Finite Group Databasebeta

← #26111119

Subgroups — #26111119

Order 12032.

608 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689325C211yesmaximal, characteristic
2#20246335C211yesmaximal, characteristic
3#20244350C211yesmaximal, characteristic, complemented
4C2×#15688850C211yesmaximal, characteristic
5#20241272C211yesmaximal, characteristic, complemented
6#20241229C211yesmaximal, characteristic, complemented
7C47×#1988C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C2×C4×D188C2211yescharacteristic
10C2×#13093982—21no—
11#15689325—22no—
12#15689325C414yescomplemented
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093948—21no—
15C2×C4×C47⋊C8C2211yescharacteristic
16#15689025—21no—
17#15689015—22no—
18#15688993C2211yescharacteristic
19#15688980—21no—
20C4×C4⋊Dic47—21no—
21#15688885—22no—
22#15688850—22no—
23C2×C4⋊Q8×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C4×C4⋊C4×C47C2211yescharacteristic
26C47×#340—21no—
27C47×#340—21no—
28C22×D188D411yescharacteristic
29C22×C4×D47—21no—
30C2×#10625336—24no—
31C2×#10625336—42no—
32C2×#10625336—42no—
33C22×C47⋊C8—21no—
34C22×C47⋊C8—21no—
35C4×D188—21no—
36C4×D188—21no—
37C4×D188C2×C412yes—
38C4×D188D412yes—
39#13093982—42no—
40#13093982—42no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47D411yescharacteristic
44C2×C4⋊Dic47D412yes—
45C2×(C2×D47)⋊C4—21no—
46C2×(C2×D47)⋊C4—22no—
47C2×(C2×D47)⋊C4—22no—
48C2×C4×Dic47—22no—
49C4⋊(C47⋊C8)—21no—
50C4⋊(C47⋊C8)C2×C412yes—
51#13093948—24no—
52C4×C47⋊C8—21no—
53C4×C47⋊C8—21no—
54C47⋊#95—22no—
55C22×Q8×C47—21no—
56C22×C8×C47—21no—
57C4⋊Q8×C47—21no—
58C4⋊Q8×C47—21no—
59C4⋊Q8×C47—21no—
60C4⋊Q8×C47C2×C412yes—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47D411yescharacteristic
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C42×C47—21no—
66C2×C42×C47C2311yescharacteristic
67C4⋊C8×C47—21no—
68C4⋊C8×C47—21no—
69#95×C47—21no—
70C23×D47—21no—
71C2×Q8×C47—22no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C2×Q8×C47—41no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—41no—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×Dic47—21no—
84C22×Dic47—22no—
85C2×D188—21no—
86C2×D188—21no—
87C2×D188—21no—
88C2×D188—21no—
89C2×D188—21no—
90C2×D188C22⋊C412yes—
91C2×C4×D47—22no—
92C2×C4×D47—22no—
93C2×C4×D47—41no—
94C2×C4×D47—41no—
95C2×C4×D47—41no—
96C2×C4×D47—41no—
97C2×C8×C47—41no—
98C2×C8×C47—41no—
99C2×C8×C47—41no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47SD1612yescomplemented
108C42×C47—41no—
109C42×C47—41no—
110C42×C47C22×C411yescharacteristic
111C42×C47C2×D411yescharacteristic
112C42×C47#4111yescharacteristic
113C42×C47#4111yescharacteristic
114#10625336—48no—
115(C2×D47)⋊C4—22no—
116(C2×D47)⋊C4—22no—
117(C2×D47)⋊C4—44no—
118(C2×D47)⋊C4—82no—
119C4⋊Dic47—21no—
120C4⋊Dic47—41no—
121C4⋊Dic47—41no—
122C4⋊Dic47—41no—
123C4⋊Dic47—41no—
124C4⋊Dic47—42no—
125C4⋊Dic47C22⋊C412yes—
126C4×Dic47—44no—
127C2×C47⋊C8—22no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8—41no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C23×C47#12011yescharacteristic
134C22×D47—22no—
135C22×D47—41no—
136C22×D47—41no—
137C22×D47—41no—
138C22×D47—41no—
139C22×D47—41no—
140C22×D47—41no—
141Q8×C47—42no—
142Q8×C47—42no—
143Q8×C47—81no—
144Q8×C47—81no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—22no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—81no—
155C2×C4×C47—81no—
