Finite Group Databasebeta

← #35308626

Subgroups — #35308626

Order 21248.

490 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128049C211yesmaximal, characteristic
2#24908390C211yesmaximal, characteristic
3#24908182C211yesmaximal, characteristic, complemented
4C2×#19127450C211yesmaximal, characteristic, complemented
5C2×#19127444C211yesmaximal, characteristic
6#24905297C211yesmaximal, characteristic, complemented
7C83×#1987C211yesmaximal, characteristic, complemented
8C2×#14581789C2211yescharacteristic
9#19128049—22no—
10#19128049—22no—
11C2×C4⋊(C83⋊C8)—21no—
12C2×C4⋊(C83⋊C8)C2211yescharacteristic
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×#14581712—21no—
15C2×#14581712—21no—
16C2×#14581712—21no—
17C2×#14581712—21no—
18C2×C4×C83⋊C8C2211yescharacteristic
19#19127456—21no—
20#19127450—22no—
21#19127444—22no—
22C2×C4⋊Q8×C83C2211yescharacteristic
23C2×C4⋊C8×C83C2211yescharacteristic
24C4×C4⋊C4×C83C2211yescharacteristic
25#340×C83—21no—
26#340×C83—21no—
27C22×D332—21no—
28C22×D332—21no—
29C22×D332—21no—
30C22×C83⋊C8—21no—
31C22×C83⋊C8—21no—
32C22×C83⋊C8D411yescharacteristic
33C22×C83⋊C8D411yescharacteristic
34#14581789—21no—
35#14581789—21no—
36#14581789—21no—
37#14581789D412yes—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)—21no—
42C4⋊(C83⋊C8)—42no—
43#14581712—24no—
44#14581712—24no—
45#14581712—42no—
46#14581712—42no—
47C4×C83⋊C8—21no—
48C4×C83⋊C8—21no—
49C22×Q8×C83—21no—
50C22×C8×C83—21no—
51C4⋊Q8×C83—21no—
52C4⋊Q8×C83—21no—
53C4⋊Q8×C83—21no—
54C4⋊Q8×C83D412yes—
55C2×C4⋊C4×C83—21no—
56C2×C4⋊C4×C83—21no—
57C2×C4⋊C4×C83D411yescharacteristic
58C2×C4⋊C4×C83D411yescharacteristic
59C2×C42×C83—21no—
60C2×C42×C83C2311yescharacteristic
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63#95×C83—21no—
64C23×D83—41no—
65C2×D332—22no—
66C2×D332—22no—
67C2×D332—22no—
68C2×D332—41no—
69C2×D332—41no—
70C2×D332—41no—
71C2×D332—41no—
72C2×D332—41no—
73C2×D332—41no—
74C2×D332—41no—
75C2×D332—41no—
76C2×D332—41no—
77C2×D332—41no—
78C2×D332—41no—
79C2×D332—81no—
80C2×D332—81no—
81C2×C83⋊C8—21no—
82C2×C83⋊C8—21no—
83C2×C83⋊C8—21no—
84C2×C83⋊C8—21no—
85C2×C83⋊C8—21no—
86C2×C83⋊C8—21no—
87C2×C83⋊C8—41no—
88C2×C83⋊C8—41no—
89C2×C83⋊C8—41no—
90C2×C83⋊C8—41no—
91C2×C83⋊C8—41no—
92C2×C83⋊C8—41no—
93C2×Q8×C83—22no—
94C2×Q8×C83—41no—
95C2×Q8×C83—81no—
96C2×Q8×C83—81no—
97C22×C4×C83—21no—
98C22×C4×C83—21no—
99C22×C4×C83—41no—
100C22×C4×C83C2×D411yescharacteristic
101C22×C4×C83C2×D411yescharacteristic
102C22×C4×C83#4111yescharacteristic
103C2×C8×C83—41no—
104C2×C8×C83—41no—
105C2×C8×C83—41no—
106C4⋊C4×C83—21no—
107C4⋊C4×C83—22no—
108C4⋊C4×C83—22no—
109C4⋊C4×C83—21no—
110C4⋊C4×C83—21no—
111C4⋊C4×C83—41no—
112C4⋊C4×C83—41no—
113C4⋊C4×C83SD1612yes—
114C42×C83—41no—
115C42×C83—41no—
116C42×C83C2×D411yescharacteristic
117C42×C83C2×D411yescharacteristic
118C42×C83#4111yescharacteristic
119C42×C83#4111yescharacteristic
120C23×C83#12111yescharacteristic
121C22×D83—42no—
122C22×D83—81no—
123C22×D83—81no—
124C22×D83—81no—
125C22×D83—81no—
126C22×D83—81no—
127C22×D83—81no—
128Q8×C83—82no—
129Q8×C83—82no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—22no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—42no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—81no—
140C2×C4×C83—81no—
141C2×C4×C83#12011yescharacteristic
142C2×C4×C83#12111yescharacteristic
143C2×C4×C83#12111yescharacteristic
