Finite Group Databasebeta

← #26117355

Subgroups — #26117355

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689497C211yesmaximal, characteristic
2C2×#15689323C211yesmaximal, characteristic
3#20244315C211yesmaximal, characteristic
4C2×#15688872C211yesmaximal, characteristic, complemented
5C2×Q8⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
6#20241229C211yesmaximal, characteristic, complemented
7C47×#2567C211yesmaximal, characteristic, complemented
8C22×#10625336—22no—
9C22×#10625336—21no—
10C2×#13094021C2211yescharacteristic
11#15689497—22no—
12#15689497—22no—
13#15689323—22no—
14#15689323—22no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×#13093944—21no—
18C2×#13093944—22no—
19C2×C4×C47⋊C8C2211yescharacteristic
20#15688872—22no—
21Q8⋊(C47⋊C8)—22no—
22C2×C4⋊Q8×C47C2211yescharacteristic
23C2×C4×Q8×C47C2211yescharacteristic
24C47×#395—22no—
25C47×#391—21no—
26C47×#389—22no—
27C47×#384—21no—
28C4×C4⋊C4×C47C2211yescharacteristic
29C22×D188—21no—
30C22×D188—21no—
31C22×D188—21no—
32C2×#10625336—28no—
33C2×#10625336—24no—
34C2×#10625336—48no—
35C2×#10625336—42no—
36C2×#10625336—42no—
37C22×C47⋊C8—21no—
38C22×C47⋊C8—21no—
39C22×C47⋊C8—21no—
40#13094021—21no—
41#13094021—21no—
42#13094021—21no—
43#13094021D412yes—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)D412yes—
48#13093944—24no—
49#13093944—44no—
50C4×C47⋊C8—21no—
51C4×C47⋊C8D412yes—
52C22×Q8×C47D411yescharacteristic
53C22×Q8×C47D412yes—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47—21no—
56C4⋊Q8×C47—21no—
57C4⋊Q8×C47—21no—
58C4×Q8×C47—21no—
59C4×Q8×C47—21no—
60C4×Q8×C47—21no—
61C4×Q8×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—24no—
66C2×C4⋊C4×C47D411yescharacteristic
67C2×C4⋊C4×C47D412yes—
68C2×C42×C47—21no—
69C2×C42×C47—21no—
70C2×C42×C47C2311yescharacteristic
71#95×C47—24no—
72#95×C47—21no—
73C23×D47—41no—
74C2×Q8×C47—21no—
75C2×Q8×C47—21no—
76C2×Q8×C47—22no—
77C2×Q8×C47—21no—
78C2×Q8×C47—22no—
79C2×Q8×C47—21no—
80C2×Q8×C47—21no—
81C2×Q8×C47—44no—
82C2×Q8×C47SD1612yescomplemented
83C22×C4×C47—22no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—22no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C2×D188—22no—
91C2×D188—22no—
92C2×D188—22no—
93C2×D188—44no—
94C2×D188—42no—
95C2×D188—41no—
96C2×D188—41no—
97C2×D188—41no—
98C2×D188—41no—
99C2×D188—42no—
100C2×D188—41no—
101C2×D188—41no—
102C2×D188—41no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—41no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—84no—
113C4⋊C4×C47SD1614yescomplemented
114C42×C47—41no—
115C42×C47—41no—
116C42×C47—41no—
117C42×C47—41no—
118C42×C47C2×D411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C42×C47#4111yescharacteristic
122#10625336—416no—
123#10625336—48no—
124C2×C47⋊C8—24no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—41no—
128C2×C47⋊C8—41no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8—42no—
131C23×C47#12011yescharacteristic
132C22×D47—42no—
133C22×D47—81no—
134C22×D47—82no—
135C22×D47—81no—
136C22×D47—82no—
137Q8×C47—22no—
