Finite Group Databasebeta

← #26112518

Subgroups — #26112518

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689315C211yesmaximal, characteristic
2C2×#15689202C211yesmaximal, characteristic, complemented
3#20243385C211yesmaximal, characteristic, complemented
4C2×#15688867C211yesmaximal, characteristic, complemented
5#20241271C211yesmaximal, characteristic, complemented
6#20241223C211yesmaximal, characteristic, complemented
7C47×#1957C211yesmaximal, characteristic, complemented
8C22×#10625334—21no—
9C22×#10625324—21no—
10C2×C4×D188C2211yescharacteristic
11#15689315—22no—
12#15689315C414yescomplemented
13#15689202—22no—
14#15689202—22no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×C4⋊(C47⋊C8)C2211yescharacteristic
18C2×#13093945—21no—
19C2×#13093944—21no—
20C2×(C2×D47)⋊C8—21no—
21#15688885—21no—
22#15688867—22no—
23C2×#127×C47C2211yescharacteristic
24C2×#102×C47—21no—
25C2×#102×C47—21no—
26C2×C4×C8×C47C2211yescharacteristic
27C47×#384C2211yescharacteristic
28C22×D188D411yescharacteristic
29C22×C4×D47—21no—
30C2×#10625334—24no—
31C2×#10625334—42no—
32C2×#10625334—42no—
33C2×#10625324—24no—
34C2×#10625324—42no—
35C2×#10625324—42no—
36C22×C47⋊C8—21no—
37C22×C47⋊C8—21no—
38C22×C47⋊C8—21no—
39C4×D188—21no—
40C4×D188—21no—
41C4×D188C2×C412yes—
42C4×D188D412yes—
43C2×C4⋊Dic47D411yescharacteristic
44C2×(C2×D47)⋊C4—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)C2×C412yes—
50C4⋊(C47⋊C8)D412yes—
51#13093945—24no—
52#13093944—42no—
53(C2×D47)⋊C8—24no—
54C22×D4×C47—21no—
55C22×D4×C47—21no—
56C22×D4×C47—21no—
57C22×C8×C47D411yescharacteristic
58C22×C8×C47D411yescharacteristic
59#127×C47—21no—
60#127×C47—21no—
61#127×C47—21no—
62#127×C47C2×C412yes—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C42×C47C2311yescharacteristic
67#102×C47—42no—
68#102×C47—42no—
69C4×C8×C47—21no—
70C4×C8×C47—21no—
71C24×C47—41no—
72C23×D47—21no—
73C2×D4×C47—22no—
74C2×D4×C47—22no—
75C2×D4×C47—22no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47—41no—
79C2×D4×C47—41no—
80C2×D4×C47—41no—
81C2×D4×C47—41no—
82C2×D4×C47—41no—
83C2×D4×C47—41no—
84C2×D4×C47—41no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—81no—
88C2×D4×C47—81no—
89C22×C4×C47—41no—
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47#4111yescharacteristic
93C22×Dic47—21no—
94C2×D188—21no—
95C2×D188—21no—
96C2×D188—21no—
97C2×D188—21no—
98C2×D188—21no—
99C2×D188C22⋊C412yes—
100C2×C4×D47—22no—
101C2×C4×D47—22no—
102C2×C4×D47—41no—
103C2×C4×D47—41no—
104C2×C4×D47—41no—
105C2×C4×D47—41no—
106C2×C8×C47—21no—
107C2×C8×C47—21no—
108C2×C8×C47—21no—
109C2×C8×C47—21no—
110C2×C8×C47—21no—
111C2×C8×C47D812yescomplemented
112C4⋊C4×C47—41no—
113C4⋊C4×C47—41no—
114C4⋊C4×C47—41no—
115C4⋊C4×C47—41no—
116C4⋊C4×C47—81no—
117C42×C47C22×C411yescharacteristic
118C42×C47C2×D411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47#4111yescharacteristic
121#10625334—48no—
122(C2×D47)⋊C4—22no—
123(C2×D47)⋊C4—22no—
124C4⋊Dic47—21no—
125C4⋊Dic47C22⋊C412yes—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—22no—
128C2×C47⋊C8—41no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8—41no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135#10625324—48no—
136C23×C47—42no—
137C23×C47—81no—
138C23×C47—81no—
139C23×C47—81no—
140C23×C47—81no—
141C23×C47—81no—
142C23×C47—81no—
143C23×C47#12111yescharacteristic
144C22×D47—22no—
145C22×D47—41no—
146C22×D47—41no—
147C22×D47—41no—
148C22×D47—41no—
149C22×D47—41no—
150C22×D47—41no—
151D4×C47—42no—
152D4×C47—42no—
153D4×C47—42no—
154D4×C47—42no—
155D4×C47—82no—
156D4×C47—82no—
