Finite Group Databasebeta

← #26111530

Subgroups — #26111530

Order 12032.

690 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×#15689471C211yesmaximal, characteristic
2#20246452C211yesmaximal, characteristic
3#20244312C211yesmaximal, characteristic, complemented
4C2×#15688866C211yesmaximal, characteristic
5#20241229C211yesmaximal, characteristic, complemented
6#20241227C211yesmaximal, characteristic, complemented
7C47×#1988C211yesmaximal, characteristic, complemented
8C2×#13094021C2211yescharacteristic
9C2×C4×D188—21no—
10C2×#13093986—21no—
11#15689471—22no—
12#15689471C414yescomplemented
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093948—21no—
15C2×#13093948—21no—
16C2×#13093944—22no—
17C2×C4×C47⋊C8C2211yescharacteristic
18#15689025—21no—
19#15689024—21no—
20#15689001—22no—
21C4×C4⋊Dic47C2211yescharacteristic
22#15688866—22no—
23C2×C4⋊Q8×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C4×C4⋊C4×C47C2211yescharacteristic
26C47×#340—21no—
27C47×#340—21no—
28C22×D188—21no—
29C22×D188—21no—
30C22×D188D412yes—
31C22×C4×D47—22no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8—21no—
34#13094021—21no—
35#13094021—21no—
36#13094021C2×C412yes—
37#13094021D412yes—
38C4×D188—42no—
39C4×D188—42no—
40#13093986—42no—
41#13093986—42no—
42C2×C4⋊Dic47D411yescharacteristic
43C2×C4⋊Dic47D411yescharacteristic
44C2×(C2×D47)⋊C4—22no—
45C2×(C2×D47)⋊C4—22no—
46C2×(C2×D47)⋊C4—22no—
47C2×C4×Dic47—21no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)—21no—
50#13093948—42no—
51#13093948—42no—
52#13093944—28no—
53C4×C47⋊C8—21no—
54C4×C47⋊C8C2×C412yescomplemented
55C47⋊#95—21no—
56C22×Q8×C47—21no—
57C22×C8×C47—21no—
58C4⋊Q8×C47—21no—
59C4⋊Q8×C47—21no—
60C4⋊Q8×C47—21no—
61C4⋊Q8×C47C2×C412yescomplemented
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C42×C47—21no—
67C2×C42×C47C2311yescharacteristic
68C4⋊C8×C47—21no—
69C4⋊C8×C47—21no—
70#95×C47—21no—
71C23×D47—22no—
72C2×Q8×C47—22no—
73C2×Q8×C47—41no—
74C2×Q8×C47—41no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C2×Q8×C47—41no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—41no—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×Dic47—21no—
85C22×Dic47—21no—
86C2×D188—22no—
87C2×D188—22no—
88C2×D188—22no—
89C2×D188—22no—
90C2×D188—22no—
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—42no—
100C2×D188—42no—
101C2×D188C22⋊C414yes—
102C2×C4×D47—44no—
103C2×C4×D47—44no—
104C2×C4×D47—82no—
105C2×C8×C47—41no—
106C2×C8×C47—41no—
107C2×C8×C47—41no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—22no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47—21no—
112C4⋊C4×C47—21no—
113C4⋊C4×C47—41no—
114C4⋊C4×C47—41no—
115C4⋊C4×C47SD1612yes—
116C42×C47—41no—
117C42×C47—41no—
118C42×C47C22×C411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47#4111yescharacteristic
121C42×C47#4111yescharacteristic
122(C2×D47)⋊C4—44no—
123(C2×D47)⋊C4—44no—
124(C2×D47)⋊C4—82no—
125C4⋊Dic47—21no—
126C4⋊Dic47—21no—
127C4⋊Dic47—21no—
128C4⋊Dic47D812yes—
129C4×Dic47—41no—
130C4×Dic47—41no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C2×C47⋊C8—41no—
137C23×C47#12011yescharacteristic
138C22×D47—24no—
139C22×D47—42no—
140C22×D47—42no—
141C22×D47—42no—
142C22×D47—42no—
143C22×D47—42no—
144C22×D47—42no—
145Q8×C47—42no—
146Q8×C47—42no—
147Q8×C47—81no—
148Q8×C47—81no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—22no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—42no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—81no—
