Finite Group Databasebeta

← #35415642

Subgroups — #35415642

Order 21248.

1212 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127851C211yesmaximal, characteristic
2C2×#19127603C211yesmaximal, characteristic
3#24908347C212yesmaximal, complemented
4#24908159C212yesmaximal, complemented
5#24908083C212yesmaximal, complemented
6#24907818C212yesmaximal, complemented
7#24907771C211yesmaximal, characteristic, complemented
8#24907650C211yesmaximal, characteristic, complemented
9#24907393C211yesmaximal, characteristic
10#24907225C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
12C23×D332C2211yescharacteristic
13C22×#13050577C2212yes—
14C22×C4⋊Dic83C2211yescharacteristic
15C22×(C2×D83)⋊C4—21no—
16C22×(C2×D83)⋊C4—21no—
17#19127851C418yescomplemented
18C2×#14581725C2212yescomplemented
19C2×#14581716C2212yescomplemented
20C2×#14581712C2212yes—
21C2×#14581712C2212yes—
22C2×#14581712C2212yes—
23C2×C22⋊(C83⋊C8)C2211yescharacteristic
24C2×C22⋊(C83⋊C8)C2211yescharacteristic
25C2×#14581703C2212yescomplemented
26#19127603—21no—
27#19127603—21no—
28#19127603C2212yes—
29#19127603C2212yes—
30#19127463C2212yescomplemented
31#19127463C2212yescomplemented
32#19127456C2212yescomplemented
33C2×C22⋊Q8×C83C2211yescharacteristic
34C2×C4×Q8×C83C2211yescharacteristic, complemented
35C22×C42×C83C2211yescharacteristic
36#391×C83C2211yescharacteristic, complemented
37#389×C83C2211yescharacteristic, complemented
38#387×C83—21no—
39#385×C83—21no—
40#380×C83C2211yescharacteristic
41C4×C4⋊C4×C83—21no—
42C4×C4⋊C4×C83C2211yescharacteristic, complemented
43C4×C22⋊C4×C83—21no—
44C24×D83—21no—
45C22×D332—21no—
46C22×D332—22no—
47C22×D332—42no—
48C22×D332—42no—
49C22×D332C2×C414yes—
50C22×D332C2312yes—
51C23×Dic83—21no—
52C2×#13050577—42no—
53C2×#13050577—42no—
54C2×#13050577C2×C418yescomplemented
55C22×C83⋊C8C2312yes—
56C22×C83⋊C8C2312yes—
57C2×C4⋊Dic83—21no—
58C2×C4⋊Dic83—42no—
59C2×C4⋊Dic83C2312yes—
60C2×(C2×D83)⋊C4—22no—
61C2×(C2×D83)⋊C4—22no—
62C2×(C2×D83)⋊C4—22no—
63C2×(C2×D83)⋊C4—22no—
64C2×(C2×D83)⋊C4—44no—
65C2×(C2×D83)⋊C4—44no—
66#14581725—42no—
67#14581716—42no—
68#14581712—42no—
69#14581712—42no—
70#14581712C2×C418yescomplemented
71C22⋊(C83⋊C8)—22no—
72C22⋊(C83⋊C8)C2×C414yes—
73#14581703—42no—
74C22×Q8×C83C2311yescharacteristic
75C23×C4×C83D411yescharacteristic
76C23×C4×C83D411yescharacteristic
77C23×C4×C83C2311yescharacteristic
78C22⋊Q8×C83—22no—
79C22⋊Q8×C83C2×C414yes—
80C4×Q8×C83—42no—
81C2×C4⋊C4×C83—21no—
82C2×C4⋊C4×C83—21no—
83C2×C4⋊C4×C83—21no—
84C2×C4⋊C4×C83C2311yescharacteristic
85C2×C4⋊C4×C83C2311yescharacteristic
86C2×C4⋊C4×C83C2311yescharacteristic
87C2×C22⋊C4×C83—21no—
88C2×C22⋊C4×C83—21no—
89C2×C42×C83—21no—
90C2×C42×C83—21no—
91C2×C42×C83—42no—
92C2×C42×C83D411yescharacteristic
93C2×C42×C83D411yescharacteristic
94C2×C42×C83C2311yescharacteristic
95C2×C42×C83C2311yescharacteristic
96#95×C83—21no—
97#95×C83—21no—
98#95×C83—21no—
99#95×C83—21no—
100#95×C83—21no—
101C23×D83—24no—
102C23×D83—22no—
103C23×D83—42no—
104C23×D83—42no—
105C23×D83—42no—
106C23×D83—42no—
107C23×D83—42no—
108C23×D83—42no—
109C2×D332—22no—
110C2×D332—22no—
111C2×D332—22no—
112C2×D332—22no—
