Finite Group Databasebeta

← #35413652

Subgroups — #35413652

Order 21248.

1442 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128213C211yesmaximal, characteristic
2C2×#19127762C211yesmaximal, characteristic, complemented
3#24908261C212yesmaximal, complemented
4#24908260C212yesmaximal, complemented
5#24908257C212yesmaximal, complemented
6#24908256C212yesmaximal, complemented
7#24907374C212yesmaximal, complemented
8#24907372C212yesmaximal, complemented
9C2×#433×C83C211yesmaximal, characteristic, complemented
10C22×#13050590C2211yescharacteristic
11C23×C4×D83C2211yescharacteristic
12#19128213—24no—
13#19128213—24no—
14C2×#14581776C412yes—
15C2×#14581772C412yes—
16C2×#14581767C412yes—
17C2×#14581762C412yes—
18C2×C4×D332C2212yes—
19C2×C4×D332C2212yes—
20#19127762—24no—
21C2×C4⋊(C83⋊C8)C2212yescomplemented
22C2×C22⋊(C83⋊C8)C2212yes—
23C2×(C2×D83)⋊C8C2212yes—
24C2×(C2×D83)⋊C8C2212yes—
25C2×(C2×D83)⋊C8C2212yes—
26C2×(C2×D83)⋊C8C2212yes—
27C2×(C2×D83)⋊C8C2212yescomplemented
28C2×(C2×D83)⋊C8C2212yescomplemented
29C2×(C2×D83)⋊C8C2212yescomplemented
30C2×(C2×D83)⋊C8C2212yescomplemented
31C2×#14581703C2211yescharacteristic, complemented
32C2×C4×C83⋊C8C2211yescharacteristic, complemented
33C2×#117×C83C2211yescharacteristic
34#433×C83—24no—
35C2×C4⋊C8×C83C2212yescomplemented
36C2×C22⋊C8×C83C2212yes—
37C2×#97×C83C2211yescharacteristic, complemented
38C2×C4×C8×C83C2211yescharacteristic, complemented
39C24×D83C2×C411yescharacteristic
40C2×#13050590—22no—
41C2×#13050590—22no—
42C2×#13050590—42no—
43C2×#13050590—42no—
44C22×Dic166C2×C411yescharacteristic
45C22×D332C2×C412yes—
46C22×D332C2×C412yes—
47C22×D332C2×C411yescharacteristic
48C22×C22⋊D83C2×C412yes—
49C23×Dic83C2×C411yescharacteristic
50C22×C4×D83—22no—
51C22×C4×D83—22no—
52C22×C4×D83—42no—
53C22×C4×D83—42no—
54C22×C4×D83C2×C412yes—
55C22×C4×D83C2312yes—
56C22×C4×D83C2312yes—
57C22×C83⋊C8C2312yes—
58C22×C83⋊C8C2312yes—
59#14581776—28no—
60#14581772—28no—
61#14581767—28no—
62#14581762—28no—
63C4×D332—24no—
64C4×D332—24no—
65C4×D332—24no—
66C4×D332—24no—
67C2×C4⋊Dic83C2×C412yes—
68C2×C4⋊Dic83C2×C412yes—
69C2×(C2×D83)⋊C4C2×C412yes—
70C2×(C2×D83)⋊C4C2×C412yes—
71C2×(C2×D83)⋊C4C2×C412yes—
72C2×(C2×D83)⋊C4C2×C412yes—
73C4⋊(C83⋊C8)—24no—
74C22⋊(C83⋊C8)—24no—
75(C2×D83)⋊C8—42no—
76(C2×D83)⋊C8—42no—
77(C2×D83)⋊C8—42no—
78(C2×D83)⋊C8—42no—
79(C2×D83)⋊C8—42no—
80(C2×D83)⋊C8—42no—
81(C2×D83)⋊C8—42no—
82(C2×D83)⋊C8—42no—
83#14581703—22no—
84C4×C83⋊C8—22no—
85C23×C4×C83C2311yescharacteristic
86C22×C8×C83C2312yes—
87C22×C8×C83C2312yes—
