Finite Group Databasebeta

← #25245317

Subgroups — #25245317

Order 11008.

3940 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 3940 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063370C211yesmaximal, characteristic
2C2×#14583076C214yesmaximal, complemented
3#19143853C212yesmaximal, complemented
4#19134548C214yesmaximal, complemented
5#19134540C212yesmaximal, complemented
6#19134516C211yesmaximal, characteristic, complemented
7C43×#2435C211yesmaximal, characteristic, complemented
8C2×#13063370—216no—
9C2×#13063370—24no—
10C2×#13063370—22no—
11C2×#13063370C2214yes—
12C22×#10621898C2212yes—
13C22×#10621900—22no—
14C23×C22⋊D43—22no—
15C22×D4×D43—22no—
16C2×#13063356C2214yescomplemented
17#14583076—216no—
18#14583059—24no—
19#14583054—22no—
20#14583049—22no—
21C2×#13063253C2218yescomplemented
22#14582898C2214yescomplemented
23C2×#13063242C2214yescomplemented
24#14582863C2212yescomplemented
25C23×D4×C43C2211yescharacteristic
26C2×#124×C43C2212yescomplemented
27C2×C22⋊Q8×C43C2211yescharacteristic, complemented
28C43×#394—22no—
29C43×#390C2212yescomplemented
30C43×#379—22no—
31C4×C22⋊C4×C43C2211yescharacteristic, complemented
32C24×D43—22no—
33#13063370—232no—
34#13063370—232no—
35#13063370—232no—
36#13063370—28no—
37#13063370—216no—
38#13063370D4116yes—
39C2×#10621898—22no—
40C2×#10621898—22no—
41C2×#10621898—216no—
42C2×#10621898D414yes—
43C2×#10621898C2314yes—
44C2×#10621900—28no—
45C2×#10621900—232no—
46C2×#10621900—28no—
47C22×Dic86D412yes—
48C22×D172D412yes—
49C22×C22⋊D43—24no—
50C22×C22⋊D43—232no—
51C22×C22⋊D43—28no—
52C22×C22⋊D43—22no—
53C22×C22⋊D43—28no—
54C2×D4×D43—232no—
55C2×D4×D43—28no—
56C2×D4×D43—28no—
57C23×Dic43—22no—
58C22×C4×D43—22no—
59#13063356—216no—
60C2×#10621885—24no—
61C2×#10621884—24no—
62C2×#10621877C2314yescomplemented
63#13063253—216no—
64#13063248—24no—
65#13063245—24no—
66#13063242—28no—
67C22⋊(C43⋊C8)—22no—
68C22⋊(C43⋊C8)—22no—
69C25×C43D412yes—
70C22×Q8×C43C2311yescharacteristic, complemented
71C22×D4×C43—28no—
72C22×D4×C43—22no—
73C22×D4×C43—22no—
74C22×D4×C43D412yes—
75C22×D4×C43C2312yes—
76C23×C4×C43C2311yescharacteristic
77#124×C43—28no—
78C22⋊Q8×C43—22no—
79C22⋊Q8×C43—22no—
80C2×C4⋊C4×C43—21no—
81C2×C4⋊C4×C43C2311yescharacteristic, complemented
82C2×C22⋊C4×C43—21no—
83C2×C22⋊C4×C43—24no—
84C2×C22⋊C4×C43—21no—
85C2×C22⋊C4×C43—24no—
86C2×C42×C43C2312yescomplemented
87#95×C43—21no—
88C24×C43—22no—
89C24×C43—28no—
90C24×C43—22no—
91C24×C43—44no—
92C24×C43—82no—
93C24×C43C2×D411yescharacteristic
94C24×C43C2×D414yes—
95C23×D43—22no—
96C23×D43—216no—
97C23×D43—28no—
98C23×D43—84no—
99C23×D43—84no—
100C2×Q8×C43—22no—
101C2×Q8×C43—24no—
102C2×D4×C43—216no—
103C2×D4×C43—216no—
104C2×D4×C43—24no—
105C2×D4×C43—24no—
106C2×D4×C43—24no—
107C2×D4×C43—28no—
108C2×D4×C43C2×D418yes—
109C22×C4×C43—24no—
110C22×C4×C43—22no—
