Finite Group Databasebeta

← #26223222

Subgroups — #26223222

Order 12032.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253104C211yesmaximal, characteristic
2#20252289C212yesmaximal, complemented
3#20251813C211yesmaximal, characteristic, complemented
4#20244287C211yesmaximal, characteristic, complemented
5#20244176C211yesmaximal, characteristic, complemented
6#20244054C212yesmaximal, complemented
7#20243892C212yesmaximal, complemented
8#20243443C211yesmaximal, characteristic, complemented
9#20243272C211yesmaximal, characteristic, complemented
10#20243188C212yesmaximal, complemented
11C2×C47×#444C211yesmaximal, characteristic, complemented
12#15689921C2212yes—
13C2×#13094030C2211yescharacteristic
14C2×#13094030C2211yescharacteristic
15C4×#10625350C2211yescharacteristic
16#15689395C412yescomplemented
17#15689390C412yescomplemented
18#15689383C414yescomplemented
19#15689243—24no—
20#15689238—24no—
21#15689171C2212yes—
22C2×#13093960C2211yescharacteristic
23#15689161C2211yescharacteristic
24#15689140C2212yes—
25#15689139C2212yes—
26#15689107C2212yes—
27#15689097C2212yes—
28#15689080—21no—
29#15689080—21no—
30#15689078C2211yescharacteristic
31#15689078C2212yes—
32#15689078C2211yescharacteristic
33C4×#10625329C2211yescharacteristic
34#15689075—21no—
35#97×D47—21no—
36#15688894C2212yes—
37#15688894C2212yes—
38#15688883C2212yes—
39C2×#135×C47C2211yescharacteristic
40C2×#131×C47C2212yes—
41C47×#444—21no—
42C47×#444—21no—
43C47×#444—21no—
44C47×#444—21no—
45C2×#108×C47C2211yescharacteristic
46C2×#104×C47C2212yes—
47C2×#97×C47C2211yescharacteristic
48C2×#10625350C2311yescharacteristic
49#13094055C2×C414yes—
50#13094054—22no—
51#13094050C2×C414yes—
52#13094049—22no—
53#13094044—24no—
54#13094030—21no—
55#13094030—21no—
56#13094030—22no—
57#13094030C2×C412yes—
58#13094030C2×C412yes—
59#13094030D414yes—
60C2×#10625329C2311yescharacteristic
61C2×#10625329C2312yes—
62C2×#10625329C2311yescharacteristic
63C2×#10625324—21no—
64C2×#10625324—21no—
65C2×C8×D47—21no—
66C2×C8×D47—21no—
67C4×Dic94—21no—
68C4×D188—21no—
69C4×C22⋊D47—22no—
70#13093983C2312yes—
71#13093983C2311yescharacteristic
72#13093981—21no—
73C42×D47—21no—
74#13093960C2×C412yescomplemented
75#13093960C2×C412yescomplemented
76#13093959—24no—
77#13093958—21no—
78#13093956—21no—
79#13093951—24no—
80#13093950—24no—
81#13093943C2×C414yescomplemented
82#13093941C2×C414yescomplemented
83#13093939—24no—
84#13093935—21no—
85#13093934—22no—
86#13093934—21no—
87#13093934—21no—
88C4×C47⋊C8—21no—
89C8×Dic47—22no—
90C8×Dic47—21no—
91C8×Dic47—21no—
92C2×#41×C47C2312yes—
93#135×C47—21no—
94#135×C47—21no—
95#135×C47—21no—
96#135×C47—21no—
97C2×Q16×C47C2×C411yescharacteristic, complemented
98C2×SD16×C47C2×C412yescomplemented
99C2×D8×C47C2×C411yescharacteristic, complemented
100#131×C47—22no—
101#131×C47—22no—
102#131×C47—22no—
103#131×C47—22no—
104C2×M16×C47—22no—
105C2×M16×C47C2312yes—
106C22×C8×C47—22no—
107C22×C8×C47C2311yescharacteristic
108C22×C8×C47C2311yescharacteristic
109C2×C42×C47C2311yescharacteristic
110#108×C47—21no—
111#108×C47—21no—
112#104×C47—22no—
113#104×C47—22no—
