Finite Group Databasebeta

← #25103856

Subgroups — #25103856

Order 11008.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583218C211yesmaximal, characteristic
2#19137288C211yesmaximal, characteristic
3#19136951C211yesmaximal, characteristic, complemented
4#19135223C211yesmaximal, characteristic, complemented
5#19135216C211yesmaximal, characteristic, complemented
6#19135181C211yesmaximal, characteristic, complemented
7#19135174C211yesmaximal, characteristic, complemented
8#19134915C211yesmaximal, characteristic
9#19134796C211yesmaximal, characteristic, complemented
10#19134682C211yesmaximal, characteristic, complemented
11#19134492C211yesmaximal, characteristic, complemented
12C4×#13063257C211yesmaximal, characteristic, complemented
13#19134421C211yesmaximal, characteristic, complemented
14C4×#13063247C211yesmaximal, characteristic, complemented
15C43×#1750C211yesmaximal, characteristic, complemented
16C2×#13063299C2211yescharacteristic
17C2×C4×D172C2211yescharacteristic
18C2×#13063284C2211yescharacteristic
19C22×C4⋊Dic43C2211yescharacteristic
20#14583218C418yescomplemented
21C2×#13063259C2211yescharacteristic
22C2×#13063257C2211yescharacteristic
23C2×#13063257C2211yescharacteristic
24C2×#13063257C2211yescharacteristic
25C2×#13063247C2211yescharacteristic
26C2×#13063247C2211yescharacteristic
27C2×#13063247C2211yescharacteristic
28C2×#13063247C2211yescharacteristic
29#14582801C2211yescharacteristic
30#14582799—21no—
31#14582797C2211yescharacteristic
32#14582789—21no—
33#14582775—21no—
34#14582770—21no—
35#14582759C2211yescharacteristic
36C4⋊C4×Dic43—21no—
37C4⋊C4×Dic43—21no—
38C4×C4⋊Dic43C2211yescharacteristic
39C4×C4⋊Dic43C2211yescharacteristic
40C4×C4⋊Dic43C2211yescharacteristic
41C4×C4⋊Dic43C2211yescharacteristic
42C4×C4⋊Dic43C2211yescharacteristic
43C22⋊C4×Dic43—21no—
44C22⋊C4×Dic43—21no—
45#14582662C2211yescharacteristic
46#14582662C2211yescharacteristic
47#14582662C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582662C2211yescharacteristic
51#14582662C2211yescharacteristic
52#14582662C2211yescharacteristic
53C2×#117×C43C2211yescharacteristic
54C2×C22⋊C8×C43C2211yescharacteristic
55C2×C22⋊C8×C43C2211yescharacteristic
56C2×#97×C43C2211yescharacteristic
57C2×C4×C8×C43C2211yescharacteristic
58C43×#336C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C22×D172C2311yescharacteristic
61C22×C22⋊D43—21no—
62C23×Dic43—21no—
63C22×C4×D43—21no—
64#13063299—22no—
65#13063299C2×C414yes—
66C4×D172—22no—
67C4×D172—22no—
68#13063284—22no—
69#13063284—22no—
70C2×C4⋊Dic43—21no—
71C2×C4⋊Dic43—22no—
72C2×C4⋊Dic43C2×C414yescomplemented
73C2×C4⋊Dic43C2311yescharacteristic
74C2×C4⋊Dic43C2311yescharacteristic
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×Dic43⋊C4—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×C4×Dic43—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43—21no—
90C2×C4×Dic43—21no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43—21no—
93#13063259—22no—
94#13063257—22no—
95#13063257C2×C414yescomplemented
96#13063257C2×C414yescomplemented
97#13063247—22no—
98#13063247—22no—
99#13063247C2×C414yes—
100#13063247C2×C414yes—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C23×C4×C43C2311yescharacteristic
109C22×C8×C43C2311yescharacteristic
110C22×C8×C43C2311yescharacteristic
111C22×C8×C43C2311yescharacteristic
112C22×C8×C43C2311yescharacteristic
113#117×C43—22no—
114#117×C43—22no—
115C2×C4⋊C4×C43D411yescharacteristic
116C2×C4⋊C4×C43D411yescharacteristic
117C2×C22⋊C4×C43D411yescharacteristic
118C2×C22⋊C4×C43D411yescharacteristic
119C2×C42×C43C2311yescharacteristic
