Finite Group Databasebeta

← #26124142

Subgroups — #26124142

Order 12032.

1042 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689087C211yesmaximal, characteristic
2C2×#15688983C211yesmaximal, characteristic
3C2×#15688891C212yesmaximal, complemented
4C2×#15688891C212yesmaximal, complemented
5C2×C47×#407C211yesmaximal, characteristic
6C22×#10625329C2212yes—
7C22×C4⋊Dic47C2211yescharacteristic
8C22×C4⋊Dic47C2211yescharacteristic
9C22×C22⋊Dic47C412yes—
10#15689087—24no—
11C2×C22⋊(C47⋊C8)—21no—
12C2×C22⋊(C47⋊C8)—21no—
13#15688983—21no—
14#15688983—21no—
15#15688983—21no—
16#15688983—21no—
17C2×C47⋊#95C412yes—
18#15688891—24no—
19#15688891—24no—
20C24×C4×C47C2211yescharacteristic
21C22×M16×C47C2212yes—
22C47×#407—24no—
23C2×C22⋊C8×C47—21no—
24C2×C22⋊C8×C47—21no—
25C23×Dic47C2×C412yes—
26C23×Dic47C2×C412yes—
27C2×#10625329—22no—
28C2×#10625329—22no—
29C2×#10625329—22no—
30C2×#10625329—22no—
31C2×#10625329—22no—
32C2×#10625329D414yes—
33C22×C47⋊C8—22no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47—21no—
38C2×C4⋊Dic47—22no—
39C2×C4⋊Dic47—22no—
40C2×C4⋊Dic47—22no—
41C2×C4⋊Dic47—22no—
42C2×C22⋊Dic47—22no—
43C2×C22⋊Dic47—22no—
44C2×C22⋊Dic47—44no—
45C22⋊(C47⋊C8)—24no—
46C22⋊(C47⋊C8)—24no—
47C47⋊#95—22no—
48C47⋊#95—22no—
49C25×C47C2×C411yescharacteristic
50C23×C4×C47—22no—
51C23×C4×C47—22no—
52C23×C4×C47—22no—
53C23×C4×C47—22no—
54C23×C4×C47—22no—
55C23×C4×C47—22no—
56C23×C4×C47C2×C411yescharacteristic
57C23×C4×C47D411yescharacteristic
58C23×C4×C47D411yescharacteristic
59C23×C4×C47D411yescharacteristic
60C23×C4×C47D411yescharacteristic
61C23×C4×C47C2311yescharacteristic
62C2×M16×C47—22no—
63C2×M16×C47—22no—
64C2×M16×C47—22no—
65C2×M16×C47—22no—
66C2×M16×C47—22no—
67C2×M16×C47D414yes—
68C22×C8×C47—22no—
69C22⋊C8×C47—24no—
70C22⋊C8×C47—24no—
71C24×C47—22no—
72C24×C47—22no—
73C24×C47—21no—
74C24×C47—21no—
75C24×C47—22no—
76C24×C47—22no—
77C24×C47—41no—
78C24×C47—41no—
79C24×C47C22⋊C411yescharacteristic
80C24×C47C22⋊C411yescharacteristic
81C24×C47C22×C411yescharacteristic
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—22no—
91C22×C4×C47—22no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—42no—
103C22×C4×C47—42no—
104C22×C4×C47—42no—
105C22×C4×C47—42no—
106C22×C4×C47—42no—
107C22×C4×C47—42no—
108C22×C4×C47—42no—
109C22×C4×C47—42no—
110C22×C4×C47C22⋊C411yescharacteristic
111C22×C4×C47C22⋊C411yescharacteristic
112C22×C4×C47C2×D412yes—
113C22×C4×C47C2×D412yes—
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47#4112yes—
117C22×C4×C47#4112yes—
118C22×C4×C47#4112yes—
119C22×C4×C47#4112yes—
120C22×Dic47—22no—
121C22×Dic47—22no—
122C22×Dic47—24no—
123C22×Dic47—24no—
124C22×Dic47—42no—
125C22×Dic47—42no—
126C22×Dic47—42no—
127C22×Dic47—42no—
128M16×C47—24no—
129M16×C47—24no—
130M16×C47—42no—
131M16×C47—42no—
132C2×C8×C47—24no—
133C2×C8×C47—42no—
134C2×C8×C47—42no—
135C22⋊Dic47—44no—
136C22⋊Dic47—44no—
137C4⋊Dic47—41no—
138C4⋊Dic47—41no—
139C4⋊Dic47—42no—
140C4⋊Dic47—42no—
141C4⋊Dic47—41no—
142C4⋊Dic47—41no—
143#10625329—24no—
144#10625329—24no—
145#10625329—42no—
146#10625329—42no—
147C2×C47⋊C8—24no—
148C2×C47⋊C8—42no—
149C2×C47⋊C8—42no—
150C23×C47—21no—
151C23×C47—21no—
152C23×C47—21no—
153C23×C47—21no—
154C23×C47—24no—
155C23×C47—24no—
156C23×C47—21no—
157C23×C47—21no—
158C23×C47—21no—
159C23×C47—21no—
160C23×C47—21no—
161C23×C47—21no—
162C23×C47—21no—
163C23×C47—21no—
164C23×C47—21no—
165C23×C47—21no—
166C23×C47—21no—
167C23×C47—21no—
168C23×C47—44no—
169C23×C47—44no—
170C23×C47—44no—
