Finite Group Databasebeta

← #26108944

Subgroups — #26108944

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic47⋊C4C211yesmaximal, characteristic
2#20243368C211yesmaximal, characteristic
3C2×#15688848C212yesmaximal
4#20241230C212yesmaximal
5C47×#1797C211yesmaximal, characteristic
6C22×Dic47⋊C4C411yescharacteristic
7C22×Dic47⋊C4C2211yescharacteristic
8C22×C4×Dic47C411yescharacteristic
9C22×C4×Dic47C411yescharacteristic
10C2×C4⋊(C47⋊C8)—21no—
11C2×C22⋊(C47⋊C8)C2212yes—
12C4×Dic47⋊C4—21no—
13C4×Dic47⋊C4—21no—
14C4×Dic47⋊C4—21no—
15C4×Dic47⋊C4—21no—
16C2×C47⋊#95C411yescharacteristic
17C2×C47⋊#95C2211yescharacteristic
18#15688878—21no—
19#15688848—24no—
20C22×C42×C47C2211yescharacteristic
21C2×C4⋊C8×C47—21no—
22C2×C22⋊C8×C47C2212yes—
23C47×#336—21no—
24C23×Dic47C2×C411yescharacteristic
25C23×Dic47C2×C411yescharacteristic
26C23×Dic47C2×C411yescharacteristic
27C23×Dic47C2×C411yescharacteristic
28C22×C47⋊C8—22no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—22no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—21no—
36C2×Dic47⋊C4C814yescomplemented
37C2×Dic47⋊C4D412yes—
38C2×Dic47⋊C4Q812yes—
39C2×C4×Dic47—21no—
40C2×C4×Dic47—21no—
41C2×C4×Dic47—22no—
42C2×C4×Dic47—22no—
43C2×C4×Dic47—21no—
44C2×C4×Dic47—21no—
45C2×C4×Dic47—42no—
46C4⋊(C47⋊C8)—42no—
47C22⋊(C47⋊C8)—22no—
48C22⋊(C47⋊C8)—22no—
49C47⋊#95—21no—
50C47⋊#95—21no—
51C47⋊#95—21no—
52C47⋊#95—21no—
53C23×C4×C47C2×C411yescharacteristic
54C23×C4×C47C2×C411yescharacteristic
55C23×C4×C47C2311yescharacteristic
56C22×C8×C47—22no—
57C2×C42×C47—42no—
58C2×C42×C47D411yescharacteristic
59C2×C42×C47D411yescharacteristic
60C2×C42×C47D411yescharacteristic
61C2×C42×C47D411yescharacteristic
62C4⋊C8×C47—42no—
63C22⋊C8×C47—22no—
64C22⋊C8×C47—22no—
65C24×C47C22×C411yescharacteristic
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C2×C812yes—
79C22×C4×C47M1612yes—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×Q811yescharacteristic
84C22×C4×C47#4111yescharacteristic
85C22×C4×C47#4111yescharacteristic
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—21no—
89C22×Dic47—21no—
90C22×Dic47—21no—
91C22×Dic47—21no—
92C22×Dic47—21no—
93C22×Dic47—21no—
94C22×Dic47—21no—
95C22×Dic47—21no—
96C22×Dic47—41no—
97C22×Dic47—41no—
98C22×Dic47—41no—
99C22×Dic47—41no—
100C22×Dic47—41no—
101C22×Dic47—41no—
102C22×Dic47C4⋊C412yes—
103C22×Dic47C4⋊C412yes—
104C22×Dic47C2×C812yes—
105C22×Dic47C2×C812yes—
106C22×Dic47M1612yes—
107C22×Dic47M1612yes—
108C2×C8×C47—42no—
109C2×C8×C47—42no—
110C2×C8×C47—42no—
111C42×C47—22no—
112C42×C47—22no—
113C42×C47—41no—
114C42×C47—41no—
115Dic47⋊C4—24no—
116Dic47⋊C4—24no—
117Dic47⋊C4—22no—
118Dic47⋊C4—22no—
119Dic47⋊C4—22no—
120Dic47⋊C4—22no—
121C4×Dic47—24no—
122C4×Dic47—24no—
123C4×Dic47—42no—
124C4×Dic47—42no—
125C2×C47⋊C8—42no—
126C2×C47⋊C8—42no—
127C2×C47⋊C8—42no—
128C23×C47—21no—
129C23×C47—21no—
130C23×C47—21no—
131C23×C47—21no—
132C23×C47C2×C22⋊C411yescharacteristic
133C23×C47C2×C4⋊C411yescharacteristic
134C23×C47#11711yescharacteristic
135C23×C47C22×C811yescharacteristic
136C23×C47C2×M1611yescharacteristic
137C23×C47C2×M1611yescharacteristic
138C23×C47C2×M1611yescharacteristic
139C2×C4×C47—24no—
140C2×C4×C47—24no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—42no—
