Finite Group Databasebeta

← #26234774

Subgroups — #26234774

Order 12032.

862 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689238C211yesmaximal, characteristic
2#20252261C212yesmaximal
3#20251985C211yesmaximal, characteristic, complemented
4#20244170C212yesmaximal, complemented
5#20243614C212yesmaximal, complemented
6#20243611C212yesmaximal, complemented
7#20243546C211yesmaximal, characteristic, complemented
8#20243453C211yesmaximal, characteristic, complemented
9#20243452C211yesmaximal, characteristic, complemented
10#20243269C211yesmaximal, characteristic, complemented
11C47×#2913C211yesmaximal, characteristic, complemented
12C2×#13094044C2211yescharacteristic
13C2×#13094030C2211yescharacteristic
14#15689603C2212yes—
15#15689238C418yescomplemented
16#15689223—22no—
17#15689221—22no—
18C4×#10625337—22no—
19C4×#10625334—22no—
20C2×#13093960C2211yescharacteristic
21C2×#13093959C2211yescharacteristic
22#15689166C2212yes—
23#15689162—22no—
24#15689160—21no—
25C2×#13093958C2211yescharacteristic
26C2×#13093957C2212yes—
27#107×D47—21no—
28C2×#13093951C2211yescharacteristic
29C2×#13093950C2211yescharacteristic
30#15689124C2212yes—
31#15689122C2212yes—
32#15689079C2212yes—
33C2×C8×Dic47C2211yescharacteristic
34#15688992C2212yes—
35#15688892C2212yes—
36#15688891C2212yes—
37#15688882C2212yes—
38C2×#131×C47C2211yescharacteristic
39C47×#520C2212yes—
40C47×#428—22no—
41C47×#428C2212yes—
42C47×#427—21no—
43C47×#427C2211yescharacteristic
44C2×#107×C47—21no—
45C2×#107×C47C2211yescharacteristic
46C2×#10625350C2311yescharacteristic
47#13094044C2×C414yes—
48#13094044Q814yes—
49C2×#10625337D411yescharacteristic
50C2×#10625335D411yescharacteristic
51C2×#10625336D411yescharacteristic
52C2×#10625334D411yescharacteristic
53#13094030—22no—
54#13094030C2×C414yescomplemented
55C2×#10625329C2312yes—
56C2×#10625329C2311yescharacteristic
57C2×#10625324—21no—
58C22×C47⋊C8C2311yescharacteristic
59C2×C8×D47—21no—
60C4×Dic94—22no—
61C4×D188—22no—
62(C22⋊D47)⋊C4—42no—
63#13093982—22no—
64C2×C4⋊Dic47C2312yes—
65C2×Dic47⋊C4—42no—
66C4⋊C4×D47—22no—
67C2×C4×Dic47C2311yescharacteristic
68#13093960—22no—
69#13093959C2×C414yescomplemented
70#13093958—21no—
71#13093958—21no—
72#13093957—22no—
73#13093957—22no—
74#13093957—22no—
75#13093955—22no—
76#13093951C2×C414yescomplemented
77#13093950C2×C414yescomplemented
78#13093948—22no—
79#13093947—22no—
80#13093945—22no—
81#13093944—22no—
82#13093935—22no—
83C4×C47⋊C8—22no—
84C8×Dic47C2×C414yescomplemented
85C2×#41×C47C2311yescharacteristic
86#131×C47—22no—
87#131×C47C2×C414yescomplemented
88C2×M16×C47—21no—
89C2×M16×C47C2311yescharacteristic
90C22×C8×C47—21no—
91C22×C8×C47C2311yescharacteristic
92C4×Q8×C47—22no—
93C4×D4×C47—22no—
94C4×D4×C47—42no—
95#117×C47—22no—
96#117×C47C2312yes—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47C2312yes—
99#107×C47—21no—
100#107×C47—21no—
101#107×C47—22no—
