Finite Group Databasebeta

← #25253490

Subgroups — #25253490

Order 11008.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19146959C211yesmaximal, characteristic, complemented
2#19146336C211yesmaximal, characteristic
3#19136828C211yesmaximal, characteristic, complemented
4#19136801C211yesmaximal, characteristic, complemented
5#19136775C211yesmaximal, characteristic, complemented
6#19136757C211yesmaximal, characteristic, complemented
7#19136691C211yesmaximal, characteristic, complemented
8#19136685C211yesmaximal, characteristic, complemented
9#19136660C211yesmaximal, characteristic, complemented
10#19136538C211yesmaximal, characteristic, complemented
11#19136529C211yesmaximal, characteristic, complemented
12#19136451C211yesmaximal, characteristic, complemented
13C16×C22⋊D43C211yesmaximal, characteristic, complemented
14#19136301C211yesmaximal, characteristic, complemented
15C2×C43×#358C211yesmaximal, characteristic, complemented
16C2×#13063348C2211yescharacteristic
17#14583584—21no—
18#14583560—21no—
19C2×#13063271C2211yescharacteristic
20C2×#13063271C2211yescharacteristic
21C2×#13063270C2211yescharacteristic
22C2×C16×D43C2211yescharacteristic
23#14583300C2211yescharacteristic
24#14583284C2211yescharacteristic
25#14583242—21no—
26#14583229—21no—
27#14583205—21no—
28#14583141C2211yescharacteristic
29#14583140C2211yescharacteristic
30#14583137C2211yescharacteristic
31#14583135C2211yescharacteristic
32#14583123—21no—
33#14583060—21no—
34#14583038—21no—
35Q16×Dic43C2211yescharacteristic
36#14582993C2211yescharacteristic
37C8×C22⋊D43C2211yescharacteristic
38#14582936C2211yescharacteristic
39#14582727C2211yescharacteristic
40#14582724C2211yescharacteristic
41#14582724C2211yescharacteristic
42Dic43⋊C16C2211yescharacteristic
43#14582704C2211yescharacteristic
44#14582703C2211yescharacteristic
45#14582703C2211yescharacteristic
46#14582701C2211yescharacteristic
47#14582699C2211yescharacteristic
48#14582699C2211yescharacteristic
49C22⋊(C43⋊C16)C2211yescharacteristic
50(C2×D43)⋊C16C2211yescharacteristic
51C16×Dic43C2211yescharacteristic
52C22×Q16×C43C2211yescharacteristic
53C22×C16×C43C2211yescharacteristic
54C2×#107×C43C2211yescharacteristic
55C43×#358C2211yescharacteristic
56C43×#358C2211yescharacteristic
57C43×#358C2211yescharacteristic
58C43×#358C2211yescharacteristic
59C2×#10621902—21no—
60#13063348—22no—
61#13063348—41no—
62C2×Dic172C2311yescharacteristic
63C2×#10621884—21no—
64C2×#10621884—21no—
65C2×D344C2311yescharacteristic
66C2×C8×D43C2311yescharacteristic
67#13063320—41no—
68#13063316—21no—
69#13063315—41no—
70#13063314—21no—
71#13063314—41no—
72#13063308—41no—
73#13063305—21no—
74#13063304—21no—
75#13063301—21no—
76#13063301—41no—
77#13063299—21no—
78#13063299—41no—
79Q8×Dic43—21no—
80#13063290—21no—
81C4×C22⋊D43—41no—
82C4×C22⋊D43D411yescharacteristic
83#13063284—41no—
84#13063283—21no—
85#13063271—22no—
86#13063271—41no—
87#13063270—41no—
88C2×C43⋊C16C2311yescharacteristic
89C2×C43⋊C16C2311yescharacteristic
90C16×D43—41no—
