Finite Group Databasebeta

← #35403890

Subgroups — #35403890

Order 21248.

1450 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D83C211yesmaximal, characteristic
2C2×#19127723C211yesmaximal, characteristic
3C2×#19127708C212yesmaximal
4#24909869C211yesmaximal, characteristic
5#24907479C211yesmaximal, characteristic, complemented
6#24907478C211yesmaximal, characteristic, complemented
7#24907475C211yesmaximal, characteristic, complemented
8#24907467C212yesmaximal, complemented
9#24907466C212yesmaximal, complemented
10#24907181C212yesmaximal
11C83×#1805C211yesmaximal, characteristic, complemented
12C22×Q8×D83C2211yescharacteristic
13C22×#13050573C2212yes—
14C2×Q8×Dic83C2211yescharacteristic
15C2×C4×Dic166—21no—
16C2×C4×Dic166C2211yescharacteristic
17C2×#14581757—21no—
18C2×#14581757C2211yescharacteristic
19C4×Q8×D83—24no—
20C4×Q8×D83C418yescomplemented
21C2×#14581750C2212yes—
22C2×C4⋊C4×D83—21no—
23C2×C4⋊C4×D83C2211yescharacteristic
24C2×C42×D83—21no—
25C2×C42×D83C2211yescharacteristic
26#19127723—24no—
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×Dic83⋊C8C2212yes—
29#19127708—28no—
30C2×#14581717C2212yes—
31C2×#14581716C2212yes—
32C2×#14581715C2212yes—
33#19127570C2212yes—
34#19127567C2211yescharacteristic
35#19127560—21no—
36#19127560—21no—
37#19127552—21no—
38#19127552—21no—
39#19127464C2212yes—
40#19127463C2212yes—
41#19127463C2212yes—
42C2×C4×Q8×C83C2211yescharacteristic
43C2×C4⋊C8×C83C2211yescharacteristic
44C2×#103×C83C2212yes—
45#382×C83C2211yescharacteristic
46#340×C83C2212yes—
47C22×Dic166—21no—
48C22×Dic166C2311yescharacteristic
49C2×Q8×D83—22no—
50C2×Q8×D83—22no—
51C2×Q8×D83—22no—
52C2×Q8×D83C2×C414yes—
53C2×Q8×D83D414yes—
54C2×Q8×D83Q814yes—
55C22×C4×D83—21no—
56C22×C4×D83—21no—
57C22×C4×D83D411yescharacteristic
58C22×C4×D83D411yescharacteristic
59C22×C4×D83C2311yescharacteristic
60C2×#13050573—24no—
61C2×#13050573—24no—
62C2×#13050573—24no—
63C22×C83⋊C8C2312yes—
64Q8×Dic83—22no—
65Q8×Dic83C2×C414yes—
66C4×Dic166—28no—
67C4×Dic166—22no—
68C4×Dic166C2×C414yes—
69#14581757—28no—
70#14581757—22no—
71#14581757C2×C414yes—
72#14581750—24no—
73#14581750—24no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83C2312yes—
76C2×C4⋊Dic83C2311yescharacteristic
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—22no—
80C2×Dic83⋊C4—22no—
81C2×Dic83⋊C4—21no—
82C4⋊C4×D83—28no—
83C4⋊C4×D83—22no—
84C4⋊C4×D83C2×C414yes—
85C2×(C2×D83)⋊C4—22no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83—22no—
89C2×C4×Dic83—21no—
90C2×C4×Dic83D411yescharacteristic
91C2×C4×Dic83D411yescharacteristic
92C2×C4×Dic83C2311yescharacteristic
93C42×D83—28no—
94C42×D83C2×C414yes—
95C42×D83D414yes—
96C4⋊(C83⋊C8)—22no—
97Dic83⋊C8—24no—
98#14581717—24no—
99#14581716—24no—
100#14581715—24no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C22×Q8×C83C2311yescharacteristic
104C22×C8×C83C2312yes—
105C4×Q8×C83—22no—
106C4×Q8×C83C2×C414yes—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83C2312yes—
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C42×C83—21no—
111C2×C42×C83C2311yescharacteristic
112C4⋊C8×C83—22no—
113#103×C83—24no—
114#95×C83—21no—
115#95×C83—21no—
116C23×D83C2×D411yescharacteristic
117C2×Dic166—21no—
118C2×Dic166—21no—
