Finite Group Databasebeta

← #35566469

Subgroups — #35566469

Order 21248.

2860 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊D83C211yesmaximal, characteristic
2C2×#19128179C211yesmaximal, characteristic, complemented
3C2×#19128177C211yesmaximal, characteristic, complemented
4#24910520C213yesmaximal, complemented
5#24910495C213yesmaximal, complemented
6#24910488C213yesmaximal, complemented
7#24910407C213yesmaximal
8Q8×Dic83⋊C4C211yesmaximal, characteristic, complemented
9Q8×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
10#24910148C213yesmaximal, complemented
11#24910141C213yesmaximal, complemented
12#24910032C213yesmaximal, complemented
13#24909926C213yesmaximal
14C4×Q8×Dic83C211yesmaximal, characteristic, complemented
15Q8×C22⋊C4×C83C211yesmaximal, characteristic, complemented
16C22×Q8×D83C2211yescharacteristic
17Q8×C22⋊D83C4116yescomplemented
18C2×#14581790C2213yes—
19C2×#14581771C2213yes—
20C2×#14581768C2213yes—
21#19128179—28no—
22#19128177—28no—
23C2×Q8×Dic83C2211yescharacteristic
24C2×Q8×Dic83C2213yes—
25C2×Q8×Dic83C2213yes—
26C2×Q8×Dic83C2211yescharacteristic
27C2×Q8×Dic83C2211yescharacteristic
28C2×C4×Dic166C2213yes—
29C2×#14581757C2213yes—
30C2×C4×C22⋊D83C2213yes—
31C2×#14581751C2213yes—
32C2×#14581751C2213yes—
33C2×#14581750C2213yes—
34C2×#14581749C2213yes—
35#19127607C2213yes—
36#19127606C2213yes—
37#19127601C2213yes—
38#19127601C2211yescharacteristic
39#19127599C2213yes—
40#19127598C2216yes—
41#19127598C2213yes—
42#19127598C2213yes—
43#19127597C2213yes—
44#19127595C2216yes—
45#19127593C2213yes—
46#19127593C2213yes—
47#19127590C2216yes—
48#19127589C2213yes—
49#19127589C2213yes—
50#19127588C2216yes—
51#19127583C2216yes—
52#19127570C2213yes—
53#19127558C2213yes—
54C4⋊C4×Dic83C2213yes—
55C4⋊C4×Dic83C2213yes—
56C4×Dic83⋊C4C2213yes—
57C4×C4⋊Dic83C2213yes—
58#19127552C2216yes—
59#19127552C2216yes—
60C4×(C2×D83)⋊C4C2213yes—
61C42×Dic83C2213yes—
62C23×Q8×C83C2211yescharacteristic
63C2×C4×Q8×C83C2211yescharacteristic
64C2×C4×Q8×C83C2211yescharacteristic
65#391×C83C2213yes—
66#391×C83C2213yes—
67#380×C83C2213yes—
68C4×C22⋊C4×C83C2213yes—
69C22×Dic166C2313yes—
70C2×Q8×D83—24no—
71C2×Q8×D83—24no—
72C2×Q8×D83C2×C418yes—
73C22×C22⋊D83C2311yescharacteristic
74C22×C4×D83C2313yes—
75#14581790C2×C4124yes—
76#14581771C2×C4124yes—
77#14581768C2×C4124yes—
78Q8×Dic83—212no—
79Q8×Dic83—212no—
80Q8×Dic83—24no—
81Q8×Dic83C2×C418yes—
