Finite Group Databasebeta

← #36268226

Subgroups — #36268226

Order 21248.

2038 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24920662C211yesmaximal, characteristic
2#24916131C213yesmaximal, complemented
3#24916128C213yesmaximal, complemented
4#24916120C213yesmaximal, complemented
5#24916120C213yesmaximal, complemented
6#24916113C213yesmaximal, complemented
7#24916106C213yesmaximal, complemented
8#24916100C213yesmaximal, complemented
9C2×#19127761C213yesmaximal, complemented
10#24916031C211yesmaximal, characteristic, complemented
11#24915910C211yesmaximal, characteristic
12#24915910C211yesmaximal, characteristic
13#24915889C211yesmaximal, characteristic, complemented
14#131×Dic83C211yesmaximal, characteristic, complemented
15C83×#2906C211yesmaximal, characteristic, complemented
16C2×#41×D83C2211yescharacteristic
17C2×#14581799C2211yescharacteristic
18C2×#14581799C2211yescharacteristic
19#19128183C411yescharacteristic
20#19128183C411yescharacteristic
21#19128180C413yes—
22#19128180C413yes—
23#19128177C411yescharacteristic
24#19128177C411yescharacteristic
25#19128173C413yes—
26#19128173C413yes—
27#41×Dic83C2211yescharacteristic
28C4×#13050590C2213yes—
29#19128169C2213yes—
30C2×#14581749C2213yes—
31#117×D83C2213yes—
32#19127761C2211yescharacteristic
33#19127761C2213yes—
34#19127761C2213yes—
35#19127761C2213yes—
36#19127761C2216yes—
37#19127761C2213yescomplemented
38#19127761C2213yescomplemented
39#19127761C2213yescomplemented
40#19127761C2213yescomplemented
41#19127761C2216yescomplemented
42#19127761C2216yescomplemented
43#19127757C2216yescomplemented
44#19127757C2216yescomplemented
45#19127753C2216yescomplemented
46#19127753C2216yescomplemented
47#19127728C2213yescomplemented
48#19127728C2213yescomplemented
49#19127725C2213yescomplemented
50C2×C4⋊(C83⋊C8)C2213yescomplemented
51#19127669C2213yes—
52#19127669C2213yes—
53#19127667C2213yes—
54#19127667C2213yes—
55#19127664C2213yes—
56#19127664C2213yes—
57C2×(C2×D83)⋊C8C2213yes—
58C2×(C2×D83)⋊C8C2213yes—
59#19127659C2213yescomplemented
60#19127659C2213yescomplemented
61#19127656C2213yescomplemented
62#19127656C2213yescomplemented
63M16×Dic83C2213yescomplemented
64M16×Dic83C2213yescomplemented
65C2×#14581702C2213yescomplemented
66C2×C8×Dic83C2213yescomplemented
67C2×#131×C83C2211yescharacteristic
68C2×#131×C83C2211yescharacteristic
69C4×#41×C83C2211yescharacteristic
70#424×C83C2213yescomplemented
71#424×C83C2213yescomplemented
72#423×C83C2213yescomplemented
73C2×C4⋊C8×C83C2213yescomplemented
74C2×#13050591C2×C411yescharacteristic
75C2×#13050590C2313yes—
76C2×#13050589C2×C413yes—
77#41×D83—24no—
78#41×D83—24no—
79C2×Q8×D83C2×C411yescharacteristic