157D4×C47—81no—
158D4×C47—81no—
159D4×C47—81no—
160D4×C47—81no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—81no—
164C2×C4×C47—81no—
165C2×C4×C47—81no—
166C2×C4×C47C2×C22⋊C411yescharacteristic
167C2×C4×C47C4×D411yescharacteristic
168C2×C4×C47C4×D411yescharacteristic
169C2×C4×C47#12011yescharacteristic
170C2×C4×C47#12111yescharacteristic
171C2×C4×C47#12311yescharacteristic
172C2×C4×C47C2×D811yescharacteristic
173C2×C4×C47#13511yescharacteristic
174C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C47#13711yescharacteristic
180C2×Dic47—22no—
181C2×Dic47—41no—
182C2×Dic47—41no—
183D188—22no—
184D188—22no—
185D188—41no—
186D188—41no—
187C4×D47—42no—
188C4×D47—42no—
189C4×D47—42no—
190C4×D47—42no—
191C8×C47—22no—
192C8×C47—41no—
193C47⋊C8—42no—
194C47⋊C8—42no—
195C47⋊C8—81no—
196#9229—471nomaximal, Sylow
197C22×C47—82no—
198C22×C47—82no—
199C22×C47—82no—
200C22×C47—161no—
201C22×C47—161no—
202C22×C47—161no—
203C22×C47—161no—
204C22×C47#38611yescharacteristic
205C22×C47#45911yescharacteristic
206C22×C47#46311yescharacteristic
207C22×C47#46811yescharacteristic
208C22×C47#46911yescharacteristic
209C22×C47#47111yescharacteristic
210C22×C47#47211yescharacteristic
211C2×D47—42no—
212C2×D47—42no—
213C2×D47—42no—
214C2×D47—81no—
215C2×D47—81no—
216C2×D47—81no—
217C2×D47—81no—
218C4×C47—21no—
219C4×C47—21no—
220C4×C47—161no—
221C4×C47#41711yescharacteristic
222C4×C47#41811yescharacteristic
223C4×C47C4×D811yescharacteristic
224C4×C47#44211yescharacteristic
225C4×C47#44711yescharacteristic
226C4×C47#44911yescharacteristic
227C4×C47#46811yescharacteristic
228C4×C47#46911yescharacteristic
229Dic47—42no—
230C2×#459—471no—
231C2×#436—471no—
232#1957D4711yescharacteristic, complemented
233#1802—471no—
234C2×#331—471no—
235#1296—471no—
236#1277—471no—
237C2×C47—162no—
238C2×C47#168511yescharacteristic
239C2×C47#168711yescharacteristic
240C2×C47#169711yescharacteristic
241C2×C47#170111yescharacteristic
242C2×C47#187911yescharacteristic
243C2×C47#189511yescharacteristic
244C2×C47#190711yescharacteristic
245D47—82no—
246C22×D8—941no—
247C22×M16—941no—
248C2×#127C2×D4711yescharacteristic
249C2×C4×D4—471no—
250#459—474no—
251#459—942no—
252#436—942no—
253#436—942no—
254C2×C4⋊C8—471no—
255C2×C4⋊C8—471no—
256C2×C4⋊C8—471no—
257C2×#102—21no—
258C2×#102—21no—
259C2×#102—941no—
260C2×#102—941no—
261C2×C22⋊C8—941no—
262C2×C4×C8C2×D4711yescharacteristic
263#384C2×D4711yescharacteristic
264#340—941no—
265#331—942no—
266C47#922911yescharacteristic, complemented, Sylow
267C22×D4—21no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4—471no—
271C23×C4—941no—
272C2×D8—944no—
273C2×D8—1882no—
274C2×D8—1882no—
275C2×M16—944no—
276C2×M16—1882no—
277C2×M16—1882no—
278C22×C8—941no—
279C22×C8—941no—
280C22×C8—941no—
281C22×C8C22⋊D4711yescharacteristic
282C22×C8C22⋊D4711yescharacteristic
283#127—21no—
284#127—21no—
285#127—21no—
286#127C4×D4712yes—
287C4×D4—472no—
288C4×D4—472no—
289C4×D4—941no—
290C4×D4—941no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—471no—
295C2×C22⋊C4—941no—
296C2×C42C22×D4711yescharacteristic
297C4⋊C8—472no—
298C4⋊C8—472no—
299C4⋊C8—941no—
300C4⋊C8—941no—
301C4⋊C8—941no—
302C4⋊C8—941no—
303#102—42no—
304#102—42no—
305#102—944no—
306#102—1882no—
307C22⋊C8—944no—
308C4×C8—21no—
309C4×C8—21no—
310C24—41no—
311C24—941no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—81no—
327C2×D4—81no—
328C2×D4—472no—
329C2×D4—941no—
330C2×D4—941no—
331C2×D4—941no—
332C2×D4—941no—
333C2×D4—941no—
334C22×C4—41no—
335C22×C4—941no—
336C22×C4—942no—
337C22×C4—942no—
338C22×C4—1881no—
