Finite Group Databasebeta

← #35408923

Subgroups — #35408923

Order 21248.

1250 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128254C211yesmaximal, characteristic
2C2×#19127724C211yesmaximal, characteristic
3C2×C4⋊C4×Dic83C211yesmaximal, characteristic
4#24908419C212yesmaximal, complemented
5#24908417C212yesmaximal, complemented
6#24908413C212yesmaximal, complemented
7#24908409C212yesmaximal, complemented
8#24907184C211yesmaximal, characteristic
9#24907183C211yesmaximal, characteristic
10#24907182C211yesmaximal, characteristic
11C83×#1736C211yesmaximal, characteristic
12C22×#13050574C2211yescharacteristic
13#19128254—24no—
14#19128254C418yescomplemented
15C2×#14581791C2211yescharacteristic
16C2×#14581791C2211yescharacteristic
17C2×#14581783C2211yescharacteristic
18C2×#14581783C2211yescharacteristic
19C2×#14581769—21no—
20C2×#14581769—21no—
21C2×C4×Dic166—21no—
22C2×C4×Dic166—21no—
23C2×#14581751C2211yescharacteristic
24C22×C4⋊Dic83C2211yescharacteristic
25C22×C4×Dic83—21no—
26C22×C4×Dic83C2211yescharacteristic
27#19127724—24no—
28C2×Dic83⋊C8C2212yes—
29C2×Dic83⋊C8C2212yes—
30C4⋊C4×Dic83—21no—
31C4⋊C4×Dic83—21no—
32C4⋊C4×Dic83C2212yes—
33C4⋊C4×Dic83C2212yes—
34C2×C83⋊#95—21no—
35#19127464C2212yes—
36#19127464C2212yes—
37#19127463C2212yes—
38#19127463C2212yes—
39#19127463C2212yes—
40#19127463C2212yes—
41C22×M16×C83C2211yescharacteristic
42C2×#117×C83C2211yescharacteristic
43C22×C4⋊C4×C83C2211yescharacteristic
44#340×C83C2212yes—
45#340×C83C2212yes—
46C22×Dic166—21no—
47C22×Dic166—21no—
48C23×Dic83—21no—
49C23×Dic83D411yescharacteristic
50C23×Dic83D411yescharacteristic
51C2×#13050574—22no—
52C2×#13050574—22no—
53C2×#13050574—22no—
54C22×C83⋊C8C2312yes—
55#14581791—22no—
56#14581791—22no—
57#14581791C2×C414yescomplemented
58#14581791C2×C414yescomplemented
59#14581783—22no—
60#14581783—22no—
61#14581783C2×C414yescomplemented
62#14581783C2×C414yescomplemented
63#14581769—28no—
64#14581769—28no—
65C4×Dic166—28no—
66C4×Dic166—28no—
67#14581751—22no—
68#14581751C2×C414yes—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—22no—
73C2×C4⋊Dic83Q814yescomplemented
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2312yes—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×C22⋊Dic83—21no—
82C2×C4×Dic83—22no—
83C2×C4×Dic83—22no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—44no—
86C2×C4×Dic83C2×C414yes—
87C2×C4×Dic83D411yescharacteristic
88C2×C4×Dic83D411yescharacteristic
89C2×C4×Dic83D414yes—
90C2×C4×Dic83C2311yescharacteristic
91C2×C4×Dic83C2311yescharacteristic
92Dic83⋊C8—24no—
93Dic83⋊C8—24no—
94C83⋊#95—22no—
95C83⋊#95—22no—
96C23×C4×C83—21no—
97C23×C4×C83C2311yescharacteristic
98C2×M16×C83—22no—
99C2×M16×C83—22no—
100C2×M16×C83—22no—
101C22×C8×C83C2312yes—
102#117×C83—22no—
103#117×C83C2×C414yes—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—22no—
106C2×C4⋊C4×C83Q814yescomplemented
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2312yes—
110C2×C22⋊C4×C83—21no—
111C2×C42×C83—21no—
112C2×Dic166—24no—
113C2×Dic166—24no—
114C2×Dic166—42no—
115C2×Dic166—42no—
116C2×Dic166—42no—
117C2×Dic166—42no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—24no—
121C22×Dic83—22no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—41no—
125C22×Dic83—41no—
126C22×Dic83—42no—
127C22×Dic83—42no—