88#117×C83D414yes—
89#117×C83D414yes—
90C2×C4⋊C4×C83C2×C412yes—
91C2×C22⋊C4×C83C2×C412yes—
92C2×C42×C83C2311yescharacteristic
93C2×C42×C83C2311yescharacteristic
94C4⋊C8×C83—24no—
95C22⋊C8×C83—24no—
96#97×C83—22no—
97C4×C8×C83—22no—
98C23×D83—22no—
99C23×D83—24no—
100C23×D83—41no—
101C23×D83—41no—
102C23×D83—42no—
103C23×D83C22×C412yes—
104C23×D83C22×C412yes—
105#13050590—48no—
106#13050590—48no—
107C2×Dic166—22no—
108C2×Dic166—41no—
109C2×Dic166—41no—
110C2×D332—24no—
111C2×D332—24no—
112C2×D332—24no—
113C2×D332—24no—
114C2×D332—22no—
115C2×D332—42no—
116C2×D332—42no—
117C2×D332—41no—
118C2×D332—41no—
119C2×C22⋊D83—24no—
120C2×C22⋊D83—42no—
121C2×C22⋊D83—42no—
122C22×Dic83—22no—
123C22×Dic83—41no—
124C22×Dic83—41no—
125C22×Dic83C22×C412yes—
126C22×Dic83C22×C412yes—
127C2×C4×D83—24no—
128C2×C4×D83—22no—
129C2×C4×D83—22no—
130C2×C4×D83—22no—
131C2×C4×D83—22no—
132C2×C4×D83—22no—
133C2×C4×D83—22no—
134C2×C4×D83—22no—
135C2×C4×D83—22no—
136C2×C4×D83—22no—
137C2×C4×D83—22no—
138C2×C4×D83—22no—
139C2×C4×D83—22no—
140C2×C4×D83—42no—
141C2×C4×D83—42no—
142C2×C4×D83—48no—
143C2×C4×D83—48no—
144C2×C83⋊C8—22no—
145C2×C83⋊C8—22no—
146C2×C83⋊C8—22no—
147C2×C83⋊C8—22no—
148C2×C83⋊C8—22no—
149C2×C83⋊C8—22no—
150C4⋊Dic83—24no—
151C4⋊Dic83—24no—
152(C2×D83)⋊C4—24no—
153(C2×D83)⋊C4—24no—
154(C2×D83)⋊C4—24no—
155(C2×D83)⋊C4—24no—
156C24×C83C22×C411yescharacteristic
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83C22×C411yescharacteristic
160C22×C4×C83C22×C412yes—
161C22×C4×C83C22×C412yes—
162C22×C4×C83C2×D412yes—
163C22×C4×C83C2×D412yes—
164C22×C4×C83C2411yescharacteristic
165C2×C8×C83—22no—
166C2×C8×C83—22no—
167C2×C8×C83—22no—
168C2×C8×C83—22no—
169C2×C8×C83—22no—
170C2×C8×C83—22no—
171C4⋊C4×C83C22⋊C418yes—
172C22⋊C4×C83C22⋊C418yes—
173C42×C83C2×D412yes—
174C42×C83C2×D412yes—
175C42×C83C2×D412yes—
176C42×C83C2×D412yes—
177C23×C83—21no—
178C23×C83—21no—
179C23×C83—22no—
180C23×C83—22no—
181C23×C83C2×C22⋊C412yes—
182C23×C83C23×C411yescharacteristic
183C22×D83—22no—
184C22×D83—22no—
185C22×D83—22no—
186C22×D83—22no—
187C22×D83—22no—
188C22×D83—22no—
189C22×D83—22no—
190C22×D83—22no—
191C22×D83—22no—
192C22×D83—22no—
193C22×D83—22no—
194C22×D83—22no—
195C22×D83—22no—
196C22×D83—44no—
197C22×D83—48no—
198C22×D83—48no—
199C22×D83—44no—
200C22×D83—44no—
201C22×D83#13114yes—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—22no—
207C2×C4×C83—22no—
208C2×C4×C83—24no—
209C2×C4×C83—24no—
210C2×C4×C83C2×C22⋊C414yes—
211C2×C4×C83C2×C22⋊C414yes—
212C2×C4×C83C2×C22⋊C412yes—
213C2×C4×C83C22×D411yescharacteristic
