Finite Group Databasebeta

← #35467430

Subgroups — #35467430

Order 21248.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127986C211yesmaximal, characteristic, complemented
2C2×C4×#13050577C211yesmaximal, characteristic
3#24911313C211yesmaximal, characteristic
4#24908997C211yesmaximal, characteristic
5#24908940C211yesmaximal, characteristic, complemented
6#24908834C211yesmaximal, characteristic
7#24908796C211yesmaximal, characteristic, complemented
8#24908188C211yesmaximal, characteristic, complemented
9#24908022C211yesmaximal, characteristic, complemented
10#24907818C211yesmaximal, characteristic
11#24907818C211yesmaximal, characteristic
12#24907731C211yesmaximal, characteristic, complemented
13#24907731C211yesmaximal, characteristic, complemented
14#24907723C211yesmaximal, characteristic
15Q8×C4⋊C4×C83C211yesmaximal, characteristic, complemented
16C22×#13050577C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581770—21no—
19C2×C4×D332C2211yescharacteristic
20#19127986—24no—
21C4×#13050577—24no—
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×#14581716C2211yescharacteristic
29C2×#14581716C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581712C2211yescharacteristic
32C2×#14581712C2211yescharacteristic
33C2×#14581712C2211yescharacteristic
34C2×C4×C83⋊C8C2211yescharacteristic
35#19127623—21no—
36#19127614—21no—
37#19127603C2211yescharacteristic
38#19127603C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127599C2211yescharacteristic
41#19127575—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4×Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53#389×C83—21no—
54#389×C83—21no—
55#389×C83C2211yescharacteristic
56#389×C83C2211yescharacteristic
57#384×C83—21no—
58#384×C83C2211yescharacteristic
59C4×C4⋊C4×C83—21no—
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×D332C2311yescharacteristic
63C22×C4×D83—21no—
64C2×#13050577—22no—
65C2×#13050577—22no—
66C2×#13050577Q814yescomplemented
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71#14581792—22no—
72#14581792—41no—
73#14581770—44no—
74C4×D332—22no—
75C4×D332—41no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85C2×(C2×D83)⋊C4—21no—
86C2×(C2×D83)⋊C4—21no—
87#14581723—22no—
88#14581723—22no—
89#14581723—22no—
90#14581723—22no—
91#14581723—41no—
92C4⋊(C83⋊C8)—22no—
93#14581716—22no—
94#14581716—22no—
95#14581716D414yes—
96#14581712—22no—
97#14581712—22no—
98#14581712—41no—
99C4×C83⋊C8—22no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C22×Q8×C83C2311yescharacteristic
103C4×Q8×C83—22no—
104C4×Q8×C83—22no—
105C4×Q8×C83—22no—
106C4×Q8×C83—22no—
107C4×Q8×C83—41no—
108C4×Q8×C83—41no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83—21no—
115C2×C4⋊C4×C83—21no—
116C2×C4⋊C4×C83D411yescharacteristic
117C2×C4⋊C4×C83D411yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C42×C83—21no—
122C2×C42×C83—21no—
123C2×C42×C83—21no—
124C2×C42×C83C2311yescharacteristic
125C2×C42×C83C2311yescharacteristic
126C2×C42×C83C2311yescharacteristic
127#95×C83—21no—
128#95×C83—21no—
129#95×C83D411yescharacteristic
130#95×C83D411yescharacteristic
131C23×D83—21no—
132C2×Dic166—42no—
133C2×Dic166—42no—
134C2×Dic166—81no—
