Finite Group Databasebeta

← #25113832

Subgroups — #25113832

Order 11008.

1010 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×#14582897C211yesmaximal, characteristic
2#19137471C211yesmaximal, characteristic, complemented
3#19137420C211yesmaximal, characteristic
4#19135286C211yesmaximal, characteristic, complemented
5#19135270C211yesmaximal, characteristic, complemented
6#19135237C211yesmaximal, characteristic, complemented
7#19135228C211yesmaximal, characteristic, complemented
8#19134844C211yesmaximal, characteristic, complemented
9#19134680C211yesmaximal, characteristic, complemented
10#19134625C211yesmaximal, characteristic, complemented
11#19134532C211yesmaximal, characteristic
12#19134422C211yesmaximal, characteristic, complemented
13C4×#13063246C211yesmaximal, characteristic, complemented
14#19134375C211yesmaximal, characteristic, complemented
15C43×#1864C211yesmaximal, characteristic, complemented
16C22×#10621900C2211yescharacteristic
17C22×#10621872C2211yescharacteristic
18C2×D4×Dic43C2211yescharacteristic
19C2×C4⋊C4×D43C2211yescharacteristic
20C2×#13063259C2211yescharacteristic
21#14582897C418yescomplemented
22C2×#13063250C2211yescharacteristic
23C2×#13063250C2211yescharacteristic
24C2×#13063248C2211yescharacteristic
25C2×#13063248C2211yescharacteristic
26C2×#13063246C2211yescharacteristic
27C2×#13063246C2211yescharacteristic
28C2×(C2×D43)⋊C8C2211yescharacteristic
29C2×#13063235C2211yescharacteristic
30C2×C4×C43⋊C8C2211yescharacteristic
31#14582805C2211yescharacteristic
32#14582804C2211yescharacteristic
33#14582798C2211yescharacteristic
34#14582786—21no—
35#14582780—21no—
36#14582778—21no—
37#14582776—21no—
38#14582760—21no—
39#14582756—21no—
40#14582756C2211yescharacteristic
41#14582756C2211yescharacteristic
42C4⋊C4×Dic43C2211yescharacteristic
43C4×C4⋊Dic43C2211yescharacteristic
44C4×(C2×D43)⋊C4C2211yescharacteristic
45#14582746—21no—
46C2×C43⋊#95—21no—
47#14582662C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582653C2211yescharacteristic
53C2×C4×D4×C43C2211yescharacteristic
54C2×#106×C43C2211yescharacteristic
55C2×#102×C43C2211yescharacteristic
56C2×#102×C43C2211yescharacteristic
57C43×#389C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C2×#10621900—22no—
61C2×#10621900—22no—
62C2×#10621900—22no—
63C2×#10621900—41no—
64C2×#10621900—41no—
65C2×#10621900C2×C414yes—
66C22×Dic86C2311yescharacteristic
67C22×C22⋊D43—21no—
68C23×Dic43—21no—
69C23×Dic43—21no—
70C22×C4×D43—21no—
71C22×C4×D43C2311yescharacteristic
72C2×#10621872—41no—
73C2×#10621872—41no—
74C2×#10621872C2×C414yescomplemented
75C22×C43⋊C8C2311yescharacteristic
76C22×C43⋊C8C2311yescharacteristic
77D4×Dic43—22no—
78D4×Dic43—41no—
79C2×C4⋊Dic43—21no—
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×Dic43⋊C4—21no—
84C4⋊C4×D43—22no—
85C4⋊C4×D43C2×C414yescomplemented
86C2×C22⋊Dic43—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4D411yescharacteristic
90C2×(C2×D43)⋊C4D411yescharacteristic
91C2×C4×Dic43—21no—
92C2×C4×Dic43—21no—
