Finite Group Databasebeta

← #35329996

Subgroups — #35329996

Order 21248.

1586 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127594C212yesmaximal
2C2×C4⋊C4×Dic83C212yesmaximal, complemented
3#24905731C212yesmaximal, complemented
4#24905712C211yesmaximal, characteristic
5#24905700C212yesmaximal
6#24905700C211yesmaximal, characteristic
7#24905697C212yesmaximal, complemented
8#24905695C212yesmaximal, complemented
9C2×#399×C83C211yesmaximal, characteristic
10C22×C4⋊Dic83C2212yes—
11C22×C22⋊Dic83C2212yes—
12C22×C22⋊Dic83C2211yescharacteristic
13C22×C22⋊Dic83C2211yescharacteristic
14C22×C4×Dic83C2212yes—
15C22×C4×Dic83C2212yes—
16C22×C4×Dic83C2211yescharacteristic
17#19127594—28no—
18C4⋊C4×Dic83—28no—
19C2×C83⋊#95C2212yes—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2211yescharacteristic
27C2×C83⋊#95C2212yes—
28C22×C4⋊C4×C83C2212yes—
29C22×C4⋊C4×C83C2211yescharacteristic
30C22×C22⋊C4×C83C2212yes—
31C22×C22⋊C4×C83C2211yescharacteristic
32#399×C83C418yescomplemented
33C2×#95×C83C2211yescharacteristic
34C23×Dic83C2312yes—
35C23×Dic83C2312yes—
36C23×Dic83C2312yes—
37C23×Dic83C2312yes—
38C2×C4⋊Dic83—24no—
39C2×C4⋊Dic83—24no—
40C2×C4⋊Dic83D418yes—
41C2×C22⋊Dic83—24no—
42C2×C22⋊Dic83—24no—
43C2×C22⋊Dic83—24no—
44C2×C22⋊Dic83—24no—
45C2×C22⋊Dic83—24no—
46C2×C22⋊Dic83—41no—
47C2×C22⋊Dic83—41no—
48C2×C4×Dic83—24no—
49C2×C4×Dic83—24no—
50C2×C4×Dic83—24no—
51C2×C4×Dic83—24no—
52C2×C4×Dic83—24no—
53C2×C4×Dic83—24no—
54C2×C4×Dic83—41no—
55C2×C4×Dic83—42no—
56C83⋊#95—24no—
57C83⋊#95—24no—
58C83⋊#95—24no—
59C83⋊#95—24no—
60C83⋊#95—24no—
61C83⋊#95—24no—
62C83⋊#95—24no—
63C83⋊#95—41no—
64C83⋊#95—42no—
65C25×C83C2311yescharacteristic
66C23×C4×C83C2312yes—
67C23×C4×C83C2311yescharacteristic
68C23×C4×C83C2311yescharacteristic
69C23×C4×C83C2312yes—
70C2×C4⋊C4×C83—24no—
71C2×C4⋊C4×C83—24no—
72C2×C4⋊C4×C83—41no—
73C2×C4⋊C4×C83—41no—
74C2×C4⋊C4×C83C2×C418yes—
75C2×C4⋊C4×C83C2×C414yes—
76C2×C22⋊C4×C83—24no—
77C2×C22⋊C4×C83—24no—
78C2×C22⋊C4×C83—41no—
79C2×C22⋊C4×C83—41no—
80C2×C22⋊C4×C83C2×C418yes—
81C2×C22⋊C4×C83C2×C414yes—
82#95×C83C2×C414yes—
83C22×Dic83—22no—
84C22×Dic83—22no—
85C22×Dic83—22no—
86C22×Dic83—22no—
87C22×Dic83—22no—
88C22×Dic83—22no—
89C22×Dic83—22no—
90C22×Dic83—22no—
91C22×Dic83—22no—
92C22×Dic83—22no—
93C22×Dic83—22no—
94C22×Dic83—22no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83C2×D414yes—
107C22×Dic83C2×D414yes—
108C22×Dic83#4114yes—
109C22×Dic83#4114yes—
110C22×Dic83#4114yes—
111C4⋊Dic83—216no—
112C22⋊Dic83—44no—
113C22⋊Dic83—44no—
114C22⋊Dic83—48no—
115C4×Dic83—216no—
116C4×Dic83—44no—
117C4×Dic83—48no—
118C24×C83—22no—
119C24×C83—22no—
120C24×C83—22no—
121C24×C83—22no—
122C24×C83—22no—
123C24×C83—21no—
124C24×C83—21no—
125C24×C83—22no—
126C24×C83C22×C412yes—
127C24×C83C2411yescharacteristic