128C22×Dic83C22⋊C414yes—
129C22×Dic83C22⋊C414yes—
130C22×Dic83C2×D411yescharacteristic
131C22×Dic83C2×D411yescharacteristic
132C22×Dic83C2×D411yescharacteristic
133C22×Dic83C2×D411yescharacteristic
134#13050574—44no—
135C2×C83⋊C8—22no—
136C2×C83⋊C8—22no—
137C2×C83⋊C8—22no—
138C4⋊Dic83—24no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—22no—
142C4⋊Dic83—24no—
143C4⋊Dic83—24no—
144C4⋊Dic83C22×C412yes—
145C4⋊Dic83C22×C412yes—
146C4⋊Dic83C2×Q814yes—
147Dic83⋊C4—24no—
148Dic83⋊C4—24no—
149Dic83⋊C4—24no—
150Dic83⋊C4—24no—
151C22⋊Dic83—24no—
152C4×Dic83—24no—
153C4×Dic83—44no—
154C4×Dic83—44no—
155C4×Dic83C22⋊C414yes—
156C4×Dic83C22⋊C414yes—
157C4×Dic83C22×C412yes—
158C4×Dic83C22×C412yes—
159C4×Dic83C2×D412yes—
160C4×Dic83C2×D412yes—
161C24×C83C2×D411yescharacteristic
162C22×C4×C83—24no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—22no—
166C22×C4×C83—42no—
167C22×C4×C83—42no—
168C22×C4×C83C22×C412yes—
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D412yes—
171C22×C4×C83C2×Q812yes—
172C22×C4×C83C2×Q812yes—
173C22×C4×C83#4112yes—
174C22×C4×C83#4112yes—
175C22×C4×C83C2411yescharacteristic
176M16×C83—44no—
177C2×C8×C83—22no—
178C2×C8×C83—22no—
179C2×C8×C83—22no—
180C4⋊C4×C83—24no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—22no—
184C4⋊C4×C83C22×C412yes—
185C4⋊C4×C83C22×C412yes—
186C4⋊C4×C83C2×Q814yes—
187C22⋊C4×C83—24no—
188C42×C83—24no—
189C23×C83—21no—
190C23×C83—21no—
191C23×C83—21no—
192C23×C83—21no—
193C23×C83C2×C22⋊C412yes—
194C23×C83C22⋊Q812yes—
195C23×C83C22⋊Q812yes—
196C23×C83C22×D411yescharacteristic
197C2×C4×C83—22no—
198C2×C4×C83—22no—
199C2×C4×C83—24no—
200C2×C4×C83—48no—
201C2×C4×C83—42no—
202C2×C4×C83—41no—
203C2×C4×C83—41no—
204C2×C4×C83—42no—
205C2×C4×C83—41no—
206C2×C4×C83—41no—
207C2×C4×C83C2×C22⋊C412yes—
208C2×C4×C83C4×Q814yes—
209C2×C4×C83C4×Q814yes—
210C2×C4×C83C22⋊Q812yes—
211C2×C4×C83C22⋊Q812yes—
212C2×C4×C83#12414yes—
213C2×C4×C83C4⋊Q814yes—
214C2×C4×C83C23×C411yescharacteristic
215C2×C4×C83C22×D411yescharacteristic
216C2×C4×C83C22×Q811yescharacteristic
217C2×C4×C83C22×Q811yescharacteristic
218C2×C4×C83C2×#4111yescharacteristic
219C2×C4×C83C2×#4111yescharacteristic
220C2×Dic83—21no—
221C2×Dic83—21no—
222C2×Dic83—21no—
223C2×Dic83—21no—
224C2×Dic83—21no—
225C2×Dic83—21no—
226C2×Dic83—28no—
227C2×Dic83—22no—
228C2×Dic83—22no—
229C2×Dic83—24no—
230C2×Dic83—44no—
231C2×Dic83—48no—
232C2×Dic83—42no—
233C2×Dic83—41no—
234C2×Dic83—41no—
235C2×Dic83—42no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83C2×C22⋊C412yes—
239C2×Dic83C2×C22⋊C412yes—
240C2×Dic83C2×C22⋊C412yes—
241C2×Dic83C2×C22⋊C412yes—
242C2×Dic83AGL(1,ℤ/8)12yes—
243C2×Dic83#13712yes—
244Dic166—48no—
245Dic166—48no—
246C8×C83—42no—
247C83⋊C8—42no—
248C22×C83—22no—
249C22×C83—22no—
250C22×C83—22no—
251C22×C83—22no—
252C22×C83—22no—
253C22×C83—22no—
254C22×C83#39114yes—
255C22×C83C22×C22⋊C411yescharacteristic
256C22×C83C2×C22⋊Q811yescharacteristic
257C22×C83C2×C22⋊Q811yescharacteristic
258C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C83C2×#13711yescharacteristic
260C22×C83#57511yescharacteristic
261C22×C83#57511yescharacteristic
262C4×C83—44no—
263C4×C83—42no—
264C4×C83—42no—
265C4×C83#39114yes—
266C4×C83C2×C4×Q811yescharacteristic
267C4×C83C2×C4×Q811yescharacteristic
268C4×C83C2×#12411yescharacteristic
