Finite Group Databasebeta

← #26131226

Subgroups — #26131226

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093939C211yesmaximal, characteristic
2#20245966C211yesmaximal, characteristic
3#20241507C214yesmaximal, complemented
4C22×#100×C47C211yesmaximal, characteristic
5C22×C22⋊Dic47—21no—
6C22×C22⋊Dic47C411yescharacteristic, complemented
7C22×C22⋊Dic47C411yescharacteristic, complemented
8C2×#13093939—21no—
9C2×#13093939—22no—
10C2×#13093939C414yescomplemented
11C2×#13093939C2212yes—
12#15689003—21no—
13#15689003—21no—
14#15689003C412yescomplemented
15#15689003C2212yes—
16#15688986—21no—
17#15688979—21no—
18C23×D4×C47C2211yescharacteristic
19C22×M16×C47—21no—
20C2×#100×C47—22no—
21C2×#100×C47—21no—
22C2×#100×C47C414yescomplemented
23C2×#100×C47C2212yes—
24C23×Dic47—21no—
25C23×Dic47—41no—
26C23×Dic47C2×C412yescomplemented
27C2×C22⋊Dic47—24no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—41no—
35C2×C22⋊Dic47—42no—
36C2×C22⋊Dic47—41no—
37C2×C22⋊Dic47—41no—
38C2×C22⋊Dic47—42no—
39C2×C22⋊Dic47C2×C412yescomplemented
40#13093939—24no—
41#13093939—22no—
42#13093939C2×C414yescomplemented
43C47⋊#95—22no—
44C47⋊#95—22no—
45C25×C47C2×C411yescharacteristic
46C25×C47D411yescharacteristic
47C22×D4×C47—41no—
48C22×D4×C47—41no—
49C22×D4×C47—42no—
50C22×D4×C47C2×C411yescharacteristic
51C22×D4×C47C2×C412yes—
52C22×D4×C47C2×C412yes—
53C22×D4×C47D411yescharacteristic
54C22×D4×C47D412yes—
55C22×D4×C47D411yescharacteristic
56C22×D4×C47Q812yes—
57C22×D4×C47C2311yescharacteristic
58C23×C4×C47D411yescharacteristic
59C2×M16×C47—24no—
60C2×M16×C47—22no—
61C2×M16×C47—42no—
62C2×M16×C47—41no—
63C22×C8×C47—41no—
64#100×C47—24no—
65#100×C47—22no—
66#100×C47C2×C414yescomplemented
67C24×C47—21no—
68C24×C47—21no—
69C24×C47—21no—
70C24×C47—22no—
71C24×C47—22no—
72C24×C47—21no—
73C24×C47—82no—
74C24×C47—82no—
75C24×C47C4212yes—
76C24×C47C22⋊C411yescharacteristic
77C24×C47C22⋊C411yescharacteristic
78C24×C47C22⋊C412yes—
79C24×C47C22⋊C411yescharacteristic
80C24×C47C4⋊C412yes—
81C24×C47C4⋊C412yes—
82C24×C47C22×C411yescharacteristic
83C24×C47C2×D411yescharacteristic
84C2×D4×C47—42no—
85C2×D4×C47—42no—
86C2×D4×C47—42no—
87C2×D4×C47—42no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47—42no—
91C2×D4×C47—42no—
92C2×D4×C47—41no—
93C2×D4×C47—41no—
94C2×D4×C47—41no—
95C2×D4×C47—41no—
96C2×D4×C47—42no—
97C2×D4×C47—42no—
98C2×D4×C47—41no—
99C2×D4×C47—41no—
100C2×D4×C47C4212yes—
101C2×D4×C47C22⋊C412yes—
102C2×D4×C47C22⋊C412yes—
103C2×D4×C47C4⋊C412yes—
104C2×D4×C47C4⋊C412yes—
105C2×D4×C47C4⋊C412yes—
106C2×D4×C47C22×C411yescharacteristic
107C2×D4×C47C22×C411yescharacteristic
108C2×D4×C47C2×D411yescharacteristic
109C2×D4×C47C2×Q811yescharacteristic
