Finite Group Databasebeta

← #35320367

Subgroups — #35320367

Order 21248.

808 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127488C212yesmaximal
2#24907455C211yesmaximal, characteristic, complemented
3#24907451C211yesmaximal, characteristic, complemented
4#24905581C212yesmaximal, complemented
5C4×#120×C83C211yesmaximal, characteristic
6#19127725—21no—
7C2×#14581709C2212yes—
8C2×#14581707C2212yes—
9#19127561—21no—
10#19127488—24no—
11#19127470—21no—
12#19127468—21no—
13C24×C4×C83C411yescharacteristic, complemented
14C2×#120×C83C2211yescharacteristic
15C2×C4×D4×C83—21no—
16C2×C4×D4×C83C2211yescharacteristic
17#386×C83—21no—
18#386×C83C2211yescharacteristic
19#385×C83—21no—
20#385×C83C411yescharacteristic, complemented
21#379×C83C411yescharacteristic, complemented
22C4×C22⋊C4×C83—21no—
23C4×C22⋊C4×C83C411yescharacteristic, complemented
24C2×#13050574—22no—
25C2×C4⋊Dic83—41no—
26C2×C22⋊Dic83—22no—
27C4⋊(C83⋊C8)—41no—
28C4⋊(C83⋊C8)—41no—
29#14581709—22no—
30#14581709—22no—
31#14581707—22no—
32#14581707—22no—
33C83⋊#95—41no—
34C25×C83C2×C411yescharacteristic
35C22×D4×C83—21no—
36C22×D4×C83C2311yescharacteristic
37C23×C4×C83—21no—
38C23×C4×C83—21no—
39C23×C4×C83—44no—
40C23×C4×C83—41no—
41C23×C4×C83—41no—
42C23×C4×C83C2×C411yescharacteristic
43C23×C4×C83C2×C411yescharacteristic
44C23×C4×C83D411yescharacteristic
45#120×C83—21no—
46#120×C83—21no—
47#120×C83D412yes—
48#120×C83Q812yes—
49C4×D4×C83—41no—
50C4×D4×C83—41no—
51C4×D4×C83—44no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83D411yescharacteristic
54C2×C22⋊C4×C83—21no—
55C2×C22⋊C4×C83—21no—
56C2×C22⋊C4×C83—21no—
57C2×C22⋊C4×C83—21no—
58C2×C22⋊C4×C83C2×C411yescharacteristic
59C2×C22⋊C4×C83C2×C411yescharacteristic
60C2×C22⋊C4×C83C2×C411yescharacteristic
61C2×C22⋊C4×C83D411yescharacteristic
62C2×C42×C83—21no—
63C2×C42×C83D411yescharacteristic
64#95×C83—21no—
65#95×C83—21no—
66#95×C83—21no—
67C22×Dic83—42no—
68#13050574—42no—
69#13050574—42no—
70C2×C83⋊C8—42no—
71C4⋊Dic83—82no—
72C22⋊Dic83—42no—
73C22⋊Dic83—42no—
74C24×C83—21no—
75C24×C83—21no—
76C24×C83—42no—
77C24×C83—41no—
78C24×C83—42no—
79C24×C83—41no—
80C24×C83C4⋊C412yescomplemented
81C24×C83C22×C411yescharacteristic
82C24×C83C2×D411yescharacteristic
83C2×D4×C83—22no—
84C2×D4×C83—22no—
85C2×D4×C83—41no—
86C2×D4×C83—41no—
87C2×D4×C83—42no—
88C2×D4×C83—42no—
89C2×D4×C83C2×D411yescharacteristic
90C2×D4×C83C2×Q811yescharacteristic
91C2×D4×C83#4111yescharacteristic
92C2×D4×C83#4111yescharacteristic
93C22×C4×C83—24no—
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83—21no—
97C22×C4×C83—21no—
98C22×C4×C83—21no—
99C22×C4×C83—21no—
100C22×C4×C83—21no—
101C22×C4×C83—42no—
102C22×C4×C83—42no—
103C22×C4×C83—42no—
104C22×C4×C83—42no—
105C22×C4×C83—42no—
106C22×C4×C83—42no—
107C22×C4×C83—42no—
108C22×C4×C83—44no—
109C22×C4×C83—44no—
110C22×C4×C83—44no—
111C22×C4×C83—41no—
112C22×C4×C83—41no—
113C22×C4×C83—41no—
114C22×C4×C83—44no—
115C22×C4×C83C22⋊C411yescharacteristic
116C22×C4×C83C22⋊C411yescharacteristic
117C22×C4×C83C22⋊C411yescharacteristic
118C22×C4×C83C22⋊C411yescharacteristic
119C22×C4×C83C2×D411yescharacteristic
120C4⋊C4×C83—41no—
121C4⋊C4×C83—42no—
122C22⋊C4×C83—21no—
123C22⋊C4×C83—22no—
124C22⋊C4×C83—22no—
125C22⋊C4×C83—41no—
126C22⋊C4×C83—41no—
127C22⋊C4×C83—42no—
128C22⋊C4×C83—42no—
129C22⋊C4×C83—42no—
130C22⋊C4×C83—41no—
131C22⋊C4×C83C4⋊C412yescomplemented
132C42×C83—21no—
133C42×C83—21no—
134C42×C83—42no—
135C23×C83—22no—
136C23×C83—22no—
137C23×C83—22no—
138C23×C83—21no—
139C23×C83—21no—
140C23×C83—21no—
141C23×C83—21no—
142C23×C83—41no—
143C23×C83—44no—
