Finite Group Databasebeta

← #25134975

Subgroups — #25134975

Order 11008.

986 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583228C211yesmaximal, characteristic
2C2×#103×D43C211yesmaximal, characteristic
3#19138236C211yesmaximal, characteristic
4#19137463C211yesmaximal, characteristic
5#19135973C211yesmaximal, characteristic, complemented
6#19135972C211yesmaximal, characteristic, complemented
7#19135840C211yesmaximal, characteristic, complemented
8#19135519C211yesmaximal, characteristic, complemented
9#19135516C211yesmaximal, characteristic, complemented
10#19135231C211yesmaximal, characteristic, complemented
11#19135230C211yesmaximal, characteristic, complemented
12#19135015C211yesmaximal, characteristic
13#19134774C211yesmaximal, characteristic
14#19134528C211yesmaximal, characteristic
15C43×#2043C211yesmaximal, characteristic, complemented
16C22×Q8×D43C2211yescharacteristic
17C22×C8×D43C2211yescharacteristic
18C2×#13063323C2211yescharacteristic
19C2×#13063301C2211yescharacteristic
20C2×#13063283—21no—
21C2×C4⋊C4×D43C2211yescharacteristic
22#14583228—24no—
23C2×#13063257C2211yescharacteristic
24C2×#13063256C2211yescharacteristic
25C2×C4⋊(C43⋊C8)C2211yescharacteristic
26C2×Dic43⋊C8C2211yescharacteristic
27C2×#13063250C2211yescharacteristic
28C2×#13063250C2211yescharacteristic
29C2×#13063249C2211yescharacteristic
30C2×#13063249C2211yescharacteristic
31C2×#13063249C2211yescharacteristic
32C2×#13063249C2211yescharacteristic
33C2×#13063248C2211yescharacteristic
34C2×#13063248C2211yescharacteristic
35#103×D43—24no—
36C2×(C2×D43)⋊C8C2211yescharacteristic
37#14582819—21no—
38#14582819—21no—
39#14582806C2211yescharacteristic
40#14582805C2211yescharacteristic
41#14582798C2211yescharacteristic
42#14582786—21no—
43#14582780—21no—
44#14582770—21no—
45#14582770—21no—
46#14582761—21no—
47#14582756C2211yescharacteristic
48C4⋊C4×Dic43C2211yescharacteristic
49C4×C4⋊Dic43C2211yescharacteristic
50C4×(C2×D43)⋊C4C2211yescharacteristic
51#14582662C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×C4⋊Q8×C43C2211yescharacteristic
55C2×#103×C43C2211yescharacteristic
56C2×#103×C43C2211yescharacteristic
57C2×#103×C43C2211yescharacteristic
58C2×#103×C43C2211yescharacteristic
59C43×#389C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C22×Dic86—21no—
62C22×Dic86C2311yescharacteristic
63C2×Q8×D43—22no—
64C2×Q8×D43—22no—
65C2×Q8×D43—22no—
66C2×Q8×D43—22no—
67C2×Q8×D43—41no—
68C2×Q8×D43—41no—
69C22×C4×D43—21no—
70C22×C4×D43—21no—
71C22×C4×D43C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74C2×C8×D43—22no—
75C2×C8×D43—41no—
76C2×C8×D43—41no—
77#13063323—22no—
78#13063323—41no—
79#13063301—22no—
80#13063301—22no—
81#13063283—44no—
82C2×C4⋊Dic43—21no—
83C2×C4⋊Dic43C2311yescharacteristic
84C2×C4⋊Dic43C2311yescharacteristic
85C2×C4⋊Dic43C2311yescharacteristic
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89C2×Dic43⋊C4—21no—
90C4⋊C4×D43—22no—
91C4⋊C4×D43D414yes—
92C2×(C2×D43)⋊C4—21no—
93C2×(C2×D43)⋊C4—21no—
94C2×(C2×D43)⋊C4D411yescharacteristic