119C2×Dic166—21no—
120C2×Dic166—24no—
121C2×Dic166—24no—
122C2×Dic166—24no—
123C2×Dic166C22×C412yes—
124C2×Dic166C2×D412yes—
125C2×Dic166C2×Q812yes—
126Q8×D83—28no—
127Q8×D83—28no—
128Q8×D83—28no—
129Q8×D83C4⋊C4116yescomplemented
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—22no—
133C22×Dic83C2×D411yescharacteristic
134C22×Dic83C2×D411yescharacteristic
135C22×Dic83C2×D411yescharacteristic
136C2×C4×D83—24no—
137C2×C4×D83—24no—
138C2×C4×D83—24no—
139C2×C4×D83—24no—
140C2×C4×D83—24no—
141C2×C4×D83—42no—
142C2×C4×D83—44no—
143C2×C4×D83C22⋊C414yes—
144C2×C4×D83C22⋊C414yes—
145C2×C4×D83C22×C412yes—
146C2×C4×D83C2×D412yes—
147C2×C4×D83C2×D412yes—
148C2×C4×D83C2×Q812yes—
149C2×C4×D83#4112yes—
150C2×C4×D83#4112yes—
151#13050573—48no—
152C2×C83⋊C8—22no—
153C2×C83⋊C8—22no—
154C2×C83⋊C8—22no—
155C4⋊Dic83—21no—
156C4⋊Dic83—24no—
157C4⋊Dic83—22no—
158C4⋊Dic83—22no—
159C4⋊Dic83C22×C412yes—
160Dic83⋊C4—24no—
161Dic83⋊C4—24no—
162Dic83⋊C4—24no—
163Dic83⋊C4—44no—
164Dic83⋊C4—44no—
165(C2×D83)⋊C4—44no—
166C4×Dic83—24no—
167C4×Dic83—24no—
168C4×Dic83—24no—
169C4×Dic83—44no—
170C4×Dic83C22⋊C414yes—
171C4×Dic83C22⋊C414yes—
172C4×Dic83C22×C412yes—
173C4×Dic83C2×D412yes—
174C2×Q8×C83—21no—
175C2×Q8×C83—21no—
176C2×Q8×C83—21no—
177C2×Q8×C83C22×C412yes—
178C2×Q8×C83C2×D412yes—
179C2×Q8×C83C2×Q812yes—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—22no—
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2411yescharacteristic
186C2×C8×C83—22no—
187C2×C8×C83—22no—
188C2×C8×C83—22no—
189C4⋊C4×C83—24no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—22no—
192C4⋊C4×C83—22no—
193C4⋊C4×C83C22×C412yes—
194C42×C83—24no—
195C42×C83C22×C412yes—
196C42×C83C2×D412yes—
197C23×C83C22×D411yescharacteristic
198C22×D83—22no—
199C22×D83—22no—
200C22×D83C2×C22⋊C412yes—
201C22×D83#12112yes—
202C22×D83C22⋊Q812yes—
203Q8×C83—22no—
204Q8×C83—22no—
205Q8×C83—22no—
206Q8×C83C2×C4⋊C414yes—
207C2×C4×C83—22no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—22no—
213C2×C4×C83—42no—
214C2×C4×C83—42no—
215C2×C4×C83—42no—
216C2×C4×C83—42no—
217C2×C4×C83C2×C22⋊C412yes—
218C2×C4×C83C2×C22⋊C412yes—
219C2×C4×C83C23×C411yescharacteristic
220C2×C4×C83C22×D411yescharacteristic
221C2×C4×C83C22×D411yescharacteristic
222C2×C4×C83C22×Q811yescharacteristic
223C2×C4×C83C2×#4111yescharacteristic
224C2×C4×C83C2×#4111yescharacteristic
225C2×Dic83—22no—
226C2×Dic83—22no—
227C2×Dic83—22no—
228C2×Dic83—22no—
229C2×Dic83—22no—
230C2×Dic83—42no—
231C2×Dic83—42no—
232C2×Dic83—42no—
233C2×Dic83—42no—
234C2×Dic83C2×C22⋊C412yes—
235C2×Dic83C2×C22⋊C412yes—
236C2×Dic83C2×C22⋊C412yes—
237C2×Dic83#12112yes—
238C2×Dic83C22⋊Q812yes—
239C2×Dic83#13714yes—
240C4×D83—216no—
241C4×D83—28no—
242C4×D83—44no—
243C4×D83—48no—
244C4×D83C2×C4⋊C414yes—
245C4×D83#11714yes—
246C4×D83#12114yes—
247C4×D83C22⋊Q814yes—
248Dic166—22no—
249Dic166—22no—
250Dic166—22no—
251Dic166—216no—
252Dic166C2×C4⋊C414yes—
253C8×C83—42no—
254C83⋊C8—42no—
255C22×C83C22×C22⋊C411yescharacteristic
256C22×C83C2×#12111yescharacteristic
257C22×C83C2×C22⋊Q811yescharacteristic
258C22×C83C2×#13712yes—
259C22×C83#57512yes—
260C2×D83—24no—
261C2×D83—22no—
262C2×D83—22no—
263C2×D83—24no—
264C2×D83#38014yes—
265C4×C83—22no—
266C4×C83—22no—