156C2×C4×C47C2×C22⋊C411yescharacteristic
157C2×C4×C47C4×D411yescharacteristic
158C2×C4×C47C4×D411yescharacteristic
159C2×C4×C47#12111yescharacteristic
160C2×C4×C47#12111yescharacteristic
161C2×C4×C47#12311yescharacteristic
162C2×C4×C47C2×SD1611yescharacteristic
163C2×C4×C47#13511yescharacteristic
164C2×C4×C47#13511yescharacteristic
165C2×C4×C47#13511yescharacteristic
166C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C47#13711yescharacteristic
170C2×Dic47—22no—
171C2×Dic47—41no—
172C2×Dic47—41no—
173C2×Dic47—42no—
174C2×Dic47—42no—
175C2×Dic47—42no—
176D188—22no—
177D188—22no—
178D188—41no—
179D188—41no—
180C4×D47—42no—
181C4×D47—42no—
182C4×D47—42no—
183C4×D47—42no—
184C8×C47—81no—
185C47⋊C8—42no—
186C47⋊C8—42no—
187#8988—471nomaximal, Sylow
188C22×C47#38611yescharacteristic
189C22×C47#44911yescharacteristic
190C22×C47#44911yescharacteristic
191C22×C47#45011yescharacteristic
192C22×C47#45211yescharacteristic
193C22×C47#45511yescharacteristic
194C22×C47#45611yescharacteristic
195C2×D47—42no—
196C2×D47—42no—
197C2×D47—42no—
198C2×D47—81no—
199C2×D47—81no—
200C2×D47—81no—
201C2×D47—81no—
202C4×C47—21no—
203C4×C47—21no—
204C4×C47—42no—
205C4×C47—82no—
206C4×C47—81no—
207C4×C47#41611yescharacteristic
208C4×C47#41811yescharacteristic
209C4×C47C4×SD1611yescharacteristic
210C4×C47#44011yescharacteristic
211C4×C47#46111yescharacteristic
212C4×C47#46311yescharacteristic
213C4×C47#48211yescharacteristic
214C4×C47#48211yescharacteristic
215Dic47—42no—
216Dic47—82no—
217C2×#461—471no—
218#2354—471no—
219#1990—471no—
220#1988D4711yescharacteristic, complemented
221C2×D4⋊C8—471no—
222#1296—471no—
223#1279—471no—
224C2×C47#164211yescharacteristic
225C2×C47#164611yescharacteristic
226C2×C47#165011yescharacteristic
227C2×C47#165411yescharacteristic
228C2×C47#187511yescharacteristic
229C2×C47#187711yescharacteristic
230C2×C47#188211yescharacteristic
231D47—82no—
232C22×SD16—941no—
233C2×C4⋊Q8C2×D4711yescharacteristic
234C2×C4×D4—471no—
235C2×#117—941no—
236#461—474no—
237#461—942no—
238C2×C4⋊C8—471no—
239C2×C4⋊C8C2×D4711yescharacteristic
240C2×#103—941no—
241C2×C4×C8—471no—
242#390—941no—
243#388—942no—
244#384—471no—
245#380—941no—
246C4×C4⋊C4—941no—
247C4×C4⋊C4C2×D4711yescharacteristic
248#340—21no—
249#340—21no—
250#340—942no—
251D4⋊C8—942no—
252C47#898811yescharacteristic, complemented, Sylow
253C22×Q8—21no—
254C22×D4—471no—
255C23×C4—941no—
256C2×SD16—944no—
257C2×SD16—1882no—
258C2×SD16—1882no—
259C22×C8—21no—
260C22×C8—941no—
261C22×C8—941no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8C4×D4712yes—
266C4×D4—472no—
267C4×D4—472no—
268C4×D4—941no—
269C4×D4—941no—
270#117—1882no—
271#117—1882no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—471no—
275C2×C4⋊C4—472no—
276C2×C4⋊C4—941no—
277C2×C4⋊C4—941no—
278C2×C4⋊C4C22⋊D4711yescharacteristic
279C2×C4⋊C4C22⋊D4711yescharacteristic
280C2×C22⋊C4—941no—
281C2×C22⋊C4—942no—
282C2×C22⋊C4—942no—
283C2×C42—21no—
284C2×C42—942no—
285C2×C42C22×D4711yescharacteristic
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—472no—
289C4⋊C8—941no—
290#103—944no—
291C4×C8—941no—
292C4×C8—941no—
293#95—21no—
294#95—942no—
295C24—941no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×D4—472no—
303C2×D4—941no—
304C2×D4—941no—
305C2×D4—941no—
306C2×D4—941no—
307C2×D4—941no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—941no—
312C22×C4—942no—