135C2×D332—21no—
136C2×D332—21no—
137C2×D332—21no—
138C2×D332—21no—
139C2×D332—21no—
140C2×D332C2×Q812yes—
141C22×Dic83—21no—
142C22×Dic83—21no—
143C22×Dic83—21no—
144C2×C4×D83—42no—
145C2×C4×D83—42no—
146C2×C4×D83—81no—
147#13050577—28no—
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⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8C2×D412yes—
159C2×C83⋊C8C2×Q812yes—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—21no—
164C4⋊Dic83—21no—
165C4⋊Dic83—42no—
166C4⋊Dic83C2×D412yes—
167Dic83⋊C4—42no—
168Dic83⋊C4—42no—
169(C2×D83)⋊C4—42no—
170(C2×D83)⋊C4—42no—
171(C2×D83)⋊C4—42no—
172(C2×D83)⋊C4—42no—
173(C2×D83)⋊C4—81no—
174C2×Q8×C83—21no—
175C2×Q8×C83—21no—
176C2×Q8×C83—21no—
177C2×Q8×C83C2×D412yes—
178C2×Q8×C83C2×Q812yes—
179C2×Q8×C83#4112yes—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2411yescharacteristic
191C4⋊C4×C83—24no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—41no—
198C4⋊C4×C83—42no—
199C4⋊C4×C83—42no—
200C4⋊C4×C83—42no—
201C4⋊C4×C83—81no—
202C4⋊C4×C83—81no—
203C4⋊C4×C83—81no—
204C4⋊C4×C83SD1614yescomplemented
205C4⋊C4×C83#4112yes—
206C42×C83—24no—
207C42×C83—21no—
208C42×C83—21no—
209C42×C83—21no—
210C42×C83—21no—
211C42×C83—21no—
212C42×C83—21no—
213C42×C83—42no—
214C42×C83—42no—
215C23×C83C22×D411yescharacteristic
216C22×D83—22no—
217C22×D83—41no—
218C22×D83—41no—
219C22×D83—41no—
220C22×D83—41no—
221C22×D83—41no—
222C22×D83—41no—
223Q8×C83—22no—
224Q8×C83—22no—
225Q8×C83—22no—
226Q8×C83C22⋊Q814yescomplemented
227C2×C4×C83—21no—
228C2×C4×C83—22no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
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—41no—
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×C83C2×SD1612yes—
255C2×C4×C83C2×SD1612yes—
256C2×C4×C83C2×SD1612yes—
257C2×C4×C83C22×D411yescharacteristic
258C2×C4×C83C22×Q811yescharacteristic
259C2×C4×C83C2×#4111yescharacteristic
260C2×C4×C83C2×#4111yescharacteristic
261C2×C4×C83H(5,𝔽2)11yescharacteristic
262C2×C4×C832−1+411yescharacteristic
263C2×Dic83—22no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272D332—22no—
273D332—22no—
274D332—41no—
275D332—41no—
276C4×D83—84no—
277Dic166—84no—
278C83⋊C8—22no—
279C83⋊C8—22no—
280C83⋊C8—41no—
281C83⋊C8—41no—
282C22×C83#52511yescharacteristic
283C22×C83D4211yescharacteristic
284C22×C83D4×Q811yescharacteristic
285C22×C83C22×SD1611yescharacteristic
286C22×C83#57511yescharacteristic
287C22×C83#57711yescharacteristic
288C22×C83#57711yescharacteristic
289C2×D83—42no—
290C2×D83—42no—
291C2×D83—42no—
292C2×D83—81no—
293C2×D83—81no—
294C2×D83—81no—
295C2×D83—81no—
296C4×C83—22no—
297C4×C83—24no—
298C4×C83—22no—
299C4×C83—22no—
300C4×C83—41no—
301C4×C83—42no—
302C4×C83—41no—
303C4×C83—41no—
304C4×C83—81no—
305C4×C83—81no—
306C4×C83C2×C22⋊Q811yescharacteristic
307C4×C83#53911yescharacteristic
308C4×C83#54711yescharacteristic
309C4×C83#55411yescharacteristic
310Dic83—42no—
311Dic83—81no—
312Dic83—81no—
313#26373—831nomaximal, Sylow
314C2×C83#244911yescharacteristic
315C2×C83#310311yescharacteristic
316C2×C83#312511yescharacteristic
317C2×C83#327111yescharacteristic
318C2×C83D4×SD1611yescharacteristic
319C2×C83Q8×SD1611yescharacteristic
320C2×C83#338811yescharacteristic
