Finite Group Databasebeta

← #35375336

Subgroups — #35375336

Order 21248.

602 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917500C211yesmaximal, characteristic
2#24916415C211yesmaximal, characteristic
3#24916405C211yesmaximal, characteristic
4#24906360C211yesmaximal, characteristic, complemented
5#24906358C211yesmaximal, characteristic, complemented
6#24906316C211yesmaximal, characteristic, complemented
7#24906316C211yesmaximal, characteristic, complemented
8#24906315C211yesmaximal, characteristic, complemented
9#24906269C211yesmaximal, characteristic, complemented
10C8×#13050580C211yesmaximal, characteristic, complemented
11#24906236C211yesmaximal, characteristic
12#24905916C211yesmaximal, characteristic
13Q8⋊C8×D83C211yesmaximal, characteristic
14#24905742C211yesmaximal, characteristic
15C83×#1584C211yesmaximal, characteristic, complemented
16#19128255C2211yescharacteristic
17C4×Q8×D83C2211yescharacteristic
18#19128038C411yescharacteristic
19#19128035C411yescharacteristic
20#19127995C411yescharacteristic
21#19127994C411yescharacteristic
22#19127910C411yescharacteristic
23#19127909C411yescharacteristic
24#19127858C411yescharacteristic
25#19127855C411yescharacteristic
26#19127832C2211yescharacteristic
27#19127832C2211yescharacteristic
28#19127831—21no—
29#19127831C2211yescharacteristic
30#19127830—21no—
31#19127830—21no—
32#19127829—21no—
33C8×Dic166C2211yescharacteristic
34#19127825C2211yescharacteristic
35#19127825C2211yescharacteristic
36C4×Dic332C2211yescharacteristic
37C4×#13050580C2211yescharacteristic
38C4×#13050580C2211yescharacteristic
39#19127752C2211yescharacteristic
40#19127726—21no—
41#19127723—21no—
42#19127723C2211yescharacteristic
43C4⋊C8×D83C2211yescharacteristic
44#19127655—21no—
45#19127653—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic
49#19127453C2211yescharacteristic
50#19127453C2211yescharacteristic
51#19127433C2211yescharacteristic
52#19127433C2211yescharacteristic
53Q8⋊(C83⋊C8)C2211yescharacteristic
54Q8⋊(C83⋊C8)C2211yescharacteristic
55#19127431C2211yescharacteristic
56#19127431C2211yescharacteristic
57#19127423C2211yescharacteristic
58C8×C83⋊C8C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60#446×C83C2211yescharacteristic
61C8×Q8×C83C2211yescharacteristic
62C4×Q16×C83C2211yescharacteristic
63#335×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65Q8⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×Q8×D83C2×C411yescharacteristic
68C2×Dic332C2×C411yescharacteristic
69C2×#13050580C2×C411yescharacteristic
70C2×#13050580C2×C411yescharacteristic
71C2×#13050573—21no—
72C2×#13050573D411yescharacteristic
73C2×#13050573D411yescharacteristic
74C2×C8×D83—21no—
75C2×C8×D83—21no—
76#14581791C2×C411yescharacteristic
77#14581790C2×C411yescharacteristic
78#14581782—21no—
79#14581772C2×C411yescharacteristic
80#14581770—21no—
81#14581768C2×C411yescharacteristic
82Q8×Dic83C2×C411yescharacteristic
83C4×Dic166—21no—
84C4×Dic166—21no—
85C4×Dic166C2311yescharacteristic
86C4×Dic166C2311yescharacteristic
87#14581757—21no—
88#14581757C2×C411yescharacteristic
89#14581749—21no—
90C4⋊C4×D83—21no—
91C4⋊C4×D83C2×C411yescharacteristic
92C42×D83—21no—
93C42×D83C2311yescharacteristic
94#14581726C2×C411yescharacteristic
95#14581725C2×C411yescharacteristic
96#14581725C2×C411yescharacteristic
97C4⋊(C83⋊C8)—21no—
98C4⋊(C83⋊C8)—21no—
99C4⋊(C83⋊C8)C2311yescharacteristic
