Finite Group Databasebeta

← #25101206

Subgroups — #25101206

Order 11008.

1134 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583526C211yesmaximal, characteristic
2C2×#14582856C211yesmaximal, characteristic
3#19137220C211yesmaximal, characteristic
4#19135566C211yesmaximal, characteristic, complemented
5#19135564C211yesmaximal, characteristic, complemented
6#19135562C211yesmaximal, characteristic, complemented
7#19135559C211yesmaximal, characteristic, complemented
8#19135558C211yesmaximal, characteristic, complemented
9#19135557C211yesmaximal, characteristic, complemented
10#19135554C211yesmaximal, characteristic, complemented
11#19134313C211yesmaximal, characteristic
12#19134313C211yesmaximal, characteristic
13#19134310C211yesmaximal, characteristic
14#340×D43C211yesmaximal, characteristic
15C43×#1993C211yesmaximal, characteristic, complemented
16C22×#10621872C2211yescharacteristic
17C22×C8×D43C2211yescharacteristic
18#14583526—24no—
19#14583526C418yescomplemented
20C2×#13063324C2211yescharacteristic
21C2×#13063323—21no—
22C2×#13063317C2211yescharacteristic
23C2×#13063316C2211yescharacteristic
24C2×#13063315—21no—
25C2×#13063303—21no—
26C2×#13063301—21no—
27C2×#13063283C2211yescharacteristic
28C2×#13063283C2211yescharacteristic
29C2×#13063282C2211yescharacteristic
30C2×C4⋊C4×D43C2211yescharacteristic
31C2×C4⋊C4×D43C2211yescharacteristic
32#14582856—24no—
33C2×#13063235C2211yescharacteristic
34C2×C4×C43⋊C8C2211yescharacteristic
35C2×C8×Dic43C2211yescharacteristic
36#14582796—21no—
37#14582796—21no—
38#14582773—21no—
39#14582773—21no—
40#14582769C2211yescharacteristic
41#14582768C2211yescharacteristic
42#14582764C2211yescharacteristic
43#14582662C2211yescharacteristic
44#14582662C2211yescharacteristic
45#14582662C2211yescharacteristic
46#14582662C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582661C2211yescharacteristic
54#14582661C2211yescharacteristic
55C2×C4⋊Q8×C43C2211yescharacteristic
56C2×#97×C43C2211yescharacteristic
57C43×#382C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C43×#340C2211yescharacteristic
62C22×Dic86—21no—
63C22×C4×D43—21no—
64C22×C4×D43—21no—
65C22×C4×D43C2311yescharacteristic
66C2×#10621872—22no—
67C2×#10621872—22no—
68C2×#10621872Q814yescomplemented
69C22×C43⋊C8C2311yescharacteristic
70C22×C43⋊C8C2311yescharacteristic
71C2×C8×D43—22no—
72C2×C8×D43—22no—
73C2×C8×D43Q814yescomplemented
74#13063324—22no—
75#13063324C2×C414yescomplemented
76#13063323—28no—
77#13063317—22no—
78#13063317C2×C414yescomplemented
79#13063316—22no—
80#13063316C2×C414yescomplemented
81#13063315—28no—
82#13063303—28no—
83#13063301—28no—
84#13063283—22no—
85#13063283—22no—
86#13063283—22no—
87#13063283C2×C414yes—
88#13063282C2×C414yes—
89#13063282D414yes—
90C2×C4⋊Dic43—21no—
91C2×C4⋊Dic43—21no—
92C2×C4⋊Dic43—21no—
93C2×C4⋊Dic43C2311yescharacteristic
94C2×C4⋊Dic43C2311yescharacteristic
95C2×C4⋊Dic43C2311yescharacteristic