138Q8×C47—22no—
139Q8×C47—44no—
140Q8×C47—41no—
141Q8×C47—41no—
142Q8×C47—82no—
143C2×C4×C47—22no—
144C2×C4×C47—24no—
145C2×C4×C47—22no—
146C2×C4×C47—41no—
147C2×C4×C47—42no—
148C2×C4×C47—41no—
149C2×C4×C47—42no—
150C2×C4×C47—41no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—44no—
154C2×C4×C47#12011yescharacteristic
155C2×C4×C47#12011yescharacteristic
156C2×C4×C47#12111yescharacteristic
157C2×C4×C47#12111yescharacteristic
158C2×C4×C47#12111yescharacteristic
159C2×C4×C47#12711yescharacteristic
160C2×C4×C47C2×SD1611yescharacteristic
161C2×C4×C47C2×SD1612yes—
162C2×C4×C47#13511yescharacteristic
163C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C47#13711yescharacteristic
165C2×C4×C47#13712yes—
166D188—44no—
167D188—42no—
168D188—42no—
169D188—44no—
170D188—82no—
171D188—81no—
172D188—81no—
173D188—81no—
174D188—81no—
175C47⋊C8—24no—
176C47⋊C8—42no—
177C47⋊C8—81no—
178#10049—471nomaximal, Sylow
179C22×C47#39211yescharacteristic
180C22×C47#44812yes—
181C22×C47#44811yescharacteristic
182C22×C47#44911yescharacteristic
183C22×C47#45312yes—
184C2×D47—82no—
185C2×D47—82no—
186C2×D47—82no—
187C2×D47—162no—
188C2×D47—162no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—44no—
192C4×C47—42no—
193C4×C47—81no—
194C4×C47—82no—
195C4×C47#45011yescharacteristic
196C4×C47#45211yescharacteristic
197C4×C47#46112yes—
198C4×C47#46511yescharacteristic
199C4×C47#46811yescharacteristic
200C4×C47#49211yescharacteristic
201C4×C47#49711yescharacteristic
202C2×#492—471no—
203C2×#460—471no—
204#2567D4711yescharacteristic, complemented
205#1982—471no—
206C2×#333—471no—
207C2×Q8⋊C8—471no—
208#1279—471no—
209C2×C47#162212yes—
210C2×C47#162312yes—
211C2×C47#200011yescharacteristic
212C2×C47#200211yescharacteristic
213C2×C47#201311yescharacteristic
214D47—162no—
215C22×SD16—942no—
216C22×SD16—941no—
217C2×C4⋊Q8C2×D4711yescharacteristic
218C2×#127—471no—
219C2×C4×Q8C2×D4711yescharacteristic
220#492—942no—
221#492—942no—
222#460—942no—
223#460—942no—
224C2×C4⋊C8—471no—
225C2×C4⋊C8—471no—
226C2×#102—941no—
227C2×#102—942no—
228C2×C4×C8—471no—
229#395—22no—
230#391—21no—
231#389—22no—
232#384—21no—
233C4×C4⋊C4C2×D4711yescharacteristic
234#333—942no—
235Q8⋊C8—942no—
236C47#1004911yescharacteristic, complemented, Sylow
237C22×Q8C22⋊D4711yescharacteristic
238C22×Q8C22⋊D4712yes—
239C22×D4—941no—
240C22×D4—941no—
241C22×D4—941no—
242C2×SD16—944no—
243C2×SD16—948no—
244C2×SD16—1888no—
245C2×SD16—1882no—
246C2×SD16—1882no—
247C22×C8—941no—
248C22×C8—941no—
249C22×C8—941no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254#127—472no—
255#127—941no—
256#127—941no—
257#127—941no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4C22⋊D4711yescharacteristic
267C2×C4⋊C4C22⋊D4712yes—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42C22×D4711yescharacteristic
271C4⋊C8—472no—
272C4⋊C8—941no—
273C4⋊C8—941no—
274C4⋊C8—941no—
275#102—944no—
276#102—1884no—
277C4×C8—472no—
278C4×C8—941no—
279#95—24no—
280#95—21no—
281C24—1881no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—21no—