128C22×C4×C83—22no—
129C22×C4×C83—22no—
130C22×C4×C83—22no—
131C22×C4×C83—22no—
132C22×C4×C83—22no—
133C22×C4×C83—22no—
134C22×C4×C83—22no—
135C22×C4×C83—22no—
136C22×C4×C83—22no—
137C22×C4×C83—22no—
138C22×C4×C83—22no—
139C22×C4×C83—22no—
140C22×C4×C83—21no—
141C22×C4×C83—21no—
142C22×C4×C83—21no—
143C22×C4×C83—21no—
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83C22×C412yes—
147C22×C4×C83C22×C414yes—
148C22×C4×C83C22×C412yes—
149C22×C4×C83C22×C414yes—
150C22×C4×C83C2×D414yes—
151C22×C4×C83#4114yes—
152C4⋊C4×C83—216no—
153C4⋊C4×C83—44no—
154C22⋊C4×C83—216no—
155C22⋊C4×C83—44no—
156C23×C83—24no—
157C23×C83—22no—
158C23×C83—24no—
159C23×C83—22no—
160C23×C83—22no—
161C23×C83—42no—
162C23×C83—42no—
163C23×C83—42no—
164C23×C83—42no—
165C23×C83—42no—
166C23×C83—42no—
167C23×C83—41no—
168C23×C83—42no—
169C23×C83—42no—
170C23×C83—42no—
171C23×C83—41no—
172C23×C83—42no—
173C23×C83—42no—
174C23×C83—42no—
175C23×C83—41no—
176C23×C83—41no—
177C23×C83C23×C411yescharacteristic
178C23×C83C22×D412yes—
179C23×C83C2×#4112yes—
180C23×C83C2×#4112yes—
181C23×C83C2×#4112yes—
182C23×C83C2×#4112yes—
183C23×C83H(5,𝔽2)11yescharacteristic
184C23×C83H(5,𝔽2)11yescharacteristic
185C23×C832−1+412yes—
186C2×C4×C83—24no—
187C2×C4×C83—24no—
188C2×C4×C83—24no—
189C2×C4×C83—24no—
190C2×C4×C83—24no—
191C2×C4×C83—24no—
192C2×C4×C83—24no—
193C2×C4×C83—24no—
194C2×C4×C83—24no—
195C2×C4×C83—24no—
196C2×C4×C83—22no—
197C2×C4×C83—22no—
198C2×C4×C83—24no—
199C2×C4×C83—42no—
200C2×C4×C83—41no—
201C2×C4×C83—42no—
202C2×C4×C83—42no—
203C2×C4×C83—42no—
204C2×C4×C83—41no—
205C2×C4×C83—42no—
206C2×C4×C83—41no—
207C2×C4×C83—42no—
208C2×C4×C83—41no—
209C2×C4×C83C4×D418yes—
210C2×Dic83—24no—
211C2×Dic83—24no—
212C2×Dic83—24no—
213C2×Dic83—24no—
214C2×Dic83—24no—
215C2×Dic83—24no—
216C2×Dic83—24no—
217C2×Dic83—24no—
218C2×Dic83—24no—
219C2×Dic83—24no—
220C2×Dic83—24no—
221C2×Dic83—24no—
222C2×Dic83—42no—
223C2×Dic83—42no—
224C2×Dic83—42no—
225C2×Dic83—42no—
226C2×Dic83—42no—
227C2×Dic83—42no—
228C2×Dic83—42no—
229C2×Dic83—42no—
230C2×Dic83—42no—
231C2×Dic83—42no—
232C2×Dic83—42no—
233C2×Dic83—42no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83—42no—
237C2×Dic83#12318yes—
238C22×C83—42no—
239C22×C83—44no—
240C22×C83—42no—
241C22×C83—42no—
242C22×C83—44no—
243C22×C83—82no—
244C22×C83—82no—
245C22×C83—82no—
246C22×C83—82no—
247C22×C83C2×C4×D412yes—
248C22×C83C4×#4111yescharacteristic
249C22×C83C4×#4112yes—
250C22×C83#51811yescharacteristic
251C22×C83#51911yescharacteristic
252C22×C83C2×#12312yes—
253C22×C83#52512yes—
254C22×C83#52912yes—
255C22×C83#52912yes—
256C22×C83#53912yes—
257C22×C83#54212yes—
258C22×C83D4211yescharacteristic
259C22×C83#54712yes—
260C22×C83#54812yes—
261C22×C83#55012yes—
262C22×C83#55111yescharacteristic
