Finite Group Databasebeta

← #26234766

Subgroups — #26234766

Order 12032.

846 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689238C211yesmaximal, characteristic
2#20252269C212yesmaximal
3#20251985C211yesmaximal, characteristic, complemented
4#20244165C212yesmaximal, complemented
5#20243621C212yesmaximal, complemented
6#20243618C212yesmaximal, complemented
7#20243551C211yesmaximal, 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#15689233—22no—
17#15689228—22no—
18C4×#10625336—22no—
19C4×#10625335—22no—
20C2×#13093960C2211yescharacteristic
21C2×#13093959C2211yescharacteristic
22#15689166C2212yes—
23#15689162—22no—
24#15689160—21no—
25C2×#13093958C2211yescharacteristic
26#107×D47—21no—
27C2×#13093955C2212yes—
28C2×#13093951C2211yescharacteristic
29C2×#13093950C2211yescharacteristic
30#15689124C2212yes—
31#15689122C2212yes—
32#15689079C2212yes—
33C2×C8×Dic47C2211yescharacteristic
34#15688988C2212yes—
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—
65C4⋊C4×D47—22no—
66C2×C4×Dic47C2311yescharacteristic
67#13093960—22no—
68#13093959C2×C414yescomplemented
69#13093958—21no—
70#13093958—21no—
71#13093957—22no—
72#13093955—22no—
73#13093955—22no—
74#13093955—22no—
75#13093951C2×C414yescomplemented
76#13093950C2×C414yescomplemented
77#13093948—22no—
78#13093947—22no—
79#13093945—22no—
80#13093944—22no—
81#13093935—22no—
82C4×C47⋊C8—22no—
83C8×Dic47C2×C414yescomplemented
84C47⋊#95—42no—
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⋊C414yes—
147#10625336C4⋊C414yes—
148#10625335C4⋊C414yes—
149#10625334C4⋊C414yes—
150(C2×D47)⋊C4—42no—
151C4⋊Dic47—22no—
152C4⋊Dic47—22no—
153Dic47⋊C4—42no—
154C4×Dic47—21no—
155C4×Dic47—42no—
156C4×Dic47C22×C412yes—
157#10625329—21no—
158#10625329—22no—
159#10625329—22no—
160#10625329C22×C412yes—
161C2×C47⋊C8—21no—
162C2×C47⋊C8—22no—
163C2×C47⋊C8C22×C412yes—
164C2×C47⋊C8C2×D411yescharacteristic
165C2×C47⋊C8C2×D411yescharacteristic
166C2×C47⋊C8C2×Q812yes—
167#10625324—24no—
168C8×D47—24no—
169C23×C47—21no—
170C23×C47H(5,𝔽2)11yescharacteristic
171C22×D47—21no—
172Q8×C47—21no—
173Q8×C47C2×C4⋊C412yes—
174D4×C47—22no—
175D4×C47—22no—
176D4×C47—21no—
177D4×C47C2×C4⋊C412yes—
178C2×C4×C47—21no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—22no—
183C2×C4×C47—42no—
184C2×C4×C47—82no—
185C2×C4×C47—162no—
186C2×C4×C47C23×C411yescharacteristic
187C2×C4×C47C22×D411yescharacteristic
188C2×C4×C47C22×D411yescharacteristic
189C2×C4×C47C22×Q811yescharacteristic
190C2×C4×C47C2×#4111yescharacteristic
191C2×C4×C472−1+411yescharacteristic
192C22⋊D47—22no—
193C22⋊D47—22no—
194C2×Dic47—21no—
195C2×Dic47—22no—
196C2×Dic47—41no—
197C2×Dic47—41no—
198C2×Dic47—42no—
199C2×Dic47—82no—
200D188—21no—
201D188C2×C4⋊C412yes—
202C4×D47—22no—
203C4×D47—22no—
204C4×D47—162no—
205Dic94—21no—
206Dic94C2×C4⋊C412yes—
207C8×C47—22no—
208C8×C47C4×D412yes—
209C8×C47C4×D412yes—
210C47⋊C8—22no—
211C47⋊C8—42no—
212C47⋊C8C2×C4⋊C412yes—
213C47⋊C8C2×C4⋊C412yes—
214#22691—471nomaximal, Sylow