214C2×C4×C83C22×D411yescharacteristic
215C2×C4×C83H(5,𝔽2)12yes—
216C2×C4×C83H(5,𝔽2)12yes—
217C22⋊D83—48no—
218C2×Dic83—22no—
219C2×Dic83—22no—
220C2×Dic83—22no—
221C2×Dic83—22no—
222C2×Dic83—22no—
223C2×Dic83—22no—
224C2×Dic83—44no—
225D332—48no—
226D332—48no—
227D332—44no—
228C4×D83—48no—
229C4×D83—42no—
230C4×D83—42no—
231C4×D83—42no—
232C4×D83—42no—
233C4×D83—42no—
234C4×D83—42no—
235C4×D83—42no—
236C4×D83—42no—
237Dic166—44no—
238C8×C83—42no—
239C8×C83—42no—
240C83⋊C8—42no—
241C83⋊C8—42no—
242C22×C83—22no—
243C22×C83—24no—
244C22×C83—42no—
245C22×C83—41no—
246C22×C83—41no—
247C22×C83C22×C22⋊C411yescharacteristic
248C22×C83#51811yescharacteristic
249C22×C83#51811yescharacteristic
250C22×C83C2×#13112yes—
251C22×C83#56812yes—
252C2×D83—24no—
253C2×D83—24no—
254C2×D83—24no—
255C2×D83—416no—
256C2×D83—42no—
257C2×D83—42no—
258C2×D83—42no—
259C2×D83—42no—
260C2×D83—42no—
261C2×D83—42no—
262C2×D83—42no—
263C2×D83—42no—
264C2×D83—42no—
265C2×D83—42no—
266C2×D83—42no—
267C4×C83—22no—
268C4×C83—24no—
269C4×C83—24no—
270C4×C83#53512yes—
271C4×C83#53512yes—
272Dic83—42no—
273Dic83—42no—
274#21792—831nomaximal, Sylow
275C2×C83—42no—
276C2×C83#230911yescharacteristic
277C2×C83#288012yes—
278C2×C83#297612yes—
279C2×C83#297711yescharacteristic
280C2×C83#297811yescharacteristic
281D83—44no—
282D83—82no—
283C2×#525—831no—
284C2×#433—831no—
285C2×#433D8311yescharacteristic, complemented
286#1955—832no—
287#1955—832no—
288#1954—832no—
289#1954—832no—
290#1780—832no—
291#1779—832no—
292C83#2179211yescharacteristic, complemented, Sylow
293C22×#41—831no—
294C24×C4—831no—
295#525—1664no—
296#525—1664no—
297C2×#123—832no—
298C2×C22⋊Q8—832no—
299C2×#121—832no—
300C2×#120—832no—
301C2×C4×D4—832no—
302C2×C4×D4—832no—
303C2×#117C2×D8311yescharacteristic
304#433—24no—
305#433—1664no—
306C2×C4⋊C8—832no—
307C2×C4⋊C8C2×D8312yescomplemented
308C2×C22⋊C8—832no—
309C2×C22⋊C8—832no—
310C2×C22⋊C8—832no—
311C2×C22⋊C8—832no—
312C2×C22⋊C8—832no—
313C2×C22⋊C8—832no—
314C2×C22⋊C8—832no—
315C2×C22⋊C8—832no—
316C2×C22⋊C8—832no—
317C2×C22⋊C8C2×D8312yes—
318C2×#97—831no—
319C2×#97C2×D8311yescharacteristic, complemented
320C2×C4×C8—831no—
321C2×C4×C8C2×D8311yescharacteristic, complemented
322C25—831no—
323C2×#41—3322no—
324C2×#41—3322no—
325C2×#41—1662no—
326C2×#41—1662no—
327C22×Q8—831no—
328C22×D4—832no—
329C22×D4—831no—
330C22×D4—832no—
331C22×D4—832no—
332C23×C4—3322no—
333C23×C4—3322no—
334C23×C4—831no—
335C23×C4—832no—
336C23×C4—832no—
337C23×C4—832no—
338C23×C4—1662no—
339C23×C4—1662no—
