Finite Group Databasebeta

← #25114884

Subgroups — #25114884

Order 11008.

1018 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582999C211yesmaximal, characteristic
2C2×#14582996C211yesmaximal, characteristic
3C2×#14582990C211yesmaximal, characteristic
4#19137509C211yesmaximal, characteristic, complemented
5D4×Dic43⋊C4C211yesmaximal, characteristic
6#19135337C211yesmaximal, characteristic, complemented
7#19135335C211yesmaximal, characteristic, complemented
8#19134852C211yesmaximal, characteristic, complemented
9#19134846C211yesmaximal, characteristic, complemented
10#19134844C211yesmaximal, characteristic
11#19134753C211yesmaximal, characteristic, complemented
12#19134599C211yesmaximal, characteristic, complemented
13#19134593C211yesmaximal, characteristic
14#19134461C211yesmaximal, characteristic, complemented
15C43×#1862C211yesmaximal, characteristic
16C22×#10621883C2211yescharacteristic
17C2×C4×Dic86C2211yescharacteristic
18C2×#13063290C2211yescharacteristic
19C2×#13063290C2211yescharacteristic
20C2×D4×Dic43C2211yescharacteristic
21C2×D4×Dic43C2211yescharacteristic
22C22×Dic43⋊C4—21no—
23#14582999—24no—
24#14582996—24no—
25#14582990C418yescomplemented
26C2×#13063259C2211yescharacteristic
27C2×#13063258C2211yescharacteristic
28C2×#13063258C2211yescharacteristic
29C2×#13063256C2211yescharacteristic
30C2×Dic43⋊C8C2211yescharacteristic
31C2×#13063250C2211yescharacteristic
32C2×#13063250C2211yescharacteristic
33C2×#13063248C2211yescharacteristic
34C2×#13063246C2211yescharacteristic
35C2×#13063236C2211yescharacteristic
36C2×#13063235C2211yescharacteristic
37C2×C8×Dic43C2211yescharacteristic
38#14582798C2211yescharacteristic
39#14582797C2211yescharacteristic
40#14582796—21no—
41#14582795C2211yescharacteristic
42#14582794—21no—
43#14582772—21no—
44#14582771—21no—
45#14582768—21no—
46#14582768C2211yescharacteristic
47#14582768C2211yescharacteristic
48#14582757—21no—
49C4×Dic43⋊C4C2211yescharacteristic
50#14582750—21no—
51#14582662C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582661C2211yescharacteristic
54#14582653C2211yescharacteristic
55C2×C4×D4×C43C2211yescharacteristic
56C2×#107×C43C2211yescharacteristic
57C2×#102×C43C2211yescharacteristic
58C2×#102×C43C2211yescharacteristic
59C43×#389C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C43×#336C2211yescharacteristic
62C22×Dic86C2311yescharacteristic
63C23×Dic43—21no—
64C23×Dic43—21no—
65C2×#10621883C2×C414yes—
66C2×#10621883D414yes—
67C2×#10621883Q814yes—
68C22×C43⋊C8C2311yescharacteristic
69C22×C43⋊C8C2311yescharacteristic
70C4×Dic86—22no—
71C4×Dic86—41no—
72#13063290—22no—
73#13063290—22no—
74#13063290—41no—
75#13063290C2×C414yescomplemented
76D4×Dic43—22no—
77D4×Dic43—22no—
78D4×Dic43—41no—
79D4×Dic43C2×C414yes—
80C2×C4⋊Dic43—21no—
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89C2×Dic43⋊C4—21no—
90C2×Dic43⋊C4—21no—
91C2×Dic43⋊C4—21no—
92C2×Dic43⋊C4—21no—
