Finite Group Databasebeta

← #25080141

Subgroups — #25080141

Order 11008.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145902C211yesmaximal, characteristic
2#19145723C211yesmaximal, characteristic, complemented
3#19143341C211yesmaximal, characteristic, complemented
4#19134167C212yesmaximal, complemented
5#19134125C212yesmaximal, complemented
6#19134043C212yesmaximal, complemented
7#19134001C212yesmaximal, complemented
8#19133347C211yesmaximal, characteristic, complemented
9C2×#14582648C211yesmaximal, characteristic, complemented
10#19133244C211yesmaximal, characteristic, complemented
11C43×#3141C211yesmaximal, characteristic, complemented
12C2×#13063333—21no—
13#14583449C2211yescharacteristic
14#14583295—21no—
15#14583247—21no—
16#14583217—22no—
17#14583186C2212yes—
18#14583186C2212yes—
19#14583179—21no—
20#14583176C2212yes—
21#14583133—22no—
22#14583094—21no—
23#14583091C2212yes—
24#14582978C2212yes—
25#14582972—22no—
26C8×D172C2212yes—
27#14582952C2211yescharacteristic
28C2×C4⋊(C43⋊C8)C2211yescharacteristic
29C2×C4⋊(C43⋊C8)C2211yescharacteristic
30#14582867—21no—
31#14582652C2212yes—
32#14582650C2212yes—
33#14582648C2212yes—
34#14582648C2212yes—
35#14582645C2211yescharacteristic
36#14582645C2211yescharacteristic
37#14582643C2211yescharacteristic
38#14582643C2211yescharacteristic
39C43×#532C2211yescharacteristic
40C2×C4⋊Q8×C43C2211yescharacteristic
41C43×#489—22no—
42C43×#487C2212yes—
43C43×#486C2212yes—
44C43×#417—21no—
45C43×#417—21no—
46C4×M16×C43C2211yescharacteristic
47C2×#10621898—21no—
48C2×#10621882—22no—
49#13063333—44no—
50C2×#10621872—22no—
51C22×C43⋊C8—21no—
52C22×C43⋊C8—21no—
53C2×C8×D43—22no—
54#13063325D411yescharacteristic
55#13063322D411yescharacteristic
56#13063313—21no—
57#13063299—21no—
58#13063299D412yes—
59C4×D172—21no—
60C4×D172C2312yes—
61#13063258—22no—
62#13063256—22no—
63#13063256—22no—
64C4⋊(C43⋊C8)—21no—
65C4⋊(C43⋊C8)—21no—
66C4⋊(C43⋊C8)C2312yes—
67C4⋊(C43⋊C8)C2311yescharacteristic
68C4⋊(C43⋊C8)C2311yescharacteristic
69C4⋊(C43⋊C8)C2311yescharacteristic
70C4⋊(C43⋊C8)C2311yescharacteristic
71#13063246—22no—
72#13063245—22no—
73#13063245—22no—
74#13063245—22no—
75C22⋊(C43⋊C8)—21no—
76(C2×D43)⋊C8—22no—
77(C2×D43)⋊C8—22no—
78C4×C43⋊C8—21no—
79C2×#41×C43—21no—
80C22×Q8×C43—21no—
81C2×M16×C43D411yescharacteristic
82C2×M16×C43D411yescharacteristic
83C4⋊Q8×C43—21no—
84C4⋊Q8×C43—21no—
85C4⋊Q8×C43—21no—
86C4⋊Q8×C43C2312yes—
87C4⋊Q8×C43C2311yescharacteristic
88#127×C43C2311yescharacteristic
89#124×C43—21no—
90#121×C43—21no—
91#121×C43—21no—
92C4×D4×C43—21no—
93C4×D4×C43—21no—
94C2×C4⋊C4×C43—21no—
95C2×C4⋊C4×C43—21no—
96C2×C42×C43C2311yescharacteristic
97#103×C43—22no—
98#103×C43—22no—
99#102×C43—22no—
100#102×C43—22no—
101#97×C43D412yes—
102C4×C8×C43C2312yes—
103#41×C43—42no—
104#41×C43—42no—
105C2×Q8×C43—21no—
106C2×Q8×C43—22no—
107C2×Q8×C43—41no—
108C2×Q8×C43—82no—
109C2×D4×C43—21no—
110C2×D4×C43—21no—
111C2×D4×C43—21no—
112C2×D4×C43—41no—
113C22×C4×C43—41no—
114C22×C4×C43—41no—
115C22×C4×C43C2×D411yescharacteristic
116C22×C4×C43C2×D411yescharacteristic
117C22×C4×C43#4111yescharacteristic
118C2×C22⋊D43—22no—
119#10621898—42no—
120#10621898—42no—
121C2×D172—21no—
122C2×D172—21no—
123C2×D172C2×D412yes—
124C2×C4×D43—22no—
125C2×Dic86—21no—
126M16×C43—22no—
127M16×C43—22no—
128C2×C8×C43C2×D412yes—
129C2×C8×C43C2×D412yes—
130C4⋊C4×C43—22no—
131C4⋊C4×C43—22no—
132C4⋊C4×C43—21no—
133C4⋊C4×C43—21no—
134C4⋊C4×C43—41no—
135C4⋊C4×C43—41no—