114#97×C47D411yescharacteristic
115#97×C47D411yescharacteristic
116#97×C47D411yescharacteristic
117#97×C47D411yescharacteristic
118#41×C47—22no—
119#41×C47—22no—
120#41×C47—22no—
121#41×C47—22no—
122C2×Q8×C47C22×C412yes—
123C2×D4×C47—22no—
124C2×D4×C47C22×C412yes—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47C2411yescharacteristic
128C2×C22⋊D47—21no—
129#10625350—21no—
130#10625350—21no—
131#10625350C22×C412yes—
132#10625350C2×D412yes—
133C2×D188#4111yescharacteristic
134C2×C4×D47—21no—
135C2×C4×D47—22no—
136C2×Dic94#4111yescharacteristic
137Q16×C47—21no—
138Q16×C47—21no—
139SD16×C47—22no—
140SD16×C47—22no—
141D8×C47—21no—
142D8×C47—21no—
143M16×C47—22no—
144M16×C47—22no—
145M16×C47—42no—
146M16×C47—42no—
147C2×C8×C47—22no—
148C2×C8×C47—42no—
149C2×C8×C47—42no—
150C2×C8×C47C22×C411yescharacteristic
151C2×C8×C47C22×C411yescharacteristic
152C2×C8×C47C2×D411yescharacteristic
153C2×C8×C47C2×D411yescharacteristic
154C2×C8×C47C2×D412yes—
155C2×C8×C47C2×D411yescharacteristic
156C2×C8×C47C2×D411yescharacteristic
157C2×C8×C47#4111yescharacteristic
158C2×C8×C47#4111yescharacteristic
159C2×C8×C47#4112yes—
160C42×C47—21no—
161C42×C47C2×D411yescharacteristic
162C42×C47C2×D411yescharacteristic
163C22⋊Dic47—22no—
164C22⋊Dic47#4112yes—
165#10625337—24no—
166#10625336—24no—
167#10625335—24no—
168#10625334—24no—
169(C2×D47)⋊C4—22no—
170C4⋊Dic47—21no—
171C4⋊Dic47—22no—
172Dic47⋊C4—22no—
173C4×Dic47—22no—
174C4×Dic47—21no—
175C4×Dic47—22no—
176#10625329—21no—
177#10625329—21no—
178#10625329—22no—
179#10625329C22×C412yes—
180#10625329C22×C414yes—
181#10625329C2×D412yes—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—22no—
185#10625324—22no—
186#10625324—22no—
187#10625324—24no—
188C8×D47—22no—
189C8×D47—22no—
190C8×D47—24no—
191C23×C47—22no—
192C23×C47C2×#4111yescharacteristic
193C22×D47—21no—
194Q8×C47—22no—
195Q8×C47—22no—
196D4×C47—22no—
197D4×C47—22no—
198D4×C47—42no—
199D4×C47—42no—
200C2×C4×C47—22no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—81no—
206C2×C4×C47C23×C411yescharacteristic
207C2×C4×C47C22×D411yescharacteristic
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47C2×#4111yescharacteristic
210C2×C4×C47C2×#4111yescharacteristic
211C2×C4×C47C2×#4111yescharacteristic
212C22⋊D47—22no—
213C22⋊D47—22no—
214C2×Dic47—21no—
215C2×Dic47—22no—
216C2×Dic47—42no—
217D188—21no—
218D188C4×D412yes—
219C4×D47—22no—
220C4×D47—22no—
221C4×D47—82no—
222Dic94—21no—
223Dic94C4×D412yes—
224C8×C47—42no—
225C8×C47C4×D411yescharacteristic
226C8×C47C4×D411yescharacteristic
227C8×C47C4×D411yescharacteristic
228C8×C47C4×D411yescharacteristic
229C8×C47#12112yes—
230C8×C47#12112yes—
231C47⋊C8—22no—
232C47⋊C8—24no—
233C47⋊C8—22no—
234#21559—471nomaximal, Sylow
235C22×C47—41no—
236C22×C47—42no—
237C22×C47—82no—
238C22×C47C4×#4111yescharacteristic
239C22×C47#52511yescharacteristic
240C22×C47#53211yescharacteristic
241C2×D47—22no—
242C2×D47—42no—
243C4×C47—41no—
244C4×C47—41no—
245C4×C47—42no—
246C4×C47C2×C4×D411yescharacteristic
247C4×C47C2×C4×D411yescharacteristic
248C4×C47C2×#12111yescharacteristic
249C4×C47#52911yescharacteristic
