Finite Group Databasebeta

← #26223229

Subgroups — #26223229

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#20252291C212yesmaximal, complemented
2C2×#15689245C211yesmaximal, characteristic, complemented
3#20251814C211yesmaximal, characteristic, complemented
4#20244287C211yesmaximal, characteristic, complemented
5#20244179C211yesmaximal, characteristic, complemented
6#20244059C212yesmaximal, complemented
7#20243891C212yesmaximal, complemented
8#20243442C211yesmaximal, characteristic, complemented
9#20243275C211yesmaximal, characteristic, complemented
10#20243191C212yesmaximal, complemented
11C47×#3067C211yesmaximal, characteristic
12C2×#13094031C2212yes—
13C2×#13094031C2211yescharacteristic
14#41×Dic47C2211yescharacteristic
15#15689245—24no—
16#15689245—21no—
17#15689245—21no—
18#15689245—21no—
19#15689245—21no—
20#15689240—24no—
21#15689172C2212yes—
22C2×#13093960C2211yescharacteristic
23#15689161C2211yescharacteristic
24#15689140C2212yes—
25#15689140C2212yes—
26C2×#13093951C2212yes—
27#15689097C2212yes—
28#15689081—21no—
29#15689081C2212yes—
30#15689081C2211yescharacteristic
31#15689078—21no—
32#15689078C2211yescharacteristic
33M16×Dic47—21no—
34M16×Dic47C2211yescharacteristic
35C2×#13093934—21no—
36C2×#13093934C2211yescharacteristic
37#15688895C2212yes—
38#15688895C2212yes—
39#15688884C2212yes—
40C47×#575C2212yes—
41C2×#135×C47C2211yescharacteristic
42C2×#131×C47C2211yescharacteristic
43C47×#473C412yescomplemented
44C47×#472C412yescomplemented
45C47×#471C414yescomplemented
46C2×#108×C47C2211yescharacteristic
47C47×#413C2212yes—
48#13094031—22no—
49#13094031—22no—
50#13094031—22no—
51#13094031—22no—
52#13094031—22no—
53#13094031D414yes—
54C2×#10625329—21no—
55C2×#10625329—22no—
56C2×#10625329C2312yes—
57C2×#10625329C2311yescharacteristic
58C22×C47⋊C8—21no—
59C22×C47⋊C8—22no—
60C22×C47⋊C8C2311yescharacteristic
61Q8×Dic47—21no—
62D4×Dic47—21no—
63D4×Dic47—22no—
64#13093983—21no—
65#13093983C2312yes—
66#13093983C2311yescharacteristic
67C2×C4×Dic47—21no—
68C2×C4×Dic47C2311yescharacteristic
69#13093960—21no—
70#13093960—21no—
71#13093960—24no—
72#13093958—21no—
73#13093956—21no—
74#13093951—24no—
75#13093951—22no—
76#13093951—22no—
77#13093951—24no—
78#13093939—24no—
79#13093934—21no—
80#13093934—21no—
81#13093934—22no—
82#13093934—21no—
83#13093934D411yescharacteristic
84#13093934D411yescharacteristic
85#13093934D411yescharacteristic
86#13093934D411yescharacteristic
87C8×Dic47—21no—
88C8×Dic47—21no—
89C8×Dic47—22no—
90C8×Dic47—21no—
91C2×#41×C47C2312yes—
92C2×#41×C47C2311yescharacteristic
93#137×C47—22no—
94#137×C47C2×C414yes—
95AGL(1,ℤ/8)×C47—22no—
96AGL(1,ℤ/8)×C47C2×C414yes—
97#135×C47—24no—
98#135×C47—21no—
99#135×C47—21no—
100#135×C47—21no—
101#135×C47—21no—
102C2×Q16×C47C2×C411yescharacteristic, complemented
103C2×SD16×C47C2×C412yescomplemented
104C2×D8×C47C2×C411yescharacteristic, complemented
105#131×C47—21no—
106#131×C47—21no—
107#131×C47C2×C412yes—
108#131×C47C2×C412yes—
109C2×M16×C47—21no—
110C2×M16×C47C2311yescharacteristic
111C2×M16×C47C2312yes—