215C22×C47—21no—
216C22×C47—21no—
217C22×C47—22no—
218C22×C47—41no—
219C22×C47—41no—
220C22×C47#52011yescharacteristic
221C22×C47D4211yescharacteristic
222C22×C47#55411yescharacteristic
223C2×D47—22no—
224C2×D47—41no—
225C2×D47—41no—
226C4×C47—22no—
227C4×C47—42no—
228C4×C47—42no—
229C4×C47—82no—
230C4×C47C22×C4⋊C411yescharacteristic
231C4×C47C2×C4×D411yescharacteristic
232C4×C47#54711yescharacteristic
233C4×C47D4×Q811yescharacteristic
234Dic47—22no—
235Dic47—42no—
236Dic47—82no—
237C2×#443—471no—
238#2952—472no—
239#2913—471no—
240#2913D4711yescharacteristic, complemented
241#1951—472no—
242#1860—472no—
243#1860—472no—
244#1846—471no—
245#1816—471no—
246#1816—471no—
247#1772—471no—
248C2×C47—21no—
249C2×C47—21no—
250C2×C47—42no—
251C2×C47D4×C4⋊C411yescharacteristic
252C2×C47#329011yescharacteristic
253C2×C47#329011yescharacteristic
254D47—42no—
255C2×#135—471no—
256C2×#131—471no—
257C2×#131C2×D4711yescharacteristic
258#520—472no—
259#520C2×D4712yes—
260#443—478no—
261#442—942no—
262#441—942no—
263C4×SD16—942no—
264C4×SD16—942no—
265C2×#108—471no—
266C2×#108—471no—
267#428—22no—
268#428—472no—
269#428—942no—
270#428C2×D4712yes—
271#427—21no—
272#427—941no—
273#427C2×D4711yescharacteristic
274C2×#107—21no—
275C2×#107—471no—
276C2×#107—941no—
277C2×#107C2×D4711yescharacteristic
278C2×#106—472no—
279C2×#104—471no—
280C2×#104—471no—
281#417—472no—
282#417—472no—
283#405—472no—
284C2×C4×C8—471no—
285#382—472no—
286#343—472no—
287#343—472no—
288#339—472no—
289C47#2269111yescharacteristic, complemented, Sylow
290C2×#41—471no—
291C2×#41C22×D4711yescharacteristic
292#135—474no—
293#135—474no—
294C2×Q16—471no—
295C2×SD16—471no—
296C2×SD16—471no—
297C2×D8—471no—
298#131—22no—
299#131—474no—
300#131—942no—
301#131C4×D4714yescomplemented
302C2×M16—21no—
303C2×M16—471no—
304C2×M16—472no—
305C2×M16—941no—
306C2×M16C22×D4711yescharacteristic
307C22×C8—21no—
308C22×C8—471no—
309C22×C8—941no—
310C22×C8C22×D4711yescharacteristic
311C4×Q8—22no—
312C4×Q8—942no—
313C4×D4—22no—
314C4×D4—42no—
315C4×D4—942no—
316C4×D4—1882no—
317#117—22no—
318#117—942no—
319#117C22×D4712yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—472no—
322C2×C4⋊C4—942no—
323C2×C4⋊C4C22×D4712yes—
324C2×C42—471no—
325#108—474no—
326#108—942no—
327#107—21no—
328#107—21no—
329#107—22no—
330#107—941no—
331#107—941no—
332#107—942no—
333#107C22⋊D4711yescharacteristic
334#107C22⋊D4711yescharacteristic
335#106—22no—
336#106—942no—
337#106—942no—
338#106—942no—
339#106C22⋊D4712yes—
340#104—474no—
341#104—474no—
342#103—942no—
343#103—942no—
344#102—942no—
345#102—942no—
346#97—942no—
347C4×C8—474no—
348C4×C8—942no—
349#95—1882no—
350#41—21no—
351#41—21no—
352#41—472no—
353#41—472no—
354#41—941no—
355#41—941no—
356#41C2×C4×D4712yes—
357#41Q8×D4712yes—
358C2×Q8—471no—
359C2×Q8D4×D4711yescharacteristic
360C2×D4—21no—
361C2×D4—471no—
362C2×D4—941no—
363C2×D4D4×D4711yescharacteristic
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—941no—
