—
19C47×#385—22no—
20C47×#385—21no—
21C2×#95×C47C412yescomplemented
22C23×Dic47—22no—
23C23×Dic47—22no—
24C2×C4⋊Dic47—24no—
25C2×C4⋊Dic47—44no—
26C2×C22⋊Dic47—22no—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—22no—
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47D414yes—
37C2×C22⋊Dic47Q814yes—
38C47⋊#95—24no—
39C47⋊#95—44no—
40C25×C47C2311yescharacteristic
41C23×C4×C47—22no—
42C23×C4×C47—22no—
43C23×C4×C47—44no—
44C23×C4×C47C2×C412yes—
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47C2×C412yes—
48C23×C4×C47D411yescharacteristic
49C23×C4×C47D412yes—
50C23×C4×C47D411yescharacteristic
51C2×C4⋊C4×C47—22no—
52C2×C4⋊C4×C47—22no—
53C2×C4⋊C4×C47—44no—
54C2×C22⋊C4×C47—21no—
55C2×C22⋊C4×C47—21no—
56C2×C22⋊C4×C47—21no—
57C2×C22⋊C4×C47—21no—
58C2×C22⋊C4×C47—22no—
59C2×C22⋊C4×C47—21no—
60C2×C22⋊C4×C47—21no—
61C2×C22⋊C4×C47—22no—
62C2×C22⋊C4×C47D412yes—
63C2×C22⋊C4×C47Q812yes—
64#95×C47—22no—
65#95×C47—22no—
66C24×C47—42no—
67C24×C47—42no—
68C24×C47C22×C411yescharacteristic
69C24×C47C2×D411yescharacteristic
70C24×C47C2×D411yescharacteristic
71C24×C47C2×D411yescharacteristic
72C24×C47C2×D412yes—
73C24×C47C2×Q811yescharacteristic
74C24×C47C2×Q812yes—
75C24×C47#4111yescharacteristic
76C24×C47#4111yescharacteristic
77C24×C47#4112yes—
78C24×C47#4112yes—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—24no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—24no—
85C22×C4×C47—21no—
86C22×C4×C47—24no—
87C22×C4×C47—21no—
88C22×C4×C47—24no—
89C22×C4×C47—24no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—42no—
95C22×C4×C47—42no—
96C22×C4×C47—42no—
97C22×C4×C47—48no—
98C22×C4×C47—48no—
99C22×C4×C47—42no—
100C22×C4×C47—42no—
101C22×C4×C47—42no—
102C22×C4×C47—42no—
103C22×C4×C47C22⋊C412yes—
104C22×C4×C47C22⋊C412yes—
105C22×C4×C47C4⋊C414yescomplemented
106C22×Dic47—24no—
107C22×Dic47—24no—
108C22×Dic47—44no—
109C22×Dic47—44no—
110C22×Dic47—44no—
111C22×Dic47—44no—
112C22×Dic47—42no—
113C22×Dic47—42no—
114C22×Dic47—42no—
115C22×Dic47—42no—
116C4⋊C4×C47—44no—
117C4⋊C4×C47—44no—
118C22⋊C4×C47—22no—
119C22⋊C4×C47—22no—
120C22⋊C4×C47—22no—
121C22⋊C4×C47—22no—
122C22⋊C4×C47—41no—
123C22⋊C4×C47—42no—
124C22⋊C4×C47—41no—
125C22⋊Dic47—24no—
126C22⋊Dic47—24no—
127C22⋊Dic47—24no—
128C22⋊Dic47—24no—
129C22⋊Dic47—42no—
130C22⋊Dic47—42no—
131C22⋊Dic47—42no—
132C22⋊Dic47—42no—
133C4⋊Dic47—48no—
134C23×C47—22no—
135C23×C47—22no—
136C23×C47—22no—
137C23×C47—22no—
138C23×C47—22no—
139C23×C47—22no—
140C23×C47—42no—
141C23×C47—42no—
142C23×C47—42no—
143C23×C47—42no—
144C23×C47—42no—
145C23×C47—44no—
146C23×C47—42no—
147C23×C47—42no—
148C23×C47—42no—
149C23×C47—44no—
150C23×C47C2×C22⋊C411yescharacteristic
151C23×C47C2×C4⋊C411yescharacteristic
152C23×C47#11711yescharacteristic
153C23×C47C4×D412yes—
154C23×C47C4×Q812yes—
155C23×C47#12011yescharacteristic
156C23×C47#12112yes—
157C23×C47#12111yescharacteristic
158C23×C47#12111yescharacteristic
159C23×C47#12112yes—
160C23×C47C22⋊Q812yes—
161C23×C47C22⋊Q812yes—
162C23×C47C22⋊Q811yescharacteristic
163C23×C47C22⋊Q811yescharacteristic
164C23×C47C22⋊Q812yes—
165C23×C47#12311yescharacteristic
166C23×C47#12312yes—
167C23×C47#12411yescharacteristic
168C23×C47#12512yes—
169C23×C47#12512yes—
170C23×C47#12612yes—
171C23×C47C4⋊Q811yescharacteristic
172C23×C47C4⋊Q812yes—