120C2×C42×C43C2311yescharacteristic
121C22⋊C8×C43—22no—
122C22⋊C8×C43C2×C414yescomplemented
123#97×C43—22no—
124C4×C8×C43—22no—
125C24×C43C2×D411yescharacteristic
126C23×D43—21no—
127C22×C4×C43—21no—
128C22×C4×C43—21no—
129C22×C4×C43—21no—
130C22×C4×C43—21no—
131C22×C4×C43C22×C412yes—
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2×D411yescharacteristic
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43#4112yes—
138C22×C4×C43C2411yescharacteristic
139C2×C22⋊D43—24no—
140C2×C22⋊D43—81no—
141C2×C22⋊D43—81no—
142C22×Dic43—24no—
143C22×Dic43—21no—
144C22×Dic43—21no—
145C22×Dic43—21no—
146C22×Dic43—21no—
147C22×Dic43—21no—
148C22×Dic43—21no—
149C22×Dic43—21no—
150C22×Dic43—81no—
151C22×Dic43—81no—
152C2×D172—21no—
153C2×D172—21no—
154C2×D172—21no—
155C2×D172—21no—
156C2×D172—21no—
157C2×D172C22×C412yes—
158C2×C4×D43—42no—
159C2×C4×D43—81no—
160C2×C4×D43—81no—
161C2×C8×C43—21no—
162C2×C8×C43—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—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43C22×C412yes—
172C2×C8×C43C22×C412yes—
173C4⋊C4×C43—22no—
174C4⋊C4×C43—22no—
175C22⋊C4×C43—22no—
176C22⋊C4×C43—22no—
177C42×C43—21no—
178C42×C43—21no—
179C42×C43#4112yes—
180C42×C43#4112yes—
181C22⋊Dic43—24no—
182C22⋊Dic43—81no—
183(C2×D43)⋊C4—24no—
184(C2×D43)⋊C4—42no—
185(C2×D43)⋊C4—81no—
186(C2×D43)⋊C4—81no—
187C4⋊Dic43—24no—
188C4⋊Dic43—21no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43C22×C412yes—
198C4⋊Dic43C22×C412yes—
199C4⋊Dic43C22×C412yes—
200C4⋊Dic43#4112yes—
201C4⋊Dic43#4112yes—
202Dic43⋊C4—81no—
203C4×Dic43—24no—
204C4×Dic43—81no—
205C4×Dic43—81no—
206C4×Dic43—81no—
207C4×Dic43—81no—
208C4×Dic43—81no—
209C23×C43—21no—
210C23×C43—21no—
211C23×C43—21no—
212C23×C43—21no—
213C23×C43—21no—
214C23×C43—21no—
215C23×C43C4×D412yes—
216C23×C43C22×D411yescharacteristic
217C22×D43—22no—
218C22×D43—41no—
219C22×D43—41no—
220C22×D43—41no—
221C22×D43—41no—
222C22×D43—41no—
223C22×D43—41no—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—22no—
231C2×C4×C43—22no—
232C2×C4×C43—22no—
233C2×C4×C43—22no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43C4×D412yes—
241C2×C4×C43#12312yes—
242C2×C4×C43#12312yes—
243C2×C4×C43C23×C411yescharacteristic
244C2×C4×C43C2×#4111yescharacteristic
245C2×C4×C43C2×#4111yescharacteristic
246C2×C4×C43C2×#4111yescharacteristic
247C2×C4×C43C2×#4111yescharacteristic
248C2×C4×C43C2×#4111yescharacteristic
249C22⋊D43—84no—
250C2×Dic43—22no—
251C2×Dic43—22no—
252C2×Dic43—22no—
253C2×Dic43—22no—
254C2×Dic43—22no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
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—84no—
272D172—22no—
273D172—22no—
274D172—41no—
275D172—41no—
276C4×D43—84no—
277C8×C43—22no—
278C8×C43—22no—
279C8×C43—41no—
280C8×C43—41no—
281#21584—431nomaximal, Sylow
282C22×C43—22no—
283C22×C43—22no—
284C22×C43—22no—
285C22×C43—41no—
286C22×C43—41no—
287C22×C43—41no—
288C22×C43—41no—
289C22×C43C2×C4×D411yescharacteristic
290C22×C43C2×#12311yescharacteristic
291C22×C43#52511yescharacteristic
292C22×C43#57611yescharacteristic
293C22×C43#57711yescharacteristic
294C22×C43#57711yescharacteristic
295C22×C43#57711yescharacteristic
296C2×D43—42no—
297C2×D43—42no—
298C2×D43—42no—
299C2×D43—81no—
300C2×D43—81no—
301C2×D43—81no—
302C2×D43—81no—
303C4×C43—22no—
304C4×C43—22no—
305C4×C43—22no—
306C4×C43—41no—
307C4×C43—41no—
308C4×C43C4×#4111yescharacteristic
309C4×C43C4×#4111yescharacteristic
310C4×C43#52911yescharacteristic
311C4×C43#52911yescharacteristic
312Dic43—42no—
313Dic43—42no—
314Dic43—42no—
315Dic43—42no—
