Finite Group Databasebeta

← #25200630

Subgroups — #25200630

Order 11008.

814 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19142963C211yesmaximal, characteristic, complemented
2#124×Dic43C211yesmaximal, characteristic, complemented
3#19136076C211yesmaximal, characteristic, complemented
4#19136010C211yesmaximal, characteristic, complemented
5#19135672C211yesmaximal, characteristic
6#19135458C211yesmaximal, characteristic
7#19135446C211yesmaximal, characteristic, complemented
8#19135280C211yesmaximal, characteristic, complemented
9#19135273C211yesmaximal, characteristic, complemented
10#19134948C211yesmaximal, characteristic, complemented
11#19134941C211yesmaximal, characteristic
12#19134854C211yesmaximal, characteristic, complemented
13#19134847C211yesmaximal, characteristic, complemented
14#19134339C211yesmaximal, characteristic, complemented
15C43×#2042C211yesmaximal, characteristic, complemented
16C22×#10621884C2211yescharacteristic
17C22×#10621882C2211yescharacteristic
18C2×#13063320—21no—
19C2×Q8×Dic43C2211yescharacteristic
20C2×D4×Dic43C2211yescharacteristic
21C2×#13063258C2211yescharacteristic
22C2×#13063256C2211yescharacteristic
23C2×#13063249C2211yescharacteristic
24C2×#13063247C2211yescharacteristic
25C2×#13063247C2211yescharacteristic
26C2×#13063246C2211yescharacteristic
27C2×C8×Dic43C2211yescharacteristic
28C2×C8×Dic43C2211yescharacteristic
29#14582806C2211yescharacteristic
30#14582802C2211yescharacteristic
31#14582801C2211yescharacteristic
32#14582800C2211yescharacteristic
33#14582800C2211yescharacteristic
34#14582798C2211yescharacteristic
35#14582795C2211yescharacteristic
36#14582792—21no—
37#14582792—21no—
38#14582791—21no—
39#14582789—21no—
40#14582788—21no—
41#14582765C2211yescharacteristic
42C22⋊C4×Dic43—21no—
43C22⋊C4×Dic43—21no—
44C42×Dic43C2211yescharacteristic
45#14582662C2211yescharacteristic
46#14582662C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582654C2211yescharacteristic
50#14582653C2211yescharacteristic
51#14582653C2211yescharacteristic
52C2×#124×C43C2211yescharacteristic
53C2×#103×C43C2211yescharacteristic
54C2×#103×C43C2211yescharacteristic
55C2×#102×C43C2211yescharacteristic
56C2×#102×C43C2211yescharacteristic
57C43×#389C2211yescharacteristic
58C43×#336C2211yescharacteristic
59C22×D172C2311yescharacteristic
60C23×Dic43—21no—
61C2×#10621884—22no—
62C2×#10621884—22no—
63C2×#10621884—41no—
64C2×#10621882—22no—
65C2×#10621882—22no—
66C2×#10621882—41no—
67C22×C43⋊C8C2311yescharacteristic
68C22×C43⋊C8C2311yescharacteristic
69#13063320—44no—
70Q8×Dic43—41no—
71Q8×Dic43—41no—
72D4×Dic43—41no—
73D4×Dic43—41no—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×Dic43⋊C4—21no—
80C2×C22⋊Dic43—21no—
81C2×C22⋊Dic43—21no—
82C2×C22⋊Dic43—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×C4×Dic43—21no—
90C2×C4×Dic43D411yescharacteristic
91C2×C4×Dic43D411yescharacteristic
92C2×C4×Dic43D411yescharacteristic
93C2×C4×Dic43D411yescharacteristic
94C2×C4×Dic43C2311yescharacteristic
95C2×C4×Dic43C2311yescharacteristic
96#13063258—41no—
97#13063256—41no—
98#13063249—22no—
99#13063247—41no—
100#13063247—41no—
101#13063246—22no—
102C8×Dic43—22no—
103C8×Dic43—22no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C22×Q8×C43C2311yescharacteristic
111C22×D4×C43C2311yescharacteristic
112C22×C8×C43C2311yescharacteristic
113C22×C8×C43C2311yescharacteristic
114#124×C43—22no—
115#124×C43—22no—
116C2×C4⋊C4×C43—21no—
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C22⋊C4×C43—21no—
120C2×C22⋊C4×C43—21no—