112C22×C8×C47—21no—
113C22×C8×C47C2311yescharacteristic
114#108×C47C2×C412yescomplemented
115#108×C47C2×C412yescomplemented
116#101×C47C2×C414yescomplemented
117#100×C47C2×C414yescomplemented
118#41×C47—21no—
119#41×C47—21no—
120#41×C47—22no—
121#41×C47—22no—
122#41×C47—22no—
123#41×C47—22no—
124#41×C47C22×C412yes—
125#41×C47C2×D412yes—
126C2×Q8×C47C22×C412yes—
127C2×Q8×C47#4111yescharacteristic
128C2×D4×C47—21no—
129C2×D4×C47—22no—
130C2×D4×C47C22×C412yes—
131C2×D4×C47#4111yescharacteristic
132C22×C4×C47—21no—
133C22×C4×C47—22no—
134C22×C4×C47C2411yescharacteristic
135C22×Dic47—22no—
136C22×Dic47—21no—
137Q16×C47—24no—
138Q16×C47—21no—
139Q16×C47—21no—
140SD16×C47—24no—
141SD16×C47—24no—
142SD16×C47—22no—
143SD16×C47—22no—
144D8×C47—24no—
145D8×C47—21no—
146D8×C47—21no—
147M16×C47—21no—
148M16×C47—22no—
149M16×C47—22no—
150M16×C47—22no—
151M16×C47C22×C414yes—
152M16×C47C22×C412yes—
153C2×C8×C47—21no—
154C2×C8×C47—22no—
155C2×C8×C47—22no—
156C2×C8×C47—22no—
157C2×C8×C47C22×C411yescharacteristic
158C2×C8×C47C22×C411yescharacteristic
159C2×C8×C47C2×D411yescharacteristic
160C2×C8×C47C2×D411yescharacteristic
161C2×C8×C47#4111yescharacteristic
162C2×C8×C47#4111yescharacteristic
163C22⋊Dic47—22no—
164C22⋊Dic47—22no—
165C22⋊Dic47#4112yes—
166C4⋊Dic47—22no—
167C4⋊Dic47—21no—
168C4⋊Dic47—22no—
169C4×Dic47—22no—
170C4×Dic47—21no—
171C4×Dic47—21no—
172C4×Dic47—22no—
173C4×Dic47C2×D411yescharacteristic
174C4×Dic47C2×D411yescharacteristic
175#10625329—21no—
176#10625329—24no—
177#10625329—22no—
178#10625329—22no—
179#10625329—42no—
180#10625329—42no—
181#10625329C2×D412yes—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—24no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—42no—
186C2×C47⋊C8—42no—
187C2×C47⋊C8C2×D412yes—
188C2×C47⋊C8C2×D411yescharacteristic
189C2×C47⋊C8C2×D411yescharacteristic
190C2×C47⋊C8#4112yes—
191C23×C47—21no—
192C23×C47—22no—
193C23×C47C2×#4111yescharacteristic
194Q8×C47—21no—
195Q8×C47—22no—
196Q8×C47—22no—
197Q8×C47C4×D412yes—
198D4×C47—22no—
199D4×C47—22no—
200D4×C47—22no—
201D4×C47—22no—
202D4×C47—21no—
203D4×C47—42no—
204D4×C47—42no—
205D4×C47C4×D412yes—
206C2×C4×C47—21no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47C23×C411yescharacteristic
213C2×C4×C47C22×D411yescharacteristic
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C2×Dic47—21no—
219C2×Dic47—22no—
220C2×Dic47—21no—
221C2×Dic47—42no—
222C2×Dic47—81no—
223C2×Dic47—82no—
224C8×C47—22no—
225C8×C47—24no—
226C8×C47C4×D411yescharacteristic
227C8×C47C4×D411yescharacteristic
228C8×C47C4×D411yescharacteristic
229C8×C47C4×D411yescharacteristic
230C47⋊C8—22no—
231C47⋊C8—42no—
232C47⋊C8#12112yes—
233C47⋊C8#12112yes—
234#21559—471nomaximal, Sylow
235C22×C47—22no—
236C22×C47—41no—
237C22×C47—42no—
238C22×C47—42no—
239C22×C47—82no—
240C22×C47C4×#4111yescharacteristic
241C22×C47#52511yescharacteristic
242C22×C47#53211yescharacteristic
243C4×C47—22no—
244C4×C47—42no—
245C4×C47C2×C4×D411yescharacteristic
246C4×C47C2×C4×D411yescharacteristic
247C4×C47C2×#12111yescharacteristic