159C2×C4×C47—81no—
160C2×C4×C47C2×C22⋊C411yescharacteristic
161C2×C4×C47C4×D411yescharacteristic
162C2×C4×C47C4×D411yescharacteristic
163C2×C4×C47#12111yescharacteristic
164C2×C4×C47#12111yescharacteristic
165C2×C4×C47#12311yescharacteristic
166C2×C4×C47C2×D811yescharacteristic
167C2×C4×C47C2×SD1611yescharacteristic
168C2×C4×C47#13511yescharacteristic
169C2×C4×C47#13511yescharacteristic
170C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C47#13711yescharacteristic
174C2×Dic47—22no—
175C2×Dic47—41no—
176C2×Dic47—41no—
177C2×Dic47—41no—
178C2×Dic47—41no—
179C2×Dic47—41no—
180D188—44no—
181D188—42no—
182D188—42no—
183D188—42no—
184D188—42no—
185D188—42no—
186D188—42no—
187D188—42no—
188D188—42no—
189D188—82no—
190C4×D47—88no—
191C8×C47—81no—
192C47⋊C8—42no—
193C47⋊C8—81no—
194#9057—471nomaximal, Sylow
195C22×C47#38611yescharacteristic
196C22×C47#44711yescharacteristic
197C22×C47#44911yescharacteristic
198C22×C47#45011yescharacteristic
199C22×C47#45211yescharacteristic
200C22×C47#45511yescharacteristic
201C22×C47#45511yescharacteristic
202C2×D47—44no—
203C2×D47—44no—
204C2×D47—44no—
205C2×D47—82no—
206C2×D47—82no—
207C2×D47—82no—
208C2×D47—82no—
209C4×C47—21no—
210C4×C47—21no—
211C4×C47—42no—
212C4×C47—82no—
213C4×C47—81no—
214C4×C47C4×D811yescharacteristic
215C4×C47C4×SD1611yescharacteristic
216C4×C47#44011yescharacteristic
217C4×C47#44211yescharacteristic
218C4×C47#45911yescharacteristic
219C4×C47#46111yescharacteristic
220C4×C47#46311yescharacteristic
221C4×C47#46311yescharacteristic
222Dic47—42no—
223Dic47—81no—
224C2×#486—471no—
225#2366—471no—
226#1988D4711yescharacteristic, complemented
227#1981—471no—
228C2×#331—471no—
229#1279—471no—
230#1279—471no—
231C2×C47#163911yescharacteristic
232C2×C47#164011yescharacteristic
233C2×C47#164111yescharacteristic
234C2×C47#164211yescharacteristic
235C2×C47#187711yescharacteristic
236C2×C47#187911yescharacteristic
237C2×C47#188511yescharacteristic
238D47—84no—
239C2×C4⋊Q8C2×D4711yescharacteristic
240C2×#127—471no—
241C2×C4×D4—941no—
242C2×C4×D4—941no—
243#486—474no—
244#486—942no—
245C2×C4⋊C8—471no—
246C2×C4⋊C8C2×D4711yescharacteristic
247C2×#103—941no—
248C2×#103—941no—
249C2×#102—942no—
250C2×C4×C8—471no—
251#390—941no—
252#390—941no—
253#386—942no—
254C4×C4⋊C4—471no—
255C4×C4⋊C4C2×D4711yescharacteristic
256#340—21no—
257#340—21no—
258#331—942no—
259C47#905711yescharacteristic, complemented, Sylow
260C22×Q8—21no—
261C22×D4—472no—
262C22×D4—941no—
263C22×D4—941no—
264C23×C4—942no—
265C22×C8—21no—
266C22×C8—941no—
267C22×C8—941no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8C4×D4712yescomplemented
272#127—472no—
273#127—472no—
274#127—941no—
275#127—941no—
276C4×D4—1882no—
277C4×D4—1882no—
278C4×D4—1882no—
279C4×D4—1882no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—471no—
283C2×C4⋊C4—471no—
284C2×C4⋊C4C22⋊D4711yescharacteristic
285C2×C4⋊C4C22⋊D4711yescharacteristic
286C2×C22⋊C4—942no—
287C2×C22⋊C4—942no—
288C2×C22⋊C4—942no—
289C2×C42—21no—
290C2×C42—941no—
291C2×C42C22×D4711yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—941no—
295C4⋊C8—941no—
296#103—1882no—
297#103—1882no—
298#102—948no—
299C4×C8—472no—
300C4×C8—941no—
301#95—21no—
302#95—941no—
303C24—942no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×D4—474no—
311C2×D4—942no—
312C2×D4—942no—
313C2×D4—942no—
314C2×D4—942no—
315C2×D4—942no—
316C2×D4—1882no—
317C2×D4—1882no—
318C2×D4—1881no—
319C2×D4—1881no—
320C2×D4—1881no—