136C22⋊C4×C43—41no—
137C22⋊C4×C43—41no—
138C42×C43C2×D411yescharacteristic
139C42×C43C2×D411yescharacteristic
140C42×C43#4111yescharacteristic
141C42×C43C2411yescharacteristic
142#10621882—82no—
143(C2×D43)⋊C4—22no—
144(C2×D43)⋊C4—22no—
145C4⋊Dic43—21no—
146C4⋊Dic43C2×D412yes—
147C2×C43⋊C8—21no—
148C2×C43⋊C8—21no—
149C2×C43⋊C8—21no—
150C2×C43⋊C8—21no—
151C2×C43⋊C8—22no—
152C2×C43⋊C8—42no—
153C2×C43⋊C8—81no—
154#10621872—44no—
155C8×D43—44no—
156C23×C43—41no—
157C23×C43#12111yescharacteristic
158C22×D43—22no—
159Q8×C43—41no—
160Q8×C43—41no—
161Q8×C43—82no—
162Q8×C43—82no—
163D4×C43—41no—
164D4×C43—41no—
165D4×C43—41no—
166D4×C43—41no—
167D4×C43—81no—
168D4×C43—82no—
169C2×C4×C43—21no—
170C2×C4×C43—21no—
171C2×C4×C43—41no—
172C2×C4×C43—41no—
173C2×C4×C43—42no—
174C2×C4×C43—82no—
175C2×C4×C43—82no—
176C2×C4×C43#12111yescharacteristic
177C2×C4×C43#12111yescharacteristic
178C2×C4×C43#12111yescharacteristic
179C2×C4×C43#13512yes—
180C2×C4×C43AGL(1,ℤ/8)12yes—
181C2×C4×C43C22×D411yescharacteristic
182C2×C4×C43C22×D411yescharacteristic
183C2×C4×C43C2×#4111yescharacteristic
184C22⋊D43—44no—
185C2×Dic43—22no—
186D172—22no—
187D172—22no—
188D172—42no—
189D172—81no—
190C4×D43—42no—
191C4×D43—42no—
192Dic86—42no—
193C8×C43—22no—
194C8×C43—22no—
195C43⋊C8—41no—
196C43⋊C8—41no—
197C43⋊C8—41no—
198C43⋊C8—42no—
199C43⋊C8—41no—
200#19236—431nomaximal, Sylow
201C22×C43—21no—
202C22×C43—21no—
203C22×C43—81no—
204C22×C43—81no—
205C22×C43—81no—
206C22×C43#46312yes—
207C22×C43C2×#12111yescharacteristic
208C2×D43—42no—
209C2×D43—42no—
210C2×D43—42no—
211C4×C43—21no—
212C4×C43—21no—
213C4×C43—21no—
214C4×C43—81no—
215C4×C43—82no—
216C4×C43#46312yes—
217C4×C43C2×#13511yescharacteristic
218C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
219C4×C43#57711yescharacteristic
220C4×C43#57811yescharacteristic
221Dic43—42no—
222#3245—431no—
223#3227—431no—
224#3141D4311yescharacteristic, complemented
225#2935—431no—
226#1703—432no—
227#1697—432no—
228#1683—432no—
229#1677—432no—
230#1565—431no—
231C2×#333—431no—
232#1540—431no—
233C2×C43—22no—
234C2×C43—161no—
235C2×C43C2×#46311yescharacteristic
236C2×C43#306311yescharacteristic
237C2×C43#306811yescharacteristic
238D43—82no—
239C2×#131—861no—
240#532—431no—
241#532C2×D4311yescharacteristic
242C2×C4⋊Q8C2×D4311yescharacteristic
243#498—861no—
244#489—22no—
245#487C2×D4312yes—
246#486—861no—
247#486C2×D4312yes—
248#481—862no—
249#476—432no—
250#476—432no—
251#474—432no—
252#474—861no—
253#466—862no—
254#459—432no—
255#459—861no—
256#436—432no—
257#435—862no—
258C8×D4—432no—
259#431—431no—
260C2×C4⋊C8—431no—
261C2×C4⋊C8—431no—
262#417—21no—
263#417—21no—
264#408—861no—
265C4×M16C2×D4311yescharacteristic
266#335—432no—
267#334—432no—
268#333—432no—
269#333—432no—
270#332—431no—
271#332—431no—
272#331—431no—
273#331—431no—
274C43#1923611yescharacteristic, complemented, Sylow
275C2×#41—21no—
276C2×#41—861no—
277C22×Q8—21no—
278C2×D8—862no—
279#131—1724no—
280C2×M16—862no—
281C2×M16C22⋊D4311yescharacteristic
282C2×M16C22⋊D4311yescharacteristic
283C22×C8—861no—
284C22×C8—861no—
285C22×C8—862no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—431no—
290C4⋊Q8C22×D4312yes—
291C4⋊Q8C22×D4311yescharacteristic
292#127—431no—
293#127C22×D4311yescharacteristic
294#124—21no—
295#124—861no—
296#121—21no—
297#121—21no—
298#121—432no—
299#121—861no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—432no—
