Finite Group Databasebeta

← #25106278

Subgroups — #25106278

Order 11008.

1222 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583453C211yesmaximal, characteristic
2C2×#14582951C211yesmaximal, characteristic
3#384×D43C211yesmaximal, characteristic
4#19135554C212yesmaximal, complemented
5#19135395C212yesmaximal, complemented
6#19134901C212yesmaximal
7#19134899C212yesmaximal
8#19134601C211yesmaximal, characteristic, complemented
9#19134600C211yesmaximal, characteristic, complemented
10#19134597C211yesmaximal, characteristic, complemented
11C43×#1803C211yesmaximal, characteristic, complemented
12#14583453—24no—
13#14583453C418yescomplemented
14C2×#13063324C2211yescharacteristic
15C2×#13063323C2211yescharacteristic
16C2×#13063315—21no—
17C2×#13063305C2211yescharacteristic
18C2×#13063303—21no—
19C2×#13063301C2211yescharacteristic
20C2×C4×Dic86—21no—
21C2×C4×Dic86C2211yescharacteristic
22C2×#13063282—21no—
23C2×C4⋊C4×D43—21no—
24C2×C4⋊C4×D43—21no—
25C2×C4⋊C4×D43C2212yes—
26C2×C42×D43C2211yescharacteristic
27#14582951—24no—
28C2×C4⋊(C43⋊C8)C2211yescharacteristic
29C2×#13063250C2212yes—
30C2×#13063249C2212yes—
31C2×#13063248C2212yes—
32C2×(C2×D43)⋊C8C2212yes—
33C2×C8×Dic43C2212yes—
34#14582769C2211yescharacteristic
35#14582768—21no—
36#14582768—21no—
37#14582768C2212yes—
38#14582662C2212yes—
39#14582661C2212yes—
40#14582661C2212yes—
41C2×C4×Q8×C43C2211yescharacteristic
42C2×C4⋊C8×C43C2211yescharacteristic
43C2×#103×C43C2212yes—
44C43×#384C2211yescharacteristic
45C43×#340C2212yes—
46C22×Dic86—21no—
47C22×Dic86C2311yescharacteristic
48C22×C4×D43—22no—
49C22×C4×D43D411yescharacteristic
50C22×C4×D43D411yescharacteristic
51C22×C4×D43C2311yescharacteristic
52C22×C43⋊C8C2312yes—
53#13063324—22no—
54#13063324C2×C414yescomplemented
55#13063323—22no—
56#13063323C2×C414yescomplemented
57#13063315—28no—
58#13063305—22no—
59#13063305C2×C414yes—
60#13063303—28no—
61#13063301—22no—
62#13063301C2×C414yes—
63C4×Dic86—22no—
64C4×Dic86—28no—
65C4×Dic86C2×C414yescomplemented
66#13063282—28no—
67C2×C4⋊Dic43—21no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43C2312yes—
71C2×C4⋊Dic43C2311yescharacteristic
72C2×Dic43⋊C4—21no—
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—21no—
75C2×Dic43⋊C4—21no—
76C2×Dic43⋊C4—21no—
77C2×Dic43⋊C4—22no—
78C2×Dic43⋊C4—22no—
79C2×Dic43⋊C4—22no—
80C4⋊C4×D43—42no—
81C4⋊C4×D43—42no—
82C4⋊C4×D43—44no—
83C4⋊C4×D43—44no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×C4×Dic43—21no—
87C2×C4×Dic43D411yescharacteristic
88C2×C4×Dic43D411yescharacteristic
89C2×C4×Dic43C2311yescharacteristic
90C42×D43C2×C414yes—
91C42×D43D414yes—
92C4⋊(C43⋊C8)—22no—
93#13063250—42no—
94#13063249—42no—