91#13063259C2311yescharacteristic
92#13063259C2311yescharacteristic
93#13063256—21no—
94#13063256—21no—
95Dic43⋊C8D411yescharacteristic
96#13063250—21no—
97#13063249—21no—
98#13063249—21no—
99#13063248—21no—
100#13063248—21no—
101#13063247—21no—
102C22⋊(C43⋊C8)D411yescharacteristic
103(C2×D43)⋊C8D411yescharacteristic
104C8×Dic43C2311yescharacteristic
105C22×Q8×C43—21no—
106C2×Q16×C43—22no—
107C2×Q16×C43—21no—
108C2×Q16×C43—41no—
109C2×Q16×C43C2311yescharacteristic
110C2×Q16×C43C2311yescharacteristic
111C22×C8×C43C2311yescharacteristic
112C2×C4⋊C4×C43—21no—
113C2×C16×C43—41no—
114C2×C16×C43C2311yescharacteristic
115C2×C16×C43C2311yescharacteristic
116#107×C43—21no—
117#107×C43C2311yescharacteristic
118#107×C43C2311yescharacteristic
119C2×Q8×C43—21no—
120C2×Q8×C43—21no—
121C2×Q8×C43—42no—
122C2×Q8×C43—41no—
123C2×Q8×C43—81no—
124C22×C4×C43—41no—
125C22×C4×C43—41no—
126C22×C4×C43C2×D411yescharacteristic
127C2×C22⋊D43—41no—
128C2×C22⋊D43D811yescharacteristic
129#10621902—42no—
130#10621902—81no—
131C2×D172—21no—
132C2×D172—41no—
133C2×C4×D43—41no—
134C2×C4×D43C2×D411yescharacteristic
135C2×Dic86—21no—
136Q16×C43—21no—
137Q16×C43—21no—
138Q16×C43—21no—
139Q16×C43—42no—
140Q16×C43#4112yes—
141C2×C8×C43—21no—
142C2×C8×C43—21no—
143C2×C8×C43C2×D411yescharacteristic
144C2×C8×C43C2411yescharacteristic
145C4⋊C4×C43—21no—
146C4⋊C4×C43—21no—
147C4⋊C4×C43—41no—
148C22⋊Dic43—41no—
149C22⋊Dic43D811yescharacteristic
150#10621884—42no—
151#10621884—81no—
152(C2×D43)⋊C4—41no—
153(C2×D43)⋊C4—41no—
154(C2×D43)⋊C4—41no—
155(C2×D43)⋊C4—41no—
156(C2×D43)⋊C4—41no—
157(C2×D43)⋊C4—41no—
158(C2×D43)⋊C4D811yescharacteristic
159C4⋊Dic43—21no—
160C4⋊Dic43—21no—
161C4⋊Dic43—41no—
162C4⋊Dic43—41no—
163C4⋊Dic43—41no—
164Dic43⋊C4—41no—
165Dic43⋊C4—41no—
166Dic43⋊C4—41no—
167Dic43⋊C4—41no—
168Dic43⋊C4D811yescharacteristic
169C4×Dic43—41no—
170C4×Dic43—41no—
171C4×Dic43—41no—
172C4×Dic43C2×D411yescharacteristic
173C2×C43⋊C8C2×D411yescharacteristic
174C2×C43⋊C8C2×D411yescharacteristic
175Dic172—21no—
176Dic172—21no—
177D344—21no—
178D344—21no—
179C8×D43—21no—
180C8×D43—21no—
181C16×C43—21no—
182C16×C43—21no—
183C43⋊C16—21no—
184C43⋊C16—21no—
185C23×C43C2×D811yescharacteristic
186C22×D43—41no—
187C22×D43C2×D811yescharacteristic
188Q8×C43—22no—
189Q8×C43—41no—
190Q8×C43—41no—
191Q8×C43—41no—
192Q8×C43—82no—
193C2×C4×C43—21no—
194C2×C4×C43—21no—
195C2×C4×C43—41no—
196C2×C4×C43—41no—
197C2×C4×C43—41no—
198C2×C4×C43—41no—
199C2×C4×C43—82no—
200C2×C4×C43—82no—
201C2×C4×C43C2×D811yescharacteristic
202C2×C4×C43C22×D411yescharacteristic
203C22⋊D43—22no—
204C22⋊D43—82no—
205C2×Dic43—41no—
206C2×Dic43—41no—
207C2×Dic43—41no—
208C2×Dic43C2×D811yescharacteristic
209C2×Dic43C2×D811yescharacteristic
210C2×Dic43C2×D811yescharacteristic
211D172—41no—
212D172—41no—
213D172—82no—
214C4×D43—21no—
215C4×D43—21no—
216C4×D43—82no—
217Dic86—41no—
218Dic86—41no—
219C8×C43—21no—
220C8×C43C2×#4111yescharacteristic
221C8×C43H(5,𝔽2)11yescharacteristic
222C43⋊C8—21no—
223C43⋊C8—21no—
224#33867—431nomaximal, Sylow
