Finite Group Databasebeta

← #26112464

Subgroups — #26112464

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689323C211yesmaximal, characteristic
2C2×Q8×C47⋊C8C211yesmaximal, characteristic, complemented
3#20244193C211yesmaximal, characteristic
4C2×#15688866C211yesmaximal, characteristic
5#20241271C211yesmaximal, characteristic, complemented
6#20241222C211yesmaximal, characteristic, complemented
7C47×#1803C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C2×#13094021C2211yescharacteristic
10#15689323—22no—
11#15689323C414yescomplemented
12Q8×C47⋊C8—22no—
13Q8×C47⋊C8—22no—
14C2×C4⋊(C47⋊C8)—21no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×#13093944—21no—
18C2×#13093944—21no—
19C2×#13093944—21no—
20C2×C4×C47⋊C8—21no—
21C2×C4×C47⋊C8C2211yescharacteristic
22#15688866—22no—
23C2×C4×Q8×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C2×#103×C47—21no—
26C47×#384C2211yescharacteristic
27C47×#340—21no—
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⋊C8D411yescharacteristic
37C22×C47⋊C8D411yescharacteristic
38#13094021—21no—
39#13094021—21no—
40#13094021—21no—
41#13094021C2×C412yes—
42C4⋊(C47⋊C8)—24no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)C2×C412yes—
46C4⋊(C47⋊C8)D412yes—
47#13093944—24no—
48#13093944—42no—
49#13093944—42no—
50C4×C47⋊C8—24no—
51C4×C47⋊C8—21no—
52C4×C47⋊C8—21no—
53C22×Q8×C47D411yescharacteristic
54C22×C8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47—21no—
57C4×Q8×C47C2×C412yes—
58C4×Q8×C47D412yes—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47D411yescharacteristic
64C2×C42×C47—21no—
65C2×C42×C47C2311yescharacteristic
66C4⋊C8×C47—21no—
67C4⋊C8×C47—21no—
68#103×C47—42no—
69C23×D47—41no—
70C2×Q8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47C22⋊C412yes—
74C2×Q8×C47D812yes—
75C2×Q8×C47SD1612yes—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—41no—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47#4111yescharacteristic
82C2×D188—22no—
83C2×D188—22no—
84C2×D188—22no—
85C2×D188—41no—
86C2×D188—41no—
87C2×D188—41no—
88C2×D188—41no—
89C2×D188—41no—
90C2×D188—41no—
91C2×D188—41no—
92C2×D188—41no—
93C2×D188—41no—
94C2×D188—41no—
95C2×D188—41no—
96C2×D188—81no—
97C2×D188—81no—
98C2×C8×C47—41no—
99C2×C8×C47—41no—
100C2×C8×C47—41no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—81no—
109C4⋊C4×C47C22⋊C412yes—
110C42×C47—22no—
111C42×C47—22no—
112C42×C47C22×C411yescharacteristic
113C42×C47C2×D411yescharacteristic
114C42×C47C2×D411yescharacteristic
115C42×C47#4111yescharacteristic
116#10625336—48no—
117C2×C47⋊C8—21no—
118C2×C47⋊C8—21no—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—21no—
121C2×C47⋊C8—22no—
122C2×C47⋊C8—21no—
123C2×C47⋊C8—21no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127C2×C47⋊C8—41no—
128C2×C47⋊C8SD1612yescomplemented
129C23×C47#12111yescharacteristic
130C22×D47—42no—
131C22×D47—81no—
132C22×D47—81no—
133C22×D47—81no—
134C22×D47—81no—
135C22×D47—81no—
136C22×D47—81no—
137Q8×C47—22no—
138Q8×C47—22no—
139Q8×C47—22no—
140Q8×C47#10214yescomplemented
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—22no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—81no—
150C2×C4×C47—81no—
151C2×C4×C47—81no—
152C2×C4×C47C2×C22⋊C411yescharacteristic
153C2×C4×C47C4×D411yescharacteristic
154C2×C4×C47C4×D411yescharacteristic
155C2×C4×C47#12011yescharacteristic