111C22×C4×C43—21no—
112C22×C4×C43—21no—
113C22×C4×C43C2×D412yes—
114C22×C4×C43C2×D412yes—
115C22×C4×C43C2×D411yescharacteristic
116C22×C4×C43C2411yescharacteristic
117C2×C22⋊D43—232no—
118C2×C22⋊D43—216no—
119C2×C22⋊D43—216no—
120C2×C22⋊D43—264no—
121C2×C22⋊D43—232no—
122C2×C22⋊D43—24no—
123C2×C22⋊D43—232no—
124C2×C22⋊D43—24no—
125C22×Dic43—216no—
126C22×Dic43—28no—
127C22×Dic43—22no—
128#10621900—264no—
129#10621900—232no—
130#10621900—232no—
131D4×D43—264no—
132D4×D43—232no—
133D4×D43—232no—
134#10621898—216no—
135#10621898—216no—
136#10621898—28no—
137#10621898—28no—
138#10621898—28no—
139#10621898C2×D4116yes—
140C2×D172—22no—
141C2×D172—28no—
142C2×D172C2×D414yes—
143C2×C4×D43—216no—
144C2×C4×D43—24no—
145C2×C4×D43—24no—
146C2×Dic86—22no—
147C2×Dic86—28no—
148C2×Dic86C2×D414yes—
149C4⋊C4×C43—22no—
150C4⋊C4×C43—42no—
151C22⋊C4×C43—42no—
152C22⋊C4×C43—41no—
153C22⋊C4×C43—41no—
154C22⋊C4×C43—48no—
155C22⋊C4×C43—84no—
156C42×C43—24no—
157#10621885—48no—
158#10621884—48no—
159#10621877—28no—
160C2×C43⋊C8—24no—
161C23×C43—28no—
162C23×C43—28no—
163C23×C43—24no—
164C23×C43—22no—
165C23×C43—22no—
166C23×C43—24no—
167C23×C43—41no—
168C23×C43—44no—
169C23×C43—44no—
170C23×C43—44no—
171C23×C43—44no—
172C23×C43—82no—
173C23×C43—82no—
174C23×C43—816no—
175C23×C43#12011yescharacteristic
176C23×C43#12011yescharacteristic
177C23×C43#12018yes—
178C23×C43C22×D411yescharacteristic
179C22×D43—28no—
180C22×D43—24no—
181C22×D43—232no—
182C22×D43—44no—
183C22×D43—44no—
184C22×D43—832no—
185C22×D43—88no—
186C22×D43—88no—
187Q8×C43—44no—
188D4×C43—216no—
189D4×C43—216no—
190C2×C4×C43—22no—
191C2×C4×C43—24no—
192C2×C4×C43—24no—
193C2×C4×C43—42no—
194C2×C4×C43—42no—
195C2×C4×C43—42no—
196C2×C4×C43—42no—
197C2×C4×C43—42no—
198C2×C4×C43—41no—
199C2×C4×C43—41no—
200C2×C4×C43#12012yes—
201C2×C4×C43C22×D412yes—
202C2×C4×C43C22×D414yes—
203C22⋊D43—216no—
204C22⋊D43—264no—
205C22⋊D43—216no—
206C2×Dic43—28no—
207C2×Dic43—232no—
208C2×Dic43—24no—
209D172—28no—
210D172—24no—
211D172#12018yes—
212C4×D43—216no—
213C4×D43—216no—
214Dic86—28no—
215Dic86—24no—
216Dic86#12018yes—
217C43⋊C8—44no—
218#32891—431nomaximal, Sylow
219C22×C43—21no—
220C22×C43—21no—
221C22×C43—21no—
222C22×C43—21no—
223C22×C43—28no—
224C22×C43—21no—
225C22×C43—21no—
226C22×C43—21no—
227C22×C43—44no—
228C22×C43—48no—
229C22×C43—416no—
230C22×C43—44no—
231C22×C43—44no—
232C22×C43—84no—
233C22×C43—88no—
234C22×C43#45512yescomplemented
235C22×C43C2×#12012yes—
236C22×C43C2×#12011yescharacteristic
237C2×D43—28no—
238C2×D43—416no—
239C2×D43—44no—
240C2×D43—44no—
241C2×D43—832no—
242C4×C43—22no—
243C4×C43—82no—
244C4×C43—81no—
245C4×C43—81no—
246C4×C43C2×#12014yes—
247Dic43—28no—
248C22×H(5,𝔽2)—431no—
249C2×#455—434no—
250#3007—432no—
251#2435D4311yescharacteristic, complemented
252#1792—434no—
253#1789—432no—