80C2×D4×D83C2×C413yes—
81C22×C4×D83C2313yes—
82#14581799—24no—
83#14581799—24no—
84C2×#13050574C2313yes—
85C2×#13050574C2313yes—
86C22×C83⋊C8C2313yes—
87C22×C83⋊C8C2313yes—
88Q8×Dic83C2×C411yescharacteristic
89Q8×Dic83C2×C411yescharacteristic
90C4×Dic166C2×C413yes—
91C4×Dic166C2×C413yes—
92#14581757C2×C413yes—
93#14581757C2×C413yes—
94C4×D332C2×C413yes—
95C4×D332C2×C413yes—
96D4×Dic83C2×C413yes—
97D4×Dic83C2×C413yes—
98#14581754C2×C413yes—
99#14581754C2×C413yes—
100C4×C22⋊D83C2×C416yes—
101C4×C22⋊D83C2×C416yes—
102(C22⋊D83)⋊C4C2×C416yes—
103(C22⋊D83)⋊C4C2×C416yes—
104#14581751C2313yes—
105#14581750C2×C413yes—
106#14581750C2×C413yes—
107#14581749C2×C413yes—
108#14581749C2×C413yes—
109#14581749C2×C413yes—
110#14581749C2×C413yes—
111#14581749C2313yes—
112#14581749C2311yescharacteristic
113#14581749C2313yes—
114#14581749C2313yes—
115#14581748C2×C416yes—
116C2×Dic83⋊C4C2×C413yes—
117C2×Dic83⋊C4C2×C413yes—
118C4⋊C4×D83C2×C413yes—
119C4⋊C4×D83C2×C413yes—
120C2×(C2×D83)⋊C4C2×C413yes—
121C2×(C2×D83)⋊C4C2×C413yes—
122C22⋊C4×D83C2×C416yes—
123C2×C4×Dic83C2313yes—
124C42×D83C2316yes—
125C4⋊(C83⋊C8)C2311yescharacteristic
126C4⋊(C83⋊C8)C2313yes—
127C4⋊(C83⋊C8)C2313yes—
128C4⋊(C83⋊C8)C2313yes—
129C4⋊(C83⋊C8)C2316yes—
130(C2×D83)⋊C8C2311yescharacteristic
131(C2×D83)⋊C8C2311yescharacteristic
132(C2×D83)⋊C8C2313yes—
133(C2×D83)⋊C8C2313yes—
134(C2×D83)⋊C8C2313yes—
135(C2×D83)⋊C8C2313yes—
136(C2×D83)⋊C8C2313yes—
137(C2×D83)⋊C8C2313yes—
138(C2×D83)⋊C8C2316yes—
139(C2×D83)⋊C8C2316yes—
140#14581702C2311yescharacteristic
141#14581702C2313yes—
142#14581702C2313yes—
143#14581702C2313yes—
144#14581702C2316yes—
145C8×Dic83C2311yescharacteristic
146C8×Dic83C2313yes—
147C8×Dic83C2313yes—
148C8×Dic83C2313yes—
149C8×Dic83C2316yes—
150C2×#41×C83C2311yescharacteristic
151#131×C83—24no—
152#131×C83—24no—
153C2×M16×C83C2313yes—
154C2×M16×C83C2313yes—
155C22×C8×C83C2313yes—
156C22×C8×C83C2313yes—
157C4×Q8×C83C2×C411yescharacteristic
158C4×Q8×C83C2×C411yescharacteristic
159C4×D4×C83C2×C413yes—
160C4×D4×C83C2×C413yes—
161#117×C83C2313yes—
162C2×C42×C83C2313yes—
163C4⋊C8×C83C2313yes—
164C4⋊C8×C83C2316yes—
165C4⋊C8×C83C2313yes—
166C4⋊C8×C83C2311yescharacteristic
167C4⋊C8×C83C2313yes—
168C23×D83C22×C413yes—
169#13050591—24no—
170#13050590—26no—
171#13050590—26no—
172#13050589—212no—
173C2×Dic166C22×C413yes—
174Q8×D83—24no—
175C2×D332C22×C413yes—
176C2×C22⋊D83C22×C416yes—
177D4×D83—212no—
178C22×Dic83C22×C413yes—
179C22×Dic83C22×C413yes—
180C22×Dic83C22×C413yes—
181C2×C4×D83—26no—
182C2×C4×D83—26no—
183C2×C4×D83C22×C413yes—
184C2×C4×D83C22×C413yes—
185C2×C4×D83C22×C416yes—
186C2×C4×D83C22×C416yes—