93C2×Dic43⋊C4—44no—
94C2×Dic43⋊C4D411yescharacteristic
95C2×Dic43⋊C4D411yescharacteristic
96C2×C22⋊Dic43—21no—
97C2×C22⋊Dic43—21no—
98C2×C4×Dic43—21no—
99C2×C4×Dic43—21no—
100C2×C4×Dic43—21no—
101C2×C4×Dic43C2311yescharacteristic
102C2×C4×Dic43C2311yescharacteristic
103#13063259—41no—
104#13063258—22no—
105#13063258—22no—
106#13063256C2×C414yescomplemented
107Dic43⋊C8—22no—
108#13063250—22no—
109#13063250C2×C414yescomplemented
110#13063248—22no—
111#13063246—22no—
112#13063236—22no—
113#13063235—22no—
114C8×Dic43C2×C414yescomplemented
115C43⋊#95—21no—
116C43⋊#95—21no—
117C43⋊#95—21no—
118C43⋊#95D411yescharacteristic
119C43⋊#95D411yescharacteristic
120C22×D4×C43C2311yescharacteristic
121C23×C4×C43—21no—
122C22×C8×C43C2311yescharacteristic
123C22×C8×C43C2311yescharacteristic
124C4×D4×C43—22no—
125C4×D4×C43—41no—
126C2×C4⋊C4×C43—21no—
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C4⋊C4×C43C2311yescharacteristic
129C2×C4⋊C4×C43C2311yescharacteristic
130C2×C22⋊C4×C43—21no—
131C2×C42×C43C2311yescharacteristic
132#107×C43—41no—
133#102×C43—22no—
134#102×C43C2×C414yes—
135#95×C43—21no—
136C24×C43—21no—
137C2×D4×C43—21no—
138C2×D4×C43—21no—
139C2×D4×C43—21no—
140C2×D4×C43C22×C412yes—
141C2×D4×C43C2×D412yes—
142C2×D4×C43C2×Q812yes—
143C22×C4×C43—21no—
144C22×C4×C43—21no—
145C22×C4×C43—21no—
146C22×C4×C43—42no—
147C22×C4×C43—42no—
148C22×C4×C43—81no—
149C22×C4×C43C2×D411yescharacteristic
150C22×C4×C43C2×D411yescharacteristic
151C22×C4×C43C2411yescharacteristic
152C22×Dic43—24no—
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43—42no—
158C22×Dic43—42no—
159C22×Dic43—42no—
160C22×Dic43—42no—
161C22×Dic43—81no—
162C22×Dic43C2×D411yescharacteristic
163C22×Dic43C2×D411yescharacteristic
164C22×Dic43C2×D411yescharacteristic
165C22×Dic43C2×D411yescharacteristic
166C2×Dic86—21no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×Dic86C22×C412yes—
170C2×Dic86C2×D412yes—
171C2×Dic86C2×Q812yes—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43—21no—
176C2×C8×C43—21no—
177C2×C8×C43C22×C412yes—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—81no—
184C4⋊C4×C43C22×C412yes—
185C22⋊C4×C43—42no—
186C42×C43—21no—
187C42×C43—21no—
188C22⋊Dic43—24no—
189C22⋊Dic43—42no—
190#10621883C4⋊C4116yescomplemented
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—81no—
197C4⋊Dic43C22×C412yes—
198Dic43⋊C4—24no—
199Dic43⋊C4—22no—
200Dic43⋊C4—22no—
201Dic43⋊C4—42no—
202Dic43⋊C4—42no—
203Dic43⋊C4—42no—
204Dic43⋊C4—42no—
205Dic43⋊C4—42no—
206Dic43⋊C4—42no—
207Dic43⋊C4—81no—
208Dic43⋊C4—81no—
209C4×Dic43—21no—
210C4×Dic43—21no—
211C4×Dic43—24no—
212C4×Dic43—42no—
213C4×Dic43—42no—
214C4×Dic43C22×C412yes—
215C4×Dic43#4112yes—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8C22×C412yes—
220C2×C43⋊C8C2×D412yes—
221C2×C43⋊C8C2×Q812yes—
222C23×C43—22no—
223C23×C43—22no—
224C23×C43—22no—
225C23×C43—41no—
226C23×C43—41no—
227C23×C43—41no—