154C2×C4×C47—42no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47C22⋊C812yes—
160C2×C4×C47C22⋊C812yes—
161C2×C4×C47#9912yes—
162C2×C4×C47#10112yes—
163C2×C4×C47#12111yescharacteristic
164C2×C4×C47#12111yescharacteristic
165C2×C4×C47C22⋊Q811yescharacteristic
166C2×C4×C47C22⋊Q811yescharacteristic
167C2×Dic47—22no—
168C2×Dic47—22no—
169C2×Dic47—22no—
170C2×Dic47—22no—
171C2×Dic47—22no—
172C2×Dic47—22no—
173C2×Dic47—22no—
174C2×Dic47—22no—
175C2×Dic47—22no—
176C2×Dic47—42no—
177C2×Dic47—42no—
178C2×Dic47—42no—
179C2×Dic47—41no—
180C2×Dic47—41no—
181C2×Dic47—41no—
182C2×Dic47—41no—
183C2×Dic47—41no—
184C2×Dic47—41no—
185C2×Dic47—42no—
186C2×Dic47—42no—
187C2×Dic47—42no—
188C2×Dic47C4⋊C814yescomplemented
189C8×C47—82no—
190C47⋊C8—82no—
191#8560—471nomaximal, Sylow
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—21no—
197C22×C47—21no—
198C22×C47—21no—
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—21no—
203C22×C47—21no—
204C22×C47#38011yescharacteristic
205C22×C47C2×C22⋊C811yescharacteristic
206C22×C47#40711yescharacteristic
207C22×C47C2×#9911yescharacteristic
208C22×C47#41011yescharacteristic
209C22×C47C2×#10111yescharacteristic
210C22×C47#41311yescharacteristic
211C22×C47C2×C4⋊C811yescharacteristic
212C22×C47#42311yescharacteristic
213C22×C47#43111yescharacteristic
214C22×C47#43211yescharacteristic
215C4×C47—21no—
216C4×C47—21no—
217C4×C47—21no—
218C4×C47—21no—
219C4×C47—22no—
220C4×C47—22no—
221C4×C47—22no—
222C4×C47—22no—
223Dic47—24no—
224Dic47—42no—
225Dic47—42no—
226Dic47—42no—
227C2×C4×C4⋊C4—471no—
228#1797—471no—
229#1797D4711yescharacteristic
230C2×#324—472no—
231#1280—472no—
232C2×C47—21no—
233C2×C47—21no—
234C2×C47—21no—
235C2×C47—21no—
236C2×C47#145911yescharacteristic
237C2×C47#146111yescharacteristic
238C2×C47#146311yescharacteristic
239C2×C47#146411yescharacteristic
240C2×C47#179711yescharacteristic
241C2×C47#179911yescharacteristic
242C2×C47#180111yescharacteristic
243C22×C4⋊C4—471no—
244C22×C4⋊C4—471no—
245C22×C42—471no—
246C22×C42—471no—
247C22×C42C2×D4711yescharacteristic
248C2×C4⋊C8—21no—
249C2×C4⋊C8—941no—
250C2×C22⋊C8—472no—
251C2×C22⋊C8C2×D4712yes—
252C4×C4⋊C4—941no—
253C4×C4⋊C4—941no—
254C4×C4⋊C4—941no—
255C4×C4⋊C4—941no—
256C2×#95—471no—
257C2×#95—471no—
258#336—21no—
259#336—941no—
260#324—944no—
261C47#856011yescharacteristic, complemented, Sylow
262C23×C4—471no—
263C23×C4—471no—
264C23×C4—471no—
265C23×C4—471no—
266C23×C4C4×D4711yescharacteristic
267C23×C4C4×D4711yescharacteristic
268C23×C4C22×D4711yescharacteristic
269C22×C8—22no—
270C22×C8—942no—
271C2×C4⋊C4—474no—
272C2×C4⋊C4—472no—
273C2×C4⋊C4—472no—
274C2×C4⋊C4—942no—
275C2×C4⋊C4—941no—
276C2×C4⋊C4—941no—
277C2×C4⋊C4—941no—
278C2×C4⋊C4—941no—
279C2×C4⋊C4—941no—
280C2×C4⋊C4—941no—
281C2×C42—42no—
282C2×C42—942no—
283C2×C42—942no—
284C2×C42—941no—
285C2×C42—941no—
286C2×C42—941no—
287C2×C42—941no—
288C2×C42—1882no—
289C2×C42D18811yescharacteristic
290C2×C42D18811yescharacteristic
291C2×C42C22⋊D4711yescharacteristic
292C2×C42C22⋊D4711yescharacteristic
293C4⋊C8—42no—
294C4⋊C8—1882no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—942no—
298C22⋊C8—942no—
299#95—941no—
300#95—941no—
301#95—941no—
302#95—941no—
303C24C2×C4×D4711yescharacteristic
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—472no—
313C22×C4—472no—