269C4×C83C2×C4⋊Q811yescharacteristic
270Dic83—22no—
271Dic83—22no—
272Dic83—22no—
273Dic83—44no—
274Dic83—42no—
275Dic83—42no—
276Dic83#41714yes—
277#21312—831nomaximal, Sylow
278C2×C83—24no—
279C2×C83C2×#39111yescharacteristic
280C2×C83C2×#41711yescharacteristic
281C2×C83#289611yescharacteristic
282C2×C83#307711yescharacteristic
283C2×C83#307711yescharacteristic
284C2×C83#307811yescharacteristic
285C2×C83#307811yescharacteristic
286C2×#533—831no—
287C2×#423—831no—
288C2×C4×C4⋊C4—831no—
289#1989—832no—
290#1989—832no—
291#1988—832no—
292#1987—832no—
293#1736—831no—
294#1736—831no—
295#1736—831no—
296#1736D8311yescharacteristic
297C83#2131211yescharacteristic, complemented, Sylow
298C22×M16—831no—
299C22×M16C2×D8311yescharacteristic
300#533—838no—
301#533—1664no—
302C2×C4⋊Q8—831no—
303C2×C4⋊Q8—831no—
304C2×#125—831no—
305C2×#125—831no—
306C2×C22⋊Q8—1661no—
307C2×C22⋊Q8—1661no—
308C2×C4×Q8—1661no—
309C2×C4×Q8—1661no—
310C2×#117—831no—
311C2×#117C2×D8311yescharacteristic
312C22×C4⋊C4—831no—
313C22×C4⋊C4C2×D8311yescharacteristic
314C22×C42—831no—
315C22×C42—1661no—
316#423—1664no—
317C2×C4⋊C8—832no—
318C2×C4⋊C8—832no—
319C4×C4⋊C4—832no—
320C4×C4⋊C4—832no—
321C4×C4⋊C4—1661no—
322C4×C4⋊C4—1661no—
323C2×#95—1661no—
324#340—832no—
325#340—832no—
326#340—832no—
327#340—832no—
328#340—832no—
329#340—832no—
330#340C2×D8312yes—
331#340C2×D8312yes—
332C22×Q8—1661no—
333C22×Q8—1661no—
334C23×C4—21no—
335C23×C4—831no—
336C23×C4—831no—
337C23×C4—1661no—
338C23×C4C22×D8311yescharacteristic
339C2×M16—22no—
340C2×M16—22no—
341C2×M16—22no—
342C2×M16—1662no—
343C2×M16—1662no—
344C2×M16—1662no—
345C22×C8—832no—
346C22×C8C22×D8312yes—
347C4⋊Q8—834no—
348C4⋊Q8—834no—
349C4⋊Q8—1662no—
350C4⋊Q8—1662no—
351#125—834no—
352#125—834no—
353#125—1662no—
354#125—1662no—
355C22⋊Q8—1668no—
356C22⋊Q8—1668no—
357C4×Q8—1668no—
358C4×Q8—1668no—
359#117—22no—
360#117—834no—
361#117—1662no—
362#117C4×D8314yes—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—834no—
366C2×C4⋊C4—832no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—1661no—
370C2×C4⋊C4—1661no—
371C2×C4⋊C4—1661no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1662no—
377C2×C4⋊C4Dic16614yescomplemented
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C4⋊C4C22×D8311yescharacteristic
380C2×C4⋊C4C22×D8312yes—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1661no—
383C2×C42—21no—
384C2×C42—3324no—
385C2×C42—834no—
386C2×C42—831no—
387C2×C42—831no—
388C2×C42—834no—
389C2×C42—831no—
390C2×C42—831no—
391C2×C42—1662no—
392C2×C42—1662no—
393C2×C42—1661no—
394C4⋊C8—1664no—
395C4⋊C8—1664no—
396#95—1662no—
397#95—1662no—
398C24D4×D8311yescharacteristic
399C2×Q8—3322no—
400C2×Q8—3322no—
401C2×Q8—3322no—
402C2×Q8—3322no—
403C2×Q8—1664no—
404C2×Q8—1664no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—3322no—
412C22×C4—3322no—
413C22×C4—3321no—
414C22×C4—3321no—
415C22×C4—834no—
416C22×C4—831no—
417C22×C4—831no—
418C22×C4—831no—
419C22×C4—831no—
420C22×C4—834no—
421C22×C4—1664no—
422C22×C4—1662no—
423C22×C4—1661no—
424C22×C4—1661no—
425C22×C4—1662no—
426C22×C4—1662no—
427C22×C4C2×C4×D8312yes—
428C22×C4D4×D8311yescharacteristic
429C22×C4D4×D8312yes—
430C22×C4Q8×D8312yes—
431C22×C4C2×Dic16612yes—
432C22×C4#1305058912yes—
433C22×C4#1305059012yes—
434C22×C4C23×D8311yescharacteristic