316Dic43—42no—
317Dic43—81no—
318Dic43—81no—
319C2×#481—431no—
320#2324—431no—
321#2274—431no—
322#1926—431no—
323#1925—431no—
324#1920—431no—
325#1919—431no—
326#1875—431no—
327#1854—431no—
328#1823—431no—
329#1776—431no—
330C4×#106—431no—
331#1762—431no—
332C4×#102—431no—
333#1750D4311yescharacteristic, complemented
334C2×C43—42no—
335C2×C43C4×#12311yescharacteristic
336C2×C43#294611yescharacteristic
337C2×C43#294911yescharacteristic
338C2×C43#309611yescharacteristic
339C2×C43#309911yescharacteristic
340C2×C43#311911yescharacteristic
341C2×C43#312511yescharacteristic
342D43—82no—
343C2×#121—431no—
344C2×C4×D4—431no—
345C2×#117—431no—
346C2×#117C2×D4311yescharacteristic
347C22×C4⋊C4—431no—
348#481—438no—
349C2×#107—431no—
350C2×#106—431no—
351C2×#106—431no—
352C2×#106—431no—
353C2×#102—431no—
354C2×#102—431no—
355C2×#102—431no—
356C2×#102—431no—
357C2×C22⋊C8C2×D4311yescharacteristic
358C2×C22⋊C8C2×D4311yescharacteristic
359C2×#97C2×D4311yescharacteristic
360C2×C4×C8C2×D4311yescharacteristic
361#390—431no—
362#390—861no—
363#389—431no—
364#388—861no—
365#386—861no—
366#385—861no—
367#380—431no—
368C4×C4⋊C4—431no—
369C4×C4⋊C4—431no—
370C4×C4⋊C4—431no—
371C4×C4⋊C4—431no—
372C4×C4⋊C4—431no—
373C4×C4⋊C4—861no—
374C4×C4⋊C4—861no—
375C4×C22⋊C4—861no—
376C4×C22⋊C4—861no—
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#336C2×D4311yescharacteristic
386#336C2×D4311yescharacteristic
387C43#2158411yescharacteristic, complemented, Sylow
388C22×D4—431no—
389C22×D4—861no—
390C23×C4—861no—
391C23×C4—861no—
392C23×C4C22×D4311yescharacteristic
393C22×C8C22×D4311yescharacteristic
394C22×C8C22×D4311yescharacteristic
395C22×C8C22×D4311yescharacteristic
396C22×C8C22×D4311yescharacteristic
397#121—434no—
398#121—862no—
399C4×D4—862no—
400C4×D4—862no—
401#117—22no—
402#117—22no—
403#117—862no—
404#117—862no—
405C2×C4⋊C4—434no—
406C2×C4⋊C4—431no—
407C2×C4⋊C4—431no—
408C2×C4⋊C4—431no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—431no—
411C2×C4⋊C4—431no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—861no—
414C2×C4⋊C4—862no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4D17211yescharacteristic
417C2×C4⋊C4D17211yescharacteristic
418C2×C22⋊C4—861no—
419C2×C22⋊C4—861no—
420C2×C22⋊C4—861no—
421C2×C22⋊C4—861no—
422C2×C22⋊C4—861no—
423C2×C22⋊C4—861no—
424C2×C22⋊C4D17211yescharacteristic
425C2×C22⋊C4D17211yescharacteristic
426C2×C42—861no—
427C2×C42—861no—
428C2×C42—861no—
429C2×C42—861no—
430C2×C42—861no—
431C2×C42—861no—
432C2×C42C22×D4311yescharacteristic
433C2×C42C22×D4311yescharacteristic
434#107—862no—
435#106—434no—
436#106—434no—
437#106—862no—
438#102—434no—
439#102—434no—
440#102—862no—
441#102—862no—
442C22⋊C8—22no—
443C22⋊C8C4×D4314yescomplemented
444#97—22no—
445C4×C8—22no—
446#95—861no—
447#95—861no—
448#95—861no—
449#95—861no—
450#95—861no—
451#95—861no—
452#95—861no—
453C24—861no—
454C24C2×D17211yescharacteristic
455C2×D4—432no—
456C2×D4—864no—
457C2×D4—861no—
458C2×D4—861no—
459C2×D4—861no—
460C2×D4—861no—
461C2×D4—861no—
462C2×D4—3441no—
463C2×D4—3441no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—864no—
469C22×C4—861no—
470C22×C4—861no—
471C22×C4—861no—
472C22×C4—861no—
473C22×C4—861no—
474C22×C4—861no—
475C22×C4—861no—
476C22×C4—3441no—
477C22×C4—3441no—
478C22×C4—3441no—
479C22×C4—3441no—
480C22×C4—1722no—
481C22×C4C2×C4×D4312yes—
482C22×C4C2×D17211yescharacteristic
483C22×C4C2×D17211yescharacteristic
484C22×C4C2×D17211yescharacteristic
485C22×C4C2×D17211yescharacteristic
486C22×C4C2×D17211yescharacteristic
487C22×C4#1062189812yes—
