Finite Group Databasebeta

← #26111046

Subgroups — #26111046

Order 12032.

616 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689323C211yesmaximal, characteristic
2#20244315C211yesmaximal, characteristic
3#20244193C211yesmaximal, characteristic
4C2×Q8⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
5#20241271C211yesmaximal, characteristic, complemented
6#20241229C211yesmaximal, characteristic, complemented
7C47×#2355C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C2×#13094021C2211yescharacteristic
10#15689323—22no—
11#15689323C414yescomplemented
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093944—21no—
15C2×#13093944—21no—
16C2×#13093944—21no—
17C2×#13093944—22no—
18C2×C4×C47⋊C8C2211yescharacteristic
19Q8⋊(C47⋊C8)—22no—
20C2×C4×Q8×C47C2211yescharacteristic
21C47×#391—21no—
22C47×#387—22no—
23C47×#384C2211yescharacteristic
24C47×#382—21no—
25C4×C4⋊C4×C47—21no—
26C4×C4⋊C4×C47—21no—
27C4×C4⋊C4×C47C2211yescharacteristic
28C22×D188—21no—
29C22×D188—21no—
30C22×D188—21no—
31C2×#10625336—24no—
32C2×#10625336—42no—
33C2×#10625336—42no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8—21no—
36C22×C47⋊C8—21no—
37#13094021—21no—
38#13094021—21no—
39#13094021—21no—
40#13094021C2×C412yes—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)C2×C412yes—
45#13093944—24no—
46#13093944—44no—
47#13093944—42no—
48#13093944—42no—
49C4×C47⋊C8—21no—
50C4×C47⋊C8—21no—
51C22×Q8×C47D411yescharacteristic
52C4×Q8×C47—21no—
53C4×Q8×C47—21no—
54C4×Q8×C47C2×C412yes—
55C4×Q8×C47D412yes—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47D411yescharacteristic
60C2×C4⋊C4×C47D411yescharacteristic
61C2×C4⋊C4×C47D411yescharacteristic
62C2×C4⋊C4×C47D412yes—
63C2×C42×C47—21no—
64C2×C42×C47—21no—
65C2×C42×C47—22no—
66C2×C42×C47C2311yescharacteristic
67#95×C47—22no—
68#95×C47—22no—
69#95×C47—22no—
70#95×C47—21no—
71C23×D47—41no—
72C2×Q8×C47—21no—
73C2×Q8×C47—21no—
74C2×Q8×C47—21no—
75C2×Q8×C47—21no—
76C2×Q8×C47—21no—
77C2×Q8×C47C22⋊C412yes—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—22no—
82C22×C4×C47—21no—
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C2×D188—22no—
87C2×D188—22no—
88C2×D188—22no—
89C2×D188—42no—
90C2×D188—42no—
91C2×D188—41no—
92C2×D188—41no—
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—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—42no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—41no—
113C4⋊C4×C47C22⋊C412yes—
114C4⋊C4×C47SD1612yescomplemented
115C42×C47—22no—
116C42×C47—22no—
117C42×C47—41no—
118C42×C47—41no—
119C42×C47—82no—
120C42×C47C22×C411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47#4111yescharacteristic
123C42×C47#4111yescharacteristic
124#10625336—48no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—41no—
128C2×C47⋊C8—41no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8—41no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C23×C47#12011yescharacteristic
135C22×D47—42no—
136C22×D47—81no—
137C22×D47—81no—
138C22×D47—81no—
139C22×D47—81no—
140C22×D47—81no—
141C22×D47—81no—
142Q8×C47—22no—
143Q8×C47—22no—
144Q8×C47—41no—
145Q8×C47—41no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—42no—
160C2×C4×C47—42no—
161C2×C4×C47—42no—
162C2×C4×C47—41no—
163C2×C4×C47C2×C22⋊C411yescharacteristic
