Finite Group Databasebeta

← C83×#8328 #35236152

Subgroups — C83×#8328 #35236152

Order 21248.

1620 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121×C83C211yesmaximal, characteristic
2C83×#1779C211yesmaximal, characteristic, complemented
3C2×#327×C83C214yesmaximal, complemented
4C83×C22⋊#95C211yesmaximal, characteristic, complemented
5C23×D4×C83—21no—
6C23×D4×C83C411yescharacteristic, complemented
7C24×C4×C83C2211yescharacteristic
8C2×#121×C83—21no—
9C2×#121×C83—21no—
10C2×#121×C83—28no—
11C2×#121×C83C412yescomplemented
12C2×#121×C83C2212yes—
13C22×C4⋊C4×C83C411yescharacteristic, complemented
14C22×C22⋊C4×C83—21no—
15C22×C22⋊C4×C83—21no—
16C2×C22⋊C8×C83—21no—
17C2×C22⋊C8×C83—21no—
18C2×C22⋊C8×C83C2212yescomplemented
19C2×#95×C83—21no—
20C2×#95×C83C2212yescomplemented
21#327×C83—28no—
22C25×C83—21no—
23C25×C83D411yescharacteristic
24C22×D4×C83—28no—
25C22×D4×C83—22no—
26C22×D4×C83—22no—
27C22×D4×C83—24no—
28C22×D4×C83—21no—
29C22×D4×C83—42no—
30C22×D4×C83—42no—
31C22×D4×C83—44no—
32C22×D4×C83—42no—
33C22×D4×C83—42no—
34C22×D4×C83C2×C412yes—
35C23×C4×C83—21no—
36C23×C4×C83—24no—
37C23×C4×C83—22no—
38C23×C4×C83—42no—
39C23×C4×C83—42no—
40C23×C4×C83C2×C411yescharacteristic
41C23×C4×C83C2×C411yescharacteristic
42C23×C4×C83D411yescharacteristic
43C23×C4×C83D411yescharacteristic
44C23×C4×C83D411yescharacteristic
45C23×C4×C83C2311yescharacteristic
46C22×C8×C83—22no—
47#121×C83—28no—
48#121×C83—28no—
49#121×C83—24no—
50#121×C83—24no—
51#121×C83—24no—
52#121×C83D418yes—
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—24no—
55C2×C4⋊C4×C83C2×C412yes—
56C2×C22⋊C4×C83—28no—
57C2×C22⋊C4×C83—22no—
58C2×C22⋊C4×C83—22no—
59C2×C22⋊C4×C83—24no—
60C2×C22⋊C4×C83—42no—
61C2×C22⋊C4×C83—42no—
62C22⋊C8×C83—24no—
63C22⋊C8×C83—42no—
64C22⋊C8×C83—42no—
65#95×C83—24no—
66#95×C83—42no—
67C24×C83—22no—
68C24×C83—22no—
69C24×C83—22no—
70C24×C83—22no—
71C24×C83—24no—
72C24×C83—22no—
73C24×C83—41no—
74C24×C83—41no—
75C24×C83—42no—
76C24×C83—42no—
77C24×C83—42no—
78C24×C83—42no—
79C24×C83—42no—
80C24×C83—42no—
81C24×C83C22⋊C411yescharacteristic
82C24×C83C22⋊C411yescharacteristic
83C24×C83D812yes—
84C24×C83SD1612yes—
85C24×C83C2×D411yescharacteristic
86C2×D4×C83—24no—
87C2×D4×C83—28no—
88C2×D4×C83—28no—
89C2×D4×C83—22no—
90C2×D4×C83—42no—
91C2×D4×C83—42no—
92C2×D4×C83—416no—
93C2×D4×C83—42no—
94C2×D4×C83—416no—
95C2×D4×C83—416no—
96C2×D4×C83—44no—
97C2×D4×C83—44no—
98C2×D4×C83—42no—
99C2×D4×C83—42no—
100C2×D4×C83—44no—
101C2×D4×C83—42no—
102C2×D4×C83C22⋊C414yescomplemented
103C22×C4×C83—22no—
104C22×C4×C83—22no—
105C22×C4×C83—22no—
106C22×C4×C83—24no—
107C22×C4×C83—22no—
108C22×C4×C83—22no—
109C22×C4×C83—22no—
110C22×C4×C83—22no—
111C22×C4×C83—22no—
112C22×C4×C83—22no—
113C22×C4×C83—22no—
114C22×C4×C83—22no—
115C22×C4×C83—24no—
116C22×C4×C83—41no—
117C22×C4×C83—41no—
118C22×C4×C83—41no—