82Q8×Dic83D418yescomplemented
83C4×Dic166—212no—
84#14581757—212no—
85C4×C22⋊D83C2×C4124yes—
86#14581751—212no—
87#14581751—212no—
88#14581750—212no—
89#14581749—212no—
90C2×C4⋊Dic83C2313yes—
91C2×C4⋊Dic83C2313yes—
92C2×C4⋊Dic83C2313yes—
93C2×C4⋊Dic83C2313yes—
94C2×C4⋊Dic83C2313yes—
95C2×Dic83⋊C4C2313yes—
96C2×Dic83⋊C4C2311yescharacteristic
97C2×Dic83⋊C4C2313yes—
98C2×Dic83⋊C4C2313yes—
99C2×Dic83⋊C4C2316yes—
100C2×Dic83⋊C4C2313yes—
101C2×Dic83⋊C4C2316yes—
102C2×C22⋊Dic83C2311yescharacteristic
103C2×C22⋊Dic83C2313yes—
104C2×C22⋊Dic83C2313yes—
105C2×(C2×D83)⋊C4C2313yes—
106C2×(C2×D83)⋊C4C2311yescharacteristic
107C2×(C2×D83)⋊C4C2311yescharacteristic
108C2×(C2×D83)⋊C4C2313yes—
109C2×(C2×D83)⋊C4C2313yes—
110C2×C4×Dic83C2313yes—
111C2×C4×Dic83C2313yes—
112C2×C4×Dic83C2316yes—
113C2×C4×Dic83C2316yes—
114C2×C4×Dic83C2313yes—
115C2×C4×Dic83C2311yescharacteristic
116C2×C4×Dic83C2313yes—
117C2×C4×Dic83C2313yes—
118C2×C4×Dic83C2313yes—
119C83⋊#95C2313yes—
120C83⋊#95C2316yes—
121C83⋊#95C2316yes—
122C83⋊#95C2313yes—
123C83⋊#95C2313yes—
124C83⋊#95C2313yes—
125C22×Q8×C83—24no—
126C22×Q8×C83—24no—
127C22×Q8×C83C2×C418yes—
128C22×Q8×C83C2313yes—
129C22×Q8×C83C2311yescharacteristic
130C23×C4×C83C2313yes—
131C4×Q8×C83—24no—
132C4×Q8×C83—24no—
133C2×C4⋊C4×C83C2313yes—
134C2×C4⋊C4×C83C2313yes—
135C2×C22⋊C4×C83C2311yescharacteristic
136C2×C22⋊C4×C83C2313yes—
137C2×C42×C83C2313yes—
138C2×C42×C83C2313yes—
139#95×C83C2313yes—
140#95×C83C2316yes—
141#95×C83C2313yes—
142C23×D83C2411yescharacteristic
143C2×Dic166—26no—
144C2×Dic166—26no—
145C2×Dic166C22×C4112yes—
146Q8×D83—232no—
147C2×C22⋊D83—22no—
148C2×C22⋊D83—22no—
149C2×C22⋊D83C22×C414yes—
150C22×Dic83C2411yescharacteristic
151C22×Dic83C2413yes—
152C22×Dic83C2413yes—
153C22×Dic83C2413yes—
154C22×Dic83C2413yes—
155C22×Dic83C2411yescharacteristic
156C22×Dic83C2411yescharacteristic
157C2×C4×D83—26no—
158C2×C4×D83—26no—
159C2×C4×D83C22×C4112yes—
160C4⋊Dic83—26no—
161C4⋊Dic83—26no—
162C4⋊Dic83—26no—
163C4⋊Dic83C22×C4112yes—
164C4⋊Dic83C2×D4112yes—
165Dic83⋊C4—26no—
166Dic83⋊C4—26no—
167Dic83⋊C4—22no—
168Dic83⋊C4—26no—
169Dic83⋊C4—46no—
170Dic83⋊C4C22×C4124yes—
171Dic83⋊C4C22×C4112yes—
172C22⋊Dic83—26no—
173C22⋊Dic83—22no—
174C22⋊Dic83C22×C4112yes—
175(C2×D83)⋊C4—22no—
176(C2×D83)⋊C4—22no—
177(C2×D83)⋊C4—26no—
178(C2×D83)⋊C4—43no—
179(C2×D83)⋊C4C22×C4112yes—
180C4×Dic83—26no—
181C4×Dic83—26no—
182C4×Dic83—212no—