254#1783—431no—
255C2×C43—22no—
256C2×C43—22no—
257C2×C43—48no—
258C2×C43—41no—
259C2×C43#284611yescharacteristic
260C2×C43C2×#45511yescharacteristic
261C2×C43#301911yescharacteristic
262D43—48no—
263C2×H(5,𝔽2)—434no—
264C2×H(5,𝔽2)—864no—
265C2×H(5,𝔽2)—8616no—
266C2×H(5,𝔽2)—862no—
267C22×#41—432no—
268C22×#41—862no—
269C23×D4—862no—
270C23×D4—862no—
271C23×D4C2×D4311yescharacteristic
272C2×#137—434no—
273C2×#124C2×D4312yescomplemented
274C2×C22⋊Q8C2×D4311yescharacteristic, complemented
275#455—8616no—
276#452—864no—
277#451—862no—
278#450—862no—
279C2×#104—438no—
280#417—434no—
281C2×#100—434no—
282#407—432no—
283#394—22no—
284#390C2×D4312yescomplemented
285#379—22no—
286C4×C22⋊C4C2×D4311yescharacteristic, complemented
287C43#3289111yescharacteristic, complemented, Sylow
288C25—862no—
289C25C22⋊D4312yes—
290H(5,𝔽2)—4316no—
291H(5,𝔽2)—8616no—
292H(5,𝔽2)—8632no—
293H(5,𝔽2)—8632no—
294H(5,𝔽2)—8632no—
295H(5,𝔽2)—868no—
296C2×#41—434no—
297C2×#41—434no—
298C2×#41—8616no—
299C2×#41—868no—
300C2×#41—8632no—
301C2×#41—868no—
302C2×#41—862no—
303C2×#41—862no—
304C22×Q8—432no—
305C22×Q8C22×D4311yescharacteristic, complemented
306C22×D4—28no—
307C22×D4—22no—
308C22×D4—22no—
309C22×D4—432no—
310C22×D4—868no—
311C22×D4—8632no—
312C22×D4—868no—
313C22×D4—864no—
314C22×D4—8632no—
315C22×D4—868no—
316C22×D4—862no—
317C22×D4—868no—
318C22×D4C22⋊D4312yes—
319C22×D4C22×D4312yes—
320C23×C4—862no—
321C23×C4—862no—
322C23×C4C22×D4311yescharacteristic
323#137—8616no—
324C2×Q16—864no—
325C2×SD16—864no—
326C2×M16—434no—
327#124—28no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4C22×D4311yescharacteristic, complemented
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C42C22×D4312yescomplemented
337#104—8616no—
338#103—864no—
339#102—864no—
340#100—868no—
341C22⋊C8—862no—
342C22⋊C8—862no—
343#95—21no—
344C24—22no—
345C24—28no—
346C24—22no—
347C24—44no—
348C24—82no—
349C24—8616no—
350C24—868no—
351C24—862no—
352C24—3444no—
353C24—3444no—
354C24C2×C22⋊D4311yescharacteristic
355C24C2×C22⋊D4314yes—
356#41—4316no—
357#41—868no—
358#41—868no—
359#41—8616no—
360#41—8616no—
361#41—8632no—
362#41—8664no—
363#41—8632no—
364#41—868no—
365C2×Q8—22no—
366C2×Q8—24no—
367C2×Q8—434no—
368C2×Q8—868no—
369C2×Q8—862no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—28no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—434no—
377C2×D4—868no—
378C2×D4—8632no—
379C2×D4—8664no—
380C2×D4—8632no—
381C2×D4—8632no—
382C2×D4—8616no—
383C2×D4—8616no—
384C2×D4—8664no—
385C2×D4—8632no—
386C2×D4—864no—
387C2×D4—8632no—
388C2×D4—864no—
389C2×D4—862no—
390C2×D4D4×D4318yes—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—8616no—
396C22×C4—8616no—
397C22×C4—868no—
398C22×C4—862no—
399C22×C4—864no—
400C22×C4—864no—
401C22×C4D4×D4312yes—
402C22×C4D4×D4312yes—
