Finite Group Databasebeta

← #26107606

Subgroups — #26107606

Order 12032.

1502 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D47C211yesmaximal, characteristic
2#20243297C212yesmaximal
3C2×#15688849C211yesmaximal, characteristic, complemented
4C2×#15688848C212yesmaximal, complemented
5C2×C47×#324C211yesmaximal, characteristic, complemented
6C23×C4×D47C411yescharacteristic
7C23×C4×D47C411yescharacteristic
8C23×C4×D47C2211yescharacteristic
9C22×C4⋊Dic47C411yescharacteristic
10C22×Dic47⋊C4C411yescharacteristic
11C22×Dic47⋊C4C2211yescharacteristic
12C2×C4⋊C4×D47—24no—
13C2×C4⋊C4×D47—24no—
14C2×C4⋊C4×D47—22no—
15C2×C4⋊C4×D47—22no—
16C2×C22⋊(C47⋊C8)C2212yes—
17C2×(C2×D47)⋊C8—24no—
18#15688849—22no—
19#15688848—24no—
20C22×C4⋊C4×C47C2211yescharacteristic
21C2×C22⋊C8×C47C2212yes—
22C47×#324—22no—
23C24×D47C2×C411yescharacteristic
24C23×Dic47C2×C411yescharacteristic
25C23×Dic47C2×C411yescharacteristic
26C23×Dic47C2×C411yescharacteristic
27C22×C4×D47—22no—
28C22×C4×D47—22no—
29C22×C4×D47—22no—
30C22×C4×D47—22no—
31C22×C4×D47—22no—
32C22×C4×D47—22no—
33C22×C4×D47—22no—
34C22×C4×D47—22no—
35C22×C4×D47—21no—
36C22×C4×D47—22no—
37C22×C4×D47—21no—
38C22×C4×D47—22no—
39C22×C4×D47—44no—
40C22×C4×D47—44no—
41C22×C4×D47C814yescomplemented
42C22×C4×D47D414yes—
43C22×C47⋊C8—22no—
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—22no—
47C2×Dic47⋊C4—22no—
48C2×Dic47⋊C4—22no—
49C2×Dic47⋊C4—21no—
50C2×Dic47⋊C4—21no—
51C2×Dic47⋊C4—22no—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4D414yes—
54C4⋊C4×D47—216no—
55C4⋊C4×D47—216no—
56C22⋊(C47⋊C8)—22no—
57C22⋊(C47⋊C8)—22no—
58(C2×D47)⋊C8—48no—
59C23×C4×C47C2×C411yescharacteristic
60C23×C4×C47C2×C411yescharacteristic
61C23×C4×C47C2311yescharacteristic
62C22×C8×C47—22no—
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47D412yes—
66C2×C4⋊C4×C47D412yes—
67C22⋊C8×C47—22no—
68C22⋊C8×C47—22no—
69C24×C47C22×C411yescharacteristic
70C23×D47—21no—
71C23×D47—21no—
72C23×D47—21no—
73C23×D47—21no—
74C23×D47—24no—
75C23×D47—24no—
76C23×D47C22⋊C412yes—
77C23×D47C2×C812yes—
78C23×D47M1612yes—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—42no—
86C22×C4×C47—42no—
87C22×C4×C47C22⋊C412yes—
88C22×C4×C47C22⋊C412yes—
89C22×C4×C47C2×C812yes—
90C22×C4×C47M1612yes—
91C22×C4×C47C2×D412yes—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D412yes—
94C22×C4×C47C2×D411yescharacteristic
95C22×Dic47—21no—
96C22×Dic47—21no—
97C22×Dic47—21no—
98C22×Dic47—21no—
99C22×Dic47—21no—
100C22×Dic47—21no—
101C22×Dic47—21no—
102C22×Dic47—21no—
103C22×Dic47—42no—
104C22×Dic47—42no—
105C22×Dic47C22⋊C412yes—
106C22×Dic47C22⋊C412yes—
107C22×Dic47C22⋊C412yes—