183C4×Dic83—212no—
184C4×Dic83—26no—
185C4×Dic83—22no—
186C4×Dic83—43no—
187C4×Dic83C22×C4112yes—
188C4×Dic83C2×D4112yes—
189C24×C83C2411yescharacteristic
190C2×Q8×C83—232no—
191C2×Q8×C83—26no—
192C2×Q8×C83—26no—
193C2×Q8×C83C22×C4112yes—
194C2×Q8×C83C22×C414yes—
195C2×Q8×C83C2×D414yes—
196C2×Q8×C83#4114yes—
197C22×C4×C83—26no—
198C22×C4×C83—26no—
199C22×C4×C83C22×C4112yes—
200C22×C4×C83C2411yescharacteristic
201C22×C4×C83C2413yes—
202C22×C4×C83C2413yes—
203C22×C4×C83C2413yes—
204C22×C4×C83C2411yescharacteristic
205C22×C4×C83C2413yes—
206C4⋊C4×C83—26no—
207C4⋊C4×C83—26no—
208C22⋊C4×C83—22no—
209C22⋊C4×C83—43no—
210C42×C83—26no—
211C42×C83—26no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83—21no—
215C23×C83—21no—
216C23×C83—21no—
217C23×C83—21no—
218C23×C83C23×C412yes—
219C23×C83C2511yescharacteristic
220C22×D83—21no—
221C22×D83—21no—
222C22×D83—21no—
223C22×D83—21no—
224C22×D83—21no—
225C22×D83—21no—
226C22×D83C23×C412yes—
227Q8×C83—224no—
228Q8×C83C4×D4116yes—
229C2×C4×C83—23no—
230C2×C4×C83—21no—
231C2×C4×C83—23no—
232C2×C4×C83—21no—
233C2×C4×C83—224no—
234C2×C4×C83—21no—
235C2×C4×C83—23no—
236C2×C4×C83—23no—
237C2×C4×C83—21no—
238C2×C4×C83—23no—
239C2×C4×C83—23no—
240C2×C4×C83—21no—
241C2×C4×C83—23no—
242C2×C4×C83—23no—
243C2×C4×C83C23×C416yes—
244C2×C4×C83C23×C416yes—
245C2×C4×C83C22×D416yes—
246C2×C4×C83C2×#4116yes—
247C2×C4×C832−1+412yes—
248C22⋊D83—28no—
249C2×Dic83—23no—
250C2×Dic83—23no—
251C2×Dic83—21no—
252C2×Dic83—23no—
253C2×Dic83—23no—
254C2×Dic83—23no—
255C2×Dic83—21no—
256C2×Dic83—23no—
257C2×Dic83—23no—
258C2×Dic83—21no—
259C2×Dic83—21no—
260C2×Dic83—23no—
261C2×Dic83—21no—
262C2×Dic83—23no—
263C2×Dic83C23×C416yes—
264C2×Dic83C23×C416yes—
265C2×Dic83C23×C412yes—
266C2×Dic83C22×D416yes—
267C2×Dic83C22×D412yes—
268C2×Dic832−1+412yes—
269C2×Dic832−1+412yes—
270C4×D83—224no—
271Dic166—224no—
272C22×C83—22no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83C24×C411yescharacteristic
280C22×C83C23×D411yescharacteristic
281C22×C83C22×#4111yescharacteristic
282C22×C83C2×2−1+411yescharacteristic
283C22×C83C2×2−1+411yescharacteristic
284C22×C83C2×2−1+411yescharacteristic
285C22×C83#58511yescharacteristic
286C2×D83—22no—
287C2×D83—22no—
288C2×D83—22no—
289C2×D83—41no—
290C2×D83—41no—
291C2×D83—41no—
292C2×D83—41no—
293C4×C83—26no—
294C4×C83—22no—
295C4×C83—43no—
296C4×C83—41no—
297C4×C83—43no—
298C4×C83C2×C4×D4112yes—