225C22×C43—21no—
226C22×C43—21no—
227C22×C43#50812yes—
228C22×C43C22×D811yescharacteristic
229C2×D43—21no—
230C2×D43—21no—
231C2×D43—21no—
232C2×D43—81no—
233C2×D43—81no—
234C2×D43—81no—
235C4×C43—21no—
236C4×C43—42no—
237C4×C43—81no—
238C4×C43—81no—
239C4×C43—81no—
240C4×C43#54611yescharacteristic
241C4×C43#57711yescharacteristic
242Dic43—21no—
243Dic43—21no—
244Dic43—21no—
245Dic43—81no—
246Dic43—81no—
247Dic43—81no—
248#3336—431no—
249#3285—431no—
250#2242—431no—
251#2237—431no—
252#2232—431no—
253#2229—431no—
254#2217—431no—
255#2216—431no—
256#2211—431no—
257#2189—431no—
258#2188—431no—
259C4×Q32—431no—
260D4×C16—431no—
261#2138—431no—
262C2×#358D4311yescharacteristic, complemented
263C2×C43—22no—
264C2×C43#329411yescharacteristic
265C2×C43C2×#50811yescharacteristic
266C2×C43#341711yescharacteristic
267D43—41no—
268D43—161no—
269C2×#135—431no—
270C22×Q16C2×D4311yescharacteristic
271#552—861no—
272#548—861no—
273C2×Q32—431no—
274C2×Q32—431no—
275C2×SD32—431no—
276C22×C16—431no—
277C22×C16C2×D4311yescharacteristic
278#499—431no—
279#495—431no—
280#485—861no—
281#483—861no—
282#479—861no—
283#467—431no—
284#467—431no—
285#467—431no—
286#466—431no—
287#464—861no—
288#452—861no—
289#448—861no—
290C4×Q16—431no—
291C4×Q16—431no—
292C8×D4—431no—
293#427—431no—
294C2×#107C2×D4311yescharacteristic
295#367—431no—
296#366—431no—
297#366—431no—
298C4⋊C16—431no—
299#358—431no—
300#358—431no—
301#358—431no—
302#358C2×D4311yescharacteristic
303#358C2×D4311yescharacteristic
304#358C2×D4311yescharacteristic
305#358C2×D4311yescharacteristic
306#357—431no—
307#357—431no—
308#357—431no—
309C22⋊C16—431no—
310C22⋊C16—431no—
311C4×C16—431no—
312C43#3386711yescharacteristic, complemented, Sylow
313C2×#41—861no—
314C22×Q8—21no—
315#135—862no—
316#135—1721no—
317C2×Q16—22no—
318C2×Q16—21no—
319C2×Q16—41no—
320C2×Q16—431no—
321C2×Q16C22×D4311yescharacteristic
322C2×Q16C22×D4311yescharacteristic
323C2×SD16—861no—
324C2×SD16—861no—
325C2×D8—431no—
326C22×C8—431no—
327C22×C8C22×D4311yescharacteristic
328#126—1721no—
329#125—861no—
330#125—1721no—
331#124—861no—
332#124—1721no—
333#123—1721no—
334C22⋊Q8—861no—
335C22⋊Q8—861no—
336C22⋊Q8—861no—
337C22⋊Q8—1721no—
338#121—861no—
339#121—1721no—
340C4×Q8—861no—
341C4×Q8—861no—
342C4×D4—431no—
343C4×D4—1721no—
344#117—861no—
345#117—1721no—
346C2×C4⋊C4—21no—
347Q32—862no—
348Q32—1721no—
349SD32—1721no—
350C2×C16—41no—
351C2×C16—431no—
352C2×C16—431no—
353C2×C16—1721no—
354C2×C16C22×D4311yescharacteristic
355C2×C16C22×D4311yescharacteristic
356#107—21no—
357#107—431no—
358#107—431no—
359#107C22×D4311yescharacteristic
360#107C22×D4311yescharacteristic
361#106—861no—
362#106—861no—
363C4⋊C8—431no—
364#103—861no—
365#103—861no—
366#103—861no—
367#103—861no—
368#103—861no—
369#102—861no—
370C22⋊C8—431no—
371C22⋊C8—431no—
372C4×C8—431no—
373#41—3441no—
374#41—1722no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—42no—
378C2×Q8—41no—
379C2×Q8—81no—
380C2×Q8—861no—
381C2×D4—431no—
382C2×D4—861no—
383C2×D4—1721no—