313C22×C4—942no—
314C22×C4—942no—
315C22×C4—1881no—
316C22×C4—1881no—
317C22×C4—1881no—
318C22×C4—1881no—
319C22×C4D4×D4711yescharacteristic
320C22×C4D4×D4711yescharacteristic
321C22×C4C2×C22⋊D4711yescharacteristic
322SD16—1888no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—942no—
327C2×C8—942no—
328C2×C8—1881no—
329C2×C8—1881no—
330C2×C8—1881no—
331C2×C8—1881no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—472no—
340C4⋊C4—941no—
341C4⋊C4—1882no—
342C4⋊C4—1881no—
343C4⋊C4—1881no—
344C4⋊C4—1881no—
345C4⋊C4—1881no—
346C4⋊C4#1062533612yescomplemented
347C22⋊C4—942no—
348C22⋊C4—942no—
349C22⋊C4—3762no—
350C22⋊C4—1884no—
351C42—41no—
352C42—41no—
353C42—1884no—
354C42C2×C4×D4711yescharacteristic
355C42#1062535011yescharacteristic
356C42D4×D4711yescharacteristic
357C42#1062535211yescharacteristic
358C23—942no—
359C23—1881no—
360C23—1881no—
361C23—1881no—
362C23—1881no—
363C23—1881no—
364C23—1881no—
365C23#1309399211yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—81no—
369Q8—81no—
370D4—942no—
371D4—942no—
372D4—1881no—
373D4—1881no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—942no—
386C2×C4—1882no—
387C2×C4—1882no—
388C2×C4—1882no—
389C2×C4—1881no—
390C2×C4—1881no—
391C2×C4—1882no—
392C2×C4—1882no—
393C2×C4—1882no—
394C2×C4—1882no—
395C2×C4C22⋊C4×D4711yescharacteristic
396C2×C4(C22⋊D47)⋊C411yescharacteristic
397C2×C4C4×C22⋊D4711yescharacteristic
398C2×C4#1309399511yescharacteristic
399C2×C4#1309399711yescharacteristic
400C2×C4#1309400611yescharacteristic
401C2×C4C2×#1062533611yescharacteristic
402C2×C4#1309404311yescharacteristic
403C2×C4#1309404411yescharacteristic
404C2×C4#1309404511yescharacteristic
405C2×C4#1309404811yescharacteristic
406C2×C4#1309404911yescharacteristic
407C2×C4#1309405111yescharacteristic
408C2×C4#1309405511yescharacteristic
409C8—81no—
410C8—1882no—
411C8—1882no—
412C22—3761no—
413C22—3761no—
414C22—3761no—
415C22—3761no—
416C22—1882no—
417C22—1882no—
418C22—1882no—
419C22#1568899911yescharacteristic
420C22#1568926511yescharacteristic
421C22#1568926711yescharacteristic
422C22#1568927311yescharacteristic
423C22#1568928311yescharacteristic
424C22#1568929911yescharacteristic
425C22#1568930411yescharacteristic
426C4—21no—
427C4—21no—
428C4—42no—
429C4—81no—
430C4—82no—
431C4—3762no—
432C4—1882no—
433C4#1568911611yescharacteristic
434C4#1568912611yescharacteristic
435C4C4×#1062533611yescharacteristic
436C4#1568922111yescharacteristic
437C4#1568932511yescharacteristic
438C4#1568933511yescharacteristic
439C4#1568944511yescharacteristic
440C4#1568944611yescharacteristic
441C2—3762no—
442C2#2024258711yescharacteristic
443C2#2024261511yescharacteristic
444C2#2024264311yescharacteristic
445C2#2024267111yescharacteristic
446C2#2024370711yescharacteristic
447C2#2024372511yescharacteristic
448C2#2024375411yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1988C2
socleC23×C47#120
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1988D47
hypocenterC47#8988
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. #20244350C2semidirect
5. #20241272C2semidirect
6. #20241229C2semidirect
7. C47×#1988C2semidirect
12. #15689325C4semidirect
107. C4⋊C4×C47SD16semidirect
220. #1988D47semidirect
252. C47#8988semidirect
346. C4⋊C4#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156893252
#202463352
#202443502
C2×#156888502
#202412722
#202412292
C47×#19882
#898847