Finite Group Databasebeta

← #25023584

Subgroups — #25023584

Order 11008.

1622 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582806C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×Dic43C211yesmaximal, characteristic
3#19132866C211yesmaximal, characteristic, complemented
4#19132860C211yesmaximal, characteristic, complemented
5#19132857C212yesmaximal, complemented
6#19132840C211yesmaximal, characteristic
7#19132840C212yesmaximal
8#19132829C212yesmaximal, complemented
9#19132820C211yesmaximal, characteristic, complemented
10#19132817C212yesmaximal
11C2×C43×#393C211yesmaximal, characteristic
12C24×Dic43C2211yescharacteristic
13C22×C4⋊Dic43C2211yescharacteristic
14C22×C4⋊Dic43C2212yes—
15C22×C22⋊Dic43C2211yescharacteristic
16C22×C22⋊Dic43C2212yes—
17C22×C4×Dic43C2211yescharacteristic
18C22×C4×Dic43C2211yescharacteristic
19#14582806—24no—
20C22⋊C4×Dic43—24no—
21C2×C43⋊#95C2212yes—
22C2×C43⋊#95C2212yes—
23C2×C43⋊#95C2212yes—
24C2×C43⋊#95C2211yescharacteristic
25C2×C43⋊#95C2211yescharacteristic
26C2×C43⋊#95C2211yescharacteristic
27C2×C43⋊#95C2212yes—
28C2×C43⋊#95C2212yes—
29C2×C43⋊#95C2212yes—
30C2×C43⋊#95C2212yes—
31C2×C43⋊#95C2212yes—
32C23×Q8×C43C2211yescharacteristic
33C22×C22⋊C4×C43C2211yescharacteristic
34C22×C22⋊C4×C43C2212yes—
35C43×#393C418yescomplemented
36C2×#95×C43C2211yescharacteristic
37C2×#95×C43C2212yes—
38C23×Dic43—22no—
39C23×Dic43—22no—
40C23×Dic43—42no—
41C23×Dic43—42no—
42C23×Dic43Q814yescomplemented
43C23×Dic43C2312yes—
44C23×Dic43C2312yes—
45C23×Dic43C2311yescharacteristic
46C23×Dic43C2311yescharacteristic
47C23×Dic43C2311yescharacteristic
48C23×Dic43C2311yescharacteristic
49C2×C4⋊Dic43—22no—
50C2×C4⋊Dic43—24no—
51C2×C4⋊Dic43—42no—
52C2×C4⋊Dic43—42no—
53C2×C4⋊Dic43—42no—
54C2×C22⋊Dic43—22no—
55C2×C22⋊Dic43—22no—
56C2×C22⋊Dic43—22no—
57C2×C22⋊Dic43—24no—
58C2×C22⋊Dic43—42no—
59C2×C22⋊Dic43—42no—
60C2×C4×Dic43—22no—
61C2×C4×Dic43—42no—
62C2×C4×Dic43—42no—
63C2×C4×Dic43D414yes—
64C43⋊#95—22no—
65C43⋊#95—22no—
66C43⋊#95—24no—
67C43⋊#95—24no—
68C43⋊#95—22no—
69C43⋊#95—24no—
70C43⋊#95—42no—
71C43⋊#95—42no—
72C43⋊#95—42no—
73C43⋊#95—42no—
74C43⋊#95—42no—
75C25×C43C2311yescharacteristic
76C22×Q8×C43—22no—
77C22×Q8×C43—22no—
78C22×Q8×C43—42no—
79C22×Q8×C43—42no—
80C22×Q8×C43C2×C414yescomplemented
81C23×C4×C43C2311yescharacteristic
82C23×C4×C43C2312yes—
83C23×C4×C43C2312yes—
84C23×C4×C43C2311yescharacteristic
85C2×C22⋊C4×C43—22no—
86C2×C22⋊C4×C43—22no—
87C2×C22⋊C4×C43—42no—
88C2×C22⋊C4×C43—42no—
89C2×C22⋊C4×C43C2×C414yes—
90C2×C22⋊C4×C43C2×C418yes—
91#95×C43C2×C418yescomplemented
92#95×C43C2×C414yescomplemented
93C24×C43—21no—
94C24×C43—21no—
95C24×C43—21no—
96C24×C43—21no—
97C24×C43—22no—
98C24×C43—22no—
99C24×C43—22no—
100C24×C43—22no—
101C24×C43C22×C412yes—
102C24×C43C2×Q812yes—
