Finite Group Databasebeta

← #26111053

Subgroups — #26111053

Order 12032.

660 conjugacy classes of subgroups.

The subgroup lattice has 464 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
7Q8×C4⋊C4×C47C211yesmaximal, 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×C47—21no—
21C2×C4×Q8×C47C2211yescharacteristic
22C47×#389—22no—
23C47×#389—21no—
24C47×#384—21no—
25C47×#384C2211yescharacteristic
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×C47—42no—
55C4×Q8×C47—42no—
56C4×Q8×C47C2×C412yes—
57C4×Q8×C47D412yes—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—22no—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—22no—
63C2×C4⋊C4×C47D411yescharacteristic
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C4⋊C4×C47D412yes—
67C2×C42×C47—21no—
68C2×C42×C47—22no—
69C2×C42×C47—21no—
70C2×C42×C47C2311yescharacteristic
71#95×C47—22no—
72#95×C47—21no—
73C23×D47—41no—
74C2×Q8×C47—21no—
75C2×Q8×C47—21no—
76C2×Q8×C47—21no—
77C2×Q8×C47C22⋊C412yes—
78C2×Q8×C47D812yes—
79C2×Q8×C47SD1612yes—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—22no—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C2×D188—22no—
89C2×D188—22no—
90C2×D188—22no—
91C2×D188—42no—
92C2×D188—42no—
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—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—22no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—44no—
111C4⋊C4×C47—42no—
112C4⋊C4×C47—41no—
113C4⋊C4×C47—41no—
114C4⋊C4×C47—41no—
115C4⋊C4×C47—41no—
116C4⋊C4×C47—82no—
117C4⋊C4×C47C22⋊C412yes—
118C4⋊C4×C47SD1612yescomplemented
119C42×C47—22no—
120C42×C47—22no—
121C42×C47—44no—
122C42×C47—41no—
123C42×C47—41no—
124C42×C47C22×C411yescharacteristic
125C42×C47C2×D411yescharacteristic
126C42×C47#4111yescharacteristic
127C42×C47#4111yescharacteristic
128#10625336—48no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C2×C47⋊C8—41no—
137C2×C47⋊C8—41no—
138C23×C47#12011yescharacteristic
139C22×D47—42no—
140C22×D47—81no—
141C22×D47—81no—
142C22×D47—81no—
143C22×D47—81no—
144C22×D47—81no—
145C22×D47—81no—
146Q8×C47—22no—
147Q8×C47—22no—
148Q8×C47—22no—
149Q8×C47#10214yescomplemented
150C2×C4×C47—22no—
151C2×C4×C47—22no—
152C2×C4×C47—22no—
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—42no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—42no—
165C2×C4×C47—42no—
166C2×C4×C47—41no—
167C2×C4×C47C2×C22⋊C411yescharacteristic
168C2×C4×C47C4×D411yescharacteristic
169C2×C4×C47C4×D411yescharacteristic
170C2×C4×C47#12111yescharacteristic
171C2×C4×C47#12111yescharacteristic
172C2×C4×C47#12311yescharacteristic
173C2×C4×C47C2×D811yescharacteristic
174C2×C4×C47C2×SD1611yescharacteristic
175C2×C4×C47C2×SD1611yescharacteristic
176C2×C4×C47#13511yescharacteristic
177C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C47#13711yescharacteristic
180C2×C4×C47#13711yescharacteristic
181D188—44no—
182D188—42no—
183D188—42no—
184D188—42no—
185D188—42no—
186D188—82no—
187D188—81no—
188D188—81no—
189D188—81no—
190D188—81no—
191C47⋊C8—42no—
192C47⋊C8—42no—
193C47⋊C8—81no—
194#8979—471nomaximal, Sylow
195C22×C47#38611yescharacteristic
196C22×C47#44711yescharacteristic
197C22×C47#44811yescharacteristic
198C22×C47#44811yescharacteristic
199C22×C47#44911yescharacteristic
200C22×C47#45311yescharacteristic
201C22×C47#45411yescharacteristic
202C2×D47—82no—
203C2×D47—82no—
204C2×D47—82no—
205C2×D47—161no—
206C2×D47—161no—
207C2×D47—161no—
208C2×D47—161no—
209C4×C47—24no—
210C4×C47—21no—
211C4×C47—21no—
212C4×C47—42no—
213C4×C47—42no—
214C4×C47—81no—
215C4×C47—82no—
216C4×C47C2×#10211yescharacteristic
217C4×C47#41711yescharacteristic
218C4×C47C4×SD1611yescharacteristic
219C4×C47#44011yescharacteristic
220C4×C47#46011yescharacteristic
221C4×C47#46411yescharacteristic
222C4×C47#48011yescharacteristic
223C4×C47#48111yescharacteristic
224C2×#460—471no—
225Q8×C4⋊C4D4711yescharacteristic, complemented
226#1982—471no—
227#1957—471no—
228C2×Q8⋊C8—471no—
229#1296—471no—
230#1279—471no—
231C2×C47#162111yescharacteristic
232C2×C47#162311yescharacteristic
233C2×C47#162611yescharacteristic
234C2×C47#162711yescharacteristic
235C2×C47#187411yescharacteristic
236C2×C47#188011yescharacteristic
237C2×C47#188311yescharacteristic
238D47—162no—
239C22×SD16—941no—
240C2×#127—471no—
241C2×C4×Q8—21no—
242C2×C4×Q8C2×D4711yescharacteristic
243#460—474no—
244#460—942no—
245C2×C4⋊C8—471no—
246C2×C4⋊C8—471no—
247C2×#102—941no—
248C2×#102—942no—
249C2×#102—941no—
250C2×#102—941no—
251C2×C4×C8—471no—
252#389—21no—
253#389—22no—
254#384—21no—