321C2×D4—1881no—
322C2×D4—1881no—
323C2×D4—1881no—
324C2×D4—1881no—
325C2×D4—1881no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—941no—
330C22×C4—941no—
331C22×C4—3762no—
332C22×C4—1884no—
333C22×C4—1884no—
334C22×C4D4×D4711yescharacteristic
335C22×C4D4×D4711yescharacteristic
336C22×C4C2×C22⋊D4711yescharacteristic
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—942no—
341C2×C8—1881no—
342C2×C8—1881no—
343C2×C8—1881no—
344C2×C8—1881no—
345C2×C8—1881no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—472no—
354C4⋊C4—941no—
355C4⋊C4—941no—
356C4⋊C4—941no—
357C4⋊C4#1062533612yes—
358C22⋊C4—3762no—
359C22⋊C4—1884no—
360C22⋊C4—1884no—
361C42—41no—
362C42—41no—
363C42—1881no—
364C42—1881no—
365C42C2×C4×D4711yescharacteristic
366C42#1062535011yescharacteristic
367C42D4×D4711yescharacteristic
368C42#1062535211yescharacteristic
369C23—944no—
370C23—1882no—
371C23—1882no—
372C23—1882no—
373C23—1882no—
374C23—1882no—
375C23—1882no—
376C23#1309399211yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
381D4—3762no—
382D4—1884no—
383D4—1882no—
384D4—1882no—
385D4—1882no—
386D4—1882no—
387D4—1882no—
388D4—1882no—
389D4—1882no—
390D4—1882no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—942no—
403C2×C4—3768no—
404C2×C4—1881no—
405C2×C4—1881no—
406C2×C4—1881no—
407C2×C4—1881no—
408C2×C4—1881no—
409C2×C4C22⋊C4×D4711yescharacteristic
410C2×C4(C22⋊D47)⋊C411yescharacteristic
411C2×C4C4×C22⋊D4711yescharacteristic
412C2×C4#1309399511yescharacteristic
413C2×C4#1309399711yescharacteristic
414C2×C4#1309400611yescharacteristic
415C2×C4D8×D4711yescharacteristic
416C2×C4C2×#1062533611yescharacteristic
417C2×C4#1309404411yescharacteristic
418C2×C4#1309404511yescharacteristic
419C2×C4#1309404811yescharacteristic
420C2×C4#1309404911yescharacteristic
421C2×C4#1309405111yescharacteristic
422C2×C4#1309405511yescharacteristic
423C8—81no—
424C8—3761no—
425C8—1882no—
426C22—3762no—
427C22—3762no—
428C22—3762no—
429C22—3762no—
430C22—1884no—
431C22—1884no—
432C22—1884no—
433C22#1568899911yescharacteristic
434C22#1568925511yescharacteristic
435C22#1568926711yescharacteristic
436C22#1568927311yescharacteristic
437C22#1568928311yescharacteristic
438C22#1568929911yescharacteristic
439C22#1568929911yescharacteristic
440C4—21no—
441C4—21no—
442C4—42no—
443C4—82no—
444C4—81no—
445C4—3761no—
446C4—1882no—
447C4#1568920311yescharacteristic
448C4C4×#1062533611yescharacteristic
449C4#1568922111yescharacteristic
450C4#1568923211yescharacteristic
451C4#1568931711yescharacteristic
452C4#1568932511yescharacteristic
453C4#1568933511yescharacteristic
454C4#1568933711yescharacteristic
455C2—3764no—
456C2#2024256611yescharacteristic
457C2#2024257311yescharacteristic
458C2#2024258011yescharacteristic
459C2#2024258711yescharacteristic
460C2#2024372511yescharacteristic
461C2#2024373311yescharacteristic
462C2#2024377311yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1988C2
socleC23×C47#120
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#1988D47
hypocenterC47#9057
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
3. #20244312C2semidirect
5. #20241229C2semidirect
6. #20241227C2semidirect
7. C47×#1988C2semidirect
12. #15689471C4semidirect
54. C4×C47⋊C8C2×C4semidirect
61. C4⋊Q8×C47C2×C4semidirect
226. #1988D47semidirect
259. C47#9057semidirect
271. C4⋊Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894712
#202464522
#202443122
C2×#156888662
#202412292
#202412272
C47×#19882
#905747