314C22×C4—472no—
315C22×C4—472no—
316C22×C4—472no—
317C22×C4—472no—
318C22×C4—941no—
319C22×C4—941no—
320C22×C4—941no—
321C22×C4—941no—
322C22×C4—941no—
323C22×C4—941no—
324C22×C4—941no—
325C22×C4—941no—
326C22×C4—941no—
327C22×C4—941no—
328C22×C4—1881no—
329C22×C4—1881no—
330C22×C4—1881no—
331C22×C4—1881no—
332C22×C4—1881no—
333C22×C4—1881no—
334C22×C4C8×D4712yes—
335C22×C4#1062532412yes—
336C22×C4(C2×D47)⋊C411yescharacteristic
337C22×C4(C2×D47)⋊C411yescharacteristic
338C22×C4(C2×D47)⋊C411yescharacteristic
339C22×C4(C2×D47)⋊C411yescharacteristic
340C22×C4C2×D18811yescharacteristic
341C22×C4D4×D4711yescharacteristic
342C22×C4#1062535211yescharacteristic
343C22×C4Q8×D4711yescharacteristic
344C22×C4#1062535411yescharacteristic
345C22×C4C2×C22⋊D4711yescharacteristic
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—1882no—
350C2×C8—1882no—
351C2×C8—1882no—
352C4⋊C4—944no—
353C4⋊C4—944no—
354C4⋊C4—942no—
355C4⋊C4—942no—
356C4⋊C4—942no—
357C4⋊C4—942no—
358C42—22no—
359C42—22no—
360C42—41no—
361C42—41no—
362C42—944no—
363C42—944no—
364C42—1882no—
365C42—1882no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23C2×(C2×D47)⋊C411yescharacteristic
371C23C4⋊C4×D4711yescharacteristic
372C23#1309398211yescharacteristic
373C23C2×C8×D4711yescharacteristic
374C23M16×D4711yescharacteristic
375C23M16×D4711yescharacteristic
376C23C2×#1062532411yescharacteristic
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—474no—
398C2×C4—942no—
399C2×C4—942no—
400C2×C4—942no—
401C2×C4—942no—
402C2×C4—942no—
403C2×C4—942no—
404C2×C4—942no—
405C2×C4—942no—
406C2×C4—942no—
407C2×C4—1882no—
408C2×C4—1882no—
409C2×C4—1882no—
410C2×C4—1881no—
411C2×C4—1881no—
412C2×C4—1881no—
413C2×C4—1881no—
414C2×C4—1881no—
415C2×C4—1881no—
416C2×C4—1882no—
417C2×C4—1882no—
418C2×C4—1882no—
419C2×C4(C2×D47)⋊C812yes—
420C2×C4(C2×D47)⋊C812yes—
421C2×C4#1309393812yes—
422C2×C4#1309394212yes—
423C2×C4#1309399511yescharacteristic
424C2×C4#1309399911yescharacteristic
425C2×C4#1309400011yescharacteristic
426C2×C4#1309400411yescharacteristic
427C8—82no—
428C8—3762no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22#1568898011yescharacteristic
442C22C2×(C2×D47)⋊C811yescharacteristic
443C22#1568908611yescharacteristic
444C22C2×#1309393811yescharacteristic
445C22#1568909611yescharacteristic
446C22C2×#1309394211yescharacteristic
447C22#1568910511yescharacteristic
448C22C4⋊C8×D4711yescharacteristic
449C22#1568914511yescharacteristic
450C22#1568917511yescharacteristic
451C22#1568917911yescharacteristic
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—944no—
461C4—1882no—
462C4—1882no—
463C4—1882no—
464C2—21no—
465C2—21no—
466C2—21no—
467C2—21no—
468C2#2024170411yescharacteristic
469C2#2024171111yescharacteristic
470C2#2024171711yescharacteristic
471C2#2024172011yescharacteristic
472C2#2024336511yescharacteristic
473C2#2024337311yescharacteristic
474C2#2024338111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22×C4
cosocleC23×C4×C47C23
hypercenter#1797D47
hypocenterC47#8560
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
36. C2×Dic47⋊C4C8semidirect
188. C2×Dic47C4⋊C8semidirect
261. C47#8560semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic47⋊C42
#202433682
C2×#156888482
#202412302
C47×#17972
#856047