368C22×C4—941no—
369C22×C4—942no—
370C22×C4—1882no—
371C22×C4C23×D4711yescharacteristic
372Q16—474no—
373SD16—474no—
374SD16—474no—
375D8—474no—
376M16—21no—
377M16—24no—
378M16—472no—
379M16—941no—
380M16—944no—
381M16—942no—
382M16—942no—
383M16C2×C4×D4712yes—
384C2×C8—21no—
385C2×C8—24no—
386C2×C8—472no—
387C2×C8—471no—
388C2×C8—471no—
389C2×C8—472no—
390C2×C8—941no—
391C2×C8—944no—
392C2×C8—942no—
393C2×C8C2×C4×D4712yes—
394C2×C8#1062535012yes—
395C2×C8C2×C22⋊D4711yescharacteristic
396C2×C8C2×C22⋊D4711yescharacteristic
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—42no—
400C4⋊C4—942no—
401C4⋊C4—942no—
402C4⋊C4—1882no—
403C4⋊C4C2×C22⋊D4712yes—
404C4⋊C4C2×C22⋊D4712yes—
405C22⋊C4—22no—
406C22⋊C4—42no—
407C22⋊C4—1882no—
408C42—22no—
409C42—42no—
410C42—472no—
411C42—941no—
412C42—1882no—
413C23—21no—
414C23—941no—
415C23#1309406911yescharacteristic
416Q8—21no—
417Q8—472no—
418Q8—941no—
419Q8C4⋊C4×D4712yes—
420D4—22no—
421D4—22no—
422D4—21no—
423D4—472no—
424D4—942no—
425D4—942no—
426D4—941no—
427D4C4⋊C4×D4712yes—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—162no—
436C2×C4—941no—
437C2×C4—942no—
438C2×C4—942no—
439C2×C4—942no—
440C2×C4—3762no—
441C2×C4—1882no—
442C2×C4—1881no—
443C2×C4—1881no—
444C2×C4—7522no—
445C2×C4C22×C4×D4711yescharacteristic
446C2×C4C2×D4×D4711yescharacteristic
447C2×C4C22×C22⋊D4711yescharacteristic
448C2×C4C2×Q8×D4711yescharacteristic
449C2×C4C2×#1062535011yescharacteristic
450C2×C4#1309407211yescharacteristic
451C8—22no—
452C8—472no—
453C8—472no—
454C8—942no—
455C8—1882no—
456C8C4×C22⋊D4712yes—
457C8C4×C22⋊D4712yes—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—942no—
464C22—1881no—
465C22—1881no—
466C22#1568960411yescharacteristic
467C22D4×C22⋊D4711yescharacteristic
468C22#1568982811yescharacteristic
469C4—22no—
470C4—42no—
471C4—42no—
472C4—82no—
473C4—942no—
474C4—3762no—
475C4—1882no—
476C4C2×C4⋊C4×D4711yescharacteristic
477C4C2×C4×C22⋊D4711yescharacteristic
478C4#1568977411yescharacteristic
479C4Q8×C22⋊D4711yescharacteristic
480C2—21no—
481C2—21no—
482C2—42no—
483C2—1882no—
484C2C4⋊C4×C22⋊D4711yescharacteristic
485C2#2025517011yescharacteristic
486C2#2025517411yescharacteristic
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#22691
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. #20244165C2semidirect
5. #20243621C2semidirect
6. #20243618C2semidirect
7. #20243551C2semidirect
8. #20243453C2semidirect
9. #20243452C2semidirect
10. #20243269C2semidirect
11. C47×#2913C2semidirect
15. #15689238C4semidirect
54. #13094030C2×C4semidirect
68. #13093959C2×C4semidirect
75. #13093951C2×C4semidirect
76. #13093950C2×C4semidirect
83. C8×Dic47C2×C4semidirect
87. #131×C47C2×C4semidirect
240. #2913D47semidirect
289. C47#22691semidirect
301. #131C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892382
#202522692
#202519852
#202441652
#202436212
#202436182
#202435512
#202434532
#202434522
#202432692
C47×#29132
#2269147