Finite Group Databasebeta

← #35422765

Subgroups — #35422765

Order 21248.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127801C211yesmaximal, characteristic
2C2×#19127801C211yesmaximal, characteristic
3C2×C4×#13050578C211yesmaximal, characteristic
4C4⋊C4×Dic166C211yesmaximal, characteristic, complemented
5#24908198C211yesmaximal, characteristic, complemented
6#24908193C211yesmaximal, characteristic, complemented
7#24907730C211yesmaximal, characteristic, complemented
8#24907724C211yesmaximal, characteristic
9#24907716C211yesmaximal, characteristic, complemented
10#24907710C211yesmaximal, characteristic
11#24907632C211yesmaximal, characteristic, complemented
12#24907475C211yesmaximal, characteristic, complemented
13#24907463C211yesmaximal, characteristic
14#24907338C211yesmaximal, characteristic, complemented
15C83×#2349C211yesmaximal, characteristic
16C22×#13050578C2211yescharacteristic
17C2×C4×Dic166C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19C2×C4×Dic166C2211yescharacteristic
20#19127801—24no—
21#19127801C418yescomplemented
22C4×#13050578—24no—
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581725C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581715C2211yescharacteristic
30C2×#14581715C2211yescharacteristic
31C2×#14581715C2211yescharacteristic
32C2×#14581713C2211yescharacteristic
33C2×#14581713C2211yescharacteristic
34C2×#14581713C2211yescharacteristic
35C2×#14581703C2211yescharacteristic
36C2×#14581703C2211yescharacteristic
37C2×C4×C83⋊C8C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127598—21no—
41#19127598—21no—
42#19127571C2211yescharacteristic
43#19127570—21no—
44C4⋊C4×Dic83—21no—
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127456C2211yescharacteristic
50#19127456C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52C2×C4×D4×C83C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56#385×C83—21no—
57#385×C83—21no—
58#382×C83C2211yescharacteristic
59#380×C83—21no—
60C4×C4⋊C4×C83C2211yescharacteristic
61C2×#95×C83—21no—
62C22×Dic166C2311yescharacteristic
63C2×#13050578C2×C414yes—
64C2×#13050578D414yes—
65C2×#13050578Q814yes—
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70C4×Dic166—22no—
71C4×Dic166—22no—
72C4×Dic166—22no—
73C4×Dic166—41no—
74C4×Dic166—41no—
75C4×Dic166C2×C414yescomplemented
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83—21no—
89#14581725—22no—
90#14581725—41no—
91#14581725C2×C414yescomplemented
92#14581723—22no—
93#14581723—41no—
94C4⋊(C83⋊C8)—22no—
95#14581715—22no—
96#14581715—22no—
97#14581715C2×C414yescomplemented
98#14581713—22no—
99#14581713—22no—
100#14581713C2×C414yes—
101#14581703—22no—
102#14581703C2×C414yescomplemented
103C4×C83⋊C8—22no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C22×D4×C83C2311yescharacteristic
107C23×C4×C83—21no—
108C23×C4×C83—21no—
109C23×C4×C83—21no—
110C4×D4×C83—22no—
111C4×D4×C83—22no—
112C4×D4×C83—22no—
113C4×D4×C83—41no—
114C4×D4×C83—41no—
115C4×D4×C83C2×C414yes—
116C2×C4⋊C4×C83—21no—
117C2×C4⋊C4×C83—21no—
118C2×C4⋊C4×C83D411yescharacteristic
119C2×C4⋊C4×C83D411yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C4⋊C4×C83C2311yescharacteristic
123C2×C22⋊C4×C83—21no—
124C2×C22⋊C4×C83—21no—
125C2×C22⋊C4×C83—21no—
126C2×C42×C83C2311yescharacteristic
127C2×C42×C83C2311yescharacteristic
128C2×C42×C83C2311yescharacteristic
129#95×C83—21no—
130#95×C83—21no—
131#95×C83—21no—
132#95×C83—21no—
133#95×C83D411yescharacteristic