187C2×C4×D83C2411yescharacteristic
188C2×C4×D83C2413yes—
189#13050574—26no—
190#13050574—26no—
191C2×C83⋊C8—26no—
192C2×C83⋊C8—26no—
193C2×C83⋊C8C2411yescharacteristic
194C2×C83⋊C8C2411yescharacteristic
195C2×C83⋊C8C2413yes—
196C2×C83⋊C8C2413yes—
197C4⋊Dic83C22×C413yes—
198C4⋊Dic83C22×C413yes—
199Dic83⋊C4C22×C413yes—
200Dic83⋊C4C22×C413yes—
201Dic83⋊C4C22×C411yescharacteristic
202Dic83⋊C4C22×C411yescharacteristic
203Dic83⋊C4C22×C413yes—
204Dic83⋊C4C22×C413yes—
205Dic83⋊C4C22×C413yes—
206Dic83⋊C4C22×C413yes—
207C22⋊Dic83C22×C416yes—
208(C2×D83)⋊C4C22×C413yes—
209(C2×D83)⋊C4C22×C413yes—
210(C2×D83)⋊C4C22×C411yescharacteristic
211(C2×D83)⋊C4C22×C411yescharacteristic
212(C2×D83)⋊C4C22×C413yes—
213(C2×D83)⋊C4C22×C413yes—
214(C2×D83)⋊C4C22×C413yes—
215(C2×D83)⋊C4C22×C413yes—
216C4×Dic83C22×C416yes—
217C4×Dic83C22×C416yes—
218C4×Dic83C22×C413yes—
219C4×Dic83C22×C413yes—
220C4×Dic83C2413yes—
221C4×Dic83C2411yescharacteristic
222#41×C83#4114yes—
223#41×C83#4114yes—
224C2×Q8×C83C22×C411yescharacteristic
225C2×D4×C83C22×C413yes—
226C22×C4×C83C22×C413yes—
227C22×C4×C83C22×C413yes—
228C22×C4×C83C2413yes—
229M16×C83—26no—
230M16×C83—26no—
231C2×C8×C83—26no—
232C2×C8×C83—26no—
233C2×C8×C83C2411yescharacteristic
234C2×C8×C83C2411yescharacteristic
235C2×C8×C83C2413yes—
236C2×C8×C83C2413yes—
237C4⋊C4×C83C22×C413yes—
238C4⋊C4×C83C22×C413yes—
239C22⋊C4×C83C22×C416yes—
240C42×C83C22×C413yes—
241C42×C83C22×C413yes—
242C42×C83C2411yescharacteristic
243C42×C83C2413yes—
244C23×C83C23×C413yes—
245C22×D83—26no—
246C22×D83—43no—
247C22×D83—43no—
248C22×D83C23×C411yescharacteristic
249C22×D83C23×C413yes—
250Q8×C83#11714yes—
251D4×C83#117112yes—
252C2×C4×C83—43no—
253C2×C4×C83—43no—
254C2×C4×C83C23×C411yescharacteristic
255C2×C4×C83C23×C411yescharacteristic
256C2×C4×C83C23×C413yes—
257C2×C4×C83C23×C413yes—
258C2×C4×C83C23×C413yes—
259C2×C4×C83C2×#4116yes—
260C2×C4×C83C2×#4116yes—
261C2×C4×C83C2511yescharacteristic
262C22⋊D83—212no—
263C2×Dic83—26no—
264C2×Dic83—43no—
265C2×Dic83—43no—
266C2×Dic83C23×C413yes—
267C2×Dic83C23×C411yescharacteristic
268C2×Dic83C23×C411yescharacteristic
269C2×Dic83C23×C413yes—
270C2×Dic83C23×C413yes—
271C2×Dic83C23×C411yescharacteristic
272D332—26no—
273C4×D83—26no—
274C4×D83—26no—
275C4×D83—21no—
276C4×D83—21no—
277C4×D83—23no—
278C4×D83—23no—
279C4×D83—46no—
280C4×D83—46no—
281Dic166—26no—
282C8×C83—21no—
283C8×C83—21no—
284C8×C83—23no—
285C8×C83—23no—
286C83⋊C8—21no—
287C83⋊C8—21no—
288C83⋊C8—23no—
289C83⋊C8—23no—
290C22×C83—23no—
291C22×C83—23no—
292C22×C83C2×#11716yes—
293C22×C83C24×C411yescharacteristic
294C2×D83—21no—
295C2×D83—23no—
296C2×D83—23no—
297C2×D83—23no—
298C2×D83—21no—
299C2×D83—21no—
