Finite Group Databasebeta

← #26111044

Subgroups — #26111044

Order 12032.

616 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689323C211yesmaximal, characteristic
2#20246345C211yesmaximal, characteristic, complemented
3#20244314C211yesmaximal, characteristic
4#20244194C211yesmaximal, characteristic
5#20241272C211yesmaximal, characteristic, complemented
6#20241228C211yesmaximal, characteristic, complemented
7C2×C47×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C2×#13094021C2211yescharacteristic
10#15689323—22no—
11#15689323C414yescomplemented
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×#13093946—21no—
14C2×#13093946—21no—
15C2×#13093946—22no—
16C2×#13093944—21no—
17#15689030—21no—
18#15689009—22no—
19#15688993C2211yescharacteristic
20#15688989—21no—
21C4⋊C4×Dic47—21no—
22C4×C4⋊Dic47—21no—
23C4×C4⋊Dic47C2211yescharacteristic
24C2×C4×Q8×C47C2211yescharacteristic
25C2×C4⋊C8×C47C2211yescharacteristic
26C2×C4×C8×C47C2211yescharacteristic
27C47×Q8⋊C8—22no—
28C22×D188—21no—
29C22×D188—21no—
30C22×D188—21no—
31C2×#10625336—24no—
32C2×#10625336—42no—
33C2×#10625336—42no—
34C22×C47⋊C8—21no—
35#13094021—21no—
36#13094021—21no—
37#13094021—21no—
38#13094021C2×C412yes—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47D411yescharacteristic
42C2×C4⋊Dic47D411yescharacteristic
43C2×C4⋊Dic47D412yes—
44C2×C4×Dic47—21no—
45C2×C4×Dic47—22no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)C2×C412yes—
48#13093946—42no—
49#13093946—44no—
50#13093946—42no—
51#13093944—24no—
52C47⋊#95—22no—
53C47⋊#95—22no—
54C47⋊#95—22no—
55C47⋊#95—21no—
56C22×Q8×C47D411yescharacteristic
57C22×C8×C47—21no—
58C22×C8×C47—21no—
59C4×Q8×C47—21no—
60C4×Q8×C47—21no—
61C4×Q8×C47C2×C412yes—
62C4×Q8×C47D412yes—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C42×C47—21no—
66C2×C42×C47C2311yescharacteristic
67C4⋊C8×C47—21no—
68C4⋊C8×C47—21no—
69C4×C8×C47—21no—
70C4×C8×C47—21no—
71C23×D47—41no—
72C2×Q8×C47—21no—
73C2×Q8×C47—21no—
74C2×Q8×C47—21no—
75C2×Q8×C47—21no—
76C2×Q8×C47—21no—
77C2×Q8×C47C22⋊C412yes—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—22no—
86C2×D188—22no—
87C2×D188—22no—
88C2×D188—22no—
89C2×D188—42no—
90C2×D188—42no—
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—41no—
100C2×D188—41no—
101C2×D188—41no—
102C2×C8×C47—22no—
103C2×C8×C47—41no—
104C2×C8×C47—41no—
105C2×C8×C47—41no—
106C2×C8×C47—41no—
107C2×C8×C47—41no—
108C4⋊C4×C47—22no—
109C4⋊C4×C47—22no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47C22⋊C412yes—
112C42×C47—22no—
113C42×C47—22no—
114C42×C47C22×C411yescharacteristic
115C42×C47C2×D411yescharacteristic
116C42×C47#4111yescharacteristic
117C42×C47#4111yescharacteristic
118#10625336—48no—
119C4⋊Dic47—21no—
120C4⋊Dic47—21no—
121C4⋊Dic47—21no—
122C4⋊Dic47—41no—
123C4⋊Dic47—41no—
124C4⋊Dic47—41no—
125C4⋊Dic47—41no—
126C4⋊Dic47—42no—
127C4⋊Dic47SD1612yescomplemented
128C4×Dic47—41no—
129C4×Dic47—41no—
130C4×Dic47—82no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C23×C47#12011yescharacteristic
135C22×D47—42no—
136C22×D47—81no—
137C22×D47—81no—
138C22×D47—81no—
139C22×D47—81no—
140C22×D47—81no—
141C22×D47—81no—
142Q8×C47—22no—
143Q8×C47—22no—
144Q8×C47—41no—
145Q8×C47—41no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47C2×C22⋊C411yescharacteristic
155C2×C4×C47C4×D411yescharacteristic
156C2×C4×C47C4×D411yescharacteristic
157C2×C4×C47#12111yescharacteristic
