Finite Group Databasebeta

← #26111560

Subgroups — #26111560

Order 12032.

618 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689471C211yesmaximal, characteristic
2#20244315C211yesmaximal, characteristic
3#20244315C211yesmaximal, characteristic
4C2×#15688869C211yesmaximal, characteristic, complemented
5#20241229C211yesmaximal, characteristic, complemented
6#20241229C211yesmaximal, characteristic, complemented
7C47×#2365C211yesmaximal, characteristic, complemented
8C2×#13094021C2211yescharacteristic
9#15689471—22no—
10#15689471C414yescomplemented
11C2×C4⋊(C47⋊C8)C2211yescharacteristic
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×#13093944—22no—
14C2×#13093944—21no—
15C2×#13093944—21no—
16C2×#13093944—21no—
17C2×#13093944—21no—
18C2×C4×C47⋊C8C2211yescharacteristic
19#15688869—22no—
20C2×C4⋊Q8×C47C2211yescharacteristic
21C2×C4×Q8×C47—21no—
22C47×#391—21no—
23C47×#389—21no—
24C47×#387—22no—
25C4×C4⋊C4×C47—21no—
26C4×C4⋊C4×C47C2211yescharacteristic
27C4×C4⋊C4×C47C2211yescharacteristic
28C22×D188—21no—
29C22×D188—21no—
30C22×D188—21no—
31C22×C47⋊C8—21no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8—21no—
34#13094021—21no—
35#13094021—21no—
36#13094021—21no—
37#13094021C2×C412yes—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42#13093944—28no—
43#13093944—42no—
44#13093944—42no—
45#13093944—42no—
46#13093944—42no—
47C4×C47⋊C8—21no—
48C4×C47⋊C8C2×C412yes—
49C22×Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47C2×C412yes—
53C4⋊Q8×C47D412yes—
54C4×Q8×C47—42no—
55C4×Q8×C47—42no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—22no—
58C2×C4⋊C4×C47D411yescharacteristic
59C2×C4⋊C4×C47D411yescharacteristic
60C2×C4⋊C4×C47D412yes—
61C2×C4⋊C4×C47D411yescharacteristic
62C2×C4⋊C4×C47D411yescharacteristic
63C2×C42×C47—22no—
64C2×C42×C47—21no—
65C2×C42×C47—21no—
66C2×C42×C47C2311yescharacteristic
67#95×C47—22no—
68#95×C47—22no—
69#95×C47—21no—
70#95×C47—21no—
71C23×D47—41no—
72C2×Q8×C47—22no—
73C2×Q8×C47—22no—
74C2×Q8×C47—41no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C2×Q8×C47—41no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—22no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
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—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—44no—
112C4⋊C4×C47C22⋊C414yes—
113C4⋊C4×C47D812yes—
114C4⋊C4×C47SD1612yes—
115C42×C47—44no—
116C42×C47—41no—
117C42×C47—41no—
118C42×C47—41no—
119C42×C47—41no—
120C42×C47C22×C411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47#4111yescharacteristic
123C42×C47#4111yescharacteristic
124C2×C47⋊C8—22no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127C2×C47⋊C8—41no—
128C2×C47⋊C8—41no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8—41no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C23×C47#12011yescharacteristic
134C22×D47—42no—
135C22×D47—81no—
136C22×D47—81no—
137C22×D47—81no—
138C22×D47—81no—
139C22×D47—81no—
140C22×D47—81no—
141Q8×C47—42no—
142Q8×C47—42no—
143Q8×C47—42no—
144Q8×C47—42no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—24no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—41no—
151C2×C4×C47—42no—
152C2×C4×C47—41no—
153C2×C4×C47—42no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47C2×C22⋊C411yescharacteristic
