Finite Group Databasebeta

← #26111029

Subgroups — #26111029

Order 12032.

592 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689410C211yesmaximal, characteristic
2#20246339C211yesmaximal, characteristic
3#20244337C211yesmaximal, characteristic, complemented
4C2×#15688852C211yesmaximal, characteristic
5#20241271C211yesmaximal, characteristic, complemented
6#20241228C211yesmaximal, characteristic, complemented
7C47×#1956C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9C2×#13093982—21no—
10#15689410—22no—
11#15689410C414yescomplemented
12C2×#13093955—21no—
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093944—21no—
16#15689024—21no—
17#15689013—22no—
18#15688980—21no—
19C4⋊C4×Dic47—21no—
20C4×C4⋊Dic47C2211yescharacteristic
21#15688885—21no—
22#15688885—21no—
23#15688852—22no—
24C2×C4⋊Q8×C47C2211yescharacteristic
25C2×#103×C47—22no—
26C2×C4×C8×C47C2211yescharacteristic
27C47×#384C2211yescharacteristic
28C22×D188D411yescharacteristic
29C22×C4×D47—21no—
30C22×C47⋊C8—21no—
31C22×C47⋊C8—21no—
32C4×D188—21no—
33C4×D188—21no—
34C4×D188C2×C412yes—
35C4×D188D412yes—
36#13093982—42no—
37#13093982—42no—
38C2×C4⋊Dic47D411yescharacteristic
39C2×C4⋊Dic47D411yescharacteristic
40C2×C4⋊Dic47D411yescharacteristic
41C2×(C2×D47)⋊C4—21no—
42C2×(C2×D47)⋊C4—22no—
43C2×(C2×D47)⋊C4—22no—
44C2×C4×Dic47—22no—
45C2×C4×Dic47—21no—
46#13093955—24no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)—21no—
50C4⋊(C47⋊C8)C2×C412yes—
51#13093944—24no—
52C47⋊#95—22no—
53C47⋊#95—21no—
54C22×Q8×C47—21no—
55C22×C8×C47—21no—
56C4⋊Q8×C47—21no—
57C4⋊Q8×C47—21no—
58C4⋊Q8×C47—21no—
59C4⋊Q8×C47C2×C412yes—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47D412yes—
65C2×C42×C47C2311yescharacteristic
66#103×C47—44no—
67C4×C8×C47—21no—
68C4×C8×C47—21no—
69C23×D47—21no—
70C2×Q8×C47—22no—
71C2×Q8×C47—42no—
72C2×Q8×C47—42no—
73C2×Q8×C47—41no—
74C22×C4×C47—22no—
75C22×C4×C47—41no—
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C2×D188—21no—
83C2×D188—21no—
84C2×D188—21no—
85C2×D188—21no—
86C2×D188—21no—
87C2×D188C22⋊C412yes—
88C2×C4×D47—22no—
89C2×C4×D47—22no—
90C2×C4×D47—41no—
91C2×C4×D47—41no—
92C2×C4×D47—41no—
93C2×C4×D47—41no—
94C2×C8×C47—22no—
95C2×C8×C47—41no—
96C2×C8×C47—41no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—42no—
100C4⋊C4×C47—41no—
101C4⋊C4×C47—41no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47—41no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106C42×C47C22×C411yescharacteristic
107C42×C47C2×D411yescharacteristic
108C42×C47#4111yescharacteristic
109C42×C47#4111yescharacteristic
110(C2×D47)⋊C4—22no—
111(C2×D47)⋊C4—22no—
112(C2×D47)⋊C4—44no—
113(C2×D47)⋊C4—82no—
114C4⋊Dic47—21no—
115C4⋊Dic47—21no—
116C4⋊Dic47—21no—
117C4⋊Dic47—21no—
118C4⋊Dic47C22⋊C412yes—
119C4⋊Dic47SD1612yes—
120C4×Dic47—44no—
121C4×Dic47—41no—
122C4×Dic47—41no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127C2×C47⋊C8—41no—
128C2×C47⋊C8—41no—
129C23×C47#12011yescharacteristic
130C22×D47—22no—
131C22×D47—41no—
132C22×D47—41no—
133C22×D47—41no—
134C22×D47—41no—
135C22×D47—41no—
136C22×D47—41no—
137Q8×C47—44no—
138Q8×C47—82no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—81no—
146C2×C4×C47—81no—
147C2×C4×C47C2×C22⋊C411yescharacteristic
148C2×C4×C47C4×D411yescharacteristic
149C2×C4×C47C4×D411yescharacteristic
