Finite Group Databasebeta

← #26112472

Subgroups — #26112472

Order 12032.

546 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689414C211yesmaximal, characteristic
2C2×#15689202C211yesmaximal, characteristic
3#20244349C211yesmaximal, characteristic, complemented
4#20243389C211yesmaximal, characteristic
5C2×#15688871C211yesmaximal, characteristic, complemented
6#20241272C211yesmaximal, characteristic, complemented
7C47×#1277C211yesmaximal, characteristic, complemented
8C22×#10625324—21no—
9C2×#13094024C2211yescharacteristic
10C2×C4×D188C2211yescharacteristic
11#15689414—22no—
12#15689414C414yescomplemented
13#15689202—22no—
14#15689202—22no—
15C2×#13093956—21no—
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×#13093946—21no—
18C2×#13093946—21no—
19C2×(C2×D47)⋊C8—21no—
20#15688993C2211yescharacteristic
21#15688886—21no—
22#15688886—21no—
23#15688886—21no—
24#15688871—22no—
25C2×C4⋊C8×C47C2211yescharacteristic
26C2×C4⋊C8×C47C2211yescharacteristic
27C2×C4×C8×C47C2211yescharacteristic
28C22×Dic94—21no—
29C22×D188D411yescharacteristic
30C22×C4×D47—21no—
31C2×#10625324—24no—
32C2×#10625324—42no—
33C2×#10625324—42no—
34C22×C47⋊C8—21no—
35#13094024—21no—
36#13094024—21no—
37#13094024—21no—
38#13094024C2×C412yes—
39C4×D188—21no—
40C4×D188—21no—
41C4×D188C2×C412yes—
42C4×D188D412yes—
43C2×C4⋊Dic47—21no—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47D411yescharacteristic
49C2×(C2×D47)⋊C4—21no—
50#13093956—24no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)D412yes—
53#13093946—24no—
54#13093946—42no—
55(C2×D47)⋊C8—24no—
56C22×C8×C47—21no—
57C22×C8×C47—21no—
58C22×C8×C47D411yescharacteristic
59C22×C8×C47D411yescharacteristic
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C4⋊C8×C47C2×C412yes—
65C4×C8×C47—21no—
66C4×C8×C47—21no—
67C23×D47—21no—
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47#4111yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—41no—
73C22×Dic47—41no—
74C2×D188—21no—
75C2×D188—21no—
76C2×D188—21no—
77C2×D188—21no—
78C2×D188—21no—
79C2×D188C22⋊C412yes—
80C2×C4×D47—22no—
81C2×C4×D47—22no—
82C2×C4×D47—41no—
83C2×C4×D47—41no—
84C2×C4×D47—41no—
85C2×C4×D47—41no—
86C2×Dic94—22no—
87C2×Dic94—41no—
88C2×Dic94—81no—
89C2×Dic94—81no—
90C2×C8×C47—21no—
91C2×C8×C47—21no—
92C2×C8×C47—22no—
93C2×C8×C47—21no—
94C2×C8×C47—21no—
95C2×C8×C47—21no—
96C2×C8×C47—41no—
97C2×C8×C47—41no—
98C2×C8×C47—41no—
99C2×C8×C47—41no—
100C2×C8×C47—41no—
101C2×C8×C47SD1612yes—
102C42×C47C22×C411yescharacteristic
103C42×C47C2×D411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C42×C47#4111yescharacteristic
106(C2×D47)⋊C4—22no—
107(C2×D47)⋊C4—22no—
108C4⋊Dic47—21no—
109C4⋊Dic47—22no—
110C4⋊Dic47—22no—
111C4⋊Dic47—41no—
112C4⋊Dic47—41no—
113C4⋊Dic47—41no—
114C4⋊Dic47—41no—
115C4⋊Dic47—41no—
116C4⋊Dic47—41no—
117C4⋊Dic47—81no—
118C4⋊Dic47C22⋊C412yes—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—41no—
121C2×C47⋊C8—41no—
122#10625324—48no—
123C23×C47#12111yescharacteristic
124C22×D47—22no—
125C22×D47—41no—
126C22×D47—41no—
127C22×D47—41no—
128C22×D47—41no—
129C22×D47—41no—
130C22×D47—41no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47C2×C22⋊C411yescharacteristic
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47C4×D411yescharacteristic
136C2×C4×C47#12011yescharacteristic
