Finite Group Databasebeta

← C2×C83×#1516 #35245392

Subgroups — C2×C83×#1516 #35245392

Order 21248.

1498 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525×C83C211yesmaximal, characteristic
2C22×C22⋊C8×C83C211yesmaximal, characteristic, complemented
3C2×#381×C83C211yesmaximal, characteristic, complemented
4C83×#1516C218yesmaximal, direct product factor
5C2×#328×C83C212yesmaximal, complemented
6C2×#327×C83C212yesmaximal, complemented
7C22×#41×C83C411yescharacteristic, complemented
8C24×C4×C83C2211yescharacteristic
9C23×C8×C83—21no—
10#525×C83—22no—
11#525×C83—22no—
12#525×C83C414yescomplemented
13#525×C83C2214yes—
14C2×#123×C83—21no—
15C2×C22⋊Q8×C83C411yescharacteristic, complemented
16C2×C22⋊Q8×C83C2211yescharacteristic
17C2×#121×C83C411yescharacteristic, complemented
18C2×#121×C83C2211yescharacteristic
19C2×#120×C83—21no—
20C2×C4×D4×C83—21no—
21C2×C4×D4×C83—21no—
22C2×#117×C83C411yescharacteristic, complemented
23C22×C22⋊C4×C83—21no—
24C2×C22⋊C8×C83—22no—
25C2×C22⋊C8×C83—21no—
26C2×C22⋊C8×C83C2212yes—
27C2×C22⋊C8×C83C2214yescomplemented
28#381×C83—22no—
29#381×C83C2214yescomplemented
30C2×#95×C83—21no—
31C2×#95×C83C2212yes—
32#328×C83C2218yescomplemented
33#327×C83C2218yescomplemented
34C25×C83D411yescharacteristic
35C2×#41×C83—22no—
36C2×#41×C83—44no—
37C2×#41×C83C2×C414yescomplemented
38C22×Q8×C83C2×C411yescharacteristic
39C22×D4×C83—21no—
40C22×D4×C83—21no—
41C22×D4×C83—21no—
42C22×D4×C83C2×C411yescharacteristic
43C23×C4×C83—21no—
44C23×C4×C83—22no—
45C23×C4×C83—44no—
46C23×C4×C83C2×C411yescharacteristic
47C23×C4×C83C2×C411yescharacteristic
48C23×C4×C83C2×C412yes—
49C23×C4×C83D411yescharacteristic
50C23×C4×C83D411yescharacteristic
51C23×C4×C83D412yes—
52C23×C4×C83D411yescharacteristic
53C23×C4×C83D412yes—
54C23×C4×C83C2311yescharacteristic
55C23×C4×C83C2312yes—
56C22×C8×C83—22no—
57C22×C8×C83—24no—
58C22×C8×C83—44no—
59#123×C83—24no—
60#123×C83—24no—
61C22⋊Q8×C83—22no—
62C22⋊Q8×C83—21no—
63C22⋊Q8×C83—21no—
64C22⋊Q8×C83C2×C412yes—
65C22⋊Q8×C83C2×C414yescomplemented
66C22⋊Q8×C83C2312yes—
67#121×C83—22no—
68#121×C83—21no—
69#121×C83—21no—
70#121×C83C2×C412yes—
71#121×C83C2×C414yescomplemented
72#121×C83C2312yes—
73#120×C83—24no—
74#120×C83—24no—
75C4×D4×C83—24no—
76C4×D4×C83—24no—
77C4×D4×C83—24no—
78C4×D4×C83—24no—
79#117×C83—22no—
80#117×C83C2×C414yescomplemented
81C2×C4⋊C4×C83—21no—
82C2×C4⋊C4×C83—21no—
83C2×C4⋊C4×C83C2×C411yescharacteristic
84C2×C4⋊C4×C83C2×C411yescharacteristic
85C2×C22⋊C4×C83—21no—
86C2×C22⋊C4×C83—21no—
87C2×C22⋊C4×C83—21no—
88C2×C22⋊C4×C83—21no—
89C2×C22⋊C4×C83—21no—
90C2×C22⋊C4×C83—24no—
91C2×C22⋊C4×C83—44no—
92C2×C42×C83—21no—
93C22⋊C8×C83—22no—
94C22⋊C8×C83—24no—
95C22⋊C8×C83C2314yes—
96#95×C83—24no—