103C24×C43#4112yes—
104C24×C43C2411yescharacteristic
105C2×Q8×C43—28no—
106C2×Q8×C43—48no—
107C2×Q8×C43—44no—
108C2×Q8×C43—44no—
109C2×Q8×C43—48no—
110C22×C4×C43—21no—
111C22×C4×C43—22no—
112C22×C4×C43—21no—
113C22×C4×C43—22no—
114C22×C4×C43—22no—
115C22×C4×C43—22no—
116C22×C4×C43—22no—
117C22×C4×C43—22no—
118C22×C4×C43—22no—
119C22×C4×C43—22no—
120C22×C4×C43—22no—
121C22×C4×C43—22no—
122C22×C4×C43—22no—
123C22×C4×C43—22no—
124C22×C4×C43—22no—
125C22×C4×C43—22no—
126C22×C4×C43—22no—
127C22×C4×C43—22no—
128C22×C4×C43C22×C412yes—
129C22×C4×C43C22×C414yes—
130C22×C4×C43C22×C414yes—
131C22×C4×C43C22×C412yes—
132C22×C4×C43C2×D412yes—
133C22×C4×C43#4112yes—
134C22×Dic43—28no—
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—22no—
138C22×Dic43—21no—
139C22×Dic43—21no—
140C22×Dic43—21no—
141C22×Dic43—21no—
142C22×Dic43—22no—
143C22×Dic43—22no—
144C22×Dic43—22no—
145C22×Dic43—22no—
146C22×Dic43—22no—
147C22×Dic43—22no—
148C22×Dic43—22no—
149C22×Dic43—22no—
150C22×Dic43—22no—
151C22×Dic43—22no—
152C22×Dic43—22no—
153C22×Dic43—22no—
154C22×Dic43—22no—
155C22×Dic43—22no—
156C22×Dic43—22no—
157C22×Dic43—22no—
158C22×Dic43—22no—
159C22×Dic43—22no—
160C22×Dic43—22no—
161C22×Dic43—22no—
162C22×Dic43—22no—
163C22×Dic43—48no—
164C22×Dic43—44no—
165C22×Dic43—44no—
166C22×Dic43—48no—
167C22×Dic43C2×D412yes—
168C22×Dic43C2×D412yes—
169C22×Dic43C2×Q812yes—
170C22×Dic43C2×Q812yes—
171C22×Dic43#4112yes—
172C22×Dic43#4112yes—
173C22⋊C4×C43—28no—
174C22⋊C4×C43—48no—
175C22⋊Dic43—28no—
176C22⋊Dic43—48no—
177C4⋊Dic43—48no—
178C4⋊Dic43—44no—
179C4×Dic43—44no—
180C4×Dic43—44no—
181C23×C43—22no—
182C23×C43—22no—
183C23×C43—24no—
184C23×C43—24no—
185C23×C43—22no—
186C23×C43—22no—
187C23×C43—22no—
188C23×C43—22no—
189C23×C43—24no—
190C23×C43—24no—
191C23×C43—24no—
192C23×C43—24no—
193C23×C43—41no—
194C23×C43—41no—
195C23×C43—41no—
196C23×C43—41no—
197C23×C43—42no—
198C23×C43—42no—
199C23×C43—42no—
200C23×C43—42no—
201C23×C43—42no—
202C23×C43—42no—
203C23×C43C4×Q814yes—
204C23×C43C23×C411yescharacteristic
205C23×C43C22×D411yescharacteristic
206C23×C43C22×Q811yescharacteristic
207C23×C43C2×#4111yescharacteristic
208C23×C43C2×#4111yescharacteristic
209C23×C43C2×#4111yescharacteristic
210C23×C43C2×#4111yescharacteristic
211C23×C43H(5,𝔽2)12yes—
212C23×C43H(5,𝔽2)12yes—
213C23×C43H(5,𝔽2)12yes—
214C23×C432−1+412yes—
215Q8×C43—416no—
216C2×C4×C43—24no—
217C2×C4×C43—22no—
218C2×C4×C43—22no—
219C2×C4×C43—24no—
220C2×C4×C43—22no—
221C2×C4×C43—24no—
222C2×C4×C43—24no—
223C2×C4×C43—24no—
224C2×C4×C43—24no—
225C2×C4×C43—24no—
226C2×C4×C43—24no—
227C2×C4×C43—24no—
228C2×C4×C43—42no—
229C2×C4×C43—41no—
230C2×C4×C43—42no—
231C2×C4×C43—41no—
232C2×C4×C43—42no—
233C2×C4×C43—42no—
234C2×C4×C43—42no—
235C2×C4×C43—42no—
236C2×C4×C43—42no—