321D83—82no—
322C2×#475—831no—
323C2×C4×SD16—831no—
324#2452—831no—
325Q8×C4⋊C4D8311yescharacteristic, complemented
326#2094—831no—
327#2084—831no—
328#2065—831no—
329#2058—831no—
330#1941—831no—
331#1916—831no—
332#1880—831no—
333#1880—831no—
334#1865—831no—
335#1865—831no—
336#1864—831no—
337C83#2637311yescharacteristic, complemented, Sylow
338C22×SD16—831no—
339C2×C4⋊Q8—831no—
340C2×C22⋊Q8—1661no—
341C2×C4×Q8C2×D8311yescharacteristic
342C2×C4×Q8C2×D8311yescharacteristic
343C2×C4×Q8C2×D8311yescharacteristic
344C2×C4×D4—831no—
345#475—1664no—
346C4×SD16—1664no—
347C2×#106—831no—
348C2×#106—831no—
349C2×#106—831no—
350C2×#106—831no—
351C2×#106—831no—
352C2×C4⋊C8—831no—
353C2×#103—831no—
354C2×#103—831no—
355C2×#103—831no—
356C2×#102—831no—
357C2×#102—831no—
358C2×#102—831no—
359C2×C4×C8—831no—
360#396—1661no—
361#393—1661no—
362#390—831no—
363#390—831no—
364#389—21no—
365#389—21no—
366#389—831no—
367#389—831no—
368#389C2×D8311yescharacteristic
369#389C2×D8311yescharacteristic
370#385—1661no—
371#384—21no—
372#384C2×D8311yescharacteristic
373C4×C4⋊C4—21no—
374C4×C4⋊C4C2×D8311yescharacteristic
375#340—831no—
376#340—831no—
377#340—831no—
378#340—831no—
379#336—831no—
380#336—831no—
381#336—831no—
382#336—831no—
383C22×Q8—1661no—
384C22×Q8C22×D8311yescharacteristic
385C22×D4—831no—
386C23×C4—1661no—
387C2×SD16—834no—
388C2×SD16—1662no—
389C2×SD16—1662no—
390C22×C8—831no—
391C22×C8—831no—
392C22×C8—831no—
393C22×C8—831no—
394C4⋊Q8—3321no—
395C4⋊Q8—1662no—
396C22⋊Q8—3324no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×Q8—41no—
403C4×D4—3321no—
404C4×D4—1662no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—831no—
413C2×C4⋊C4—831no—
414C2×C4⋊C4—831no—
415C2×C4⋊C4—1661no—
416C2×C4⋊C4—1661no—
417C2×C4⋊C4—1661no—
418C2×C4⋊C4C22⋊D8311yescharacteristic
419C2×C4⋊C4C22⋊D8311yescharacteristic
420C2×C4⋊C4C22×D8311yescharacteristic
421C2×C4⋊C4C22×D8311yescharacteristic
422C2×C4⋊C4C22×D8311yescharacteristic
423C2×C22⋊C4—1661no—
424C2×C22⋊C4—1661no—
425C2×C22⋊C4—1661no—
426C2×C22⋊C4—1661no—
427C2×C22⋊C4—1661no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42C22×D8311yescharacteristic
432C2×C42C22×D8311yescharacteristic
433C2×C42C22×D8311yescharacteristic
434#106—3321no—
435#106—1662no—
436#106—1662no—
437#106—1662no—
438#106—1662no—
439C4⋊C8—1662no—
440#103—834no—
441#103—1662no—
442#103—1662no—
443#102—3321no—
444#102—1662no—
445#102—1662no—
446C4×C8—1662no—
447#95—21no—
448#95—21no—
449#95—1661no—
450#95—1661no—
451#95C22⋊D8311yescharacteristic
452#95C22⋊D8311yescharacteristic
453C24—1661no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—3322no—
458C2×Q8—3322no—
459C2×Q8—6641no—
460C2×Q8D4×D8312yes—
461C2×Q8Q8×D8312yes—
462C2×Q8#1305059012yes—
463C2×D4—832no—
464C2×D4—1661no—
465C2×D4—1661no—
466C2×D4—1661no—
467C2×D4—1661no—
468C2×D4—1661no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—3322no—
474C22×C4—3322no—
475C22×C4—6641no—
476C22×C4—1661no—
477C22×C4—1661no—
478C22×C4—1661no—
479C22×C4C2×C22⋊D8311yescharacteristic
480C22×C4C2×C22⋊D8311yescharacteristic
481C22×C4C2×C22⋊D8311yescharacteristic
482C22×C4C2×C22⋊D8311yescharacteristic
483C22×C4C2×C22⋊D8311yescharacteristic