100C4⋊(C83⋊C8)C2311yescharacteristic
101Dic83⋊C8—21no—
102Dic83⋊C8—21no—
103Dic83⋊C8—21no—
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
106Dic83⋊C8—21no—
107Dic83⋊C8C2×C411yescharacteristic
108Dic83⋊C8C2×C411yescharacteristic
109#14581717C2×C411yescharacteristic
110#14581717C2×C411yescharacteristic
111#14581716C2×C411yescharacteristic
112#14581716C2×C411yescharacteristic
113#14581715C2×C411yescharacteristic
114#14581715C2×C411yescharacteristic
115(C2×D83)⋊C8C2×C411yescharacteristic
116(C2×D83)⋊C8C2×C411yescharacteristic
117#14581703—21no—
118#14581703—21no—
119#14581702—21no—
120#14581702D411yescharacteristic
121#14581702D411yescharacteristic
122C4×C83⋊C8C2311yescharacteristic
123C4×C83⋊C8C2311yescharacteristic
124C4×C83⋊C8C2311yescharacteristic
125C8×Dic83—21no—
126C8×Dic83—21no—
127C8×Dic83—21no—
128C8×Dic83—21no—
129C2×Q16×C83C2×C411yescharacteristic
130C4×Q8×C83C2311yescharacteristic
131C4×Q8×C83C2311yescharacteristic
132#107×C83C2×C411yescharacteristic
133C4⋊C8×C83—21no—
134C4⋊C8×C83—21no—
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137#103×C83C2×C411yescharacteristic
138#103×C83C2×C411yescharacteristic
139#97×C83—21no—
140C4×C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C2×Dic166—21no—
145C2×Dic166—21no—
146C2×Dic166C22×C411yescharacteristic
147C2×Dic166C22×C411yescharacteristic
148Q8×D83—22no—
149Q8×D83—41no—
150C2×C4×D83—21no—
151C2×C4×D83—21no—
152C2×C4×D83C22×C411yescharacteristic
153C2×C4×D83C2×D411yescharacteristic
154C2×C4×D83#4111yescharacteristic
155Dic332—22no—
156#13050580—22no—
157#13050580—41no—
158#13050573—22no—
159#13050573—42no—
160#13050573Q1614yescomplemented
161C2×C83⋊C8—21no—
162C2×C83⋊C8—21no—
163C2×C83⋊C8—21no—
164C2×C83⋊C8C22×C411yescharacteristic
165C2×C83⋊C8C22×C411yescharacteristic
166C2×C83⋊C8C2×D411yescharacteristic
167C2×C83⋊C8C2×D411yescharacteristic
168C8×D83—42no—
169C8×D83—42no—
170C4⋊Dic83—21no—
171C4⋊Dic83—21no—
172C4⋊Dic83C22×C411yescharacteristic
173C4⋊Dic83C22×C411yescharacteristic
174Dic83⋊C4—21no—
175Dic83⋊C4—21no—
176Dic83⋊C4—21no—
177Dic83⋊C4—21no—
178Dic83⋊C4—21no—
179Dic83⋊C4—21no—
180Dic83⋊C4—21no—
181Dic83⋊C4—21no—
182(C2×D83)⋊C4—21no—
183(C2×D83)⋊C4—21no—
184C4×Dic83—21no—
185C4×Dic83—21no—
186C4×Dic83—21no—
187C4×Dic83—21no—
188C4×Dic83C22×C411yescharacteristic
189C4×Dic83C2×D411yescharacteristic
190C4×Dic83#4111yescharacteristic
191C2×Q8×C83C22×C411yescharacteristic
192C2×Q8×C83C22×C411yescharacteristic
193Q16×C83—41no—
194C2×C8×C83—21no—
195C2×C8×C83—21no—
196C2×C8×C83—21no—
197C2×C8×C83C22×C411yescharacteristic
198C2×C8×C83C22×C411yescharacteristic
199C2×C8×C83C2×D411yescharacteristic
200C2×C8×C83C2×D411yescharacteristic
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83C22×C411yescharacteristic
204C4⋊C4×C83C22×C411yescharacteristic
205C42×C83—21no—
206C42×C83—21no—
207C42×C83C2411yescharacteristic
208C22×D83C4×D411yescharacteristic
209Q8×C83—21no—
210Q8×C83—21no—
211Q8×C83—21no—
212Q8×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83C23×C411yescharacteristic
218C2×C4×C83C22×D411yescharacteristic
219C2×C4×C83C2×#4111yescharacteristic
220C2×Dic83—21no—
221C2×Dic83—21no—
222C2×Dic83—21no—
223C2×Dic83—21no—
224C2×Dic83C4×D411yescharacteristic
225C2×Dic83C4×D411yescharacteristic
226C2×Dic83C4×D411yescharacteristic
227C4×D83—21no—
228C4×D83—21no—
229C4×D83—41no—
230C4×D83—42no—
231C4×D83—81no—