163C2×C4×C47C4×D411yescharacteristic
164C2×C4×C47C4×D411yescharacteristic
165C2×C4×C47#12111yescharacteristic
166C2×C4×C47#12111yescharacteristic
167C2×C4×C47#12311yescharacteristic
168C2×C4×C47C2×D811yescharacteristic
169C2×C4×C47C2×SD1611yescharacteristic
170C2×C4×C47#13511yescharacteristic
171C2×C4×C47#13511yescharacteristic
172C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C47#13711yescharacteristic
176D188—44no—
177D188—42no—
178D188—42no—
179D188—42no—
180D188—42no—
181D188—82no—
182D188—81no—
183D188—81no—
184D188—81no—
185D188—81no—
186C47⋊C8—42no—
187C47⋊C8—81no—
188C47⋊C8—81no—
189#9062—471nomaximal, Sylow
190C22×C47#38611yescharacteristic
191C22×C47#44711yescharacteristic
192C22×C47#44811yescharacteristic
193C22×C47#44911yescharacteristic
194C22×C47#44911yescharacteristic
195C22×C47#45411yescharacteristic
196C22×C47#45411yescharacteristic
197C2×D47—82no—
198C2×D47—82no—
199C2×D47—82no—
200C2×D47—161no—
201C2×D47—161no—
202C2×D47—161no—
203C2×D47—161no—
204C4×C47—21no—
205C4×C47—21no—
206C4×C47—44no—
207C4×C47—42no—
208C4×C47—42no—
209C4×C47—81no—
210C4×C47—81no—
211C4×C47C4×D811yescharacteristic
212C4×C47C4×SD1611yescharacteristic
213C4×C47#44011yescharacteristic
214C4×C47#44211yescharacteristic
215C4×C47#45911yescharacteristic
216C4×C47#46011yescharacteristic
217C4×C47#46311yescharacteristic
218C4×C47#46411yescharacteristic
219C2×#486—471no—
220#2365D4711yescharacteristic, complemented
221#1982—471no—
222#1982—471no—
223C2×#332—471no—
224#1279—471no—
225#1279—471no—
226C2×C47#162511yescharacteristic
227C2×C47#162611yescharacteristic
228C2×C47#162711yescharacteristic
229C2×C47#162811yescharacteristic
230C2×C47#187911yescharacteristic
231C2×C47#188011yescharacteristic
232C2×C47#188811yescharacteristic
233D47—162no—
234C2×C4⋊Q8C2×D4711yescharacteristic
235C2×#127—471no—
236C2×C4×Q8—21no—
237#486—474no—
238#486—942no—
239C2×C4⋊C8—471no—
240C2×C4⋊C8—471no—
241C2×#102—942no—
242C2×#102—941no—
243C2×#102—941no—
244C2×#102—941no—
245C2×#102—941no—
246C2×C4×C8—471no—
247#391—21no—
248#389—21no—
249#387—22no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4C2×D4711yescharacteristic
252C4×C4⋊C4C2×D4711yescharacteristic
253#332—942no—
254C47#906211yescharacteristic, complemented, Sylow
255C22×Q8—21no—
256C22×D4—941no—
257C22×D4—941no—
258C22×D4—941no—
259C22×C8—941no—
260C22×C8—941no—
261C22×C8—941no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8C4×D4712yes—
265C4⋊Q8D18812yes—
266#127—472no—
267#127—941no—
268#127—941no—
269#127—941no—
270C4×Q8—42no—
271C4×Q8—42no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4C22⋊D4711yescharacteristic
275C2×C4⋊C4C22⋊D4711yescharacteristic
276C2×C4⋊C4C22⋊D4712yes—
277C2×C4⋊C4C22⋊D4711yescharacteristic
278C2×C4⋊C4C22⋊D4711yescharacteristic
279C2×C42—22no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42C22×D4711yescharacteristic
283C4⋊C8—941no—
284C4⋊C8—941no—
285C4⋊C8—941no—
286C4⋊C8—941no—
287#102—948no—
288#102—1882no—
289#102—1882no—
290#102—1882no—
291#102—1882no—
292C4×C8—472no—
293C4×C8—941no—
294#95—22no—
295#95—22no—
296#95—21no—
297#95—21no—
298C24—1881no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×D4—942no—
306C2×D4—942no—
307C2×D4—942no—
308C2×D4—1882no—
309C2×D4—1882no—
310C2×D4—1881no—
