Finite Group Databasebeta

← #26109628

Subgroups — #26109628

Order 12032.

680 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊C4×D47C211yesmaximal, characteristic
2#20243365C211yesmaximal, characteristic
3C4×(C2×D47)⋊C8C211yesmaximal, characteristic
4#20241229C211yesmaximal, characteristic, complemented
5#20241228C211yesmaximal, characteristic, complemented
6#20241227C211yesmaximal, characteristic, complemented
7C47×#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊C4×D47C411yescharacteristic
9C2×C4⋊C4×D47C411yescharacteristic
10C2×C42×D47—21no—
11C2×C42×D47C2211yescharacteristic
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×(C2×D47)⋊C8—21no—
14C2×(C2×D47)⋊C8—21no—
15C2×C4×C47⋊C8C2211yescharacteristic
16C4⋊C4×Dic47C411yescharacteristic
17C4⋊C4×Dic47C411yescharacteristic
18C4×Dic47⋊C4—21no—
19C4×C4⋊Dic47C2211yescharacteristic
20#15688974—21no—
21#95×D47—21no—
22#15688877—21no—
23#15688877—21no—
24C2×C4⋊C8×C47C2211yescharacteristic
25C2×C4×C8×C47C2211yescharacteristic
26C4×C4⋊C4×C47C2211yescharacteristic
27C22×C4×D47—21no—
28C22×C4×D47—21no—
29C22×C4×D47C2×C411yescharacteristic
30C22×C4×D47D411yescharacteristic
31C22×C4×D47D411yescharacteristic
32C22×C47⋊C8—21no—
33C22×C47⋊C8—21no—
34C2×C4⋊Dic47C2×C411yescharacteristic
35C2×C4⋊Dic47C2×C411yescharacteristic
36C2×Dic47⋊C4—21no—
37C2×Dic47⋊C4—21no—
38C4⋊C4×D47—22no—
39C4⋊C4×D47—22no—
40C4⋊C4×D47—22no—
41C4⋊C4×D47C814yescomplemented
42C2×C4×Dic47—21no—
43C2×C4×Dic47—21no—
44C2×C4×Dic47—21no—
45C2×C4×Dic47C2×C411yescharacteristic
46C2×C4×Dic47D411yescharacteristic
47C2×C4×Dic47D411yescharacteristic
48C42×D47—21no—
49C42×D47—21no—
50C42×D47—21no—
51C42×D47—21no—
52C42×D47—42no—
53C42×D47—42no—
54C4⋊(C47⋊C8)—21no—
55C4⋊(C47⋊C8)—21no—
56(C2×D47)⋊C8—24no—
57(C2×D47)⋊C8—42no—
58C4×C47⋊C8—21no—
59C4×C47⋊C8—21no—
60C47⋊#95—21no—
61C47⋊#95—21no—
62C47⋊#95—21no—
63C22×C8×C47—21no—
64C22×C8×C47—21no—
65C2×C4⋊C4×C47C2×C411yescharacteristic
66C2×C4⋊C4×C47C2×C411yescharacteristic
67C2×C42×C47—21no—
68C2×C42×C47C2311yescharacteristic
69C4⋊C8×C47—21no—
70C4⋊C8×C47—21no—
71C4×C8×C47—21no—
72C4×C8×C47—21no—
73#95×C47—21no—
74C23×D47C22⋊C411yescharacteristic
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47C22⋊C411yescharacteristic
83C22×Dic47C22⋊C411yescharacteristic
84C22×Dic47C22⋊C411yescharacteristic
85C2×C4×D47—21no—
86C2×C4×D47—21no—
87C2×C4×D47—21no—
88C2×C4×D47—21no—
89C2×C4×D47—21no—
90C2×C4×D47—21no—
91C2×C4×D47—21no—
92C2×C4×D47—21no—
93C2×C4×D47—24no—
94C2×C4×D47—24no—
95C2×C4×D47—22no—
96C2×C4×D47—21no—
97C2×C4×D47—21no—
98C2×C4×D47—41no—
99C2×C4×D47—42no—
100C2×C4×D47—42no—
101C2×C4×D47—42no—
102C2×C4×D47—42no—
103C2×C4×D47C2×C812yes—
104C2×C4×D47M1612yes—
105C2×C4×D47SD1612yes—
106C2×C4×D47Q1612yes—
107C2×C8×C47—22no—
108C2×C8×C47—41no—
109C2×C8×C47—41no—
110C2×C8×C47—41no—
111C2×C8×C47—41no—
