Finite Group Databasebeta

← #35328119

Subgroups — #35328119

Order 21248.

2006 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127574C211yesmaximal, characteristic
2C2×#19127574C211yesmaximal, characteristic
3C2×#19127574C211yesmaximal, characteristic
4C2×C4×C22⋊Dic83C211yesmaximal, characteristic
5#24905718C212yesmaximal, complemented
6#24905714C212yesmaximal, complemented
7#24905698C212yesmaximal, complemented
8Dic83⋊#95C212yesmaximal, complemented
9#24905686C211yesmaximal, characteristic
10#24905686C211yesmaximal, characteristic
11C22×#95×C83C211yesmaximal, characteristic
12C22×Dic83⋊C4C2211yescharacteristic
13C22×Dic83⋊C4C2212yes—
14C22×Dic83⋊C4C2211yescharacteristic
15C22×Dic83⋊C4C2212yes—
16C22×C22⋊Dic83C2211yescharacteristic
17C22×C22⋊Dic83C2211yescharacteristic
18C22×C22⋊Dic83C2211yescharacteristic
19C22×C22⋊Dic83C2211yescharacteristic
20C22×C4×Dic83C2212yes—
21#19127574—24no—
22#19127574—24no—
23#19127574C418yescomplemented
24C4×C22⋊Dic83—24no—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2211yescharacteristic
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34C24×C4×C83C2211yescharacteristic
35C24×C4×C83C2211yescharacteristic
36C24×C4×C83C2211yescharacteristic
37C2×#95×C83—24no—
38C2×#95×C83C2212yes—
39C2×#95×C83C2212yes—
40C23×Dic83C2311yescharacteristic
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2311yescharacteristic
44C23×Dic83C2312yes—
45C23×Dic83C2312yes—
46C2×Dic83⋊C4—24no—
47C2×Dic83⋊C4—22no—
48C2×Dic83⋊C4—22no—
49C2×Dic83⋊C4—24no—
50C2×Dic83⋊C4—24no—
51C2×Dic83⋊C4—42no—
52C2×Dic83⋊C4—42no—
53C2×Dic83⋊C4—42no—
54C2×Dic83⋊C4—42no—
55C2×Dic83⋊C4C2×C418yescomplemented
56C2×C22⋊Dic83—22no—
57C2×C22⋊Dic83—22no—
58C2×C22⋊Dic83—22no—
59C2×C22⋊Dic83—22no—
60C2×C22⋊Dic83—22no—
61C2×C22⋊Dic83—22no—
62C2×C22⋊Dic83—22no—
63C2×C22⋊Dic83C2×C414yes—
64C2×C22⋊Dic83C2×C414yes—
65C2×C22⋊Dic83D414yes—
66C2×C22⋊Dic83Q814yes—
67C2×C22⋊Dic83Q814yes—
68C2×C4×Dic83—24no—
69C2×C4×Dic83—24no—
70C2×C4×Dic83—42no—
71C83⋊#95—24no—
72C83⋊#95—24no—
73C83⋊#95—22no—
74C83⋊#95—22no—
75C83⋊#95—24no—
76C83⋊#95—42no—
77C83⋊#95—42no—
78C83⋊#95—42no—
79C83⋊#95C2×C418yescomplemented
80C25×C83C2311yescharacteristic
81C23×C4×C83—22no—
82C23×C4×C83—22no—
83C23×C4×C83—22no—
84C23×C4×C83—22no—
85C23×C4×C83—22no—
86C23×C4×C83—24no—
87C23×C4×C83—24no—
88C23×C4×C83—22no—
89C23×C4×C83—22no—
90C23×C4×C83—42no—
91C23×C4×C83—42no—
92C23×C4×C83—42no—
93C23×C4×C83—42no—
94C23×C4×C83C2×C414yes—
95C23×C4×C83D414yes—
96C23×C4×C83C2312yes—
97C23×C4×C83C2311yescharacteristic
98C23×C4×C83C2312yes—
99C23×C4×C83C2311yescharacteristic
100#95×C83—24no—
101#95×C83—24no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—21no—
108C22×Dic83—21no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—21no—