108C22×Dic47C2×C812yes—
109C22×Dic47M1612yes—
110C2×C4×D47—28no—
111C2×C4×D47—28no—
112C2×C4×D47—28no—
113C2×C4×D47—28no—
114C2×C4×D47—28no—
115C2×C4×D47—28no—
116C2×C4×D47—28no—
117C2×C4×D47—28no—
118C2×C4×D47—28no—
119C2×C4×D47—24no—
120C2×C4×D47—24no—
121C2×C4×D47—44no—
122C2×C4×D47—44no—
123C2×C4×D47—48no—
124C2×C4×D47—44no—
125C2×C4×D47—48no—
126C2×C4×D47—44no—
127C2×C4×D47—48no—
128C2×C4×D47—44no—
129C2×C8×C47—44no—
130C2×C8×C47—42no—
131C4⋊C4×C47—24no—
132C4⋊C4×C47—24no—
133C4⋊Dic47—28no—
134Dic47⋊C4—28no—
135Dic47⋊C4—24no—
136Dic47⋊C4—24no—
137C2×C47⋊C8—44no—
138C2×C47⋊C8—42no—
139C23×C47—22no—
140C23×C47—22no—
141C23×C47C2×C22⋊C411yescharacteristic
142C23×C47C2×C22⋊C411yescharacteristic
143C23×C47C2×C22⋊C411yescharacteristic
144C23×C47C22×C811yescharacteristic
145C23×C47C2×M1611yescharacteristic
146C23×C47C2×M1611yescharacteristic
147C23×C47C2×M1611yescharacteristic
148C22×D47—24no—
149C22×D47—24no—
150C22×D47—28no—
151C22×D47—28no—
152C22×D47—24no—
153C22×D47—24no—
154C22×D47—22no—
155C22×D47—22no—
156C22×D47—22no—
157C22×D47—22no—
158C22×D47—22no—
159C22×D47—22no—
160C22×D47—22no—
161C22×D47—22no—
162C22×D47—28no—
163C22×D47—28no—
164C22×D47C22⋊C814yes—
165C22×D47#9914yes—
166C22×D47#10114yes—
167C2×C4×C47—22no—
168C2×C4×C47—22no—
169C2×C4×C47—22no—
170C2×C4×C47—22no—
171C2×C4×C47—22no—
172C2×C4×C47—22no—
173C2×C4×C47—24no—
174C2×C4×C47—22no—
175C2×C4×C47—41no—
176C2×C4×C47—42no—
177C2×C4×C47—41no—
178C2×C4×C47—42no—
179C2×C4×C47—42no—
180C2×C4×C47—42no—
181C2×C4×C47C22⋊C814yescomplemented
182C2×C4×C47#12012yes—
183C2×C4×C47#12012yes—
184C2×Dic47—22no—
185C2×Dic47—22no—
186C2×Dic47—22no—
187C2×Dic47—22no—
188C2×Dic47—22no—
189C2×Dic47—22no—
190C2×Dic47—22no—
191C2×Dic47—22no—
192C2×Dic47—24no—
193C2×Dic47—42no—
194C2×Dic47—42no—
195C2×Dic47—42no—
196C2×Dic47—42no—
197C2×Dic47—42no—
198C2×Dic47C22⋊C814yescomplemented
199C4×D47—232no—
200C4×D47—48no—
201C4×D47—48no—
202C4×D47—416no—
203C8×C47—82no—
204C47⋊C8—82no—
205#8299—471nomaximal, Sylow
206C22×C47—22no—
207C22×C47—22no—
208C22×C47—22no—
209C22×C47—22no—
210C22×C47—21no—
211C22×C47—21no—
212C22×C47—21no—
213C22×C47—21no—
214C22×C47#37911yescharacteristic
215C22×C47C2×C22⋊C811yescharacteristic
216C22×C47C2×C22⋊C811yescharacteristic
217C22×C47C2×C22⋊C811yescharacteristic
218C22×C47#40711yescharacteristic
219C22×C47#40711yescharacteristic
220C22×C47#40711yescharacteristic
221C22×C47C2×#9912yes—
222C22×C47C2×#10112yes—
223C2×D47—28no—
224C2×D47—28no—
225C2×D47—28no—
226C2×D47—28no—
227C2×D47—24no—
228C2×D47—24no—
229C2×D47—24no—
230C2×D47—24no—
231C2×D47—24no—
232C2×D47—24no—
233C2×D47#32418yescomplemented
234C4×C47—24no—
