Finite Group Databasebeta

← #25133211

Subgroups — #25133211

Order 11008.

1064 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583628C211yesmaximal, characteristic, complemented
2C2×#14583198C211yesmaximal, characteristic
3C2×#14583002C211yesmaximal, characteristic
4#19139516C211yesmaximal, characteristic
5#19135935C211yesmaximal, characteristic, complemented
6#19135923C211yesmaximal, characteristic, complemented
7#19135740C211yesmaximal, characteristic
8#19135732C211yesmaximal, characteristic
9#19135562C211yesmaximal, characteristic, complemented
10#19135533C211yesmaximal, characteristic, complemented
11#19134950C211yesmaximal, characteristic
12#19134948C211yesmaximal, characteristic
13#19134847C211yesmaximal, characteristic, complemented
14#19134598C211yesmaximal, characteristic, complemented
15C43×#1865C211yesmaximal, characteristic, complemented
16C22×#10621884C2211yescharacteristic
17#14583628—24no—
18#14583628—24no—
19C2×#13063325—21no—
20C2×#13063325C2211yescharacteristic
21C2×#13063316—21no—
22C2×#13063316—21no—
23C2×#13063316C2211yescharacteristic
24C2×#13063316C2211yescharacteristic
25C2×#13063303—21no—
26C2×Q8×Dic43C2211yescharacteristic
27C2×Q8×Dic43C2211yescharacteristic
28C2×C4×Dic86—21no—
29C2×C4×Dic86C2211yescharacteristic
30C2×#13063287C2211yescharacteristic
31#14583198—24no—
32#14583002—24no—
33C2×#13063258C2211yescharacteristic
34C2×#13063258C2211yescharacteristic
35C2×Dic43⋊C8C2211yescharacteristic
36C2×#13063249C2211yescharacteristic
37C2×#13063247C2211yescharacteristic
38C2×#13063247C2211yescharacteristic
39C2×#13063245C2211yescharacteristic
40C2×#13063235C2211yescharacteristic
41#14582823—21no—
42#14582821—21no—
43#14582801C2211yescharacteristic
44#14582800C2211yescharacteristic
45#14582795C2211yescharacteristic
46#14582768C2211yescharacteristic
47#14582758—21no—
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582654C2211yescharacteristic
54#14582653C2211yescharacteristic
55#14582653C2211yescharacteristic
56C2×C4×Q8×C43C2211yescharacteristic
57C2×#106×C43C2211yescharacteristic
58C2×#103×C43C2211yescharacteristic
59C2×#103×C43C2211yescharacteristic
60C43×#389C2211yescharacteristic
61C43×#340C2211yescharacteristic
62C43×#336C2211yescharacteristic
63C22×Dic86—21no—
64C22×Dic86D411yescharacteristic
65C22×D172C2311yescharacteristic
66C22×C4×D43—21no—
67C2×#10621884—22no—
68C2×#10621884—22no—
69C2×#10621884Q814yescomplemented
70C22×C43⋊C8C2311yescharacteristic
71C22×C43⋊C8C2311yescharacteristic
72#13063325—28no—
73#13063325—22no—
74#13063325—22no—
75#13063316—28no—
76#13063316—28no—
77#13063316—22no—
78#13063316—22no—
79#13063316—22no—
80#13063316D414yes—
81#13063303—44no—
82Q8×Dic43—22no—
83Q8×Dic43—22no—
84Q8×Dic43—22no—
85Q8×Dic43D414yes—
86C4×Dic86—28no—
87C4×Dic86—22no—
88C4×Dic86—22no—
89#13063287—22no—
90#13063287—22no—
91C2×C4⋊Dic43—21no—
92C2×C4⋊Dic43—21no—
93C2×C4⋊Dic43—21no—
94C2×C4⋊Dic43—21no—
95C2×C4⋊Dic43—21no—
96C2×C4⋊Dic43D411yescharacteristic
97C2×C4⋊Dic43C2311yescharacteristic