300C2×D83—86no—
301C4×C83—23no—
302C4×C83—23no—
303C4×C83—21no—
304C4×C83—21no—
305C4×C83C2×#11716yes—
306C4×C83C22×#4111yescharacteristic
307C4×C83C22×#4111yescharacteristic
308Dic83—23no—
309Dic83—21no—
310Dic83—23no—
311Dic83—23no—
312Dic83—21no—
313Dic83—21no—
314#60129—831nomaximal, Sylow
315C2×C83—26no—
316C2×C83C22×#11711yescharacteristic
317C2×C83#357311yescharacteristic
318C2×C83#357311yescharacteristic
319D83—43no—
320D83—41no—
321#3428—831no—
322#2925—833no—
323#2925—833no—
324#2924—833no—
325#2924—833no—
326#2923—833no—
327#2922—833no—
328#2921—833no—
329C2×#433—833no—
330#2906—831no—
331#2906D8311yescharacteristic, complemented
332#2880—831no—
333#2880—831no—
334#2875—831no—
335C4×#131—831no—
336C83#6012911yescharacteristic, complemented, Sylow
337C22×#41—831no—
338C2×#131—831no—
339C2×#131—831no—
340C2×#131C2×D8311yescharacteristic
341C2×#131C2×D8311yescharacteristic
342#520—831no—
343#520—831no—
344#520—833no—
345#520—833no—
346#519—831no—
347#519—831no—
348#518—833no—
349#518—833no—
350C4×#41—831no—
351C4×#41—833no—
352C4×#41—833no—
353C4×#41C2×D8311yescharacteristic
354C2×#117—833no—
355C2×#117—833no—
356#433—831no—
357#433—833no—
358#433—833no—
359#433—833no—
360#433—833no—
361#433—833no—
362#433—833no—
363#433—833no—
364#433—836no—
365#433—836no—
366#433—836no—
367#432—836no—
368#432—836no—
369#431—836no—
370#431—836no—
371#424—833no—
372#424—833no—
373#424C2×D8313yescomplemented
374#424C2×D8313yescomplemented
375#423—833no—
376#423C2×D8313yescomplemented
377C2×C4⋊C8—833no—
378C2×C4⋊C8C2×D8313yescomplemented
379#408—833no—
380#408—833no—
381#408—833no—
382#408—833no—
383#407—833no—
384#407—833no—
385C2×C22⋊C8—833no—
386C2×C22⋊C8—833no—
387#405—833no—
388#405—833no—
389#405—833no—
390#405—833no—
391C4×M16—833no—
392C4×M16—833no—
393C2×#97—833no—
394C2×C4×C8—833no—
395C2×#41—831no—
396C2×#41—833no—
397C2×#41—833no—
398C2×#41—1664no—
399C2×#41—1664no—
400C2×#41C22×D8311yescharacteristic
401C22×Q8—831no—
402C22×D4—833no—
403C23×C4—833no—
404#131—24no—
405#131—24no—
406#131—1664no—
407#131—1664no—
408C2×M16—833no—
409C2×M16—833no—
410C2×M16C22×D8313yes—
411C2×M16C22×D8313yes—
412C22×C8—833no—
413C22×C8—833no—
414C22×C8C22×D8313yes—
415C22×C8C22×D8313yes—
416C4×Q8—831no—
417C4×Q8—831no—
418C4×Q8—833no—
419C4×Q8—833no—
420C4×Q8—833no—
421C4×Q8—833no—
422C4×Q8C4×D8311yescharacteristic
423C4×Q8C4×D8311yescharacteristic
424C4×D4—833no—
425C4×D4—833no—
426C4×D4—836no—
427C4×D4—836no—
428C4×D4—833no—
429C4×D4—833no—
430C4×D4—836no—
431C4×D4—836no—
432C4×D4—833no—
433C4×D4—833no—
434C4×D4C4×D8313yes—
435C4×D4C4×D8313yes—
436#117—833no—
437#117—833no—
438#117—833no—