110C22×C4×C47—21no—
111C22×C4×C47—22no—
112C22×C4×C47—21no—
113C22×C4×C47C22⋊C411yescharacteristic
114C22×C4×C47C22⋊C412yes—
115C22×C4×C47C4⋊C412yes—
116C22×C4×C47C2×D411yescharacteristic
117C22×Dic47—22no—
118C22×Dic47—22no—
119C22×Dic47—44no—
120C22×Dic47—42no—
121C22×Dic47—42no—
122C22×Dic47—42no—
123C22×Dic47—82no—
124C22×Dic47—82no—
125C22×Dic47—84no—
126M16×C47—24no—
127M16×C47—44no—
128M16×C47—42no—
129C2×C8×C47—44no—
130C2×C8×C47—42no—
131C22⋊Dic47—24no—
132C22⋊Dic47—44no—
133C22⋊Dic47—42no—
134C22⋊Dic47—42no—
135C22⋊Dic47—41no—
136C22⋊Dic47—41no—
137C22⋊Dic47—42no—
138C22⋊Dic47—42no—
139C22⋊Dic47—82no—
140C22⋊Dic47—82no—
141C23×C47—22no—
142C23×C47—22no—
143C23×C47—21no—
144C23×C47—21no—
145C23×C47—21no—
146C23×C47—21no—
147C23×C47—21no—
148C23×C47—21no—
149C23×C47—22no—
150C23×C47—22no—
151C23×C47—22no—
152C23×C47—22no—
153C23×C47—22no—
154C23×C47—22no—
155C23×C47—21no—
156C23×C47—21no—
157C23×C47—42no—
158C23×C47—42no—
159C23×C47—42no—
160C23×C47—44no—
161C23×C47—42no—
162C23×C47—42no—
163C23×C47—82no—
164C23×C47—82no—
165C23×C47—82no—
166C23×C47—82no—
167C23×C47—82no—
168C23×C47—82no—
169C23×C47—82no—
170C23×C47—84no—
171C23×C47—84no—
172C23×C47#9512yes—
173C23×C47#9512yes—
174C23×C47#9512yes—
175C23×C47#9911yescharacteristic
176C23×C47#9911yescharacteristic
177C23×C47#9912yes—
178C23×C47C2×C4211yescharacteristic
179C23×C47C2×C22⋊C411yescharacteristic
180C23×C47C2×C22⋊C411yescharacteristic
181C23×C47C2×C4⋊C411yescharacteristic
182C23×C47C2×C4⋊C411yescharacteristic
183D4×C47—42no—
184D4×C47—42no—
185D4×C47—42no—
186D4×C47—42no—
187D4×C47—42no—
188D4×C47—42no—
189D4×C47—41no—
190D4×C47—41no—
191D4×C47—41no—
192D4×C47—41no—
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47#9512yes—
202C2×C4×C47C2×C22⋊C411yescharacteristic
203C2×C4×C47C2×C4⋊C411yescharacteristic
204C2×Dic47—44no—
205C2×Dic47—42no—
206C2×Dic47—42no—
207C2×Dic47—42no—
208C2×Dic47—42no—
209C2×Dic47—42no—
210C2×Dic47—42no—
211C2×Dic47—84no—
212C2×Dic47—88no—
213C2×Dic47—84no—
214C2×Dic47—84no—
215C8×C47—44no—
216#12574—471nomaximal, Sylow
217C22×C47—22no—
218C22×C47—22no—
219C22×C47—22no—
220C22×C47—21no—
221C22×C47—22no—
222C22×C47—22no—
223C22×C47—21no—
224C22×C47—21no—
225C22×C47—21no—
226C22×C47—42no—
227C22×C47—42no—
228C22×C47—42no—
229C22×C47—42no—
230C22×C47—41no—
231C22×C47—41no—
232C22×C47—41no—
233C22×C47—41no—
234C22×C47—42no—
235C22×C47—44no—
236C22×C47—42no—
237C22×C47—44no—
238C22×C47—42no—
239C22×C47—42no—
240C22×C47—42no—
241C22×C47—42no—
242C22×C47—81no—
243C22×C47—81no—
244C22×C47—82no—
245C22×C47—82no—
246C22×C47—82no—
247C22×C47—82no—
248C22×C47—82no—
249C22×C47—82no—
250C22×C47#34212yes—