435M16—44no—
436M16—3324no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—1662no—
441C2×C8—1662no—
442C2×C8—1662no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—832no—
448C4⋊C4—832no—
449C4⋊C4—834no—
450C4⋊C4—1664no—
451C4⋊C4—1664no—
452C4⋊C4—1664no—
453C4⋊C4—1664no—
454C4⋊C4—1664no—
455C4⋊C4—1664no—
456C4⋊C4—1664no—
457C4⋊C4—1661no—
458C4⋊C4—1661no—
459C4⋊C4—1662no—
460C4⋊C4C2×C4×D8312yes—
461C4⋊C4C2×C4×D8312yes—
462C4⋊C4C2×Dic16614yes—
463C22⋊C4—24no—
464C22⋊C4—1664no—
465C42—24no—
466C42—3324no—
467C42—3324no—
468C42—832no—
469C42—832no—
470C42—832no—
471C42—832no—
472C42—834no—
473C42—834no—
474C42—1664no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23C22⋊C4×D8312yes—
480C23#1458176912yes—
481C23#1458177012yes—
482C23C2×D4×D8311yescharacteristic
483Q8—3328no—
484Q8—3328no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—3328no—
496C2×C4—3322no—
497C2×C4—3321no—
498C2×C4—3321no—
499C2×C4—3321no—
500C2×C4—3322no—
501C2×C4—3321no—
502C2×C4—3324no—
503C2×C4—832no—
504C2×C4—832no—
505C2×C4—832no—
506C2×C4—832no—
507C2×C4—832no—
508C2×C4—832no—
509C2×C4—1662no—
510C2×C4—1662no—
511C2×C4—1664no—
512C2×C4—1668no—
513C2×C4—1661no—
514C2×C4—1661no—
515C2×C4—1661no—
516C2×C4—1661no—
517C2×C4—1661no—
518C2×C4—1661no—
519C2×C4C22⋊C4×D8312yes—
520C2×C4#1458175714yes—
521C2×C4C4×Dic16614yes—
522C2×C4#1458176912yes—
523C2×C4#1458177012yes—
524C2×C4#1458177814yes—
525C2×C4#1458179114yes—
526C2×C4C22×C4×D8311yescharacteristic
527C2×C4C2×D4×D8311yescharacteristic
528C2×C4C2×Q8×D8311yescharacteristic
529C2×C4C22×Dic16611yescharacteristic
530C2×C4C2×#1305058911yescharacteristic
531C2×C4C2×#1305059011yescharacteristic
532C8—42no—
533C8—3322no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22#1912760514yes—
541C22C2×C22⋊C4×D8311yescharacteristic
542C22C2×#1458176911yescharacteristic
543C22C2×#1458177011yescharacteristic
544C22AGL(1,ℤ/8)×D8311yescharacteristic
545C22#137×D8311yescharacteristic
546C22#1912849711yescharacteristic
547C22#1912850011yescharacteristic
548C4—42no—
549C4—44no—
550C4—42no—
551C4—3322no—
552C4—3322no—
553C4—3324no—
554C4—834no—
555C4—1662no—
556C4—1662no—
557C4—1662no—
558C4#1912760514yes—
559C4C2×#1458175711yescharacteristic
560C4C2×C4×Dic16611yescharacteristic
561C4C2×#1458177811yescharacteristic
562C4C2×#1458179111yescharacteristic
563C2—24no—
564C2C2×#1912760511yescharacteristic
565C2#417×D8311yescharacteristic
566C2#2491597911yescharacteristic
567C2#2491725711yescharacteristic
568C2#2491725911yescharacteristic
569C2#2491726511yescharacteristic
570C2#2491726711yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1736C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#1736D83
hypocenterC83#21312
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #24908419C2semidirect
5. #24908417C2semidirect
6. #24908413C2semidirect
7. #24908409C2semidirect
14. #19128254C4semidirect
57. #14581791C2×C4semidirect
58. #14581791C2×C4semidirect
61. #14581783C2×C4semidirect
62. #14581783C2×C4semidirect
73. C2×C4⋊Dic83Q8semidirect
106. C2×C4⋊C4×C83Q8semidirect
297. C83#21312semidirect
377. C2×C4⋊C4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191282542
C2×#191277242
C2×C4⋊C4×Dic832
#249084192
#249084172
#249084132
#249084092
#249071842
#249071832
#249071822
C83×#17362
#2131283