113C2×D332—24no—
114C2×D332—22no—
115C2×D332—48no—
116C2×D332—42no—
117C2×D332—42no—
118C2×D332—48no—
119C2×D332C22×C414yes—
120C22×Dic83—22no—
121C22×Dic83—42no—
122C22×Dic83—42no—
123C22×Dic83—42no—
124#13050577—48no—
125C2×C83⋊C8—22no—
126C2×C83⋊C8—22no—
127C2×C83⋊C8—22no—
128C2×C83⋊C8—22no—
129C2×C83⋊C8—22no—
130C2×C83⋊C8C22×C414yes—
131C4⋊Dic83—22no—
132C4⋊Dic83—22no—
133C4⋊Dic83—42no—
134(C2×D83)⋊C4—44no—
135(C2×D83)⋊C4—44no—
136(C2×D83)⋊C4—44no—
137(C2×D83)⋊C4—44no—
138C24×C83C2×D411yescharacteristic
139C2×Q8×C83—21no—
140C2×Q8×C83—22no—
141C2×Q8×C83—22no—
142C2×Q8×C83C22×C412yes—
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—21no—
148C22×C4×C83—21no—
149C22×C4×C83—22no—
150C22×C4×C83—21no—
151C22×C4×C83—21no—
152C22×C4×C83—41no—
153C22×C4×C83—41no—
154C22×C4×C83—41no—
155C22×C4×C83C22×C412yes—
156C22×C4×C83C2×D411yescharacteristic
157C22×C4×C83C2×D411yescharacteristic
158C22×C4×C83C2×D411yescharacteristic
159C22×C4×C83C2×D411yescharacteristic
160C22×C4×C83C2×D411yescharacteristic
161C22×C4×C83#4112yes—
162C22×C4×C83C2411yescharacteristic
163C4⋊C4×C83—24no—
164C4⋊C4×C83—21no—
165C4⋊C4×C83—22no—
166C4⋊C4×C83—22no—
167C4⋊C4×C83—42no—
168C4⋊C4×C83—42no—
169C4⋊C4×C83C22×C412yes—
170C22⋊C4×C83—24no—
171C22⋊C4×C83—42no—
172C42×C83—22no—
173C42×C83—22no—
174C42×C83—41no—
175C42×C83—41no—
176C42×C83—42no—
177C42×C83—42no—
178C23×C83—21no—
179C23×C83—21no—
180C23×C83—21no—
181C23×C83—21no—
182C23×C83C4×D412yes—
183C23×C83AGL(1,ℤ/8)12yes—
184C23×C83#13712yes—
185C23×C83C22×D411yescharacteristic
186C22×D83—24no—
187C22×D83—48no—
188C22×D83—48no—
189C22×D83—42no—
190C22×D83—42no—
191C22×D83—42no—
192C22×D83—42no—
193C22×D83—42no—
194C22×D83—42no—
195C22×D83—48no—
196C22×D83—84no—
197C22×D83—84no—
198C22×D83—84no—
199C22×D83—84no—
200Q8×C83—24no—
201Q8×C83—42no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—44no—
220C2×C4×C83—44no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83C4×D412yes—
232C2×C4×C83AGL(1,ℤ/8)12yes—
233C2×C4×C83C23×C411yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83H(5,𝔽2)12yes—
236C2×C4×C83H(5,𝔽2)12yes—
237C2×Dic83—42no—
238C2×Dic83—42no—
239C2×Dic83—42no—
240C2×Dic83—84no—
241D332—24no—
242D332—24no—
243D332—48no—
244D332—42no—
245D332—42no—
246C83⋊C8—24no—
247C83⋊C8—42no—
248C22×C83—22no—
249C22×C83—22no—
250C22×C83—22no—
251C22×C83—22no—
252C22×C83—22no—
253C22×C83—22no—
254C22×C83#44014yes—
255C22×C83C2×C4×D411yescharacteristic
256C22×C83#53411yescharacteristic
257C22×C83#53511yescharacteristic
258C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C83C2×#13711yescharacteristic
261C22×C83#57511yescharacteristic
262C2×D83—44no—
263C2×D83—44no—
264C2×D83—44no—
265C2×D83—816no—
266C2×D83—82no—
267C2×D83—82no—
268C2×D83—82no—
269C2×D83—82no—
270C4×C83—22no—
271C4×C83—22no—
272C4×C83—42no—
273C4×C83—41no—
274C4×C83—41no—
275C4×C83—41no—
276C4×C83—42no—
277C4×C83—81no—
278C4×C83—81no—
279C4×C83#51812yes—
280C4×C83#53812yes—
281Dic83—82no—
282#22049—831nomaximal, Sylow
283C2×C83—24no—
284C2×C83#232111yescharacteristic
285C2×C83C2×#44011yescharacteristic
286C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
287C2×C83#319311yescharacteristic