112C2×C8×C47—41no—
113C4⋊C4×C47—21no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47C2×C812yes—
116C4⋊C4×C47M1612yes—
117C42×C47—41no—
118C42×C47—41no—
119C42×C47#4111yescharacteristic
120C42×C47#4111yescharacteristic
121C42×C47#4111yescharacteristic
122C42×C47#4111yescharacteristic
123C4⋊Dic47—21no—
124C4⋊Dic47—21no—
125C4⋊Dic47C2×C812yes—
126C4⋊Dic47M1612yes—
127Dic47⋊C4—24no—
128Dic47⋊C4—24no—
129C4×Dic47—21no—
130C4×Dic47—21no—
131C4×Dic47—21no—
132C4×Dic47—21no—
133C4×Dic47—21no—
134C4×Dic47—21no—
135C4×Dic47—42no—
136C4×Dic47—42no—
137C4×Dic47—41no—
138C4×Dic47—41no—
139C2×C47⋊C8—22no—
140C2×C47⋊C8—41no—
141C2×C47⋊C8—41no—
142C2×C47⋊C8—41no—
143C2×C47⋊C8—41no—
144C2×C47⋊C8—41no—
145C23×C47C2×C22⋊C411yescharacteristic
146C22×D47—21no—
147C22×D47—21no—
148C22×D47—21no—
149C22×D47—21no—
150C22×D47C22⋊C812yes—
151C22×D47#10312yes—
152C22×D47#10412yes—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47#11711yescharacteristic
162C2×C4×C47#11711yescharacteristic
163C2×C4×C47#12311yescharacteristic
164C2×C4×C47#12311yescharacteristic
165C2×C4×C47#12311yescharacteristic
166C2×C4×C47#12311yescharacteristic
167C2×C4×C47C22×C811yescharacteristic
168C2×C4×C47C2×M1611yescharacteristic
169C2×C4×C47C2×M1611yescharacteristic
170C2×C4×C47C2×M1611yescharacteristic
171C2×C4×C47C2×SD1611yescharacteristic
172C2×C4×C47C2×Q1611yescharacteristic
173C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
175C2×Dic47—21no—
176C2×Dic47—21no—
177C2×Dic47—21no—
178C2×Dic47—21no—
179C2×Dic47—21no—
180C2×Dic47—22no—
181C2×Dic47—22no—
182C2×Dic47—21no—
183C2×Dic47—21no—
184C2×Dic47—41no—
185C2×Dic47—41no—
186C2×Dic47—41no—
187C2×Dic47—41no—
188C2×Dic47C22⋊C812yes—
189C2×Dic47#10312yes—
190C4×D47—22no—
191C4×D47—22no—
192C4×D47—22no—
193C4×D47—22no—
194C4×D47—22no—
195C4×D47—22no—
196C4×D47—22no—
197C4×D47—22no—
198C4×D47—42no—
199C4×D47—42no—
200C4×D47—48no—
201C4×D47—48no—
202C8×C47—42no—
203C8×C47—81no—
204C47⋊C8—42no—
205C47⋊C8—81no—
206#8707—471nomaximal, Sylow
207C22×C47#38111yescharacteristic
208C22×C47C2×C22⋊C811yescharacteristic
209C22×C47#40711yescharacteristic
210C22×C47C2×#10311yescharacteristic
211C22×C47#41811yescharacteristic
212C22×C47C2×#10411yescharacteristic
213C22×C47#42111yescharacteristic
214C2×D47—22no—
215C2×D47—22no—
216C2×D47—22no—
217C2×D47—22no—
218C2×D47—22no—
219C2×D47—22no—
220C2×D47Q8⋊C814yes—
221C4×C47—21no—
222C4×C47—21no—
223C4×C47—42no—
224C4×C47—42no—
225C4×C47#43111yescharacteristic
226C4×C47#43111yescharacteristic
227C4×C47#43211yescharacteristic
228C4×C47#43211yescharacteristic
229C4×C47#48111yescharacteristic
230C4×C47#48111yescharacteristic
231C4×C47#48411yescharacteristic
232C4×C47#48411yescharacteristic
233Dic47—22no—
234Dic47—22no—
235Dic47—41no—
236Dic47—42no—
237Dic47—42no—
238C2×C4×C4⋊C4—471no—
239#1797—471no—
240C4×C22⋊C8—471no—
241#1279—471no—