484C22×C4C2×C22⋊D8311yescharacteristic
485C22×C4C23×D8311yescharacteristic
486SD16—1668no—
487C2×C8—832no—
488C2×C8—832no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8—1661no—
496C2×C8—1661no—
497C2×C8—1661no—
498C2×C8—1661no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—3322no—
513C4⋊C4—3322no—
514C4⋊C4—3322no—
515C4⋊C4—832no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4—1661no—
519C4⋊C4—1661no—
520C4⋊C4—1661no—
521C4⋊C4#1305057714yescomplemented
522C4⋊C4#1305059012yes—
523C22⋊C4—3322no—
524C22⋊C4—3322no—
525C22⋊C4—3322no—
526C22⋊C4—3322no—
527C22⋊C4—6641no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—42no—
537C23—3321no—
538C23—3321no—
539C23—3321no—
540C23—3321no—
541C23—3321no—
542C23—3321no—
543C23—1662no—
544C23C22×C22⋊D8311yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—6644no—
549Q8#1458177014yescomplemented
550D4—3321no—
551D4—3321no—
552D4—1662no—
553D4—1662no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—3321no—
586C2×C4—3321no—
587C2×C4—3321no—
588C2×C4—3321no—
589C2×C4—6644no—
590C2×C4—1662no—
591C2×C4C2×#1305057712yes—
592C2×C4C2×#1305057712yes—
593C2×C4C2×#1305057712yes—
594C2×C4C2×D4×D8311yescharacteristic
595C2×C4C2×Q8×D8311yescharacteristic
596C2×C4C2×#1305059011yescharacteristic
597C2×C4C2×#1305059011yescharacteristic
598C2×C4#1458183711yescharacteristic
599C2×C4#1458184011yescharacteristic
600C8—3321no—
601C8—3321no—
602C8—1662no—
603C8—1662no—
604C22—3322no—
605C22—3322no—
606C22—3322no—
607C22—6641no—
608C22—6641no—
609C22—6641no—
610C22—6641no—
611C22#1912821211yescharacteristic
612C22D4×C22⋊D8311yescharacteristic
613C22Q8×C22⋊D8311yescharacteristic
614C22C22×#1305057711yescharacteristic
615C22#1912849911yescharacteristic
616C22#1912850911yescharacteristic
617C22#1912851211yescharacteristic
618C4—22no—
619C4—22no—
620C4—24no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—3322no—
629C4—6641no—
630C4—6641no—
631C4C2×#1458177011yescharacteristic
632C4#1912829811yescharacteristic
633C4#1912835211yescharacteristic
634C4#1912840611yescharacteristic
635C2—6642no—
636C2#2491127211yescharacteristic
637C2#2491749411yescharacteristic
638C2#2491774011yescharacteristic
639C2#2491904411yescharacteristic
640C2D4×#1305057711yescharacteristic
641C2Q8×#1305057711yescharacteristic
642C2#2492040211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingQ8×C4⋊C4×C83C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenterQ8×C4⋊C4D83
hypocenterC83#26373
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19127986C2semidirect
5. #24908940C2semidirect
7. #24908796C2semidirect
8. #24908188C2semidirect
9. #24908022C2semidirect
12. #24907731C2semidirect
13. #24907731C2semidirect
15. Q8×C4⋊C4×C83C2semidirect
66. C2×#13050577Q8semidirect
204. C4⋊C4×C83SD16semidirect
226. Q8×C83C22⋊Q8semidirect
325. Q8×C4⋊C4D83semidirect
337. C83#26373semidirect
521. C4⋊C4#13050577semidirect
549. Q8#14581770semidirect

Maximal subgroups

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

subgroupindex
C2×#191279862
C2×C4×#130505772
#249113132
#249089972
#249089402
#249088342
#249087962
#249081882
#249080222
#249078182
#249077312
#249077232
Q8×C4⋊C4×C832
#2637383