173C2×C4×C47—24no—
174C2×C4×C47—24no—
175C2×C4×C47—22no—
176C2×C4×C47—22no—
177C2×C4×C47—24no—
178C2×C4×C47—22no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—48no—
182C2×C4×C47—48no—
183C2×C4×C47—44no—
184C2×C4×C47—42no—
185C2×C4×C47—42no—
186C2×C4×C47—42no—
187C2×C4×C47—42no—
188C2×C4×C47—44no—
189C2×C4×C47—41no—
190C2×C4×C47—41no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—82no—
194C2×C4×C47—82no—
195C2×C4×C47#9914yes—
196C2×Dic47—44no—
197C2×Dic47—44no—
198C2×Dic47—48no—
199C2×Dic47—44no—
200C2×Dic47—44no—
201C2×Dic47—84no—
202C2×Dic47—84no—
203C2×Dic47—82no—
204C2×Dic47—82no—
205C2×Dic47—84no—
206#11203—471nomaximal, Sylow
207C22×C47—22no—
208C22×C47—22no—
209C22×C47—22no—
210C22×C47—22no—
211C22×C47—22no—
212C22×C47—22no—
213C22×C47—21no—
214C22×C47—22no—
215C22×C47—21no—
216C22×C47—22no—
217C22×C47—21no—
218C22×C47—22no—
219C22×C47—22no—
220C22×C47—21no—
221C22×C47—24no—
222C22×C47—22no—
223C22×C47—24no—
224C22×C47—22no—
225C22×C47—42no—
226C22×C47—42no—
227C22×C47—42no—
228C22×C47—42no—
229C22×C47—42no—
230C22×C47—42no—
231C22×C47—42no—
232C22×C47—42no—
233C22×C47#38011yescharacteristic
234C22×C47#38611yescharacteristic
235C22×C47#38712yes—
236C22×C47#38912yes—
237C22×C47#39111yescharacteristic
238C22×C47#39411yescharacteristic
239C22×C47#39511yescharacteristic
240C22×C47#39612yes—
241C22×C47#39811yescharacteristic
242C22×C47#39812yes—
243C22×C47#40011yescharacteristic
244C22×C47C2×#9912yes—
245C22×C47#41012yes—
246C4×C47—24no—
247C4×C47—44no—
248C4×C47—42no—
249C4×C47—81no—
250C4×C47—81no—
251Dic47—84no—
252Dic47—84no—
253C2×#385D4711yescharacteristic
254C2×#380—471no—
255C2×#342—472no—
256C2×#342—472no—
257C22⋊#95—471no—
258C2×C47—21no—
259C2×C47—22no—
260C2×C47—22no—
261C2×C47—22no—
262C2×C47—21no—
263C2×C47—21no—
264C2×C47—22no—
265C2×C47—21no—
266C2×C47#144111yescharacteristic
267C2×C47#179912yes—
268C2×C47#189811yescharacteristic
269C2×C47#189911yescharacteristic
270C2×C47#199612yes—
271C24×C4C2×D4711yescharacteristic
272C22×C4⋊C4—941no—
273C22×C4⋊C4Dic4712yescomplemented
274C22×C22⋊C4—472no—
275C22×C22⋊C4—472no—
276C22×C22⋊C4C2×D4711yescharacteristic
277C22×C22⋊C4C2×D4711yescharacteristic
278#385—21no—
279#385—22no—
280#385—21no—
281#380—944no—
282C2×#95—941no—
283C2×#95—942no—
284C2×#95Dic4712yescomplemented
285#342—944no—
286#342—944no—
287C47#1120311yescharacteristic, complemented, Sylow
288C25C22×D4711yescharacteristic
289C23×C4—22no—
290C23×C4—22no—
291C23×C4—44no—
292C23×C4—942no—
293C23×C4—942no—
294C23×C4C2×Dic4712yes—
295C23×C4C2×Dic4711yescharacteristic
296C23×C4C2×Dic4711yescharacteristic
297C23×C4C2×Dic4712yes—
298C23×C4C22⋊D4711yescharacteristic
299C23×C4C22⋊D4712yes—
300C23×C4C22⋊D4711yescharacteristic
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—944no—
305C2×C4⋊C4—1884no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—474no—
315C2×C22⋊C4—474no—
316C2×C22⋊C4—942no—
317C2×C22⋊C4—942no—
318C2×C22⋊C4—942no—
319C2×C22⋊C4—942no—
320C2×C22⋊C4—942no—
321C2×C22⋊C4—942no—
322C2×C22⋊C4—942no—
323C2×C22⋊C4—942no—
324C2×C22⋊C4—942no—
325C2×C22⋊C4—942no—
326C2×C22⋊C4Dic9412yes—
327C2×C22⋊C4D18812yes—
328#95—22no—
329#95—22no—
330#95—944no—
331#95—1884no—
332C24—42no—
333C24—42no—
334C24C2×Dic9411yescharacteristic
335C24C2×D18811yescharacteristic
