Finite Group Databasebeta

← #25082021

Subgroups — #25082021

Order 11008.

620 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145540C211yesmaximal, characteristic, complemented
2#19144411C211yesmaximal, characteristic
3#19144377C211yesmaximal, characteristic
4#19143704C211yesmaximal, characteristic
5#19134056C212yesmaximal, complemented
6#19134049C212yesmaximal, complemented
7#19133964C212yesmaximal, complemented
8#19133959C212yesmaximal, complemented
9#19133371C211yesmaximal, characteristic, complemented
10C2×#14582646C211yesmaximal, characteristic, complemented
11C43×#1550C211yesmaximal, characteristic, complemented
12C2×#13063325C2211yescharacteristic
13C4×#10621898C2211yescharacteristic
14#14583278C2212yes—
15#14583251C2212yes—
16#14583203—21no—
17#14583196—21no—
18#14583195—21no—
19#14583191C2211yescharacteristic
20#14583190—21no—
21#14583186C2211yescharacteristic
22#14583185C2212yes—
23#14583181C2211yescharacteristic
24#14583180C2212yes—
25#14583176C2211yescharacteristic
26#14583109—21no—
27#14583109—21no—
28#14583054—22no—
29#14583023C2212yes—
30#14582964C2212yes—
31C8×C22⋊D43—22no—
32#14582952C2211yescharacteristic
33#14582951—21no—
34#14582941—21no—
35#14582935—21no—
36C2×C4⋊(C43⋊C8)C2211yescharacteristic
37C4⋊C8×D43—21no—
38#14582901—21no—
39#14582893—21no—
40#14582651C2212yes—
41#14582651C2212yes—
42#14582646C2212yes—
43#14582646C2212yes—
44C43×#533C2211yescharacteristic
45C43×#423C2211yescharacteristic
46C2×C4⋊C8×C43C2211yescharacteristic
47C43×#333C2212yes—
48C43×#332C2212yes—
49C2×#10621898D411yescharacteristic
50C22×Dic86—21no—
51C2×#10621885—22no—
52C2×#10621885—22no—
53C22×C43⋊C8—21no—
54C2×C8×D43—22no—
55#13063325—21no—
56#13063325—21no—
57#13063325—21no—
58#13063325C2312yes—
59C4×Dic86D411yescharacteristic
60C4×Dic86C2311yescharacteristic
61C4×D172D411yescharacteristic
62C4×D172C2311yescharacteristic
63C4×C22⋊D43—21no—
64C4×C22⋊D43—21no—
65#13063284D411yescharacteristic
66#13063282—21no—
67C2×C4⋊Dic43—21no—
68C2×C4⋊Dic43—21no—
69C42×D43—21no—
70#13063259—22no—
71#13063258—21no—
72#13063258—21no—
73#13063257—22no—
74#13063256—21no—
75#13063256—21no—
76C4⋊(C43⋊C8)—21no—
77C4⋊(C43⋊C8)D411yescharacteristic
78C4⋊(C43⋊C8)D411yescharacteristic
79C4⋊(C43⋊C8)C2311yescharacteristic
80C4⋊(C43⋊C8)C2311yescharacteristic
81Dic43⋊C8—22no—
82#13063250—22no—
83#13063248—21no—
84#13063248—21no—
85#13063248—22no—
86#13063248—22no—
87#13063247—22no—
88#13063245—21no—
89#13063245—21no—
90C22⋊(C43⋊C8)D412yes—
91(C2×D43)⋊C8—22no—
92C4×C43⋊C8C2312yes—
93C8×Dic43—22no—
94C2×M16×C43—21no—
95C22×C8×C43—21no—
96C4⋊Q8×C43C2311yescharacteristic
97C4⋊Q8×C43C2311yescharacteristic
98#125×C43—21no—
99C22⋊Q8×C43—21no—
100C22⋊Q8×C43—21no—
101C4×Q8×C43—21no—
102C4×Q8×C43—21no—
103#117×C43—21no—
104C2×C42×C43C2311yescharacteristic
105C4⋊C8×C43—21no—
106C4⋊C8×C43—21no—
107C4⋊C8×C43C2312yes—
108C4⋊C8×C43C2312yes—
109C2×Q8×C43—21no—
110C2×Q8×C43—21no—
111C22×C4×C43C2×D411yescharacteristic
112C22×C4×C43C2×D411yescharacteristic
113C22×C4×C43#4111yescharacteristic
114C2×C22⋊D43—21no—
115C22×Dic43—41no—
116#10621898—22no—
117#10621898Q1614yescomplemented
118C2×D172C2×D411yescharacteristic
119C2×C4×D43—21no—
120C2×C4×D43—21no—
121C2×C4×D43—21no—
122C2×Dic86—22no—
123C2×Dic86—41no—
124C2×Dic86—81no—
125C2×Dic86—81no—
126C2×Dic86C2×D411yescharacteristic
127M16×C43—42no—
128C2×C8×C43—22no—
129C2×C8×C43—22no—
130C2×C8×C43—81no—
131C4⋊C4×C43—21no—
132C4⋊C4×C43—21no—
133C4⋊C4×C43—21no—
134C4⋊C4×C43—21no—
135C4⋊C4×C43—41no—
136C4⋊C4×C43—41no—