158C2×C4×C47#12111yescharacteristic
159C2×C4×C47#12311yescharacteristic
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×C47AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C47#13711yescharacteristic
168C2×Dic47—22no—
169C2×Dic47—41no—
170C2×Dic47—41no—
171C2×Dic47—41no—
172C2×Dic47—41no—
173C2×Dic47—42no—
174C2×Dic47—42no—
175C2×Dic47—42no—
176C2×Dic47—41no—
177D188—44no—
178D188—42no—
179D188—42no—
180D188—42no—
181D188—42no—
182D188—82no—
183D188—81no—
184D188—81no—
185D188—81no—
186D188—81no—
187C8×C47—42no—
188C8×C47—81no—
189C47⋊C8—42no—
190#8978—471nomaximal, Sylow
191C22×C47#38611yescharacteristic
192C22×C47#44811yescharacteristic
193C22×C47#44911yescharacteristic
194C22×C47#44911yescharacteristic
195C22×C47#44911yescharacteristic
196C22×C47#45311yescharacteristic
197C22×C47#45411yescharacteristic
198C2×D47—82no—
199C2×D47—82no—
200C2×D47—82no—
201C2×D47—161no—
202C2×D47—161no—
203C2×D47—161no—
204C2×D47—161no—
205C4×C47—21no—
206C4×C47—21no—
207C4×C47—42no—
208C4×C47—42no—
209C4×C47#41611yescharacteristic
210C4×C47#41811yescharacteristic
211C4×C47C4×SD1611yescharacteristic
212C4×C47#44011yescharacteristic
213C4×C47#46011yescharacteristic
214C4×C47#46411yescharacteristic
215C4×C47#48211yescharacteristic
216C4×C47#48211yescharacteristic
217Dic47—42no—
218Dic47—82no—
219Dic47—81no—
220C2×#460—471no—
221#2355—471no—
222#1982—471no—
223#1957—471no—
224C2×Q8⋊C8D4711yescharacteristic, complemented
225#1296—471no—
226#1279—471no—
227C2×C47#162011yescharacteristic
228C2×C47#162211yescharacteristic
229C2×C47#162611yescharacteristic
230C2×C47#162811yescharacteristic
231C2×C47#187511yescharacteristic
232C2×C47#188011yescharacteristic
233C2×C47#188311yescharacteristic
234D47—162no—
235C22×SD16—941no—
236C2×#127—471no—
237C2×C4×Q8C2×D4711yescharacteristic
238#460—474no—
239#460—942no—
240C2×C4⋊C8—471no—
241C2×C4⋊C8C2×D4711yescharacteristic
242C2×#102—941no—
243C2×#102—942no—
244C2×#102—941no—
245C2×#102—941no—
246C2×C4×C8C2×D4711yescharacteristic
247#391—941no—
248#387—942no—
249#384—471no—
250#382—941no—
251C4×C4⋊C4—471no—
252C4×C4⋊C4—941no—
253C4×C4⋊C4—941no—
254Q8⋊C8—22no—
255C47#897811yescharacteristic, complemented, Sylow
256C22×Q8D18811yescharacteristic
257C22×D4—941no—
258C22×D4—941no—
259C22×D4—941no—
260C2×SD16—944no—
261C2×SD16—1882no—
262C2×SD16—1882no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—941no—
266#127—472no—
267#127—941no—
268#127—941no—
269#127—941no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8C4×D4712yes—
273C4×Q8C22⋊D4712yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—471no—
276C2×C4⋊C4—471no—
277C2×C4⋊C4—472no—
278C2×C4⋊C4—941no—
279C2×C4⋊C4—941no—
280C2×C4⋊C4D18811yescharacteristic
281C2×C42—21no—
282C2×C42—942no—
283C2×C42—941no—
284C2×C42C22×D4711yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—472no—
288C4⋊C8—941no—
289#102—944no—
290#102—1882no—
291#102—1882no—
292#102—1884no—
293C4×C8—21no—
294C4×C8—21no—
295#95—942no—
296#95—942no—
297#95—942no—
298#95—941no—
299C24—1881no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8(C2×D47)⋊C412yes—
306C2×D4—942no—
307C2×D4—942no—
308C2×D4—942no—
309C2×D4—1882no—
310C2×D4—1882no—
311C2×D4—1881no—
312C2×D4—1881no—
313C2×D4—1881no—
314C2×D4—1881no—
315C2×D4—1881no—
316C2×D4—1881no—
317C2×D4—1881no—
318C2×D4—1881no—
319C2×D4—1881no—
320C2×D4—1881no—