267C4×C83—24no—
268C4×C83—42no—
269C4×C83—82no—
270C4×C83C22×C4⋊C411yescharacteristic
271C4×C83C2×#11711yescharacteristic
272C4×C83C2×#12111yescharacteristic
273C4×C83C2×C22⋊Q811yescharacteristic
274Dic83—22no—
275Dic83—24no—
276Dic83—42no—
277Dic83—82no—
278Dic83#38014yes—
279Dic83#41914yescomplemented
280#20647—831nomaximal, Sylow
281C2×C83C2×#38011yescharacteristic
282C2×C83C2×#41911yescharacteristic
283C2×C83#289611yescharacteristic
284C2×C83#303712yes—
285C2×C83#307812yes—
286D83—24no—
287C22×C4×Q8—831no—
288C2×#423—831no—
289C2×#419—832no—
290#2285—831no—
291#1805—831no—
292#1805—831no—
293#1805—831no—
294#1805D8311yescharacteristic, complemented
295#1803—832no—
296#1803—832no—
297#1736—832no—
298C83#2064711yescharacteristic, complemented, Sylow
299C23×Q8—831no—
300C22×M16—832no—
301C2×C4×Q8—831no—
302C2×C4×Q8—838no—
303C2×C4×Q8—831no—
304C2×C4×Q8—831no—
305C2×C4×Q8—1661no—
306C2×C4×Q8—1661no—
307C2×C4×Q8—1664no—
308C2×C4×Q8C2×D8311yescharacteristic
309C2×#117—832no—
310C22×C4⋊C4—831no—
311C22×C4⋊C4—1661no—
312C22×C42—831no—
313C22×C42—1661no—
314#423—1664no—
315C2×C4⋊C8—832no—
316C2×C4⋊C8—831no—
317C2×C4⋊C8C2×D8311yescharacteristic
318#419—1668no—
319C2×#103—832no—
320C2×#103—832no—
321C2×#103—832no—
322C2×#103C2×D8312yes—
323#384—832no—
324#382—831no—
325#382C2×D8311yescharacteristic
326#380—1661no—
327#380—1661no—
328C4×C4⋊C4—1661no—
329C4×C4⋊C4—1661no—
330#340—832no—
331#340—832no—
332#340—832no—
333#340C2×D8312yes—
334C22×Q8—834no—
335C22×Q8—834no—
336C22×Q8—834no—
337C22×Q8—831no—
338C22×Q8—1661no—
339C22×Q8—1662no—
340C22×Q8—1662no—
341C22×Q8—1662no—
342C22×Q8C22×D8311yescharacteristic
343C23×C4—831no—
344C23×C4—831no—
345C23×C4—831no—
346C23×C4—1661no—
347C23×C4—1661no—
348C2×M16—1664no—
349C2×M16—1664no—
350C2×M16—1664no—
351C22×C8—832no—
352C22×C8C22×D8312yes—
353C4×Q8—22no—
354C4×Q8—834no—
355C4×Q8—834no—
356C4×Q8—834no—
357C4×Q8—1668no—
358C4×Q8—1662no—
359C4×Q8—1668no—
360C4×Q8—1662no—
361C4×Q8—1662no—
362C4×Q8C4×D8314yes—
363#117—1664no—
364#117—1664no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—831no—
367C2×C4⋊C4—834no—
368C2×C4⋊C4—832no—
369C2×C4⋊C4—1662no—
370C2×C4⋊C4—1668no—
371C2×C4⋊C4—1662no—
372C2×C4⋊C4—1662no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4C22×D8312yes—
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C22⋊C4—1662no—
380C2×C42—21no—
381C2×C42—831no—
382C2×C42—831no—
383C2×C42—831no—
384C2×C42—834no—
385C2×C42—834no—
386C2×C42—1668no—
387C2×C42—1662no—
388C2×C42—1661no—
389C2×C42—1661no—
390C2×C42—1661no—
391C2×C42C22×D8311yescharacteristic
392C4⋊C8—22no—
393C4⋊C8—1664no—
394C4⋊C8—1662no—
395#103—24no—
396#103—1664no—
397#103—1664no—
398#103—1664no—
399#95—21no—
400#95—21no—
401#95—1661no—
402#95—1661no—
403C24—831no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—8316no—
408C2×Q8—832no—
409C2×Q8—832no—
410C2×Q8—832no—
411C2×Q8—1664no—
412C2×Q8—1664no—
413C2×Q8—1664no—
414C2×Q8—1668no—
415C2×Q8—1668no—
416C2×Q8—1668no—
417C2×Q8—1661no—
418C2×Q8—1661no—
419C2×Q8—1661no—
420C2×Q8C2×C4×D8312yes—
421C2×Q8D4×D8312yes—
422C2×Q8Q8×D8312yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—3322no—
427C22×C4—3324no—
428C22×C4—834no—
429C22×C4—831no—
430C22×C4—831no—