235C4×C47—22no—
236C4×C47—22no—
237C4×C47—42no—
238Dic47—24no—
239Dic47—42no—
240Dic47—42no—
241C23×C4⋊C4—471no—
242#1779—472no—
243C2×#324—472no—
244C2×#324—471no—
245C2×#324D4711yescharacteristic, complemented
246C2×C47—21no—
247C2×C47—21no—
248C2×C47—21no—
249C2×C47—21no—
250C2×C47C2×#32412yes—
251C2×C47#146312yes—
252C2×C47#177911yescharacteristic
253C2×C47#178111yescharacteristic
254C2×C47#178411yescharacteristic
255D47—28no—
256C24×C4—471no—
257C24×C4—471no—
258C24×C4—471no—
259C22×C4⋊C4—471no—
260C22×C4⋊C4—471no—
261C22×C4⋊C4—471no—
262C22×C4⋊C4—944no—
263C22×C4⋊C4—944no—
264C22×C4⋊C4—942no—
265C22×C4⋊C4—942no—
266C22×C4⋊C4C2×D4711yescharacteristic
267C2×C22⋊C8—472no—
268C2×C22⋊C8—944no—
269C2×C22⋊C8C2×D4712yes—
270#324—22no—
271#324—944no—
272#324—942no—
273C47#829911yescharacteristic, complemented, Sylow
274C25—471no—
275C23×C4—471no—
276C23×C4—471no—
277C23×C4—471no—
278C23×C4—474no—
279C23×C4—474no—
280C23×C4—942no—
281C23×C4—942no—
282C23×C4—942no—
283C23×C4—942no—
284C23×C4—942no—
285C23×C4—942no—
286C23×C4—942no—
287C23×C4—942no—
288C23×C4—942no—
289C23×C4—941no—
290C23×C4—942no—
291C23×C4—941no—
292C23×C4—1884no—
293C23×C4—1884no—
294C23×C4C4×D4711yescharacteristic
295C23×C4C4×D4711yescharacteristic
296C23×C4C22×D4711yescharacteristic
297C22×C8—22no—
298C22×C8—942no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—474no—
302C2×C4⋊C4—9416no—
303C2×C4⋊C4—9416no—
304C2×C4⋊C4—942no—
305C2×C4⋊C4—942no—
306C2×C4⋊C4—942no—
307C2×C4⋊C4—942no—
308C2×C4⋊C4—942no—
309C2×C4⋊C4—942no—
310C2×C4⋊C4—941no—
311C2×C4⋊C4—942no—
312C2×C4⋊C4—941no—
313C2×C4⋊C4D18812yes—
314C2×C4⋊C4C22⋊D4712yes—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—942no—
318C22⋊C8—942no—
319C22⋊C8—1888no—
320C24—472no—
321C24—472no—
322C24—472no—
323C24—944no—
324C24—944no—
325C24—941no—
326C24—941no—
327C24—941no—
328C24—941no—
329C24C2×C4×D4711yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—472no—
339C22×C4—472no—
340C22×C4—472no—
341C22×C4—472no—
342C22×C4—472no—
343C22×C4—948no—
344C22×C4—948no—
345C22×C4—948no—
346C22×C4—948no—
347C22×C4—948no—
348C22×C4—948no—
349C22×C4—948no—
350C22×C4—948no—
351C22×C4—941no—
352C22×C4—941no—
353C22×C4—941no—
354C22×C4—941no—
355C22×C4—948no—
356C22×C4—941no—
357C22×C4—941no—
358C22×C4—941no—
359C22×C4—941no—
360C22×C4—944no—
361C22×C4—944no—
362C22×C4—1884no—
363C22×C4—1884no—
364C22×C4—1884no—
365C22×C4—1888no—
366C22×C4—1884no—
367C22×C4—1888no—
368C22×C4—1884no—
369C22×C4—1888no—
370C22×C4—1882no—
371C22×C4—1882no—
372C22×C4C8×D4712yes—
373C22×C4#1062532412yes—
374C22×C4(C2×D47)⋊C412yes—
375C22×C4(C2×D47)⋊C412yes—
376C22×C4C2×D18811yescharacteristic