237C2×C4×C43—42no—
238C2×C4×C43—42no—
239C2×C4×C43C4×D414yes—
240C2×Dic43—22no—
241C2×Dic43—22no—
242C2×Dic43—24no—
243C2×Dic43—24no—
244C2×Dic43—22no—
245C2×Dic43—22no—
246C2×Dic43—22no—
247C2×Dic43—22no—
248C2×Dic43—22no—
249C2×Dic43—24no—
250C2×Dic43—24no—
251C2×Dic43—416no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—42no—
257C2×Dic43—42no—
258C2×Dic43—42no—
259C2×Dic43—42no—
260C2×Dic43—42no—
261C2×Dic43—42no—
262C2×Dic43—42no—
263C2×Dic43—42no—
264C2×Dic43—42no—
265C2×Dic43—42no—
266C2×Dic43—42no—
267C2×Dic43—42no—
268C2×Dic43—42no—
269C2×Dic43—42no—
270C2×Dic43—42no—
271C2×Dic43—42no—
272C2×Dic43—42no—
273C2×Dic43—42no—
274C2×Dic43C22⋊Q814yes—
275#14319—431nomaximal, Sylow
276C22×C43—28no—
277C22×C43—24no—
278C22×C43—42no—
279C22×C43—42no—
280C22×C43—44no—
281C22×C43—42no—
282C22×C43—42no—
283C22×C43—42no—
284C22×C43—42no—
285C22×C43—44no—
286C22×C43—44no—
287C22×C43C2×C4×D411yescharacteristic
288C22×C43C2×C4×Q811yescharacteristic
289C22×C43C4×#4111yescharacteristic
290C22×C43#51812yes—
291C22×C43#52012yes—
292C22×C43C2×C22⋊Q811yescharacteristic
293C22×C43#52511yescharacteristic
294C22×C43#52911yescharacteristic
295C22×C43#53311yescharacteristic
296C22×C43#53411yescharacteristic
297C22×C43#53511yescharacteristic
298C22×C43#53511yescharacteristic
299C22×C43#53611yescharacteristic
300C22×C43#53812yes—
301C22×C43#53812yes—
302C22×C43#53812yes—
303C22×C43#53812yes—
304C22×C43#53912yes—
305C22×C43#54212yes—
306C22×C43#54212yes—
307C22×C43#54212yes—
308C22×C43#54312yes—
309C22×C43#54412yes—
310C4×C43—24no—
311C4×C43—44no—
312C4×C43—42no—
313C4×C43—44no—
314Dic43—24no—
315Dic43—42no—
316Dic43—42no—
317Dic43—42no—
318Dic43—82no—
319Dic43—82no—
320C2×#393D4311yescharacteristic
321C2×#391—431no—
322C2×C4×C22⋊C4—431no—
323#1446—431no—
324#1444—431no—
325#1443—432no—
326#1439—432no—
327#1439—431no—
328#1436—432no—
329#1434—431no—
330C22⋊#95—432no—
331C2×C43—44no—
332C2×C43C4×C22⋊Q811yescharacteristic
333C2×C43#232111yescharacteristic
334C2×C43#232611yescharacteristic
335C2×C43#232912yes—
336C2×C43#233712yes—
337C2×C43#262611yescharacteristic
338C2×C43#262811yescharacteristic
339C2×C43#263111yescharacteristic
340C2×C43#263811yescharacteristic
341C2×C43#265312yes—
342C2×C43#266512yes—
343C23×Q8C2×D4311yescharacteristic
344C24×C4—431no—
345C22×C4⋊C4—432no—
346C22×C4⋊C4—431no—
347C22×C22⋊C4—431no—
348C22×C22⋊C4—432no—
349C22×C22⋊C4C2×D4311yescharacteristic
350C22×C22⋊C4C2×D4312yes—
351C22×C42—431no—
352C22×C42—431no—
353#393Dic4318yescomplemented
354#391—864no—
355C4×C22⋊C4—864no—
356C2×#95—432no—
357C2×#95—432no—
358C2×#95—431no—
359C2×#95—432no—
360C2×#95—431no—
361C2×#95—431no—
362C2×#95—432no—
363C2×#95—432no—
364C2×#95—432no—
365C2×#95—432no—
366C2×#95—432no—
367C2×#95C2×D4311yescharacteristic