150C2×C4×C47#12111yescharacteristic
151C2×C4×C47#12111yescharacteristic
152C2×C4×C47#12311yescharacteristic
153C2×C4×C47C2×SD1611yescharacteristic
154C2×C4×C47#13511yescharacteristic
155C2×C4×C47#13511yescharacteristic
156C2×C4×C47#13511yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47#13711yescharacteristic
159C2×C4×C47#13711yescharacteristic
160C2×C4×C47#13711yescharacteristic
161C2×Dic47—22no—
162C2×Dic47—22no—
163C2×Dic47—41no—
164C2×Dic47—41no—
165C2×Dic47—41no—
166C2×Dic47—41no—
167C2×Dic47—41no—
168C2×Dic47—41no—
169C2×Dic47—41no—
170D188—22no—
171D188—22no—
172D188—41no—
173D188—41no—
174C4×D47—42no—
175C4×D47—42no—
176C4×D47—42no—
177C4×D47—42no—
178C8×C47—42no—
179C47⋊C8—42no—
180C47⋊C8—81no—
181#8976—471nomaximal, Sylow
182C22×C47#38611yescharacteristic
183C22×C47#45011yescharacteristic
184C22×C47#45211yescharacteristic
185C22×C47#45211yescharacteristic
186C22×C47#45211yescharacteristic
187C22×C47#45311yescharacteristic
188C22×C47#45411yescharacteristic
189C2×D47—42no—
190C2×D47—42no—
191C2×D47—42no—
192C2×D47—81no—
193C2×D47—81no—
194C2×D47—81no—
195C2×D47—81no—
196C4×C47—21no—
197C4×C47—21no—
198C4×C47—82no—
199C4×C47—82no—
200C4×C47#41611yescharacteristic
201C4×C47#41911yescharacteristic
202C4×C47C4×SD1611yescharacteristic
203C4×C47#44011yescharacteristic
204C4×C47#46111yescharacteristic
205C4×C47#46311yescharacteristic
206C4×C47#48511yescharacteristic
207C4×C47#48511yescharacteristic
208Dic47—42no—
209Dic47—42no—
210Dic47—81no—
211C2×#476—471no—
212#2354—471no—
213#1987—471no—
214#1956D4711yescharacteristic, complemented
215C2×D4⋊C8—471no—
216#1296—471no—
217#1279—471no—
218C2×C47#163111yescharacteristic
219C2×C47#163411yescharacteristic
220C2×C47#163511yescharacteristic
221C2×C47#163811yescharacteristic
222C2×C47#187311yescharacteristic
223C2×C47#187711yescharacteristic
224C2×C47#188311yescharacteristic
225D47—82no—
226C2×C4⋊Q8C2×D4711yescharacteristic
227C2×C4×D4—471no—
228C2×#117—941no—
229#476—474no—
230#476—942no—
231C2×#106—941no—
232C2×C4⋊C8—471no—
233C2×C4⋊C8—471no—
234C2×#103—22no—
235C2×#102—941no—
236C2×C4×C8C2×D4711yescharacteristic
237#390—941no—
238#388—942no—
239#384C2×D4711yescharacteristic
240#380—941no—
241C4×C4⋊C4—471no—
242C4×C4⋊C4—941no—
243#340—941no—
244#340—941no—
245D4⋊C8—942no—
246C47#897611yescharacteristic, complemented, Sylow
247C22×Q8—21no—
248C22×D4—471no—
249C23×C4—941no—
250C22×C8—21no—
251C22×C8—941no—
252C22×C8—941no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8C4×D4712yes—
257C4×D4—472no—
258C4×D4—472no—
259C4×D4—941no—
260C4×D4—941no—
261#117—1882no—
262#117—1882no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—471no—
268C2×C4⋊C4—471no—
269C2×C4⋊C4—471no—
270C2×C4⋊C4C22⋊D4712yes—
271C2×C22⋊C4—941no—
272C2×C22⋊C4—942no—
273C2×C22⋊C4—942no—
274C2×C42—942no—
275C2×C42—941no—
276C2×C42C22×D4711yescharacteristic
277#106—944no—
278C4⋊C8—472no—
279C4⋊C8—941no—
280C4⋊C8—941no—
281C4⋊C8—941no—
282#103—44no—
283#102—944no—
284C4×C8—21no—
285C4×C8—21no—
286#95—942no—
287#95—941no—
288C24—941no—
289C2×Q8—22no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—41no—
293C2×D4—472no—
294C2×D4—941no—
295C2×D4—941no—
296C2×D4—941no—
297C2×D4—941no—
298C2×D4—941no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—941no—
302C22×C4—941no—
303C22×C4—942no—
304C22×C4—942no—