97#95×C83—22no—
98#95×C83C2314yes—
99C24×C83—24no—
100C24×C83—24no—
101C24×C83—21no—
102C24×C83—42no—
103C24×C83C22⋊C411yescharacteristic
104C24×C83C22⋊C412yes—
105C24×C83C22⋊C411yescharacteristic
106C24×C83C2×D412yes—
107C24×C83C2×D411yescharacteristic
108#41×C83—48no—
109#41×C83—48no—
110C2×Q8×C83—21no—
111C2×Q8×C83—42no—
112C2×Q8×C83C22×C412yes—
113C2×D4×C83—22no—
114C2×D4×C83—24no—
115C2×D4×C83—24no—
116C2×D4×C83—22no—
117C2×D4×C83—21no—
118C2×D4×C83—42no—
119C2×D4×C83—44no—
120C2×D4×C83—44no—
121C2×D4×C83—44no—
122C2×D4×C83C22×C412yes—
123C22×C4×C83—24no—
124C22×C4×C83—24no—
125C22×C4×C83—21no—
126C22×C4×C83—21no—
127C22×C4×C83—21no—
128C22×C4×C83—24no—
129C22×C4×C83—24no—
130C22×C4×C83—42no—
131C22×C4×C83—44no—
132C22×C4×C83—42no—
133C22×C4×C83—44no—
134C22×C4×C83—42no—
135C22×C4×C83—44no—
136C22×C4×C83—42no—
137C22×C4×C83—44no—
138C22×C4×C83—44no—
139C22×C4×C83—44no—
140C22×C4×C83—44no—
141C22×C4×C83—44no—
142C22×C4×C83C22⋊C412yes—
143C22×C4×C83C22⋊C412yes—
144C22×C4×C83C22⋊C411yescharacteristic
145C22×C4×C83C22⋊C411yescharacteristic
146C22×C4×C83C22⋊C412yes—
147C22×C4×C83C22⋊C412yes—
148C22×C4×C83C22⋊C412yes—
149C22×C4×C83C22×C412yes—
150C22×C4×C83C22×C411yescharacteristic
151C22×C4×C83C22×C411yescharacteristic
152C22×C4×C83C22×C412yes—
153C22×C4×C83C2×D412yes—
154C22×C4×C83C2×D411yescharacteristic
155C22×C4×C83C2×D412yes—
156C22×C4×C83C2×D411yescharacteristic
157C22×C4×C83C2×D412yes—
158C22×C4×C83C2×D411yescharacteristic
159C22×C4×C83C2411yescharacteristic
160C2×C8×C83—24no—
161C2×C8×C83—44no—
162C2×C8×C83—48no—
163C4⋊C4×C83—22no—
164C4⋊C4×C83—24no—
165C4⋊C4×C83—21no—
166C4⋊C4×C83—21no—
167C4⋊C4×C83—22no—
168C4⋊C4×C83C22×C412yes—
169C4⋊C4×C83C22×C412yes—
170C22⋊C4×C83—24no—
171C22⋊C4×C83—22no—
172C22⋊C4×C83—24no—
173C22⋊C4×C83—22no—
174C22⋊C4×C83—22no—
175C22⋊C4×C83—24no—
176C22⋊C4×C83—22no—
177C22⋊C4×C83—48no—
178C42×C83—24no—
179C23×C83—22no—
180C23×C83—22no—
181C23×C83—22no—
182C23×C83—22no—
183C23×C83—24no—
184C23×C83—24no—
185C23×C83—24no—
186C23×C83—22no—
187C23×C83—24no—
188C23×C83—22no—
189C23×C83—22no—
190C23×C83—22no—
191C23×C83—42no—
192C23×C83—42no—
193C23×C83—416no—
194C23×C83—42no—
195C23×C83—42no—
196C23×C83—42no—
197C23×C83#9914yes—
198C23×C83C2×C22⋊C412yes—
199C23×C83C2×C22⋊C412yes—
200C23×C83C2×C22⋊C411yescharacteristic
201C23×C83C2×C22⋊C411yescharacteristic
202C23×C83#13514yes—
203C23×C83C22×D411yescharacteristic
204Q8×C83—42no—
205Q8×C83—42no—
206D4×C83—42no—
207D4×C83—42no—
208D4×C83—48no—
209D4×C83—44no—
210D4×C83—48no—
211D4×C83—44no—
212C2×C4×C83—24no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—24no—
216C2×C4×C83—22no—
217C2×C4×C83—24no—
218C2×C4×C83—22no—
219C2×C4×C83—24no—