248C4×C47#52911yescharacteristic
249Dic47—42no—
250Dic47—41no—
251Dic47—41no—
252Dic47—82no—
253#3067D4711yescharacteristic
254#2955—472no—
255C2×#444—471no—
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—42no—
268C2×C47—82no—
269C2×C47C4×#12111yescharacteristic
270#575C2×D4712yes—
271C2×#135C2×D4711yescharacteristic
272C2×#131—471no—
273C2×#131—472no—
274C2×#131C2×D4711yescharacteristic
275C4×#41—471no—
276#473Dic4712yescomplemented
277#472Dic4712yescomplemented
278#471Dic4714yescomplemented
279#444—941no—
280#444—941no—
281#444—941no—
282#444—941no—
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×#104—472no—
292#413C2×D4712yes—
293#410—472no—
294#405—471no—
295#405—472no—
296#405—471no—
297#405—941no—
298#405—941no—
299C4×M16—471no—
300C4×M16—941no—
301C2×#97—471no—
302C2×#97—941no—
303#344—472no—
304#344—472no—
305#339—472no—
306C47#2155911yescharacteristic, complemented, Sylow
307C2×#41C22×D4711yescharacteristic
308C2×#41C22×D4712yes—
309#137—22no—
310#137C2×Dic4714yes—
311AGL(1,ℤ/8)—22no—
312AGL(1,ℤ/8)C2×Dic4714yes—
313#135—24no—
314#135—21no—
315#135—21no—
316#135—21no—
317#135—21no—
318C2×Q16C2×Dic4711yescharacteristic, complemented
319C2×SD16C2×Dic4712yescomplemented
320C2×D8C2×Dic4711yescharacteristic, complemented
321#131—21no—
322#131—21no—
323#131—474no—
324#131—942no—
325#131—942no—
326#131—942no—
327#131—942no—
328#131—942no—
329#131C2×Dic4712yes—
330#131C2×Dic4712yes—
331C2×M16—21no—
332C2×M16—471no—
333C2×M16—472no—
334C2×M16—941no—
335C2×M16—942no—
336C2×M16C22×D4711yescharacteristic
337C2×M16C22×D4712yes—
338C22×C8—21no—
339C22×C8—471no—
340C22×C8—941no—
341C22×C8—942no—
342C22×C8C22×D4711yescharacteristic
343C4×Q8—941no—
344C4×D4—942no—
345C4×D4—941no—
346#117—471no—
347#117—472no—
348#117—941no—
349C2×C42—471no—
350C2×C42—941no—
351#108—944no—
352#108—941no—
353#108—941no—
354#108C2×Dic4712yescomplemented
355#108C2×Dic4712yescomplemented
356#107—941no—
357#106—941no—
358#104—942no—
359#104—942no—
360#104—944no—
361#104—944no—
362#101C2×Dic4714yescomplemented
363#100C2×Dic4714yescomplemented
364#99—944no—
365#97—471no—
366#97—471no—
367#97—471no—
368#97—471no—
369#97—941no—
370#97—941no—
371#97—942no—
372#97—941no—
373C4×C8—941no—
374C4×C8—941no—
375C4×C8—942no—
376C4×C8—941no—
377#41—21no—
378#41—21no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41—22no—
383#41D4×D4712yes—
384#41C22×Dic4712yes—
385C2×Q8#1062535211yescharacteristic
386C2×Q8C22×Dic4712yes—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4#1062535211yescharacteristic
390C2×D4C22×Dic4712yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—942no—
394C22×C4—941no—
395C22×C4C23×D4711yescharacteristic
396Q16—24no—
397Q16—21no—
398Q16—21no—
399SD16—24no—
400SD16—24no—
401SD16—22no—
402SD16—22no—
403D8—24no—
404D8—21no—
405D8—21no—
406M16—21no—
407M16—22no—
408M16—22no—
409M16—22no—
410M16—472no—
411M16—944no—
412M16—941no—
413M16—942no—
414M16—942no—
415M16—1882no—
416M16—1882no—
417M16C22×Dic4712yes—
418M16C22×Dic4714yes—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—471no—