95#13063248—42no—
96(C2×D43)⋊C8—24no—
97C8×Dic43—24no—
98C22×Q8×C43C2311yescharacteristic
99C22×C8×C43C2312yes—
100C4×Q8×C43—22no—
101C4×Q8×C43C2×C414yescomplemented
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43—21no—
105C2×C4⋊C4×C43C2312yes—
106C2×C4⋊C4×C43C2311yescharacteristic
107C2×C42×C43—21no—
108C2×C42×C43C2311yescharacteristic
109C4⋊C8×C43—22no—
110#103×C43—42no—
111C23×D43C2×D411yescharacteristic
112C2×Q8×C43—21no—
113C2×Q8×C43—21no—
114C2×Q8×C43—21no—
115C2×Q8×C43—21no—
116C2×Q8×C43—21no—
117C2×Q8×C43C22×C412yes—
118C22×C4×C43—21no—
119C22×C4×C43—21no—
120C22×C4×C43—22no—
121C22×C4×C43C2×D411yescharacteristic
122C22×C4×C43C2×D411yescharacteristic
123C22×C4×C43C2411yescharacteristic
124C22×Dic43—21no—
125C22×Dic43—21no—
126C22×Dic43—22no—
127C22×Dic43C2×D411yescharacteristic
128C22×Dic43C2×D411yescharacteristic
129C22×Dic43C2×D411yescharacteristic
130C2×C4×D43—21no—
131C2×C4×D43—21no—
132C2×C4×D43—21no—
133C2×C4×D43—21no—
134C2×C4×D43—22no—
135C2×C4×D43—22no—
136C2×C4×D43—44no—
137C2×C4×D43—44no—
138C2×C4×D43—44no—
139C2×C4×D43C22⋊C414yes—
140C2×C4×D43C22⋊C414yes—
141C2×C4×D43SD1614yes—
142C2×C4×D43Q1614yes—
143C2×C4×D43C22×C412yes—
144C2×C4×D43C2×D412yes—
145C2×Dic86—21no—
146C2×Dic86—21no—
147C2×Dic86—21no—
148C2×Dic86—21no—
149C2×Dic86—21no—
150C2×Dic86—24no—
151C2×Dic86—42no—
152C2×Dic86—42no—
153C2×Dic86C22×C412yes—
154C2×C8×C43—22no—
155C2×C8×C43—22no—
156C2×C8×C43—22no—
157C4⋊C4×C43—24no—
158C4⋊C4×C43—21no—
159C4⋊C4×C43—22no—
160C4⋊C4×C43—22no—
161C4⋊C4×C43—42no—
162C4⋊C4×C43—42no—
163C4⋊C4×C43C22×C412yes—
164C42×C43—24no—
165C42×C43C22×C412yes—
166C42×C43C2×D412yes—
167(C2×D43)⋊C4—24no—
168(C2×D43)⋊C4—24no—
169C4⋊Dic43—21no—
170C4⋊Dic43—24no—
171C4⋊Dic43—22no—
172C4⋊Dic43—22no—
173C4⋊Dic43—42no—
174C4⋊Dic43—42no—
175C4⋊Dic43C22×C412yes—
176Dic43⋊C4—24no—
177Dic43⋊C4—24no—
178Dic43⋊C4—24no—
179Dic43⋊C4—24no—
180Dic43⋊C4—24no—
181Dic43⋊C4—44no—
182Dic43⋊C4—44no—
183Dic43⋊C4—44no—
184C4×Dic43—24no—
185C4×Dic43C22⋊C414yes—
186C4×Dic43C22⋊C414yes—
187C4×Dic43C22×C412yes—
188C4×Dic43C2×D412yes—
189C2×C43⋊C8—22no—
190C2×C43⋊C8—22no—
191C2×C43⋊C8—22no—
192C23×C43C22×D411yescharacteristic
193C22×D43—21no—
194C22×D43—21no—
195C22×D43—21no—
196C22×D43—21no—
197C22×D43C2×C22⋊C412yes—
198C22×D43C2×SD1612yes—
199C22×D43C2×Q1612yes—
200Q8×C43—22no—
201Q8×C43—22no—
202Q8×C43—41no—
203Q8×C43—41no—
204C2×C4×C43—21no—
205C2×C4×C43—21no—
206C2×C4×C43—22no—
207C2×C4×C43—22no—
208C2×C4×C43—41no—
209C2×C4×C43—41no—
210C2×C4×C43—41no—
211C2×C4×C43—41no—
212C2×C4×C43—42no—
213C2×C4×C43—42no—
214C2×C4×C43—42no—
215C2×C4×C43C2×C22⋊C412yes—
216C2×C4×C43C2×C22⋊C412yes—
217C2×C4×C43C2×SD1612yes—
218C2×C4×C43C2×Q1612yes—
