Finite Group Databasebeta

← #25257699

Subgroups — #25257699

Order 11008.

2198 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583375C211yesmaximal, characteristic
2#19137562C213yesmaximal, complemented
3#19137550C213yesmaximal, complemented
4Q8×C4⋊Dic43C211yesmaximal, characteristic, complemented
5#19137487C213yesmaximal
6#19137479C213yesmaximal
7Q8×(C2×D43)⋊C4C211yesmaximal, characteristic
8Q8×(C2×D43)⋊C4C211yesmaximal, characteristic
9#19137242C213yesmaximal, complemented
10#19137205C213yesmaximal
11#19137088C213yesmaximal
12#19137066C213yesmaximal, complemented
13#19136960C211yesmaximal, characteristic, complemented
14C4×Q8×Dic43C211yesmaximal, characteristic, complemented
15C43×#2351C211yesmaximal, characteristic, complemented
16C22×Q8×D43C2211yescharacteristic
17#14583375C4116yescomplemented
18C2×Q8×Dic43C2211yescharacteristic
19C2×Q8×Dic43C2211yescharacteristic
20C2×Q8×Dic43C2211yescharacteristic
21C2×C4×Dic86C2213yes—
22C2×#13063290C2213yes—
23C2×#13063283C2213yes—
24C2×#13063282C2213yes—
25#14582805C2213yes—
26#14582805C2213yes—
27#14582804C2213yes—
28#14582804C2213yes—
29#14582798C2213yes—
30#14582798C2213yes—
31#14582791C2211yescharacteristic
32#14582791C2213yes—
33#14582791C2216yes—
34#14582789C2213yes—
35#14582789C2213yes—
36#14582786C2213yes—
37#14582786C2216yes—
38#14582786C2213yes—
39#14582786C2216yes—
40#14582780C2213yes—
41#14582780C2213yes—
42#14582769C2213yes—
43#14582766C2216yes—
44#14582764C2216yes—
45#14582756C2213yes—
46#14582756C2213yes—
47C4⋊C4×Dic43C2216yes—
48C4⋊C4×Dic43C2213yes—
49C4⋊C4×Dic43C2213yes—
50C4⋊C4×Dic43C2213yes—
51C4⋊C4×Dic43C2213yes—
52C4×C4⋊Dic43C2213yes—
53C4×C4⋊Dic43C2213yes—
54C4×C4⋊Dic43C2213yes—
55C4×(C2×D43)⋊C4C2213yes—
56C4×(C2×D43)⋊C4C2213yes—
57#14582745C2213yes—
58#14582744C2216yes—
59C42×Dic43C2213yes—
60C2×C4×Q8×C43C2211yescharacteristic
61C2×C4×Q8×C43C2211yescharacteristic
62C2×C4×Q8×C43C2211yescharacteristic
63C43×#389C2213yes—
64C43×#389C2213yes—
65C43×#382C2213yes—
66C4×C4⋊C4×C43C2213yes—
67C22×Dic86C2313yes—
68C2×Q8×D43—24no—
69C2×Q8×D43—24no—
70C2×Q8×D43C2×C418yes—
71C22×C4×D43C2313yes—
72Q8×Dic43—24no—
73Q8×Dic43—24no—
74Q8×Dic43C2×C418yes—
75C4×Dic86C2×C4124yes—
76#13063290C2×C4124yes—
77#13063283C2×C4124yes—
78#13063282C2×C4124yes—
79C2×C4⋊Dic43C2311yescharacteristic
80C2×C4⋊Dic43C2313yes—
81C2×C4⋊Dic43C2313yes—
82C2×C4⋊Dic43C2313yes—
83C2×C4⋊Dic43C2316yes—
84C2×C4⋊Dic43C2313yes—
85C2×Dic43⋊C4C2313yes—
86C2×Dic43⋊C4C2313yes—
87C2×(C2×D43)⋊C4C2311yescharacteristic
88C2×(C2×D43)⋊C4C2311yescharacteristic
