Finite Group Databasebeta

← #35303549

Subgroups — #35303549

Order 21248.

650 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907443C211yesmaximal, characteristic, complemented
2C2×#19127438C212yesmaximal, complemented
3C2×#19127435C212yesmaximal
4#24905690C211yesmaximal, characteristic, complemented
5C83×#2850C211yesmaximal, characteristic
6C2×C4⋊(C83⋊C8)—21no—
7C2×C22⋊(C83⋊C8)C2212yes—
8C2×C83⋊#95—21no—
9C2×C83⋊#95—21no—
10C2×C83⋊#95C2212yes—
11#19127456—21no—
12#19127438—24no—
13#19127435—24no—
14C2×#124×C83C411yescharacteristic, complemented
15C2×C22⋊Q8×C83C2211yescharacteristic
16C2×#121×C83C2211yescharacteristic
17C22×C42×C83C2211yescharacteristic
18#391×C83C411yescharacteristic, complemented
19#390×C83C411yescharacteristic, complemented
20#388×C83—21no—
21#388×C83—21no—
22#386×C83—21no—
23#386×C83—21no—
24#382×C83C411yescharacteristic, complemented
25C23×Dic83—22no—
26C22×C83⋊C8—22no—
27C4⋊(C83⋊C8)—42no—
28C22⋊(C83⋊C8)—22no—
29C22⋊(C83⋊C8)—22no—
30C83⋊#95—22no—
31C83⋊#95—42no—
32C83⋊#95—42no—
33C83⋊#95D414yes—
34C22×Q8×C83C2×C411yescharacteristic
35C22×D4×C83—21no—
36C22×D4×C83C2×C411yescharacteristic
37C23×C4×C83D411yescharacteristic
38C23×C4×C83D411yescharacteristic
39C23×C4×C83C2311yescharacteristic
40#124×C83—42no—
41C22⋊Q8×C83—22no—
42C22⋊Q8×C83—22no—
43#121×C83—22no—
44#121×C83—22no—
45C2×C4⋊C4×C83—21no—
46C2×C4⋊C4×C83C2×C411yescharacteristic
47C2×C4⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83—21no—
49C2×C22⋊C4×C83—21no—
50C2×C22⋊C4×C83—21no—
51C2×C22⋊C4×C83—21no—
52C2×C22⋊C4×C83—21no—
53C2×C22⋊C4×C83—21no—
54C2×C42×C83—42no—
55C2×C42×C83C2×C411yescharacteristic
56C2×C42×C83C2×C411yescharacteristic
57C2×C42×C83D411yescharacteristic
58C2×C42×C83D411yescharacteristic
59#95×C83—21no—
60#95×C83—21no—
61#95×C83—21no—
62#95×C83—21no—
63C22×Dic83—24no—
64C22×Dic83—42no—
65C22×Dic83—42no—
66C22×Dic83—42no—
67C22×Dic83—42no—
68C22×Dic83—42no—
69C22×Dic83—42no—
70C2×C83⋊C8—42no—
71C2×C83⋊C8—42no—
72C2×C83⋊C8—42no—
73C24×C83—21no—
74C24×C83C2×D411yescharacteristic
75C2×Q8×C83—22no—
76C2×Q8×C83—41no—
77C2×Q8×C83—41no—
78C2×D4×C83—22no—
79C2×D4×C83—41no—
80C2×D4×C83—41no—
81C2×D4×C83—42no—
82C2×D4×C83—42no—
83C2×D4×C83—41no—
84C2×D4×C83—41no—
85C22×C4×C83—22no—
86C22×C4×C83—21no—
87C22×C4×C83—21no—
88C22×C4×C83—22no—
89C22×C4×C83—21no—
90C22×C4×C83—21no—
91C22×C4×C83—21no—
92C22×C4×C83—41no—
93C22×C4×C83—41no—
94C22×C4×C83—41no—
95C22×C4×C83—41no—
96C22×C4×C83—41no—
97C22×C4×C83C22⋊C411yescharacteristic
98C22×C4×C83C22⋊C411yescharacteristic
99C22×C4×C83C22⋊C411yescharacteristic
100C22×C4×C83C22⋊C411yescharacteristic
101C22×C4×C83C22×C411yescharacteristic
102C22×C4×C83C2×D412yes—
103C22×C4×C83C2×D411yescharacteristic
104C22×C4×C83#4112yes—
105C4⋊C4×C83—22no—
106C4⋊C4×C83—22no—
107C4⋊C4×C83—41no—
108C4⋊C4×C83—41no—
109C22⋊C4×C83—42no—
110C22⋊C4×C83—42no—
111C22⋊C4×C83—42no—
112C22⋊C4×C83—42no—
113C22⋊C4×C83—42no—
114C22⋊C4×C83—41no—
115C22⋊C4×C83—41no—
116C42×C83—22no—
117C42×C83—22no—
118C42×C83—41no—
119C42×C83—41no—
120C23×C83—21no—
121C23×C83—21no—
122C23×C83—21no—
123C23×C83—21no—