368C2×#95C2×D4312yes—
369C43#1431911yescharacteristic, complemented, Sylow
370C25C22×D4311yescharacteristic
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—42no—
374C22×Q8—42no—
375C22×Q8C2×Dic4314yescomplemented
376C23×C4—431no—
377C23×C4—431no—
378C23×C4—431no—
379C23×C4—431no—
380C23×C4—434no—
381C23×C4—432no—
382C23×C4—432no—
383C23×C4—862no—
384C23×C4—862no—
385C23×C4—1722no—
386C23×C4—1722no—
387C23×C4C22×D4312yes—
388C23×C4C22×D4312yes—
389C23×C4C22×D4311yescharacteristic
390C23×C4C22×D4311yescharacteristic
391C2×C4⋊C4—864no—
392C2×C4⋊C4—862no—
393C2×C4⋊C4—1722no—
394C2×C4⋊C4—1722no—
395C2×C4⋊C4—1722no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—862no—
401C2×C22⋊C4—862no—
402C2×C22⋊C4—862no—
403C2×C22⋊C4—864no—
404C2×C22⋊C4—1722no—
405C2×C22⋊C4—1722no—
406C2×C22⋊C4C2×Dic4314yes—
407C2×C22⋊C4C2×Dic4318yes—
408C2×C42—434no—
409C2×C42—862no—
410C2×C42—1722no—
411C2×C42—1722no—
412#95—864no—
413#95—862no—
414#95—864no—
415#95—864no—
416#95—862no—
417#95—862no—
418#95—1722no—
419#95—1722no—
420#95—1722no—
421#95—1722no—
422#95—1722no—
423#95C2×Dic4318yescomplemented
424#95C2×Dic4314yescomplemented
425C24—21no—
426C24—21no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24Q8×D4312yes—
434C24#1062190212yes—
435C24C22×Dic4312yes—
436C24C23×D4311yescharacteristic
437C2×Q8—28no—
438C2×Q8—48no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—48no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—432no—
461C22×C4—432no—
462C22×C4—432no—
463C22×C4—432no—
464C22×C4—432no—
465C22×C4—432no—
466C22×C4—868no—
467C22×C4—862no—
468C22×C4—862no—
469C22×C4—861no—
470C22×C4—861no—
471C22×C4—861no—
472C22×C4—861no—
473C22×C4—861no—
474C22×C4—861no—
475C22×C4—862no—
476C22×C4—862no—
477C22×C4—862no—
478C22×C4—862no—
479C22×C4—862no—
480C22×C4—862no—
481C22×C4—862no—
482C22×C4—862no—
483C22×C4—862no—
484C22×C4—862no—
485C22×C4—862no—
486C22×C4—862no—
487C22×C4—862no—
488C22×C4—862no—
489C22×C4—862no—
490C22×C4—862no—
491C22×C4—862no—
492C22×C4—862no—
493C22×C4—862no—
494C22×C4—862no—
495C22×C4—1728no—
496C22×C4—1724no—
497C22×C4—1724no—
498C22×C4—1728no—
499C22×C4D4×D4312yes—
500C22×C4#1062190012yes—
501C22×C4C22×Dic4312yes—
502C22×C4C22×Dic4314yes—
503C22×C4C22×Dic4314yes—
504C22×C4C22×Dic4312yes—
505C4⋊C4—1724no—
506C4⋊C4—1728no—
507C22⋊C4—28no—
508C22⋊C4—48no—
509C22⋊C4—868no—
510C22⋊C4—1728no—
511C42—1724no—
512C42—1724no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23Q8×Dic4314yes—
536C23C23×Dic4311yescharacteristic
537C23C2×D4×D4311yescharacteristic
538C23C2×Q8×D4311yescharacteristic
539C23#41×D4311yescharacteristic
540C23#41×D4311yescharacteristic
541C23C2×#1062190011yescharacteristic
542C23C2×#1062190211yescharacteristic
543C23#1306337012yes—
544C23#1306337012yes—
545C23#1306337112yes—