232C4×D83C2×Q1612yes—
233C4×D83#13512yes—
234Dic166—21no—
235Dic166—21no—
236Dic166—21no—
237Dic166—42no—
238Dic166—81no—
239Dic166#13112yes—
240C8×C83—21no—
241C8×C83—21no—
242C8×C83—21no—
243C8×C83—41no—
244C8×C83—41no—
245C8×C83—41no—
246C8×C83C2×Q1612yes—
247C83⋊C8—22no—
248C83⋊C8—21no—
249C83⋊C8—21no—
250C83⋊C8—21no—
251C83⋊C8—41no—
252C83⋊C8—41no—
253C83⋊C8C2×Q1612yes—
254C22×C83C2×C4×D411yescharacteristic
255C2×D83—21no—
256C2×D83—21no—
257C2×D83C4×Q1612yes—
258C4×C83—21no—
259C4×C83—41no—
260C4×C83—41no—
261C4×C83—41no—
262C4×C83—41no—
263C4×C83C2×#13111yescharacteristic
264C4×C83#56811yescharacteristic
265C4×C83C22×Q1611yescharacteristic
266C4×C83C2×#13511yescharacteristic
267Dic83—22no—
268Dic83—21no—
269Dic83—21no—
270Dic83—41no—
271Dic83—41no—
272Dic83—41no—
273Dic83C4×Q1612yes—
274#17998—831nomaximal, Sylow
275C2×C83#293311yescharacteristic
276C2×C83C2×C4×Q1611yescharacteristic
277C2×C83#295611yescharacteristic
278D83—22no—
279#3104—831no—
280Q8×M16—831no—
281#2962—831no—
282#1591—831no—
283#1591—831no—
284#1584—831no—
285#1584—831no—
286#1584—831no—
287#1584D8311yescharacteristic, complemented
288C8×Q16—831no—
289C8×Q16—831no—
290#1571—831no—
291#1493—831no—
292C2×Q8⋊C8—831no—
293C8×M16—831no—
294C83#1799811yescharacteristic, complemented, Sylow
295#533—831no—
296C2×C4×Q8—831no—
297#484—831no—
298#484—831no—
299#477—831no—
300#477—831no—
301#462—831no—
302#462—831no—
303#451—831no—
304#451—831no—
305#446—831no—
306#446—831no—
307#446—831no—
308#446—1661no—
309#446—1661no—
310#446—1661no—
311#446—1661no—
312#446C2×D8311yescharacteristic
313C8×Q8—831no—
314C8×Q8—831no—
315C8×Q8—831no—
316C8×Q8C2×D8311yescharacteristic
317C4×Q16—831no—
318C4×Q16—831no—
319C4×Q16—831no—
320C4×Q16C2×D8311yescharacteristic
321#431—831no—
322#424—1661no—
323#423—831no—
324#423—1661no—
325C2×C4⋊C8—831no—
326#405—1661no—
327C4×M16—831no—
328C4×M16—1661no—
329C2×C4×C8—831no—
330#335—831no—
331#335—831no—
332#335—831no—
333#335C2×D8311yescharacteristic
334Q8⋊C8—831no—
335Q8⋊C8—831no—
336Q8⋊C8—831no—
337Q8⋊C8—831no—
338Q8⋊C8—831no—
339Q8⋊C8—831no—
340Q8⋊C8C2×D8311yescharacteristic
341Q8⋊C8C2×D8311yescharacteristic
342#322—831no—
343#322C2×D8311yescharacteristic
344C82—831no—
345C82—831no—
346C22×Q8—831no—
347C2×Q16—831no—
348C2×Q16—831no—
349C2×Q16—831no—
350C2×Q16C4×D8311yescharacteristic
351C2×M16—831no—
352C2×M16—831no—
353C2×M16—1661no—
354C22×C8—1661no—
355C22×C8—1661no—
356C4⋊Q8—831no—
357C4⋊Q8—831no—
358#125—1661no—
359C22⋊Q8—831no—
360C22⋊Q8—831no—
361C22⋊Q8—1661no—
362C4×Q8—831no—
363C4×Q8—831no—
364C4×Q8—831no—
365C4×Q8—831no—
366C4×Q8—1661no—
367C4×Q8—1661no—
368C4×Q8—1661no—
369C4×Q8C22×D8311yescharacteristic
370C4×Q8C22×D8311yescharacteristic
371#117—1661no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—1661no—
374C2×C42—831no—
375C2×C42—1661no—
376#107—831no—
377#107—831no—
378#107—831no—
379#107C4×D8311yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—831no—
383C4⋊C8—831no—
384C4⋊C8—831no—
385C4⋊C8—831no—
386C4⋊C8—1661no—
387C4⋊C8—1661no—
388C4⋊C8—1661no—
389C4⋊C8—1661no—
390C4⋊C8—1661no—
391C4⋊C8—1661no—
392C4⋊C8—1661no—
393C4⋊C8—1661no—
394C4⋊C8C22×D8311yescharacteristic
395C4⋊C8C22×D8311yescharacteristic
396#103—831no—
397#103—831no—
398#103—831no—
399#103—831no—
400#103—831no—
401#103—831no—
402#103C4×D8311yescharacteristic
403#103C4×D8311yescharacteristic
404C22⋊C8—831no—