156C2×C4×C47#12111yescharacteristic
157C2×C4×C47#12311yescharacteristic
158C2×C4×C47C2×D811yescharacteristic
159C2×C4×C47C2×SD1611yescharacteristic
160C2×C4×C47C2×SD1611yescharacteristic
161C2×C4×C47#13511yescharacteristic
162C2×C4×C47#13511yescharacteristic
163C2×C4×C47#13511yescharacteristic
164C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C47#13711yescharacteristic
166D188—42no—
167D188—42no—
168D188—42no—
169D188—42no—
170D188—82no—
171D188—82no—
172D188—81no—
173D188—81no—
174D188—81no—
175D188—81no—
176C8×C47—81no—
177C47⋊C8—22no—
178C47⋊C8—42no—
179C47⋊C8—41no—
180C47⋊C8—42no—
181#9222—471nomaximal, Sylow
182C22×C47#38611yescharacteristic
183C22×C47#46011yescharacteristic
184C22×C47#46411yescharacteristic
185C22×C47#46511yescharacteristic
186C22×C47#46611yescharacteristic
187C22×C47#47111yescharacteristic
188C22×C47#47211yescharacteristic
189C2×D47—82no—
190C2×D47—82no—
191C2×D47—82no—
192C2×D47—161no—
193C2×D47—161no—
194C2×D47—161no—
195C2×D47—161no—
196C4×C47—24no—
197C4×C47—21no—
198C4×C47—21no—
199C4×C47—42no—
200C4×C47—161no—
201C4×C47C2×#10211yescharacteristic
202C4×C47#41611yescharacteristic
203C4×C47C4×SD1611yescharacteristic
204C4×C47#44011yescharacteristic
205C4×C47#44811yescharacteristic
206C4×C47#44911yescharacteristic
207C4×C47#46511yescharacteristic
208C4×C47#46611yescharacteristic
209C2×#460—471no—
210C2×C8×Q8—471no—
211#1957—471no—
212#1803D4711yescharacteristic, complemented
213C2×#331—471no—
214#1296—471no—
215#1277—471no—
216C2×C47#166611yescharacteristic
217C2×C47#166811yescharacteristic
218C2×C47#167811yescharacteristic
219C2×C47#168211yescharacteristic
220C2×C47#188011yescharacteristic
221C2×C47#189311yescharacteristic
222C2×C47#190711yescharacteristic
223D47—162no—
224C22×SD16—941no—
225C2×#127—471no—
226C2×C4×Q8C2×D4711yescharacteristic
227#460—474no—
228#460—942no—
229C8×Q8—942no—
230C8×Q8—942no—
231C2×C4⋊C8—471no—
232C2×C4⋊C8—471no—
233C2×C4⋊C8—941no—
234C2×C4⋊C8C2×D4711yescharacteristic
235C2×#103—21no—
236C2×#102—941no—
237C2×#102—941no—
238C2×#102—941no—
239C2×C4×C8—471no—
240C2×C4×C8—941no—
241#384C2×D4711yescharacteristic
242#340—21no—
243#331—942no—
244C47#922211yescharacteristic, complemented, Sylow
245C22×Q8C22⋊D4711yescharacteristic
246C22×D4—941no—
247C22×D4—941no—
248C22×D4—941no—
249C2×SD16—944no—
250C2×SD16—1882no—
251C2×SD16—1882no—
252C22×C8—21no—
253C22×C8—471no—
254C22×C8—471no—
255C22×C8—941no—
256C22×C8—941no—
257#127—472no—
258#127—941no—
259#127—941no—
260#127—941no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8C4×D4712yes—
264C4×Q8D18812yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C22⋊D4711yescharacteristic
270C2×C42—21no—
271C2×C42C22×D4711yescharacteristic
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—472no—
275C4⋊C8—472no—
276C4⋊C8—944no—
277C4⋊C8—941no—
278C4⋊C8—941no—
279#103—42no—
280#102—944no—
281#102—1882no—
282#102—1882no—
283C4×C8—944no—
284C4×C8—941no—
285C4×C8—941no—
286C24—1881no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8(C2×D47)⋊C412yes—
291C2×Q8#1062533412yes—
292C2×Q8#1062533612yes—
293C2×D4—942no—
294C2×D4—942no—
295C2×D4—942no—
296C2×D4—3761no—
297C2×D4—3761no—
298C2×D4—1881no—
299C2×D4—1881no—
300C2×D4—1881no—
301C2×D4—1881no—
302C2×D4—1881no—
303C2×D4—1881no—
304C2×D4—1881no—
305C2×D4—1881no—
306C2×D4—1881no—
