Finite Group Databasebeta

← #35310917

Subgroups — #35310917

Order 21248.

1578 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581716C212yesmaximal, complemented
2C22×C22⋊(C83⋊C8)C211yesmaximal, characteristic
3C2×#19127561C211yesmaximal, characteristic
4C2×#19127463C212yesmaximal, complemented
5C22×C22⋊Q8×C83C211yesmaximal, characteristic
6C23×C83⋊C8C2212yes—
7C22×C4⋊Dic83C2212yes—
8C22×C22⋊Dic83—21no—
9C2×#14581716—24no—
10C2×#14581716—24no—
11C2×#14581716—24no—
12C2×C22⋊(C83⋊C8)—22no—
13C2×C22⋊(C83⋊C8)—24no—
14#19127561—24no—
15C2×C83⋊#95—21no—
16#19127463—24no—
17C23×Q8×C83C2211yescharacteristic
18C24×C4×C83C2211yescharacteristic
19C2×C22⋊Q8×C83—24no—
20C2×C22⋊Q8×C83—22no—
21C2×C22⋊Q8×C83—24no—
22C2×C22⋊Q8×C83C414yescomplemented
23C22×C4⋊C4×C83—21no—
24C22×C4⋊C4×C83C2211yescharacteristic
25C22×C22⋊C4×C83—21no—
26C23×Dic83—22no—
27C22×C83⋊C8—22no—
28C22×C83⋊C8—22no—
29C22×C83⋊C8—22no—
30C22×C83⋊C8—22no—
31C22×C83⋊C8—22no—
32C22×C83⋊C8—22no—
33C22×C83⋊C8D414yes—
34C2×C4⋊Dic83—22no—
35C2×C4⋊Dic83—22no—
36C2×C4⋊Dic83—22no—
37C2×C4⋊Dic83—22no—
38C2×C4⋊Dic83—22no—
39C2×C4⋊Dic83D414yes—
40C2×C22⋊Dic83—24no—
41C2×C22⋊Dic83—42no—
42C2×C22⋊Dic83—42no—
43#14581716—216no—
44C22⋊(C83⋊C8)—28no—
45C83⋊#95—24no—
46C25×C83D411yescharacteristic
47C22×Q8×C83—21no—
48C22×Q8×C83—22no—
49C22×Q8×C83—22no—
50C22×Q8×C83—22no—
51C22×Q8×C83—22no—
52C22×Q8×C83—22no—
53C22×Q8×C83C2×C412yes—
54C22×Q8×C83D414yes—
55C23×C4×C83—21no—
56C23×C4×C83—21no—
57C23×C4×C83—22no—
58C23×C4×C83—22no—
59C23×C4×C83—22no—
60C23×C4×C83—22no—
61C23×C4×C83—22no—
62C23×C4×C83—22no—
63C23×C4×C83C2×C412yes—
64C23×C4×C83D411yescharacteristic
65C23×C4×C83D412yes—
66C23×C4×C83C2311yescharacteristic
67C22⋊Q8×C83—216no—
68C22⋊Q8×C83—28no—
69C22⋊Q8×C83—28no—
70C2×C4⋊C4×C83—28no—
71C2×C4⋊C4×C83—24no—
72C2×C4⋊C4×C83—22no—
73C2×C4⋊C4×C83—21no—
74C2×C4⋊C4×C83C2×C412yes—
75C2×C4⋊C4×C83D414yes—
76C2×C22⋊C4×C83—28no—
77C2×C22⋊C4×C83—24no—
78C22×Dic83—24no—
79C22×Dic83—42no—
80C22×Dic83—42no—
81C22×Dic83—42no—
82C22×Dic83—42no—
83C22×Dic83—42no—
84C22×Dic83—42no—
85C2×C83⋊C8—24no—
86C2×C83⋊C8—24no—
87C2×C83⋊C8—24no—
88C2×C83⋊C8—42no—
89C2×C83⋊C8—42no—
90C2×C83⋊C8—42no—
91C2×C83⋊C8—42no—
92C4⋊Dic83—24no—
93C4⋊Dic83—24no—
94C4⋊Dic83—42no—
95C4⋊Dic83—42no—
96C22⋊Dic83—48no—
97C24×C83—22no—
98C24×C83—21no—
99C24×C83—22no—
100C24×C83—21no—
101C24×C83—22no—
102C24×C83—21no—
103C24×C83—21no—
104C24×C83—22no—
105C24×C83C22⋊C412yes—
106C24×C83SD1612yes—
107C24×C83Q1612yes—
108C24×C83C2×D411yescharacteristic
109C2×Q8×C83—22no—
110C2×Q8×C83—24no—
111C2×Q8×C83—24no—
112C2×Q8×C83—24no—
113C2×Q8×C83—24no—
114C2×Q8×C83—24no—
115C2×Q8×C83—24no—
116C2×Q8×C83—24no—
117C2×Q8×C83—41no—
118C2×Q8×C83—42no—
119C2×Q8×C83—42no—
120C2×Q8×C83—42no—
121C2×Q8×C83—41no—
122C2×Q8×C83—42no—
123C2×Q8×C83—42no—
124C2×Q8×C83C22⋊C414yescomplemented
125C22×C4×C83—22no—
126C22×C4×C83—22no—
127C22×C4×C83—24no—
128C22×C4×C83—24no—
129C22×C4×C83—22no—
130C22×C4×C83—21no—
131C22×C4×C83—22no—
132C22×C4×C83—21no—
133C22×C4×C83—22no—
134C22×C4×C83—22no—
135C22×C4×C83—24no—
136C22×C4×C83—24no—
137C22×C4×C83—24no—
138C22×C4×C83—22no—
139C22×C4×C83—24no—
140C22×C4×C83—22no—