377C22×C4D4×D4712yes—
378C22×C4D4×D4712yes—
379C22×C4C2×C22⋊D4711yescharacteristic
380C2×C8—44no—
381C2×C8—42no—
382C2×C8—1884no—
383C2×C8—1882no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—948no—
387C4⋊C4—948no—
388C4⋊C4—944no—
389C4⋊C4—944no—
390C23—22no—
391C23—22no—
392C23—474no—
393C23—474no—
394C23—474no—
395C23—948no—
396C23—948no—
397C23—944no—
398C23—944no—
399C23—948no—
400C23—948no—
401C23—944no—
402C23—944no—
403C23—942no—
404C23—942no—
405C23—942no—
406C23—942no—
407C23—942no—
408C23—942no—
409C23—942no—
410C23—942no—
411C23C22⋊C4×D4711yescharacteristic
412C23C22⋊C4×D4711yescharacteristic
413C23C2×(C2×D47)⋊C411yescharacteristic
414C23C2×C8×D4711yescharacteristic
415C23M16×D4711yescharacteristic
416C23M16×D4711yescharacteristic
417C23C2×#1062532411yescharacteristic
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—474no—
433C2×C4—9432no—
434C2×C4—942no—
435C2×C4—942no—
436C2×C4—942no—
437C2×C4—942no—
438C2×C4—942no—
439C2×C4—942no—
440C2×C4—942no—
441C2×C4—942no—
442C2×C4—944no—
443C2×C4—1888no—
444C2×C4—1888no—
445C2×C4—18816no—
446C2×C4—1882no—
447C2×C4—1882no—
448C2×C4—1882no—
449C2×C4—1882no—
450C2×C4—1882no—
451C2×C4(C2×D47)⋊C814yescomplemented
452C2×C4#1309399212yes—
453C2×C4#1309399412yes—
454C8—82no—
455C8—3762no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—478no—
465C22—948no—
466C22—948no—
467C22—948no—
468C22—948no—
469C22—944no—
470C22—944no—
471C22—944no—
472C22—944no—
473C22—944no—
474C22—944no—
475C22#1568897811yescharacteristic
476C22C22⋊C8×D4711yescharacteristic
477C22C22⋊C8×D4711yescharacteristic
478C22C2×(C2×D47)⋊C811yescharacteristic
479C22#1568908511yescharacteristic
480C22#1568908511yescharacteristic
481C22#1568908611yescharacteristic
482C22#99×D4712yes—
483C22#101×D4712yes—
484C4—24no—
485C4—22no—
486C4—22no—
487C4—42no—
488C4—944no—
489C4—1882no—
490C4—1882no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—948no—
496C2#324×D4712yes—
497C2#2024171612yes—
498C2#2024329711yescharacteristic
499C2#2024330111yescharacteristic
500C2#2024331011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC23×C47C2×C22⋊C4
prime radicalC24×D47C2×C4
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#8299
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. C2×#15688849C2semidirect
4. C2×#15688848C2semidirect
5. C2×C47×#324C2semidirect
41. C22×C4×D47C8semidirect
181. C2×C4×C47C22⋊C8semidirect
198. C2×Dic47C22⋊C8semidirect
233. C2×D47#324semidirect
245. C2×#324D47semidirect
273. C47#8299semidirect
451. C2×C4(C2×D47)⋊C8semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D472
#202432972
C2×#156888492
C2×#156888482
C2×C47×#3242
#829947