250Dic47—22no—
251Dic47—42no—
252Dic47—82no—
253#3067—471no—
254#2955—472no—
255C2×#444D4711yescharacteristic, complemented
256#2875—471no—
257#1976—471no—
258#1953—471no—
259#1932—472no—
260#1906—472no—
261#1814—471no—
262#1773—471no—
263#1757—472no—
264C2×C47—21no—
265C2×C47—21no—
266C2×C47—21no—
267C2×C47—82no—
268C2×C47C4×#12111yescharacteristic
269D47—42no—
270#575—472no—
271C2×#135C2×D4711yescharacteristic
272C2×#131—471no—
273C2×#131—471no—
274C2×#131C2×D4712yes—
275C4×#41—471no—
276#473—472no—
277#472—472no—
278#471—474no—
279#444—21no—
280#444—21no—
281#444—21no—
282#444—21no—
283#444—944no—
284#443—944no—
285#430—472no—
286C2×#108—471no—
287C2×#108C2×D4711yescharacteristic
288#427—471no—
289#421—472no—
290#421—472no—
291C2×#104C2×D4712yes—
292#413—472no—
293#410—472no—
294#405—471no—
295#405—471no—
296#405—472no—
297#405—941no—
298#405—941no—
299C4×M16—471no—
300C4×M16—941no—
301C2×#97—941no—
302C2×#97C2×D4711yescharacteristic
303#344—472no—
304#344—472no—
305#339—472no—
306C47#2155911yescharacteristic, complemented, Sylow
307C2×#41—471no—
308C2×#41C22×D4712yes—
309#137—474no—
310#137—942no—
311AGL(1,ℤ/8)—474no—
312AGL(1,ℤ/8)—942no—
313#135—21no—
314#135—21no—
315#135—21no—
316#135—21no—
317#135—944no—
318C2×Q16C4×D4711yescharacteristic, complemented
319C2×SD16C4×D4712yescomplemented
320C2×D8C4×D4711yescharacteristic, complemented
321#131—22no—
322#131—22no—
323#131—22no—
324#131—22no—
325#131—472no—
326#131—472no—
327#131—474no—
328#131—941no—
329#131—941no—
330#131—942no—
331C2×M16—22no—
332C2×M16—471no—
333C2×M16—471no—
334C2×M16—472no—
335C2×M16—941no—
336C2×M16—941no—
337C2×M16C22×D4712yes—
338C22×C8—22no—
339C22×C8—941no—
340C22×C8—941no—
341C22×C8C22×D4711yescharacteristic
342C22×C8C22×D4711yescharacteristic
343C4×Q8—941no—
344C4×D4—941no—
345C4×D4—942no—
346#117—471no—
347#117—472no—
348#117—941no—
349C2×C42—941no—
350C2×C42C22×D4711yescharacteristic
351#108—21no—
352#108—21no—
353#108—472no—
354#108—472no—
355#108—944no—
356#107—941no—
357#106—941no—
358#104—22no—
359#104—22no—
360#104—944no—
361#104—944no—
362#101—474no—
363#100—474no—
364#99—944no—
365#97—942no—
366#97—941no—
367#97—941no—
368#97—941no—
369#97C22⋊D4711yescharacteristic
370#97C22⋊D4711yescharacteristic
371#97C22⋊D4711yescharacteristic
372#97C22⋊D4711yescharacteristic
373C4×C8—942no—
374C4×C8—941no—
375C4×C8—941no—
376C4×C8—941no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—472no—
382#41—472no—
383#41—941no—
384#41—941no—
385C2×Q8—471no—
386C2×Q8C2×C4×D4712yes—
387C2×D4—22no—
388C2×D4—471no—
389C2×D4—941no—
390C2×D4C2×C4×D4712yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—942no—
394C22×C4—941no—
395C22×C4C23×D4711yescharacteristic
396Q16—21no—
397Q16—21no—
398Q16—944no—
399SD16—22no—
400SD16—22no—
401SD16—944no—
402SD16—944no—
403D8—21no—
404D8—21no—
405D8—944no—
406M16—22no—
407M16—22no—
408M16—42no—
409M16—42no—
410M16—472no—
411M16—474no—
412M16—472no—
413M16—942no—
414M16—941no—
415M16—941no—
416M16—942no—
417M16—942no—
418M16—944no—
419C2×C8—22no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—942no—