144C23×C83—42no—
145C23×C83—41no—
146C23×C83—41no—
147C23×C83—41no—
148C23×C83—44no—
149C23×C83—42no—
150C23×C83—41no—
151C23×C83—41no—
152C23×C83—42no—
153C23×C83—42no—
154C23×C83—42no—
155C23×C83—48no—
156C23×C83—42no—
157C23×C83—41no—
158C23×C83—41no—
159C23×C83C2×C22⋊C411yescharacteristic
160C23×C83C2×C4⋊C411yescharacteristic
161C23×C83#11711yescharacteristic
162C23×C83#12111yescharacteristic
163C23×C83C22⋊Q811yescharacteristic
164D4×C83—41no—
165D4×C83—41no—
166D4×C83—41no—
167D4×C83—41no—
168D4×C83—44no—
169D4×C83—44no—
170C2×C4×C83—21no—
171C2×C4×C83—21no—
172C2×C4×C83—22no—
173C2×C4×C83—22no—
174C2×C4×C83—22no—
175C2×C4×C83—21no—
176C2×C4×C83—21no—
177C2×C4×C83—21no—
178C2×C4×C83—44no—
179C2×C4×C83—42no—
180C2×C4×C83—41no—
181C2×C4×C83—41no—
182C2×C4×C83—41no—
183C2×C4×C83—48no—
184C2×C4×C83—42no—
185C2×C4×C83—41no—
186C2×C4×C83—44no—
187C2×C4×C83—42no—
188C2×C4×C83—42no—
189C2×C4×C83—41no—
190C2×C4×C83—41no—
191C2×C4×C83—41no—
192C2×C4×C83—42no—
193C2×C4×C83—41no—
194C2×C4×C83—41no—
195C2×C4×C83—41no—
196C2×C4×C83—81no—
197C2×C4×C83#9911yescharacteristic
198C2×C4×C83#9911yescharacteristic
199C2×C4×C83#12111yescharacteristic
200C2×C4×C83C22⋊Q811yescharacteristic
201C2×Dic83—82no—
202C2×Dic83—82no—
203C2×Dic83—82no—
204C83⋊C8—82no—
205C22×C83—21no—
206C22×C83—21no—
207C22×C83—24no—
208C22×C83—21no—
209C22×C83—21no—
210C22×C83—21no—
211C22×C83—21no—
212C22×C83—41no—
213C22×C83—41no—
214C22×C83—41no—
215C22×C83—41no—
216C22×C83—42no—
217C22×C83—42no—
218C22×C83—42no—
219C22×C83—42no—
220C22×C83—44no—
221C22×C83—44no—
222C22×C83—44no—
223C22×C83—44no—
224C22×C83—42no—
225C22×C83—41no—
226C22×C83—42no—
227C22×C83—41no—
228C22×C83—41no—
229C22×C83—42no—
230C22×C83—41no—
231C22×C83—81no—
232C22×C83#38011yescharacteristic
233C22×C83C2×#9911yescharacteristic
234C22×C83#41011yescharacteristic
235C4×C83—21no—
236C4×C83—21no—
237C4×C83—21no—
238C4×C83—42no—
239C4×C83—42no—
240C4×C83—41no—
241C4×C83—41no—
242C4×C83—41no—
243C4×C83—81no—
244C4×C83AΣL(1,𝔽16)12yescomplemented
245Dic83—162no—
246#12793—831nomaximal, Sylow
247C2×C83—21no—
248C2×C83—21no—
249C2×C83—41no—
250C2×C83—44no—
251C2×C83—41no—
252C2×C83—41no—
253C2×C83—41no—
254C2×C83#179911yescharacteristic
255C2×C83C2×AΣL(1,𝔽16)11yescharacteristic
256C2×C83#212011yescharacteristic
257C4×#120D8311yescharacteristic
258C2×AΣL(1,𝔽16)—832no—
259#1800—831no—
260#1799—831no—
261#1390—832no—
262C83#1279311yescharacteristic, complemented, Sylow
263C24×C4Dic8311yescharacteristic, complemented
264C2×#120C2×D8311yescharacteristic
265C2×C4×D4—21no—
266C2×C4×D4C2×D8311yescharacteristic
267#423—1661no—
268C2×#100—832no—
269C2×#99—832no—
270#386—21no—
271#386C2×D8311yescharacteristic
272#385—21no—
273#385Dic8311yescharacteristic, complemented
274#380—1661no—
275#379Dic8311yescharacteristic, complemented
276C4×C22⋊C4—21no—
277C4×C22⋊C4Dic8311yescharacteristic, complemented
278AΣL(1,𝔽16)—1664no—
279#343—1661no—
280#342—1661no—
281C25C2×Dic8311yescharacteristic
282C22×D4—21no—
283C22×D4C22×D8311yescharacteristic
284C23×C4—21no—
285C23×C4—21no—
286C23×C4—44no—
287C23×C4—41no—
288C23×C4—41no—
289C23×C4C2×Dic8311yescharacteristic
290C23×C4C2×Dic8311yescharacteristic
291C23×C4C22⋊D8311yescharacteristic
292C2×M16—1662no—
293#120—21no—
294#120—21no—
295#120Dic16612yes—
296#120D33212yes—
297C4×D4—41no—
298C4×D4—41no—
299C4×D4—44no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—3321no—
302C2×C4⋊C4C22⋊D8311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—1662no—
308C2×C22⋊C4C2×Dic8311yescharacteristic
309C2×C22⋊C4C2×Dic8311yescharacteristic