340C23×C4C22×D8311yescharacteristic
341C22×C8—832no—
342C22×C8—832no—
343C22×C8C22×D8312yes—
344C22×C8C22×D8312yes—
345#123—1668no—
346C22⋊Q8—1668no—
347#121—1668no—
348#120—1668no—
349C4×D4—1664no—
350C4×D4—1664no—
351C4×D4—1664no—
352C4×D4—1664no—
353#117D33214yes—
354#117C22⋊D8314yes—
355C2×C4⋊C4—832no—
356C2×C4⋊C4—832no—
357C2×C4⋊C4C4×D8312yes—
358C2×C22⋊C4—832no—
359C2×C22⋊C4—832no—
360C2×C22⋊C4—832no—
361C2×C22⋊C4—832no—
362C2×C22⋊C4C4×D8312yes—
363C2×C42C22×D8311yescharacteristic
364C2×C42C22×D8311yescharacteristic
365C4⋊C8—24no—
366C4⋊C8—1664no—
367C22⋊C8—24no—
368C22⋊C8—3322no—
369C22⋊C8—3322no—
370C22⋊C8—3322no—
371C22⋊C8—3322no—
372C22⋊C8—3322no—
373C22⋊C8—3322no—
374C22⋊C8—3322no—
375C22⋊C8—3322no—
376C22⋊C8—1664no—
377#97—22no—
378#97—1662no—
379C4×C8—22no—
380C4×C8—1662no—
381C24—3321no—
382C24—3321no—
383C24—3322no—
384C24—832no—
385C24—832no—
386C24—1664no—
387C24—1662no—
388C24C2×C4×D8311yescharacteristic
389#41—3328no—
390#41—3328no—
391C2×Q8—3321no—
392C2×Q8—3321no—
393C2×Q8—1662no—
394C2×D4—3321no—
395C2×D4—3321no—
396C2×D4—3322no—
397C2×D4—3322no—
398C2×D4—3322no—
399C2×D4—3322no—
400C2×D4—1664no—
401C2×D4—1664no—
402C2×D4—1664no—
403C2×D4—1664no—
404C2×D4—1664no—
405C2×D4—1662no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—3328no—
409C22×C4—3328no—
410C22×C4—3322no—
411C22×C4—3322no—
412C22×C4—3321no—
413C22×C4—3321no—
414C22×C4—832no—
415C22×C4—832no—
416C22×C4—1662no—
417C22×C4—1662no—
418C22×C4—1662no—
419C22×C4—1662no—
420C22×C4—1662no—
421C22×C4—1662no—
422C22×C4—1662no—
423C22×C4—1662no—
424C22×C4—1662no—
425C22×C4—1662no—
426C22×C4—1662no—
427C22×C4—1662no—
428C22×C4—1662no—
429C22×C4—1664no—
430C22×C4C2×C4×D8312yes—
431C22×C4C2×C4×D8312yes—
432C22×C4C2×C4×D8311yescharacteristic
433C22×C4C2×C22⋊D8312yes—
434C22×C4C2×D33212yes—
435C22×C4C23×D8311yescharacteristic
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—1662no—
443C2×C8—1662no—
444C2×C8—1662no—
445C2×C8—1662no—
446C2×C8—1662no—
447C2×C8—1662no—
448C4⋊C4—1664no—
449C4⋊C4—1664no—
450C4⋊C4(C2×D83)⋊C418yes—
451C22⋊C4—1664no—
452C22⋊C4—1664no—
453C22⋊C4—1664no—
454C22⋊C4—1664no—
455C22⋊C4(C2×D83)⋊C418yes—
456C42C2×C22⋊D8312yes—
457C42C2×C22⋊D8312yes—
458C42C2×D33212yes—
459C42C2×D33212yes—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—22no—
464C23—3324no—
465C23—3328no—
466C23—3324no—
467C23—3328no—
468C23—3324no—
469C23—834no—
470C23—1662no—
471C23—1662no—
472C23—1662no—
473C23—1662no—
474C23—1662no—
475C23—1662no—
476C23—1662no—
477C23—1662no—
478C23—1662no—
479C23—1662no—