299Dic83—26no—
300Dic83—26no—
301Dic83—22no—
302Dic83—26no—
303Dic83—43no—
304Dic83—41no—
305Dic83#51914yescomplemented
306#34311—831nomaximal, Sylow
307C2×C83—42no—
308C2×C83C22×C4×D411yescharacteristic
309C2×C83C2×#51911yescharacteristic
310C2×C83C4×2−1+411yescharacteristic
311C2×C83#349711yescharacteristic
312C2×C83#349911yescharacteristic
313C2×C83#351811yescharacteristic
314C2×C83#352311yescharacteristic
315D83—42no—
316C2×D4×Q8—831no—
317C2×#519—831no—
318C2×#519—831no—
319#2365—833no—
320#2362—833no—
321#2362—833no—
322#2354—833no—
323Q8×C4⋊C4—831no—
324Q8×C22⋊C4—831no—
325Q8×C22⋊C4D8311yescharacteristic, complemented
326#2323—833no—
327#2323—833no—
328C4×C4⋊Q8—833no—
329#2293—833no—
330C42×Q8—831no—
331C83#3431111yescharacteristic, complemented, Sylow
332C23×Q8—831no—
333C23×Q8C2×D8311yescharacteristic
334D4×Q8—8316no—
335C2×C4⋊Q8—833no—
336C2×C22⋊Q8—833no—
337C2×C22⋊Q8—833no—
338#519—1668no—
339#519—1668no—
340C2×C4×Q8—831no—
341C2×C4×Q8—831no—
342C2×C4×Q8—833no—
343C2×C4×Q8—831no—
344C2×C4×Q8—833no—
345C2×C4×Q8—833no—
346C2×C4×Q8—833no—
347C2×C4×Q8C2×D8311yescharacteristic
348C2×C4×Q8C2×D8311yescharacteristic
349C2×C4×D4—833no—
350C2×#117—833no—
351C2×#117—833no—
352C2×#117—833no—
353C2×#117—833no—
354#391—833no—
355#391—833no—
356#391C2×D8313yes—
357#391C2×D8313yes—
358#390—833no—
359#390—831no—
360#389—833no—
361#389—833no—
362#389—833no—
363#389—836no—
364#389—833no—
365#388—833no—
366#388—836no—
367#388—833no—
368#388—833no—
369#388—836no—
370#388—833no—
371#387—836no—
372#387—836no—
373#384—833no—
374#380—833no—
375#380C2×D8313yes—
376C4×C4⋊C4—833no—
377C4×C4⋊C4—833no—
378C4×C4⋊C4—833no—
379C4×C4⋊C4—833no—
380C4×C4⋊C4—836no—
381C4×C4⋊C4—836no—
382C4×C22⋊C4—833no—
383C4×C22⋊C4C2×D8313yes—
384C43—833no—
385C22×Q8—24no—
386C22×Q8—24no—
387C22×Q8—838no—
388C22×Q8—833no—
389C22×Q8—1664no—
390C22×Q8—1664no—
391C22×Q8C4×D8318yes—
392C22×Q8C22×D8313yes—
393C22×Q8C22×D8311yescharacteristic
394C22×D4—831no—
395C23×C4—833no—
396C23×C4C22×D8313yes—
397C4⋊Q8—8324no—
398C22⋊Q8—8324no—
399C22⋊Q8—8324no—
400C4×Q8—24no—
401C4×Q8—24no—
402C4×Q8—838no—
403C4×Q8—838no—
404C4×Q8—16612no—
405C4×Q8—16612no—
406C4×Q8—16612no—
407C4×Q8—16612no—
408C4×Q8—1664no—
409C4×D4—8324no—
410#117—16612no—
411#117—16612no—
412#117—16612no—
413#117—16612no—
414C2×C4⋊C4—836no—
415C2×C4⋊C4—833no—
416C2×C4⋊C4—833no—
417C2×C4⋊C4—833no—
418C2×C4⋊C4—836no—