336C24#1062535011yescharacteristic
337C24#1062535011yescharacteristic
338C24D4×D4712yes—
339C24#1062535212yes—
340C24Q8×D4712yes—
341C24#1062535412yes—
342C24C22×Dic4711yescharacteristic
343C24C2×C22⋊D4711yescharacteristic
344C24C2×C22⋊D4711yescharacteristic
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—944no—
370C22×C4—944no—
371C22×C4—1884no—
372C22×C4—1884no—
373C22×C4—1884no—
374C22×C4—1884no—
375C22×C4—1882no—
376C22×C4—1882no—
377C22×C4—1882no—
378C22×C4—1882no—
379C22×C4C4⋊Dic4714yescomplemented
380C22×C4C22⋊Dic4712yes—
381C22×C4C22⋊Dic4712yes—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—1888no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—41no—
390C22⋊C4—42no—
391C22⋊C4—41no—
392C22⋊C4—944no—
393C22⋊C4—944no—
394C22⋊C4—944no—
395C22⋊C4—944no—
396C22⋊C4—1882no—
397C22⋊C4—1882no—
398C22⋊C4—1882no—
399C22⋊C4—1882no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23C2×C22⋊Dic4711yescharacteristic
417C23C2×C4⋊Dic4711yescharacteristic
418C23#1309398311yescharacteristic
419C23D4×Dic4712yes—
420C23Q8×Dic4712yes—
421C23#1309399211yescharacteristic
422C23#1309399612yes—
423C23#1309399811yescharacteristic
424C23#1309399811yescharacteristic
425C23#1309399912yes—
426C23#1309400012yes—
427C23#1309400212yes—
428C23#1309400311yescharacteristic
429C23#1309400311yescharacteristic
430C23#1309400412yes—
431C23#1309400511yescharacteristic
432C23#1309400912yes—
433C23#1309401311yescharacteristic
434C23#1309401412yes—
435C23#1309401512yes—
436C23#1309401912yes—
437C23#1309402211yescharacteristic
438C23#1309402312yes—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—3762no—
462C2×C4—3762no—
463C2×C4—3764no—
464C2×C4—3764no—
465C2×C4—3764no—
466C2×C4—1884no—
467C2×C4—1884no—
468C2×C4—1888no—
469C2×C4—1884no—
470C2×C4—1884no—
471C2×C4#1309393914yes—
472C22—21no—
473C22—21no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—21no—
479C22—21no—
480C22—22no—
481C22—24no—
482C22—24no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22#1568898311yescharacteristic
499C22#1568900311yescharacteristic
500C22#1568900912yes—
501C22#1568902212yes—
502C22#1568903011yescharacteristic
503C22#1568904011yescharacteristic
504C22#1568904311yescharacteristic
505C22#1568904512yes—
506C22#1568905311yescharacteristic
507C22#1568905612yes—
508C22#1568906211yescharacteristic
509C22C2×#1309393912yes—
510C22#1568909712yes—
511C4—24no—
512C4—42no—
513C4—44no—
514C4—81no—
515C4—81no—
516C4—3764no—
517C4—3764no—
518C2—21no—
519C2—22no—
520C2—22no—
521C2—22no—
522C2—21no—
523C2—21no—
524C2—22no—
525C2—21no—
526C2#2024164111yescharacteristic
527C2#2024337612yes—
528C2#2024384211yescharacteristic
529C2#2024384711yescharacteristic
530C2#2024437412yes—
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC24×C47C22×C4
FrattiniC25C22×D47
FittingC2×C47×#385C2
socleC23×C47C2×C22⋊C4
prime radical, cosocleC25×C47C23
hypercenterC2×#385D47
hypocenterC47#11203
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
15. C22×C4⋊C4×C47C4semidirect
21. C2×#95×C47C4semidirect
105. C22×C4×C47C4⋊C4semidirect
273. C22×C4⋊C4Dic47semidirect
284. C2×#95Dic47semidirect
287. C47#11203semidirect
379. C22×C4C4⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889832
C2×#156888902
#202416112
C2×C47×#3852
#1120347