255#384C2×D4711yescharacteristic
256C4×C4⋊C4—21no—
257C4×C4⋊C4C2×D4711yescharacteristic
258Q8⋊C8—942no—
259C47#897911yescharacteristic, complemented, Sylow
260C22×Q8C22⋊D4711yescharacteristic
261C22×D4—941no—
262C22×D4—941no—
263C22×D4—941no—
264C2×SD16—944no—
265C2×SD16—1882no—
266C2×SD16—1882no—
267C22×C8—941no—
268C22×C8—941no—
269C22×C8—941no—
270#127—472no—
271#127—941no—
272#127—941no—
273#127—941no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8C4×D4712yes—
279C4×Q8D18812yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C22⋊D4711yescharacteristic
286C2×C4⋊C4C22⋊D4711yescharacteristic
287C2×C4⋊C4C22⋊D4711yescharacteristic
288C2×C4⋊C4C22⋊D4712yes—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42C22×D4711yescharacteristic
293C4⋊C8—472no—
294C4⋊C8—941no—
295C4⋊C8—941no—
296C4⋊C8—941no—
297#102—944no—
298#102—1882no—
299#102—1884no—
300#102—1882no—
301C4×C8—941no—
302C4×C8—941no—
303#95—22no—
304#95—21no—
305C24—1881no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8(C2×D47)⋊C412yes—
310C2×Q8#1062533412yes—
311C2×Q8#1062533612yes—
312C2×D4—942no—
313C2×D4—942no—
314C2×D4—942no—
315C2×D4—1882no—
316C2×D4—1882no—
317C2×D4—1881no—
318C2×D4—1881no—
319C2×D4—1881no—
320C2×D4—1881no—
321C2×D4—1881no—
322C2×D4—1881no—
323C2×D4—1881no—
324C2×D4—1881no—
325C2×D4—1881no—
326C2×D4—1881no—
327C2×D4—1881no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4C2×C22⋊D4711yescharacteristic
334C22×C4C2×C22⋊D4711yescharacteristic
335C22×C4C2×C22⋊D4711yescharacteristic
336SD16—1888no—
337C2×C8—942no—
338C2×C8—942no—
339C2×C8—1881no—
340C2×C8—1881no—
341C2×C8—1881no—
342C2×C8—1881no—
343C2×C8—1881no—
344C2×C8—1881no—
345C2×C8—1881no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—82no—
359C4⋊C4(C2×D47)⋊C412yes—
360C4⋊C4#1062533612yescomplemented
361C42—22no—
362C42—22no—
363C42—44no—
364C42—41no—
365C42—41no—
366C42C2×C4×D4711yescharacteristic
367C42C2×D18811yescharacteristic
368C42#1062535011yescharacteristic
369C42#1062535011yescharacteristic
370C23—3761no—
371C23—3761no—
372C23—3761no—
373C23—3761no—
374C23—3761no—
375C23—3761no—
376C23—1882no—
377C23#1309399311yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—22no—
381Q8#1309394414yescomplemented
382D4—3762no—
383D4—3761no—
384D4—3761no—
385D4—3761no—
386D4—3761no—
387D4—1884no—
388D4—1882no—
389D4—1882no—
390D4—1882no—
391D4—1882no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4C2×(C2×D47)⋊C411yescharacteristic
410C2×C4C4×C22⋊D4711yescharacteristic
411C2×C4C4×C22⋊D4711yescharacteristic
412C2×C4#1309399811yescharacteristic
413C2×C4#1309399811yescharacteristic
414C2×C4#1309400711yescharacteristic
415C2×C4C2×#1062533411yescharacteristic
416C2×C4C2×#1062533611yescharacteristic
417C2×C4C2×#1062533611yescharacteristic
418C2×C4#1309404411yescharacteristic
419C2×C4#1309404911yescharacteristic
420C2×C4#1309405011yescharacteristic
421C2×C4#1309405511yescharacteristic
422C2×C4#1309405511yescharacteristic
423C8—3761no—
424C8—1882no—
425C8—1882no—
426C22—3762no—
427C22—3762no—
428C22—3762no—
429C22—7521no—
430C22—7521no—
431C22—7521no—
432C22—7521no—
433C22#1568900211yescharacteristic
434C22#1568925711yescharacteristic
435C22#1568926211yescharacteristic
436C22#1568926211yescharacteristic
437C22#1568926811yescharacteristic
438C22#1568929111yescharacteristic
439C22#1568929611yescharacteristic
440C4—21no—
441C4—21no—
442C4—24no—
443C4—42no—
444C4—42no—
445C4—81no—
446C4—82no—
447C4C2×#1309394411yescharacteristic
448C4#1568912211yescharacteristic
449C4C4×#1062533611yescharacteristic
450C4#1568922111yescharacteristic
451C4#1568932311yescharacteristic
452C4#1568934511yescharacteristic
453C4#1568943611yescharacteristic
454C4#1568944111yescharacteristic
455C2—7522no—
456C2#2024245311yescharacteristic
457C2#2024246611yescharacteristic
458C2#2024248411yescharacteristic
459C2#2024249111yescharacteristic
460C2#2024370511yescharacteristic
461C2#2024374311yescharacteristic
462C2#2024376411yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingQ8×C4⋊C4×C47C2
socleC23×C47#120
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenterQ8×C4⋊C4D47
hypocenterC47#8979
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ Q8×C4⋊C4

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. Q8×C4⋊C4×C47C2semidirect
11. #15689323C4semidirect
118. C4⋊C4×C47SD16semidirect
149. Q8×C47#102semidirect
225. Q8×C4⋊C4D47semidirect
259. C47#8979semidirect
360. C4⋊C4#10625336semidirect
381. Q8#13093944semidirect

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
Q8×C4⋊C4×C472
#897947