171C23×C47—44no—
172C23×C47—42no—
173C23×C47—42no—
174C23×C47—42no—
175C23×C47—42no—
176C23×C47#10012yes—
177C23×C47#10112yes—
178C23×C47C2×C22⋊C411yescharacteristic
179C23×C47#11711yescharacteristic
180C23×C47#11711yescharacteristic
181C23×C47C4×D412yes—
182C23×C47C4×D412yes—
183C2×C4×C47—21no—
184C2×C4×C47—21no—
185C2×C4×C47—21no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—21no—
189C2×C4×C47—42no—
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×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47#12011yescharacteristic
200C2×C4×C47#12011yescharacteristic
201C2×C4×C47#12112yes—
202C2×C4×C47#12112yes—
203C2×C4×C47#12112yes—
204C2×C4×C47#12112yes—
205C2×C4×C47#12312yes—
206C2×C4×C47#12312yes—
207C2×C4×C47#12312yes—
208C2×C4×C47#12311yescharacteristic
209C2×C4×C47#12312yes—
210C2×C4×C47#12311yescharacteristic
211C2×C4×C47#12412yes—
212C2×C4×C47#12412yes—
213C2×C4×C47#12612yes—
214C2×C4×C47#12612yes—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—84no—
222C2×Dic47—84no—
223C8×C47—44no—
224C47⋊C8—44no—
225#11198—471nomaximal, Sylow
226C22×C47—21no—
227C22×C47—21no—
228C22×C47—21no—
229C22×C47—21no—
230C22×C47—22no—
231C22×C47—22no—
232C22×C47—22no—
233C22×C47—22no—
234C22×C47—22no—
235C22×C47—22no—
236C22×C47—22no—
237C22×C47—22no—
238C22×C47—22no—
239C22×C47—22no—
240C22×C47—48no—
241C22×C47—48no—
242C22×C47—41no—
243C22×C47—41no—
244C22×C47—41no—
245C22×C47—41no—
246C22×C47—41no—
247C22×C47—41no—
248C22×C47—41no—
249C22×C47—41no—
250C22×C47#38111yescharacteristic
251C22×C47#38611yescharacteristic
252C22×C47#38611yescharacteristic
253C22×C47#38812yes—
254C22×C47#38812yes—
255C22×C47C2×#10011yescharacteristic
256C22×C47C2×#10111yescharacteristic
257C22×C47#41311yescharacteristic
258C22×C47#41311yescharacteristic
259C4×C47—41no—
260C4×C47—41no—
261C4×C47#39412yes—
262C4×C47#39412yes—
263C4×C47#39712yes—
264C4×C47#39712yes—
265Dic47—82no—
266Dic47—82no—
267C2×#407—471no—
268C2×#407D4711yescharacteristic
269C2×#380—471no—
270C2×#343—472no—
271C2×#343—472no—
272C2×C47—22no—
273C2×C47—22no—
274C2×C47—42no—
275C2×C47—42no—
276C2×C47#144011yescharacteristic
277C2×C47#182111yescharacteristic
278C2×C47#182211yescharacteristic
279C2×C47#190011yescharacteristic
280C2×C47#190011yescharacteristic
281C2×C47#190111yescharacteristic
282C2×C47#190111yescharacteristic
283C24×C4C2×D4711yescharacteristic
284C22×M16—472no—
285C22×M16C2×D4712yes—
286C22×C4⋊C4—471no—
287C22×C4⋊C4—471no—
288C22×C22⋊C4—472no—
289#407—24no—
290#407—944no—
291C2×C22⋊C8—21no—
292C2×C22⋊C8—21no—
293C2×C22⋊C8—941no—
294C2×C22⋊C8—941no—
295#380—941no—
296#380—941no—
297#380—941no—
298#380—941no—
299C2×#95—472no—
300#343—944no—
301#343—944no—
302C47#1119811yescharacteristic, complemented, Sylow
303C25C4×D4711yescharacteristic
304C23×C4—22no—
305C23×C4—22no—
306C23×C4—22no—
307C23×C4—22no—
308C23×C4—22no—
309C23×C4—22no—
310C23×C4—472no—
311C23×C4—472no—
312C23×C4C4×D4711yescharacteristic
313C23×C4D18811yescharacteristic
314C23×C4D18811yescharacteristic
315C23×C4C22⋊D4711yescharacteristic
316C23×C4C22⋊D4711yescharacteristic
317C23×C4C22×D4711yescharacteristic
318C2×M16—22no—
319C2×M16—22no—
320C2×M16—22no—
321C2×M16—22no—
322C2×M16—22no—
323C2×M16—474no—
324C2×M16—942no—
325C2×M16—942no—
326C2×M16—942no—
327C2×M16—942no—
328C2×M16—942no—
329C2×M16C22⋊D4714yes—
330C22×C8—22no—
331C22×C8—942no—
332C2×C4⋊C4—941no—
333C2×C4⋊C4—941no—
334C2×C4⋊C4—942no—
335C2×C4⋊C4—942no—
336C2×C4⋊C4—942no—
337C2×C4⋊C4—942no—
338C2×C4⋊C4—941no—