419C2×C4⋊C4—833no—
420C2×C4⋊C4—833no—
421C2×C4⋊C4—833no—
422C2×C4⋊C4—833no—
423C2×C4⋊C4—833no—
424C2×C4⋊C4—833no—
425C2×C4⋊C4—831no—
426C2×C4⋊C4C22×D8313yes—
427C2×C4⋊C4C22×D8313yes—
428C2×C22⋊C4—833no—
429C2×C22⋊C4—833no—
430C2×C22⋊C4—833no—
431C2×C22⋊C4—833no—
432C2×C22⋊C4—831no—
433C2×C22⋊C4—833no—
434C2×C22⋊C4—831no—
435C2×C22⋊C4—831no—
436C2×C22⋊C4C22×D8311yescharacteristic
437C2×C22⋊C4C22×D8313yes—
438C2×C42—833no—
439C2×C42—833no—
440C2×C42—833no—
441C2×C42—833no—
442C2×C42—836no—
443C2×C42—836no—
444C2×C42—831no—
445C2×C42—833no—
446C2×C42—833no—
447C2×C42C22×D8313yes—
448C2×C42C22×D8313yes—
449#95—833no—
450#95—833no—
451#95—836no—
452#95—836no—
453#95—833no—
454#95—833no—
455#95C22×D8313yes—
456#95C22×D8313yes—
457#95C22×D8316yes—
458C24—831no—
459C24C23×D8311yescharacteristic
460C2×Q8—232no—
461C2×Q8—26no—
462C2×Q8—26no—
463C2×Q8—8312no—
464C2×Q8—16632no—
465C2×Q8—1666no—
466C2×Q8—1666no—
467C2×Q8C2×C4×D83112yes—
468C2×Q8C2×C4×D8314yes—
469C2×Q8D4×D8314yes—
470C2×Q8#1305058914yes—
471C2×D4—834no—
472C2×D4—1662no—
473C2×D4—1662no—
474C22×C4—26no—
475C22×C4—26no—
476C22×C4—831no—
477C22×C4—833no—
478C22×C4—833no—
479C22×C4—833no—
480C22×C4—831no—
481C22×C4—8312no—
482C22×C4—831no—
483C22×C4—833no—
484C22×C4—1666no—
485C22×C4—1666no—
486C22×C4C2×C4×D83112yes—
487C22×C4C23×D8313yes—
488C22×C4C23×D8311yescharacteristic
489C22×C4C23×D8313yes—
490C22×C4C23×D8313yes—
491C22×C4C23×D8311yescharacteristic
492C22×C4C23×D8313yes—
493C4⋊C4—26no—
494C4⋊C4—26no—
495C4⋊C4—3326no—
496C4⋊C4—8312no—
497C4⋊C4—8312no—
498C4⋊C4—8324no—
499C4⋊C4—8312no—
500C4⋊C4—1666no—
501C4⋊C4—1666no—
502C4⋊C4—1666no—
503C4⋊C4—1666no—
504C4⋊C4—1662no—
505C4⋊C4—1666no—
506C4⋊C4—1666no—
507C22⋊C4—22no—
508C22⋊C4—43no—
509C22⋊C4—3323no—
510C22⋊C4—8312no—
511C22⋊C4—8312no—
512C22⋊C4—1662no—
513C22⋊C4—1666no—
514C22⋊C4—1662no—
515C22⋊C4—1666no—
516C22⋊C4—1662no—
517C42—26no—
518C42—26no—
519C42—3323no—
520C42—8312no—
521C42—8312no—
522C42—16612no—
523C42—16612no—
524C42—1662no—
525C42—1666no—
526C42—1666no—
527C42—1666no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—832no—
535C23—1661no—
536C23—1661no—
537C23—1661no—
538C23—1661no—
539C23—1661no—
540C23—1661no—
541C23C22×C4×D8312yes—
542C23C24×D8311yescharacteristic
543Q8—224no—
544Q8—16624no—
545Q8(C22⋊D83)⋊C4116yes—
546D4—1668no—
547C2×C4—23no—
548C2×C4—21no—
549C2×C4—23no—
550C2×C4—21no—
551C2×C4—224no—
552C2×C4—21no—
