Finite Group Databasebeta

← #25012164

Subgroups — #25012164

Order 11008.

976 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19136280C211yesmaximal, characteristic, complemented
2#19136260C211yesmaximal, characteristic, complemented
3#19136245C212yesmaximal, complemented
4#19134570C212yesmaximal, complemented
5C43×#3500C211yesmaximal, characteristic
6#14582916—22no—
7#14582908—22no—
8#14582883C2212yescomplemented
9#14582873C2212yescomplemented
10#14582698—21no—
11#14582696—21no—
12#14582694—21no—
13#14582692—21no—
14#14582686—24no—
15C2×2−1+4×C43C2211yescharacteristic
16C2×H(5,𝔽2)×C43C411yescharacteristic, complemented
17C43×#548—22no—
18C43×#546—22no—
19C43×#546—24no—
20C43×#536—21no—
21C43×#536C411yescharacteristic, complemented
22C43×#535—21no—
23C43×#535C2211yescharacteristic
24C43×#532—21no—
25C43×#532C2211yescharacteristic
26C2×#124×C43—21no—
27C2×#124×C43C411yescharacteristic, complemented
28C4×#41×C43C411yescharacteristic, complemented
29C2×#10621877—22no—
30#13063284—22no—
31#13063252—44no—
32#13063249—44no—
33#13063244—22no—
34#13063242—22no—
35#13063240—22no—
36#13063240—22no—
372−1+4×C43—42no—
382−1+4×C43—42no—
39H(5,𝔽2)×C43—42no—
40H(5,𝔽2)×C43—42no—
41C2×#41×C43—21no—
42C2×#41×C43—21no—
43C2×#41×C43—21no—
44C2×#41×C43—22no—
45C2×#41×C43C2×C411yescharacteristic
46C2×#41×C43C2×C411yescharacteristic
47C2×#41×C43D412yes—
48C2×#41×C43C2311yescharacteristic
49C22×Q8×C43—21no—
50C22×Q8×C43C2×C411yescharacteristic
51C22×Q8×C43D412yes—
52C22×D4×C43—22no—
53C22×D4×C43—21no—
54C22×D4×C43—21no—
55C22×D4×C43C2×C411yescharacteristic
56C4⋊Q8×C43—21no—
57C4⋊Q8×C43D411yescharacteristic
58#127×C43—21no—
59#127×C43D411yescharacteristic
60#124×C43—24no—
61#124×C43—22no—
62#124×C43—21no—
63#124×C43—21no—
64#124×C43—21no—
65#124×C43—21no—
66#124×C43D411yescharacteristic
67#124×C43D411yescharacteristic
68#123×C43—24no—
69#123×C43—22no—
70C22⋊Q8×C43—24no—
71C22⋊Q8×C43—22no—
72#121×C43—24no—
73#121×C43—24no—
74#121×C43—22no—
75#121×C43—22no—
76#120×C43—24no—
77#120×C43D414yes—
78C4×Q8×C43—21no—
79C4×D4×C43—24no—
80C4×D4×C43—21no—
81C4×D4×C43—22no—
82C4×D4×C43—22no—
83#117×C43—21no—
84#117×C43C2×C411yescharacteristic
85C2×C22⋊C4×C43—24no—
86C2×C22⋊C4×C43—22no—
87C2×C42×C43—21no—
88C2×C42×C43C2×C411yescharacteristic
89C24×C43—22no—
90C24×C43C22⋊C412yes—
91#41×C43—22no—
92#41×C43—22no—
93#41×C43—24no—
94#41×C43—41no—
95#41×C43—41no—
96#41×C43—42no—
97#41×C43—44no—
98#41×C43—44no—
99#41×C43—41no—
100#41×C43—41no—
101#41×C43—48no—
102#41×C43—42no—
103#41×C43—42no—
104C2×Q8×C43—21no—
105C2×Q8×C43—22no—
106C2×Q8×C43—24no—
107C2×Q8×C43—41no—
108C2×Q8×C43—41no—
109C2×Q8×C43—42no—
110C2×Q8×C43—44no—