431C22×C4—832no—
432C22×C4—832no—
433C22×C4—832no—
434C22×C4—832no—
435C22×C4—832no—
436C22×C4—832no—
437C22×C4—831no—
438C22×C4—834no—
439C22×C4—1664no—
440C22×C4—1664no—
441C22×C4—1664no—
442C22×C4—1664no—
443C22×C4—1664no—
444C22×C4—1662no—
445C22×C4—1661no—
446C22×C4—1661no—
447C22×C4D4×D8311yescharacteristic
448C22×C4D4×D8311yescharacteristic
449C22×C4C23×D8311yescharacteristic
450M16—3328no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—1662no—
455C2×C8—1662no—
456C2×C8—1662no—
457C4⋊C4—21no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—3324no—
462C4⋊C4—3324no—
463C4⋊C4—832no—
464C4⋊C4—1664no—
465C4⋊C4—1664no—
466C4⋊C4—1664no—
467C4⋊C4—1662no—
468C4⋊C4—1662no—
469C4⋊C4—1661no—
470C4⋊C4—1664no—
471C4⋊C4C2×C4×D8312yes—
472C22⋊C4—3324no—
473C42—24no—
474C42—3324no—
475C42—832no—
476C42—832no—
477C42—834no—
478C42—834no—
479C42—1664no—
480C42—1664no—
481C42—1664no—
482C42C2×C4×D8312yes—
483C42D4×D8312yes—
484C23—832no—
485C23—832no—
486C23—832no—
487C23—1662no—
488C23—1662no—
489C23C2×D4×D8311yescharacteristic
490Q8—22no—
491Q8—22no—
492Q8—22no—
493Q8—834no—
494Q8—16616no—
495Q8—1662no—
496Q8—1662no—
497Q8—1662no—
498Q8C4⋊C4×D8314yes—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—3324no—
510C2×C4—3328no—
511C2×C4—3322no—
512C2×C4—3322no—
513C2×C4—3322no—
514C2×C4—3322no—
515C2×C4—832no—
516C2×C4—832no—
517C2×C4—832no—
518C2×C4—832no—
519C2×C4—834no—
520C2×C4—834no—
521C2×C4—834no—
522C2×C4—834no—
523C2×C4—834no—
524C2×C4—832no—
525C2×C4—1668no—
526C2×C4—16616no—
527C2×C4—1662no—
528C2×C4—1662no—
529C2×C4—1662no—
530C2×C4—1662no—
531C2×C4—1662no—
532C2×C4C22⋊C4×D8312yes—
533C2×C4C22⋊C4×D8312yes—
534C2×C4C22×C4×D8311yescharacteristic
535C2×C4C2×D4×D8311yescharacteristic
536C2×C4C2×D4×D8311yescharacteristic
537C2×C4C2×Q8×D8311yescharacteristic
538C2×C4#41×D8311yescharacteristic
539C2×C4#41×D8311yescharacteristic
540C8—42no—
541C8—3322no—
542C22—834no—
543C22—1662no—
544C22—1664no—
545C22—1664no—
546C22—1662no—
547C22C2×C22⋊C4×D8311yescharacteristic
548C22#121×D8311yescharacteristic
549C22C22⋊Q8×D8311yescharacteristic
550C22#137×D8312yes—
551C22#1912849812yes—
552C4—24no—
553C4—22no—
554C4—22no—
555C4—42no—
556C4—82no—
557C4—3322no—
558C4—834no—
559C4—834no—
560C4—6642no—
561C4—1664no—
562C4—1662no—
563C4C2×C4⋊C4×D8311yescharacteristic
564C4#117×D8311yescharacteristic
565C4#121×D8311yescharacteristic
566C4C22⋊Q8×D8311yescharacteristic
567C2—1664no—
568C2#380×D8311yescharacteristic
569C2#419×D8311yescharacteristic
570C2#2491597811yescharacteristic
571C2#2491693312yes—
572C2#2491726612yes—
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1805C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1805D83
hypocenterC83#20647
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
5. #24907479C2semidirect
6. #24907478C2semidirect
7. #24907475C2semidirect
8. #24907467C2semidirect
9. #24907466C2semidirect
11. C83×#1805C2semidirect
20. C4×Q8×D83C4semidirect
129. Q8×D83C4⋊C4semidirect
279. Dic83#419semidirect
294. #1805D83semidirect
298. C83#20647semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D832
C2×#191277232
C2×#191277082
#249098692
#249074792
#249074782
#249074752
#249074672
#249074662
#249071812
C83×#18052
#2064783