98C2×C4⋊Dic43C2311yescharacteristic
99C2×C4⋊Dic43C2311yescharacteristic
100C2×Dic43⋊C4—21no—
101C2×Dic43⋊C4—21no—
102C2×Dic43⋊C4—21no—
103C2×Dic43⋊C4—21no—
104C2×Dic43⋊C4—21no—
105C2×Dic43⋊C4D411yescharacteristic
106C2×Dic43⋊C4D411yescharacteristic
107C2×(C2×D43)⋊C4—21no—
108C2×(C2×D43)⋊C4—21no—
109C2×(C2×D43)⋊C4—21no—
110C2×(C2×D43)⋊C4—21no—
111C2×(C2×D43)⋊C4—21no—
112C2×C4×Dic43—21no—
113C2×C4×Dic43—21no—
114C2×C4×Dic43C2311yescharacteristic
115C2×C4×Dic43C2311yescharacteristic
116#13063258—22no—
117#13063258—22no—
118Dic43⋊C8—22no—
119#13063249—22no—
120#13063247—22no—
121#13063247—22no—
122#13063245—22no—
123#13063235—22no—
124C43⋊#95—21no—
125C22×Q8×C43C2311yescharacteristic
126C22×C8×C43C2311yescharacteristic
127C22×C8×C43C2311yescharacteristic
128C4×Q8×C43—22no—
129C4×Q8×C43—22no—
130C2×C4⋊C4×C43—21no—
131C2×C4⋊C4×C43—21no—
132C2×C4⋊C4×C43C2311yescharacteristic
133C2×C4⋊C4×C43C2311yescharacteristic
134C2×C4⋊C4×C43C2311yescharacteristic
135C2×C42×C43—21no—
136C2×C42×C43C2311yescharacteristic
137#106×C43—22no—
138#103×C43—22no—
139#103×C43—22no—
140#95×C43—21no—
141C23×D43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43—21no—
145C2×Q8×C43C2×D412yes—
146C2×Q8×C43C2×Q812yes—
147C2×Q8×C43#4112yes—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43C2×D411yescharacteristic
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2411yescharacteristic
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43—21no—
158C22×Dic43C2×D411yescharacteristic
159C22×Dic43C2×D411yescharacteristic
160C22×Dic43C2×D411yescharacteristic
161C22×Dic43C2×D411yescharacteristic
162C2×D172—21no—
163C2×D172—21no—
164C2×D172—21no—
165C2×D172—21no—
166C2×D172—21no—
167C2×D172C2×Q812yes—
168C2×C4×D43—42no—
169C2×C4×D43—42no—
170C2×C4×D43—42no—
171C2×Dic86—24no—
172C2×Dic86—22no—
173C2×Dic86—22no—
174C2×Dic86—41no—
175C2×Dic86—42no—
176C2×Dic86—42no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C2×C8×C43—21no—
181C2×C8×C43—21no—
182C2×C8×C43—21no—
183C4⋊C4×C43—24no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—21no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43—21no—
189C4⋊C4×C43—42no—
190C4⋊C4×C43C2×D412yes—
191C42×C43—24no—
192C42×C43—21no—
193C42×C43—21no—
194#10621884—28no—
195(C2×D43)⋊C4—42no—
196(C2×D43)⋊C4—42no—
197(C2×D43)⋊C4—42no—
198(C2×D43)⋊C4—81no—
199(C2×D43)⋊C4—81no—
200C4⋊Dic43—24no—
201C4⋊Dic43—24no—
202C4⋊Dic43—24no—
203C4⋊Dic43—21no—
204C4⋊Dic43—21no—
205C4⋊Dic43—24no—
206C4⋊Dic43—24no—
207C4⋊Dic43—22no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210C4⋊Dic43C2×D412yes—
211C4⋊Dic43C2×D412yes—
212Dic43⋊C4—24no—
213Dic43⋊C4—24no—
214Dic43⋊C4—22no—
215Dic43⋊C4—22no—
216Dic43⋊C4—42no—
217Dic43⋊C4—42no—
218Dic43⋊C4—81no—
219C4×Dic43—24no—
220C4×Dic43—21no—
221C4×Dic43—21no—
222C4×Dic43—24no—
223C4×Dic43C2×D412yes—
224C4×Dic43C2×D412yes—
225C2×C43⋊C8—21no—
226C2×C43⋊C8—21no—
227C2×C43⋊C8—21no—
228C2×C43⋊C8—21no—
229C2×C43⋊C8—21no—
230C2×C43⋊C8C2×Q812yes—
231C23×C43C22×D411yescharacteristic
232C22×D43—22no—
233C22×D43—41no—