423C2×C8—941no—
424C2×C8—941no—
425C2×C8—942no—
426C2×C8—942no—
427C2×C8—944no—
428C2×C8C2×C4×D4711yescharacteristic
429C2×C8C2×C4×D4711yescharacteristic
430C2×C8#1062535011yescharacteristic
431C2×C8#1062535011yescharacteristic
432C2×C8D4×D4712yes—
433C2×C8#1062535212yes—
434C2×C8C2×C22⋊D4711yescharacteristic
435C2×C8C2×C22⋊D4711yescharacteristic
436C2×C8C2×C22⋊D4711yescharacteristic
437C2×C8C2×C22⋊D4711yescharacteristic
438C4⋊C4—942no—
439C4⋊C4—941no—
440C4⋊C4—942no—
441C22⋊C4—472no—
442C22⋊C4—942no—
443C22⋊C4—942no—
444C42—21no—
445C42—942no—
446C42—941no—
447C42—942no—
448C42C2×C22⋊D4711yescharacteristic
449C42C2×C22⋊D4711yescharacteristic
450C23—22no—
451C23—941no—
452C23#41×D4711yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—472no—
456Q8—941no—
457D4—22no—
458D4—22no—
459D4—42no—
460D4—42no—
461D4—472no—
462D4—942no—
463D4—942no—
464D4—941no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—81no—
471C2×C4—941no—
472C2×C4—942no—
473C2×C4—942no—
474C2×C4—942no—
475C2×C4—3762no—
476C2×C4—1882no—
477C2×C4C22×C4×D4711yescharacteristic
478C2×C4C2×D4×D4711yescharacteristic
479C2×C4C22×C22⋊D4711yescharacteristic
480C2×C4#41×D4711yescharacteristic
481C2×C4C2×#1062535211yescharacteristic
482C2×C4C2×#1062535011yescharacteristic
483C8—42no—
484C8—944no—
485C8—942no—
486C8—942no—
487C8C4×C22⋊D4711yescharacteristic
488C8C4×C22⋊D4711yescharacteristic
489C8C4×C22⋊D4711yescharacteristic
490C8C4×C22⋊D4711yescharacteristic
491C8#1309399512yes—
492C8#1309399512yes—
493C22—41no—
494C22—42no—
495C22—82no—
496C22—942no—
497C22—1882no—
498C22C4×#1062535211yescharacteristic
499C22#1568963011yescharacteristic
500C22#1568967011yescharacteristic
501C4—41no—
502C4—41no—
503C4—42no—
504C4—942no—
505C4—3762no—
506C4—1882no—
507C4C4×D4×D4711yescharacteristic
508C4C2×C4×C22⋊D4711yescharacteristic
509C4C2×#1309399511yescharacteristic
510C4#1568965511yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—82no—
515C2—1882no—
516C2C4×#1309399511yescharacteristic
subgroupG/H
centerC4C2×#13093995
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#444C2
socleC2×C47C4×#121
prime radical#20253104C2
cosocleC22×C4×C47C24
hypercenterC2×#444D47
hypocenterC47#21559
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. #20252289C2semidirect
3. #20251813C2semidirect
4. #20244287C2semidirect
5. #20244176C2semidirect
6. #20244054C2semidirect
7. #20243892C2semidirect
8. #20243443C2semidirect
9. #20243272C2semidirect
10. #20243188C2semidirect
11. C2×C47×#444C2semidirect
16. #15689395C4semidirect
17. #15689390C4semidirect
18. #15689383C4semidirect
74. #13093960C2×C4semidirect
75. #13093960C2×C4semidirect
81. #13093943C2×C4semidirect
82. #13093941C2×C4semidirect
97. C2×Q16×C47C2×C4semidirect
98. C2×SD16×C47C2×C4semidirect
99. C2×D8×C47C2×C4semidirect
255. C2×#444D47semidirect
306. C47#21559semidirect
318. C2×Q16C4×D47semidirect
319. C2×SD16C4×D47semidirect
320. C2×D8C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202531042
#202522892
#202518132
#202442872
#202441762
#202440542
#202438922
#202434432
#202432722
#202431882
C2×C47×#4442
#2155947