288C2×C83#319411yescharacteristic
289C2×C83#321511yescharacteristic
290C2×C83#321711yescharacteristic
291D83—84no—
292C2×#450—831no—
293C4×C22⋊Q8D8311yescharacteristic, complemented
294C2×#390—831no—
295#1973—832no—
296#1936—832no—
297#1925—832no—
298#1880—832no—
299#1872—831no—
300#1852—831no—
301#1786—831no—
302#1749—831no—
303C83#2204911yescharacteristic, complemented, Sylow
304C23×D4—831no—
305C22×SD16—832no—
306C2×C22⋊Q8C2×D8311yescharacteristic
307C2×C4×Q8C2×D8311yescharacteristic, complemented
308C22×C4⋊C4—831no—
309C22×C22⋊C4—1661no—
310C22×C22⋊C4—1661no—
311C22×C42C2×D8311yescharacteristic
312#450—838no—
313C2×#107—832no—
314C2×#103—832no—
315C2×#102—832no—
316C2×#102—832no—
317C2×#102—832no—
318C2×C22⋊C8—831no—
319C2×C22⋊C8—831no—
320C2×#97—832no—
321#391C2×D8311yescharacteristic, complemented
322#390—832no—
323#390—832no—
324#390—1661no—
325#390—1661no—
326#389C2×D8311yescharacteristic, complemented
327#387—21no—
328#385—21no—
329#380C2×D8311yescharacteristic
330C4×C4⋊C4—21no—
331C4×C4⋊C4C2×D8311yescharacteristic, complemented
332C4×C22⋊C4—21no—
333#340—832no—
334#340—832no—
335#336—832no—
336C25—1661no—
337C22×Q8C22×D8311yescharacteristic
338C22×D4—3322no—
339C22×D4—3322no—
340C22×D4—834no—
341C22×D4—832no—
342C22×D4—1661no—
343C22×D4—1662no—
344C23×C4—1661no—
345C23×C4C22⋊D8311yescharacteristic
346C23×C4C22⋊D8311yescharacteristic
347C23×C4C22×D8311yescharacteristic
348C2×SD16—3322no—
349C2×SD16—3322no—
350C2×SD16—838no—
351C22×C8—832no—
352C22×C8—832no—
353C22⋊Q8—22no—
354C22⋊Q8C4×D8314yes—
355C4×Q8—42no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—3322no—
360C2×C4⋊C4—832no—
361C2×C4⋊C4—1661no—
362C2×C4⋊C4C22×D8311yescharacteristic
363C2×C4⋊C4C22×D8311yescharacteristic
364C2×C4⋊C4C22×D8311yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—3324no—
368C2×C22⋊C4—3324no—
369C2×C22⋊C4—1662no—
370C2×C22⋊C4—1662no—
371C2×C22⋊C4—1662no—
372C2×C22⋊C4—1662no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—42no—
376C2×C42C22⋊D8311yescharacteristic
377C2×C42C22⋊D8311yescharacteristic
378C2×C42C22×D8311yescharacteristic
379C2×C42C22×D8311yescharacteristic
380#107—3322no—
381#103—3322no—
382#102—3322no—
383#102—3322no—
384#102—838no—
385C22⋊C8—834no—
386C22⋊C8—1662no—
387#97—3322no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393C24—3322no—
394C24—3322no—
395C24—3322no—
396C24—3322no—
397C24—3322no—
398C24—3322no—
399C24—1662no—
400C24—1664no—
401C24C2×C22⋊D8311yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C2×C4×D8312yes—
406C2×D4—3322no—
407C2×D4—3322no—
408C2×D4—3328no—
409C2×D4—3328no—
410C2×D4—834no—
411C2×D4—1662no—
412C2×D4—1662no—
413C2×D4—1662no—
414C2×D4—1662no—
415C2×D4—1662no—
416C2×D4—1664no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—3322no—
430C22×C4—3322no—
431C22×C4—3322no—
432C22×C4—1662no—
433C22×C4C2×C4×D8312yes—
434C22×C4C2×C22⋊D8311yescharacteristic
435C22×C4C2×C22⋊D8311yescharacteristic
436C22×C4C2×C22⋊D8311yescharacteristic
437C22×C4C2×C22⋊D8311yescharacteristic
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4#1305059012yes—
440C22×C4C23×D8311yescharacteristic
441SD16—3328no—