121C2×C42×C43C2311yescharacteristic
122#103×C43—22no—
123#103×C43—41no—
124#102×C43—22no—
125#102×C43—41no—
126#95×C43—21no—
127C24×C43—21no—
128C23×D43—21no—
129C2×Q8×C43—21no—
130C2×Q8×C43—21no—
131C2×Q8×C43—21no—
132C2×Q8×C43—21no—
133C2×Q8×C43—21no—
134C2×Q8×C43#4112yes—
135C2×D4×C43—21no—
136C2×D4×C43—21no—
137C2×D4×C43—21no—
138C2×D4×C43—21no—
139C2×D4×C43—21no—
140C2×D4×C43#4112yes—
141C22×C4×C43—21no—
142C22×C4×C43—21no—
143C22×C4×C43—21no—
144C22×C4×C43C2×D411yescharacteristic
145C22×C4×C43C2×D411yescharacteristic
146C22×C4×C43C2411yescharacteristic
147C22×Dic43—21no—
148C22×Dic43—21no—
149C22×Dic43—21no—
150C22×Dic43—42no—
151C22×Dic43—42no—
152C22×Dic43—42no—
153C22×Dic43C2×D411yescharacteristic
154C22×Dic43C2×D411yescharacteristic
155C22×Dic43C2×D411yescharacteristic
156C22×Dic43C2×D411yescharacteristic
157C2×D172—21no—
158C2×D172—21no—
159C2×D172—21no—
160C2×D172—21no—
161C2×D172—21no—
162C2×D172—21no—
163C2×C8×C43—21no—
164C2×C8×C43—21no—
165C2×C8×C43—21no—
166C2×C8×C43—21no—
167C2×C8×C43—21no—
168C2×C8×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—21no—
171C4⋊C4×C43—21no—
172C4⋊C4×C43—21no—
173C4⋊C4×C43—42no—
174C22⋊C4×C43—24no—
175C22⋊C4×C43—24no—
176C42×C43—21no—
177C42×C43—21no—
178C22⋊Dic43—42no—
179C22⋊Dic43—42no—
180C22⋊Dic43—42no—
181#10621884—44no—
182#10621882—44no—
183(C2×D43)⋊C4—42no—
184(C2×D43)⋊C4—42no—
185(C2×D43)⋊C4—42no—
186(C2×D43)⋊C4—42no—
187(C2×D43)⋊C4—42no—
188(C2×D43)⋊C4—42no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—42no—
196C4⋊Dic43—42no—
197Dic43⋊C4—42no—
198C4×Dic43—21no—
199C4×Dic43—21no—
200C4×Dic43—21no—
201C4×Dic43—21no—
202C4×Dic43—22no—
203C4×Dic43—22no—
204C4×Dic43—22no—
205C4×Dic43—41no—
206C4×Dic43—42no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8#4112yes—
212C2×C43⋊C8#4112yes—
213C23×C43—22no—
214C23×C43—41no—
215C23×C43—41no—
216C23×C43—41no—
217C23×C43—41no—
218C23×C43—41no—
219C23×C43—41no—
220C23×C43C22×D411yescharacteristic
221C22×D43—41no—
222C22×D43—41no—
223C22×D43—41no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228Q8×C43—22no—
229Q8×C43—22no—
230Q8×C43—41no—
231Q8×C43—41no—
232D4×C43—22no—
233D4×C43—22no—
234D4×C43—41no—
235D4×C43—41no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—22no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—41no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43C2×#4111yescharacteristic
252C2×C4×C43C2×#4111yescharacteristic
253C2×C4×C43C2×#4111yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×C4×C43H(5,𝔽2)11yescharacteristic
256C2×C4×C43H(5,𝔽2)11yescharacteristic
257C2×Dic43—21no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—84no—
276C2×Dic43#13512yes—
277C2×Dic43#13512yes—
278C2×Dic43AGL(1,ℤ/8)12yes—
279D172—41no—
280D172—41no—
281D172—41no—
282D172—41no—
283C8×C43—41no—
284C8×C43—41no—
285C43⋊C8—22no—
286C43⋊C8—22no—
287#29497—431nomaximal, Sylow
288C22×C43—42no—
289C22×C43—42no—
290C22×C43—42no—
291C22×C43—81no—
292C22×C43—81no—
293C22×C43—81no—
294C22×C43—81no—
295C22×C43#52511yescharacteristic
296C22×C43#53211yescharacteristic
297C22×C43#53511yescharacteristic
298C22×C43C2×#13511yescharacteristic
299C22×C43C2×#13511yescharacteristic
300C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C43#57511yescharacteristic
302C2×D43—81no—
303C2×D43—81no—
304C2×D43—81no—
305C2×D43—81no—
306C2×D43—81no—
307C2×D43—81no—
308C2×D43—81no—
309C4×C43—41no—
310C4×C43—41no—
311C4×C43—42no—
312C4×C43—81no—
313C4×C43—81no—
314C4×C43#55111yescharacteristic
315C4×C43#55111yescharacteristic