234C22×D43—41no—
235C22×D43—41no—
236C22×D43—41no—
237C22×D43—41no—
238C22×D43—41no—
239Q8×C43—22no—
240Q8×C43—22no—
241Q8×C43—22no—
242Q8×C43C22⋊Q814yescomplemented
243C2×C4×C43—22no—
244C2×C4×C43—21no—
245C2×C4×C43—22no—
246C2×C4×C43—22no—
247C2×C4×C43—21no—
248C2×C4×C43—21no—
249C2×C4×C43—22no—
250C2×C4×C43—21no—
251C2×C4×C43—21no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43—41no—
258C2×C4×C43—41no—
259C2×C4×C43—41no—
260C2×C4×C43AGL(1,ℤ/8)12yes—
261C2×C4×C43C22×D411yescharacteristic
262C2×C4×C43C22×D411yescharacteristic
263C2×C4×C43C22×Q811yescharacteristic
264C2×C4×C43C2×#4111yescharacteristic
265C2×C4×C432−1+411yescharacteristic
266C2×C4×C432−1+411yescharacteristic
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—21no—
270C2×Dic43—21no—
271C2×Dic43—21no—
272C2×Dic43—21no—
273C2×Dic43—22no—
274C2×Dic43—22no—
275C2×Dic43—21no—
276C2×Dic43—21no—
277C2×Dic43—21no—
278C2×Dic43—21no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—41no—
286C2×Dic43#12012yes—
287C2×Dic43#12012yes—
288D172—22no—
289D172—22no—
290D172—41no—
291D172—41no—
292C4×D43—84no—
293Dic86—44no—
294Dic86—48no—
295C8×C43—41no—
296C8×C43—41no—
297C43⋊C8—22no—
298C43⋊C8—41no—
299#24346—431nomaximal, Sylow
300C22×C43C2×#12011yescharacteristic
301C22×C43#54811yescharacteristic
302C22×C43D4×Q811yescharacteristic
303C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C43#57511yescharacteristic
305C22×C43#57711yescharacteristic
306C22×C43#57811yescharacteristic
307C2×D43—42no—
308C2×D43—42no—
309C2×D43—42no—
310C2×D43—81no—
311C2×D43—81no—
312C2×D43—81no—
313C2×D43—81no—
314C4×C43—24no—
315C4×C43—22no—
316C4×C43—41no—
317C4×C43—42no—
318C4×C43—81no—
319C4×C43—81no—
320C4×C43C2×C22⋊Q811yescharacteristic
321C4×C43#53611yescharacteristic
322C4×C43#54811yescharacteristic
323C4×C43D4×Q811yescharacteristic
324Dic43—22no—
325Dic43—22no—
326Dic43—41no—
327Dic43—42no—
328Dic43—42no—
329Dic43—41no—
330Dic43—81no—
331C2×#557—431no—
332C2×#478—431no—
333C2×#440—431no—
334#2538—431no—
335#2059—431no—
336#2057—431no—
337#2025—431no—
338#2023—431no—
339#1992—431no—
340#1985—431no—
341#1880—431no—
342#1880—431no—
343#1865D4311yescharacteristic, complemented
344#1863—431no—
345#1803—431no—
346C2×C43#240911yescharacteristic
347C2×C43#300611yescharacteristic
348C2×C43#301311yescharacteristic
349C2×C43#336811yescharacteristic
350C2×C43#337711yescharacteristic
351C2×C43#338611yescharacteristic
352C2×C43#338811yescharacteristic
353D43—82no—
354C22×SD16—431no—
355#557—864no—
356#557—864no—
357C2×C4⋊Q8—431no—
358C2×C4⋊Q8—861no—
359C2×#125—431no—
360C2×#125—431no—
361C2×#125—861no—
362C2×#125—861no—
363C2×C22⋊Q8—861no—
364C2×C4×Q8—431no—
365C2×C4×Q8—431no—
366C2×C4×Q8—431no—
367C2×C4×Q8—861no—
368C2×C4×Q8C2×D4311yescharacteristic
369C2×C4×D4—431no—
370#478—864no—
371#440—864no—
372C2×#107—431no—
373C2×#107—431no—
374C2×#106C2×D4311yescharacteristic
375C2×C4⋊C8—431no—
376C2×#103—431no—
377C2×#103C2×D4311yescharacteristic
378C2×#103C2×D4311yescharacteristic
379C2×#102—431no—
380C2×#102—431no—