111C2×Q8×C43C22⋊C412yes—
112C2×Q8×C43C22⋊C411yescharacteristic
113C2×Q8×C43C22⋊C411yescharacteristic
114C2×Q8×C43C2×D411yescharacteristic
115C2×D4×C43—22no—
116C2×D4×C43—22no—
117C2×D4×C43—21no—
118C2×D4×C43—21no—
119C2×D4×C43—21no—
120C2×D4×C43—21no—
121C2×D4×C43—22no—
122C2×D4×C43—22no—
123C2×D4×C43—24no—
124C2×D4×C43—41no—
125C2×D4×C43—41no—
126C2×D4×C43—44no—
127C2×D4×C43—42no—
128C2×D4×C43—48no—
129C2×D4×C43—42no—
130C2×D4×C43C22⋊C411yescharacteristic
131C2×D4×C43C22⋊C411yescharacteristic
132C2×D4×C43C2×D412yes—
133C2×D4×C43C2×D411yescharacteristic
134C22×C4×C43—21no—
135C22×C4×C43—22no—
136C22×C4×C43—22no—
137C22×C4×C43—21no—
138C22×C4×C43—22no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—22no—
142C22×C4×C43C22×C411yescharacteristic
143C22×C4×C43C2×D412yes—
144C4⋊C4×C43—22no—
145C4⋊C4×C43—24no—
146C4⋊C4×C43—21no—
147C4⋊C4×C43—22no—
148C22⋊C4×C43—24no—
149C22⋊C4×C43—24no—
150C22⋊C4×C43—24no—
151C22⋊C4×C43—22no—
152C22⋊C4×C43—22no—
153C22⋊C4×C43—24no—
154C22⋊C4×C43—22no—
155C22⋊C4×C43C22⋊C412yes—
156C42×C43—22no—
157C42×C43—21no—
158C42×C43—21no—
159C42×C43C22⋊C411yescharacteristic
160C42×C43C22⋊C411yescharacteristic
161C22⋊Dic43—42no—
162C4⋊Dic43—42no—
163C4×Dic43—42no—
164#10621877—42no—
165#10621877—42no—
166C2×C43⋊C8—42no—
167C23×C43—21no—
168C23×C43—22no—
169C23×C43—21no—
170C23×C43—44no—
171C23×C43—42no—
172C23×C43—82no—
173C23×C43—84no—
174C23×C43C2×C22⋊C411yescharacteristic
175C23×C43#12012yes—
176Q8×C43—21no—
177Q8×C43—22no—
178Q8×C43—22no—
179Q8×C43—41no—
180Q8×C43—44no—
181Q8×C43—41no—
182Q8×C43—42no—
183Q8×C43#9912yes—
184D4×C43—24no—
185D4×C43—21no—
186D4×C43—42no—
187D4×C43—41no—
188D4×C43—41no—
189D4×C43—42no—
190D4×C43—44no—
191D4×C43—44no—
192D4×C43—44no—
193D4×C43—42no—
194D4×C43—41no—
195D4×C43—41no—
196D4×C43—82no—
197D4×C43—84no—
198D4×C43#9912yes—
199C2×C4×C43—24no—
200C2×C4×C43—21no—
201C2×C4×C43—22no—
202C2×C4×C43—22no—
203C2×C4×C43—21no—
204C2×C4×C43—21no—
205C2×C4×C43—21no—
206C2×C4×C43—22no—
207C2×C4×C43—21no—
208C2×C4×C43—22no—
209C2×C4×C43—21no—
210C2×C4×C43—44no—
211C2×C4×C43—44no—
212C2×C4×C43—44no—
213C2×C4×C43—42no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—81no—
217C2×C4×C43—82no—
218C2×C4×C43C2×C22⋊C411yescharacteristic
219C2×C4×C43C2×C22⋊C411yescharacteristic
220C2×C4×C43#12012yes—
221C2×Dic43—82no—
222C43⋊C8—82no—
223#12715—431nomaximal, Sylow
224C22×C43—22no—
225C22×C43—21no—
226C22×C43—22no—
227C22×C43—41no—
228C22×C43—42no—
229C22×C43—42no—
230C22×C43—44no—
231C22×C43—42no—
232C22×C43—41no—
233C22×C43—42no—
234C22×C43—162no—
235C22×C43—164no—
236C22×C43#37911yescharacteristic
237C4×C43—22no—
238C4×C43—21no—
239C4×C43—22no—