339C2×C4⋊C4—941no—
340C2×C22⋊C4—942no—
341C2×C22⋊C4—942no—
342C2×C22⋊C4—1884no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345C22⋊C8—944no—
346C22⋊C8—944no—
347#95—942no—
348#95—942no—
349C24—22no—
350C24—22no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—41no—
356C24—41no—
357C24(C2×D47)⋊C411yescharacteristic
358C24(C2×D47)⋊C411yescharacteristic
359C24C2×C4×D4711yescharacteristic
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—944no—
389C22×C4—944no—
390C22×C4—942no—
391C22×C4—942no—
392C22×C4—1882no—
393C22×C4—1882no—
394C22×C4—1882no—
395C22×C4—1882no—
396C22×C4(C2×D47)⋊C411yescharacteristic
397C22×C4(C2×D47)⋊C411yescharacteristic
398C22×C4C2×D18811yescharacteristic
399C22×C4#1062535012yes—
400C22×C4D4×D4712yes—
401C22×C4#1062535212yes—
402C22×C4#1062535212yes—
403C22×C4#1062535212yes—
404C22×C4C2×C22⋊D4712yes—
405C22×C4C2×C22⋊D4711yescharacteristic
406M16—24no—
407M16—24no—
408M16—42no—
409M16—42no—
410M16—944no—
411M16—944no—
412M16—1882no—
413M16—1882no—
414C2×C8—24no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—944no—
418C2×C8—1882no—
419C2×C8—1882no—
420C4⋊C4—1881no—
421C4⋊C4—1881no—
422C4⋊C4—1882no—
423C4⋊C4—1882no—
424C4⋊C4—1881no—
425C4⋊C4—1881no—
426C22⋊C4—1884no—
427C22⋊C4—1884no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—24no—
433C23—24no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23#1309394012yes—
455C23#1309394212yes—
456C23C2×(C2×D47)⋊C411yescharacteristic
457C23#1309398011yescharacteristic
458C23#1309398011yescharacteristic
459C23(C22⋊D47)⋊C412yes—
460C23C4×C22⋊D4712yes—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—3764no—
478C2×C4—3764no—
479C2×C4—1882no—
480C2×C4—1882no—
481C2×C4—1882no—
482C2×C4—1882no—
483C2×C4—1882no—
484C2×C4—1882no—
485C2×C4#1309399311yescharacteristic
486C2×C4#1309399411yescharacteristic
487C2×C4#1309399512yes—
488C2×C4#1309399612yes—
489C2×C4#1309399612yes—
490C2×C4#1309399812yes—
491C2×C4#1309400512yes—
492C2×C4#1309400512yes—
493C2×C4#1309400612yes—
494C2×C4#1309400711yescharacteristic
495C2×C4#1309400812yes—
496C2×C4#1309400811yescharacteristic
497C2×C4#1309401012yes—
498C2×C4#1309401112yes—
499C2×C4#1309401712yes—
500C2×C4#1309401712yes—
501C8—44no—
502C8—1884no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—48no—
518C22—48no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22#1568898411yescharacteristic
528C22#1568900011yescharacteristic
529C22#1568900211yescharacteristic
530C22#1568901112yes—
531C22#1568901712yes—
532C22C2×#1309394011yescharacteristic
533C22C2×#1309394211yescharacteristic
534C22#1568910511yescharacteristic
535C22#1568910511yescharacteristic
536C4—41no—
537C4—41no—
538C4—3762no—
539C4—3762no—
540C4#1568903712yes—
541C4#1568903912yes—
542C4#1568904812yes—
543C4#1568905112yes—
544C2—22no—
545C2—22no—
546C2—42no—
547C2—42no—
548C2#2024163511yescharacteristic
549C2#2024346611yescharacteristic
550C2#2024346911yescharacteristic
551C2#2024384911yescharacteristic
552C2#2024385111yescharacteristic
553C2#2024385411yescharacteristic
554C2#2024385611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#407C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenterC2×#407D47
hypocenterC47#11198
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
3. C2×#15688891C2semidirect
4. C2×#15688891C2semidirect
302. C47#11198semidirect

Maximal subgroups

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

subgroupindex
C2×#156890872
C2×#156889832
C2×#156888912
C2×C47×#4072
#1119847