384C2×D4—1721no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—431no—
388C22×C4—1721no—
389C22×C4D4×D4311yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—21no—
393Q16—42no—
394Q16—861no—
395Q16—861no—
396Q16#1062189812yes—
397SD16—3441no—
398SD16—1722no—
399D8—861no—
400D8—861no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—431no—
404C2×C8—431no—
405C2×C8—861no—
406C2×C8—861no—
407C2×C8D4×D4311yescharacteristic
408C2×C8C23×D4311yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—41no—
412C4⋊C4—431no—
413C4⋊C4—861no—
414C4⋊C4—861no—
415C4⋊C4—1721no—
416C4⋊C4—1721no—
417C4⋊C4—1721no—
418C4⋊C4—1721no—
419C4⋊C4—1721no—
420C4⋊C4—1721no—
421C4⋊C4—1721no—
422C22⋊C4—431no—
423C22⋊C4—431no—
424C22⋊C4—1721no—
425C22⋊C4—1721no—
426C22⋊C4—1721no—
427C22⋊C4—1721no—
428C22⋊C4—1721no—
429C22⋊C4—1721no—
430C22⋊C4—1721no—
431C42—431no—
432C42—1721no—
433C42—1721no—
434C42—1721no—
435C16—21no—
436C16—21no—
437C16—861no—
438C16—861no—
439C23—431no—
440C23—1721no—
441C23D8×D4311yescharacteristic
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—82no—
447Q8—1721no—
448Q8—1721no—
449D4—862no—
450D4—3442no—
451D4—3442no—
452D4—1721no—
453D4—1721no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—431no—
463C2×C4—431no—
464C2×C4—431no—
465C2×C4—861no—
466C2×C4—861no—
467C2×C4—3442no—
468C2×C4—1721no—
469C2×C4—1721no—
470C2×C4—1721no—
471C2×C4D8×D4311yescharacteristic
472C2×C4C2×D4×D4311yescharacteristic
473C8—21no—
474C8—861no—
475C8—861no—
476C8C2×#1062189811yescharacteristic
477C8#1306337011yescharacteristic
478C22—21no—
479C22—21no—
480C22—861no—
481C22—861no—
482C22—861no—
483C22—3441no—
484C22—3441no—
485C22—3441no—
486C22#1458333012yes—
487C22C2×D8×D4311yescharacteristic
488C4—21no—
489C4—42no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—861no—
494C4—861no—
495C4—861no—
496C4—3441no—
497C4—3441no—
498C4—3441no—
499C4#1458353911yescharacteristic
500C4#1458370511yescharacteristic
501C2—22no—
502C2—1721no—
503C2—6881no—
504C2#1914640411yescharacteristic
505C2C2×#1458333011yescharacteristic
506C2#1914774411yescharacteristic
subgroupG/H
centerC22C2×D8×D43
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingC2×C43×#358C2
socleC22×C43C22×D8
prime radical#19146336C2
hypercenterC2×#358D43
hypocenterC43#33867
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
1. #19146959C2semidirect
3. #19136828C2semidirect
4. #19136801C2semidirect
5. #19136775C2semidirect
6. #19136757C2semidirect
7. #19136691C2semidirect
8. #19136685C2semidirect
9. #19136660C2semidirect
10. #19136538C2semidirect
11. #19136529C2semidirect
12. #19136451C2semidirect
13. C16×C22⋊D43C2semidirect
14. #19136301C2semidirect
15. C2×C43×#358C2semidirect
262. C2×#358D43semidirect
312. C43#33867semidirect

Maximal subgroups

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

subgroupindex
#191469592
#191463362
#191368282
#191368012
#191367752
#191367572
#191366912
#191366852
#191366602
#191365382
#191365292
#191364512
C16×C22⋊D432
#191363012
C2×C43×#3582
#3386743