141C22×C4×C83—41no—
142C22×C4×C83—42no—
143C22×C4×C83—41no—
144C22×C4×C83—42no—
145C22×C4×C83—41no—
146C22×C4×C83—41no—
147C22×C4×C83—42no—
148C22×C4×C83—42no—
149C22×C4×C83—42no—
150C22×C4×C83—42no—
151C22×C4×C83—42no—
152C22×C4×C83—42no—
153C22×C4×C83—42no—
154C22×C4×C83—42no—
155C22×C4×C83C22⋊C412yes—
156C22×C4×C83C22×C411yescharacteristic
157C22×C4×C83C2×D412yes—
158C22×C4×C83C2×D412yes—
159C22×C4×C83C2×D412yes—
160C22×C4×C83C2×D412yes—
161C22×C4×C83C2×D411yescharacteristic
162C22×C4×C83#4112yes—
163C22×C4×C83#4112yes—
164C4⋊C4×C83—216no—
165C4⋊C4×C83—22no—
166C4⋊C4×C83—24no—
167C4⋊C4×C83C22⋊C414yescomplemented
168C22⋊C4×C83—216no—
169C23×C83—22no—
170C23×C83—22no—
171C23×C83—22no—
172C23×C83—22no—
173C23×C83—22no—
174C23×C83—22no—
175C23×C83—24no—
176C23×C83—22no—
177C23×C83—22no—
178C23×C83—24no—
179C23×C83—24no—
180C23×C83—24no—
181C23×C83—22no—
182C23×C83—22no—
183C23×C83—42no—
184C23×C83—42no—
185C23×C83—42no—
186C23×C83—42no—
187C23×C83—42no—
188C23×C83—42no—
189C23×C83—42no—
190C23×C83—42no—
191C23×C83#10314yescomplemented
192C23×C83C2×C22⋊C411yescharacteristic
193C23×C83#12011yescharacteristic
194C23×C83#12012yes—
195C23×C83#12112yes—
196C23×C83#12311yescharacteristic
197C23×C83C2×SD1611yescharacteristic
198C23×C83C2×Q1611yescharacteristic
199C23×C83#13512yes—
200C23×C83AGL(1,ℤ/8)11yescharacteristic
201C23×C83#13711yescharacteristic
202C23×C83#13712yes—
203Q8×C83—28no—
204Q8×C83—44no—
205Q8×C83—44no—
206Q8×C83—44no—
207C2×C4×C83—24no—
208C2×C4×C83—24no—
209C2×C4×C83—22no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—42no—
213C2×C4×C83—42no—
214C2×C4×C83—42no—
215C2×C4×C83—44no—
216C2×C4×C83—42no—
217C2×C4×C83—44no—
218C2×C4×C83—42no—
219C2×C4×C83—42no—
220C2×C4×C83—44no—
221C2×C4×C83—44no—
222C2×C4×C83—44no—
223C2×C4×C83—44no—
224C2×C4×C83—44no—
225C2×C4×C83—44no—
226C2×C4×C83C2×C22⋊C411yescharacteristic
227C2×C4×C83C2×C22⋊C411yescharacteristic
228C2×C4×C83C4×D412yes—
229C2×C4×C83C4×D412yes—
230C2×C4×C83#12012yes—
231C2×C4×C83#12011yescharacteristic
232C2×C4×C83#12012yes—
233C2×C4×C83#12112yes—
234C2×C4×C83#12112yes—
235C2×C4×C83#12112yes—
236C2×C4×C83#12112yes—
237C2×C4×C83#12112yes—
238C2×C4×C83#12312yes—
239C2×C4×C83#12311yescharacteristic
240C2×C4×C83#12412yes—
241C2×C4×C83#12712yes—
242C2×Dic83—44no—
243C2×Dic83—44no—
244C2×Dic83—44no—
245C2×Dic83—82no—
246C2×Dic83—82no—
247C2×Dic83—82no—
248C2×Dic83—82no—
249C83⋊C8—44no—
250C22×C83—24no—
251C22×C83—24no—
252C22×C83—24no—
253C22×C83—42no—
254C22×C83—42no—
255C22×C83—42no—
256C22×C83—44no—
257C22×C83—44no—
258C22×C83—44no—
259C22×C83—44no—
260C22×C83—42no—
261C22×C83#37911yescharacteristic
262C22×C83#38611yescharacteristic
263C22×C83#38611yescharacteristic
264C22×C83#39212yes—
265C22×C83#39412yes—
266C22×C83C2×#10311yescharacteristic
267C22×C83#41611yescharacteristic
268C22×C83#41711yescharacteristic
269C22×C83#41911yescharacteristic
270C22×C83#44812yes—
271C22×C83#44912yes—
272C22×C83#45011yescharacteristic
273C22×C83#45111yescharacteristic
274C22×C83#45112yes—
275C22×C83#45212yes—
276C22×C83#45212yes—
277C22×C83#46512yes—
278C22×C83#46712yes—
279C22×C83#46812yes—
280C22×C83#47012yes—
281C22×C83#48311yescharacteristic