134#95×C83D411yescharacteristic
135C2×Dic166—21no—
136C2×Dic166—21no—
137C2×Dic166—21no—
138C2×Dic166C22×C412yes—
139C2×Dic166C2×D412yes—
140C2×Dic166C2×Q812yes—
141C22×Dic83—21no—
142C22×Dic83—21no—
143C22×Dic83—21no—
144C22×Dic83—21no—
145#13050578C4⋊C4116yescomplemented
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8C22×C412yes—
155C2×C83⋊C8C22×C412yes—
156C2×C83⋊C8C2×D412yes—
157C2×C83⋊C8C2×Q812yes—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—42no—
164C4⋊Dic83—81no—
165C4⋊Dic83C22×C412yes—
166Dic83⋊C4—24no—
167Dic83⋊C4—42no—
168Dic83⋊C4—42no—
169Dic83⋊C4—81no—
170Dic83⋊C4—81no—
171C4×Dic83—24no—
172C4×Dic83—42no—
173C4×Dic83—42no—
174C24×C83—21no—
175C2×D4×C83—21no—
176C2×D4×C83—21no—
177C2×D4×C83—21no—
178C2×D4×C83C22×C412yes—
179C2×D4×C83C2×D412yes—
180C2×D4×C83C2×Q812yes—
181C22×C4×C83—21no—
182C22×C4×C83—24no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—42no—
186C22×C4×C83—42no—
187C22×C4×C83—42no—
188C22×C4×C83—42no—
189C22×C4×C83—81no—
190C22×C4×C83—81no—
191C22×C4×C83—81no—
192C22×C4×C83—81no—
193C22×C4×C83C2×D411yescharacteristic
194C22×C4×C83C2×D411yescharacteristic
195C22×C4×C83C2×D411yescharacteristic
196C22×C4×C83C2×D411yescharacteristic
197C22×C4×C83C2×D411yescharacteristic
198C22×C4×C83C2×D411yescharacteristic
199C22×C4×C83C2411yescharacteristic
200C4⋊C4×C83—22no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—21no—
206C4⋊C4×C83—41no—
207C4⋊C4×C83—81no—
208C4⋊C4×C83—81no—
209C4⋊C4×C83C22×C412yes—
210C22⋊C4×C83—24no—
211C22⋊C4×C83—42no—
212C22⋊C4×C83—42no—
213C42×C83—21no—
214C42×C83—21no—
215C42×C83—21no—
216C42×C83—21no—
217C42×C83C22×C412yes—
218C42×C83#4112yes—
219C23×C83—22no—
220C23×C83—22no—
221C23×C83—22no—
222C23×C83—41no—
223C23×C83—41no—
224C23×C83—41no—
225C23×C83—41no—
226C23×C83C22×D411yescharacteristic
227D4×C83—22no—
228D4×C83—22no—
229D4×C83—22no—
230D4×C83C2×C4⋊C414yes—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—22no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—21no—
246C2×C4×C83—21no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83—41no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83—84no—
256C2×C4×C83—84no—
257C2×C4×C83—84no—
258C2×C4×C83C4×D412yes—
259C2×C4×C83C4×D412yes—
260C2×C4×C83#13712yes—
261C2×C4×C83C23×C411yescharacteristic
262C2×C4×C83C22×D411yescharacteristic
263C2×C4×C83C22×Q811yescharacteristic
264C2×C4×C83C2×#4111yescharacteristic
265C2×C4×C83H(5,𝔽2)11yescharacteristic
266C2×C4×C832−1+411yescharacteristic
267C2×Dic83—22no—
268C2×Dic83—22no—
269C2×Dic83—22no—
270C2×Dic83—22no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275C2×Dic83—41no—
276C2×Dic83—41no—
277C2×Dic83—41no—
278C2×Dic83—41no—
279Dic166—22no—
280Dic166—22no—
281Dic166—22no—
282Dic166C2×C4⋊C414yes—
283C83⋊C8—22no—
284C83⋊C8—41no—
285C83⋊C8—41no—
286C83⋊C8C2×C4⋊C414yes—
287C22×C83—24no—
288C22×C83—42no—
289C22×C83—42no—
290C22×C83—42no—
291C22×C83—42no—
292C22×C83—42no—
293C22×C83—42no—
294C22×C83C2×C4×D411yescharacteristic
295C22×C83D4211yescharacteristic
296C22×C83D4×Q811yescharacteristic
297C22×C83C2×#13711yescharacteristic
298C22×C83#57511yescharacteristic
299C22×C83#57611yescharacteristic
300C22×C83#57711yescharacteristic
301C4×C83—22no—
302C4×C83—22no—
303C4×C83—22no—
304C4×C83—42no—
305C4×C83—41no—
306C4×C83—41no—
307C4×C83—41no—
308C4×C83—81no—
309C4×C83—81no—
310C4×C83C22×C4⋊C411yescharacteristic
311C4×C83#52011yescharacteristic
312C4×C83#54711yescharacteristic