137C22⋊C4×C43—41no—
138C42×C43—41no—
139C42×C43C2×D411yescharacteristic
140C42×C43C2×D411yescharacteristic
141C42×C43#4111yescharacteristic
142C42×C43C2411yescharacteristic
143C22⋊Dic43—21no—
144#10621885—82no—
145#10621885—82no—
146(C2×D43)⋊C4—21no—
147(C2×D43)⋊C4—21no—
148C4⋊Dic43—22no—
149C4⋊Dic43—22no—
150C4⋊Dic43—41no—
151C4⋊Dic43—41no—
152C4⋊Dic43C2×D411yescharacteristic
153C4⋊Dic43C2×D411yescharacteristic
154Dic43⋊C4—21no—
155Dic43⋊C4—21no—
156C4×Dic43—21no—
157C4×Dic43—21no—
158C4×Dic43—21no—
159C2×C43⋊C8—21no—
160C2×C43⋊C8—21no—
161C2×C43⋊C8—42no—
162C2×C43⋊C8C2×D412yes—
163C2×C43⋊C8C2×D412yes—
164C8×D43—82no—
165C23×C43#12111yescharacteristic
166C22×D43—21no—
167Q8×C43—41no—
168Q8×C43—41no—
169Q8×C43—81no—
170C2×C4×C43—21no—
171C2×C4×C43—21no—
172C2×C4×C43—41no—
173C2×C4×C43—41no—
174C2×C4×C43—41no—
175C2×C4×C43#12111yescharacteristic
176C2×C4×C43#12111yescharacteristic
177C2×C4×C43#12111yescharacteristic
178C2×C4×C43C2×Q1612yes—
179C2×C4×C43#13512yes—
180C2×C4×C43C22×D411yescharacteristic
181C2×C4×C43C22×D411yescharacteristic
182C2×C4×C43C2×#4111yescharacteristic
183C22⋊D43—24no—
184C2×Dic43—21no—
185C2×Dic43—21no—
186C2×Dic43—21no—
187C2×Dic43—42no—
188C2×Dic43—82no—
189D172—21no—
190D172C2×Q1612yes—
191C4×D43—24no—
192C4×D43—41no—
193C4×D43—41no—
194C4×D43—81no—
195Dic86—21no—
196Dic86—42no—
197Dic86—42no—
198Dic86—82no—
199Dic86C2×Q1612yes—
200C8×C43—42no—
201C8×C43—42no—
202C43⋊C8—22no—
203C43⋊C8—22no—
204C43⋊C8—41no—
205C43⋊C8—41no—
206#19411—431nomaximal, Sylow
207C22×C43—21no—
208C22×C43—21no—
209C22×C43#46712yes—
210C22×C43C2×#12111yescharacteristic
211C2×D43—22no—
212C2×D43—41no—
213C2×D43—41no—
214C4×C43—21no—
215C4×C43—21no—
216C4×C43—21no—
217C4×C43—81no—
218C4×C43—81no—
219C4×C43#46712yes—
220C4×C43C22×Q1611yescharacteristic
221C4×C43C2×#13511yescharacteristic
222C4×C43#57611yescharacteristic
223C4×C43#57711yescharacteristic
224Dic43—22no—
225Dic43—41no—
226Dic43—41no—
227Dic43—82no—
228#3209—431no—
229#3079—431no—
230#3076—431no—
231#2988—431no—
232#1684—432no—
233#1683—432no—
234#1670—432no—
235#1669—432no—
236#1571—431no—
237#1550D4311yescharacteristic, complemented
238C2×#332—431no—
239C2×C43—22no—
240C2×C43#304511yescharacteristic
241C2×C43C2×#46711yescharacteristic
242C2×C43#306911yescharacteristic
243D43—42no—
244#533C2×D4311yescharacteristic
245C2×C4⋊Q8—431no—
246C4×#41—431no—
247#494—432no—
248#487—432no—
249#479—861no—
250#478—861no—
251#477—431no—
252#477—861no—
253#476—431no—
254#476—861no—
255#475—431no—
256#475—432no—
257#474—431no—
258#474—432no—
259#462—861no—
260#462—861no—
261#451—862no—
262C8×Q8—432no—
263C8×D4—432no—
264C8×D4—862no—
265#431—431no—
266#431—861no—
267#428—861no—
268#427—861no—
269#423C2×D4311yescharacteristic
270C2×C4⋊C8—431no—
271C2×C4⋊C8—861no—
272C2×C4⋊C8C2×D4311yescharacteristic
273#417—861no—
274#416—861no—
275#335—432no—
276#335—432no—
277#333C2×D4312yes—
278#332—432no—
279#332—432no—
280#332C2×D4312yes—
281C43#1941111yescharacteristic, complemented, Sylow
282C2×#41—431no—
283C22×Q8—861no—
284C2×Q16—862no—
285C2×Q16—862no—
286C2×M16—21no—
287C22×C8—21no—
288C22×C8—862no—
289C22×C8—861no—
290C4⋊Q8—432no—
291C4⋊Q8—861no—
292C4⋊Q8—861no—
293C4⋊Q8—861no—
294C4⋊Q8C22×D4311yescharacteristic
295C4⋊Q8C22×D4311yescharacteristic
296#125—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—431no—
302C4×Q8—431no—
303C4×D4—431no—
304C4×D4—431no—
305C4×D4—861no—
306C4×D4—861no—
307#117—21no—
308#117—431no—
309#117—861no—
310C2×C4⋊C4—861no—
311C2×C4⋊C4—861no—
312C2×C42—861no—