305C22×C4—941no—
306C22×C4—1881no—
307C22×C4—1881no—
308C22×C4—1881no—
309C22×C4—1881no—
310C22×C4D4×D4711yescharacteristic
311C22×C4D4×D4711yescharacteristic
312C22×C4C2×C22⋊D4711yescharacteristic
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—942no—
317C2×C8—1881no—
318C2×C8—1881no—
319C2×C8—1881no—
320C2×C8—1881no—
321C2×C8—1881no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—472no—
332C4⋊C4—472no—
333C4⋊C4—941no—
334C4⋊C4—941no—
335C4⋊C4—941no—
336C4⋊C4—941no—
337C22⋊C4—942no—
338C22⋊C4—942no—
339C22⋊C4—3762no—
340C22⋊C4—1884no—
341C42—1884no—
342C42—1881no—
343C42—1881no—
344C42C2×C4×D4711yescharacteristic
345C42#1062535011yescharacteristic
346C42D4×D4711yescharacteristic
347C42#1062535211yescharacteristic
348C23—942no—
349C23—1881no—
350C23—1881no—
351C23—1881no—
352C23—1881no—
353C23—1881no—
354C23—1881no—
355C23#1309399211yescharacteristic
356Q8—44no—
357Q8—82no—
358D4—942no—
359D4—942no—
360D4—1881no—
361D4—1881no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—942no—
371C2×C4—942no—
372C2×C4—1881no—
373C2×C4—1881no—
374C2×C4—1881no—
375C2×C4—1881no—
376C2×C4—1881no—
377C2×C4—1881no—
378C2×C4—1881no—
379C2×C4—1882no—
380C2×C4—1882no—
381C2×C4—1882no—
382C2×C4—1882no—
383C2×C4C22⋊C4×D4711yescharacteristic
384C2×C4(C22⋊D47)⋊C411yescharacteristic
385C2×C4C4×C22⋊D4711yescharacteristic
386C2×C4#1309399511yescharacteristic
387C2×C4#1309399711yescharacteristic
388C2×C4#1309400611yescharacteristic
389C2×C4SD16×D4711yescharacteristic
390C2×C4#1309404511yescharacteristic
391C2×C4#1309404711yescharacteristic
392C2×C4#1309404711yescharacteristic
393C2×C4#1309405111yescharacteristic
394C2×C4#1309405311yescharacteristic
395C2×C4#1309405611yescharacteristic
396C2×C4#1309405611yescharacteristic
397C8—42no—
398C8—3761no—
399C8—1882no—
400C22—3761no—
401C22—3761no—
402C22—3761no—
403C22—3761no—
404C22—1882no—
405C22—1882no—
406C22—1882no—
407C22#1568899911yescharacteristic
408C22#1568927211yescharacteristic
409C22#1568928211yescharacteristic
410C22#1568928411yescharacteristic
411C22#1568928411yescharacteristic
412C22#1568929011yescharacteristic
413C22#1568929511yescharacteristic
414C4—21no—
415C4—21no—
416C4—82no—
417C4—82no—
418C4—3761no—
419C4—1882no—
420C4—1882no—
421C4#1568911811yescharacteristic
422C4#1568913011yescharacteristic
423C4#1568920811yescharacteristic
424C4#1568922611yescharacteristic
425C4#1568932711yescharacteristic
426C4#1568934111yescharacteristic
427C4#1568946211yescharacteristic
428C4#1568946311yescharacteristic
429C2—3762no—
430C2#2024251211yescharacteristic
431C2#2024253111yescharacteristic
432C2#2024253611yescharacteristic
433C2#2024255711yescharacteristic
434C2#2024369711yescharacteristic
435C2#2024371911yescharacteristic
436C2#2024376711yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1956C2
socleC23×C47#120
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1956D47
hypocenterC47#8976
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
3. #20244337C2semidirect
5. #20241271C2semidirect
6. #20241228C2semidirect
7. C47×#1956C2semidirect
11. #15689410C4semidirect
214. #1956D47semidirect
246. C47#8976semidirect

Maximal subgroups

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

subgroupindex
C2×#156894102
#202463392
#202443372
C2×#156888522
#202412712
#202412282
C47×#19562
#897647