240C4×C43—41no—
241C4×C43—42no—
242C4×C43—41no—
243C4×C43—41no—
244C4×C43—42no—
245C4×C43C2×#9911yescharacteristic
246C4×C43C2×#9911yescharacteristic
247Dic43—162no—
248#3500D4311yescharacteristic
249#2132—431no—
250#2126—431no—
251#2122—432no—
252#1796—432no—
253C2×C43—21no—
254C2×C43—42no—
255C2×C43—42no—
256C2×C43—41no—
257C2×C43—82no—
258C2×C43—81no—
259C2×C43#178111yescharacteristic
260C2×2−1+4C2×D4311yescharacteristic
261C2×H(5,𝔽2)Dic4311yescharacteristic, complemented
262#548—22no—
263#546—22no—
264#546—24no—
265#536—21no—
266#536Dic4311yescharacteristic, complemented
267#535—21no—
268#535C2×D4311yescharacteristic
269#532—21no—
270#532C2×D4311yescharacteristic
271C2×#124—21no—
272C2×#124Dic4311yescharacteristic, complemented
273C4×#41Dic4311yescharacteristic, complemented
274#421—862no—
275#419—862no—
276#413—432no—
277#410—432no—
278#356—861no—
279#355—861no—
280#354—861no—
281#353—861no—
282AΣL(1,𝔽16)—864no—
283C43#1271511yescharacteristic, complemented, Sylow
2842−1+4—42no—
2852−1+4—42no—
286H(5,𝔽2)—42no—
287H(5,𝔽2)—42no—
288C2×#41—21no—
289C2×#41—21no—
290C2×#41—21no—
291C2×#41—22no—
292C2×#41C2×Dic4311yescharacteristic
293C2×#41C2×Dic4311yescharacteristic
294C2×#41C22⋊D4312yes—
295C2×#41C22×D4311yescharacteristic
296C22×Q8—21no—
297C22×Q8C2×Dic4311yescharacteristic
298C22×Q8C22⋊D4312yes—
299C22×D4—22no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4C2×Dic4311yescharacteristic
303C2×M16—862no—
304C4⋊Q8—21no—
305C4⋊Q8C22⋊D4311yescharacteristic
306#127—21no—
307#127C22⋊D4311yescharacteristic
308#124—24no—
309#124—22no—
310#124—21no—
311#124—21no—
312#124—21no—
313#124—21no—
314#124C22⋊D4311yescharacteristic
315#124C22⋊D4311yescharacteristic
316#123—24no—
317#123—22no—
318C22⋊Q8—24no—
319C22⋊Q8—22no—
320#121—24no—
321#121—24no—
322#121—22no—
323#121—22no—
324#120—24no—
325#120C22⋊D4314yes—
326C4×Q8—21no—
327C4×D4—24no—
328C4×D4—21no—
329C4×D4—22no—
330C4×D4—22no—
331#117—21no—
332#117—862no—
333#117C2×Dic4311yescharacteristic
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C42—21no—
337C2×C42C2×Dic4311yescharacteristic
338#104—1724no—
339#103—1724no—
340#101—862no—
341#100—862no—
342#99—862no—
343#99—862no—
344C24—22no—
345C24C22⋊Dic4312yes—
346#41—22no—
347#41—22no—
348#41—24no—
349#41—41no—
350#41—41no—
351#41—42no—
352#41—44no—
353#41—44no—
354#41—41no—
355#41—41no—
356#41—48no—
357#41—42no—
358#41—42no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—42no—
365C2×Q8—44no—
366C2×Q8C22⋊Dic4311yescharacteristic
367C2×Q8C22⋊Dic4312yes—
368C2×Q8C22⋊Dic4311yescharacteristic
369C2×Q8C2×C22⋊D4311yescharacteristic