480C23—1662no—
481C23—1662no—
482C23—1662no—
483C23C2×(C2×D83)⋊C412yes—
484C23C22×C4×D8311yescharacteristic
485Q8—3324no—
486D4—3324no—
487D4—3328no—
488D4—3328no—
489D4—3328no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—3328no—
499C2×C4—3324no—
500C2×C4—3322no—
501C2×C4—3322no—
502C2×C4—3322no—
503C2×C4—3322no—
504C2×C4—3322no—
505C2×C4—3322no—
506C2×C4—3322no—
507C2×C4—3322no—
508C2×C4—1662no—
509C2×C4—1662no—
510C2×C4—1662no—
511C2×C4—1662no—
512C2×C4—1662no—
513C2×C4—1662no—
514C2×C4C2×(C2×D83)⋊C414yes—
515C2×C4C2×(C2×D83)⋊C414yes—
516C2×C4C2×(C2×D83)⋊C412yes—
517C2×C4C22×C22⋊D8311yescharacteristic
518C2×C4C22×D33211yescharacteristic
519C2×C4#1458183812yes—
520C2×C4#1458183812yes—
521C8—42no—
522C8—42no—
523C8—3322no—
524C8—3322no—
525C22—24no—
526C22—22no—
527C22—42no—
528C22—41no—
529C22—41no—
530C22—33216no—
531C22—3322no—
532C22—3322no—
533C22—3322no—
534C22—3322no—
535C22—3322no—
536C22—3322no—
537C22—3322no—
538C22—3322no—
539C22—3322no—
540C22—3322no—
541C22—3322no—
542C22—1664no—
543C22—1664no—
544C22—1664no—
545C22C22×(C2×D83)⋊C411yescharacteristic
546C22#1912817511yescharacteristic
547C22#1912817511yescharacteristic
548C22#131×D8312yes—
549C22#1912846612yes—
550C4—24no—
551C4—24no—
552C4—22no—
553C4—3322no—
554C4—3322no—
555C4#1912826512yes—
556C4#1912827012yes—
557C2—42no—
558C2—3324no—
559C2—6642no—
560C2#2491004311yescharacteristic
561C2#2491591012yes—
562C2#2491650912yes—
563C2#2491651811yescharacteristic
564C2#2491652411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D83
commutator, socleC23×C83C23×C4
FittingC2×#433×C83C2
prime radicalC2×#19128213C2
cosocleC22×C4×C83C24
hypercenterC2×#433D83
hypocenterC83#21792
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
2. C2×#19127762C2semidirect
3. #24908261C2semidirect
4. #24908260C2semidirect
5. #24908257C2semidirect
6. #24908256C2semidirect
7. #24907374C2semidirect
8. #24907372C2semidirect
9. C2×#433×C83C2semidirect
21. C2×C4⋊(C83⋊C8)C22semidirect
27. C2×(C2×D83)⋊C8C22semidirect
28. C2×(C2×D83)⋊C8C22semidirect
29. C2×(C2×D83)⋊C8C22semidirect
30. C2×(C2×D83)⋊C8C22semidirect
31. C2×#14581703C22semidirect
32. C2×C4×C83⋊C8C22semidirect
35. C2×C4⋊C8×C83C22semidirect
37. C2×#97×C83C22semidirect
38. C2×C4×C8×C83C22semidirect
285. C2×#433D83semidirect
292. C83#21792semidirect
307. C2×C4⋊C8C2×D83semidirect
319. C2×#97C2×D83semidirect
321. C2×C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282132
C2×#191277622
#249082612
#249082602
#249082572
#249082562
#249073742
#249073722
C2×#433×C832
#2179283