403C22×C4C2×C22⋊D4311yescharacteristic
404C22×C4C23×D4311yescharacteristic
405Q16—1728no—
406SD16—1728no—
407M16—868no—
408C2×C8—864no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—48no—
415C22⋊C4—84no—
416C42—24no—
417C23—28no—
418C23—28no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—24no—
423C23—41no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—82no—
429C23—82no—
430C23—816no—
431C23—8632no—
432C23—868no—
433C23—864no—
434C23—34432no—
435C23—3448no—
436C23—3448no—
437C23—1724no—
438C23—1724no—
439C23#1306329311yescharacteristic
440C23#1306329311yescharacteristic
441C23#1306329318yes—
442C23C22×C22⋊D4311yescharacteristic
443Q8—44no—
444Q8—438no—
445Q8—868no—
446Q8—864no—
447D4—216no—
448D4—216no—
449D4—438no—
450D4—868no—
451D4—8616no—
452D4—8664no—
453D4—8616no—
454D4—864no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—8616no—
466C2×C4—868no—
467C2×C4—8632no—
468C2×C4—864no—
469C2×C4—8616no—
470C2×C4#1306329312yes—
471C2×C4C2×D4×D4314yes—
472C2×C4C2×D4×D4312yes—
473C8—1724no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—28no—
481C22—21no—
482C22—44no—
483C22—48no—
484C22—416no—
485C22—44no—
486C22—44no—
487C22—84no—
488C22—88no—
489C22—868no—
490C22—34432no—
491C22—17216no—
492C22—1724no—
493C22—1724no—
494C22#1458307612yescomplemented
495C22C2×#1306329312yes—
496C22C2×#1306329311yescharacteristic
497C4—22no—
498C4—82no—
499C4—81no—
500C4—81no—
501C4—868no—
502C4#120×D4314yes—
503C2—22no—
504C2—22no—
505C2—41no—
506C2—48no—
507C2—1728no—
508C2#1914287111yescharacteristic
509C2C2×#1458307611yescharacteristic
510C2#1914395111yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2435C2
socleC22×C43C2×#120
prime radicalC22×#13063370C2
hypercenter#2435D43
hypocenterC43#32891
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
2. C2×#14583076C2semidirect
3. #19143853C2semidirect
4. #19134548C2semidirect
5. #19134540C2semidirect
6. #19134516C2semidirect
7. C43×#2435C2semidirect
16. C2×#13063356C22semidirect
21. C2×#13063253C22semidirect
22. #14582898C22semidirect
23. C2×#13063242C22semidirect
24. #14582863C22semidirect
26. C2×#124×C43C22semidirect
27. C2×C22⋊Q8×C43C22semidirect
29. C43×#390C22semidirect
31. C4×C22⋊C4×C43C22semidirect
62. C2×#10621877C23semidirect
70. C22×Q8×C43C23semidirect
81. C2×C4⋊C4×C43C23semidirect
86. C2×C42×C43C23semidirect
234. C22×C43#455semidirect
251. #2435D43semidirect
273. C2×#124C2×D43semidirect
274. C2×C22⋊Q8C2×D43semidirect
284. #390C2×D43semidirect
286. C4×C22⋊C4C2×D43semidirect
287. C43#32891semidirect
305. C22×Q8C22×D43semidirect
331. C2×C4⋊C4C22×D43semidirect
336. C2×C42C22×D43semidirect
494. C22#14583076semidirect

Maximal subgroups

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

subgroupindex
C22×#130633702
C2×#145830762
#191438532
#191345482
#191345402
#191345162
C43×#24352
#3289143