102#107×C47D411yescharacteristic
103#107×C47D411yescharacteristic
104#106×C47—22no—
105#106×C47D412yes—
106#41×C47—21no—
107#41×C47—21no—
108#41×C47C22×C412yes—
109#41×C47C2×Q812yes—
110C2×Q8×C47C2×D411yescharacteristic
111C2×D4×C47—21no—
112C2×D4×C47C2×D411yescharacteristic
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—42no—
116C22×C4×C47C2411yescharacteristic
117C2×C22⋊D47—21no—
118C22×Dic47—21no—
119C22×Dic47—22no—
120#10625350—21no—
121#10625350—21no—
122#10625350C22×C412yes—
123#10625350C2×Q812yes—
124C2×D188C2×D411yescharacteristic
125C2×C4×D47—21no—
126C2×C4×D47—42no—
127C2×Dic94C2×D411yescharacteristic
128M16×C47—21no—
129M16×C47—24no—
130M16×C47C22×C412yes—
131C2×C8×C47—21no—
132C2×C8×C47—24no—
133C2×C8×C47C22×C412yes—
134C2×C8×C47C2×D411yescharacteristic
135C2×C8×C47C2×D411yescharacteristic
136C2×C8×C47#4112yes—
137C4⋊C4×C47—22no—
138C4⋊C4×C47—22no—
139C4⋊C4×C47—42no—
140C4⋊C4×C47C2×D412yes—
141C4⋊C4×C47C2×D412yes—
142C22⋊C4×C47—22no—
143C22⋊C4×C47—42no—
144C42×C47—22no—
145C42×C47—42no—
146#10625337C4⋊C414yescomplemented
147#10625336C4⋊C414yescomplemented
148#10625335C4⋊C414yescomplemented
149#10625334C4⋊C414yescomplemented
150(C2×D47)⋊C4—42no—
151C4⋊Dic47—22no—
152C4⋊Dic47—22no—
153Dic47⋊C4—42no—
154Dic47⋊C4—84no—
155C4×Dic47—21no—
156C4×Dic47—42no—
157C4×Dic47C22×C412yes—
158#10625329—21no—
159#10625329—22no—
160#10625329—22no—
161#10625329C22×C412yes—
162C2×C47⋊C8—21no—
163C2×C47⋊C8—22no—
164C2×C47⋊C8C22×C412yes—
165C2×C47⋊C8C2×D411yescharacteristic
166C2×C47⋊C8C2×D411yescharacteristic
167C2×C47⋊C8C2×Q812yes—
168#10625324—24no—
169C8×D47—24no—
170C23×C47—21no—
171C23×C47H(5,𝔽2)11yescharacteristic
172C22×D47—21no—
173Q8×C47—21no—
174Q8×C47C2×C4⋊C412yes—
175D4×C47—22no—
176D4×C47—22no—
177D4×C47—21no—
178D4×C47C2×C4⋊C412yes—
179C2×C4×C47—21no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—42no—
185C2×C4×C47—82no—
186C2×C4×C47—162no—
187C2×C4×C47C23×C411yescharacteristic
188C2×C4×C47C22×D411yescharacteristic
189C2×C4×C47C22×D411yescharacteristic
190C2×C4×C47C22×Q811yescharacteristic
191C2×C4×C47C2×#4111yescharacteristic
192C2×C4×C472−1+411yescharacteristic
193C22⋊D47—22no—
194C22⋊D47—22no—
195C2×Dic47—22no—
196C2×Dic47—22no—
197C2×Dic47—22no—
198C2×Dic47—21no—
199C2×Dic47—42no—
200C2×Dic47—82no—
201D188—21no—
202D188C2×C4⋊C412yes—
203C4×D47—22no—
204C4×D47—22no—
205C4×D47—162no—
206Dic94—21no—
207Dic94C2×C4⋊C412yes—
208C8×C47—22no—
209C8×C47C4×D412yes—
210C8×C47C4×D412yes—
211C47⋊C8—22no—
212C47⋊C8—42no—
213C47⋊C8C2×C4⋊C412yes—
214C47⋊C8C2×C4⋊C412yes—
215#22692—471nomaximal, Sylow
216C22×C47—21no—
217C22×C47—21no—
218C22×C47—22no—
219C22×C47—41no—
220C22×C47—41no—
221C22×C47#52011yescharacteristic
222C22×C47D4211yescharacteristic
223C22×C47#55411yescharacteristic
224C2×D47—22no—
225C2×D47—41no—
226C2×D47—41no—
227C4×C47—22no—
228C4×C47—42no—
229C4×C47—42no—
230C4×C47—82no—
231C4×C47C22×C4⋊C411yescharacteristic