339C22×C4—1881no—
340C22×C4—1881no—
341C22×C4—1881no—
342C22×C4D4×D4711yescharacteristic
343C22×C4#1062535211yescharacteristic
344C22×C4C2×C22⋊D4711yescharacteristic
345D8—1888no—
346M16—1888no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—942no—
353C2×C8—942no—
354C2×C8—1881no—
355C2×C8—1881no—
356C2×C8—1881no—
357C2×C8—1881no—
358C2×C8—1881no—
359C2×C8—1881no—
360C2×C8—1881no—
361C2×C8#1062533412yescomplemented
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—81no—
367C4⋊C4—472no—
368C4⋊C4—941no—
369C22⋊C4—942no—
370C22⋊C4—942no—
371C42C2×C4×D4711yescharacteristic
372C42#1062535011yescharacteristic
373C42D4×D4711yescharacteristic
374C42D4×D4711yescharacteristic
375C23—42no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—942no—
383C23—1881no—
384C23—1881no—
385C23—1881no—
386C23—1881no—
387C23—1881no—
388C23—1881no—
389C23#1309399511yescharacteristic
390D4—42no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—82no—
395D4—82no—
396D4—81no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—942no—
401D4—942no—
402D4—1881no—
403D4—1881no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—942no—
410C2×C4—1881no—
411C2×C4—1881no—
412C2×C4—1882no—
413C2×C4—1882no—
414C2×C4—1882no—
415C2×C4—1882no—
416C2×C4C22⋊C4×D4711yescharacteristic
417C2×C4(C22⋊D47)⋊C411yescharacteristic
418C2×C4C4×C22⋊D4711yescharacteristic
419C2×C4#1309399211yescharacteristic
420C2×C4#1309399711yescharacteristic
421C2×C4#1309400611yescharacteristic
422C2×C4C2×#1062533411yescharacteristic
423C2×C4#1309404411yescharacteristic
424C2×C4#1309404811yescharacteristic
425C2×C4#1309404811yescharacteristic
426C2×C4#1309404911yescharacteristic
427C2×C4#1309405011yescharacteristic
428C2×C4#1309405111yescharacteristic
429C2×C4#1309405311yescharacteristic
430C8—22no—
431C8—41no—
432C8—3761no—
433C8—1882no—
434C8—1882no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—3761no—
443C22—3761no—
444C22—3761no—
445C22—3761no—
446C22—1882no—
447C22—1882no—
448C22—1882no—
449C22#1568899911yescharacteristic
450C22#1568931511yescharacteristic
451C22#1568933511yescharacteristic
452C22#1568936611yescharacteristic
453C22#1568937311yescharacteristic
454C22#1568938311yescharacteristic
455C22#1568939011yescharacteristic
456C4—21no—
457C4—21no—
458C4—161no—
459C4—1882no—
460C4#1568912111yescharacteristic
461C4#1568912611yescharacteristic
462C4C4×#1062533411yescharacteristic
463C4#1568923311yescharacteristic
464C4#1568925611yescharacteristic
465C4#1568926611yescharacteristic
466C4#1568936811yescharacteristic
467C4#1568937511yescharacteristic
468C2—162no—
469C2—3762no—
470C2#2024285011yescharacteristic
471C2#2024286411yescharacteristic
472C2#2024291811yescharacteristic
473C2#2024294611yescharacteristic
474C2#2024373911yescharacteristic
475C2#2024382311yescharacteristic
476C2#2024389911yescharacteristic
subgroupG/H
centerC23#13093995
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1957C2
socleC23×C47#121
prime radicalC2×#15689315C2
cosocleC2×C42×C47C23
hypercenter#1957D47
hypocenterC47#9229
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. C2×#15689202C2semidirect
3. #20243385C2semidirect
4. C2×#15688867C2semidirect
5. #20241271C2semidirect
6. #20241223C2semidirect
7. C47×#1957C2semidirect
12. #15689315C4semidirect
111. C2×C8×C47D8semidirect
232. #1957D47semidirect
266. C47#9229semidirect
361. C2×C8#10625334semidirect

Maximal subgroups

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

subgroupindex
C2×#156893152
C2×#156892022
#202433852
C2×#156888672
#202412712
#202412232
C47×#19572
#922947