89C2×(C2×D43)⋊C4C2313yes—
90C2×(C2×D43)⋊C4C2313yes—
91C2×(C2×D43)⋊C4C2313yes—
92C2×(C2×D43)⋊C4C2311yescharacteristic
93C2×C4×Dic43C2313yes—
94C2×C4×Dic43C2311yescharacteristic
95C2×C4×Dic43C2313yes—
96C2×C4×Dic43C2313yes—
97C2×C4×Dic43C2313yes—
98C2×C4×Dic43C2313yes—
99C2×C4×Dic43C2316yes—
100C2×C4×Dic43C2313yes—
101C2×C4×Dic43C2316yes—
102C43⋊#95C2313yes—
103C43⋊#95C2316yes—
104C43⋊#95C2316yes—
105C43⋊#95C2313yes—
106C43⋊#95C2313yes—
107C43⋊#95C2316yes—
108C43⋊#95C2313yes—
109C43⋊#95C2313yes—
110C43⋊#95C2313yes—
111C43⋊#95C2313yes—
112C43⋊#95C2313yes—
113C43⋊#95C2313yes—
114C43⋊#95C2311yescharacteristic
115C22×Q8×C43C2311yescharacteristic
116C4×Q8×C43—24no—
117C4×Q8×C43—24no—
118C4×Q8×C43C2×C418yes—
119C2×C4⋊C4×C43C2313yes—
120C2×C4⋊C4×C43C2313yes—
121C2×C4⋊C4×C43C2313yes—
122C2×C4⋊C4×C43C2313yes—
123C2×C4⋊C4×C43C2313yes—
124C2×C42×C43C2313yes—
125C2×C42×C43C2313yes—
126C2×C42×C43C2313yes—
127#95×C43C2311yescharacteristic
128#95×C43C2313yes—
129#95×C43C2316yes—
130C23×D43C2411yescharacteristic
131C2×Q8×C43C22×C414yes—
132C2×Q8×C43#4114yes—
133C2×Q8×C43#4114yes—
134C22×C4×C43C2413yes—
135C22×C4×C43C2413yes—
136C22×C4×C43C2411yescharacteristic
137C22×C4×C43C2411yescharacteristic
138C22×C4×C43C2413yes—
139C22×C4×C43C2411yescharacteristic
140C22×C4×C43C2413yes—
141C22×Dic43C2411yescharacteristic
142C22×Dic43C2413yes—
143C22×Dic43C2413yes—
144C22×Dic43C2413yes—
145C22×Dic43C2411yescharacteristic
146C22×Dic43C2413yes—
147C22×Dic43C2411yescharacteristic
148Q8×D43—232no—
149C2×C4×D43—26no—
150C2×C4×D43—26no—
151C2×C4×D43C22×C4112yes—
152C2×Dic86—26no—
153C2×Dic86—26no—
154C2×Dic86C22×C4112yes—
155C4⋊C4×C43—26no—
156C4⋊C4×C43—26no—
157C4⋊C4×C43—26no—
158C4⋊C4×C43—43no—
159C4⋊C4×C43C22×C4112yes—
160C42×C43—26no—
161C42×C43—26no—
162C42×C43C22×C4112yes—
163(C2×D43)⋊C4—22no—
164(C2×D43)⋊C4—22no—
165(C2×D43)⋊C4—43no—
166(C2×D43)⋊C4—43no—
167(C2×D43)⋊C4C22×C414yes—
168(C2×D43)⋊C4C22×C4112yes—
169C4⋊Dic43—26no—
170C4⋊Dic43—26no—
171C4⋊Dic43—22no—
172C4⋊Dic43—43no—
173C4⋊Dic43—46no—
174C4⋊Dic43C22×C4112yes—
175Dic43⋊C4C22×C4112yes—
176Dic43⋊C4C22×C4112yes—
177C4×Dic43—26no—
178C4×Dic43—26no—
179C4×Dic43—22no—
180C4×Dic43—26no—
181C4×Dic43—46no—
182C4×Dic43—43no—
183C4×Dic43—43no—
184C4×Dic43C22×C4112yes—
185C4×Dic43C22×C4124yes—
186C23×C43C2511yescharacteristic
187C22×D43—21no—
188C22×D43—21no—
189C22×D43—21no—
190C22×D43—21no—
191C22×D43—21no—
192C22×D43—21no—
193C22×D43C23×C412yes—
194Q8×C43#117116yes—
195C2×C4×C43—21no—
196C2×C4×C43—21no—
197C2×C4×C43—23no—