95C2×(C2×D43)⋊C4D411yescharacteristic
96C2×C4×Dic43—21no—
97C2×C4×Dic43—21no—
98C2×C4×Dic43C2311yescharacteristic
99#13063257—22no—
100#13063256—22no—
101C4⋊(C43⋊C8)—41no—
102Dic43⋊C8—41no—
103#13063250—22no—
104#13063250—22no—
105#13063249—22no—
106#13063249—22no—
107#13063249—41no—
108#13063249—41no—
109#13063248—22no—
110#13063248—41no—
111(C2×D43)⋊C8—22no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95—21no—
116C43⋊#95—21no—
117C43⋊#95—21no—
118C43⋊#95—21no—
119C43⋊#95—21no—
120C43⋊#95D411yescharacteristic
121C43⋊#95D411yescharacteristic
122C22×Q8×C43C2311yescharacteristic
123C22×Q8×C43C2311yescharacteristic
124C22×C8×C43C2311yescharacteristic
125C22×C8×C43C2311yescharacteristic
126C4⋊Q8×C43—22no—
127C4⋊Q8×C43—41no—
128C2×C4⋊C4×C43—21no—
129C2×C4⋊C4×C43—21no—
130C2×C4⋊C4×C43—21no—
131C2×C4⋊C4×C43C2311yescharacteristic
132C2×C4⋊C4×C43C2311yescharacteristic
133C2×C42×C43C2311yescharacteristic
134#103×C43—22no—
135#103×C43—22no—
136#103×C43—22no—
137#103×C43—41no—
138#95×C43—21no—
139C23×D43C2×D411yescharacteristic
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43—21no—
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43—21no—
148C2×Q8×C43—21no—
149C2×Q8×C43—21no—
150C2×Q8×C43—21no—
151C2×Q8×C43#4112yes—
152C22×C4×C43—21no—
153C22×C4×C43—21no—
154C22×C4×C43—21no—
155C22×C4×C43—21no—
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43C2411yescharacteristic
159C22×Dic43—21no—
160C22×Dic43—21no—
161C22×Dic43—21no—
162C22×Dic43—21no—
163C22×Dic43C2×D411yescharacteristic
164C22×Dic43C2×D411yescharacteristic
165C22×Dic43C2×D411yescharacteristic
166Q8×D43—28no—
167Q8×D43—44no—
168Q8×D43—44no—
169Q8×D43—44no—
170C2×C4×D43—21no—
171C2×C4×D43—21no—
172C2×C4×D43—21no—
173C2×C4×D43—21no—
174C2×C4×D43—24no—
175C2×C4×D43—42no—
176C2×C4×D43—42no—
177C2×C4×D43—42no—
178C2×C4×D43—42no—
179C2×C4×D43—42no—
180C2×C4×D43C2×D412yes—
181C2×C4×D43#4112yes—
182C2×Dic86—21no—
183C2×Dic86—21no—
184C2×Dic86—21no—
185C2×Dic86—21no—
186C2×Dic86—21no—
187C2×Dic86—42no—
188C2×Dic86—42no—
189C2×Dic86—42no—
190C2×Dic86#4112yes—
191C2×C8×C43—21no—
192C2×C8×C43—21no—
193C2×C8×C43—21no—
194C2×C8×C43—21no—
195C2×C8×C43—21no—
196C2×C8×C43—21no—
197C4⋊C4×C43—21no—
198C4⋊C4×C43—21no—
199C4⋊C4×C43—21no—
200C4⋊C4×C43—42no—
201C4⋊C4×C43—42no—
202C4⋊C4×C43—42no—
203C4⋊C4×C43C2×D412yes—
204C42×C43—21no—
205C42×C43—21no—
206(C2×D43)⋊C4—22no—
207(C2×D43)⋊C4—42no—
208(C2×D43)⋊C4—42no—
209(C2×D43)⋊C4SD1614yes—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—21no—
213C4⋊Dic43—21no—
214C4⋊Dic43—21no—
215C4⋊Dic43—42no—
216C4⋊Dic43C2×D412yes—
217Dic43⋊C4—24no—
218Dic43⋊C4—42no—
219Dic43⋊C4—42no—
220Dic43⋊C4—42no—
221C4×Dic43—21no—
222C4×Dic43—21no—
223C4×Dic43—42no—
224C4×Dic43—42no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8—21no—
227C2×C43⋊C8—21no—
228C2×C43⋊C8—21no—
229C2×C43⋊C8—21no—
230C2×C43⋊C8—21no—
231C8×D43—44no—
232C23×C43C22×D411yescharacteristic
233C22×D43—21no—
234C22×D43—21no—
235C22×D43—21no—
236C22×D43—21no—
237C22×D43#12112yes—
238C22×D43C2×SD1612yes—