219C2×C4×C43C23×C411yescharacteristic
220C2×C4×C43C22×D411yescharacteristic
221C2×C4×C43H(5,𝔽2)11yescharacteristic
222C2×C4×C43H(5,𝔽2)11yescharacteristic
223C2×C4×C432−1+411yescharacteristic
224C2×C4×C432−1+411yescharacteristic
225C2×Dic43—21no—
226C2×Dic43—21no—
227C2×Dic43—22no—
228C2×Dic43—22no—
229C2×Dic43—41no—
230C2×Dic43—41no—
231C2×Dic43—41no—
232C2×Dic43—41no—
233C2×Dic43—42no—
234C2×Dic43—42no—
235C2×Dic43—42no—
236C2×Dic43C2×C22⋊C412yes—
237C2×Dic43C2×C22⋊C412yes—
238C2×Dic43C2×C22⋊C412yes—
239C2×Dic43C2×SD1612yes—
240C2×Dic43C2×Q1612yes—
241C2×Dic43AGL(1,ℤ/8)12yes—
242C2×Dic43#13712yes—
243C4×D43—22no—
244C4×D43—22no—
245C4×D43—22no—
246C4×D43—22no—
247C4×D43—28no—
248C4×D43—88no—
249C4×D43#103116yescomplemented
250Dic86—22no—
251Dic86—22no—
252Dic86—41no—
253Dic86—41no—
254Dic86—48no—
255C8×C43—42no—
256C43⋊C8—42no—
257#21860—431nomaximal, Sylow
258C22×C43C22×C22⋊C411yescharacteristic
259C22×C43#53511yescharacteristic
260C22×C43#53611yescharacteristic
261C22×C43C22×SD1611yescharacteristic
262C22×C43C22×Q1611yescharacteristic
263C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C43C2×#13711yescharacteristic
265C2×D43—22no—
266C2×D43—22no—
267C2×D43—22no—
268C2×D43—22no—
269C2×D43—22no—
270C2×D43—22no—
271C2×D43C2×#10314yes—
272C4×C43—22no—
273C4×C43—42no—
274C4×C43—42no—
275C4×C43—82no—
276C4×C43C2×#10314yes—
277C4×C43#52011yescharacteristic
278C4×C43#52011yescharacteristic
279C4×C43#53711yescharacteristic
280C4×C43#53711yescharacteristic
281Dic43—22no—
282Dic43—42no—
283Dic43—42no—
284Dic43—82no—
285Dic43C2×#10314yes—
286Dic43#41714yes—
287C2×#533—431no—
288C2×#431—431no—
289C2×#384—431no—
290#1990—432no—
291#1956—432no—
292#1872—432no—
293#1872—432no—
294#1803—431no—
295#1803—431no—
296#1803—431no—
297#1803D4311yescharacteristic, complemented
298C2×C43#231111yescharacteristic
299C2×C43C22×#10311yescharacteristic
300C2×C43C2×#41711yescharacteristic
301C2×C43#320811yescharacteristic
302C2×C43#320911yescharacteristic
303C2×C43#322911yescharacteristic
304C2×C43#323011yescharacteristic
305D43—24no—
306#533—438no—
307#533—864no—
308C2×C4⋊Q8—431no—
309C2×C4⋊Q8—431no—
310C2×#125—861no—
311C2×C22⋊Q8—431no—
312C2×C22⋊Q8—431no—
313C2×C22⋊Q8—861no—
314C2×C4×Q8—431no—
315C2×C4×Q8—861no—
316C2×C4×Q8C2×D4311yescharacteristic
317C2×#117—861no—
318C22×C4⋊C4—432no—
319C22×C4⋊C4—861no—
320C22×C4⋊C4—861no—
321C22×C42—431no—
322#431—864no—
323C2×C4⋊C8—431no—
324C2×C4⋊C8C2×D4311yescharacteristic
325C2×#103—432no—
326C2×#103—432no—
327C2×#103—432no—
328C2×#103C2×D4312yes—
329C2×C22⋊C8—432no—
330C2×C4×C8—432no—
331#384—432no—
332#384—431no—
333#384—861no—
334#384—861no—
335#384C2×D4311yescharacteristic