198C2×C4×C43—21no—
199C2×C4×C43—21no—
200C2×C4×C43—23no—
201C2×C4×C43—21no—
202C2×C4×C43—23no—
203C2×C4×C43—23no—
204C2×C4×C43—23no—
205C2×C4×C43—21no—
206C2×C4×C43—23no—
207C2×C4×C43—23no—
208C2×C4×C43—23no—
209C2×C4×C43—21no—
210C2×C4×C43C23×C416yes—
211C2×C4×C43C23×C412yes—
212C2×C4×C43C23×C416yes—
213C2×C4×C43C2×#4116yes—
214C2×C4×C43C2×#4116yes—
215C2×C4×C432−1+412yes—
216C2×Dic43—21no—
217C2×Dic43—23no—
218C2×Dic43—23no—
219C2×Dic43—23no—
220C2×Dic43—21no—
221C2×Dic43—21no—
222C2×Dic43—21no—
223C2×Dic43—23no—
224C2×Dic43—21no—
225C2×Dic43—23no—
226C2×Dic43—23no—
227C2×Dic43—21no—
228C2×Dic43—21no—
229C2×Dic43—23no—
230C2×Dic43—23no—
231C2×Dic43—23no—
232C2×Dic43—23no—
233C2×Dic43C23×C412yes—
234C2×Dic43C23×C416yes—
235C2×Dic43C23×C416yes—
236C2×Dic432−1+412yes—
237C4×D43—224no—
238Dic86—224no—
239#34262—431nomaximal, Sylow
240C22×C43C24×C411yescharacteristic
241C22×C43C22×#4111yescharacteristic
242C22×C43C22×#4111yescharacteristic
243C22×C43C2×2−1+411yescharacteristic
244C22×C43C2×2−1+411yescharacteristic
245C22×C43#58511yescharacteristic
246C22×C43#58511yescharacteristic
247C2×D43—22no—
248C2×D43—22no—
249C2×D43—22no—
250C2×D43—41no—
251C2×D43—41no—
252C2×D43—41no—
253C2×D43—41no—
254C4×C43—22no—
255C4×C43—26no—
256C4×C43—22no—
257C4×C43—43no—
258C4×C43—41no—
259C4×C43—43no—
260C4×C43C2×#117112yes—
261Dic43—26no—
262Dic43—26no—
263Dic43—22no—
264Dic43—22no—
265Dic43—41no—
266Dic43—43no—
267Dic43—43no—
268C2×#519—431no—
269#2356—433no—
270#2355—433no—
271#2351D4311yescharacteristic, complemented
272Q8×C4⋊C4—431no—
273#2347—433no—
274#2346—433no—
275Q8×C22⋊C4—431no—
276Q8×C22⋊C4—431no—
277#2320—433no—
278#2314—433no—
279#2298—433no—
280#2295—433no—
281#2276—431no—
282C42×Q8—431no—
283C2×C43C22×#11711yescharacteristic
284C2×C43C4×2−1+411yescharacteristic
285C2×C43C4×2−1+411yescharacteristic
286C2×C43#351611yescharacteristic
287C2×C43#351811yescharacteristic
288C2×C43#352311yescharacteristic
289C2×C43#352411yescharacteristic
290D43—42no—
291C23×Q8—431no—
292#519—4316no—
293C2×C4×Q8—431no—
294C2×C4×Q8—431no—
295C2×C4×Q8—431no—
296C2×C4×Q8—433no—
297C2×C4×Q8—433no—
298C2×C4×Q8C2×D4311yescharacteristic
299C2×C4×Q8C2×D4311yescharacteristic
300C2×C4×Q8C2×D4311yescharacteristic
301C2×#117—433no—
302C2×#117—433no—
303#391—433no—
304#391—433no—
305#391—433no—
306#391—433no—
307#389—433no—
308#389—433no—
309#389C2×D4313yes—
310#389C2×D4313yes—
311#388—433no—
312#388—433no—
313#388—433no—
314#388—436no—
315#388—431no—
316#387—433no—
317#387—433no—
318#387—436no—