124C23×C83—41no—
125C23×C83—41no—
126C23×C83—41no—
127C23×C83—41no—
128C23×C83—41no—
129C23×C83—41no—
130C23×C83—41no—
131C23×C83C2×C22⋊C411yescharacteristic
132C23×C83#12112yes—
133C23×C83#13512yes—
134C23×C83AGL(1,ℤ/8)11yescharacteristic
135C23×C83#13711yescharacteristic
136Q8×C83—82no—
137D4×C83—82no—
138D4×C83—82no—
139D4×C83—82no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—21no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83—21no—
147C2×C4×C83—21no—
148C2×C4×C83—21no—
149C2×C4×C83—21no—
150C2×C4×C83—21no—
151C2×C4×C83—21no—
152C2×C4×C83—21no—
153C2×C4×C83—21no—
154C2×C4×C83—21no—
155C2×C4×C83—42no—
156C2×C4×C83—42no—
157C2×C4×C83—44no—
158C2×C4×C83—42no—
159C2×C4×C83—42no—
160C2×C4×C83—41no—
161C2×C4×C83—41no—
162C2×C4×C83—41no—
163C2×C4×C83—41no—
164C2×C4×C83—41no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—41no—
168C2×C4×C83—41no—
169C2×C4×C83#9912yes—
170C2×C4×C83C4×D412yes—
171C2×C4×C83#12411yescharacteristic
172C2×C4×C83#12711yescharacteristic
173C2×Dic83—44no—
174C2×Dic83—44no—
175C2×Dic83—44no—
176C2×Dic83—82no—
177C2×Dic83—82no—
178C2×Dic83—82no—
179C2×Dic83—82no—
180C83⋊C8—82no—
181C22×C83—22no—
182C22×C83—22no—
183C22×C83—21no—
184C22×C83—21no—
185C22×C83—21no—
186C22×C83—21no—
187C22×C83—22no—
188C22×C83—22no—
189C22×C83—81no—
190C22×C83—81no—
191C22×C83—81no—
192C22×C83—81no—
193C22×C83—81no—
194C22×C83—81no—
195C22×C83—81no—
196C22×C83#39011yescharacteristic
197C22×C83C2×#9911yescharacteristic
198C22×C83#41011yescharacteristic
199C22×C83#41611yescharacteristic
200C22×C83#41711yescharacteristic
201C22×C83#42111yescharacteristic
202C22×C83#42111yescharacteristic
203C22×C83#46312yes—
204C22×C83#46412yes—
205C4×C83—21no—
206C4×C83—22no—
207C4×C83—21no—
208C4×C83—22no—
209C4×C83—41no—
210C4×C83—41no—
211C4×C83—41no—
212C4×C83—81no—
213C4×C83—81no—
214C4×C83—81no—
215Dic83—84no—
216#9716—831nomaximal, Sylow
217C2×C83—22no—
218C2×C83—22no—
219C2×C83—161no—
220C2×C83#151211yescharacteristic
221C2×C83#151311yescharacteristic
222C2×C83#151611yescharacteristic
223C2×C83#152111yescharacteristic
224C2×C83#196311yescharacteristic
225C2×C83#196411yescharacteristic
226C2×C83#198311yescharacteristic
227#2850D8311yescharacteristic
228#1797—831no—
229C2×#328—832no—
230C2×#327—832no—
231#1434—831no—
232C83#971611yescharacteristic, complemented, Sylow
233C2×#124Dic8311yescharacteristic, complemented
234C2×C22⋊Q8C2×D8311yescharacteristic
235C2×#121C2×D8311yescharacteristic
236C22×C42C2×D8311yescharacteristic
237C2×C4⋊C8—1661no—
238C2×C22⋊C8—832no—
239#391Dic8311yescharacteristic, complemented
240#390Dic8311yescharacteristic, complemented
241#388—21no—
242#388—21no—
243#386—21no—
244#386—21no—
245#382Dic8311yescharacteristic, complemented
246C2×#95—832no—
247C2×#95—1661no—
248C2×#95—1661no—
249#336—1661no—
250#328—1664no—
251#327—1664no—
252C22×Q8C2×Dic8311yescharacteristic
253C22×D4—21no—
254C22×D4C2×Dic8311yescharacteristic
255C23×C4—1662no—
256C23×C4C22⋊D8311yescharacteristic
257C23×C4C22⋊D8311yescharacteristic
258C23×C4C22×D8311yescharacteristic
259C22×C8—1662no—