251C22×C47#34212yes—
252C22×C47C2×#9511yescharacteristic
253C22×C47C2×#9911yescharacteristic
254C22×C47C2×#9911yescharacteristic
255C4×C47—21no—
256C4×C47—22no—
257C4×C47—21no—
258Dic47—82no—
259Dic47—84no—
260Dic47—82no—
261C22×#100D4711yescharacteristic
262C22×#99—471no—
263#2308—471no—
264#1390—474no—
265C2×C47—21no—
266C2×C47—21no—
267C2×C47—22no—
268C2×C47—22no—
269C2×C47—41no—
270C2×C47—42no—
271C2×C47—41no—
272C2×C47—42no—
273C2×C47—41no—
274C2×C47—41no—
275C2×C47C2×#34211yescharacteristic
276C2×C47#212112yes—
277C23×D4C2×D4711yescharacteristic
278C22×M16—21no—
279C22×C22⋊C4—471no—
280C22×C22⋊C4—471no—
281C22×C22⋊C4—941no—
282C2×#100—21no—
283C2×#100—22no—
284C2×#100Dic4714yescomplemented
285C2×#100C2×D4712yes—
286C2×#99—474no—
287C2×#99—472no—
288C2×#99—941no—
289C2×#99—942no—
290#386—472no—
291#386—472no—
292#386—941no—
293#386—941no—
294#381—941no—
295#379—941no—
296C47#1257411yescharacteristic, complemented, Sylow
297C25C4×D4711yescharacteristic
298C25D18811yescharacteristic
299C22×D4—41no—
300C22×D4—41no—
301C22×D4—42no—
302C22×D4Dic9412yes—
303C22×D4C4×D4712yes—
304C22×D4C4×D4711yescharacteristic
305C22×D4D18811yescharacteristic
306C22×D4C2×Dic4712yes—
307C22×D4C22⋊D4712yes—
308C22×D4C22⋊D4711yescharacteristic
309C22×D4C22×D4711yescharacteristic
310C23×C4—472no—
311C23×C4—941no—
312C23×C4—1881no—
313C23×C4C22⋊D4711yescharacteristic
314C2×M16—24no—
315C2×M16—22no—
316C2×M16—41no—
317C2×M16—42no—
318C22×C8—41no—
319C2×C22⋊C4—472no—
320C2×C22⋊C4—944no—
321C2×C22⋊C4—942no—
322C2×C22⋊C4—942no—
323C2×C22⋊C4—942no—
324C2×C22⋊C4—941no—
325C2×C22⋊C4—941no—
326C2×C22⋊C4—941no—
327C2×C22⋊C4—1882no—
328C2×C22⋊C4—1881no—
329C2×C22⋊C4—1881no—
330C2×C22⋊C4—1881no—
331C2×C22⋊C4—1882no—
332#100—24no—
333#100—22no—
334#100C2×Dic4714yescomplemented
335#99—474no—
336#99—944no—
337#99—942no—
338#95—942no—
339#95—942no—
340C24—21no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—82no—
347C24—82no—
348C24C4×Dic4712yes—
349C24Dic47⋊C412yes—
350C24C4⋊Dic4712yes—
351C24(C2×D47)⋊C411yescharacteristic
352C24(C2×D47)⋊C411yescharacteristic
353C24(C2×D47)⋊C412yes—
354C24(C2×D47)⋊C411yescharacteristic
355C24C2×C4×D4711yescharacteristic
356C24C2×D18811yescharacteristic
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4C4×Dic4712yes—
374C2×D4Dic47⋊C412yes—
375C2×D4Dic47⋊C412yes—
376C2×D4C4⋊Dic4712yes—
377C2×D4(C2×D47)⋊C412yes—
378C2×D4C22⋊Dic4712yes—
379C2×D4C2×Dic9411yescharacteristic
380C2×D4C2×C4×D4711yescharacteristic
381C2×D4C22×Dic4711yescharacteristic
382C2×D4C2×C22⋊D4711yescharacteristic
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—942no—
387C22×C4—942no—
388C22×C4—3764no—
389C22×C4—3762no—
390C22×C4—3762no—