93C2×C4×Dic43C2311yescharacteristic
94#13063259—41no—
95#13063250—41no—
96#13063250C2×C414yescomplemented
97#13063248—41no—
98#13063248C2×C414yescomplemented
99#13063246—22no—
100#13063246C2×C414yes—
101(C2×D43)⋊C8—22no—
102#13063235—41no—
103C4×C43⋊C8—41no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95D411yescharacteristic
114C43⋊#95D411yescharacteristic
115C22×D4×C43C2311yescharacteristic
116C23×C4×C43—21no—
117C22×C8×C43C2311yescharacteristic
118C22×C8×C43C2311yescharacteristic
119C4×D4×C43—22no—
120C4×D4×C43—41no—
121C2×C4⋊C4×C43—21no—
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C22⋊C4×C43—21no—
126C2×C42×C43C2311yescharacteristic
127#106×C43—41no—
128#102×C43—22no—
129#102×C43C2×C414yes—
130#95×C43—21no—
131C24×C43—21no—
132C23×D43C2×D411yescharacteristic
133C2×D4×C43—21no—
134C2×D4×C43—21no—
135C2×D4×C43—21no—
136C2×D4×C43C22×C412yes—
137C2×D4×C43#4112yes—
138C2×D4×C43#4112yes—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43—42no—
143C22×C4×C43—42no—
144C22×C4×C43—81no—
145C22×C4×C43C2×D411yescharacteristic
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2411yescharacteristic
148C2×C22⋊D43—24no—
149C2×C22⋊D43—42no—
150C2×C22⋊D43—42no—
151C22×Dic43—24no—
152C22×Dic43—21no—
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43—21no—
156C22×Dic43—42no—
157C22×Dic43—42no—
158C22×Dic43—42no—
159C22×Dic43—42no—
160C22×Dic43—81no—
161C22×Dic43C2×D411yescharacteristic
162C22×Dic43C2×D411yescharacteristic
163C22×Dic43C2×D411yescharacteristic
164#10621900—28no—
165#10621900—44no—
166#10621900—44no—
167#10621900—44no—
168C2×C4×D43—21no—
169C2×C4×D43—21no—
170C2×C4×D43—21no—
171C2×C4×D43—21no—
172C2×C4×D43—24no—
173C2×C4×D43—81no—
174C2×C4×D43—81no—
175C2×C4×D43C22×C412yes—
176C2×C4×D43#4112yes—
177C2×Dic86—21no—
178C2×Dic86—21no—
179C2×Dic86—21no—
180C2×Dic86—21no—
181C2×Dic86—21no—
182C2×Dic86C22×C412yes—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C2×C8×C43—21no—
186C2×C8×C43—21no—
187C2×C8×C43—21no—
188C2×C8×C43C22×C412yes—
189C4⋊C4×C43—21no—
190C4⋊C4×C43—21no—
191C4⋊C4×C43—21no—
192C4⋊C4×C43—21no—
193C4⋊C4×C43—21no—
194C4⋊C4×C43—81no—
195C4⋊C4×C43C22×C412yes—
196C22⋊C4×C43—42no—
197C42×C43—21no—
198C42×C43—21no—
199C22⋊Dic43—42no—
200(C2×D43)⋊C4—22no—
201(C2×D43)⋊C4—41no—
202(C2×D43)⋊C4—42no—
203(C2×D43)⋊C4—81no—
204C4⋊Dic43—21no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209C4⋊Dic43—81no—
210C4⋊Dic43C22×C412yes—
211Dic43⋊C4—24no—
212C4×Dic43—21no—
213C4×Dic43—21no—
214C4×Dic43—81no—
215C4×Dic43—81no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8C22×C412yes—
222#10621872—44no—
223C23×C43—22no—
224C23×C43—22no—
225C23×C43—22no—
226C23×C43—41no—
227C23×C43—41no—
228C23×C43—41no—
229C23×C43—41no—
230C23×C43C22×D411yescharacteristic
231C22×D43—21no—
232C22×D43—21no—
233C22×D43—21no—
234C22×D43—21no—
235C22×D43—21no—
236C22×D43—21no—
237C22×D43C4×D412yes—
238D4×C43—22no—
239D4×C43—22no—
240D4×C43—22no—
241D4×C43#11714yes—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43—22no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—84no—