313C4×C83#55411yescharacteristic
314Dic83—24no—
315Dic83—42no—
316Dic83—81no—
317Dic83—81no—
318#22678—831nomaximal, Sylow
319C2×C83—44no—
320C2×C83D4×C4⋊C411yescharacteristic
321C2×C83C4×#13711yescharacteristic
322C2×C83#294611yescharacteristic
323C2×C83#326611yescharacteristic
324C2×C83#326811yescharacteristic
325C2×C83#338511yescharacteristic
326C2×C83#338811yescharacteristic
327C2×#440—831no—
328C2×#440—831no—
329C2×C4×SD16—831no—
330Q8×C4⋊C4—831no—
331#2349D8311yescharacteristic
332#1943—831no—
333#1942—831no—
334#1865—831no—
335#1864—831no—
336#1863—831no—
337#1862—831no—
338#1847—831no—
339#1805—831no—
340#1802—831no—
341#1771—831no—
342C83#2267811yescharacteristic, complemented, Sylow
343C22×SD16—831no—
344C2×C4×Q8—831no—
345C2×C4×Q8—831no—
346C2×C4×Q8—831no—
347C2×C4×D4C2×D8311yescharacteristic
348C2×C4×D4C2×D8311yescharacteristic
349C2×C4×D4C2×D8311yescharacteristic
350#440—838no—
351#440—1664no—
352C4×SD16—1664no—
353C2×#107—831no—
354C2×#107—831no—
355C2×#107—831no—
356C2×#106—831no—
357C2×#106—831no—
358C2×C4⋊C8—831no—
359C2×#103—831no—
360C2×#103—831no—
361C2×#103—831no—
362C2×#102—831no—
363C2×#102—831no—
364C2×#102—831no—
365C2×#97—831no—
366C2×#97—831no—
367C2×C4×C8—831no—
368#389—831no—
369#389—831no—
370#389—1661no—
371#389—1661no—
372#389C2×D8311yescharacteristic
373#389C2×D8311yescharacteristic
374#385—21no—
375#385—21no—
376#384—831no—
377#384—1661no—
378#382C2×D8311yescharacteristic
379#380—21no—
380C4×C4⋊C4—1661no—
381C4×C4⋊C4C2×D8311yescharacteristic
382C2×#95—21no—
383#340—831no—
384#340—831no—
385#340—831no—
386#340—831no—
387#336—831no—
388#336—831no—
389C22×Q8—831no—
390C22×D4C22×D8311yescharacteristic
391C23×C4—21no—
392C23×C4—21no—
393C23×C4—21no—
394C2×SD16—834no—
395C2×SD16—834no—
396C2×SD16—834no—
397C22×C8—831no—
398C22×C8—831no—
399C22×C8—831no—
400C22×C8—831no—
401C4×Q8—3321no—
402C4×Q8—3321no—
403C4×Q8—834no—
404C4×Q8—1662no—
405C4×Q8—1662no—
406C4×Q8—1662no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—41no—
411C4×D4—41no—
412C4×D4C4×D8314yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—831no—
416C2×C4⋊C4—831no—
417C2×C4⋊C4—831no—
418C2×C4⋊C4—1661no—
419C2×C4⋊C4—1661no—
420C2×C4⋊C4—1661no—
421C2×C4⋊C4—1661no—
422C2×C4⋊C4—1661no—
423C2×C4⋊C4—1661no—
424C2×C4⋊C4—1661no—
425C2×C4⋊C4C22⋊D8311yescharacteristic
426C2×C4⋊C4C22⋊D8311yescharacteristic
427C2×C4⋊C4C22×D8311yescharacteristic
428C2×C4⋊C4C22×D8311yescharacteristic
429C2×C4⋊C4C22×D8311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C42—1661no—
434C2×C42—1661no—
435C2×C42—1661no—
436C2×C42C22×D8311yescharacteristic
437C2×C42C22×D8311yescharacteristic
438C2×C42C22×D8311yescharacteristic
439#107—3321no—
440#107—834no—
441#107—1662no—
442#106—3321no—
443#106—1662no—
444C4⋊C8—1662no—
445#103—834no—
446#103—1662no—
447#103—1662no—
448#102—834no—
449#102—1662no—
450#102—1662no—
451#97—834no—
452#97—1662no—
453C4×C8—1662no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—1661no—
459#95—1661no—
460#95C22⋊D8311yescharacteristic
461#95C22⋊D8311yescharacteristic
462C24—21no—
463C2×Q8—832no—
464C2×Q8—832no—
465C2×Q8—832no—
466C2×Q8—1661no—
467C2×Q8—1661no—
468C2×Q8—1661no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D8312yes—
473C2×D4D4×D8312yes—
474C2×D4Q8×D8312yes—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—81no—
484C22×C4—81no—
485C22×C4—81no—
486C22×C4—81no—
487C22×C4—1661no—
488C22×C4—1661no—
489C22×C4—1661no—