119C22×C4×C83—41no—
120C22×C4×C83—41no—
121C22×C4×C83—41no—
122C22×C4×C83—41no—
123C22×C4×C83—416no—
124C22×C4×C83—42no—
125C22×C4×C83—42no—
126C22×C4×C83—42no—
127C22×C4×C83—42no—
128C22×C4×C83—42no—
129C22×C4×C83—42no—
130C22×C4×C83—42no—
131C22×C4×C83—42no—
132C22×C4×C83C22⋊C411yescharacteristic
133C22×C4×C83C22⋊C411yescharacteristic
134C22×C4×C83C22×C411yescharacteristic
135C22×C4×C83C2×D411yescharacteristic
136C22×C4×C83C2×D414yes—
137C2×C8×C83—42no—
138C2×C8×C83—42no—
139C2×C8×C83—42no—
140C4⋊C4×C83—24no—
141C4⋊C4×C83—22no—
142C4⋊C4×C83C22⋊C414yescomplemented
143C22⋊C4×C83—28no—
144C22⋊C4×C83—28no—
145C22⋊C4×C83—48no—
146C23×C83—21no—
147C23×C83—21no—
148C23×C83—21no—
149C23×C83—21no—
150C23×C83—21no—
151C23×C83—21no—
152C23×C83—21no—
153C23×C83—21no—
154C23×C83—24no—
155C23×C83—24no—
156C23×C83—21no—
157C23×C83—21no—
158C23×C83—21no—
159C23×C83—21no—
160C23×C83—21no—
161C23×C83—21no—
162C23×C83—24no—
163C23×C83—24no—
164C23×C83—24no—
165C23×C83—42no—
166C23×C83—42no—
167C23×C83—44no—
168C23×C83—44no—
169C23×C83—44no—
170C23×C83—44no—
171C23×C83—48no—
172C23×C83—48no—
173C23×C83—48no—
174C23×C83—42no—
175C23×C83—42no—
176C23×C83—42no—
177C23×C83—42no—
178C23×C83—42no—
179C23×C83—42no—
180C23×C83—84no—
181C23×C83—84no—
182C23×C83—84no—
183C23×C83—84no—
184C23×C83#9912yes—
185C23×C83#10212yes—
186C23×C83#10212yes—
187C23×C83#10412yes—
188C23×C83C2×C22⋊C411yescharacteristic
189C23×C83#12012yes—
190C23×C83C2×D811yescharacteristic
191C23×C83C2×SD1611yescharacteristic
192C23×C83AGL(1,ℤ/8)12yes—
193D4×C83—44no—
194D4×C83—48no—
195D4×C83—44no—
196D4×C83—416no—
197D4×C83—48no—
198D4×C83—48no—
199C2×C4×C83—28no—
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—24no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—44no—
211C2×C4×C83—44no—
212C2×C4×C83—44no—
213C2×C4×C83—44no—
214C2×C4×C83—44no—
215C2×C4×C83—48no—
216C2×C4×C83—42no—
217C2×C4×C83—42no—
218C2×C4×C83—44no—
219C2×C4×C83—44no—
220C2×C4×C83—44no—
221C2×C4×C83—82no—
222C2×C4×C83—82no—
223C2×C4×C83—82no—
224C2×C4×C83—82no—
225C2×C4×C83#9912yes—
226C2×C4×C83C2×C22⋊C412yes—
227C2×C4×C83#12012yes—
228C8×C83—82no—
229C22×C83—21no—
230C22×C83—21no—
231C22×C83—22no—
232C22×C83—22no—
233C22×C83—22no—
234C22×C83—22no—
235C22×C83—21no—
236C22×C83—22no—
237C22×C83—21no—
238C22×C83—22no—
239C22×C83—22no—
240C22×C83—22no—
241C22×C83—22no—
242C22×C83—22no—
243C22×C83—22no—
244C22×C83—22no—
245C22×C83—22no—
246C22×C83—48no—
247C22×C83—48no—
248C22×C83—41no—
249C22×C83—41no—
250C22×C83—41no—
251C22×C83—41no—
252C22×C83—44no—
253C22×C83—44no—
254C22×C83—44no—
255C22×C83—816no—
256C22×C83—82no—
257C22×C83—82no—
258C22×C83—82no—
259C22×C83—82no—
260C22×C83#32714yescomplemented
261C22×C83#37911yescharacteristic
262C22×C83C2×#9911yescharacteristic
263C22×C83C2×#9911yescharacteristic
264C22×C83C2×#10211yescharacteristic