424C2×C8—472no—
425C2×C8—471no—
426C2×C8—472no—
427C2×C8—944no—
428C2×C8—941no—
429C2×C8—942no—
430C2×C8—1882no—
431C2×C8—1882no—
432C2×C8D4×D4711yescharacteristic
433C2×C8D4×D4711yescharacteristic
434C2×C8#1062535211yescharacteristic
435C2×C8#1062535211yescharacteristic
436C2×C8C22×Dic4711yescharacteristic
437C2×C8C22×Dic4711yescharacteristic
438C4⋊C4—942no—
439C4⋊C4—941no—
440C4⋊C4—942no—
441C22⋊C4—472no—
442C22⋊C4—942no—
443C22⋊C4—942no—
444C42—471no—
445C42—471no—
446C42—942no—
447C42—941no—
448C42—941no—
449C42—942no—
450C23—21no—
451C23—22no—
452C23#41×D4711yescharacteristic
453Q8—21no—
454Q8—22no—
455Q8—22no—
456Q8D4×Dic4712yes—
457D4—22no—
458D4—22no—
459D4—22no—
460D4—22no—
461D4—21no—
462D4—42no—
463D4—42no—
464D4D4×Dic4712yes—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—941no—
472C2×C4—941no—
473C2×C4—942no—
474C2×C4—3761no—
475C2×C4—3762no—
476C2×C4—1882no—
477C2×C4C23×Dic4711yescharacteristic
478C2×C4C2×D4×D4711yescharacteristic
479C2×C4C2×D4×D4711yescharacteristic
480C2×C4#41×D4711yescharacteristic
481C2×C4C2×#1062535211yescharacteristic
482C2×C4C2×#1062535211yescharacteristic
483C8—22no—
484C8—24no—
485C8—472no—
486C8—472no—
487C8—942no—
488C8—1882no—
489C8D4×Dic4711yescharacteristic
490C8D4×Dic4711yescharacteristic
491C8D4×Dic4711yescharacteristic
492C8D4×Dic4711yescharacteristic
493C22—22no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—82no—
498C22#41×Dic4711yescharacteristic
499C22#1568963311yescharacteristic
500C22#1568966811yescharacteristic
501C4—22no—
502C4—42no—
503C4—3762no—
504C4—1881no—
505C4—1881no—
506C4—1882no—
507C4C2×D4×Dic4711yescharacteristic
508C4C2×D4×Dic4711yescharacteristic
509C4#121×D4711yescharacteristic
510C4#1568965411yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—82no—
516C2#121×Dic4711yescharacteristic
subgroupG/H
centerC4#121×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#3067C2
socleC2×C47C4×#121
cosocleC22×C4×C47C24
hypercenter#3067D47
hypocenterC47#21559
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #20252291C2semidirect
2. C2×#15689245C2semidirect
3. #20251814C2semidirect
4. #20244287C2semidirect
5. #20244179C2semidirect
6. #20244059C2semidirect
7. #20243891C2semidirect
8. #20243442C2semidirect
9. #20243275C2semidirect
10. #20243191C2semidirect
43. C47×#473C4semidirect
44. C47×#472C4semidirect
45. C47×#471C4semidirect
102. C2×Q16×C47C2×C4semidirect
103. C2×SD16×C47C2×C4semidirect
104. C2×D8×C47C2×C4semidirect
114. #108×C47C2×C4semidirect
115. #108×C47C2×C4semidirect
116. #101×C47C2×C4semidirect
117. #100×C47C2×C4semidirect
276. #473Dic47semidirect
277. #472Dic47semidirect
278. #471Dic47semidirect
306. C47#21559semidirect
318. C2×Q16C2×Dic47semidirect
319. C2×SD16C2×Dic47semidirect
320. C2×D8C2×Dic47semidirect
354. #108C2×Dic47semidirect
355. #108C2×Dic47semidirect
362. #101C2×Dic47semidirect
363. #100C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202522912
C2×#156892452
#202518142
#202442872
#202441792
#202440592
#202438912
#202434422
#202432752
#202431912
C47×#30672
#2155947