164C2×C4×C47C4×D411yescharacteristic
165C2×C4×C47C4×D411yescharacteristic
166C2×C4×C47#12111yescharacteristic
167C2×C4×C47#12111yescharacteristic
168C2×C4×C47#12311yescharacteristic
169C2×C4×C47C2×SD1611yescharacteristic
170C2×C4×C47#13511yescharacteristic
171C2×C4×C47#13511yescharacteristic
172C2×C4×C47#13511yescharacteristic
173C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C47#13711yescharacteristic
177D188—44no—
178D188—42no—
179D188—42no—
180D188—42no—
181D188—42no—
182D188—82no—
183D188—81no—
184D188—81no—
185D188—81no—
186D188—81no—
187C47⋊C8—42no—
188C47⋊C8—42no—
189C47⋊C8—81no—
190#8978—471nomaximal, Sylow
191C22×C47#38611yescharacteristic
192C22×C47#44811yescharacteristic
193C22×C47#44911yescharacteristic
194C22×C47#44911yescharacteristic
195C22×C47#44911yescharacteristic
196C22×C47#45311yescharacteristic
197C22×C47#45411yescharacteristic
198C2×D47—82no—
199C2×D47—82no—
200C2×D47—82no—
201C2×D47—161no—
202C2×D47—161no—
203C2×D47—161no—
204C2×D47—161no—
205C4×C47—21no—
206C4×C47—21no—
207C4×C47—42no—
208C4×C47—42no—
209C4×C47—42no—
210C4×C47—82no—
211C4×C47—81no—
212C4×C47#41611yescharacteristic
213C4×C47#41811yescharacteristic
214C4×C47C4×SD1611yescharacteristic
215C4×C47#44011yescharacteristic
216C4×C47#46011yescharacteristic
217C4×C47#46411yescharacteristic
218C4×C47#48211yescharacteristic
219C4×C47#48211yescharacteristic
220C2×#460—471no—
221#2355D4711yescharacteristic, complemented
222#1982—471no—
223#1957—471no—
224C2×Q8⋊C8—471no—
225#1296—471no—
226#1279—471no—
227C2×C47#162011yescharacteristic
228C2×C47#162211yescharacteristic
229C2×C47#162611yescharacteristic
230C2×C47#162811yescharacteristic
231C2×C47#187511yescharacteristic
232C2×C47#188011yescharacteristic
233C2×C47#188311yescharacteristic
234D47—162no—
235C22×SD16—941no—
236C2×#127—471no—
237C2×C4×Q8C2×D4711yescharacteristic
238#460—474no—
239#460—942no—
240C2×C4⋊C8—471no—
241C2×C4⋊C8—471no—
242C2×#102—941no—
243C2×#102—942no—
244C2×#102—941no—
245C2×#102—941no—
246C2×C4×C8—471no—
247#391—21no—
248#387—22no—
249#384C2×D4711yescharacteristic
250#382—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4C2×D4711yescharacteristic
254Q8⋊C8—942no—
255C47#897811yescharacteristic, complemented, Sylow
256C22×Q8C22⋊D4711yescharacteristic
257C22×D4—941no—
258C22×D4—941no—
259C22×D4—941no—
260C2×SD16—944no—
261C2×SD16—1882no—
262C2×SD16—1882no—
263C22×C8—941no—
264C22×C8—941no—
265C22×C8—941no—
266#127—472no—
267#127—941no—
268#127—941no—
269#127—941no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8C4×D4712yes—
273C4×Q8D18812yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C22⋊D4711yescharacteristic
278C2×C4⋊C4C22⋊D4711yescharacteristic
279C2×C4⋊C4C22⋊D4711yescharacteristic
280C2×C4⋊C4C22⋊D4712yes—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42C22×D4711yescharacteristic
285C4⋊C8—472no—
286C4⋊C8—941no—
287C4⋊C8—941no—
288C4⋊C8—941no—
289#102—944no—
290#102—1882no—
291#102—1884no—
292#102—1882no—
293C4×C8—941no—
294C4×C8—941no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299C24—1881no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8(C2×D47)⋊C412yes—
306C2×D4—942no—
307C2×D4—942no—
308C2×D4—942no—
309C2×D4—1882no—
310C2×D4—1882no—
311C2×D4—1881no—
312C2×D4—1881no—
