Finite Group Databasebeta

← #26112459

Subgroups — #26112459

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689318C211yesmaximal, characteristic
2#20244194C211yesmaximal, characteristic, complemented
3#20243388C211yesmaximal, characteristic, complemented
4C2×#15688866C211yesmaximal, characteristic, complemented
5#20241272C211yesmaximal, characteristic, complemented
6#20241223C211yesmaximal, characteristic, complemented
7C2×C8×D4×C47C211yesmaximal, characteristic, complemented
8C22×#10625334—21no—
9C2×#13094021C2211yescharacteristic
10#15689318—22no—
11#15689318C414yescomplemented
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093946—21no—
15C2×#13093946—21no—
16C2×#13093945—21no—
17C2×#13093944—21no—
18#15688993C2211yescharacteristic
19#15688885—21no—
20#15688866—22no—
21C23×C8×C47—21no—
22C2×C4×D4×C47C2211yescharacteristic
23C8×D4×C47—22no—
24C8×D4×C47—22no—
25C2×C4⋊C8×C47C2211yescharacteristic
26C2×C22⋊C8×C47—21no—
27C2×C4×C8×C47C2211yescharacteristic
28C22×D188—21no—
29C22×D188—21no—
30C22×D188—21no—
31C2×#10625334—24no—
32C2×#10625334—42no—
33C2×#10625334—42no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8—21no—
36#13094021—21no—
37#13094021—21no—
38#13094021—21no—
39#13094021C2×C412yes—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)C2×C412yes—
47#13093946—42no—
48#13093946—42no—
49#13093945—42no—
50#13093944—24no—
51C22×D4×C47D411yescharacteristic
52C23×C4×C47—21no—
53C22×C8×C47—24no—
54C22×C8×C47—21no—
55C22×C8×C47—42no—
56C22×C8×C47—42no—
57C22×C8×C47D411yescharacteristic
58C22×C8×C47D411yescharacteristic
59C4×D4×C47—21no—
60C4×D4×C47—21no—
61C4×D4×C47C2×C412yes—
62C4×D4×C47D412yes—
63C2×C4⋊C4×C47D411yescharacteristic
64C2×C22⋊C4×C47—21no—
65C2×C42×C47C2311yescharacteristic
66C4⋊C8×C47—21no—
67C4⋊C8×C47D412yes—
68C22⋊C8×C47—24no—
69C4×C8×C47—21no—
70C4×C8×C47—21no—
71C24×C47—21no—
72C23×D47—41no—
73C2×D4×C47—21no—
74C2×D4×C47—21no—
75C2×D4×C47—21no—
76C2×D4×C47C22⋊C412yes—
77C2×D4×C47D812yes—
78C2×D4×C47SD1612yes—
79C22×C4×C47—21no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—41no—
83C22×C4×C47—41no—
84C22×C4×C47—41no—
85C22×C4×C47—41no—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47#4111yescharacteristic
89C22×Dic47—41no—
90C2×D188—22no—
91C2×D188—22no—
92C2×D188—22no—
93C2×D188—41no—
94C2×D188—41no—
95C2×D188—41no—
96C2×D188—41no—
97C2×D188—41no—
98C2×D188—41no—
99C2×D188—41no—
100C2×D188—41no—
101C2×D188—41no—
102C2×D188—41no—
103C2×D188—41no—
104C2×D188—81no—
105C2×D188—81no—
106C2×C8×C47—21no—
107C2×C8×C47—21no—
108C2×C8×C47—22no—
109C2×C8×C47—21no—
110C2×C8×C47—21no—
111C2×C8×C47—21no—
112C2×C8×C47—48no—
113C2×C8×C47—41no—
114C2×C8×C47—41no—
115C2×C8×C47D812yescomplemented
116C4⋊C4×C47—21no—
117C4⋊C4×C47C22⋊C412yes—
118C22⋊C4×C47—22no—
119C22⋊C4×C47—22no—
120C42×C47C22×C411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47C2×D411yescharacteristic
123C42×C47#4111yescharacteristic
124#10625334—48no—
125C4⋊Dic47—41no—
126C4⋊Dic47—41no—
127C4⋊Dic47—41no—
128C4⋊Dic47—41no—
129C4⋊Dic47—81no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C23×C47—22no—
137C23×C47—22no—
138C23×C47—22no—
139C23×C47—41no—
140C23×C47—41no—
141C23×C47—41no—
142C23×C47—41no—
143C23×C47#12111yescharacteristic
144C22×D47—42no—
145C22×D47—81no—
146C22×D47—81no—
147C22×D47—81no—
148C22×D47—81no—
149C22×D47—81no—
150C22×D47—81no—
151D4×C47—22no—
152D4×C47—22no—
153D4×C47—22no—
154D4×C47#10214yescomplemented
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—22no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—42no—
161C2×C4×C47—42no—
162C2×C4×C47—42no—
163C2×C4×C47—42no—
164C2×C4×C47C2×C22⋊C411yescharacteristic
165C2×C4×C47C4×D411yescharacteristic