260#124—42no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263#121—22no—
264#121—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×Dic8311yescharacteristic
267C2×C4⋊C4C2×Dic8311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—42no—
275C2×C42C2×Dic8311yescharacteristic
276C2×C42C2×Dic8311yescharacteristic
277C2×C42C22⋊D8311yescharacteristic
278C2×C42C22⋊D8311yescharacteristic
279C4⋊C8—3322no—
280C22⋊C8—1662no—
281C22⋊C8—1662no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—3322no—
287#95—3322no—
288#95—834no—
289#95—1662no—
290C24—21no—
291C24C2×C22⋊D8311yescharacteristic
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—3322no—
315C22×C4—3322no—
316C22×C4—3322no—
317C22×C4—3322no—
318C22×C4—3322no—
319C22×C4—3322no—
320C22×C4—1664no—
321C22×C4C22⋊Dic8311yescharacteristic
322C22×C4C22⋊Dic8311yescharacteristic
323C22×C4C22⋊Dic8311yescharacteristic
324C22×C4C22⋊Dic8311yescharacteristic
325C22×C4C22×Dic8311yescharacteristic
326C22×C4D4×D8312yes—
327C22×C4C2×C22⋊D8311yescharacteristic
328C22×C4#1305058912yes—
329C2×C8—3322no—
330C2×C8—3322no—
331C2×C8—3322no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—42no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23C2×C22⋊Dic8311yescharacteristic
359C23#1458176312yes—
360C23#1458181212yes—
361C23#1458181711yescharacteristic
362C23#1458182211yescharacteristic
363Q8—82no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—3324no—
397C2×C4—3324no—
398C2×C4—3324no—
399C2×C4—6642no—
400C2×C4—6642no—
401C2×C4—6642no—
402C2×C4—6642no—
403C2×C4#1458170712yes—
404C2×C4D4×Dic8312yes—
405C2×C4#1458177911yescharacteristic
406C2×C4#1458178811yescharacteristic
407C8—6642no—
408C22—22no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22#1912760411yescharacteristic
424C22C2×#1458170711yescharacteristic
425C22#1912767511yescharacteristic
426C22#1912769711yescharacteristic
427C22#1912770211yescharacteristic
428C22#1912771811yescharacteristic
429C22#1912771811yescharacteristic
430C22#1912791312yes—
431C22#1912792012yes—
432C4—21no—
433C4—21no—
434C4—22no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—6644no—
443C2—22no—
444C2—22no—
445C2—161no—
446C2#2490600111yescharacteristic
447C2#2490600611yescharacteristic
448C2#2490601911yescharacteristic
449C2#2490604011yescharacteristic
450C2#2490829611yescharacteristic
451C2#2490830011yescharacteristic
452C2#2490839611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2850C2
socleC23×C83C2×C22⋊C4
prime radicalC2×#121×C83C22
cosocleC23×C4×C83C23
hypercenter#2850D83
hypocenterC83#9716
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. #24907443C2semidirect
2. C2×#19127438C2semidirect
4. #24905690C2semidirect
14. C2×#124×C83C4semidirect
18. #391×C83C4semidirect
19. #390×C83C4semidirect
24. #382×C83C4semidirect
232. C83#9716semidirect
233. C2×#124Dic83semidirect
239. #391Dic83semidirect
240. #390Dic83semidirect
245. #382Dic83semidirect

Maximal subgroups

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

subgroupindex
#249074432
C2×#191274382
C2×#191274352
#249056902
C83×#28502
#971683