Finite Group Databasebeta

← #35299656

Subgroups — #35299656

Order 21248.

690 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128049C211yesmaximal, characteristic
2#24910527C211yesmaximal, characteristic
3#24908387C211yesmaximal, characteristic, complemented
4C2×#19127444C211yesmaximal, characteristic
5#24905304C211yesmaximal, characteristic, complemented
6#24905302C211yesmaximal, characteristic, complemented
7C83×#1988C211yesmaximal, characteristic, complemented
8C2×#14581789C2211yescharacteristic
9C2×C4×D332—21no—
10C2×#14581754—21no—
11#19128049—22no—
12#19128049C414yescomplemented
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×#14581716—21no—
15C2×#14581716—21no—
16C2×#14581712—22no—
17C2×C4×C83⋊C8C2211yescharacteristic
18#19127603—21no—
19#19127602—21no—
20#19127579—22no—
21C4×C4⋊Dic83C2211yescharacteristic
22#19127444—22no—
23C2×C4⋊Q8×C83C2211yescharacteristic
24C2×C4⋊C8×C83C2211yescharacteristic
25C4×C4⋊C4×C83C2211yescharacteristic
26#340×C83—21no—
27#340×C83—21no—
28C22×D332—21no—
29C22×D332—21no—
30C22×D332D412yes—
31C22×C4×D83—22no—
32C22×C83⋊C8—21no—
33C22×C83⋊C8—21no—
34#14581789—21no—
35#14581789—21no—
36#14581789C2×C412yes—
37#14581789D412yes—
38C4×D332—42no—
39C4×D332—42no—
40#14581754—42no—
41#14581754—42no—
42C2×C4⋊Dic83D411yescharacteristic
43C2×C4⋊Dic83D411yescharacteristic
44C2×(C2×D83)⋊C4—22no—
45C2×(C2×D83)⋊C4—22no—
46C2×(C2×D83)⋊C4—22no—
47C2×C4×Dic83—21no—
48C4⋊(C83⋊C8)—21no—
49C4⋊(C83⋊C8)—21no—
50#14581716—42no—
51#14581716—42no—
52#14581712—28no—
53C4×C83⋊C8—21no—
54C4×C83⋊C8C2×C412yescomplemented
55C83⋊#95—21no—
56C22×Q8×C83—21no—
57C22×C8×C83—21no—
58C4⋊Q8×C83—21no—
59C4⋊Q8×C83—21no—
60C4⋊Q8×C83—21no—
61C4⋊Q8×C83C2×C412yescomplemented
62C2×C4⋊C4×C83—21no—
63C2×C4⋊C4×C83—21no—
64C2×C4⋊C4×C83D411yescharacteristic
65C2×C4⋊C4×C83D411yescharacteristic
66C2×C42×C83—21no—
67C2×C42×C83C2311yescharacteristic
68C4⋊C8×C83—21no—
69C4⋊C8×C83—21no—
70#95×C83—21no—
71C23×D83—22no—
72C2×D332—22no—
73C2×D332—22no—
74C2×D332—22no—
75C2×D332—22no—
76C2×D332—22no—
77C2×D332—41no—
78C2×D332—41no—
79C2×D332—41no—
80C2×D332—41no—
81C2×D332—41no—
82C2×D332—41no—
83C2×D332—41no—
84C2×D332—41no—
85C2×D332—42no—
86C2×D332—42no—
87C2×D332C22⋊C414yes—
88C22×Dic83—21no—
89C22×Dic83—21no—
90C2×C4×D83—44no—
91C2×C4×D83—44no—
92C2×C4×D83—82no—
93C2×C83⋊C8—22no—
94C2×C83⋊C8—41no—
95C2×C83⋊C8—41no—
96C2×C83⋊C8—41no—
97C2×C83⋊C8—41no—
98C2×C83⋊C8—41no—
99C4⋊Dic83—21no—
100C4⋊Dic83—21no—
101C4⋊Dic83—21no—
102C4⋊Dic83D812yes—
103(C2×D83)⋊C4—44no—
104(C2×D83)⋊C4—44no—
105(C2×D83)⋊C4—82no—
106C4×Dic83—41no—
107C4×Dic83—41no—
108C2×Q8×C83—22no—
109C2×Q8×C83—41no—
110C2×Q8×C83—41no—
111C2×Q8×C83—41no—
112C2×Q8×C83—41no—
113C2×Q8×C83—41no—
114C22×C4×C83—21no—
115C22×C4×C83—21no—
116C22×C4×C83—41no—
117C22×C4×C83C2×D411yescharacteristic
118C22×C4×C83C2×D411yescharacteristic
119C22×C4×C83C2×D411yescharacteristic
120C2×C8×C83—41no—
121C2×C8×C83—41no—
122C2×C8×C83—41no—
123C4⋊C4×C83—21no—
124C4⋊C4×C83—22no—
125C4⋊C4×C83—22no—
126C4⋊C4×C83—21no—
127C4⋊C4×C83—21no—
128C4⋊C4×C83—41no—
129C4⋊C4×C83—41no—
130C4⋊C4×C83SD1612yes—
131C42×C83—41no—
132C42×C83—41no—
133C42×C83C22×C411yescharacteristic
134C42×C83C2×D411yescharacteristic
135C42×C83#4111yescharacteristic
136C42×C83#4111yescharacteristic
137C23×C83#12011yescharacteristic
138C22×D83—24no—
139C22×D83—42no—
140C22×D83—42no—
141C22×D83—42no—
142C22×D83—42no—
143C22×D83—42no—
144C22×D83—42no—
145Q8×C83—42no—
146Q8×C83—42no—
147Q8×C83—81no—
148Q8×C83—81no—
149C2×C4×C83—21no—
150C2×C4×C83—21no—
151C2×C4×C83—22no—
152C2×C4×C83—41no—