488C22×C4C23×D4311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8C2×C4×D4312yes—
500C2×C8C2×C4×D4312yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—432no—
504C4⋊C4—432no—
505C4⋊C4—432no—
506C4⋊C4—432no—
507C4⋊C4—432no—
508C4⋊C4—864no—
509C4⋊C4—861no—
510C4⋊C4—861no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4—861no—
514C4⋊C4—861no—
515C4⋊C4—861no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C4⋊C4—3441no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—864no—
522C22⋊C4—864no—
523C22⋊C4—3441no—
524C22⋊C4—3441no—
525C22⋊C4—3441no—
526C22⋊C4—1722no—
527C42—21no—
528C42—21no—
529C42—864no—
530C42—3441no—
531C42—3441no—
532C42—3441no—
533C42—3441no—
534C42—3441no—
535C42#1062190012yes—
536C42#1062190012yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—862no—
544C23—1721no—
545C23—1721no—
546C23—1721no—
547C23—1721no—
548C23—1721no—
549C23—1721no—
550C23C4×D17212yes—
551C23C22×D17211yescharacteristic
552D4—862no—
553D4—862no—
554D4—3444no—
555D4—1721no—
556D4—1721no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—862no—
574C2×C4—862no—
575C2×C4—862no—
576C2×C4—862no—
577C2×C4—862no—
578C2×C4—3444no—
579C2×C4—3444no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4—1721no—
593C2×C4—1721no—
594C2×C4—1721no—
595C2×C4—1721no—
596C2×C4C4×D17212yes—
597C2×C4#1306330912yes—
598C2×C4#1306330912yes—
599C2×C4C22×C4×D4311yescharacteristic
600C2×C4#41×D4311yescharacteristic
601C2×C4#41×D4311yescharacteristic
602C2×C4C2×#1062190011yescharacteristic
603C2×C4C2×#1062190011yescharacteristic
604C2×C4C2×#1062189811yescharacteristic
605C8—22no—
606C8—22no—
607C8—41no—
608C8—41no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—3441no—
617C22—3441no—
618C22—3441no—
619C22—3441no—
620C22—1722no—
621C22—1722no—
622C22—1722no—
623C22C2×C4×D17211yescharacteristic
624C22C2×#1306330911yescharacteristic
625C22#1458341111yescharacteristic
626C22#1458370411yescharacteristic
627C22#1458371111yescharacteristic
628C22#1458371111yescharacteristic
629C22#1458371111yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—3441no—
636C4—3441no—
637C4—1722no—
638C4—1722no—
639C4—1722no—
640C4—1722no—
641C4—1722no—
642C4C4×#1062190011yescharacteristic
643C4C4×#1062190011yescharacteristic
644C4#1458342911yescharacteristic
645C4#1458342911yescharacteristic
646C2—42no—
647C2—3442no—
648C2C4×#1306330911yescharacteristic
649C2#1914342911yescharacteristic
650C2#1914345611yescharacteristic
651C2#1914455411yescharacteristic
652C2#1914458611yescharacteristic
653C2#1914479611yescharacteristic
654C2#1914487411yescharacteristic
subgroupG/H
centerC23C22×D172
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1750C2
socleC23×C43C22×D4
prime radicalC2×#13063299C22
cosocleC22×C4×C43C24
hypercenter#1750D43
hypocenterC43#21584
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
3. #19136951C2semidirect
4. #19135223C2semidirect
5. #19135216C2semidirect
6. #19135181C2semidirect
7. #19135174C2semidirect
9. #19134796C2semidirect
10. #19134682C2semidirect
11. #19134492C2semidirect
12. C4×#13063257C2semidirect
13. #19134421C2semidirect
14. C4×#13063247C2semidirect
15. C43×#1750C2semidirect
20. #14583218C4semidirect
72. C2×C4⋊Dic43C2×C4semidirect
95. #13063257C2×C4semidirect
96. #13063257C2×C4semidirect
122. C22⋊C8×C43C2×C4semidirect
333. #1750D43semidirect
387. C43#21584semidirect
443. C22⋊C8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832182
#191372882
#191369512
#191352232
#191352162
#191351812
#191351742
#191349152
#191347962
#191346822
#191344922
C4×#130632572
#191344212
C4×#130632472
C43×#17502
#2158443