310C2×C22⋊C4C2×Dic8311yescharacteristic
311C2×C22⋊C4C22⋊D8311yescharacteristic
312C2×C42—21no—
313C2×C42C22⋊D8311yescharacteristic
314C4⋊C8—3321no—
315C4⋊C8—3321no—
316#100—1662no—
317#100—1662no—
318#99—1662no—
319#99—1662no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—3321no—
324C24—21no—
325C24—21no—
326C24—42no—
327C24—41no—
328C24—42no—
329C24—41no—
330C24C4⋊Dic8312yescomplemented
331C24C22×Dic8311yescharacteristic
332C24C2×C22⋊D8311yescharacteristic
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C2×D33211yescharacteristic
340C2×D4C2×Dic16611yescharacteristic
341C2×D4#1305059011yescharacteristic
342C2×D4#1305059011yescharacteristic
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—44no—
365C22×C4—3322no—
366C22×C4C22⋊Dic8311yescharacteristic
367C22×C4C22⋊Dic8311yescharacteristic
368C22×C4C22⋊Dic8311yescharacteristic
369C22×C4C22⋊Dic8311yescharacteristic
370C22×C4C2×C22⋊D8311yescharacteristic
371M16—3322no—
372M16—3322no—
373C2×C8—3322no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—6642no—
377C22⋊C4—21no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—41no—
386C22⋊C4—3322no—
387C22⋊C4—3322no—
388C22⋊C4C4⋊Dic8312yescomplemented
389C42—21no—
390C42—21no—
391C42—42no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—41no—
400C23—44no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—44no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—48no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23C2×C22⋊Dic8311yescharacteristic
417C23C2×C4⋊Dic8311yescharacteristic
418C23#1458175111yescharacteristic
419C23#1458176611yescharacteristic
420C23#1458177111yescharacteristic
421D4—41no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—44no—
426D4—44no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—81no—
454C2×C4—6642no—
455C2×C4—6642no—
456C2×C4—6642no—
457C2×C4#1458170711yescharacteristic
458C2×C4#1458170711yescharacteristic
459C2×C4#1458176611yescharacteristic
460C2×C4#1458177111yescharacteristic
461C8—6642no—
462C22—21no—
463C22—21no—
464C22—24no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—42no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—42no—
486C22—41no—
487C22—42no—
488C22—81no—
489C22#1912756111yescharacteristic
490C22C2×#1458170711yescharacteristic
491C22#1912767511yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—42no—
496C4—42no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—81no—
501C4—13282no—
502C4#1912748812yescomplemented
503C2—21no—
504C2—21no—
505C2—41no—
506C2—44no—
507C2—41no—
508C2—41no—
509C2—41no—
510C2#2490745111yescharacteristic
511C2C2×#1912748811yescharacteristic
512C2#2490910411yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC24×C83C22×C4
FrattiniC22×D4C22×D83
FittingC4×#120×C83C2
socleC22×C83C2×#99
prime radicalC2×#120×C83C22
cosocleC22×D4×C83C23
hypercenterC4×#120D83
hypocenterC83#12793
LCSG ▷ C24×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
2. #24907455C2semidirect
3. #24907451C2semidirect
4. #24905581C2semidirect
13. C24×C4×C83C4semidirect
20. #385×C83C4semidirect
21. #379×C83C4semidirect
23. C4×C22⋊C4×C83C4semidirect
80. C24×C83C4⋊C4semidirect
131. C22⋊C4×C83C4⋊C4semidirect
244. C4×C83AΣL(1,𝔽16)semidirect
262. C83#12793semidirect
263. C24×C4Dic83semidirect
273. #385Dic83semidirect
275. #379Dic83semidirect
277. C4×C22⋊C4Dic83semidirect
330. C24C4⋊Dic83semidirect
388. C22⋊C4C4⋊Dic83semidirect
502. C4#19127488semidirect

Maximal subgroups

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

subgroupindex
C2×#191274882
#249074552
#249074512
#249055812
C4×#120×C832
#1279383