96C2×C4⋊Dic43C2311yescharacteristic
97C2×Dic43⋊C4—21no—
98C2×Dic43⋊C4—21no—
99C2×Dic43⋊C4—21no—
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
107C4⋊C4×D43—22no—
108C4⋊C4×D43—22no—
109C4⋊C4×D43—22no—
110C4⋊C4×D43C2×C414yes—
111C2×(C2×D43)⋊C4—21no—
112C2×(C2×D43)⋊C4—21no—
113C2×(C2×D43)⋊C4D411yescharacteristic
114C2×(C2×D43)⋊C4D411yescharacteristic
115C2×C4×Dic43—21no—
116C2×C4×Dic43—21no—
117C2×C4×Dic43C2311yescharacteristic
118#13063235—22no—
119C4×C43⋊C8—22no—
120C8×Dic43—22no—
121C43⋊#95—21no—
122C43⋊#95—21no—
123C22×Q8×C43—21no—
124C22×C8×C43C2311yescharacteristic
125C22×C8×C43C2311yescharacteristic
126C4⋊Q8×C43—22no—
127C4⋊Q8×C43C2×C414yescomplemented
128C2×C4⋊C4×C43—21no—
129C2×C4⋊C4×C43C2311yescharacteristic
130C2×C4⋊C4×C43C2311yescharacteristic
131C2×C4⋊C4×C43C2311yescharacteristic
132C2×C4⋊C4×C43C2311yescharacteristic
133C2×C42×C43C2311yescharacteristic
134#97×C43—22no—
135#95×C43—21no—
136#95×C43—21no—
137C23×D43C2×D411yescharacteristic
138C2×Q8×C43—24no—
139C2×Q8×C43—81no—
140C2×Q8×C43—81no—
141C22×C4×C43—21no—
142C22×C4×C43—21no—
143C22×C4×C43—21no—
144C22×C4×C43—21no—
145C22×C4×C43C2×D411yescharacteristic
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2411yescharacteristic
148C22×Dic43—21no—
149C22×Dic43—21no—
150C22×Dic43—21no—
151C22×Dic43—21no—
152C22×Dic43C2×D411yescharacteristic
153C22×Dic43C2×D411yescharacteristic
154C22×Dic43C2×D411yescharacteristic
155C2×C4×D43—24no—
156C2×C4×D43—42no—
157C2×C4×D43—81no—
158C2×C4×D43—81no—
159C2×C4×D43—81no—
160C2×C4×D43—81no—
161C2×C4×D43C22×C412yes—
162C2×C4×D43C2×D412yes—
163C2×C4×D43C2×Q812yes—
164C2×C4×D43C2×Q812yes—
165C2×C4×D43#4112yes—
166C2×C4×D43#4112yes—
167C2×Dic86—24no—
168C2×Dic86—81no—
169C2×Dic86—81no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43C2×Q812yes—
175C2×C8×C43C2×Q812yes—
176C4⋊C4×C43—24no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43C22×C412yes—
184C4⋊C4×C43C22×C412yes—
185C42×C43C22×C412yes—
186C42×C43C2×D412yes—
187(C2×D43)⋊C4—24no—
188(C2×D43)⋊C4—42no—
189(C2×D43)⋊C4C22⋊C414yes—
190(C2×D43)⋊C4C22⋊C414yes—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—24no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—81no—
199C4⋊Dic43—81no—
200C4⋊Dic43C22×C412yes—
201C4⋊Dic43C22×C412yes—
202Dic43⋊C4—24no—
203Dic43⋊C4—24no—
204Dic43⋊C4—24no—
205Dic43⋊C4—24no—
206Dic43⋊C4—24no—
207Dic43⋊C4—42no—
208Dic43⋊C4—81no—
209Dic43⋊C4—81no—
210Dic43⋊C4C22⋊C414yes—
211Dic43⋊C4C22⋊C414yes—
212C4×Dic43—24no—
213C4×Dic43—42no—
214C4×Dic43C22×C412yes—
215C4×Dic43C2×D412yes—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8C2×Q812yes—
221C2×C43⋊C8C2×Q812yes—
222#10621872—28no—
223C8×D43—28no—
224C23×C43C22×D411yescharacteristic
225C22×D43—21no—
226C22×D43—21no—
227C22×D43—21no—
228C22×D43—21no—
229C22×D43C2×C22⋊C412yes—
230C22×D43#12412yes—
231C22×D43C4⋊Q812yes—
232Q8×C43—84no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—22no—