439#117—836no—
440#117—833no—
441#117—833no—
442#117—833no—
443#117—833no—
444#117—833no—
445#117—831no—
446#117—833no—
447#117—833no—
448#117C22×D8313yes—
449C2×C4⋊C4—833no—
450C2×C4⋊C4—833no—
451C2×C4⋊C4—833no—
452C2×C4⋊C4—833no—
453C2×C22⋊C4—833no—
454C2×C22⋊C4—833no—
455C2×C22⋊C4—836no—
456C2×C42—833no—
457C2×C42—836no—
458C2×C42C22×D8313yes—
459C4⋊C8—831no—
460C4⋊C8—833no—
461C4⋊C8—833no—
462C4⋊C8—833no—
463C4⋊C8—836no—
464C4⋊C8C22×D8311yescharacteristic
465C4⋊C8C22×D8313yes—
466C4⋊C8C22×D8313yes—
467C4⋊C8C22×D8313yes—
468C4⋊C8C22×D8316yes—
469C22⋊C8—831no—
470C22⋊C8—831no—
471C22⋊C8—833no—
472C22⋊C8—833no—
473C22⋊C8—833no—
474C22⋊C8—833no—
475C22⋊C8—833no—
476C22⋊C8—833no—
477C22⋊C8—836no—
478C22⋊C8—836no—
479#97—831no—
480#97—833no—
481#97—833no—
482#97—833no—
483#97—836no—
484C4×C8—831no—
485C4×C8—833no—
486C4×C8—833no—
487C4×C8—833no—
488C4×C8—836no—
489C24—833no—
490#41—1664no—
491#41—16612no—
492#41—1666no—
493#41—1666no—
494#41#1305058914yes—
495#41#1305059114yes—
496C2×Q8—833no—
497C2×Q8—1664no—
498C2×Q8C2×C4×D8311yescharacteristic
499C2×D4—836no—
500C2×D4—833no—
501C2×D4—16612no—
502C2×D4C2×C4×D8313yes—
503C22×C4—833no—
504C22×C4—833no—
505C22×C4—836no—
506C22×C4—836no—
507C22×C4—833no—
508C22×C4—833no—
509C22×C4—833no—
510C22×C4—833no—
511C22×C4—831no—
512C22×C4—1666no—
513C22×C4—1666no—
514C22×C4C2×C4×D8313yes—
515C22×C4C2×C4×D8313yes—
516C22×C4C23×D8313yes—
517M16—26no—
518M16—26no—
519M16—1666no—
520M16—1666no—
521C2×C8—26no—
522C2×C8—26no—
523C2×C8—831no—
524C2×C8—833no—
525C2×C8—833no—
526C2×C8—831no—
527C2×C8—1666no—
528C2×C8—1666no—
529C2×C8C23×D8311yescharacteristic
530C2×C8C23×D8313yes—
531C2×C8C23×D8313yes—
532C2×C8C23×D8311yescharacteristic
533C4⋊C4—833no—
534C4⋊C4—833no—
535C4⋊C4—831no—
536C4⋊C4—831no—
537C4⋊C4—833no—
538C4⋊C4—833no—
539C4⋊C4—833no—
540C4⋊C4—833no—
541C4⋊C4—833no—
542C4⋊C4—833no—
543C4⋊C4C2×C4×D8313yes—
544C4⋊C4C2×C4×D8313yes—
545C22⋊C4—836no—
546C22⋊C4—833no—
547C22⋊C4—833no—
548C22⋊C4—831no—
549C22⋊C4—831no—
550C22⋊C4—833no—
551C22⋊C4—833no—
552C22⋊C4—833no—
553C22⋊C4—833no—
554C22⋊C4C2×C4×D8316yes—
555C42—833no—
556C42—833no—
557C42—836no—
558C42—836no—
559C42—831no—
560C42—833no—
561C42C2×C4×D8313yes—
562C42C2×C4×D8313yes—
563C42C23×D8313yes—
564C42C23×D8311yescharacteristic
565C23—3323no—
566C23—3323no—
567C23—831no—
568C23—833no—
569C23—1666no—
570C23C22×C4×D8313yes—
571Q8—1666no—
572Q8#1458175014yes—
573D4—16612no—
574D4—1666no—
575D4#14581750112yes—
576C2×C4—43no—
577C2×C4—43no—
578C2×C4—3326no—
579C2×C4—3326no—
580C2×C4—3323no—
581C2×C4—3323no—
582C2×C4—833no—
583C2×C4—831no—
584C2×C4—833no—
585C2×C4—831no—
586C2×C4—833no—
587C2×C4—831no—