313C2×D4—1881no—
314C2×D4—1881no—
315C2×D4—1881no—
316C2×D4—1881no—
317C2×D4—1881no—
318C2×D4—1881no—
319C2×D4—1881no—
320C2×D4—1881no—
321C2×D4—1881no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4C2×C22⋊D4711yescharacteristic
328C22×C4C2×C22⋊D4711yescharacteristic
329C22×C4C2×C22⋊D4711yescharacteristic
330SD16—1888no—
331C2×C8—942no—
332C2×C8—942no—
333C2×C8—1881no—
334C2×C8—1881no—
335C2×C8—1881no—
336C2×C8—1881no—
337C2×C8—1881no—
338C2×C8—1881no—
339C2×C8—1881no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4(C2×D47)⋊C412yes—
352C4⋊C4#1062533612yescomplemented
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—82no—
358C42C2×C4×D4711yescharacteristic
359C42C2×D18811yescharacteristic
360C42#1062535011yescharacteristic
361C42#1062535011yescharacteristic
362C23—3761no—
363C23—3761no—
364C23—3761no—
365C23—3761no—
366C23—3761no—
367C23—3761no—
368C23—1882no—
369C23#1309399311yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374D4—3762no—
375D4—3761no—
376D4—3761no—
377D4—3761no—
378D4—3761no—
379D4—1884no—
380D4—1882no—
381D4—1882no—
382D4—1882no—
383D4—1882no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4C2×(C2×D47)⋊C411yescharacteristic
402C2×C4C4×C22⋊D4711yescharacteristic
403C2×C4C4×C22⋊D4711yescharacteristic
404C2×C4#1309399811yescharacteristic
405C2×C4#1309399811yescharacteristic
406C2×C4#1309400711yescharacteristic
407C2×C4C2×#1062533611yescharacteristic
408C2×C4#1309404411yescharacteristic
409C2×C4#1309404411yescharacteristic
410C2×C4#1309404411yescharacteristic
411C2×C4#1309404911yescharacteristic
412C2×C4#1309404911yescharacteristic
413C2×C4#1309404911yescharacteristic
414C2×C4#1309405511yescharacteristic
415C8—3761no—
416C8—1882no—
417C8—1882no—
418C22—3762no—
419C22—3762no—
420C22—3762no—
421C22—7521no—
422C22—7521no—
423C22—7521no—
424C22—7521no—
425C22#1568900211yescharacteristic
426C22#1568926211yescharacteristic
427C22#1568926811yescharacteristic
428C22#1568926811yescharacteristic
429C22#1568926811yescharacteristic
430C22#1568929111yescharacteristic
431C22#1568929611yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4#1568911711yescharacteristic
440C4#1568912711yescharacteristic
441C4C4×#1062533611yescharacteristic
442C4#1568922111yescharacteristic
443C4#1568932311yescharacteristic
444C4#1568934511yescharacteristic
445C4#1568944711yescharacteristic
446C4#1568944711yescharacteristic
447C2—7522no—
448C2#2024244711yescharacteristic
449C2#2024246011yescharacteristic
450C2#2024248411yescharacteristic
451C2#2024249811yescharacteristic
452C2#2024371011yescharacteristic
453C2#2024374311yescharacteristic
454C2#2024376411yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2355C2
socleC23×C47#120
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#2355D47
hypocenterC47#8978
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2355

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊(C47⋊C8)C2semidirect
5. #20241271C2semidirect
6. #20241229C2semidirect
7. C47×#2355C2semidirect
11. #15689323C4semidirect
114. C4⋊C4×C47SD16semidirect
221. #2355D47semidirect
255. C47#8978semidirect
352. C4⋊C4#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156893232
#202443152
#202441932
C2×Q8⋊(C47⋊C8)2
#202412712
#202412292
C47×#23552
#897847