286C2×Q8—22no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—44no—
290C2×Q8#1062533612yescomplemented
291C2×D4—942no—
292C2×D4—942no—
293C2×D4—942no—
294C2×D4—1884no—
295C2×D4—1881no—
296C2×D4—1881no—
297C2×D4—1881no—
298C2×D4—1882no—
299C2×D4—1881no—
300C2×D4—1881no—
301C2×D4—1881no—
302C2×D4—1882no—
303C2×D4—1881no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4C2×C22⋊D4711yescharacteristic
309C22×C4C2×C22⋊D4711yescharacteristic
310C22×C4C2×C22⋊D4711yescharacteristic
311SD16—18816no—
312SD16—1888no—
313C2×C8—944no—
314C2×C8—942no—
315C2×C8—942no—
316C2×C8—1881no—
317C2×C8—1882no—
318C2×C8—1881no—
319C2×C8—1881no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—84no—
330C4⋊C4#1062533614yescomplemented
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42D4×D4711yescharacteristic
336C42D4×D4711yescharacteristic
337C42D4×D4711yescharacteristic
338C42#1062535211yescharacteristic
339C23—3761no—
340C23—3762no—
341C23—3761no—
342C23—3762no—
343C23—1882no—
344C23#1309399311yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—44no—
348Q8—41no—
349Q8—41no—
350Q8—82no—
351D4—3762no—
352D4—3761no—
353D4—3761no—
354D4—3761no—
355D4—3761no—
356D4—1884no—
357D4—1882no—
358D4—1882no—
359D4—1884no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4#1309399211yescharacteristic
372C2×C4#1309399211yescharacteristic
373C2×C4#1309399511yescharacteristic
374C2×C4#1309399611yescharacteristic
375C2×C4#1309399611yescharacteristic
376C2×C4#1309402011yescharacteristic
377C2×C4C2×#1062533612yes—
378C2×C4C2×#1062533611yescharacteristic
379C2×C4#1309404411yescharacteristic
380C2×C4#1309404911yescharacteristic
381C2×C4#1309405512yes—
382C2×C4#1309405511yescharacteristic
383C8—944no—
384C8—3761no—
385C8—1882no—
386C22—3762no—
387C22—3762no—
388C22—3762no—
389C22—7522no—
390C22—7522no—
391C22#1568903111yescharacteristic
392C22#1568926212yes—
393C22#1568926211yescharacteristic
394C22#1568926811yescharacteristic
395C22#1568929112yes—
396C4—21no—
397C4—21no—
398C4—44no—
399C4—42no—
400C4—82no—
401C4—81no—
402C4#1568927311yescharacteristic
403C4#1568928311yescharacteristic
404C4#1568932512yes—
405C4#1568935411yescharacteristic
406C4#1568937111yescharacteristic
407C4#1568949711yescharacteristic
408C4#1568951511yescharacteristic
409C2—7522no—
410C2#2024246012yes—
411C2#2024246612yes—
412C2#2024438911yescharacteristic
413C2#2024440111yescharacteristic
414C2#2024447111yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2567C2
socleC23×C47#120
prime radicalC2×#13094021C22
hypercenter#2567D47
hypocenterC47#10049
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
4. C2×#15688872C2semidirect
5. C2×Q8⋊(C47⋊C8)C2semidirect
6. #20241229C2semidirect
7. C47×#2567C2semidirect
82. C2×Q8×C47SD16semidirect
113. C4⋊C4×C47SD16semidirect
204. #2567D47semidirect
236. C47#10049semidirect
290. C2×Q8#10625336semidirect
330. C4⋊C4#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156894972
C2×#156893232
#202443152
C2×#156888722
C2×Q8⋊(C47⋊C8)2
#202412292
C47×#25672
#1004947