228C23×C43—41no—
229C23×C43C22×D411yescharacteristic
230D4×C43—22no—
231D4×C43—22no—
232D4×C43—22no—
233D4×C43C2×C4⋊C414yes—
234C2×C4×C43—22no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
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—84no—
250C2×C4×C43C23×C411yescharacteristic
251C2×C4×C43C22×D411yescharacteristic
252C2×C4×C43C22×Q811yescharacteristic
253C2×C4×C43C2×#4111yescharacteristic
254C2×C4×C43H(5,𝔽2)11yescharacteristic
255C2×C4×C432−1+411yescharacteristic
256C2×Dic43—22no—
257C2×Dic43—22no—
258C2×Dic43—22no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—22no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—48no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—84no—
278C2×Dic43C4×D412yes—
279C2×Dic43C4×D412yes—
280C2×Dic43AGL(1,ℤ/8)12yes—
281C2×Dic43#13712yes—
282Dic86—22no—
283Dic86—22no—
284Dic86—22no—
285Dic86C2×C4⋊C414yescomplemented
286C8×C43—22no—
287C8×C43—41no—
288C43⋊C8—41no—
289C43⋊C8C2×C4⋊C414yescomplemented
290#22671—431nomaximal, Sylow
291C22×C43—24no—
292C22×C43—42no—
293C22×C43—42no—
294C22×C43—42no—
295C22×C43—42no—
296C22×C43—42no—
297C22×C43—42no—
298C22×C43C2×C4×D411yescharacteristic
299C22×C43D4211yescharacteristic
300C22×C43D4×Q811yescharacteristic
301C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C43C2×#13711yescharacteristic
303C22×C43#57611yescharacteristic
304C22×C43#57811yescharacteristic
305C4×C43—42no—
306C4×C43—41no—
307C4×C43—41no—
308C4×C43—81no—
309C4×C43—81no—
310C4×C43C22×C4⋊C411yescharacteristic
311C4×C43#52011yescharacteristic
312C4×C43#54711yescharacteristic
313C4×C43#55411yescharacteristic
314Dic43—24no—
315Dic43—22no—
316Dic43—42no—
317Dic43—41no—
318Dic43—41no—
319Dic43—81no—
320Dic43—81no—
321Dic43#44014yescomplemented
322C2×#440—431no—
323C2×#440—431no—
324C2×C4×SD16—431no—
325Q8×C4⋊C4—431no—
326D4×C4⋊C4—431no—
327#1944—431no—
328#1944—431no—
329#1863—431no—
330#1863—431no—
331#1862—431no—
332#1862D4311yescharacteristic
333#1845—431no—
334#1803—431no—
335#1802—431no—
336#1770—431no—
337C2×C43—44no—
338C2×C43D4×C4⋊C411yescharacteristic
339C2×C43C2×#44011yescharacteristic
340C2×C43#295111yescharacteristic
341C2×C43#327411yescharacteristic
342C2×C43#327611yescharacteristic
343C2×C43#338511yescharacteristic
344C2×C43#338611yescharacteristic
345C22×SD16—431no—
346C2×C4×Q8—431no—
347C2×C4×Q8—431no—
348C2×C4×Q8—431no—
349C2×C4×D4—431no—
350C2×C4×D4—431no—
351C2×C4×D4C2×D4311yescharacteristic
352C22×C4⋊C4—861no—
353#440—864no—
354#440—864no—
355C4×SD16—438no—
356C2×#107—431no—
357C2×#107—431no—
358C2×#107—431no—
359C2×#107C2×D4311yescharacteristic
360C2×#106—431no—
361C2×C4⋊C8—431no—
362C2×#103—431no—
363C2×#103—431no—
364C2×#103—431no—
365C2×#102—431no—
366C2×#102C2×D4311yescharacteristic
367C2×#102C2×D4311yescharacteristic
368C2×#97—431no—
369C2×#97—431no—
370C2×C4×C8—431no—
371#389—431no—
372#389—431no—
373#389—431no—
374#389—861no—
375#389—861no—