490C22×C4—1661no—
491C22×C4C2×C22⋊D8311yescharacteristic
492C22×C4C2×C22⋊D8311yescharacteristic
493C22×C4C2×C22⋊D8311yescharacteristic
494C22×C4C2×C22⋊D8311yescharacteristic
495C22×C4C2×C22⋊D8311yescharacteristic
496C22×C4C2×C22⋊D8311yescharacteristic
497C22×C4C23×D8311yescharacteristic
498SD16—8316no—
499C2×C8—832no—
500C2×C8—832no—
501C2×C8—832no—
502C2×C8—832no—
503C2×C8—1661no—
504C2×C8—1661no—
505C2×C8—1661no—
506C2×C8—1661no—
507C2×C8—1661no—
508C2×C8—1661no—
509C2×C8—1661no—
510C2×C8—1661no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—41no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—3322no—
521C4⋊C4—3322no—
522C4⋊C4—3322no—
523C4⋊C4—832no—
524C4⋊C4—6641no—
525C4⋊C4—6641no—
526C4⋊C4—6641no—
527C4⋊C4—1661no—
528C4⋊C4—1661no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1664no—
533C4⋊C4C2×C4×D8312yes—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—3322no—
542C42—3322no—
543C42—1664no—
544C42C2×C4×D8312yes—
545C42#1305059012yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C22×C22⋊D8311yescharacteristic
554Q8—834no—
555Q8—1662no—
556Q8—1662no—
557Q8—1662no—
558D4—22no—
559D4—22no—
560D4—22no—
561D4C4⋊C4×D8314yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—3321no—
590C2×C4—3321no—
591C2×C4—3321no—
592C2×C4—3321no—
593C2×C4—3321no—
594C2×C4—3321no—
595C2×C4—3321no—
596C2×C4—3321no—
597C2×C4—1662no—
598C2×C4—1662no—
599C2×C4—1662no—
600C2×C4—1662no—
601C2×C4C4×C22⋊D8312yes—
602C2×C4C4×C22⋊D8312yes—
603C2×C4#1458182212yes—
604C2×C4C22×C4×D8311yescharacteristic
605C2×C4C2×D4×D8311yescharacteristic
606C2×C4C2×Q8×D8311yescharacteristic
607C2×C4C2×#1305059011yescharacteristic
608C2×C4#1458183711yescharacteristic
609C2×C4#1458184011yescharacteristic
610C8—3321no—
611C8—3321no—
612C8—834no—
613C8—1662no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C2×C4×C22⋊D8311yescharacteristic
622C22D4×C22⋊D8311yescharacteristic
623C22Q8×C22⋊D8311yescharacteristic
624C22C2×#1458182211yescharacteristic
625C22#1912849911yescharacteristic
626C22#1912850411yescharacteristic
627C22#1912851211yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—81no—
636C4—81no—
637C4—3322no—
638C4—6641no—
639C4—6641no—
640C4—1664no—
641C4C2×C4⋊C4×D8311yescharacteristic
642C4#1912818211yescharacteristic
643C4#1912835211yescharacteristic
644C4#1912840611yescharacteristic
645C2—44no—
646C2C4⋊C4×C22⋊D8311yescharacteristic
647C2C4×#1458182211yescharacteristic
648C2#2491629511yescharacteristic
649C2#2491896911yescharacteristic
650C2#2491899911yescharacteristic
651C2#2492037311yescharacteristic
652C2#2492040811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2349C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenter#2349D83
hypocenterC83#22678
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Dic166C2semidirect
5. #24908198C2semidirect
6. #24908193C2semidirect
7. #24907730C2semidirect
9. #24907716C2semidirect
11. #24907632C2semidirect
12. #24907475C2semidirect
14. #24907338C2semidirect
21. #19127801C4semidirect
75. C4×Dic166C2×C4semidirect
91. #14581725C2×C4semidirect
97. #14581715C2×C4semidirect
102. #14581703C2×C4semidirect
145. #13050578C4⋊C4semidirect
342. C83#22678semidirect

Maximal subgroups

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

subgroupindex
C2×#191278012
C2×C4×#130505782
C4⋊C4×Dic1662
#249081982
#249081932
#249077302
#249077242
#249077162
#249077102
#249076322
#249074752
#249074632
#249073382
C83×#23492
#2267883