137C2×C4×C47#12111yescharacteristic
138C2×C4×C47#12311yescharacteristic
139C2×C4×C47C2×SD1611yescharacteristic
140C2×C4×C47#13511yescharacteristic
141C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C47#13711yescharacteristic
144C2×C4×C47#13711yescharacteristic
145C2×C4×C47#13711yescharacteristic
146C2×C4×C47#13711yescharacteristic
147C2×Dic47—22no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—42no—
151C2×Dic47—81no—
152C2×Dic47—81no—
153C2×Dic47—81no—
154C2×Dic47—81no—
155C2×Dic47—81no—
156D188—22no—
157D188—22no—
158D188—41no—
159D188—41no—
160C4×D47—42no—
161C4×D47—42no—
162C4×D47—42no—
163C4×D47—42no—
164Dic94—82no—
165Dic94—82no—
166C8×C47—22no—
167C8×C47—42no—
168C8×C47—41no—
169C8×C47—81no—
170C47⋊C8—42no—
171#9223—471nomaximal, Sylow
172C22×C47#38611yescharacteristic
173C22×C47#46111yescharacteristic
174C22×C47#46311yescharacteristic
175C22×C47#46811yescharacteristic
176C22×C47#47011yescharacteristic
177C22×C47#47111yescharacteristic
178C22×C47#47211yescharacteristic
179C2×D47—42no—
180C2×D47—42no—
181C2×D47—42no—
182C2×D47—81no—
183C2×D47—81no—
184C2×D47—81no—
185C2×D47—81no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47#41711yescharacteristic
189C4×C47#41911yescharacteristic
190C4×C47C4×SD1611yescharacteristic
191C4×C47#44011yescharacteristic
192C4×C47#45011yescharacteristic
193C4×C47#45211yescharacteristic
194C4×C47#46811yescharacteristic
195C4×C47#47011yescharacteristic
196Dic47—42no—
197Dic47—82no—
198Dic47—161no—
199C2×#476—471no—
200C2×#436—471no—
201#1990—471no—
202#1802—471no—
203C2×#333—471no—
204#1296—471no—
205#1277D4711yescharacteristic, complemented
206C2×C47#169111yescharacteristic
207C2×C47#169511yescharacteristic
208C2×C47#169711yescharacteristic
209C2×C47#170311yescharacteristic
210C2×C47#187711yescharacteristic
211C2×C47#189411yescharacteristic
212C2×C47#190711yescharacteristic
213D47—82no—
214C22×M16—941no—
215C2×C4⋊Q8—471no—
216C2×C4×D4—471no—
217#476—474no—
218#476—942no—
219#436—942no—
220#436—942no—
221C2×#106—941no—
222C2×C4⋊C8—471no—
223C2×C4⋊C8C2×D4711yescharacteristic
224C2×C4⋊C8C2×D4711yescharacteristic
225C2×#102—941no—
226C2×#102—941no—
227C2×C22⋊C8—941no—
228C2×C4×C8C2×D4711yescharacteristic
229#384—471no—
230#340—941no—
231#340—941no—
232#340—941no—
233#333—942no—
234C47#922311yescharacteristic, complemented, Sylow
235C22×Q8—941no—
236C22×D4—471no—
237C23×C4—941no—
238C2×M16—944no—
239C2×M16—1882no—
240C2×M16—1882no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—941no—
244C22×C8D18811yescharacteristic
245C22×C8D18811yescharacteristic
246C4⋊Q8—472no—
247C4⋊Q8—941no—
248C4⋊Q8—941no—
249C4⋊Q8—941no—
250C4×D4—472no—
251C4×D4—472no—
252C4×D4—941no—
253C4×D4—941no—
254C2×C4⋊C4—471no—
255C2×C4⋊C4—941no—
256C2×C4⋊C4—941no—
257C2×C4⋊C4—941no—
258C2×C4⋊C4—941no—
259C2×C4⋊C4—941no—
260C2×C22⋊C4—941no—
261C2×C42C22×D4711yescharacteristic
262#106—944no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—472no—
267C4⋊C8—941no—
268C4⋊C8C4×D4712yes—
269#102—944no—
270#102—1882no—
271C22⋊C8—944no—
272C4×C8—21no—
273C4×C8—21no—
274C24—941no—
275C2×Q8—942no—
276C2×Q8—3761no—
277C2×Q8—3761no—
278C2×Q8—1881no—
279C2×D4—472no—
280C2×D4—941no—
281C2×D4—941no—
282C2×D4—941no—
283C2×D4—941no—
284C2×D4—941no—
285C22×C4—941no—
286C22×C4—942no—
287C22×C4—942no—
288C22×C4—1881no—
289C22×C4—1881no—