265C22×C83#41811yescharacteristic
266C22×C83C2×#10411yescharacteristic
267C22×C83#42111yescharacteristic
268C22×C83#44712yes—
269C22×C83#45012yes—
270C4×C83—22no—
271C4×C83—22no—
272C4×C83—22no—
273C4×C83—41no—
274C4×C83—44no—
275C4×C83—84no—
276#8328C8311yesmaximal, characteristic, direct product factor, Sylow
277C2×C83—22no—
278C2×C83—24no—
279C2×C83—22no—
280C2×C83—42no—
281C2×C83—84no—
282C2×C83C2×#32711yescharacteristic
283C2×C83#150711yescharacteristic
284C2×C83#151011yescharacteristic
285C2×C83#151911yescharacteristic
286C2×C83#178111yescharacteristic
287C2×C83#178611yescharacteristic
288C2×C83#178911yescharacteristic
289C22×#121C2×C8311yescharacteristic
290#1779C2×C8311yescharacteristic, complemented
291C2×#327C2×C8314yescomplemented
292C22⋊#95C2×C8311yescharacteristic, complemented
293C83#832811yescharacteristic, direct product factor, Sylow
294C23×D4—21no—
295C23×D4C4×C8311yescharacteristic, complemented
296C24×C4C22×C8311yescharacteristic
297C2×#121—21no—
298C2×#121—21no—
299C2×#121—28no—
300C2×#121C4×C8312yescomplemented
301C2×#121C22×C8312yes—
302C22×C4⋊C4C4×C8311yescharacteristic, complemented
303C22×C22⋊C4—21no—
304C22×C22⋊C4—21no—
305C2×C22⋊C8—21no—
306C2×C22⋊C8—21no—
307C2×C22⋊C8C22×C8312yescomplemented
308C2×#95—21no—
309C2×#95C22×C8312yescomplemented
310#327—28no—
311C25—21no—
312C25D4×C8311yescharacteristic
313C22×D4—28no—
314C22×D4—22no—
315C22×D4—22no—
316C22×D4—24no—
317C22×D4—21no—
318C22×D4—42no—
319C22×D4—42no—
320C22×D4—44no—
321C22×D4—42no—
322C22×D4—42no—
323C22×D4C2×C4×C8312yes—
324C23×C4—21no—
325C23×C4—24no—
326C23×C4—22no—
327C23×C4—42no—
328C23×C4—42no—
329C23×C4C2×C4×C8311yescharacteristic
330C23×C4C2×C4×C8311yescharacteristic
331C23×C4D4×C8311yescharacteristic
332C23×C4D4×C8311yescharacteristic
333C23×C4D4×C8311yescharacteristic
334C23×C4C23×C8311yescharacteristic
335C22×C8—22no—
336#121—28no—
337#121—28no—
338#121—24no—
339#121—24no—
340#121—24no—
341#121D4×C8318yes—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4C2×C4×C8312yes—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4—42no—
351C22⋊C8—24no—
352C22⋊C8—42no—
353C22⋊C8—42no—
354#95—24no—
355#95—42no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—41no—
363C24—41no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24C22⋊C4×C8311yescharacteristic
371C24C22⋊C4×C8311yescharacteristic
372C24D8×C8312yes—
373C24SD16×C8312yes—
374C24C2×D4×C8311yescharacteristic
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—416no—
382C2×D4—42no—
383C2×D4—416no—
384C2×D4—416no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4C22⋊C4×C8314yescomplemented
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—416no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C22⋊C4×C8311yescharacteristic
422C22×C4C22⋊C4×C8311yescharacteristic
423C22×C4C22×C4×C8311yescharacteristic