113C22×Dic83—21no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—22no—
129C22×Dic83—22no—
130C22×Dic83C22×C412yes—
131C22×Dic83C22×C412yes—
132C22×Dic83C22×C414yes—
133C22×Dic83C2×D414yes—
134C22×Dic83C2×Q814yes—
135C22×Dic83C2×Q814yes—
136Dic83⋊C4—48no—
137Dic83⋊C4—48no—
138Dic83⋊C4—44no—
139Dic83⋊C4—44no—
140C22⋊Dic83—28no—
141C22⋊Dic83—28no—
142C22⋊Dic83—44no—
143C22⋊Dic83C4⋊C4116yescomplemented
144C4×Dic83—48no—
145C24×C83—22no—
146C24×C83—22no—
147C24×C83—22no—
148C24×C83—22no—
149C24×C83C22×C412yes—
150C24×C83C2×D412yes—
151C24×C83C2×D412yes—
152C24×C83C2×Q812yes—
153C24×C83C2×Q812yes—
154C24×C83#4112yes—
155C24×C83#4112yes—
156C24×C83C2411yescharacteristic
157C22×C4×C83—21no—
158C22×C4×C83—21no—
159C22×C4×C83—22no—
160C22×C4×C83—28no—
161C22×C4×C83—28no—
162C22×C4×C83—28no—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—21no—
166C22×C4×C83—22no—
167C22×C4×C83—21no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83—22no—
178C22×C4×C83—22no—
179C22×C4×C83—28no—
180C22×C4×C83—48no—
181C22×C4×C83—44no—
182C22×C4×C83—48no—
183C22×C4×C83—48no—
184C22×C4×C83—48no—
185C22×C4×C83—48no—
186C22×C4×C83—44no—
187C22×C4×C83—44no—
188C22×C4×C83—44no—
189C22×C4×C83—48no—
190C22×C4×C83—44no—
191C22×C4×C83C22×C414yes—
192C22×C4×C83C2×D412yes—
193C22×C4×C83C2×D412yes—
194C22×C4×C83#4114yes—
195C22×C4×C83#4114yes—
196C23×C83—24no—
197C23×C83—24no—
198C23×C83—22no—
199C23×C83—22no—
200C23×C83—24no—
201C23×C83—22no—
202C23×C83—22no—
203C23×C83—24no—
204C23×C83—24no—
205C23×C83—24no—
206C23×C83—24no—
207C23×C83—24no—
208C23×C83—24no—
209C23×C83—24no—
210C23×C83—24no—
211C23×C83—24no—
212C23×C83C2×C4⋊C414yes—
213C23×C83C4×D414yes—
214C23×C83C4×Q814yes—
215C23×C83#12114yes—
216C23×C83C22⋊Q814yes—
217C23×C83#12514yes—
218C23×C83C4⋊Q814yes—
219C23×C83C23×C411yescharacteristic
220C23×C83C22×D411yescharacteristic
221C23×C83C22×D411yescharacteristic
222C23×C83C22×Q811yescharacteristic
223C23×C83C22×Q811yescharacteristic
224C23×C83C2×#4111yescharacteristic
225C23×C83C2×#4111yescharacteristic
226C23×C83C2×#4112yes—
227C23×C83C2×#4112yes—
228C23×C83H(5,𝔽2)12yes—
229C23×C83H(5,𝔽2)12yes—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—24no—
236C2×C4×C83—24no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—24no—
240C2×C4×C83—22no—
241C2×C4×C83—24no—
242C2×C4×C83—24no—
243C2×C4×C83—416no—
244C2×C4×C83—416no—
245C2×C4×C83—416no—
246C2×C4×C83—42no—
247C2×C4×C83—42no—
248C2×C4×C83—42no—
249C2×C4×C83—42no—
250C2×C4×C83—42no—
251C2×C4×C83—42no—
252C2×C4×C83—42no—
253C2×C4×C83—42no—
254C2×C4×C83—42no—
255C2×C4×C83—42no—
256C2×C4×C83#12314yes—
257C2×Dic83—24no—
258C2×Dic83—24no—
259C2×Dic83—24no—
260C2×Dic83—24no—
261C2×Dic83—24no—
262C2×Dic83—24no—
263C2×Dic83—22no—