242#1279—471no—
243#1279—471no—
244#1279D4711yescharacteristic, complemented
245C2×C47C2×Q8⋊C811yescharacteristic
246C2×C47#147811yescharacteristic
247C2×C47#148711yescharacteristic
248C2×C47#149011yescharacteristic
249C2×C47#181711yescharacteristic
250C2×C47#182411yescharacteristic
251C2×C47#182611yescharacteristic
252D47—24no—
253C22×C4⋊C4—471no—
254C22×C4⋊C4—471no—
255C22×C42—471no—
256C22×C42—941no—
257C2×C4⋊C8—471no—
258C2×C4⋊C8C2×D4711yescharacteristic
259C2×C22⋊C8—941no—
260C2×C22⋊C8—941no—
261C2×C4×C8—471no—
262C2×C4×C8C2×D4711yescharacteristic
263C4×C4⋊C4—471no—
264C4×C4⋊C4—471no—
265C4×C4⋊C4—471no—
266C4×C4⋊C4—941no—
267C4×C4⋊C4—941no—
268C4×C4⋊C4C2×D4711yescharacteristic
269C2×#95—941no—
270#336—941no—
271#336—941no—
272C47#870711yescharacteristic, complemented, Sylow
273C23×C4—471no—
274C23×C4—471no—
275C23×C4—471no—
276C23×C4—941no—
277C23×C4—941no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—941no—
281C22×C8—941no—
282C2×C4⋊C4—471no—
283C2×C4⋊C4—471no—
284C2×C4⋊C4—474no—
285C2×C4⋊C4—942no—
286C2×C4⋊C4—942no—
287C2×C4⋊C4—941no—
288C2×C4⋊C4—941no—
289C2×C4⋊C4—942no—
290C2×C4⋊C4C4×D4711yescharacteristic
291C2×C4⋊C4C4×D4711yescharacteristic
292C2×C42—21no—
293C2×C42—471no—
294C2×C42—471no—
295C2×C42—471no—
296C2×C42—941no—
297C2×C42—941no—
298C2×C42—941no—
299C2×C42—941no—
300C2×C42—941no—
301C2×C42—941no—
302C2×C42—941no—
303C2×C42—1882no—
304C2×C42—1882no—
305C2×C42C22×D4711yescharacteristic
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—941no—
309C4⋊C8—941no—
310C22⋊C8—944no—
311C22⋊C8—1882no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—941no—
315C4×C8—941no—
316#95—21no—
317#95—941no—
318#95—941no—
319#95—941no—
320C24—471no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—471no—
324C22×C4—471no—
325C22×C4—472no—
326C22×C4—472no—
327C22×C4—472no—
328C22×C4—472no—
329C22×C4—471no—
330C22×C4—942no—
331C22×C4—944no—
332C22×C4—944no—
333C22×C4—941no—
334C22×C4—941no—
335C22×C4—941no—
336C22×C4—941no—
337C22×C4—941no—
338C22×C4—941no—
339C22×C4—941no—
340C22×C4—941no—
341C22×C4—941no—
342C22×C4—941no—
343C22×C4—941no—
344C22×C4—941no—
345C22×C4—1881no—
346C22×C4—1882no—
347C22×C4—1882no—
348C22×C4—1882no—
349C22×C4—1882no—
350C22×C4C2×C4×D4711yescharacteristic
351C22×C4D4×D4711yescharacteristic
352C22×C4D4×D4711yescharacteristic
353C2×C8—22no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—942no—
360C2×C8—1881no—
361C2×C8—1881no—
362C2×C8—1881no—
363C2×C8—1881no—
364C2×C8—1881no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—472no—
368C4⋊C4—472no—
369C4⋊C4—944no—
370C4⋊C4—944no—
371C4⋊C4—941no—
372C4⋊C4—941no—
373C4⋊C4C8×D4712yes—
374C4⋊C4#1062532412yes—
375C42—41no—
376C42—41no—
377C42—941no—
378C42—941no—
379C42—941no—
380C42—941no—
381C42—941no—
382C42—941no—
383C42—1882no—
384C42—1882no—
385C42—1881no—