391C22×C4—1882no—
392C22×C4—1884no—
393C22×C4—1882no—
394C22×C4—1882no—
395C22×C4Dic47⋊C412yes—
396C22×C4(C2×D47)⋊C411yescharacteristic
397C22×C4C22⋊Dic4712yes—
398C22×C4C2×C22⋊D4711yescharacteristic
399M16—24no—
400M16—42no—
401M16—44no—
402C2×C8—42no—
403C2×C8—44no—
404C22⋊C4—944no—
405C22⋊C4—3762no—
406C22⋊C4—3762no—
407C22⋊C4—1882no—
408C22⋊C4—1884no—
409C22⋊C4—1882no—
410C22⋊C4—1882no—
411C22⋊C4—1881no—
412C22⋊C4—1881no—
413C22⋊C4—1882no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—84no—
444C23—84no—
445C23C47⋊#9512yes—
446C23C47⋊#9512yes—
447C23C47⋊#9512yes—
448C23#1309393811yescharacteristic
449C23#1309393811yescharacteristic
450C23#1309393812yes—
451C23C2×C4×Dic4711yescharacteristic
452C23C2×(C2×D47)⋊C411yescharacteristic
453C23C2×(C2×D47)⋊C411yescharacteristic
454C23C2×Dic47⋊C411yescharacteristic
455C23C2×C4⋊Dic4711yescharacteristic
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—3764no—
475C2×C4—3768no—
476C2×C4—3764no—
477C2×C4—3764no—
478C2×C4—1884no—
479C2×C4—1882no—
480C2×C4—1882no—
481C2×C4—1882no—
482C2×C4—1882no—
483C2×C4—1882no—
484C2×C4—1882no—
485C2×C4C47⋊#9512yes—
486C2×C4C2×C22⋊Dic4711yescharacteristic
487C2×C4C2×Dic47⋊C411yescharacteristic
488C8—44no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—42no—
507C22—44no—
508C22—42no—
509C22—44no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—81no—
515C22—81no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22#1568888912yes—
523C22#1568888912yes—
524C22C2×C47⋊#9511yescharacteristic
525C22C2×#1309393811yescharacteristic
526C22C2×#1309393811yescharacteristic
527C4—21no—
528C4—22no—
529C4—21no—
530C4—3762no—
531C4—3764no—
532C4—3762no—
533C2—21no—
534C2—21no—
535C2—22no—
536C2—22no—
537C2—41no—
538C2—42no—
539C2—41no—
540C2—42no—
541C2—41no—
542C2—41no—
543C2C2×#1568888911yescharacteristic
544C2#2024503512yes—
subgroupG/H
centerC22C2×#13093938
commutatorC23×C47C2×C42
FrattiniC22×D4C22×D47
FittingC22×#100×C47C2
socleC22×C47C2×#99
prime radicalC23×D4×C47C22
cosocleC22×D4×C47C23
hypercenterC22×#100D47
hypocenterC47#12574
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C22×#100

Complements

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

NH ≅ G/Ntype
3. #20241507C2semidirect
6. C22×C22⋊Dic47C4semidirect
7. C22×C22⋊Dic47C4semidirect
10. C2×#13093939C4semidirect
14. #15689003C4semidirect
22. C2×#100×C47C4semidirect
26. C23×Dic47C2×C4semidirect
39. C2×C22⋊Dic47C2×C4semidirect
42. #13093939C2×C4semidirect
66. #100×C47C2×C4semidirect
284. C2×#100Dic47semidirect
296. C47#12574semidirect
334. #100C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C22×#130939392
#202459662
#202415072
C22×#100×C472
#1257447