307C2×D4—1881no—
308C2×D4—1881no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4D4×D4711yescharacteristic
313C22×C4#1062535211yescharacteristic
314C22×C4C2×C22⋊D4711yescharacteristic
315SD16—1888no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—472no—
320C2×C8—941no—
321C2×C8—941no—
322C2×C8—942no—
323C2×C8—942no—
324C2×C8—941no—
325C2×C8—941no—
326C2×C8—941no—
327C2×C8—1881no—
328C2×C8—1881no—
329C2×C8—1881no—
330C2×C8—1881no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—81no—
339C4⋊C4(C2×D47)⋊C412yes—
340C42—22no—
341C42—22no—
342C42C2×C4×D4711yescharacteristic
343C42C2×D18811yescharacteristic
344C42D4×D4711yescharacteristic
345C42#1062535211yescharacteristic
346C23—3761no—
347C23—3761no—
348C23—3761no—
349C23—3761no—
350C23—3761no—
351C23—3761no—
352C23—1882no—
353C23#1309399611yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—22no—
357Q8#1309394414yescomplemented
358D4—3762no—
359D4—3762no—
360D4—3761no—
361D4—3761no—
362D4—3761no—
363D4—3761no—
364D4—1882no—
365D4—1882no—
366D4—1882no—
367D4—1882no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4C2×(C2×D47)⋊C411yescharacteristic
380C2×C4(C22⋊D47)⋊C411yescharacteristic
381C2×C4(C22⋊D47)⋊C411yescharacteristic
382C2×C4#1309399411yescharacteristic
383C2×C4#1309399611yescharacteristic
384C2×C4#1309400811yescharacteristic
385C2×C4C2×#1062533411yescharacteristic
386C2×C4SD16×D4711yescharacteristic
387C2×C4C2×#1062533611yescharacteristic
388C2×C4#1309404411yescharacteristic
389C2×C4#1309404411yescharacteristic
390C2×C4#1309404711yescharacteristic
391C2×C4#1309405111yescharacteristic
392C2×C4#1309405611yescharacteristic
393C8—81no—
394C8—942no—
395C8—1881no—
396C8—1882no—
397C8—1882no—
398C22—3762no—
399C22—3762no—
400C22—3762no—
401C22—7521no—
402C22—7521no—
403C22—7521no—
404C22—7521no—
405C22#1568900011yescharacteristic
406C22#1568932111yescharacteristic
407C22#1568934711yescharacteristic
408C22#1568935411yescharacteristic
409C22#1568935711yescharacteristic
410C22#1568938811yescharacteristic
411C22#1568939111yescharacteristic
412C4—24no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—161no—
417C4C2×#1309394411yescharacteristic
418C4#1568911711yescharacteristic
419C4#1568920811yescharacteristic
420C4#1568922611yescharacteristic
421C4#1568926311yescharacteristic
422C4#1568927111yescharacteristic
423C4#1568935411yescharacteristic
424C4#1568935711yescharacteristic
425C2—7522no—
426C2#2024273811yescharacteristic
427C2#2024274611yescharacteristic
428C2#2024280611yescharacteristic
429C2#2024283411yescharacteristic
430C2#2024374111yescharacteristic
431C2#2024381411yescharacteristic
432C2#2024389411yescharacteristic
subgroupG/H
centerC23#13093996
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1803C2
socleC23×C47#121
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#1803D47
hypocenterC47#9222
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C47⋊C8C2semidirect
5. #20241271C2semidirect
6. #20241222C2semidirect
7. C47×#1803C2semidirect
11. #15689323C4semidirect
128. C2×C47⋊C8SD16semidirect
140. Q8×C47#102semidirect
212. #1803D47semidirect
244. C47#9222semidirect
357. Q8#13093944semidirect

Maximal subgroups

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

subgroupindex
C2×#156893232
C2×Q8×C47⋊C82
#202441932
C2×#156888662
#202412712
#202412222
C47×#18032
#922247