263C22×C83#55212yes—
264C22×C83#55312yes—
265C22×C83#55612yes—
266C22×C83#55911yescharacteristic
267C22×C83#55911yescharacteristic
268C4×C83—28no—
269C4×C83—44no—
270C4×C83—42no—
271C4×C83—42no—
272Dic83—28no—
273Dic83—44no—
274Dic83—44no—
275Dic83—82no—
276#14194—831nomaximal, Sylow
277C2×C83—82no—
278C2×C83C4×#12312yes—
279C2×C83#233211yescharacteristic
280C2×C83#236212yes—
281C2×C83#236611yescharacteristic
282C2×C83#237411yescharacteristic
283C2×C83#244312yes—
284C2×C83#244812yes—
285C2×C83#247912yes—
286C2×C83#259612yes—
287C2×#399D8311yescharacteristic
288C2×#388—832no—
289C2×C4×C4⋊C4—832no—
290#1445—832no—
291#1440—831no—
292#1437—832no—
293#1437—831no—
294#1436—832no—
295#1436—832no—
296C83#1419411yescharacteristic, complemented, Sylow
297C22×C4⋊C4—832no—
298C22×C4⋊C4C2×D8312yes—
299C22×C4⋊C4C2×D8311yescharacteristic
300C22×C22⋊C4—831no—
301C22×C22⋊C4—831no—
302C22×C22⋊C4—832no—
303C22×C22⋊C4C2×D8312yes—
304C22×C22⋊C4C2×D8311yescharacteristic
305C22×C42—832no—
306C22×C42—831no—
307C22×C42—832no—
308#399Dic8318yescomplemented
309#388—1668no—
310C4×C4⋊C4—1668no—
311C2×#95—832no—
312C2×#95—831no—
313C2×#95—832no—
314C2×#95—832no—
315C2×#95—832no—
316C2×#95—832no—
317C2×#95—832no—
318C2×#95—832no—
319C2×#95—832no—
320C2×#95C2×D8311yescharacteristic
321C25C22×D8311yescharacteristic
322C23×C4—832no—
323C23×C4—832no—
324C23×C4—832no—
325C23×C4—832no—
326C23×C4C22×D8312yes—
327C23×C4C22×D8311yescharacteristic
328C23×C4C22×D8311yescharacteristic
329C23×C4C22×D8312yes—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4—838no—
335C2×C4⋊C4—1664no—
336C2×C4⋊C4—1664no—
337C2×C4⋊C4C2×Dic8318yes—
338C2×C4⋊C4C2×Dic8314yes—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—41no—
342C2×C22⋊C4—41no—
343C2×C22⋊C4—3321no—
344C2×C22⋊C4—3321no—
345C2×C22⋊C4—1664no—
346C2×C22⋊C4—1664no—
347C2×C22⋊C4—1664no—
348C2×C22⋊C4—1664no—
349C2×C22⋊C4—1664no—
350C2×C22⋊C4C2×Dic8318yes—
351C2×C22⋊C4C2×Dic8314yes—
352C2×C42—3321no—
353C2×C42—3322no—
354C2×C42—1664no—
355C2×C42—1664no—
356C2×C42—1664no—
357C2×C42—1664no—
358C2×C42—1664no—
359C2×C42—1664no—
360#95—3321no—
361#95—3322no—
362#95—1664no—
363#95—1664no—
364#95—1664no—
365#95—1664no—
366#95—1664no—
367#95—1664no—
368#95—1664no—
369#95C2×Dic8314yes—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—21no—
376C24—21no—
377C24—22no—
378C24C22×Dic8312yes—
379C24C23×D8311yescharacteristic
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—834no—
399C22×C4—834no—
400C22×C4—834no—
401C22×C4—834no—
402C22×C4—834no—
403C22×C4—1662no—
404C22×C4—1662no—
405C22×C4—1662no—
406C22×C4—1662no—
407C22×C4—1662no—
408C22×C4—1662no—
409C22×C4—1662no—
410C22×C4—1662no—
411C22×C4—1662no—
412C22×C4—1662no—
413C22×C4—1662no—
414C22×C4—1662no—
415C22×C4—1662no—
416C22×C4—1662no—
417C22×C4—1662no—