336#340—432no—
337#340—432no—
338#340—432no—
339#340C2×D4312yes—
340C43#2186011yescharacteristic, complemented, Sylow
341C22×Q8—431no—
342C22×Q8—861no—
343C22×Q8C22×D4311yescharacteristic
344C23×C4—431no—
345C23×C4—431no—
346C23×C4—431no—
347C23×C4—862no—
348C22×C8—432no—
349C22×C8C22×D4312yes—
350C4⋊Q8—434no—
351C4⋊Q8—434no—
352C4⋊Q8—862no—
353C4⋊Q8—862no—
354#125—868no—
355C22⋊Q8—434no—
356C22⋊Q8—434no—
357C22⋊Q8—862no—
358C22⋊Q8—868no—
359C22⋊Q8—862no—
360C4×Q8—22no—
361C4×Q8—434no—
362C4×Q8—868no—
363C4×Q8—862no—
364C4×Q8C4×D4314yescomplemented
365#117—868no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—431no—
370C2×C4⋊C4—432no—
371C2×C4⋊C4—861no—
372C2×C4⋊C4—861no—
373C2×C4⋊C4—861no—
374C2×C4⋊C4—861no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4—861no—
377C2×C4⋊C4—862no—
378C2×C4⋊C4—862no—
379C2×C4⋊C4—862no—
380C2×C4⋊C4—861no—
381C2×C4⋊C4—861no—
382C2×C4⋊C4—1722no—
383C2×C4⋊C4—1722no—
384C2×C4⋊C4—1724no—
385C2×C4⋊C4—1724no—
386C2×C4⋊C4C22×D4312yes—
387C2×C4⋊C4C22×D4311yescharacteristic
388C2×C22⋊C4—861no—
389C2×C22⋊C4—861no—
390C2×C42—21no—
391C2×C42—431no—
392C2×C42—431no—
393C2×C42—431no—
394C2×C42—434no—
395C2×C42—434no—
396C2×C42—861no—
397C2×C42C22×D4311yescharacteristic
398C4⋊C8—22no—
399C4⋊C8—862no—
400#103—42no—
401#103—1722no—
402#103—1722no—
403#103—1722no—
404C22⋊C8—864no—
405C4×C8—864no—
406C24—431no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—432no—
413C2×Q8—864no—
414C2×Q8—861no—
415C2×Q8—861no—
416C2×Q8—861no—
417C2×Q8—861no—
418C2×Q8—861no—
419C2×Q8—1722no—
420C2×Q8—1722no—
421C2×Q8C2×C4×D4312yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—431no—
426C22×C4—431no—
427C22×C4—434no—
428C22×C4—434no—
429C22×C4—434no—
430C22×C4—431no—
431C22×C4—434no—
432C22×C4—432no—
433C22×C4—432no—
434C22×C4—862no—
435C22×C4—862no—
436C22×C4—861no—
437C22×C4—861no—
438C22×C4—861no—
439C22×C4—861no—
440C22×C4—861no—
441C22×C4—861no—
442C22×C4—862no—
443C22×C4—1724no—
444C22×C4—1724no—
445C22×C4—1724no—
446C22×C4D4×D4311yescharacteristic
447C22×C4D4×D4311yescharacteristic
448C22×C4C23×D4311yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—862no—
453C2×C8—862no—
454C2×C8—862no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—432no—
462C4⋊C4—864no—
463C4⋊C4—864no—
464C4⋊C4—864no—
465C4⋊C4—864no—
466C4⋊C4—864no—
467C4⋊C4—864no—
468C4⋊C4—861no—
469C4⋊C4—862no—
470C4⋊C4—862no—
471C4⋊C4—1722no—
472C4⋊C4—1724no—
473C4⋊C4—1724no—
474C4⋊C4—1724no—
475C4⋊C4—1722no—
476C4⋊C4C2×C4×D4312yes—
477C22⋊C4—864no—
478C22⋊C4—864no—
479C42—24no—
480C42—434no—
481C42—434no—
482C42—432no—
483C42—432no—
484C42—864no—
485C42C2×C4×D4312yes—
486C42D4×D4312yes—
487C23—432no—
488C23—432no—