290C22×C4—1881no—
291C22×C4—1881no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4C2×D18811yescharacteristic
295C22×C4D4×D4711yescharacteristic
296C22×C4#1062535411yescharacteristic
297M16—1888no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—942no—
310C2×C8—1881no—
311C2×C8—1881no—
312C2×C8#1062532512yes—
313C4⋊C4—472no—
314C4⋊C4—941no—
315C4⋊C4—942no—
316C4⋊C4—942no—
317C4⋊C4—3761no—
318C4⋊C4—1881no—
319C4⋊C4—1881no—
320C4⋊C4—1881no—
321C4⋊C4—1881no—
322C4⋊C4—1881no—
323C4⋊C4—1881no—
324C22⋊C4—942no—
325C22⋊C4—942no—
326C42C2×C4×D4711yescharacteristic
327C42#1062535011yescharacteristic
328C42D4×D4711yescharacteristic
329C42D4×D4711yescharacteristic
330C23—942no—
331C23—1881no—
332C23—1881no—
333C23—1881no—
334C23—1881no—
335C23—1881no—
336C23—1881no—
337C23#1309399911yescharacteristic
338Q8—3762no—
339Q8—3762no—
340D4—942no—
341D4—942no—
342D4—1881no—
343D4—1881no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—942no—
347C2×C4—3761no—
348C2×C4—3761no—
349C2×C4—3761no—
350C2×C4—3761no—
351C2×C4—3761no—
352C2×C4—1882no—
353C2×C4—1881no—
354C2×C4—1881no—
355C2×C4—1882no—
356C2×C4—1882no—
357C2×C4—1882no—
358C2×C4—1882no—
359C2×C4C22⋊C4×D4711yescharacteristic
360C2×C4#1309398611yescharacteristic
361C2×C4C4×D18811yescharacteristic
362C2×C4#1309399411yescharacteristic
363C2×C4#1309399711yescharacteristic
364C2×C4#1309400911yescharacteristic
365C2×C4C2×#1062532511yescharacteristic
366C2×C4#1309404611yescharacteristic
367C2×C4#1309404811yescharacteristic
368C2×C4#1309405211yescharacteristic
369C2×C4#1309405311yescharacteristic
370C2×C4#1309405311yescharacteristic
371C2×C4#1309405611yescharacteristic
372C2×C4#1309405711yescharacteristic
373C8—22no—
374C8—41no—
375C8—42no—
376C8—81no—
377C8—1882no—
378C22—3761no—
379C22—3761no—
380C22—3761no—
381C22—3761no—
382C22—1882no—
383C22—1882no—
384C22—1882no—
385C22#1568900111yescharacteristic
386C22#1568932911yescharacteristic
387C22#1568933911yescharacteristic
388C22#1568937011yescharacteristic
389C22#1568938211yescharacteristic
390C22#1568938711yescharacteristic
391C22#1568939411yescharacteristic
392C4—21no—
393C4—21no—
394C4—3762no—
395C4—1882no—
396C4—7521no—
397C4#1568912111yescharacteristic
398C4#1568913011yescharacteristic
399C4C4×#1062532511yescharacteristic
400C4#1568922511yescharacteristic
401C4#1568927611yescharacteristic
402C4#1568928711yescharacteristic
403C4#1568936811yescharacteristic
404C4#1568938011yescharacteristic
405C2—3762no—
406C2#2024288811yescharacteristic
407C2#2024291211yescharacteristic
408C2#2024292211yescharacteristic
409C2#2024296411yescharacteristic
410C2#2024372111yescharacteristic
411C2#2024382011yescharacteristic
412C2#2024389511yescharacteristic
subgroupG/H
centerC23#13093999
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1277C2
socleC23×C47#121
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1277D47
hypocenterC47#9223
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
3. #20244349C2semidirect
5. C2×#15688871C2semidirect
6. #20241272C2semidirect
7. C47×#1277C2semidirect
12. #15689414C4semidirect
205. #1277D47semidirect
234. C47#9223semidirect

Maximal subgroups

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

subgroupindex
C2×#156894142
C2×#156892022
#202443492
#202433892
C2×#156888712
#202412722
C47×#12772
#922347