303C4×D4—861no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C42C22×D4311yescharacteristic
307#107—862no—
308#106—862no—
309#106—862no—
310C4⋊C8—431no—
311C4⋊C8—431no—
312C4⋊C8—431no—
313C4⋊C8—431no—
314C4⋊C8—432no—
315C4⋊C8—861no—
316C4⋊C8—861no—
317#103—22no—
318#103—22no—
319#102—22no—
320#102—22no—
321#102—862no—
322#102—862no—
323#102—862no—
324#102—862no—
325C22⋊C8—861no—
326C22⋊C8—862no—
327C22⋊C8—862no—
328#97C22⋊D4312yes—
329C4×C8—861no—
330C4×C8C22×D4312yes—
331#41—42no—
332#41—42no—
333#41—1722no—
334#41—1722no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—41no—
338C2×Q8—82no—
339C2×Q8—861no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—41no—
344C2×D4—432no—
345C2×D4—862no—
346C2×D4—861no—
347C2×D4—861no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—862no—
351C22×C4D4×D4311yescharacteristic
352C22×C4#1062190011yescharacteristic
353C22×C4C2×C22⋊D4311yescharacteristic
354D8—3442no—
355M16—22no—
356M16—22no—
357M16—1724no—
358C2×C8—861no—
359C2×C8—861no—
360C2×C8—862no—
361C2×C8—861no—
362C2×C8—861no—
363C2×C8—3441no—
364C2×C8—1724no—
365C2×C8—1722no—
366C2×C8C2×C22⋊D4312yes—
367C2×C8C2×C22⋊D4312yes—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—432no—
375C4⋊C4—861no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—862no—
379C22⋊C4—862no—
380C42D4×D4311yescharacteristic
381C42#1062190011yescharacteristic
382C42C2×C22⋊D4311yescharacteristic
383C42C23×D4311yescharacteristic
384C23—41no—
385C23—862no—
386C23#1306329611yescharacteristic
387Q8—41no—
388Q8—41no—
389Q8—82no—
390Q8—82no—
391Q8—1722no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—81no—
397D4—82no—
398D4—862no—
399D4—862no—
400D4—3441no—
401D4—1724no—
402D4—1722no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—862no—
411C2×C4—1722no—
412C2×C4—1722no—
413C2×C4#1306329611yescharacteristic
414C2×C4#1306329611yescharacteristic
415C2×C4#1306329611yescharacteristic
416C2×C4#1306334512yes—
417C2×C4#1306335012yes—
418C2×C4C2×D4×D4311yescharacteristic
419C2×C4C22×C22⋊D4311yescharacteristic
420C2×C4C2×#1062190011yescharacteristic
421C8—22no—
422C8—22no—
423C8—1721no—
424C8—1721no—
425C8—1722no—
426C8—1721no—
427C8—1721no—
428C22—21no—
429C22—21no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—1722no—
434C22—1722no—
435C22—1722no—
436C22#1458311112yes—
437C22C2×#1306329611yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—82no—
443C4—1722no—
444C4#1458311112yes—
445C4C2×#1306334511yescharacteristic
446C4C2×#1306335011yescharacteristic
447C4#1458370811yescharacteristic
448C4#1458371511yescharacteristic
449C2—22no—
450C2—161no—
451C2—3442no—
452C2C2×#1458311111yescharacteristic
453C2#1914428011yescharacteristic
454C2#1914431911yescharacteristic
subgroupG/H
centerC22C2×#13063296
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3141C2
socleC22×C43C2×#121
prime radical#19145902C2
hypercenter#3141D43
hypocenterC43#19236
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
2. #19145723C2semidirect
3. #19143341C2semidirect
4. #19134167C2semidirect
5. #19134125C2semidirect
6. #19134043C2semidirect
7. #19134001C2semidirect
8. #19133347C2semidirect
9. C2×#14582648C2semidirect
10. #19133244C2semidirect
11. C43×#3141C2semidirect
224. #3141D43semidirect
274. C43#19236semidirect

Maximal subgroups

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

subgroupindex
#191459022
#191457232
#191433412
#191341672
#191341252
#191340432
#191340012
#191333472
C2×#145826482
#191332442
C43×#31412
#1923643