424C22×C4C2×D4×C8311yescharacteristic
425C22×C4C2×D4×C8314yes—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4C22⋊C4×C8314yescomplemented
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—48no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—24no—
444C23—24no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—48no—
461C23—48no—
462C23—48no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23#99×C8312yes—
474C23#102×C8312yes—
475C23#102×C8312yes—
476C23#104×C8312yes—
477C23C2×C22⋊C4×C8311yescharacteristic
478C23#120×C8312yes—
479C23C2×D8×C8311yescharacteristic
480C23C2×SD16×C8311yescharacteristic
481C23AGL(1,ℤ/8)×C8312yes—
482D4—44no—
483D4—48no—
484D4—44no—
485D4—416no—
486D4—48no—
487D4—48no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4#99×C8312yes—
515C2×C4C2×C22⋊C4×C8312yes—
516C2×C4#120×C8312yes—
517C8—82no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—21no—
525C22—22no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—48no—
536C22—48no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—816no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22#327×C8314yescomplemented
550C22#379×C8311yescharacteristic
551C22C2×#99×C8311yescharacteristic
552C22C2×#99×C8311yescharacteristic
553C22C2×#102×C8311yescharacteristic
554C22#418×C8311yescharacteristic
555C22C2×#104×C8311yescharacteristic
556C22#421×C8311yescharacteristic
557C22#447×C8312yes—
558C22#450×C8312yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—41no—
563C4—44no—
564C4—84no—
565C2—22no—
566C2—24no—
567C2—22no—
568C2—42no—
569C2—84no—
570C2C2×#327×C8311yescharacteristic
571C2C83×#150711yescharacteristic
572C2C83×#151011yescharacteristic
573C2C83×#151911yescharacteristic
574C2C83×#178111yescharacteristic
575C2C83×#178611yescharacteristic
576C2C83×#178911yescharacteristic
subgroupG/H
center, socleC23×C83C2×C22⋊C4
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC22×#121×C83C2
LCSG ▷ C22×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. C83×#1779C2semidirect
3. C2×#327×C83C2semidirect
4. C83×C22⋊#95C2semidirect
6. C23×D4×C83C4semidirect
11. C2×#121×C83C4semidirect
13. C22×C4⋊C4×C83C4semidirect
18. C2×C22⋊C8×C83C22semidirect
20. C2×#95×C83C22semidirect
102. C2×D4×C83C22⋊C4semidirect
142. C4⋊C4×C83C22⋊C4semidirect
260. C22×C83#327semidirect
276. #8328C83direct (N × H)
290. #1779C2×C83semidirect
291. C2×#327C2×C83semidirect
292. C22⋊#95C2×C83semidirect
293. C83#8328direct (N × H)
295. C23×D4C4×C83semidirect
300. C2×#121C4×C83semidirect
302. C22×C4⋊C4C4×C83semidirect
307. C2×C22⋊C8C22×C83semidirect
309. C2×#95C22×C83semidirect
391. C2×D4C22⋊C4×C83semidirect
431. C4⋊C4C22⋊C4×C83semidirect
549. C22#327×C83semidirect

Maximal subgroups

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

subgroupindex
C22×#121×C832
C83×#17792
C2×#327×C832
C83×C22⋊#952
#832883