442C2×C8—834no—
443C2×C8—1662no—
444C2×C8—1662no—
445C2×C8—1662no—
446C2×C8—1662no—
447C2×C8—1662no—
448C4⋊C4—21no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—3322no—
455C4⋊C4—1662no—
456C4⋊C4—1662no—
457C4⋊C4C2×C4×D8312yes—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—3324no—
461C22⋊C4—3324no—
462C22⋊C4—3324no—
463C22⋊C4—3324no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—42no—
469C42—42no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—3328no—
475C23—3328no—
476C23—3322no—
477C23—3322no—
478C23—3322no—
479C23—3322no—
480C23—3322no—
481C23—3322no—
482C23—3328no—
483C23—6644no—
484C23—6644no—
485C23—6644no—
486C23—6644no—
487C23—1664no—
488C23C4×C22⋊D8312yes—
489C23#1458181712yes—
490C23#1458182312yes—
491C23C22×C22⋊D8311yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—3328no—
495D4—3322no—
496D4—3322no—
497D4—1664no—
498D4—1664no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—3322no—
529C2×C4—3322no—
530C2×C4—3322no—
531C2×C4—6644no—
532C2×C4C4×C22⋊D8312yes—
533C2×C4#1458181712yes—
534C2×C4C22×C4×D8311yescharacteristic
535C2×C4C2×#1305059011yescharacteristic
536C2×C4#1458183712yes—
537C2×C4#1458183812yes—
538C8—3322no—
539C8—1664no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—3324no—
547C22—3324no—
548C22—3324no—
549C22—66416no—
550C22—6642no—
551C22—6642no—
552C22—6642no—
553C22—6642no—
554C22#1912779914yes—
555C22C2×C4×C22⋊D8311yescharacteristic
556C22#1912826011yescharacteristic
557C22#1912826511yescharacteristic
558C22C2×#1458181711yescharacteristic
559C22C2×#1458181711yescharacteristic
560C22C2×#1458182311yescharacteristic
561C22#1912849911yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—6642no—
572C4#1912817312yes—
573C4#1912828412yes—
574C2—24no—
575C2—6644no—
576C2#2491011711yescharacteristic
577C2C2×#1912779911yescharacteristic
578C2C4×#1458181711yescharacteristic
579C2#2491826511yescharacteristic
580C2#2491827311yescharacteristic
581C2#2491846211yescharacteristic
582C2#2491847811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC4×C22⋊Q8×C83C2
socleC23×C83C22×D4
prime radicalC23×D332C22
cosocleC22×C4×C83C24
hypercenterC4×C22⋊Q8D83
hypocenterC83#22049
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #24908347C2semidirect
4. #24908159C2semidirect
5. #24908083C2semidirect
6. #24907818C2semidirect
7. #24907771C2semidirect
8. #24907650C2semidirect
10. #24907225C2semidirect
11. C4×C22⋊Q8×C83C2semidirect
17. #19127851C4semidirect
18. C2×#14581725C22semidirect
19. C2×#14581716C22semidirect
25. C2×#14581703C22semidirect
30. #19127463C22semidirect
31. #19127463C22semidirect
32. #19127456C22semidirect
34. C2×C4×Q8×C83C22semidirect
36. #391×C83C22semidirect
37. #389×C83C22semidirect
42. C4×C4⋊C4×C83C22semidirect
54. C2×#13050577C2×C4semidirect
70. #14581712C2×C4semidirect
293. C4×C22⋊Q8D83semidirect
303. C83#22049semidirect
307. C2×C4×Q8C2×D83semidirect
321. #391C2×D83semidirect
326. #389C2×D83semidirect
331. C4×C4⋊C4C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191278512
C2×#191276032
#249083472
#249081592
#249080832
#249078182
#249077712
#249076502
#249073932
#249072252
C4×C22⋊Q8×C832
#2204983