588C2×C4—1666no—
589C2×C4—1666no—
590C2×C4—1666no—
591C2×C4—1661no—
592C2×C4—1663no—
593C2×C4—1661no—
594C2×C4—1663no—
595C2×C4C22×C4×D8313yes—
596C2×C4C22×C4×D8313yes—
597C2×C4C22×C4×D8311yescharacteristic
598C2×C4C22×C4×D8311yescharacteristic
599C2×C4C22×C4×D8313yes—
600C2×C4C2×#1305058916yes—
601C2×C4C2×#1305059116yes—
602C2×C4C24×D8311yescharacteristic
603C8—21no—
604C8—21no—
605C8—23no—
606C8—23no—
607C8—1661no—
608C8—1661no—
609C8—1663no—
610C8—1663no—
611C22—23no—
612C22—23no—
613C22—6646no—
614C22—1661no—
615C22—1663no—
616C22—1663no—
617C22—1661no—
618C22—1663no—
619C22—1661no—
620C22C2×#1458175016yes—
621C22C23×C4×D8311yescharacteristic
622C4—21no—
623C4—23no—
624C4—21no—
625C4—23no—
626C4—1661no—
627C4—1663no—
628C4—1663no—
629C4—1661no—
630C4—1663no—
631C4—1661no—
632C4C2×#1458175016yes—
633C4C22×#1305058911yescharacteristic
634C4C22×#1305059111yescharacteristic
635C2—26no—
636C2—3321no—
637C2—3323no—
638C2C22×#1458175011yescharacteristic
639C2#2492217411yescharacteristic
640C2#2492217611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D83
commutator, socleC22×C83C24×C4
FittingC83×#2906C2
prime radicalC2×#41×D83C22
cosocleC2×C4×C83C25
hypercenter#2906D83
hypocenterC83#60129
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #24916131C2semidirect
3. #24916128C2semidirect
4. #24916120C2semidirect
5. #24916120C2semidirect
6. #24916113C2semidirect
7. #24916106C2semidirect
8. #24916100C2semidirect
9. C2×#19127761C2semidirect
10. #24916031C2semidirect
13. #24915889C2semidirect
14. #131×Dic83C2semidirect
15. C83×#2906C2semidirect
37. #19127761C22semidirect
38. #19127761C22semidirect
39. #19127761C22semidirect
40. #19127761C22semidirect
41. #19127761C22semidirect
42. #19127761C22semidirect
43. #19127757C22semidirect
44. #19127757C22semidirect
45. #19127753C22semidirect
46. #19127753C22semidirect
47. #19127728C22semidirect
48. #19127728C22semidirect
49. #19127725C22semidirect
50. C2×C4⋊(C83⋊C8)C22semidirect
59. #19127659C22semidirect
60. #19127659C22semidirect
61. #19127656C22semidirect
62. #19127656C22semidirect
63. M16×Dic83C22semidirect
64. M16×Dic83C22semidirect
65. C2×#14581702C22semidirect
66. C2×C8×Dic83C22semidirect
70. #424×C83C22semidirect
71. #424×C83C22semidirect
72. #423×C83C22semidirect
73. C2×C4⋊C8×C83C22semidirect
331. #2906D83semidirect
336. C83#60129semidirect
373. #424C2×D83semidirect
374. #424C2×D83semidirect
376. #423C2×D83semidirect
378. C2×C4⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249206622
#249161312
#249161282
#249161202
#249161132
#249161062
#249161002
C2×#191277612
#249160312
#249159102
#249158892
#131×Dic832
C83×#29062
#6012983