166C2×C4×C47C4×D411yescharacteristic
167C2×C4×C47#12011yescharacteristic
168C2×C4×C47#12111yescharacteristic
169C2×C4×C47#12311yescharacteristic
170C2×C4×C47C2×D811yescharacteristic
171C2×C4×C47C2×D811yescharacteristic
172C2×C4×C47C2×SD1611yescharacteristic
173C2×C4×C47#13511yescharacteristic
174C2×C4×C47#13511yescharacteristic
175C2×C4×C47#13511yescharacteristic
176C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
178C2×Dic47—81no—
179C2×Dic47—81no—
180C2×Dic47—81no—
181D188—42no—
182D188—42no—
183D188—42no—
184D188—42no—
185D188—82no—
186D188—82no—
187D188—81no—
188D188—81no—
189D188—81no—
190D188—81no—
191C8×C47—22no—
192C8×C47—42no—
193C8×C47—41no—
194C47⋊C8—42no—
195C47⋊C8—81no—
196#9221—471nomaximal, Sylow
197C22×C47—24no—
198C22×C47—42no—
199C22×C47—42no—
200C22×C47—42no—
201C22×C47—42no—
202C22×C47—42no—
203C22×C47—42no—
204C22×C47#38611yescharacteristic
205C22×C47#45911yescharacteristic
206C22×C47#46311yescharacteristic
207C22×C47#46511yescharacteristic
208C22×C47#46611yescharacteristic
209C22×C47#47111yescharacteristic
210C22×C47#47211yescharacteristic
211C2×D47—82no—
212C2×D47—82no—
213C2×D47—82no—
214C2×D47—161no—
215C2×D47—161no—
216C2×D47—161no—
217C2×D47—161no—
218C4×C47—21no—
219C4×C47—21no—
220C4×C47—42no—
221C4×C47C2×#10211yescharacteristic
222C4×C47#41611yescharacteristic
223C4×C47C4×D811yescharacteristic
224C4×C47#44211yescharacteristic
225C4×C47#44711yescharacteristic
226C4×C47#44911yescharacteristic
227C4×C47#46511yescharacteristic
228C4×C47#46611yescharacteristic
229Dic47—161no—
230C2×#459—471no—
231C2×C8×D4D4711yescharacteristic, complemented
232#1957—471no—
233#1802—471no—
234C2×#331—471no—
235#1296—471no—
236#1277—471no—
237C2×C47—44no—
238C2×C47#166511yescharacteristic
239C2×C47#166711yescharacteristic
240C2×C47#167711yescharacteristic
241C2×C47#168111yescharacteristic
242C2×C47#187911yescharacteristic
243C2×C47#189311yescharacteristic
244C2×C47#190711yescharacteristic
245D47—162no—
246C22×D8—941no—
247C23×C8—21no—
248C2×#127—471no—
249C2×C4×D4C2×D4711yescharacteristic
250#459—474no—
251#459—942no—
252C8×D4—22no—
253C8×D4—22no—
254C2×C4⋊C8—471no—
255C2×C4⋊C8—471no—
256C2×C4⋊C8C2×D4711yescharacteristic
257C2×#102—941no—
258C2×#102—941no—
259C2×#102—941no—
260C2×#102—941no—
261C2×C22⋊C8—21no—
262C2×C4×C8C2×D4711yescharacteristic
263#384—471no—
264#340—941no—
265#331—942no—
266C47#922111yescharacteristic, complemented, Sylow
267C22×D4—941no—
268C22×D4—941no—
269C22×D4—941no—
270C22×D4D18811yescharacteristic
271C23×C4—21no—
272C2×D8—944no—
273C2×D8—1882no—
274C2×D8—1882no—
275C22×C8—24no—
276C22×C8—21no—
277C22×C8—42no—
278C22×C8—42no—
279C22×C8—941no—
280C22×C8—941no—
281C22×C8C22⋊D4711yescharacteristic
282C22×C8C22⋊D4711yescharacteristic
283#127—472no—
284#127—941no—
285#127—941no—
286#127—941no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C4×D4712yes—
290C4×D4C22⋊D4712yes—
291C2×C4⋊C4—941no—
292C2×C4⋊C4—941no—
293C2×C4⋊C4—941no—
294C2×C4⋊C4D18811yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C22×D4711yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—472no—
299C4⋊C8—941no—
300C4⋊C8—941no—
301C4⋊C8—941no—
302C4⋊C8C22⋊D4712yes—
303#102—944no—
304#102—1882no—
305#102—1882no—
306#102—1882no—
307C22⋊C8—24no—
308C4×C8—21no—
309C4×C8—21no—
310C24—21no—
311C24—1881no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—942no—
316C2×D4—942no—
317C2×D4—942no—
318C2×D4—3761no—
319C2×D4—3761no—
320C2×D4—1881no—
321C2×D4—1881no—
322C2×D4—1881no—
323C2×D4—1881no—
324C2×D4—1881no—
325C2×D4—1881no—
326C2×D4—1881no—
327C2×D4—1881no—
328C2×D4—1881no—
329C2×D4—1881no—
330C2×D4—1881no—
331C2×D4#1062532512yes—
332C2×D4D37612yes—