546C23#1306337212yes—
547Q8—416no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—434no—
572C2×C4—862no—
573C2×C4—862no—
574C2×C4—864no—
575C2×C4—864no—
576C2×C4—862no—
577C2×C4—862no—
578C2×C4—862no—
579C2×C4—862no—
580C2×C4—862no—
581C2×C4—864no—
582C2×C4—864no—
583C2×C4—17216no—
584C2×C4—1722no—
585C2×C4—1722no—
586C2×C4—1722no—
587C2×C4—1722no—
588C2×C4—1722no—
589C2×C4—1722no—
590C2×C4—1722no—
591C2×C4—1722no—
592C2×C4—1722no—
593C2×C4—1722no—
594C2×C4—1722no—
595C2×C4—1722no—
596C2×C4—1722no—
597C2×C4—1722no—
598C2×C4—1721no—
599C2×C4—1721no—
600C2×C4—1721no—
601C2×C4—1721no—
602C2×C4—1722no—
603C2×C4—1722no—
604C2×C4—1722no—
605C2×C4—1722no—
606C2×C4D4×Dic4314yes—
607C22—28no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22C2×D4×Dic4311yescharacteristic
619C22C2×Q8×Dic4311yescharacteristic
620C22#41×Dic4311yescharacteristic
621C22#1458337412yes—
622C22#1458338212yes—
623C22C22⋊Q8×D4311yescharacteristic
624C22#1458341211yescharacteristic
625C22#1458343511yescharacteristic
626C22#1458345111yescharacteristic
627C22#1458345911yescharacteristic
628C22#1458346411yescharacteristic
629C22#1458346711yescharacteristic
630C22#1458346911yescharacteristic
631C22#1458348312yes—
632C22#1458348312yes—
633C22#1458348412yes—
634C22#1458348912yes—
635C22#1458349812yes—
636C22#1458351312yes—
637C22#1458351312yes—
638C22#1458351812yes—
639C22#1458352412yes—
640C22#1458352812yes—
641C4—24no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—864no—
646C4—3442no—
647C4—3442no—
648C4—1722no—
649C4—1722no—
650C4—1722no—
651C2—44no—
652C2C22⋊Q8×Dic4311yescharacteristic
653C2#1913725611yescharacteristic
654C2#1913730811yescharacteristic
655C2#1913733012yes—
656C2#1913741012yes—
657C2#1914039911yescharacteristic
658C2#1914041511yescharacteristic
659C2#1914045311yescharacteristic
660C2#1914051811yescharacteristic
661C2#1914066112yes—
662C2#1914076412yes—
subgroupG/H
center, FrattiniC24C23×D43
commutatorC23×C43C23×C4
FittingC2×C43×#393C2
socle, cosocleC24×C43C24
prime radicalC25×C43C23
hypercenterC2×#393D43
hypocenterC43#14319
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C2×#14582806C2semidirect
3. #19132866C2semidirect
4. #19132860C2semidirect
5. #19132857C2semidirect
8. #19132829C2semidirect
9. #19132820C2semidirect
35. C43×#393C4semidirect
42. C23×Dic43Q8semidirect
80. C22×Q8×C43C2×C4semidirect
91. #95×C43C2×C4semidirect
92. #95×C43C2×C4semidirect
353. #393Dic43semidirect
369. C43#14319semidirect
375. C22×Q8C2×Dic43semidirect
423. #95C2×Dic43semidirect
424. #95C2×Dic43semidirect

Maximal subgroups

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

subgroupindex
C2×#145828062
C2×C22⋊C4×Dic432
#191328662
#191328602
#191328572
#191328402
#191328292
#191328202
#191328172
C2×C43×#3932
#1431943