238C2×C4×C43—22no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
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×C43C2×C22⋊C412yes—
250C2×C4×C43C2×C22⋊C412yes—
251C2×C4×C43C23×C411yescharacteristic
252C2×C4×C43C22×D411yescharacteristic
253C2×C4×C43C22×Q811yescharacteristic
254C2×C4×C43C22×Q811yescharacteristic
255C2×C4×C43C2×#4111yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×Dic43—21no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—22no—
262C2×Dic43—22no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43C2×C22⋊C412yes—
274C2×Dic43C2×C22⋊C412yes—
275C2×Dic43C2×C22⋊C412yes—
276C2×Dic43#12412yes—
277C2×Dic43C4⋊Q812yes—
278C4×D43—84no—
279C4×D43—84no—
280C4×D43C4×Q814yes—
281C4×D43C4×Q814yes—
282C4×D43C22⋊Q814yes—
283C4×D43C22⋊Q814yes—
284Dic86—84no—
285C8×C43—22no—
286C8×C43—22no—
287C43⋊C8—22no—
288C43⋊C8—22no—
289#21326—431nomaximal, Sylow
290C22×C43C22×C22⋊C411yescharacteristic
291C22×C43C2×#12411yescharacteristic
292C22×C43C2×C4⋊Q811yescharacteristic
293C22×C43#57611yescharacteristic
294C22×C43#57711yescharacteristic
295C22×C43#57711yescharacteristic
296C22×C43#57811yescharacteristic
297C2×D43—22no—
298C2×D43—22no—
299C2×D43—22no—
300C2×D43—22no—
301C2×D43—22no—
302C2×D43—22no—
303C2×D43#39114yes—
304C4×C43—22no—
305C4×C43—22no—
306C4×C43—42no—
307C4×C43—42no—
308C4×C43—81no—
309C4×C43—81no—
310C4×C43C2×C4×Q811yescharacteristic
311C4×C43C2×C4×Q811yescharacteristic
312C4×C43C2×C22⋊Q811yescharacteristic
313C4×C43C2×C22⋊Q811yescharacteristic
314Dic43—22no—
315Dic43—22no—
316Dic43—42no—
317Dic43—42no—
318Dic43—81no—
319Dic43—81no—
320Dic43#39114yes—
321C2×#544—431no—
322C2×#405—431no—
323#2317—431no—
324#1993—431no—
325#1993D4311yescharacteristic, complemented
326#1992—431no—
327#1992—431no—
328#1991—431no—
329#1991—431no—
330#1990—431no—
331#1990—431no—
332#1736—431no—
333#1736—431no—
334#1735—431no—
335C2×#340—431no—
336C2×C43C2×#39111yescharacteristic
337C2×C43#289711yescharacteristic
338C2×C43#289811yescharacteristic
339C2×C43#313611yescharacteristic
340C2×C43#313711yescharacteristic
341C2×C43#316311yescharacteristic
342C2×C43#316411yescharacteristic
343D43—24no—
344C22×M16—431no—
345C23×C8—431no—
346#544—438no—
347#544—864no—
348C2×C4⋊Q8—431no—
349C2×C4⋊Q8—861no—
350C2×C4⋊Q8C2×D4311yescharacteristic
351C2×#125—431no—
352C2×#125—431no—
353C2×#125—861no—
354C2×C22⋊Q8—861no—
355C2×C22⋊Q8—861no—
356C2×#117—431no—
357C2×#117—431no—
358C2×#117—431no—
359C22×C4⋊C4—431no—
360C22×C4⋊C4—431no—
361#405—864no—
362C2×#97—431no—
363C2×#97C2×D4311yescharacteristic
364C2×C4×C8—431no—
365C2×C4×C8—431no—
366#389—861no—
367#389—861no—
368#385—861no—
369#385—861no—
370#384—431no—
371#384—431no—
372#382—431no—
373#382C2×D4311yescharacteristic
374#340—431no—
375#340—431no—
376#340—431no—
377#340—431no—
378#340—431no—
379#340—431no—
380#340—431no—
381#340—431no—
382#340—431no—
383#340—431no—
384#340—431no—
385#340—431no—