376#389C2×D4311yescharacteristic
377#385—861no—
378#385—861no—
379#384—431no—
380#384—431no—
381#384—861no—
382#380—861no—
383C4×C4⋊C4—431no—
384C4×C4⋊C4—861no—
385#340—431no—
386#340—431no—
387#340—431no—
388#340C2×D4311yescharacteristic
389#336—431no—
390#336C2×D4311yescharacteristic
391C43#2267111yescharacteristic, complemented, Sylow
392C22×Q8—431no—
393C22×D4C22×D4311yescharacteristic
394C23×C4—21no—
395C23×C4—861no—
396C23×C4—861no—
397C2×SD16—434no—
398C2×SD16—434no—
399C2×SD16—434no—
400C22×C8—431no—
401C22×C8—431no—
402C22×C8C22×D4311yescharacteristic
403C22×C8C22×D4311yescharacteristic
404C4×Q8—434no—
405C4×Q8—862no—
406C4×Q8—862no—
407C4×Q8—862no—
408C4×Q8—1721no—
409C4×Q8—1721no—
410C4×D4—22no—
411C4×D4—41no—
412C4×D4—434no—
413C4×D4—862no—
414C4×D4—862no—
415C4×D4—1721no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—431no—
418C2×C4⋊C4—431no—
419C2×C4⋊C4—431no—
420C2×C4⋊C4—431no—
421C2×C4⋊C4—431no—
422C2×C4⋊C4—861no—
423C2×C4⋊C4—861no—
424C2×C4⋊C4—861no—
425C2×C4⋊C4—861no—
426C2×C4⋊C4—861no—
427C2×C4⋊C4—861no—
428C2×C4⋊C4—861no—
429C2×C4⋊C4—861no—
430C2×C4⋊C4—861no—
431C2×C4⋊C4—861no—
432C2×C4⋊C4—1724no—
433C2×C4⋊C4C22×D4311yescharacteristic
434C2×C4⋊C4C22×D4311yescharacteristic
435C2×C4⋊C4C22×D4311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—861no—
438C2×C22⋊C4—861no—
439C2×C42—431no—
440C2×C42—431no—
441C2×C42—861no—
442C2×C42—861no—
443C2×C42—861no—
444C2×C42C22×D4311yescharacteristic
445#107—41no—
446#107—862no—
447#107—862no—
448#107—1721no—
449#106—434no—
450C4⋊C8—862no—
451#103—434no—
452#103—862no—
453#103—862no—
454#102—22no—
455#102—862no—
456#102C4×D4314yes—
457#97—862no—
458#97—862no—
459C4×C8—434no—
460#95—21no—
461#95—431no—
462#95—431no—
463#95—861no—
464#95—861no—
465#95—861no—
466C24—21no—
467C2×Q8—432no—
468C2×Q8—432no—
469C2×Q8—432no—
470C2×Q8—861no—
471C2×Q8—861no—
472C2×Q8—861no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4C2×C4×D4312yes—
477C2×D4D4×D4312yes—
478C2×D4Q8×D4312yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—431no—
486C22×C4—431no—
487C22×C4—431no—
488C22×C4—431no—
489C22×C4—864no—
490C22×C4—861no—
491C22×C4—861no—
492C22×C4—861no—
493C22×C4—861no—
494C22×C4—3441no—
495C22×C4—1722no—
496C22×C4—1722no—
497C22×C4—1722no—
498C22×C4—1722no—
499C22×C4D4×D4311yescharacteristic
500C22×C4D4×D4311yescharacteristic
501C22×C4C23×D4311yescharacteristic
502SD16—4316no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—432no—
509C2×C8—432no—
510C2×C8—432no—
511C2×C8—861no—
512C2×C8—861no—
513C2×C8—861no—
514C2×C8C2×C4×D4312yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—81no—
521C4⋊C4—432no—
522C4⋊C4—862no—
523C4⋊C4—862no—
524C4⋊C4—864no—
525C4⋊C4—861no—
526C4⋊C4—861no—
527C4⋊C4—861no—
528C4⋊C4—861no—
529C4⋊C4—861no—
530C4⋊C4—3441no—
531C4⋊C4—3441no—
532C4⋊C4—3441no—
533C4⋊C4—1722no—
534C4⋊C4—1722no—
535C4⋊C4—1722no—
536C4⋊C4—1722no—
537C4⋊C4—1722no—
538C4⋊C4—1722no—