264C2×Dic83—22no—
265C2×Dic83—24no—
266C2×Dic83—24no—
267C2×Dic83—24no—
268C2×Dic83—24no—
269C2×Dic83—24no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—42no—
273C2×Dic83—42no—
274C2×Dic83—41no—
275C2×Dic83—41no—
276C2×Dic83—42no—
277C2×Dic83—42no—
278C2×Dic83—42no—
279C2×Dic83—42no—
280C2×Dic83—41no—
281C2×Dic83—41no—
282C2×Dic83—41no—
283C2×Dic83—41no—
284C2×Dic83—41no—
285C2×Dic83—41no—
286C2×Dic83—42no—
287C2×Dic83—42no—
288C2×Dic83—42no—
289C2×Dic83—42no—
290C2×Dic83C2×C4⋊C418yes—
291C22×C83—24no—
292C22×C83—24no—
293C22×C83—28no—
294C22×C83—28no—
295C22×C83—24no—
296C22×C83—24no—
297C22×C83—28no—
298C22×C83—24no—
299C22×C83—28no—
300C22×C83—24no—
301C22×C83#38918yes—
302C22×C83C22×C4⋊C411yescharacteristic
303C22×C83C2×C4×D411yescharacteristic
304C22×C83C2×C4×Q811yescharacteristic
305C22×C83C4×#4112yes—
306C22×C83#51812yes—
307C22×C83C2×#12111yescharacteristic
308C22×C83C2×C22⋊Q811yescharacteristic
309C22×C83C2×#12311yescharacteristic
310C22×C83#52511yescharacteristic
311C22×C83C2×#12511yescharacteristic
312C22×C83#52912yes—
313C22×C83C2×C4⋊Q811yescharacteristic
314C22×C83#53411yescharacteristic
315C22×C83#53511yescharacteristic
316C22×C83#53812yes—
317C22×C83#54612yes—
318C22×C83#54612yes—
319C22×C83#55112yes—
320C22×C83#55112yes—
321C22×C83#55412yes—
322C22×C83#55412yes—
323C22×C83#55412yes—
324C22×C83#55412yes—
325C4×C83—24no—
326C4×C83—44no—
327C4×C83—44no—
328C4×C83—42no—
329Dic83—28no—
330Dic83—42no—
331Dic83—44no—
332Dic83—42no—
333Dic83—81no—
334Dic83—81no—
335#14044—831nomaximal, Sylow
336C2×C83—28no—
337C2×C83C2×#38911yescharacteristic
338C2×C83C4×#12311yescharacteristic
339C2×C83#232111yescharacteristic
340C2×C83#234912yes—
341C2×C83#236812yes—
342C2×C83#241111yescharacteristic
343C2×C83#241211yescharacteristic
344C2×C83#245012yes—
345C2×C83#250211yescharacteristic
346C2×C83#250512yes—
347C2×C83#258311yescharacteristic
348C2×#385—831no—
349C2×#385—831no—
350C2×#385—831no—
351C2×C4×C22⋊C4—831no—
352C22×#95D8311yescharacteristic
353#1442—832no—
354#1441—832no—
355#1436—832no—
356C4⋊#95—832no—
357C22⋊#95—831no—
358C22⋊#95—831no—
359C83#1404411yescharacteristic, complemented, Sylow
360C24×C4C2×D8311yescharacteristic
361C24×C4C2×D8311yescharacteristic
362C24×C4C2×D8311yescharacteristic
363C22×C4⋊C4—832no—
364C22×C4⋊C4—831no—
365C22×C4⋊C4—832no—
366C22×C4⋊C4—831no—
367C22×C22⋊C4—831no—
368C22×C22⋊C4—831no—
369C22×C22⋊C4—831no—
370C22×C22⋊C4—831no—
371C22×C42—832no—
372#385—838no—
373#385—1664no—
374#385—1664no—
375C4×C22⋊C4—1664no—
376C2×#95—24no—
377C2×#95—832no—
378C2×#95—832no—
379C2×#95—832no—
380C2×#95—832no—
381C2×#95—832no—
382C2×#95—831no—
383C2×#95—832no—
384C2×#95—832no—
385C2×#95—831no—
386C2×#95C2×D8312yes—
387C2×#95C2×D8312yes—