232C4×C47C2×C4×D411yescharacteristic
233C4×C47#54711yescharacteristic
234C4×C47D4×Q811yescharacteristic
235Dic47—24no—
236Dic47—22no—
237Dic47—82no—
238C2×#443—471no—
239#2951—472no—
240#2913—471no—
241#2913D4711yescharacteristic, complemented
242#1952—472no—
243#1859—472no—
244#1859—472no—
245#1845—471no—
246#1816—471no—
247#1816—471no—
248#1772—471no—
249C2×C47—21no—
250C2×C47—21no—
251C2×C47—42no—
252C2×C47D4×C4⋊C411yescharacteristic
253C2×C47#329211yescharacteristic
254C2×C47#329311yescharacteristic
255D47—42no—
256C2×#135—471no—
257C2×#131—471no—
258C2×#131C2×D4711yescharacteristic
259#520—472no—
260#520C2×D4712yes—
261#443—478no—
262#440—942no—
263#440—942no—
264C4×Q16—942no—
265C4×D8—942no—
266C2×#108—471no—
267C2×#108—471no—
268#428—22no—
269#428—472no—
270#428—942no—
271#428C2×D4712yes—
272#427—21no—
273#427—941no—
274#427C2×D4711yescharacteristic
275C2×#107—21no—
276C2×#107—471no—
277C2×#107—472no—
278C2×#107—941no—
279C2×#107C2×D4711yescharacteristic
280C2×#104—471no—
281C2×#104—471no—
282#417—472no—
283#417—472no—
284#405—472no—
285C2×C4×C8—471no—
286#384—472no—
287#343—472no—
288#343—472no—
289#339—472no—
290C47#2269211yescharacteristic, complemented, Sylow
291C2×#41—471no—
292C2×#41C22×D4711yescharacteristic
293#135—474no—
294#135—474no—
295C2×Q16—471no—
296C2×SD16—471no—
297C2×SD16—471no—
298C2×D8—471no—
299#131—22no—
300#131—474no—
301#131—942no—
302#131C4×D4714yescomplemented
303C2×M16—21no—
304C2×M16—471no—
305C2×M16—472no—
306C2×M16—941no—
307C2×M16C22×D4711yescharacteristic
308C22×C8—21no—
309C22×C8—471no—
310C22×C8—941no—
311C22×C8C22×D4711yescharacteristic
312C4×Q8—22no—
313C4×Q8—942no—
314C4×D4—22no—
315C4×D4—42no—
316C4×D4—942no—
317C4×D4—1882no—
318#117—22no—
319#117—942no—
320#117C22×D4712yes—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—472no—
323C2×C4⋊C4—942no—
324C2×C4⋊C4—1882no—
325C2×C4⋊C4C22×D4712yes—
326C2×C42—471no—
327#108—474no—
328#108—942no—
329#107—21no—
330#107—21no—
331#107—22no—
332#107—942no—
333#107—942no—
334#107—941no—
335#107—941no—
336#107—942no—
337#107C22⋊D4711yescharacteristic
338#107C22⋊D4711yescharacteristic
339#106—22no—
340#106—942no—
341#106C22⋊D4712yes—
342#104—474no—
343#104—474no—
344#103—942no—
345#103—942no—
346#102—942no—
347#102—942no—
348#97—942no—
349C4×C8—474no—
350C4×C8—942no—
351#41—21no—
352#41—21no—
353#41—472no—
354#41—472no—
355#41—941no—
356#41—941no—
357#41C2×C4×D4712yes—
358#41Q8×D4712yes—
359C2×Q8—471no—
360C2×Q8D4×D4711yescharacteristic
361C2×D4—21no—
362C2×D4—471no—
363C2×D4—941no—
364C2×D4D4×D4711yescharacteristic
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—941no—
369C22×C4—941no—
370C22×C4—942no—
371C22×C4—1882no—
372C22×C4C23×D4711yescharacteristic
373Q16—474no—
374SD16—474no—
375SD16—474no—
376D8—474no—
377M16—21no—
378M16—24no—
379M16—472no—
380M16—941no—
381M16—944no—
382M16—942no—
383M16—942no—
384M16C2×C4×D4712yes—
385C2×C8—21no—