220C2×C4×C83—22no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—42no—
224C2×C4×C83—42no—
225C2×C4×C83—42no—
226C2×C4×C83—41no—
227C2×C4×C83—42no—
228C2×C4×C83—42no—
229C2×C4×C83—41no—
230C2×C4×C83—42no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—48no—
234C2×C4×C83—42no—
235C2×C4×C83—42no—
236C2×C4×C83—42no—
237C2×C4×C83—44no—
238C2×C4×C83—44no—
239C2×C4×C83—44no—
240C2×C4×C83—88no—
241C2×C4×C83C2×C22⋊C412yes—
242C2×C4×C83C2×C22⋊C411yescharacteristic
243C2×C4×C83C2×C22⋊C412yes—
244C2×C4×C83C2×C22⋊C411yescharacteristic
245C2×C4×C83C23×C411yescharacteristic
246C2×C4×C83C22×D411yescharacteristic
247C8×C83—44no—
248C22×C83—21no—
249C22×C83—21no—
250C22×C83—22no—
251C22×C83—21no—
252C22×C83—22no—
253C22×C83—21no—
254C22×C83—28no—
255C22×C83—28no—
256C22×C83—22no—
257C22×C83—22no—
258C22×C83—22no—
259C22×C83—48no—
260C22×C83—48no—
261C22×C83—44no—
262C22×C83—42no—
263C22×C83—42no—
264C22×C83—42no—
265C22×C83—84no—
266C22×C83C2×#9914yes—
267C22×C83C2×#9912yes—
268C22×C83#41612yes—
269C22×C83#41612yes—
270C22×C83#42112yes—
271C22×C83C22×C22⋊C411yescharacteristic
272C22×C83C2×#13512yes—
273C4×C83—21no—
274C4×C83—22no—
275C4×C83—22no—
276C4×C83—21no—
277C4×C83—21no—
278C4×C83—21no—
279C4×C83—42no—
280C4×C83—42no—
281C4×C83—42no—
282C4×C83—84no—
283C2×#1516C8311yesmaximal, characteristic, direct product factor, Sylow
284C2×C83—22no—
285C2×C83—24no—
286C2×C83—22no—
287C2×C83—42no—
288C2×C83—82no—
289C2×C83#151614yesdirect product factor
290C2×C83C22×#9911yescharacteristic
291C2×C83C2×#41611yescharacteristic
292C2×C83C2×#42111yescharacteristic
293C2×#525C2×C8311yescharacteristic
294C22×C22⋊C8C2×C8311yescharacteristic, complemented
295C2×#381C2×C8311yescharacteristic, complemented
296#1516C2×C8318yesdirect product factor
297C2×#328C2×C8312yescomplemented
298C2×#327C2×C8312yescomplemented
299C83C2×#151611yescharacteristic, direct product factor, Sylow
300C22×#41C4×C8311yescharacteristic, complemented
301C24×C4C22×C8311yescharacteristic
302C23×C8—21no—
303#525—22no—
304#525—22no—
305#525C4×C8314yescomplemented
306#525C22×C8314yes—
307C2×#123—21no—
308C2×C22⋊Q8C4×C8311yescharacteristic, complemented
309C2×C22⋊Q8C22×C8311yescharacteristic
310C2×#121C4×C8311yescharacteristic, complemented
311C2×#121C22×C8311yescharacteristic
312C2×#120—21no—
313C2×C4×D4—21no—
314C2×C4×D4—21no—
315C2×#117C4×C8311yescharacteristic, complemented
316C22×C22⋊C4—21no—
317C2×C22⋊C8—22no—
318C2×C22⋊C8—21no—
319C2×C22⋊C8C22×C8312yes—
320C2×C22⋊C8C22×C8314yescomplemented
321#381—22no—
322#381C22×C8314yescomplemented
323C2×#95—21no—
324C2×#95C22×C8312yes—
325#328C22×C8318yescomplemented
326#327C22×C8318yescomplemented
327C25D4×C8311yescharacteristic
328C2×#41—22no—
329C2×#41—44no—
330C2×#41C2×C4×C8314yescomplemented