381C2×#102—431no—
382C2×#97—431no—
383#396—861no—
384#396—861no—
385#390—431no—
386#390—431no—
387#389—431no—
388#389C2×D4311yescharacteristic
389#384—431no—
390#380—861no—
391#340—431no—
392#340—431no—
393#340—431no—
394#340—431no—
395#340—431no—
396#340C2×D4311yescharacteristic
397#336—431no—
398#336—431no—
399#336—431no—
400#336C2×D4311yescharacteristic
401C43#2434611yescharacteristic, complemented, Sylow
402C22×Q8—431no—
403C22×Q8—861no—
404C22×Q8C22×D4311yescharacteristic
405C22×D4—431no—
406C23×C4—861no—
407C2×SD16—434no—
408C2×SD16—862no—
409C2×SD16—862no—
410C22×C8—431no—
411C22×C8—431no—
412C22×C8C22×D4311yescharacteristic
413C22×C8C22×D4311yescharacteristic
414C4⋊Q8—862no—
415C4⋊Q8—862no—
416C4⋊Q8—868no—
417#125—434no—
418#125—862no—
419#125—862no—
420#125—868no—
421#125—868no—
422#125—862no—
423C22⋊Q8—1724no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×Q8—434no—
427C4×Q8—862no—
428C4×Q8—862no—
429C4×Q8—868no—
430C4×Q8—862no—
431C4×Q8—862no—
432C4×Q8—862no—
433C4×D4—862no—
434C4×D4—862no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—431no—
438C2×C4⋊C4—431no—
439C2×C4⋊C4—431no—
440C2×C4⋊C4—431no—
441C2×C4⋊C4—431no—
442C2×C4⋊C4—431no—
443C2×C4⋊C4—861no—
444C2×C4⋊C4—861no—
445C2×C4⋊C4—861no—
446C2×C4⋊C4—861no—
447C2×C4⋊C4—861no—
448C2×C4⋊C4—861no—
449C2×C4⋊C4—861no—
450C2×C4⋊C4—861no—
451C2×C4⋊C4—861no—
452C2×C4⋊C4—861no—
453C2×C4⋊C4C22×D4311yescharacteristic
454C2×C4⋊C4C22×D4311yescharacteristic
455C2×C4⋊C4C22×D4311yescharacteristic
456C2×C22⋊C4—861no—
457C2×C22⋊C4—861no—
458C2×C22⋊C4—861no—
459C2×C22⋊C4—861no—
460C2×C22⋊C4—861no—
461C2×C42—21no—
462C2×C42—431no—
463C2×C42—431no—
464C2×C42—861no—
465C2×C42—861no—
466C2×C42C22×D4311yescharacteristic
467#107—862no—
468#107—862no—
469#106—22no—
470C4⋊C8—862no—
471#103—22no—
472#103—22no—
473#103—862no—
474#102—862no—
475#102—862no—
476#102—862no—
477#97—862no—
478#95—21no—
479#95—861no—
480C24—861no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—862no—
485C2×Q8—864no—
486C2×Q8—862no—
487C2×Q8—1721no—
488C2×Q8—1722no—
489C2×Q8—1722no—
490C2×Q8#1062189812yes—
491C2×Q8D4×D4312yes—
492C2×Q8Q8×D4312yes—
493C2×D4—432no—
494C2×D4—861no—
495C2×D4—861no—
496C2×D4—861no—
497C2×D4—861no—
498C2×D4—861no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—431no—
504C22×C4—431no—
505C22×C4—431no—
506C22×C4—431no—
507C22×C4—861no—
508C22×C4—861no—
509C22×C4—861no—
510C22×C4—1722no—
511C22×C4—1722no—
512C22×C4—1722no—
513C22×C4D4×D4311yescharacteristic
514C22×C4D4×D4311yescharacteristic
515C22×C4C23×D4311yescharacteristic
516SD16—868no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—432no—
524C2×C8—861no—
525C2×C8—861no—
526C2×C8—861no—
527C2×C8—861no—
528C2×C8—861no—
529C4⋊C4—24no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—432no—
537C4⋊C4—432no—
538C4⋊C4—862no—
539C4⋊C4—862no—
540C4⋊C4—862no—
541C4⋊C4—864no—
542C4⋊C4—864no—
543C4⋊C4—864no—
544C4⋊C4—864no—
545C4⋊C4—864no—
546C4⋊C4—864no—
547C4⋊C4—864no—
548C4⋊C4—861no—
549C4⋊C4—861no—
550C4⋊C4—861no—