239C22×D43#13712yes—
240Q8×C43—22no—
241Q8×C43—22no—
242Q8×C43—41no—
243Q8×C43—41no—
244Q8×C43—41no—
245Q8×C43—41no—
246Q8×C43—41no—
247Q8×C43—41no—
248C2×C4×C43—21no—
249C2×C4×C43—21no—
250C2×C4×C43—22no—
251C2×C4×C43—21no—
252C2×C4×C43—22no—
253C2×C4×C43—21no—
254C2×C4×C43—21no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43—41no—
258C2×C4×C43—41no—
259C2×C4×C43—41no—
260C2×C4×C43—41no—
261C2×C4×C43—41no—
262C2×C4×C43—41no—
263C2×C4×C43—41no—
264C2×C4×C43—41no—
265C2×C4×C43C2×SD1612yes—
266C2×C4×C43C22×D411yescharacteristic
267C2×C4×C43C2×#4111yescharacteristic
268C2×C4×C43C2×#4111yescharacteristic
269C2×C4×C43C2×#4111yescharacteristic
270C2×C4×C43H(5,𝔽2)11yescharacteristic
271C2×C4×C43H(5,𝔽2)11yescharacteristic
272C2×Dic43—21no—
273C2×Dic43—21no—
274C2×Dic43—21no—
275C2×Dic43—21no—
276C2×Dic43—22no—
277C2×Dic43—21no—
278C2×Dic43—22no—
279C2×Dic43—21no—
280C2×Dic43—21no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—41no—
286C2×Dic43—41no—
287C2×Dic43—41no—
288C2×Dic43—41no—
289C2×Dic43—41no—
290C2×Dic43—41no—
291C2×Dic43#12112yes—
292C2×Dic43C2×SD1612yes—
293C4×D43—22no—
294C4×D43—22no—
295C4×D43—22no—
296C4×D43—22no—
297C4×D43—48no—
298C4×D43—48no—
299Dic86—22no—
300Dic86—22no—
301Dic86—48no—
302Dic86—41no—
303Dic86—41no—
304C8×C43—41no—
305C8×C43—41no—
306C43⋊C8—41no—
307C43⋊C8—41no—
308#24480—431nomaximal, Sylow
309C22×C43C2×#12111yescharacteristic
310C22×C43#54611yescharacteristic
311C22×C43#54611yescharacteristic
312C22×C43C22×SD1611yescharacteristic
313C22×C43C2×#13711yescharacteristic
314C22×C43#57811yescharacteristic
315C22×C43#57811yescharacteristic
316C2×D43—22no—
317C2×D43—22no—
318C2×D43—22no—
319C2×D43—22no—
320C2×D43—22no—
321C2×D43—22no—
322C2×D43#46114yes—
323C4×C43—22no—
324C4×C43—42no—
325C4×C43—42no—
326C4×C43—41no—
327C4×C43—81no—
328C4×C43—81no—
329C4×C43#52511yescharacteristic
330C4×C43#53511yescharacteristic
331C4×C43#55111yescharacteristic
332C4×C43#55111yescharacteristic
333Dic43—22no—
334Dic43—22no—
335Dic43—42no—
336Dic43—42no—
337Dic43—41no—
338Dic43—81no—
339Dic43—81no—
340C2×#483—431no—
341C22×#103—431no—
342#2434—431no—
343#2344—431no—
344#2066—431no—
345#2066—431no—
346#2043—431no—
347#2043D4311yescharacteristic, complemented
348#1981—431no—
349#1981—431no—
350#1927—431no—
351#1927—431no—
352#1892—431no—
353#1850—431no—
354#1787—431no—
355C2×C43#241111yescharacteristic
356C2×C43C2×#46111yescharacteristic
357C2×C43#304411yescharacteristic
358C2×C43#329811yescharacteristic
359C2×C43#329811yescharacteristic
360C2×C43#331211yescharacteristic
361C2×C43#331211yescharacteristic
362D43—24no—
363C23×Q8—431no—
364C23×C8—431no—
365C2×C4⋊Q8—431no—
366C2×C4⋊Q8C2×D4311yescharacteristic
367C2×C22⋊Q8—431no—
368C2×#117—861no—
369C22×C4⋊C4—431no—
370#483—864no—
371C2×#106—431no—
372C2×#106—431no—
373C2×C4⋊C8—431no—
374C2×C4⋊C8—431no—
375C2×#103—431no—
376C2×#103—431no—
377C2×#103—431no—
378C2×#103—431no—
379C2×#103—431no—
380C2×#103—431no—
381C2×#103—431no—
382C2×#103—431no—
383C2×#103—864no—