331C22×Q8C2×C4×C8311yescharacteristic
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4C2×C4×C8311yescharacteristic
336C23×C4—21no—
337C23×C4—22no—
338C23×C4—44no—
339C23×C4C2×C4×C8311yescharacteristic
340C23×C4C2×C4×C8311yescharacteristic
341C23×C4C2×C4×C8312yes—
342C23×C4D4×C8311yescharacteristic
343C23×C4D4×C8311yescharacteristic
344C23×C4D4×C8312yes—
345C23×C4D4×C8311yescharacteristic
346C23×C4D4×C8312yes—
347C23×C4C23×C8311yescharacteristic
348C23×C4C23×C8312yes—
349C22×C8—22no—
350C22×C8—24no—
351C22×C8—44no—
352#123—24no—
353#123—24no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8C2×C4×C8312yes—
358C22⋊Q8C2×C4×C8314yescomplemented
359C22⋊Q8C23×C8312yes—
360#121—22no—
361#121—21no—
362#121—21no—
363#121C2×C4×C8312yes—
364#121C2×C4×C8314yescomplemented
365#121C23×C8312yes—
366#120—24no—
367#120—24no—
368C4×D4—24no—
369C4×D4—24no—
370C4×D4—24no—
371C4×D4—24no—
372#117—22no—
373#117C2×C4×C8314yescomplemented
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C2×C4×C8311yescharacteristic
377C2×C4⋊C4C2×C4×C8311yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—44no—
385C2×C42—21no—
386C22⋊C8—22no—
387C22⋊C8—24no—
388C22⋊C8C23×C8314yes—
389#95—24no—
390#95—22no—
391#95C23×C8314yes—
392C24—24no—
393C24—24no—
394C24—21no—
395C24—42no—
396C24C22⋊C4×C8311yescharacteristic
397C24C22⋊C4×C8312yes—
398C24C22⋊C4×C8311yescharacteristic
399C24C2×D4×C8312yes—
400C24C2×D4×C8311yescharacteristic
401#41—48no—
402#41—48no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8C22×C4×C8312yes—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4C22×C4×C8312yes—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4C22⋊C4×C8311yescharacteristic
436C22×C4C22⋊C4×C8311yescharacteristic
437C22×C4C22⋊C4×C8312yes—
438C22×C4C22⋊C4×C8312yes—
439C22×C4C22⋊C4×C8312yes—
440C22×C4C22⋊C4×C8312yes—
441C22×C4C22⋊C4×C8312yes—
442C22×C4C22×C4×C8311yescharacteristic
443C22×C4C22×C4×C8312yes—
444C22×C4C22×C4×C8311yescharacteristic
445C22×C4C22×C4×C8312yes—
446C22×C4C2×D4×C8311yescharacteristic
447C22×C4C2×D4×C8312yes—
448C22×C4C2×D4×C8312yes—
449C22×C4C2×D4×C8311yescharacteristic
450C22×C4C2×D4×C8312yes—
451C22×C4C2×D4×C8311yescharacteristic
452C22×C4C24×C8311yescharacteristic
453C2×C8—24no—
454C2×C8—44no—
455C2×C8—48no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4C22×C4×C8312yes—
462C4⋊C4C22×C4×C8312yes—
463C22⋊C4—24no—
464C22⋊C4—22no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—48no—
471C42—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—416no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23#99×C8314yes—
491C23C2×C22⋊C4×C8312yes—
492C23C2×C22⋊C4×C8312yes—
493C23C2×C22⋊C4×C8311yescharacteristic
494C23C2×C22⋊C4×C8311yescharacteristic