418C22×C4—1662no—
419C22×C4—1662no—
420C22×C4—1662no—
421C22×C4—1662no—
422C22×C4—1662no—
423C22×C4—1662no—
424C22×C4—1662no—
425C22×C4—1662no—
426C22×C4C22×Dic8312yes—
427C22×C4C22×Dic8314yes—
428C22×C4C22×Dic8312yes—
429C22×C4C22×Dic8314yes—
430C22×C4D4×D8314yes—
431C22×C4#1305058914yes—
432C4⋊C4—216no—
433C4⋊C4—44no—
434C4⋊C4—16616no—
435C22⋊C4—216no—
436C22⋊C4—44no—
437C22⋊C4—3328no—
438C22⋊C4—3324no—
439C22⋊C4—3324no—
440C42—3328no—
441C42—3324no—
442C42—16616no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C23×Dic8311yescharacteristic
465C23C2×D4×D8312yes—
466C23#41×D8312yes—
467C23#41×D8312yes—
468C23#41×D8312yes—
469C23C2×#1305058912yes—
470C23#1458183711yescharacteristic
471C23#1458183711yescharacteristic
472C23#1458184012yes—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—3322no—
497C2×C4—3322no—
498C2×C4—3322no—
499C2×C4—3322no—
500C2×C4—3322no—
501C2×C4—3322no—
502C2×C4—3322no—
503C2×C4—3322no—
504C2×C4—3322no—
505C2×C4—3322no—
506C2×C4—3322no—
507C2×C4—3322no—
508C2×C4—3322no—
509C2×C4—3322no—
510C2×C4—3322no—
511C2×C4—838no—
512C2×C4—1664no—
513C2×C4—1664no—
514C2×C4—1664no—
515C2×C4—1664no—
516C2×C4—1664no—
517C2×C4—1664no—
518C2×C4—1664no—
519C2×C4—1664no—
520C2×C4—1664no—
521C2×C4—1664no—
522C2×C4—1664no—
523C2×C4—1664no—
524C2×C4D4×Dic8318yes—
525C22—42no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22C2×D4×Dic8312yes—
535C22#41×Dic8312yes—
536C22#41×Dic8311yescharacteristic
537C22#1912817611yescharacteristic
538C22#1912817911yescharacteristic
539C22#123×D8312yes—
540C22#1912820912yes—
541C22#1912822812yes—
542C22#1912822812yes—
543C22#1912829712yes—
544C22#1912832112yes—
545C22#1912833611yescharacteristic
546C22#1912835312yes—
547C22#1912836112yes—
548C22#1912837112yes—
549C22#1912837911yescharacteristic
550C22#1912838512yes—
551C22#1912839612yes—
552C22#1912842212yes—
553C22#1912843511yescharacteristic
554C22#1912843711yescharacteristic
555C4—28no—
556C4—42no—
557C4—44no—
558C4—42no—
559C4—3324no—
560C4—3324no—
561C4—6642no—
562C4—1668no—
563C2—82no—
564C2#123×Dic8312yes—
565C2#2491022111yescharacteristic
566C2#2491049412yes—
567C2#2491052811yescharacteristic
568C2#2491059911yescharacteristic
569C2#2491120112yes—
570C2#2491125612yes—
571C2#2491168912yes—
572C2#2491300012yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#399×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#399D83
hypocenterC83#14194
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic83C2semidirect
3. #24905731C2semidirect
7. #24905697C2semidirect
8. #24905695C2semidirect
32. #399×C83C4semidirect
296. C83#14194semidirect
308. #399Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275942
C2×C4⋊C4×Dic832
#249057312
#249057122
#249057002
#249056972
#249056952
C2×#399×C832
#1419483