384C2×#103C2×D4311yescharacteristic
385C2×#103C2×D4311yescharacteristic
386C2×#103C2×D4311yescharacteristic
387C2×#103C2×D4311yescharacteristic
388C2×C22⋊C8—431no—
389#395—861no—
390#395—861no—
391#391—431no—
392#391—431no—
393#389—431no—
394#389C2×D4311yescharacteristic
395#387—861no—
396#387—861no—
397#385—861no—
398#385—861no—
399#381—861no—
400#380—431no—
401C4×C4⋊C4—431no—
402C4×C4⋊C4—431no—
403C4×C22⋊C4—431no—
404#340—431no—
405#340—431no—
406#336—431no—
407#336C2×D4311yescharacteristic
408C43#2448011yescharacteristic, complemented, Sylow
409C22×Q8—431no—
410C22×Q8—861no—
411C22×Q8—862no—
412C22×Q8—862no—
413C22×Q8—862no—
414C22×Q8—862no—
415C22×Q8—1721no—
416C22×Q8—1721no—
417C22×Q8C22×D4311yescharacteristic
418C22×Q8C22×D4311yescharacteristic
419C23×C4—431no—
420C23×C4—861no—
421C23×C4—861no—
422C22×C8—431no—
423C22×C8—431no—
424C22×C8—862no—
425C22×C8—1721no—
426C22×C8—1721no—
427C22×C8C22×D4311yescharacteristic
428C22×C8C22×D4311yescharacteristic
429C4⋊Q8—22no—
430C4⋊Q8—41no—
431C4⋊Q8—862no—
432C4⋊Q8—1721no—
433C22⋊Q8—862no—
434C22⋊Q8—862no—
435#117—1724no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—431no—
440C2×C4⋊C4—434no—
441C2×C4⋊C4—431no—
442C2×C4⋊C4—431no—
443C2×C4⋊C4—862no—
444C2×C4⋊C4—861no—
445C2×C4⋊C4—861no—
446C2×C4⋊C4—861no—
447C2×C4⋊C4—861no—
448C2×C4⋊C4—861no—
449C2×C4⋊C4C22×D4311yescharacteristic
450C2×C4⋊C4C22×D4311yescharacteristic
451C2×C22⋊C4—431no—
452C2×C22⋊C4—431no—
453C2×C22⋊C4—861no—
454C2×C22⋊C4—861no—
455C2×C42—431no—
456C2×C42—861no—
457C2×C42—861no—
458C2×C42C22×D4311yescharacteristic
459#106—862no—
460#106—862no—
461C4⋊C8—1721no—
462C4⋊C8—1721no—
463#103—22no—
464#103—22no—
465#103—22no—
466#103—41no—
467#103—862no—
468#103—862no—
469#103—862no—
470#103—862no—
471#103—862no—
472#103—1721no—
473#103—1721no—
474#103—1721no—
475C22⋊C8—862no—
476#95—21no—
477#95—431no—
478#95—431no—
479#95—861no—
480#95—861no—
481#95—861no—
482#95—861no—
483#95—861no—
484#95—861no—
485#95—861no—
486#95—861no—
487C24—431no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—432no—
500C2×Q8—861no—
501C2×Q8—868no—
502C2×Q8—861no—
503C2×Q8—861no—
504C2×Q8—861no—
505C2×Q8—861no—
506C2×Q8—1722no—
507C2×Q8—1722no—
508C2×Q8—1722no—
509C2×Q8—1724no—
510C2×Q8—1724no—
511C2×Q8—1724no—
512C2×Q8#1062190012yes—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—431no—
518C22×C4—431no—
519C22×C4—432no—
520C22×C4—432no—
521C22×C4—431no—
522C22×C4—864no—
523C22×C4—861no—
524C22×C4—861no—
525C22×C4—861no—
526C22×C4—861no—
527C22×C4—861no—
528C22×C4—861no—
529C22×C4—861no—
530C22×C4—861no—
531C22×C4—1722no—
532C22×C4—1722no—
533C22×C4—1722no—
534C22×C4—1722no—
535C22×C4—1722no—
536C22×C4D4×D4311yescharacteristic
537C22×C4D4×D4311yescharacteristic
538C22×C4C23×D4311yescharacteristic
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—861no—
546C2×C8—861no—
547C2×C8—861no—
548C2×C8—861no—
549C2×C8—861no—
550C2×C8—861no—
551C2×C8—1724no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—432no—
559C4⋊C4—864no—
560C4⋊C4—861no—
561C4⋊C4—861no—