551C4⋊C4—861no—
552C4⋊C4—3441no—
553C4⋊C4—1722no—
554C4⋊C4—1722no—
555C4⋊C4D4×D4312yes—
556C22⋊C4—3441no—
557C22⋊C4—3441no—
558C22⋊C4—1722no—
559C22⋊C4—1722no—
560C22⋊C4—1722no—
561C42—24no—
562C42—21no—
563C42—21no—
564C42—432no—
565C42—432no—
566C42—864no—
567C42—864no—
568C42—861no—
569C42—861no—
570C23—862no—
571C23—1721no—
572C23—1721no—
573C23—1721no—
574C23—1721no—
575C23—1721no—
576C23—1721no—
577C23C2×D4×D4311yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—1724no—
582Q8—1728no—
583Q8#1306330314yescomplemented
584D4—862no—
585D4—862no—
586D4—1721no—
587D4—1721no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—432no—
606C2×C4—432no—
607C2×C4—861no—
608C2×C4—861no—
609C2×C4—861no—
610C2×C4—861no—
611C2×C4—861no—
612C2×C4—862no—
613C2×C4—862no—
614C2×C4—861no—
615C2×C4—861no—
616C2×C4—861no—
617C2×C4—861no—
618C2×C4—861no—
619C2×C4—3444no—
620C2×C4—1721no—
621C2×C4—1721no—
622C2×C4—1721no—
623C2×C4—1721no—
624C2×C4—1721no—
625C2×C4—1721no—
626C2×C4—1721no—
627C2×C4#1306335212yes—
628C2×C4C2×D4×D4311yescharacteristic
629C2×C4C2×D4×D4311yescharacteristic
630C2×C4C2×Q8×D4311yescharacteristic
631C2×C4C2×#1062189811yescharacteristic
632C2×C4#1306337211yescharacteristic
633C2×C4#1306337211yescharacteristic
634C8—41no—
635C8—41no—
636C8—862no—
637C8—1721no—
638C22—3441no—
639C22—3441no—
640C22—3441no—
641C22—3441no—
642C22—1722no—
643C22—1722no—
644C22—1722no—
645C22#120×D4311yescharacteristic
646C22#1458355811yescharacteristic
647C22#1458356211yescharacteristic
648C22C2×#1306335211yescharacteristic
649C22#1458369811yescharacteristic
650C22#1458370611yescharacteristic
651C22#1458371511yescharacteristic
652C4—22no—
653C4—24no—
654C4—41no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—862no—
659C4—862no—
660C4—3441no—
661C4—1721no—
662C4—1722no—
663C4—1722no—
664C4—1721no—
665C4C2×#1306330311yescharacteristic
666C4#1458347111yescharacteristic
667C4#1458355811yescharacteristic
668C4#1458356211yescharacteristic
669C2—3442no—
670C2#1913795911yescharacteristic
671C2#1914384611yescharacteristic
672C2#1914391411yescharacteristic
673C2#1914732211yescharacteristic
674C2#1914744211yescharacteristic
675C2#1914751511yescharacteristic
676C2#1914754311yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1865C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#1865D43
hypocenterC43#24346
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
1. C2×#14583628C2semidirect
5. #19135935C2semidirect
6. #19135923C2semidirect
9. #19135562C2semidirect
10. #19135533C2semidirect
13. #19134847C2semidirect
14. #19134598C2semidirect
15. C43×#1865C2semidirect
69. C2×#10621884Q8semidirect
242. Q8×C43C22⋊Q8semidirect
343. #1865D43semidirect
401. C43#24346semidirect
583. Q8#13063303semidirect

Maximal subgroups

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

subgroupindex
C2×#145836282
C2×#145831982
C2×#145830022
#191395162
#191359352
#191359232
#191357402
#191357322
#191355622
#191355332
#191349502
#191349482
#191348472
#191345982
C43×#18652
#2434643