386C2×C8—24no—
387C2×C8—472no—
388C2×C8—471no—
389C2×C8—471no—
390C2×C8—472no—
391C2×C8—941no—
392C2×C8—944no—
393C2×C8—942no—
394C2×C8C2×C4×D4712yes—
395C2×C8#1062535012yes—
396C2×C8C2×C22⋊D4711yescharacteristic
397C2×C8C2×C22⋊D4711yescharacteristic
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—942no—
402C4⋊C4—942no—
403C4⋊C4—3764no—
404C4⋊C4—1882no—
405C4⋊C4C2×C22⋊D4712yes—
406C4⋊C4C2×C22⋊D4712yes—
407C22⋊C4—22no—
408C22⋊C4—42no—
409C22⋊C4—1882no—
410C42—22no—
411C42—42no—
412C42—472no—
413C42—941no—
414C42—1882no—
415C23—21no—
416C23—941no—
417C23#1309406911yescharacteristic
418Q8—21no—
419Q8—472no—
420Q8—941no—
421Q8C4⋊C4×D4712yes—
422D4—22no—
423D4—22no—
424D4—21no—
425D4—472no—
426D4—942no—
427D4—942no—
428D4—941no—
429D4C4⋊C4×D4712yes—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—162no—
438C2×C4—942no—
439C2×C4—942no—
440C2×C4—941no—
441C2×C4—942no—
442C2×C4—942no—
443C2×C4—942no—
444C2×C4—3762no—
445C2×C4—1882no—
446C2×C4—7522no—
447C2×C4C22×C4×D4711yescharacteristic
448C2×C4C2×D4×D4711yescharacteristic
449C2×C4C22×C22⋊D4711yescharacteristic
450C2×C4C2×Q8×D4711yescharacteristic
451C2×C4C2×#1062535011yescharacteristic
452C2×C4#1309407211yescharacteristic
453C8—22no—
454C8—472no—
455C8—472no—
456C8—942no—
457C8—1882no—
458C8C4×C22⋊D4712yes—
459C8C4×C22⋊D4712yes—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—942no—
466C22—1881no—
467C22—1881no—
468C22#1568960411yescharacteristic
469C22D4×C22⋊D4711yescharacteristic
470C22#1568982811yescharacteristic
471C4—22no—
472C4—42no—
473C4—42no—
474C4—82no—
475C4—944no—
476C4—942no—
477C4—3762no—
478C4C2×C4⋊C4×D4711yescharacteristic
479C4C2×C4×C22⋊D4711yescharacteristic
480C4#1568977411yescharacteristic
481C4Q8×C22⋊D4711yescharacteristic
482C2—21no—
483C2—21no—
484C2—42no—
485C2—1882no—
486C2C4⋊C4×C22⋊D4711yescharacteristic
487C2#2025518511yescharacteristic
488C2#2025519011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2913C2
socleC22×C47D42
prime radicalC2×#13094044C22
cosocleC22×C4×C47C24
hypercenter#2913D47
hypocenterC47#22692
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
3. #20251985C2semidirect
4. #20244170C2semidirect
5. #20243614C2semidirect
6. #20243611C2semidirect
7. #20243546C2semidirect
8. #20243453C2semidirect
9. #20243452C2semidirect
10. #20243269C2semidirect
11. C47×#2913C2semidirect
15. #15689238C4semidirect
54. #13094030C2×C4semidirect
69. #13093959C2×C4semidirect
76. #13093951C2×C4semidirect
77. #13093950C2×C4semidirect
84. C8×Dic47C2×C4semidirect
87. #131×C47C2×C4semidirect
146. #10625337C4⋊C4semidirect
147. #10625336C4⋊C4semidirect
148. #10625335C4⋊C4semidirect
149. #10625334C4⋊C4semidirect
241. #2913D47semidirect
290. C47#22692semidirect
302. #131C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892382
#202522612
#202519852
#202441702
#202436142
#202436112
#202435462
#202434532
#202434522
#202432692
C47×#29132
#2269247