333C2×D4(C2×D47)⋊C412yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—1881no—
342C22×C4C2×D18811yescharacteristic
343C22×C4#1062535011yescharacteristic
344C22×C4C2×C22⋊D4711yescharacteristic
345D8—1888no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—48no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—942no—
356C2×C8—1881no—
357C2×C8—1881no—
358C2×C8—1881no—
359C2×C8—1881no—
360C2×C8—1881no—
361C2×C8#1062533412yescomplemented
362C4⋊C4—21no—
363C4⋊C4—3761no—
364C4⋊C4—1881no—
365C4⋊C4—1881no—
366C4⋊C4—1881no—
367C4⋊C4—1881no—
368C4⋊C4(C2×D47)⋊C412yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C42C2×C4×D4711yescharacteristic
372C42#1062535011yescharacteristic
373C42C2×C22⋊D4711yescharacteristic
374C42C2×C22⋊D4711yescharacteristic
375C23—22no—
376C23—22no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—3761no—
383C23—3761no—
384C23—3761no—
385C23—3761no—
386C23—3761no—
387C23—3761no—
388C23—1882no—
389C23#1309399811yescharacteristic
390D4—22no—
391D4—22no—
392D4—22no—
393D4—3762no—
394D4—3762no—
395D4—3761no—
396D4—3761no—
397D4—3761no—
398D4—3761no—
399D4—1882no—
400D4—1882no—
401D4—1882no—
402D4—1882no—
403D4#1309394614yescomplemented
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—3761no—
414C2×C4—3761no—
415C2×C4—3761no—
416C2×C4C2×(C2×D47)⋊C411yescharacteristic
417C2×C4C4×C22⋊D4711yescharacteristic
418C2×C4C4×C22⋊D4711yescharacteristic
419C2×C4#1309399311yescharacteristic
420C2×C4#1309399811yescharacteristic
421C2×C4#1309400711yescharacteristic
422C2×C4C2×#1062533411yescharacteristic
423C2×C4C2×D37611yescharacteristic
424C2×C4C2×#1062532511yescharacteristic
425C2×C4#1309404411yescharacteristic
426C2×C4#1309404611yescharacteristic
427C2×C4#1309404611yescharacteristic
428C2×C4#1309404911yescharacteristic
429C2×C4#1309405011yescharacteristic
430C8—22no—
431C8—42no—
432C8—41no—
433C8—3761no—
434C8—1882no—
435C22—24no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—3762no—
443C22—3762no—
444C22—3762no—
445C22—7521no—
446C22—7521no—
447C22—7521no—
448C22—7521no—
449C22#1568900211yescharacteristic
450C22#1568931811yescharacteristic
451C22#1568933811yescharacteristic
452C22#1568935211yescharacteristic
453C22#1568935911yescharacteristic
454C22#1568938611yescharacteristic
455C22#1568939311yescharacteristic
456C4—21no—
457C4—21no—
458C4—42no—
459C4—7521no—
460C4C2×#1309394611yescharacteristic
461C4#1568912011yescharacteristic
462C4C4×#1062533411yescharacteristic
463C4#1568923311yescharacteristic
464C4#1568925711yescharacteristic
465C4#1568926811yescharacteristic
466C4#1568935211yescharacteristic
467C4#1568935911yescharacteristic
468C2—44no—
469C2—7522no—
470C2#2024272911yescharacteristic
471C2#2024274311yescharacteristic
472C2#2024279711yescharacteristic
473C2#2024282511yescharacteristic
474C2#2024373611yescharacteristic
475C2#2024381611yescharacteristic
476C2#2024389611yescharacteristic
subgroupG/H
centerC23#13093998
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C8×D4×C47C2
socleC23×C47#121
prime radicalC2×#15689318C2
cosocleC2×C42×C47C23
hypercenterC2×C8×D4D47
hypocenterC47#9221
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
2. #20244194C2semidirect
3. #20243388C2semidirect
4. C2×#15688866C2semidirect
5. #20241272C2semidirect
6. #20241223C2semidirect
7. C2×C8×D4×C47C2semidirect
11. #15689318C4semidirect
115. C2×C8×C47D8semidirect
154. D4×C47#102semidirect
231. C2×C8×D4D47semidirect
266. C47#9221semidirect
361. C2×C8#10625334semidirect
403. D4#13093946semidirect

Maximal subgroups

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

subgroupindex
C2×#156893182
#202441942
#202433882
C2×#156888662
#202412722
#202412232
C2×C8×D4×C472
#922147