282C22×C83#48411yescharacteristic
283C22×C83#48512yes—
284C4×C83—42no—
285C4×C83—44no—
286C4×C83—44no—
287C4×C83#38612yes—
288C4×C83#39012yes—
289C4×C83#39212yes—
290C4×C83#39412yes—
291Dic83—84no—
292#10970—831nomaximal, Sylow
293C2×C83—44no—
294C2×C83#143811yescharacteristic
295C2×C83#178711yescharacteristic
296C2×C83#178811yescharacteristic
297C2×C83#187211yescharacteristic
298C2×C83#187311yescharacteristic
299C2×C83#189211yescharacteristic
300C2×C83#189411yescharacteristic
301C2×C83#200012yes—
302C2×C83#200112yes—
303C2×C83#203512yes—
304C2×C83#203612yes—
305C22×C22⋊Q8D8311yescharacteristic
306C22×#103—832no—
307C22×C22⋊C8—831no—
308C2×#380—831no—
309C2×#340—832no—
310C83#1097011yescharacteristic, complemented, Sylow
311C23×Q8C2×D8311yescharacteristic
312C24×C4C2×D8311yescharacteristic
313C23×C8—832no—
314C2×C22⋊Q8—24no—
315C2×C22⋊Q8—22no—
316C2×C22⋊Q8—24no—
317C2×C22⋊Q8Dic8314yescomplemented
318C22×C4⋊C4—21no—
319C22×C4⋊C4—832no—
320C22×C4⋊C4C2×D8311yescharacteristic
321C22×C22⋊C4—21no—
322C22×C22⋊C4—1661no—
323C2×#103—1664no—
324C2×#103—1664no—
325C2×#103—1664no—
326C2×C22⋊C8—1664no—
327C2×C22⋊C8—1662no—
328#380—1664no—
329C2×#95—1661no—
330#340—1664no—
331C25C22⋊D8311yescharacteristic
332C22×Q8—22no—
333C22×Q8—21no—
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8C2×Dic8312yes—
339C22×Q8C22⋊D8314yes—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—22no—
343C23×C4—22no—
344C23×C4—22no—
345C23×C4—22no—
346C23×C4—22no—
347C23×C4—22no—
348C23×C4—1662no—
349C23×C4C2×Dic8312yes—
350C23×C4C22⋊D8311yescharacteristic
351C23×C4C22⋊D8312yes—
352C23×C4C22×D8311yescharacteristic
353C22×C8—834no—
354C22×C8—1662no—
355C22×C8—1662no—
356C22×C8—1662no—
357C22×C8—1662no—
358C22×C8—1662no—
359C22×C8—1662no—
360C22⋊Q8—216no—
361C22⋊Q8—28no—
362C22⋊Q8—28no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—834no—
368C2×C4⋊C4—1662no—
369C2×C4⋊C4—1662no—
370C2×C4⋊C4—1662no—
371C2×C4⋊C4—1662no—
372C2×C4⋊C4—1662no—
373C2×C4⋊C4C2×Dic8312yes—
374C2×C4⋊C4C22⋊D8314yes—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—3322no—
378C2×C22⋊C4—3322no—
379C2×C22⋊C4—1664no—
380#103—16616no—
381C22⋊C8—1668no—
382#95—1664no—
383C24—21no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24C22⋊Dic8312yes—
392C24#1305057712yes—
393C24#1305058012yes—
394C24C2×C22⋊D8311yescharacteristic
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—41no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—41no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8C22⋊Dic8314yescomplemented
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—3322no—
442C22×C4—3322no—
443C22×C4—3322no—
444C22×C4—3322no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—1664no—
448C22×C4C22⋊Dic8312yes—
449C22×C4C22×Dic8311yescharacteristic
450C22×C4D4×D8312yes—
451C22×C4D4×D8312yes—
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C2×C22⋊D8312yes—
454C22×C4C2×C22⋊D8312yes—
455C22×C4#1305058912yes—
456C22×C4#1305058912yes—
457C2×C8—3322no—
458C2×C8—3322no—
459C2×C8—3322no—
460C2×C8—3322no—
461C2×C8—1664no—
462C2×C8—1664no—
463C2×C8—1664no—
464C4⋊C4—216no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—3322no—