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—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4C2×C22⋊D4711yescharacteristic
327C22×C4C2×C22⋊D4711yescharacteristic
328C22×C4C2×C22⋊D4711yescharacteristic
329C2×C8—942no—
330C2×C8—1881no—
331C2×C8—1881no—
332C2×C8—1881no—
333C2×C8—1881no—
334C2×C8—1881no—
335C2×C8—1881no—
336C2×C8—1881no—
337C2×C8—1881no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—44no—
348C4⋊C4(C2×D47)⋊C414yes—
349C4⋊C4#1062533412yes—
350C4⋊C4#1062533612yes—
351C42—44no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42C2×C4×D4711yescharacteristic
357C42C2×D18811yescharacteristic
358C42#1062535011yescharacteristic
359C42#1062535011yescharacteristic
360C23—3761no—
361C23—3761no—
362C23—3761no—
363C23—3761no—
364C23—3761no—
365C23—3761no—
366C23—1882no—
367C23#1309399311yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—42no—
371Q8—42no—
372D4—3762no—
373D4—3761no—
374D4—3761no—
375D4—3761no—
376D4—3761no—
377D4—1884no—
378D4—1882no—
379D4—1882no—
380D4—1882no—
381D4—1882no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4C2×(C2×D47)⋊C411yescharacteristic
400C2×C4C4×C22⋊D4711yescharacteristic
401C2×C4C4×C22⋊D4711yescharacteristic
402C2×C4#1309399811yescharacteristic
403C2×C4#1309399811yescharacteristic
404C2×C4#1309400711yescharacteristic
405C2×C4C2×#1062533411yescharacteristic
406C2×C4C2×#1062533611yescharacteristic
407C2×C4#1309404411yescharacteristic
408C2×C4#1309404411yescharacteristic
409C2×C4#1309404911yescharacteristic
410C2×C4#1309404911yescharacteristic
411C2×C4#1309405011yescharacteristic
412C2×C4#1309405511yescharacteristic
413C8—3761no—
414C8—3761no—
415C8—1882no—
416C22—3762no—
417C22—3762no—
418C22—3762no—
419C22—7521no—
420C22—7521no—
421C22—7521no—
422C22—7521no—
423C22#1568900211yescharacteristic
424C22#1568925711yescharacteristic
425C22#1568926211yescharacteristic
426C22#1568926811yescharacteristic
427C22#1568926811yescharacteristic
428C22#1568929611yescharacteristic
429C22#1568929611yescharacteristic
430C4—21no—
431C4—21no—
432C4—44no—
433C4—42no—
434C4—42no—
435C4—81no—
436C4—81no—
437C4C4×#1062533411yescharacteristic
438C4C4×#1062533611yescharacteristic
439C4#1568922111yescharacteristic
440C4#1568923311yescharacteristic
441C4#1568931811yescharacteristic
442C4#1568932311yescharacteristic
443C4#1568933811yescharacteristic
444C4#1568934511yescharacteristic
445C2—7522no—
446C2#2024247711yescharacteristic
447C2#2024248411yescharacteristic
448C2#2024249111yescharacteristic
449C2#2024249811yescharacteristic
450C2#2024373611yescharacteristic
451C2#2024374311yescharacteristic
452C2#2024379111yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2365C2
socleC23×C47#120
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#2365D47
hypocenterC47#9062
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2365

Complements

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

NH ≅ G/Ntype
4. C2×#15688869C2semidirect
5. #20241229C2semidirect
6. #20241229C2semidirect
7. C47×#2365C2semidirect
10. #15689471C4semidirect
220. #2365D47semidirect
254. C47#9062semidirect

Maximal subgroups

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

subgroupindex
C2×#156894712
#202443152
C2×#156888692
#202412292
C47×#23652
#906247