153C2×C4×C83—41no—
154C2×C4×C83—42no—
155C2×C4×C83—41no—
156C2×C4×C83—41no—
157C2×C4×C83—41no—
158C2×C4×C83—81no—
159C2×C4×C83—81no—
160C2×C4×C83C2×C22⋊C411yescharacteristic
161C2×C4×C83C4×D411yescharacteristic
162C2×C4×C83C4×D411yescharacteristic
163C2×C4×C83#12111yescharacteristic
164C2×C4×C83#12111yescharacteristic
165C2×C4×C83#12311yescharacteristic
166C2×C4×C83C2×D811yescharacteristic
167C2×C4×C83C2×SD1611yescharacteristic
168C2×C4×C83#13511yescharacteristic
169C2×C4×C83#13511yescharacteristic
170C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C83#13711yescharacteristic
174C2×Dic83—22no—
175C2×Dic83—41no—
176C2×Dic83—41no—
177C2×Dic83—41no—
178C2×Dic83—41no—
179C2×Dic83—41no—
180D332—44no—
181D332—42no—
182D332—42no—
183D332—42no—
184D332—42no—
185D332—42no—
186D332—42no—
187D332—42no—
188D332—42no—
189D332—82no—
190C4×D83—88no—
191C8×C83—81no—
192C83⋊C8—42no—
193C83⋊C8—81no—
194C22×C83#38611yescharacteristic
195C22×C83#44711yescharacteristic
196C22×C83#44911yescharacteristic
197C22×C83#45011yescharacteristic
198C22×C83#45211yescharacteristic
199C22×C83#45511yescharacteristic
200C22×C83#45511yescharacteristic
201C2×D83—44no—
202C2×D83—44no—
203C2×D83—44no—
204C2×D83—82no—
205C2×D83—82no—
206C2×D83—82no—
207C2×D83—82no—
208C4×C83—21no—
209C4×C83—21no—
210C4×C83—42no—
211C4×C83—82no—
212C4×C83—81no—
213C4×C83C4×D811yescharacteristic
214C4×C83C4×SD1611yescharacteristic
215C4×C83#44011yescharacteristic
216C4×C83#44211yescharacteristic
217C4×C83#45911yescharacteristic
218C4×C83#46111yescharacteristic
219C4×C83#46311yescharacteristic
220C4×C83#46311yescharacteristic
221Dic83—42no—
222Dic83—81no—
223#9057—831nomaximal, Sylow
224C2×C83#163911yescharacteristic
225C2×C83#164011yescharacteristic
226C2×C83#164111yescharacteristic
227C2×C83#164211yescharacteristic
228C2×C83#187711yescharacteristic
229C2×C83#187911yescharacteristic
230C2×C83#188511yescharacteristic
231D83—84no—
232C2×#486—831no—
233#2366—831no—
234#1988D8311yescharacteristic, complemented
235#1981—831no—
236C2×#331—831no—
237#1279—831no—
238#1279—831no—
239C83#905711yescharacteristic, complemented, Sylow
240C2×C4⋊Q8C2×D8311yescharacteristic
241C2×#127—831no—
242C2×C4×D4—1661no—
243C2×C4×D4—1661no—
244#486—834no—
245#486—1662no—
246C2×C4⋊C8—831no—
247C2×C4⋊C8C2×D8311yescharacteristic
248C2×#103—1661no—
249C2×#103—1661no—
250C2×#102—1662no—
251C2×C4×C8—831no—
252#390—1661no—
253#390—1661no—
254#386—1662no—
255C4×C4⋊C4—831no—
256C4×C4⋊C4C2×D8311yescharacteristic
257#340—21no—
258#340—21no—
259#331—1662no—
260C22×Q8—21no—
261C22×D4—832no—
262C22×D4—1661no—
263C22×D4—1661no—
264C23×C4—1662no—
265C22×C8—21no—
266C22×C8—1661no—
267C22×C8—1661no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8C4×D8312yescomplemented
272#127—832no—
273#127—832no—
274#127—1661no—
275#127—1661no—
276C4×D4—3322no—
277C4×D4—3322no—
278C4×D4—3322no—
279C4×D4—3322no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—831no—
283C2×C4⋊C4—831no—
284C2×C4⋊C4C22⋊D8311yescharacteristic
285C2×C4⋊C4C22⋊D8311yescharacteristic
286C2×C22⋊C4—1662no—
287C2×C22⋊C4—1662no—
288C2×C22⋊C4—1662no—
289C2×C42—21no—
290C2×C42—1661no—
291C2×C42C22×D8311yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—1661no—
295C4⋊C8—1661no—
296#103—3322no—
297#103—3322no—
298#102—1668no—
299C4×C8—832no—
300C4×C8—1661no—
301#95—21no—
302#95—1661no—
303C24—1662no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×D4—3322no—
311C2×D4—3322no—
312C2×D4—3321no—
313C2×D4—3321no—
314C2×D4—3321no—
315C2×D4—3321no—
316C2×D4—3321no—
317C2×D4—3321no—
318C2×D4—3321no—