489C23—432no—
490C23—861no—
491C23—861no—
492C23—861no—
493C23—861no—
494C23C2×D4×D4311yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499Q8—862no—
500Q8—862no—
501Q8—1728no—
502Q8—1721no—
503Q8—1721no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—432no—
516C2×C4—432no—
517C2×C4—432no—
518C2×C4—432no—
519C2×C4—432no—
520C2×C4—432no—
521C2×C4—4316no—
522C2×C4—432no—
523C2×C4—868no—
524C2×C4—861no—
525C2×C4—861no—
526C2×C4—862no—
527C2×C4—862no—
528C2×C4—862no—
529C2×C4—862no—
530C2×C4—862no—
531C2×C4—862no—
532C2×C4—3448no—
533C2×C4—1722no—
534C2×C4—1722no—
535C2×C4—1722no—
536C2×C4—1721no—
537C2×C4—1721no—
538C2×C4—1721no—
539C2×C4—1721no—
540C2×C4C22⋊C4×D4312yes—
541C2×C4C22⋊C4×D4312yes—
542C2×C4SD16×D4312yes—
543C2×C4Q16×D4312yes—
544C2×C4C22×C4×D4311yescharacteristic
545C2×C4C2×D4×D4311yescharacteristic
546C2×C4#1306337011yescharacteristic
547C2×C4#1306337111yescharacteristic
548C2×C4#1306337211yescharacteristic
549C2×C4#1306337311yescharacteristic
550C8—42no—
551C8—1722no—
552C22—434no—
553C22—862no—
554C22—862no—
555C22—862no—
556C22—862no—
557C22—862no—
558C22—862no—
559C22C2×C22⋊C4×D4311yescharacteristic
560C22#1458346511yescharacteristic
561C22#1458347211yescharacteristic
562C22C2×SD16×D4311yescharacteristic
563C22C2×Q16×D4311yescharacteristic
564C22AGL(1,ℤ/8)×D4311yescharacteristic
565C22#137×D4311yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—434no—
571C4—434no—
572C4—862no—
573C4—3442no—
574C4—1722no—
575C4—1722no—
576C4#103×D4314yes—
577C4#1458337911yescharacteristic
578C4#1458338011yescharacteristic
579C4#1458347811yescharacteristic
580C4#1458347911yescharacteristic
581C2—864no—
582C2#1913718411yescharacteristic
583C2C2×#103×D4311yescharacteristic
584C2#417×D4311yescharacteristic
585C2#1914553511yescharacteristic
586C2#1914554211yescharacteristic
587C2#1914573811yescharacteristic
588C2#1914574511yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1803C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
cosocleC22×C4×C43C24
hypercenter#1803D43
hypocenterC43#21860
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #19135554C2semidirect
5. #19135395C2semidirect
8. #19134601C2semidirect
9. #19134600C2semidirect
10. #19134597C2semidirect
11. C43×#1803C2semidirect
13. #14583453C4semidirect
54. #13063324C2×C4semidirect
56. #13063323C2×C4semidirect
65. C4×Dic86C2×C4semidirect
101. C4×Q8×C43C2×C4semidirect
249. C4×D43#103semidirect
297. #1803D43semidirect
340. C43#21860semidirect
364. C4×Q8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145834532
C2×#145829512
#384×D432
#191355542
#191353952
#191349012
#191348992
#191346012
#191346002
#191345972
C43×#18032
#2186043