321C2×D4—1881no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—941no—
325C22×C4—941no—
326C22×C4—942no—
327C22×C4C2×D18811yescharacteristic
328C22×C4D4×D4711yescharacteristic
329C22×C4D4×D4711yescharacteristic
330SD16—1888no—
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—942no—
338C2×C8—1881no—
339C2×C8—1881no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—472no—
344C4⋊C4—941no—
345C4⋊C4—941no—
346C4⋊C4—941no—
347C4⋊C4—1882no—
348C4⋊C4—1881no—
349C4⋊C4—1881no—
350C4⋊C4—1881no—
351C4⋊C4—1881no—
352C4⋊C4(C2×D47)⋊C412yes—
353C42—22no—
354C42—22no—
355C42—3762no—
356C42—1881no—
357C42—1881no—
358C42C2×C4×D4711yescharacteristic
359C42#1062535211yescharacteristic
360C42#1062535211yescharacteristic
361C42C2×C22⋊D4711yescharacteristic
362C23—3761no—
363C23—3761no—
364C23—3761no—
365C23—3761no—
366C23—3761no—
367C23—3761no—
368C23—1882no—
369C23#1309399411yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374D4—3762no—
375D4—3761no—
376D4—3761no—
377D4—3761no—
378D4—3761no—
379D4—1884no—
380D4—1882no—
381D4—1882no—
382D4—1882no—
383D4—1882no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—942no—
393C2×C4—1881no—
394C2×C4—1882no—
395C2×C4—1881no—
396C2×C4—1882no—
397C2×C4—1882no—
398C2×C4—1881no—
399C2×C4—1881no—
400C2×C4—1881no—
401C2×C4C2×(C2×D47)⋊C411yescharacteristic
402C2×C4(C22⋊D47)⋊C411yescharacteristic
403C2×C4(C22⋊D47)⋊C411yescharacteristic
404C2×C4#1309399611yescharacteristic
405C2×C4#1309399611yescharacteristic
406C2×C4#1309400811yescharacteristic
407C2×C4SD16×D4711yescharacteristic
408C2×C4#1309404611yescharacteristic
409C2×C4#1309404611yescharacteristic
410C2×C4#1309404711yescharacteristic
411C2×C4#1309405111yescharacteristic
412C2×C4#1309405211yescharacteristic
413C2×C4#1309405211yescharacteristic
414C2×C4#1309405611yescharacteristic
415C8—42no—
416C8—81no—
417C8—1882no—
418C22—3762no—
419C22—3762no—
420C22—3762no—
421C22—7521no—
422C22—7521no—
423C22—7521no—
424C22—7521no—
425C22#1568900011yescharacteristic
426C22#1568926311yescharacteristic
427C22#1568927011yescharacteristic
428C22#1568927011yescharacteristic
429C22#1568927111yescharacteristic
430C22#1568929211yescharacteristic
431C22#1568929711yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—3762no—
437C4—3761no—
438C4—1882no—
439C4#1568912011yescharacteristic
440C4#1568912911yescharacteristic
441C4#1568920811yescharacteristic
442C4#1568922611yescharacteristic
443C4#1568932111yescharacteristic
444C4#1568934711yescharacteristic
445C4#1568944811yescharacteristic
446C4#1568944811yescharacteristic
447C2—7522no—
448C2#2024244911yescharacteristic
449C2#2024246311yescharacteristic
450C2#2024248611yescharacteristic
451C2#2024250111yescharacteristic
452C2#2024370811yescharacteristic
453C2#2024374111yescharacteristic
454C2#2024376211yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×Q8⋊C8C2
socleC23×C47#120
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenterC2×Q8⋊C8D47
hypocenterC47#8978
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #20246345C2semidirect
5. #20241272C2semidirect
6. #20241228C2semidirect
7. C2×C47×Q8⋊C8C2semidirect
11. #15689323C4semidirect
127. C4⋊Dic47SD16semidirect
224. C2×Q8⋊C8D47semidirect
255. C47#8978semidirect

Maximal subgroups

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

subgroupindex
C2×#156893232
#202463452
#202443142
#202441942
#202412722
#202412282
C2×C47×Q8⋊C82
#897847