386#340C2×D4311yescharacteristic
387#340C2×D4311yescharacteristic
388#340C2×D4311yescharacteristic
389#340C2×D4311yescharacteristic
390C43#2132611yescharacteristic, complemented, Sylow
391C22×Q8—21no—
392C22×Q8—861no—
393C23×C4—431no—
394C23×C4—861no—
395C23×C4—861no—
396C2×M16—434no—
397C2×M16—862no—
398C2×M16—862no—
399C22×C8—431no—
400C22×C8—431no—
401C22×C8—434no—
402C22×C8—862no—
403C22×C8—862no—
404C22×C8C22×D4311yescharacteristic
405C22×C8C22×D4311yescharacteristic
406C4⋊Q8—22no—
407C4⋊Q8—434no—
408C4⋊Q8—862no—
409C4⋊Q8—868no—
410C4⋊Q8C4×D4314yescomplemented
411#125—434no—
412#125—434no—
413#125—862no—
414#125—868no—
415#125—862no—
416C22⋊Q8—868no—
417C22⋊Q8—868no—
418#117—434no—
419#117—434no—
420#117—434no—
421#117—862no—
422#117—862no—
423#117—862no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—431no—
426C2×C4⋊C4—431no—
427C2×C4⋊C4—431no—
428C2×C4⋊C4—431no—
429C2×C4⋊C4—434no—
430C2×C4⋊C4—431no—
431C2×C4⋊C4—431no—
432C2×C4⋊C4—862no—
433C2×C4⋊C4—862no—
434C2×C4⋊C4—862no—
435C2×C4⋊C4—861no—
436C2×C4⋊C4—861no—
437C2×C4⋊C4—861no—
438C2×C4⋊C4—861no—
439C2×C4⋊C4—861no—
440C2×C4⋊C4—861no—
441C2×C4⋊C4—861no—
442C2×C4⋊C4—861no—
443C2×C4⋊C4—861no—
444C2×C4⋊C4—861no—
445C2×C4⋊C4—861no—
446C2×C4⋊C4C22×D4311yescharacteristic
447C2×C4⋊C4C22×D4311yescharacteristic
448C2×C4⋊C4C22×D4311yescharacteristic
449C2×C4⋊C4C22×D4311yescharacteristic
450C2×C22⋊C4—431no—
451C2×C22⋊C4—431no—
452C2×C22⋊C4—861no—
453C2×C22⋊C4—861no—
454C2×C42—431no—
455C2×C42—861no—
456C2×C42—861no—
457C2×C42C22×D4311yescharacteristic
458#97—22no—
459#97—862no—
460C4×C8—862no—
461C4×C8—862no—
462#95—21no—
463#95—21no—
464#95—861no—
465#95—861no—
466C24—431no—
467C2×Q8—24no—
468C2×Q8—81no—
469C2×Q8—81no—
470C2×Q8—864no—
471C2×Q8—3441no—
472C2×Q8—3441no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—431no—
478C22×C4—432no—
479C22×C4—432no—
480C22×C4—432no—
481C22×C4—432no—
482C22×C4—432no—
483C22×C4—432no—
484C22×C4—431no—
485C22×C4—431no—
486C22×C4—864no—
487C22×C4—861no—
488C22×C4—861no—
489C22×C4—861no—
490C22×C4—861no—
491C22×C4—3441no—
492C22×C4—3441no—
493C22×C4—3441no—
494C22×C4—3441no—
495C22×C4—1722no—
496C22×C4D4×D4311yescharacteristic
497C22×C4D4×D4311yescharacteristic
498C22×C4C23×D4311yescharacteristic
499M16—868no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—432no—
505C2×C8—432no—
506C2×C8—868no—
507C2×C8—861no—
508C2×C8—861no—
509C2×C8—861no—
510C2×C8—861no—
511C2×C8Q8×D4312yes—
512C2×C8Q8×D4312yes—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—434no—
521C4⋊C4—434no—
522C4⋊C4—432no—
523C4⋊C4—432no—
524C4⋊C4—864no—
525C4⋊C4—864no—
526C4⋊C4—864no—
527C4⋊C4—864no—
528C4⋊C4—864no—
529C4⋊C4—864no—
530C4⋊C4—861no—
531C4⋊C4—861no—
532C4⋊C4—861no—
533C4⋊C4—861no—
534C4⋊C4—861no—
535C4⋊C4—861no—
536C4⋊C4—3441no—
537C4⋊C4—3441no—
538C4⋊C4—3441no—
539C4⋊C4—3441no—
540C4⋊C4—1722no—
541C4⋊C4C2×C4×D4312yes—
542C4⋊C4C2×C4×D4312yes—