319C2×D4—3321no—
320C2×D4—834no—
321C2×D4—1662no—
322C2×D4—1662no—
323C2×D4—1662no—
324C2×D4—1662no—
325C2×D4—1662no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—3324no—
330C22×C4—3324no—
331C22×C4—6642no—
332C22×C4—1661no—
333C22×C4—1661no—
334C22×C4D4×D8311yescharacteristic
335C22×C4D4×D8311yescharacteristic
336C22×C4C2×C22⋊D8311yescharacteristic
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—3321no—
341C2×C8—3321no—
342C2×C8—3321no—
343C2×C8—3321no—
344C2×C8—3321no—
345C2×C8—1662no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—832no—
354C4⋊C4—1661no—
355C4⋊C4—1661no—
356C4⋊C4—1661no—
357C4⋊C4#1305057712yes—
358C22⋊C4—3324no—
359C22⋊C4—3324no—
360C22⋊C4—6642no—
361C42—41no—
362C42—41no—
363C42—3321no—
364C42—3321no—
365C42C2×C4×D8311yescharacteristic
366C42D4×D8311yescharacteristic
367C42#1305058911yescharacteristic
368C42#1305059011yescharacteristic
369C23—3322no—
370C23—3322no—
371C23—3322no—
372C23—3322no—
373C23—3322no—
374C23—3322no—
375C23—1664no—
376C23#1458176011yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
381D4—3324no—
382D4—3322no—
383D4—3322no—
384D4—3322no—
385D4—3322no—
386D4—3322no—
387D4—3322no—
388D4—3322no—
389D4—3322no—
390D4—6642no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—3321no—
403C2×C4—3321no—
404C2×C4—3321no—
405C2×C4—3321no—
406C2×C4—3321no—
407C2×C4—6648no—
408C2×C4—1662no—
409C2×C4C22⋊C4×D8311yescharacteristic
410C2×C4(C22⋊D83)⋊C411yescharacteristic
411C2×C4C4×C22⋊D8311yescharacteristic
412C2×C4#1458176311yescharacteristic
413C2×C4#1458176511yescharacteristic
414C2×C4#1458177411yescharacteristic
415C2×C4D8×D8311yescharacteristic
416C2×C4C2×#1305057711yescharacteristic
417C2×C4#1458181211yescharacteristic
418C2×C4#1458181311yescharacteristic
419C2×C4#1458181611yescharacteristic
420C2×C4#1458181711yescharacteristic
421C2×C4#1458181911yescharacteristic
422C2×C4#1458182311yescharacteristic
423C8—81no—
424C8—3322no—
425C8—6641no—
426C22—3324no—
427C22—3324no—
428C22—3324no—
429C22—6642no—
430C22—6642no—
431C22—6642no—
432C22—6642no—
433C22#1912757711yescharacteristic
434C22#1912783311yescharacteristic
435C22#1912784511yescharacteristic
436C22#1912785111yescharacteristic
437C22#1912786111yescharacteristic
438C22#1912787711yescharacteristic
439C22#1912787711yescharacteristic
440C4—21no—
441C4—21no—
442C4—42no—
443C4—82no—
444C4—81no—
445C4—3322no—
446C4—6641no—
447C4#1912778111yescharacteristic
448C4C4×#1305057711yescharacteristic
449C4#1912779911yescharacteristic
450C4#1912781011yescharacteristic
451C4#1912789511yescharacteristic
452C4#1912790311yescharacteristic
453C4#1912791311yescharacteristic
454C4#1912791511yescharacteristic
455C2—6644no—
456C2#2490664111yescharacteristic
457C2#2490664811yescharacteristic
458C2#2490665511yescharacteristic
459C2#2490666211yescharacteristic
460C2#2490780011yescharacteristic
461C2#2490780811yescharacteristic
462C2#2490784811yescharacteristic
subgroupG/H
centerC23#14581760
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1988C2
socleC23×C83#120
prime radicalC2×#14581789C22
cosocleC2×C42×C83C23
hypercenter#1988D83
hypocenterC83#9057
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
3. #24908387C2semidirect
5. #24905304C2semidirect
6. #24905302C2semidirect
7. C83×#1988C2semidirect
12. #19128049C4semidirect
54. C4×C83⋊C8C2×C4semidirect
61. C4⋊Q8×C83C2×C4semidirect
234. #1988D83semidirect
239. C83#9057semidirect
271. C4⋊Q8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191280492
#249105272
#249083872
C2×#191274442
#249053042
#249053022
C83×#19882
#905783