495C23#135×C8314yes—
496C23C22×D4×C8311yescharacteristic
497Q8—42no—
498Q8—42no—
499D4—42no—
500D4—42no—
501D4—48no—
502D4—44no—
503D4—48no—
504D4—44no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—48no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—88no—
534C2×C4C2×C22⋊C4×C8312yes—
535C2×C4C2×C22⋊C4×C8311yescharacteristic
536C2×C4C2×C22⋊C4×C8312yes—
537C2×C4C2×C22⋊C4×C8311yescharacteristic
538C2×C4C23×C4×C8311yescharacteristic
539C2×C4C22×D4×C8311yescharacteristic
540C8—44no—
541C22—28no—
542C22—28no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—21no—
547C22—22no—
548C22—21no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—48no—
553C22—48no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—84no—
559C22C2×#99×C8314yes—
560C22C2×#99×C8312yes—
561C22#416×C8312yes—
562C22#416×C8312yes—
563C22#421×C8312yes—
564C22C22×C22⋊C4×C8311yescharacteristic
565C22C2×#135×C8312yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—84no—
576C2—22no—
577C2—24no—
578C2—22no—
579C2—42no—
580C2—82no—
581C2C83×#151614yesdirect product factor
582C2C22×#99×C8311yescharacteristic
583C2C2×#416×C8311yescharacteristic
584C2C2×#421×C8311yescharacteristic
subgroupG/H
center, socleC23×C83C2×C22⋊C4
commutatorC2×C4C23×C4×C83
Frattini, cosocleC22×C4C24×C83
prime radicalC2×#525×C83C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C83 ◁ C24×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊C8×C83C2semidirect
3. C2×#381×C83C2semidirect
4. C83×#1516C2direct (N × H)
5. C2×#328×C83C2semidirect
6. C2×#327×C83C2semidirect
7. C22×#41×C83C4semidirect
12. #525×C83C4semidirect
15. C2×C22⋊Q8×C83C4semidirect
17. C2×#121×C83C4semidirect
22. C2×#117×C83C4semidirect
27. C2×C22⋊C8×C83C22semidirect
29. #381×C83C22semidirect
32. #328×C83C22semidirect
33. #327×C83C22semidirect
37. C2×#41×C83C2×C4semidirect
65. C22⋊Q8×C83C2×C4semidirect
71. #121×C83C2×C4semidirect
80. #117×C83C2×C4semidirect
283. C2×#1516C83direct (N × H)
289. C2×C83#1516direct (N × H)
294. C22×C22⋊C8C2×C83semidirect
295. C2×#381C2×C83semidirect
296. #1516C2×C83direct (N × H)
297. C2×#328C2×C83semidirect
298. C2×#327C2×C83semidirect
299. C83C2×#1516direct (N × H)
300. C22×#41C4×C83semidirect
305. #525C4×C83semidirect
308. C2×C22⋊Q8C4×C83semidirect
310. C2×#121C4×C83semidirect
315. C2×#117C4×C83semidirect
320. C2×C22⋊C8C22×C83semidirect
322. #381C22×C83semidirect
325. #328C22×C83semidirect
326. #327C22×C83semidirect
330. C2×#41C2×C4×C83semidirect
358. C22⋊Q8C2×C4×C83semidirect
364. #121C2×C4×C83semidirect
373. #117C2×C4×C83semidirect
581. C2C83×#1516direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#525×C832
C22×C22⋊C8×C832
C2×#381×C832
C83×#15162
C2×#328×C832
C2×#327×C832
C2×#151683