388C25C22×D8311yescharacteristic
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—22no—
394C23×C4—24no—
395C23×C4—24no—
396C23×C4—22no—
397C23×C4—22no—
398C23×C4—42no—
399C23×C4—42no—
400C23×C4—42no—
401C23×C4—42no—
402C23×C4—832no—
403C23×C4—831no—
404C23×C4—831no—
405C23×C4—832no—
406C23×C4—831no—
407C23×C4—831no—
408C23×C4C4×D8314yes—
409C23×C4C22⋊D8314yes—
410C23×C4C22×D8312yes—
411C23×C4C22×D8311yescharacteristic
412C23×C4C22×D8312yes—
413C23×C4C22×D8311yescharacteristic
414C2×C4⋊C4—3322no—
415C2×C4⋊C4—3322no—
416C2×C4⋊C4—3322no—
417C2×C4⋊C4—3322no—
418C2×C4⋊C4—838no—
419C2×C4⋊C4—1664no—
420C2×C4⋊C4—1662no—
421C2×C4⋊C4—1662no—
422C2×C4⋊C4—1664no—
423C2×C4⋊C4—1664no—
424C2×C22⋊C4—834no—
425C2×C22⋊C4—834no—
426C2×C22⋊C4—834no—
427C2×C22⋊C4—834no—
428C2×C22⋊C4—834no—
429C2×C22⋊C4—1662no—
430C2×C22⋊C4—1662no—
431C2×C22⋊C4—1662no—
432C2×C22⋊C4—1662no—
433C2×C22⋊C4—1662no—
434C2×C22⋊C4—1662no—
435C2×C22⋊C4—1662no—
436C2×C42—3322no—
437C2×C42—1664no—
438C2×C42—1664no—
439#95—24no—
440#95—24no—
441#95—3322no—
442#95—3322no—
443#95—3322no—
444#95—838no—
445#95—1664no—
446#95—1664no—
447#95—1662no—
448#95—1662no—
449#95—1664no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24C2×C4×D8312yes—
455C24D4×D8312yes—
456C24C2×C22⋊D8312yes—
457C24Q8×D8312yes—
458C24Q8×D8312yes—
459C24#1305058912yes—
460C24#1305059012yes—
461C24C23×D8311yescharacteristic
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—44no—
496C22×C4—834no—
497C22×C4—834no—
498C22×C4—834no—
499C22×C4—832no—
500C22×C4—832no—
501C22×C4—834no—
502C22×C4—1661no—
503C22×C4—1661no—
504C22×C4—1661no—
505C22×C4—1661no—
506C22×C4—1661no—
507C22×C4—1661no—
508C22×C4—1662no—
509C22×C4—1662no—
510C22×C4—1662no—
511C22×C4—1662no—
512C22×C4—1662no—
513C22×C4—1661no—
514C22×C4—1661no—
515C22×C4—1662no—
516C22×C4—1662no—
517C22×C4—1662no—
518C22×C4—1661no—
519C22×C4—1662no—
520C22×C4—1661no—
521C22×C4—1662no—
522C22×C4—1662no—
523C22×C4—1662no—
524C22×C4—1662no—
525C22×C4—1662no—
526C22×C4—1662no—
527C22×C4—1662no—
528C22×C4—1662no—
529C22×C4—1662no—
530C22×C4C2×C4×D8314yes—
531C22×C4C2×C22⋊D8312yes—
532C22×C4C2×C22⋊D8312yes—
533C22×C4#1305059014yes—
534C22×C4#1305059014yes—
535C4⋊C4—3324no—
536C4⋊C4—3328no—
537C4⋊C4—3328no—
538C4⋊C4—3324no—
539C22⋊C4—3324no—
540C22⋊C4—8316no—
541C22⋊C4—1668no—
542C22⋊C4—1668no—
543C42—3328no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23C4⋊C4×D8314yes—
561C23C4×C22⋊D8314yes—
562C23#1458175714yes—
563C23#1458176414yes—
564C23#1458177014yes—
565C23#1458178214yes—
566C23#1458179014yes—
567C23C22×C4×D8311yescharacteristic
568C23C2×D4×D8311yescharacteristic
569C23C22×C22⋊D8311yescharacteristic
570C23C2×Q8×D8311yescharacteristic
571C23C2×Q8×D8311yescharacteristic
572C23C2×#1305058911yescharacteristic