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—44no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—48no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4C22⋊Dic4311yescharacteristic
386C2×D4C22⋊Dic4311yescharacteristic
387C2×D4C2×C22⋊D4311yescharacteristic
388C2×D4C2×C22⋊D4312yes—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4C22×Dic4311yescharacteristic
398C22×C4C2×C22⋊D4312yes—
399M16—1722no—
400M16—1722no—
401C2×C8—1722no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—1722no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—1722no—
415C22⋊C4C22⋊Dic4312yes—
416C42—22no—
417C42—21no—
418C42—21no—
419C42—1722no—
420C42C22⋊Dic4311yescharacteristic
421C42C22⋊Dic4311yescharacteristic
422C23—21no—
423C23—22no—
424C23—21no—
425C23—44no—
426C23—42no—
427C23—82no—
428C23—84no—
429C23C2×C22⋊Dic4311yescharacteristic
430C23#1306329412yes—
431Q8—21no—
432Q8—22no—
433Q8—22no—
434Q8—41no—
435Q8—44no—
436Q8—41no—
437Q8—42no—
438Q8#1306324012yes—
439D4—24no—
440D4—21no—
441D4—42no—
442D4—41no—
443D4—41no—
444D4—42no—
445D4—44no—
446D4—44no—
447D4—42no—
448D4—44no—
449D4—41no—
450D4—41no—
451D4—82no—
452D4—84no—
453D4#1306324012yes—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—3442no—
474C2×C4C2×C22⋊Dic4311yescharacteristic
475C2×C4C2×C22⋊Dic4311yescharacteristic
476C2×C4#1306329412yes—
477C8—3442no—
478C22—22no—
479C22—21no—
480C22—22no—
481C22—41no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—41no—
488C22—162no—
489C22—164no—
490C22#1458275511yescharacteristic
491C4—22no—
492C4—21no—
493C4—22no—
494C4—41no—
495C4—42no—
496C4—41no—
497C4—41no—
498C4—42no—
499C4—6882no—
500C4C2×#1306324011yescharacteristic
501C4C2×#1306324011yescharacteristic
502C2—21no—
503C2—42no—
504C2—42no—
505C2—41no—
506C2—81no—
507C2—82no—
508C2#1913451011yescharacteristic
subgroupG/H
centerC2#19134510
commutatorC22×C4×C43C22×C4
FrattiniC2×#41C22×D43
Fitting, prime radicalC43×#3500C2
socleC2×C43#1781
cosocleC2×#41×C43C23
hypercenter#3500D43
hypocenterC43#12715
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3500

Complements

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

NH ≅ G/Ntype
1. #19136280C2semidirect
2. #19136260C2semidirect
3. #19136245C2semidirect
4. #19134570C2semidirect
8. #14582883C22semidirect
9. #14582873C22semidirect
16. C2×H(5,𝔽2)×C43C4semidirect
21. C43×#536C4semidirect
27. C2×#124×C43C4semidirect
28. C4×#41×C43C4semidirect
261. C2×H(5,𝔽2)Dic43semidirect
266. #536Dic43semidirect
272. C2×#124Dic43semidirect
273. C4×#41Dic43semidirect
283. C43#12715semidirect

Maximal subgroups

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

subgroupindex
#191362802
#191362602
#191362452
#191345702
C43×#35002
#1271543