319#387—436no—
320#387—433no—
321#387—433no—
322#384—433no—
323#383—436no—
324#382—436no—
325#382C2×D4313yes—
326#380—433no—
327#380—433no—
328C4×C4⋊C4—433no—
329C4×C4⋊C4—433no—
330C4×C4⋊C4—433no—
331C4×C4⋊C4—436no—
332C4×C4⋊C4—433no—
333C4×C4⋊C4—433no—
334C4×C4⋊C4—433no—
335C4×C4⋊C4—433no—
336C4×C4⋊C4C2×D4313yes—
337C4×C22⋊C4—433no—
338C4×C22⋊C4—433no—
339#376—433no—
340#376—436no—
341C43—433no—
342C43#3426211yescharacteristic, complemented, Sylow
343C22×Q8—438no—
344C22×Q8—433no—
345C22×Q8—864no—
346C22×Q8—864no—
347C22×Q8C22×D4311yescharacteristic
348C23×C4—433no—
349C4×Q8—24no—
350C4×Q8—24no—
351C4×Q8—438no—
352C4×Q8—4324no—
353C4×Q8—4324no—
354C4×Q8—864no—
355C4×Q8—864no—
356C4×Q8C4×D4318yes—
357#117—4324no—
358#117—4324no—
359C2×C4⋊C4—433no—
360C2×C4⋊C4—433no—
361C2×C4⋊C4—433no—
362C2×C4⋊C4—433no—
363C2×C4⋊C4—433no—
364C2×C4⋊C4—431no—
365C2×C4⋊C4—436no—
366C2×C4⋊C4—433no—
367C2×C4⋊C4C22×D4313yes—
368C2×C4⋊C4C22×D4313yes—
369C2×C4⋊C4C22×D4313yes—
370C2×C4⋊C4C22×D4313yes—
371C2×C4⋊C4C22×D4313yes—
372C2×C22⋊C4—433no—
373C2×C22⋊C4—431no—
374C2×C22⋊C4—431no—
375C2×C22⋊C4—431no—
376C2×C22⋊C4—433no—
377C2×C22⋊C4—433no—
378C2×C42—433no—
379C2×C42—436no—
380C2×C42—433no—
381C2×C42—433no—
382C2×C42—431no—
383C2×C42—433no—
384C2×C42—433no—
385C2×C42—433no—
386C2×C42—436no—
387C2×C42C22×D4313yes—
388C2×C42C22×D4313yes—
389C2×C42C22×D4313yes—
390#95—431no—
391#95—433no—
392#95—433no—
393#95—433no—
394#95—433no—
395#95—433no—
396#95—436no—
397#95—433no—
398#95—433no—
399#95—433no—
400#95—436no—
401#95—436no—
402#95—433no—
403#95C22×D4316yes—
404#95C22×D4311yescharacteristic
405#95C22×D4313yes—
406C24—431no—
407C2×Q8—4312no—
408C2×Q8—8632no—
409C2×Q8—866no—
410C2×Q8—866no—
411C2×Q8C2×C4×D4314yes—
412C2×Q8#1062190014yes—
413C2×Q8#1062190214yes—
414C22×C4—433no—
415C22×C4—431no—
416C22×C4—433no—
417C22×C4—431no—
418C22×C4—4312no—
419C22×C4—431no—
420C22×C4—433no—
421C22×C4—433no—
422C22×C4—866no—
423C22×C4—866no—
424C22×C4C23×D4311yescharacteristic
425C22×C4C23×D4313yes—
426C22×C4C23×D4313yes—
427C22×C4C23×D4313yes—
428C22×C4C23×D4311yescharacteristic
429C22×C4C23×D4311yescharacteristic
430C22×C4C23×D4313yes—
431C4⋊C4—26no—
432C4⋊C4—26no—
433C4⋊C4—26no—
434C4⋊C4—43no—
435C4⋊C4—4312no—
436C4⋊C4—4312no—
437C4⋊C4—4312no—
438C4⋊C4—862no—
439C4⋊C4—866no—
440C4⋊C4—866no—
441C4⋊C4—1726no—
442C4⋊C4—1723no—
443C4⋊C4C2×C4×D43112yes—
444C22⋊C4—4312no—
445C22⋊C4—434no—
446C22⋊C4—862no—
447C22⋊C4—862no—
448C22⋊C4—1723no—
449C22⋊C4—1723no—
450C42—26no—
451C42—26no—
452C42—4312no—