573C23C2×#1305059012yes—
574C23C2×#1305059012yes—
575C23C2×#1305059011yescharacteristic
576C23#1458183712yes—
577C23#1458183712yes—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—3322no—
605C2×C4—3322no—
606C2×C4—3321no—
607C2×C4—3321no—
608C2×C4—3322no—
609C2×C4—3322no—
610C2×C4—3321no—
611C2×C4—3321no—
612C2×C4—3322no—
613C2×C4—3322no—
614C2×C4—3322no—
615C2×C4—3322no—
616C2×C4—3322no—
617C2×C4—3322no—
618C2×C4—3321no—
619C2×C4—3321no—
620C2×C4—3321no—
621C2×C4—3321no—
622C2×C4—3321no—
623C2×C4—3321no—
624C2×C4—838no—
625C2×C4—1664no—
626C2×C4—1664no—
627C2×C4—1662no—
628C2×C4—1662no—
629C2×C4—1664no—
630C2×C4—1664no—
631C2×C4—1664no—
632C2×C4—1664no—
633C2×C4—1664no—
634C2×C4—1664no—
635C2×C4—1664no—
636C2×C4—1664no—
637C2×C4—1664no—
638C2×C4#1458177514yes—
639C22—24no—
640C22—24no—
641C22—28no—
642C22—24no—
643C22—24no—
644C22—28no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—28no—
649C22#1912759618yes—
650C22C2×C4⋊C4×D8311yescharacteristic
651C22C2×C4×C22⋊D8311yescharacteristic
652C22C2×#1458175711yescharacteristic
653C22C4×#1305059012yes—
654C22#1912817412yes—
655C22C2×#1458176411yescharacteristic
656C22C2×#1458177011yescharacteristic
657C22C2×#1458177511yescharacteristic
658C22#1912821211yescharacteristic
659C22C2×#1458178211yescharacteristic
660C22#1912823512yes—
661C22C2×#1458179011yescharacteristic
662C22#1912826011yescharacteristic
663C22#1912826811yescharacteristic
664C22#1912828912yes—
665C22#1912834112yes—
666C22#1912834112yes—
667C22#1912837712yes—
668C22#1912837712yes—
669C22#1912840612yes—
670C22#1912840612yes—
671C22#1912840612yes—
672C22#1912840612yes—
673C4—24no—
674C4—44no—
675C4—42no—
676C4—44no—
677C4—3322no—
678C4—3322no—
679C4—3324no—
680C4—6641no—
681C4—6641no—
682C4—1668no—
683C2—28no—
684C2C2×#1912759611yescharacteristic
685C2C4×#1458177511yescharacteristic
686C2#2491012111yescharacteristic
687C2#2491037012yes—
688C2#2491054312yes—
689C2#2491084811yescharacteristic
690C2#2491085911yescharacteristic
691C2#2491128712yes—
692C2#2491193711yescharacteristic
693C2#2491196412yes—
694C2#2491284311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC22×#95×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC22×#95D83
hypocenterC83#14044
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
5. #24905718C2semidirect
6. #24905714C2semidirect
7. #24905698C2semidirect
8. Dic83⋊#95C2semidirect
23. #19127574C4semidirect
55. C2×Dic83⋊C4C2×C4semidirect
79. C83⋊#95C2×C4semidirect
143. C22⋊Dic83C4⋊C4semidirect
359. C83#14044semidirect

Maximal subgroups

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

subgroupindex
C2×#191275742
C2×C4×C22⋊Dic832
#249057182
#249057142
#249056982
Dic83⋊#952
#249056862
C22×#95×C832
#1404483