313C2×C42C22×D4311yescharacteristic
314#107—861no—
315#107—861no—
316#107—862no—
317#106—861no—
318#106—861no—
319#106—862no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—431no—
323C4⋊C8—431no—
324C4⋊C8—431no—
325C4⋊C8—431no—
326C4⋊C8—862no—
327C4⋊C8—861no—
328C4⋊C8C22×D4312yes—
329C4⋊C8C22×D4312yes—
330#103—862no—
331#103—862no—
332#103—861no—
333#103—861no—
334#103—862no—
335#102—861no—
336#102—861no—
337#102—862no—
338C22⋊C8—432no—
339C22⋊C8—862no—
340C4×C8—432no—
341C4×C8—862no—
342#41—434no—
343#41—862no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—431no—
347C2×Q8—862no—
348C2×Q8—3441no—
349C2×Q8—3441no—
350C2×Q8—1721no—
351C2×D4—431no—
352C2×D4—861no—
353C22×C4—861no—
354C22×C4—861no—
355C22×C4—861no—
356C22×C4—1721no—
357C22×C4#1062189811yescharacteristic
358C22×C4D4×D4311yescharacteristic
359C22×C4D4×D4311yescharacteristic
360Q16—3442no—
361Q16—3442no—
362M16—42no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—81no—
366C2×C8—432no—
367C2×C8—432no—
368C2×C8—861no—
369C2×C8—861no—
370C2×C8—3442no—
371C2×C8—1722no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—431no—
379C4⋊C4—431no—
380C4⋊C4—861no—
381C4⋊C4—861no—
382C4⋊C4—862no—
383C4⋊C4—862no—
384C4⋊C4—1721no—
385C4⋊C4—1721no—
386C22⋊C4—41no—
387C22⋊C4—861no—
388C22⋊C4—861no—
389C22⋊C4—861no—
390C42—41no—
391C42—861no—
392C42—861no—
393C42—861no—
394C42#1062189811yescharacteristic
395C42D4×D4311yescharacteristic
396C42D4×D4311yescharacteristic
397C42C23×D4311yescharacteristic
398C23—861no—
399C23#1306329811yescharacteristic
400Q8—41no—
401Q8—41no—
402Q8—81no—
403Q8—432no—
404Q8—861no—
405Q8—3442no—
406Q8—1722no—
407Q8—1722no—
408D4—432no—
409D4—864no—
410D4—861no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—861no—
417C2×C4—864no—
418C2×C4—861no—
419C2×C4—861no—
420C2×C4—3441no—
421C2×C4—3442no—
422C2×C4—1722no—
423C2×C4—1721no—
424C2×C4—1721no—
425C2×C4#1306329811yescharacteristic
426C2×C4#1306329811yescharacteristic
427C2×C4#1306329811yescharacteristic
428C2×C4Q16×D4312yes—
429C2×C4#1306334612yes—
430C2×C4C2×D4×D4311yescharacteristic
431C2×C4C2×D4×D4311yescharacteristic
432C2×C4C2×#1062189811yescharacteristic
433C8—42no—
434C8—42no—
435C8—862no—
436C8—862no—
437C8—1721no—
438C8—1721no—
439C22—21no—
440C22—21no—
441C22—862no—
442C22—1721no—
443C22—1721no—
444C22#1458313912yes—
445C22C2×#1306329811yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—81no—
450C4—81no—
451C4—862no—
452C4—3442no—
453C4—1721no—
454C4—1721no—
455C4#1458313912yes—
456C4C2×Q16×D4311yescharacteristic
457C4C2×#1306334611yescharacteristic
458C4#1458370311yescharacteristic
459C4#1458370811yescharacteristic
460C2—22no—
461C2—1722no—
462C2#1914413311yescharacteristic
463C2C2×#1458313911yescharacteristic
464C2#1914433511yescharacteristic
subgroupG/H
centerC22C2×#13063298
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#1550C2
socleC22×C43C2×#121
prime radicalC2×#10621898D4
hypercenter#1550D43
hypocenterC43#19411
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #19145540C2semidirect
5. #19134056C2semidirect
6. #19134049C2semidirect
7. #19133964C2semidirect
8. #19133959C2semidirect
9. #19133371C2semidirect
10. C2×#14582646C2semidirect
11. C43×#1550C2semidirect
117. #10621898Q16semidirect
237. #1550D43semidirect
281. C43#19411semidirect

Maximal subgroups

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

subgroupindex
#191455402
#191444112
#191443772
#191437042
#191340562
#191340492
#191339642
#191339592
#191333712
C2×#145826462
C43×#15502
#1941143