316C4×C43#55311yescharacteristic
317C4×C43#55311yescharacteristic
318Dic43—22no—
319Dic43—22no—
320Dic43—22no—
321Dic43—41no—
322Dic43—81no—
323Dic43—81no—
324Dic43—81no—
325#2860—431no—
326C4×#124—431no—
327#2085—431no—
328#2073—431no—
329#2042D4311yescharacteristic, complemented
330#2012—431no—
331#1970—431no—
332#1968—431no—
333#1934—431no—
334#1933—431no—
335#1880—431no—
336#1879—431no—
337#1864—431no—
338#1863—431no—
339#1744—431no—
340C2×C43—82no—
341C2×C43#260311yescharacteristic
342C2×C43#310211yescharacteristic
343C2×C43#310611yescharacteristic
344C2×C43#316811yescharacteristic
345C2×C43#318011yescharacteristic
346C2×C43#320711yescharacteristic
347C2×C43#321111yescharacteristic
348D43—161no—
349C22×SD16—431no—
350C22×D8—431no—
351C2×#126—861no—
352C2×#124C2×D4311yescharacteristic
353C2×C4×Q8—431no—
354C2×C4×D4—431no—
355C2×#107—431no—
356C2×#106—431no—
357C2×#103—431no—
358C2×#103C2×D4311yescharacteristic
359C2×#103C2×D4311yescharacteristic
360C2×#102—431no—
361C2×#102—431no—
362C2×#102—431no—
363C2×#102C2×D4311yescharacteristic
364C2×#102C2×D4311yescharacteristic
365C2×C4×C8—431no—
366C2×C4×C8—431no—
367#391—431no—
368#390—431no—
369#390—431no—
370#390—431no—
371#390—431no—
372#389—431no—
373#389—431no—
374#389C2×D4311yescharacteristic
375#388—861no—
376#388—861no—
377#388—861no—
378#388—861no—
379#388—861no—
380#382—431no—
381C4×C22⋊C4—861no—
382C4×C22⋊C4—861no—
383C43—431no—
384#340—431no—
385#340—431no—
386#340—431no—
387#340—431no—
388#336—431no—
389#336—431no—
390#336—431no—
391#336C2×D4311yescharacteristic
392C43#2949711yescharacteristic, complemented, Sylow
393C22×Q8C22×D4311yescharacteristic
394C22×D4—431no—
395C22×D4C22×D4311yescharacteristic
396C23×C4—861no—
397C2×SD16—862no—
398C2×SD16—862no—
399C2×SD16—1721no—
400C2×D8—862no—
401C2×D8—862no—
402C2×D8—1721no—
403C22×C8—431no—
404C22×C8—431no—
405C22×C8C22×D4311yescharacteristic
406C22×C8C22×D4311yescharacteristic
407#126—1724no—
408#124—22no—
409#124—22no—
410C4×Q8—1721no—
411C4×Q8—1721no—
412C4×D4—1721no—
413C4×D4—1721no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—431no—
416C2×C4⋊C4—431no—
417C2×C4⋊C4—431no—
418C2×C4⋊C4—861no—
419C2×C4⋊C4—861no—
420C2×C4⋊C4—861no—
421C2×C4⋊C4C22×D4311yescharacteristic
422C2×C4⋊C4C22×D4311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—861no—
426C2×C22⋊C4—861no—
427C2×C22⋊C4—861no—
428C2×C22⋊C4—861no—
429C2×C22⋊C4—861no—
430C2×C22⋊C4—861no—
431C2×C22⋊C4—861no—
432C2×C22⋊C4—861no—
433C2×C22⋊C4—861no—
434C2×C42—431no—
435C2×C42—431no—
436C2×C42—431no—
437C2×C42—431no—
438C2×C42—431no—
439C2×C42—431no—
440C2×C42—861no—
441C2×C42C22×D4311yescharacteristic
442#107—1721no—
443#106—1721no—
444#103—22no—
445#103—41no—
446#103—862no—
447#102—22no—
448#102—41no—
449#102—862no—
450#102—1721no—
451#102—1721no—
452C4×C8—862no—
453C4×C8—862no—
454#95—21no—
455#95—861no—
456#95—861no—
457#95—861no—
458#95—861no—
459#95—861no—
460#95—861no—
461C24—21no—
462C24—861no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8#1062190012yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—861no—
475C2×D4—861no—
476C2×D4—861no—
477C2×D4—861no—
478C2×D4—861no—
479C2×D4—861no—
480C2×D4#1062190012yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—431no—
485C22×C4—431no—
486C22×C4—431no—
487C22×C4—431no—
488C22×C4—861no—
489C22×C4—861no—
490C22×C4—861no—
491C22×C4—1722no—
492C22×C4—1722no—
493C22×C4—1722no—
494C22×C4D4×D4311yescharacteristic
495C22×C4D4×D4311yescharacteristic
496C22×C4C23×D4311yescharacteristic