539C4⋊C4C2×C4×D4312yes—
540C22⋊C4—42no—
541C22⋊C4—864no—
542C22⋊C4—1722no—
543C42—21no—
544C42—21no—
545C42—432no—
546C42—432no—
547C42—864no—
548C42—861no—
549C42—861no—
550C42—1722no—
551C42—1722no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D4311yescharacteristic
560Q8—434no—
561Q8—862no—
562Q8—862no—
563Q8—862no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4C4⋊C4×D4314yes—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—432no—
585C2×C4—432no—
586C2×C4—432no—
587C2×C4—432no—
588C2×C4—862no—
589C2×C4—862no—
590C2×C4—861no—
591C2×C4—861no—
592C2×C4—862no—
593C2×C4—862no—
594C2×C4—861no—
595C2×C4—861no—
596C2×C4—861no—
597C2×C4—861no—
598C2×C4—861no—
599C2×C4—861no—
600C2×C4—3444no—
601C2×C4—1728no—
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×C4C22×C4×D4311yescharacteristic
611C2×C4C2×D4×D4311yescharacteristic
612C2×C4C2×Q8×D4311yescharacteristic
613C2×C4#41×D4311yescharacteristic
614C2×C4#1306337011yescharacteristic
615C2×C4#1306337211yescharacteristic
616C8—22no—
617C8—41no—
618C8—434no—
619C8—1721no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C4×D4×D4311yescharacteristic
628C22#1458353111yescharacteristic
629C22#1458356211yescharacteristic
630C22AGL(1,ℤ/8)×D4311yescharacteristic
631C22#137×D4311yescharacteristic
632C22#1458370011yescharacteristic
633C22#1458371311yescharacteristic
634C4—42no—
635C4—41no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—434no—
640C4—862no—
641C4—864no—
642C4—3441no—
643C4—3441no—
644C4—1722no—
645C4—1721no—
646C4—1721no—
647C4C2×C4⋊C4×D4311yescharacteristic
648C4#1458337811yescharacteristic
649C4#1458354711yescharacteristic
650C4#1458360311yescharacteristic
651C2—44no—
652C2#1913749311yescharacteristic
653C2#440×D4311yescharacteristic
654C2#1914346511yescharacteristic
655C2#1914621311yescharacteristic
656C2#1914623911yescharacteristic
657C2#1914750011yescharacteristic
658C2#1914751111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1862C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
cosocleC22×C4×C43C24
hypercenter#1862D43
hypocenterC43#22671
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
4. #19137509C2semidirect
6. #19135337C2semidirect
7. #19135335C2semidirect
8. #19134852C2semidirect
9. #19134846C2semidirect
11. #19134753C2semidirect
12. #19134599C2semidirect
14. #19134461C2semidirect
25. #14582990C4semidirect
75. #13063290C2×C4semidirect
106. #13063256C2×C4semidirect
109. #13063250C2×C4semidirect
114. C8×Dic43C2×C4semidirect
190. #10621883C4⋊C4semidirect
285. Dic86C2×C4⋊C4semidirect
289. C43⋊C8C2×C4⋊C4semidirect
321. Dic43#440semidirect
391. C43#22671semidirect

Maximal subgroups

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

subgroupindex
C2×#145829992
C2×#145829962
C2×#145829902
#191375092
D4×Dic43⋊C42
#191353372
#191353352
#191348522
#191348462
#191348442
#191347532
#191345992
#191345932
#191344612
C43×#18622
#2267143