405C22⋊C8—831no—
406#97—21no—
407#97—831no—
408#97—831no—
409#97—1661no—
410#97—1661no—
411#97—1661no—
412C4×C8—21no—
413C4×C8—831no—
414C4×C8—831no—
415C4×C8—831no—
416C4×C8—1661no—
417C4×C8—1661no—
418C4×C8—1661no—
419C4×C8—1661no—
420C4×C8C22×D8311yescharacteristic
421C4×C8C22×D8311yescharacteristic
422C4×C8C22×D8311yescharacteristic
423C2×Q8—3321no—
424C2×Q8—831no—
425C2×Q8—831no—
426C2×Q8—1661no—
427C2×Q8—1661no—
428C2×Q8—1662no—
429C2×Q8C2×C4×D8311yescharacteristic
430C2×Q8C2×C4×D8311yescharacteristic
431C22×C4—831no—
432C22×C4—831no—
433C22×C4—831no—
434C22×C4—1661no—
435C22×C4—1661no—
436Q16—41no—
437Q16—3321no—
438Q16—1662no—
439Q16—1662no—
440M16—3322no—
441M16—834no—
442M16—1662no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—3322no—
447C2×C8—3322no—
448C2×C8—831no—
449C2×C8—831no—
450C2×C8—831no—
451C2×C8—831no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8C2×C4×D8311yescharacteristic
456C2×C8C2×C4×D8311yescharacteristic
457C2×C8D4×D8311yescharacteristic
458C2×C8D4×D8311yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—831no—
462C4⋊C4—831no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4C2×C4×D8311yescharacteristic
474C4⋊C4C2×C4×D8311yescharacteristic
475C22⋊C4—1661no—
476C22⋊C4—1661no—
477C42—21no—
478C42—21no—
479C42—831no—
480C42—831no—
481C42—831no—
482C42—1661no—
483C42—1661no—
484C42—1661no—
485C42—1661no—
486C42C23×D8311yescharacteristic
487C23—831no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—3322no—
493Q8—832no—
494Q8—6641no—
495Q8—1661no—
496Q8—1661no—
497Q8—1661no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—3321no—
503C2×C4—3322no—
504C2×C4—831no—
505C2×C4—832no—
506C2×C4—832no—
507C2×C4—831no—
508C2×C4—831no—
509C2×C4—6641no—
510C2×C4—1661no—
511C2×C4—1661no—
512C2×C4—1661no—
513C2×C4—1661no—
514C2×C4—1661no—
515C2×C4—1661no—
516C2×C4C22×C4×D8311yescharacteristic
517C2×C4C2×D4×D8311yescharacteristic
518C2×C4#41×D8311yescharacteristic
519C8—21no—
520C8—21no—
521C8—21no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—3321no—
526C8—3321no—
527C8—832no—
528C8—1662no—
529C8—1661no—
530C8—1661no—
531C8—1661no—
532C8Q16×D8312yes—
533C22—832no—
534C22—1661no—
535C22—1661no—
536C22C4×D4×D8311yescharacteristic
537C4—21no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—3321no—
543C4—3321no—
544C4—3321no—
545C4—832no—
546C4—1661no—
547C4—1662no—
548C4—1661no—
549C4#131×D8311yescharacteristic
550C4#1912846711yescharacteristic
551C4C2×Q16×D8311yescharacteristic
552C4#135×D8311yescharacteristic
553C2—1662no—
554C2#2491618611yescharacteristic
555C2C4×Q16×D8311yescharacteristic
556C2#2491637111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1584C2
socleC22×C83C2×C4×D4
prime radicalC22×D83C4×D4
cosocleC42×C83C24
hypercenter#1584D83
hypocenterC83#17998
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #24906360C2semidirect
5. #24906358C2semidirect
6. #24906316C2semidirect
7. #24906316C2semidirect
8. #24906315C2semidirect
9. #24906269C2semidirect
10. C8×#13050580C2semidirect
15. C83×#1584C2semidirect
160. #13050573Q16semidirect
287. #1584D83semidirect
294. C83#17998semidirect

Maximal subgroups

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

subgroupindex
#249175002
#249164152
#249164052
#249063602
#249063582
#249063162
#249063152
#249062692
C8×#130505802
#249062362
#249059162
Q8⋊C8×D832
#249057422
C83×#15842
#1799883