257C2×C4×C43#13712yes—
258C2×C4×C43C23×C411yescharacteristic
259C2×C4×C43C2×#4111yescharacteristic
260C2×C4×C43C2×#4111yescharacteristic
261C2×C4×C43C2×#4111yescharacteristic
262C2×C4×C43H(5,𝔽2)11yescharacteristic
263C2×C4×C43H(5,𝔽2)11yescharacteristic
264C22⋊D43—48no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—22no—
268C2×Dic43—22no—
269C2×Dic43—21no—
270C2×Dic43—21no—
271C2×Dic43—21no—
272C2×Dic43—21no—
273C2×Dic43—21no—
274C2×Dic43—21no—
275C2×Dic43—48no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—41no—
286C2×Dic43—84no—
287C2×Dic43C4×D412yes—
288C4×D43—22no—
289C4×D43—22no—
290C4×D43—22no—
291C4×D43—22no—
292C4×D43—84no—
293Dic86—22no—
294Dic86—22no—
295Dic86—41no—
296Dic86—41no—
297C8×C43—22no—
298C8×C43—41no—
299C43⋊C8—22no—
300C43⋊C8—41no—
301#22578—431nomaximal, Sylow
302C22×C43—24no—
303C22×C43—42no—
304C22×C43—42no—
305C22×C43—42no—
306C22×C43—42no—
307C22×C43—42no—
308C22×C43—42no—
309C22×C43C2×C4×D411yescharacteristic
310C22×C43#54611yescharacteristic
311C22×C43#54611yescharacteristic
312C22×C43C2×#13711yescharacteristic
313C22×C43#57511yescharacteristic
314C22×C43#57611yescharacteristic
315C22×C43#57711yescharacteristic
316C2×D43—22no—
317C2×D43—22no—
318C2×D43—22no—
319C2×D43—41no—
320C2×D43—41no—
321C2×D43—41no—
322C2×D43—41no—
323C4×C43—22no—
324C4×C43—42no—
325C4×C43—41no—
326C4×C43—81no—
327C4×C43—81no—
328C4×C43C2×#11711yescharacteristic
329C4×C43#51811yescharacteristic
330C4×C43#55111yescharacteristic
331C4×C43#55111yescharacteristic
332Dic43—22no—
333Dic43—41no—
334Dic43—42no—
335Dic43—42no—
336Dic43—41no—
337Dic43—81no—
338Dic43—81no—
339C2×#417—431no—
340#2345—431no—
341#2339—431no—
342#1935—431no—
343#1933—431no—
344#1928—431no—
345#1927—431no—
346#1864D4311yescharacteristic, complemented
347#1862—431no—
348#1823—431no—
349#1809—431no—
350#1788—431no—
351#1762—431no—
352C4×#102—431no—
353#1753—431no—
354C2×C43—44no—
355C2×C43#234211yescharacteristic
356C2×C43C4×#13711yescharacteristic
357C2×C43#294611yescharacteristic
358C2×C43#330711yescharacteristic
359C2×C43#331411yescharacteristic
360C2×C43#332211yescharacteristic
361C2×C43#333111yescharacteristic
362D43—42no—
363C22×#41—431no—
364C22×M16—431no—
365C2×C4×D4—431no—
366C2×C4×D4C2×D4311yescharacteristic
367C22×C4⋊C4—431no—
368C2×#107—431no—
369C2×#106C2×D4311yescharacteristic
370#417—438no—
371C2×#103—431no—
372C2×#103—431no—
373C2×#103—431no—
374C2×#103—431no—
375C2×#102—431no—
376C2×#102—431no—
377C2×#102C2×D4311yescharacteristic
378C2×#102C2×D4311yescharacteristic
379C2×C22⋊C8—431no—
380C2×#97—431no—
381C2×C4×C8—431no—
382#391—431no—
383#391—431no—
384#389—431no—
385#389C2×D4311yescharacteristic
386#387—861no—
387#387—861no—
388#386—861no—
389#386—861no—
390#381—861no—
391#380—431no—
392#380—431no—
393#380—861no—
394C4×C4⋊C4—431no—
395C4×C4⋊C4—431no—
396C4×C22⋊C4—431no—
397C4×C22⋊C4—861no—
398C2×#95—861no—
399#340—431no—
400#340—431no—
401#340—431no—
402#340—431no—
403#340—431no—
404#340C2×D4311yescharacteristic