543C22⋊C4—434no—
544C22⋊C4—434no—
545C22⋊C4—864no—
546C22⋊C4—1722no—
547C42—432no—
548C42—432no—
549C42—864no—
550C42—1722no—
551C42C2×C4×D4312yes—
552C42D4×D4312yes—
553C23—432no—
554C23—432no—
555C23—432no—
556C23—861no—
557C23—861no—
558C23—861no—
559C23—861no—
560C23C2×D4×D4311yescharacteristic
561Q8—84no—
562Q8—3444no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—432no—
580C2×C4—432no—
581C2×C4—432no—
582C2×C4—434no—
583C2×C4—434no—
584C2×C4—434no—
585C2×C4—434no—
586C2×C4—432no—
587C2×C4—432no—
588C2×C4—861no—
589C2×C4—861no—
590C2×C4—861no—
591C2×C4—861no—
592C2×C4—862no—
593C2×C4—862no—
594C2×C4—3444no—
595C2×C4—3444no—
596C2×C4—1721no—
597C2×C4—1721no—
598C2×C4—1721no—
599C2×C4—1721no—
600C2×C4—1721no—
601C2×C4—1721no—
602C2×C4—1721no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4—1721no—
606C2×C4C22⋊C4×D4312yes—
607C2×C4C22⋊C4×D4312yes—
608C2×C4C22×C4×D4311yescharacteristic
609C2×C4C2×D4×D4311yescharacteristic
610C2×C4C2×Q8×D4311yescharacteristic
611C2×C4C2×Q8×D4311yescharacteristic
612C2×C4#41×D4311yescharacteristic
613C2×C4#41×D4311yescharacteristic
614C8—22no—
615C8—22no—
616C8—862no—
617C8—862no—
618C22—434no—
619C22—862no—
620C22—862no—
621C22—862no—
622C22—862no—
623C22—862no—
624C22—862no—
625C22C2×C22⋊C4×D4311yescharacteristic
626C22#124×D4311yescharacteristic
627C22C4⋊Q8×D4311yescharacteristic
628C22#1458370311yescharacteristic
629C22#1458370811yescharacteristic
630C22#1458370911yescharacteristic
631C22#1458371511yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—434no—
639C4—862no—
640C4—862no—
641C4—3441no—
642C4—3441no—
643C4—1722no—
644C4—1722no—
645C4C4×Q8×D4311yescharacteristic
646C4C4×Q8×D4311yescharacteristic
647C4C22⋊Q8×D4311yescharacteristic
648C4C22⋊Q8×D4311yescharacteristic
649C2—864no—
650C2#391×D4311yescharacteristic
651C2#1914312011yescharacteristic
652C2#1914312711yescharacteristic
653C2#1914494611yescharacteristic
654C2#1914495711yescharacteristic
655C2#1914512211yescharacteristic
656C2#1914513011yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1993C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
cosocleC22×C4×C43C24
hypercenter#1993D43
hypocenterC43#21326
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
4. #19135566C2semidirect
5. #19135564C2semidirect
6. #19135562C2semidirect
7. #19135559C2semidirect
8. #19135558C2semidirect
9. #19135557C2semidirect
10. #19135554C2semidirect
15. C43×#1993C2semidirect
19. #14583526C4semidirect
68. C2×#10621872Q8semidirect
73. C2×C8×D43Q8semidirect
75. #13063324C2×C4semidirect
78. #13063317C2×C4semidirect
80. #13063316C2×C4semidirect
127. C4⋊Q8×C43C2×C4semidirect
325. #1993D43semidirect
390. C43#21326semidirect
410. C4⋊Q8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145835262
C2×#145828562
#191372202
#191355662
#191355642
#191355622
#191355592
#191355582
#191355572
#191355542
#191343132
#191343102
#340×D432
C43×#19932
#2132643