453C42—4324no—
454C42—866no—
455C42—866no—
456C42—866no—
457C42—862no—
458C42—1723no—
459C42—1723no—
460C42—1726no—
461C42C2×C4×D43112yes—
462C23—432no—
463C23—861no—
464C23—861no—
465C23—861no—
466C23—861no—
467C23—861no—
468C23—861no—
469C23C24×D4311yescharacteristic
470Q8—8624no—
471Q8#13063283116yes—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—23no—
482C2×C4—21no—
483C2×C4—23no—
484C2×C4—23no—
485C2×C4—21no—
486C2×C4—23no—
487C2×C4—436no—
488C2×C4—432no—
489C2×C4—436no—
490C2×C4—432no—
491C2×C4—863no—
492C2×C4—861no—
493C2×C4—863no—
494C2×C4—863no—
495C2×C4—861no—
496C2×C4—861no—
497C2×C4—861no—
498C2×C4—863no—
499C2×C4—8624no—
500C2×C4—861no—
501C2×C4—863no—
502C2×C4—863no—
503C2×C4—861no—
504C2×C4—861no—
505C2×C4—863no—
506C2×C4—863no—
507C2×C4—863no—
508C2×C4—863no—
509C2×C4C22×C4×D4316yes—
510C2×C4C22×C4×D4312yes—
511C2×C4C22×C4×D4316yes—
512C2×C4C2×#1062190016yes—
513C2×C4C2×#1062190216yes—
514C2×C4#1306337212yes—
515C22—862no—
516C22—862no—
517C22—862no—
518C22—1721no—
519C22—1721no—
520C22—1721no—
521C22—1721no—
522C22C23×C4×D4311yescharacteristic
523C22C22×#1062190011yescharacteristic
524C22C22×#1062190211yescharacteristic
525C222−1+4×D4311yescharacteristic
526C22C2×#1306337211yescharacteristic
527C22#1458373511yescharacteristic
528C22#1458373611yescharacteristic
529C4—22no—
530C4—26no—
531C4—22no—
532C4—43no—
533C4—41no—
534C4—43no—
535C4—866no—
536C4—862no—
537C4—862no—
538C4—866no—
539C4—1721no—
540C4—1723no—
541C4—1723no—
542C4C2×#13063283112yes—
543C2—1722no—
544C2C22×#1306328311yescharacteristic
545C2#1914780411yescharacteristic
546C2C4×#1306337211yescharacteristic
547C2#1914857711yescharacteristic
548C2#1914860311yescharacteristic
549C2#1914869211yescharacteristic
550C2#1914870711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D43
commutatorC22×C43C24×C4
FittingC43×#2351C2
socle, cosocleC23×C43C25
prime radicalC23×D43C24
hypercenter#2351D43
hypocenterC43#34262
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #19137562C2semidirect
3. #19137550C2semidirect
4. Q8×C4⋊Dic43C2semidirect
9. #19137242C2semidirect
12. #19137066C2semidirect
13. #19136960C2semidirect
14. C4×Q8×Dic43C2semidirect
15. C43×#2351C2semidirect
17. #14583375C4semidirect
271. #2351D43semidirect
342. C43#34262semidirect

Maximal subgroups

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

subgroupindex
C2×#145833752
#191375622
#191375502
Q8×C4⋊Dic432
#191374872
#191374792
Q8×(C2×D43)⋊C42
#191372422
#191372052
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