405#336—431no—
406#336C2×D4311yescharacteristic
407C43#2257811yescharacteristic, complemented, Sylow
408C2×#41—434no—
409C2×#41—862no—
410C2×#41—862no—
411C2×#41—862no—
412C2×#41—1721no—
413C2×#41—1721no—
414C22×Q8—431no—
415C22×D4—861no—
416C22×D4C22×D4311yescharacteristic
417C23×C4—21no—
418C23×C4—431no—
419C23×C4—861no—
420C23×C4—861no—
421C23×C4—861no—
422C2×M16—434no—
423C2×M16—1721no—
424C2×M16—1721no—
425C22×C8—431no—
426C22×C8—431no—
427C22×C8C22×D4311yescharacteristic
428C22×C8C22×D4311yescharacteristic
429C4×D4—22no—
430C4×D4—41no—
431C4×D4—862no—
432C4×D4—1721no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—431no—
435C2×C4⋊C4—434no—
436C2×C4⋊C4—431no—
437C2×C4⋊C4—431no—
438C2×C4⋊C4—862no—
439C2×C4⋊C4—861no—
440C2×C4⋊C4—861no—
441C2×C4⋊C4C22×D4311yescharacteristic
442C2×C4⋊C4C22×D4311yescharacteristic
443C2×C4⋊C4C22×D4311yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—431no—
446C2×C22⋊C4—431no—
447C2×C22⋊C4—861no—
448C2×C22⋊C4—861no—
449C2×C22⋊C4—861no—
450C2×C42—431no—
451C2×C42—861no—
452C2×C42—861no—
453C2×C42C22×D4311yescharacteristic
454#107—1721no—
455#106—41no—
456#103—434no—
457#103—434no—
458#103—1721no—
459#103—1721no—
460#102—22no—
461#102—434no—
462#102—862no—
463#102C4×D4314yes—
464C22⋊C8—862no—
465#97—1721no—
466C4×C8—1721no—
467#95—21no—
468#95—431no—
469#95—431no—
470#95—861no—
471#95—861no—
472#95—861no—
473#95—861no—
474#95—861no—
475#95—861no—
476#95—861no—
477#95—861no—
478#95—861no—
479C24—21no—
480C24—431no—
481#41—868no—
482#41—1724no—
483#41—1724no—
484#41—1724no—
485C2×Q8—432no—
486C2×Q8—861no—
487C2×Q8—861no—
488C2×Q8—861no—
489C2×Q8—861no—
490C2×Q8—861no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—864no—
495C2×D4—1722no—
496C2×D4—1722no—
497C2×D4C2×C4×D4312yes—
498C2×D4#1062190012yes—
499C2×D4#1062190212yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—81no—
506C22×C4—431no—
507C22×C4—432no—
508C22×C4—432no—
509C22×C4—431no—
510C22×C4—431no—
511C22×C4—864no—
512C22×C4—864no—
513C22×C4—861no—
514C22×C4—861no—
515C22×C4—861no—
516C22×C4—861no—
517C22×C4—861no—
518C22×C4—861no—
519C22×C4—861no—
520C22×C4—861no—
521C22×C4—3441no—
522C22×C4—3441no—
523C22×C4—3441no—
524C22×C4—1722no—
525C22×C4—1722no—
526C22×C4—1722no—
527C22×C4—1722no—
528C22×C4D4×D4311yescharacteristic
529C22×C4D4×D4311yescharacteristic
530C22×C4C23×D4311yescharacteristic
531M16—1724no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—432no—
538C2×C8—861no—
539C2×C8—861no—
540C2×C8—861no—
541C2×C8—861no—
542C2×C8—861no—
543C2×C8C2×C4×D4312yes—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—81no—
550C4⋊C4—432no—
551C4⋊C4—864no—
552C4⋊C4—861no—
553C4⋊C4—861no—
554C4⋊C4—861no—
555C4⋊C4—861no—
556C4⋊C4—861no—
557C4⋊C4—3441no—
558C4⋊C4C2×C4×D4312yes—
559C22⋊C4—42no—
560C22⋊C4—862no—
561C22⋊C4—3441no—
562C22⋊C4—1721no—
563C22⋊C4—1722no—
564C22⋊C4—1722no—
565C42—21no—
566C42—21no—
567C42—861no—
568C42—861no—
569C42—3441no—
570C42—3441no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—432no—