468C4⋊C4—3322no—
469C4⋊C4—1664no—
470C4⋊C4—1664no—
471C4⋊C4C22⋊Dic8314yescomplemented
472C22⋊C4—216no—
473C22⋊C4—3328no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#1458171614yescomplemented
497C23C2×C22⋊Dic8311yescharacteristic
498C23#1458176011yescharacteristic
499C23#1458176112yes—
500C23#1458176412yes—
501C23#1458177311yescharacteristic
502C23C2×#1305057711yescharacteristic
503C23C2×#1305058011yescharacteristic
504C23#1458181212yes—
505C23#1458181711yescharacteristic
506C23#1458182211yescharacteristic
507C23#1458182312yes—
508Q8—28no—
509Q8—44no—
510Q8—44no—
511Q8—44no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—3324no—
532C2×C4—3324no—
533C2×C4—3324no—
534C2×C4—6642no—
535C2×C4—6642no—
536C2×C4—6642no—
537C2×C4—6642no—
538C2×C4C2×C22⋊Dic8311yescharacteristic
539C2×C4C2×C22⋊Dic8311yescharacteristic
540C2×C4D4×Dic8312yes—
541C2×C4D4×Dic8312yes—
542C2×C4#1458176012yes—
543C2×C4#1458176011yescharacteristic
544C2×C4#1458176112yes—
545C2×C4#1458176312yes—
546C2×C4#1458176312yes—
547C2×C4#1458176412yes—
548C2×C4#1458176412yes—
549C2×C4#1458176412yes—
550C2×C4#1458177312yes—
551C2×C4#1458177311yescharacteristic
552C2×C4#1458177912yes—
553C2×C4#1458178812yes—
554C8—3324no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22#1912755711yescharacteristic
567C22#1912758111yescharacteristic
568C22#1912758111yescharacteristic
569C22#1912760912yes—
570C22#1912761512yes—
571C22C2×#1458171611yescharacteristic
572C22#1912769711yescharacteristic
573C22#1912770211yescharacteristic
574C22#1912771011yescharacteristic
575C22#1912784012yes—
576C22#1912784612yes—
577C22#1912785111yescharacteristic
578C22#1912785611yescharacteristic
579C22#1912785712yes—
580C22#1912786112yes—
581C22#1912786312yes—
582C22#1912793212yes—
583C22#1912794012yes—
584C22#1912794912yes—
585C22#1912795712yes—
586C22#1912802911yescharacteristic
587C22#1912803411yescharacteristic
588C22#1912803912yes—
589C4—44no—
590C4—42no—
591C4—44no—
592C4—6644no—
593C4#1912758112yes—
594C4#1912760412yes—
595C4#1912760912yes—
596C4#1912761512yes—
597C2—44no—
598C2#2490570411yescharacteristic
599C2#2490739811yescharacteristic
600C2#2490740311yescharacteristic
601C2#2490777111yescharacteristic
602C2#2490777611yescharacteristic
603C2#2490788711yescharacteristic
604C2#2490789711yescharacteristic
605C2#2490846412yes—
606C2#2490847112yes—
607C2#2490865612yes—
608C2#2490866312yes—
subgroupG/H
centerC24C2×C22⋊D83
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC22×C22⋊Q8×C83C2
socleC24×C83C2×D4
prime radicalC25×C83D4
cosocleC23×C4×C83C23
hypercenterC22×C22⋊Q8D83
hypocenterC83#10970
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14581716C2semidirect
4. C2×#19127463C2semidirect
22. C2×C22⋊Q8×C83C4semidirect
124. C2×Q8×C83C22⋊C4semidirect
167. C4⋊C4×C83C22⋊C4semidirect
191. C23×C83#103semidirect
310. C83#10970semidirect
317. C2×C22⋊Q8Dic83semidirect
410. C2×Q8C22⋊Dic83semidirect
471. C4⋊C4C22⋊Dic83semidirect
496. C23#14581716semidirect

Maximal subgroups

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

subgroupindex
C22×#145817162
C22×C22⋊(C83⋊C8)2
C2×#191275612
C2×#191274632
C22×C22⋊Q8×C832
#1097083