497SD16—1724no—
498D8—1724no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—432no—
506C2×C8—432no—
507C2×C8—861no—
508C2×C8—861no—
509C2×C8—861no—
510C2×C8—861no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C4⋊C4—861no—
519C4⋊C4—861no—
520C4⋊C4—861no—
521C4⋊C4—861no—
522C4⋊C4—1722no—
523C4⋊C4—1722no—
524C4⋊C4—1722no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—1722no—
528C22⋊C4—1722no—
529C22⋊C4—1722no—
530C22⋊C4—1722no—
531C22⋊C4—1722no—
532C22⋊C4—1722no—
533C22⋊C4—1722no—
534C22⋊C4—1722no—
535C22⋊C4—1722no—
536C42—21no—
537C42—21no—
538C42—862no—
539C42—862no—
540C42—862no—
541C42—861no—
542C42—861no—
543C42—861no—
544C42—861no—
545C42—1721no—
546C42—1722no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—1721no—
555C23—1721no—
556C23—1721no—
557C23—1721no—
558C23—1721no—
559C23—1721no—
560C23—1721no—
561C23C2×D4×D4311yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—41no—
565Q8—41no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—1721no—
571D4—1721no—
572D4—1721no—
573D4—1721no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—432no—
590C2×C4—432no—
591C2×C4—432no—
592C2×C4—861no—
593C2×C4—861no—
594C2×C4—861no—
595C2×C4—861no—
596C2×C4—861no—
597C2×C4—861no—
598C2×C4—861no—
599C2×C4—861no—
600C2×C4—861no—
601C2×C4—3444no—
602C2×C4—1721no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4—1721no—
606C2×C4—1721no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4—1721no—
610C2×C4—1721no—
611C2×C4#41×D4311yescharacteristic
612C2×C4#41×D4311yescharacteristic
613C2×C4C2×#1062190011yescharacteristic
614C2×C4C2×#1062190011yescharacteristic
615C2×C4#1306337011yescharacteristic
616C2×C4#1306337111yescharacteristic
617C8—41no—
618C8—41no—
619C8—862no—
620C8—862no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—3441no—
629C22—3441no—
630C22—3441no—
631C22—3441no—
632C22—3441no—
633C22—3441no—
634C22—3441no—
635C22#1458340511yescharacteristic
636C22#1458344811yescharacteristic
637C22#1458346511yescharacteristic
638C22#135×D4311yescharacteristic
639C22#135×D4311yescharacteristic
640C22AGL(1,ℤ/8)×D4311yescharacteristic
641C22#1458369811yescharacteristic
642C4—42no—
643C4—41no—
644C4—41no—
645C4—81no—
646C4—81no—
647C4—862no—
648C4—862no—
649C4—862no—
650C4—3441no—
651C4—3441no—
652C4—3441no—
653C4—1721no—
654C4#1458357211yescharacteristic
655C4#1458357211yescharacteristic
656C4#1458359211yescharacteristic
657C4#1458359211yescharacteristic
658C2—82no—
659C2—6881no—
660C2#1914021711yescharacteristic
661C2#1914461411yescharacteristic
662C2#1914464811yescharacteristic
663C2#1914516411yescharacteristic
664C2#1914529611yescharacteristic
665C2#1914552711yescharacteristic
666C2#1914556411yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2042C2
socleC23×C43C22×D4
prime radicalC22×#10621882C22
hypercenter#2042D43
hypocenterC43#29497
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
1. #19142963C2semidirect
2. #124×Dic43C2semidirect
3. #19136076C2semidirect
4. #19136010C2semidirect
7. #19135446C2semidirect
8. #19135280C2semidirect
9. #19135273C2semidirect
10. #19134948C2semidirect
12. #19134854C2semidirect
13. #19134847C2semidirect
14. #19134339C2semidirect
15. C43×#2042C2semidirect
329. #2042D43semidirect
392. C43#29497semidirect

Maximal subgroups

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

subgroupindex
#191429632
#124×Dic432
#191360762
#191360102
#191356722
#191354582
#191354462
#191352802
#191352732
#191349482
#191349412
#191348542
#191348472
#191343392
C43×#20422
#2949743