579C23—861no—
580C23—861no—
581C23—861no—
582C23—861no—
583C23—861no—
584C23—861no—
585C23C2×D4×D4311yescharacteristic
586Q8—862no—
587Q8—862no—
588Q8—1721no—
589Q8—1721no—
590D4—22no—
591D4—22no—
592D4—22no—
593D4—1728no—
594D4#1306328314yes—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—432no—
611C2×C4—861no—
612C2×C4—861no—
613C2×C4—862no—
614C2×C4—862no—
615C2×C4—862no—
616C2×C4—862no—
617C2×C4—862no—
618C2×C4—862no—
619C2×C4—861no—
620C2×C4—861no—
621C2×C4—861no—
622C2×C4—861no—
623C2×C4—861no—
624C2×C4—861no—
625C2×C4—3444no—
626C2×C4—3444no—
627C2×C4—1728no—
628C2×C4—1721no—
629C2×C4—1721no—
630C2×C4—1721no—
631C2×C4—1721no—
632C2×C4—1721no—
633C2×C4—1721no—
634C2×C4—1721no—
635C2×C4—1721no—
636C2×C4—1721no—
637C2×C4—1721no—
638C2×C4#1306335412yes—
639C2×C4C22×C4×D4311yescharacteristic
640C2×C4#41×D4311yescharacteristic
641C2×C4C2×#1062190011yescharacteristic
642C2×C4C2×#1062190211yescharacteristic
643C2×C4#1306337011yescharacteristic
644C2×C4#1306337111yescharacteristic
645C8—22no—
646C8—41no—
647C8—862no—
648C8—1721no—
649C22—24no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—862no—
657C22—862no—
658C22—862no—
659C22—1721no—
660C22—1721no—
661C22—1721no—
662C22—1721no—
663C22C4×D4×D4311yescharacteristic
664C22#1458353611yescharacteristic
665C22#1458354011yescharacteristic
666C22C2×#1306335411yescharacteristic
667C22#1458369611yescharacteristic
668C22#1458370311yescharacteristic
669C22#1458370911yescharacteristic
670C4—22no—
671C4—42no—
672C4—41no—
673C4—81no—
674C4—81no—
675C4—862no—
676C4—3441no—
677C4—3441no—
678C4—1722no—
679C4—1721no—
680C4—1721no—
681C4—1722no—
682C4C2×#1306328311yescharacteristic
683C4#1458337111yescharacteristic
684C4#1458357211yescharacteristic
685C4#1458357611yescharacteristic
686C2—44no—
687C2—1722no—
688C2#1913744211yescharacteristic
689C2C4×#1306335411yescharacteristic
690C2#1914342411yescharacteristic
691C2#1914657511yescharacteristic
692C2#1914668011yescharacteristic
693C2#1914679311yescharacteristic
694C2#1914690811yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1864C2
socleC23×C43C22×D4
prime radicalC22×#10621900C22
cosocleC22×C4×C43C24
hypercenter#1864D43
hypocenterC43#22578
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #19137471C2semidirect
4. #19135286C2semidirect
5. #19135270C2semidirect
6. #19135237C2semidirect
7. #19135228C2semidirect
8. #19134844C2semidirect
9. #19134680C2semidirect
10. #19134625C2semidirect
12. #19134422C2semidirect
13. C4×#13063246C2semidirect
14. #19134375C2semidirect
15. C43×#1864C2semidirect
21. #14582897C4semidirect
74. C2×#10621872C2×C4semidirect
85. C4⋊C4×D43C2×C4semidirect
96. #13063250C2×C4semidirect
98. #13063248C2×C4semidirect
346. #1864D43semidirect
407. C43#22578semidirect

Maximal subgroups

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

subgroupindex
C2×#145828972
#191374712
#191374202
#191352862
#191352702
#191352372
#191352282
#191348442
#191346802
#191346252
#191345322
#191344222
C4×#130632462
#191343752
C43×#18642
#2257843