553C2×C4—23no—
554C2×C4—23no—
555C2×C4—23no—
556C2×C4—23no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—23no—
560C2×C4—23no—
561C2×C4—832no—
562C2×C4—832no—
563C2×C4—832no—
564C2×C4—836no—
565C2×C4—836no—
566C2×C4—832no—
567C2×C4—836no—
568C2×C4—1663no—
569C2×C4—1661no—
570C2×C4—1663no—
571C2×C4—1661no—
572C2×C4—1661no—
573C2×C4—1663no—
574C2×C4—1663no—
575C2×C4—1663no—
576C2×C4—1661no—
577C2×C4—1663no—
578C2×C4—1663no—
579C2×C4—1663no—
580C2×C4—1661no—
581C2×C4—1663no—
582C2×C4—16624no—
583C2×C4C22×C4×D8316yes—
584C2×C4C22×C4×D8316yes—
585C2×C4C2×D4×D8316yes—
586C2×C4C2×#1305058916yes—
587C2×C4#1458184012yes—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—3321no—
596C22—3321no—
597C22—3321no—
598C22—3321no—
599C22—1662no—
600C22—1662no—
601C22—1662no—
602C22C23×C4×D8311yescharacteristic
603C22C22×D4×D8311yescharacteristic
604C22C22×#1305058911yescharacteristic
605C222−1+4×D8311yescharacteristic
606C222−1+4×D8311yescharacteristic
607C22C2×#1458184011yescharacteristic
608C22#1912853711yescharacteristic
609C4—26no—
610C4—22no—
611C4—43no—
612C4—43no—
613C4—41no—
614C4—3323no—
615C4—3321no—
616C4—834no—
617C4—1666no—
618C4—1666no—
619C4—1662no—
620C4—1666no—
621C4C2×(C22⋊D83)⋊C4112yes—
622C2—42no—
623C2—3322no—
624C2C22×(C22⋊D83)⋊C411yescharacteristic
625C2#519×D8311yescharacteristic
626C2C4×#1458184011yescharacteristic
627C2#2492123111yescharacteristic
628C2#2492124711yescharacteristic
629C2#2492147111yescharacteristic
630C2#2492155011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingQ8×C22⋊C4×C83C2
socle, cosocleC23×C83C25
prime radicalC22×C22⋊D83C23
hypercenterQ8×C22⋊C4D83
hypocenterC83#34311
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#19128179C2semidirect
3. C2×#19128177C2semidirect
4. #24910520C2semidirect
5. #24910495C2semidirect
6. #24910488C2semidirect
8. Q8×Dic83⋊C4C2semidirect
9. Q8×(C2×D83)⋊C4C2semidirect
10. #24910148C2semidirect
11. #24910141C2semidirect
12. #24910032C2semidirect
14. C4×Q8×Dic83C2semidirect
15. Q8×C22⋊C4×C83C2semidirect
17. Q8×C22⋊D83C4semidirect
82. Q8×Dic83D4semidirect
305. Dic83#519semidirect
325. Q8×C22⋊C4D83semidirect
331. C83#34311semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊D832
C2×#191281792
C2×#191281772
#249105202
#249104952
#249104882
#249104072
Q8×Dic83⋊C42
Q8×(C2×D83)⋊C42
#249101482
#249101412
#249100322
#249099262
C4×Q8×Dic832
Q8×C22⋊C4×C832
#3431183