562C4⋊C4—861no—
563C4⋊C4—861no—
564C4⋊C4—861no—
565C4⋊C4—1722no—
566C4⋊C4—1722no—
567C4⋊C4—1722no—
568C4⋊C4—1722no—
569C4⋊C4D4×D4312yes—
570C22⋊C4—434no—
571C22⋊C4—862no—
572C22⋊C4—1722no—
573C22⋊C4—1722no—
574C42—21no—
575C42—21no—
576C42—861no—
577C42—861no—
578C42—1722no—
579C42—1722no—
580C23—432no—
581C23—432no—
582C23—432no—
583C23—861no—
584C23—861no—
585C23—861no—
586C23—861no—
587C23C2×D4×D4311yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—41no—
592Q8—41no—
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—862no—
597Q8—862no—
598Q8—1728no—
599Q8—1721no—
600Q8—1721no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—432no—
619C2×C4—432no—
620C2×C4—861no—
621C2×C4—861no—
622C2×C4—861no—
623C2×C4—861no—
624C2×C4—862no—
625C2×C4—862no—
626C2×C4—862no—
627C2×C4—862no—
628C2×C4—862no—
629C2×C4—861no—
630C2×C4—861no—
631C2×C4—861no—
632C2×C4—862no—
633C2×C4—1728no—
634C2×C4—1728no—
635C2×C4—1721no—
636C2×C4—1721no—
637C2×C4—1721no—
638C2×C4—1721no—
639C2×C4—1721no—
640C2×C4—1721no—
641C2×C4—1721no—
642C2×C4—1721no—
643C2×C4—1721no—
644C2×C4—1721no—
645C2×C4SD16×D4312yes—
646C2×C4C2×D4×D4311yescharacteristic
647C2×C4#41×D4311yescharacteristic
648C2×C4#41×D4311yescharacteristic
649C2×C4C2×#1062190011yescharacteristic
650C2×C4#1306337011yescharacteristic
651C2×C4#1306337111yescharacteristic
652C8—41no—
653C8—41no—
654C8—1721no—
655C8—1721no—
656C22—434no—
657C22—862no—
658C22—862no—
659C22—862no—
660C22—862no—
661C22—862no—
662C22—862no—
663C22#121×D4311yescharacteristic
664C22#1458353611yescharacteristic
665C22#1458354211yescharacteristic
666C22C2×SD16×D4311yescharacteristic
667C22#137×D4311yescharacteristic
668C22#1458371211yescharacteristic
669C22#1458371511yescharacteristic
670C4—22no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—81no—
675C4—81no—
676C4—862no—
677C4—862no—
678C4—3441no—
679C4—3441no—
680C4—1722no—
681C4—1721no—
682C4—1722no—
683C4#1458340911yescharacteristic
684C4#1458346511yescharacteristic
685C4#1458357211yescharacteristic
686C4#1458357711yescharacteristic
687C2—864no—
688C2#1913797311yescharacteristic
689C2#461×D4311yescharacteristic
690C2#1914412011yescharacteristic
691C2#1914645311yescharacteristic
692C2#1914645911yescharacteristic
693C2#1914663911yescharacteristic
694C2#1914664511yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2043C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
hypercenter#2043D43
hypocenterC43#24480
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
5. #19135973C2semidirect
6. #19135972C2semidirect
7. #19135840C2semidirect
8. #19135519C2semidirect
9. #19135516C2semidirect
10. #19135231C2semidirect
11. #19135230C2semidirect
15. C43×#2043C2semidirect
347. #2043D43semidirect
408. C43#24480semidirect

Maximal subgroups

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

subgroupindex
C2×#145832282
C2×#103×D432
#191382362
#191374632
#191359732
#191359722
#191358402
#191355192
#191355162
#191352312
#191352302
#191350152
#191347742
#191345282
C43×#20432
#2448043