386C42—1881no—
387C42#1062535011yescharacteristic
388C42#1062535011yescharacteristic
389C42#1062535211yescharacteristic
390C42#1062535411yescharacteristic
391C23—472no—
392C23—472no—
393C23—472no—
394C23—941no—
395C23—941no—
396C23—941no—
397C23—941no—
398C23C22⋊C4×D4711yescharacteristic
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—472no—
408C2×C4—472no—
409C2×C4—941no—
410C2×C4—941no—
411C2×C4—941no—
412C2×C4—941no—
413C2×C4—942no—
414C2×C4—942no—
415C2×C4—942no—
416C2×C4—942no—
417C2×C4—942no—
418C2×C4—942no—
419C2×C4—942no—
420C2×C4—942no—
421C2×C4—941no—
422C2×C4—942no—
423C2×C4—942no—
424C2×C4—941no—
425C2×C4—941no—
426C2×C4—1882no—
427C2×C4—1882no—
428C2×C4—1888no—
429C2×C4—1888no—
430C2×C4—1881no—
431C2×C4—1881no—
432C2×C4—1881no—
433C2×C4—1881no—
434C2×C4#1309398111yescharacteristic
435C2×C4#1309398211yescharacteristic
436C2×C4#1309400611yescharacteristic
437C2×C4#1309400611yescharacteristic
438C2×C4#1309400911yescharacteristic
439C2×C4#1309400911yescharacteristic
440C2×C4C2×C8×D4711yescharacteristic
441C2×C4M16×D4711yescharacteristic
442C2×C4M16×D4711yescharacteristic
443C2×C4C2×#1062532411yescharacteristic
444C2×C4SD16×D4711yescharacteristic
445C2×C4Q16×D4711yescharacteristic
446C2×C4#1309404811yescharacteristic
447C2×C4#1309405111yescharacteristic
448C8—42no—
449C8—81no—
450C8—3761no—
451C8—1882no—
452C22—474no—
453C22—942no—
454C22—942no—
455C22—942no—
456C22—942no—
457C22—942no—
458C22—942no—
459C22#1568898511yescharacteristic
460C22C22⋊C8×D4711yescharacteristic
461C22#1568908511yescharacteristic
462C22#103×D4711yescharacteristic
463C22#1568912611yescharacteristic
464C22#104×D4711yescharacteristic
465C22#1568913811yescharacteristic
466C4—21no—
467C4—21no—
468C4—42no—
469C4—42no—
470C4—942no—
471C4—942no—
472C4—1881no—
473C4—1882no—
474C4—1882no—
475C4#1568917411yescharacteristic
476C4#1568917511yescharacteristic
477C4#1568917811yescharacteristic
478C4#1568917911yescharacteristic
479C4#1568944011yescharacteristic
480C4#1568944311yescharacteristic
481C4#1568945711yescharacteristic
482C4#1568946011yescharacteristic
483C2—944no—
484C2Q8⋊C8×D4711yescharacteristic
485C2#2024177711yescharacteristic
486C2#2024181511yescharacteristic
487C2#2024182911yescharacteristic
488C2#2024345511yescharacteristic
489C2#2024347711yescharacteristic
490C2#2024348611yescharacteristic
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1279C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1279D47
hypocenterC47#8707
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
4. #20241229C2semidirect
5. #20241228C2semidirect
6. #20241227C2semidirect
7. C47×#1279C2semidirect
41. C4⋊C4×D47C8semidirect
244. #1279D47semidirect
272. C47#8707semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊C4×D472
#202433652
C4×(C2×D47)⋊C82
#202412292
#202412282
#202412272
C47×#12792
#870747