144C2×C4×C83#12311yescharacteristic
145C2×C4×C83#12311yescharacteristic
146C2×C4×C83#12411yescharacteristic
147C2×C4×C83C2×SD1611yescharacteristic
148C2×C4×C83#13511yescharacteristic
149C2×C4×C83#13511yescharacteristic
150C2×C4×C83#13511yescharacteristic
151C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C83#13711yescharacteristic
154C2×C4×C83#13711yescharacteristic
155D332—42no—
156D332—42no—
157D332—42no—
158D332—42no—
159D332—82no—
160D332—82no—
161D332—81no—
162D332—81no—
163D332—81no—
164D332—81no—
165C8×C83—81no—
166C83⋊C8—41no—
167C83⋊C8—41no—
168C83⋊C8—81no—
169C83⋊C8—81no—
170C22×C83#39411yescharacteristic
171C22×C83#46311yescharacteristic
172C22×C83#46411yescharacteristic
173C22×C83#46511yescharacteristic
174C22×C83#46811yescharacteristic
175C22×C83#47211yescharacteristic
176C22×C83#47211yescharacteristic
177C2×D83—82no—
178C2×D83—82no—
179C2×D83—82no—
180C2×D83—161no—
181C2×D83—161no—
182C2×D83—161no—
183C2×D83—161no—
184C4×C83—21no—
185C4×C83—21no—
186C4×C83—42no—
187C4×C83—81no—
188C4×C83—82no—
189C4×C83#44911yescharacteristic
190C4×C83#45211yescharacteristic
191C4×C83#46011yescharacteristic
192C4×C83#46111yescharacteristic
193C4×C83#46311yescharacteristic
194C4×C83#46411yescharacteristic
195C4×C83#48811yescharacteristic
196C4×C83#48911yescharacteristic
197#10539—831nomaximal, Sylow
198C2×C83#167711yescharacteristic
199C2×C83#167811yescharacteristic
200C2×C83#169711yescharacteristic
201C2×C83#169811yescharacteristic
202C2×C83#204211yescharacteristic
203C2×C83#204511yescharacteristic
204C2×C83#205511yescharacteristic
205D83—162no—
206C2×#486—831no—
207#1987D8311yescharacteristic, complemented
208#1982—831no—
209#1939—831no—
210C2×#333—831no—
211C2×#331—831no—
212#1277—831no—
213C83#1053911yescharacteristic, complemented, Sylow
214C2×C4⋊Q8C2×D8311yescharacteristic
215C2×#127—831no—
216#486—1662no—
217#486—1662no—
218C2×C4⋊C8—831no—
219C2×C4⋊C8—831no—
220C2×C4⋊C8—1661no—
221C2×C4⋊C8C2×D8311yescharacteristic
222C2×#102—1661no—
223C2×#102—1661no—
224C2×#102—1661no—
225C2×#102—1661no—
226C2×C4×C8—831no—
227C4×C4⋊C4C2×D8311yescharacteristic
228#340—21no—
229#340—21no—
230#336—1661no—
231#333—1662no—
232#331—1662no—
233C22×Q8—21no—
234C22×D4—1661no—
235C22×D4—1661no—
236C22×D4—1661no—
237C22×C8—21no—
238C22×C8—831no—
239C22×C8—831no—
240C22×C8—1661no—
241C22×C8—1661no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8D33212yes—
246#127—832no—
247#127—1661no—
248#127—1661no—
249#127—1661no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C22⋊D8311yescharacteristic
253C2×C4⋊C4C22⋊D8311yescharacteristic
254C2×C42—21no—
255C2×C42C22×D8311yescharacteristic
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—3322no—
259C4⋊C8—1661no—
260C4⋊C8—1661no—
261C4⋊C8—1661no—
262C4⋊C8—1661no—
263#102—3322no—
264#102—3322no—
265#102—1664no—
266#102—1664no—
267C4×C8—1661no—
268C4×C8—1661no—
269#95—21no—
270C24—3321no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—81no—
274C2×Q8—81no—
275C2×D4—3321no—
276C2×D4—3321no—
277C2×D4—3321no—
278C2×D4—3321no—
279C2×D4—3321no—
280C2×D4—3321no—
281C2×D4—3321no—
282C2×D4—3321no—
283C2×D4—3321no—
284C2×D4—3321no—
285C2×D4—3321no—
286C2×D4—6641no—
287C2×D4—6641no—
288C2×D4—1662no—
289C2×D4—1662no—
290C2×D4—1662no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4D4×D8311yescharacteristic
295C22×C4C2×C22⋊D8311yescharacteristic
296C22×C4#1305058911yescharacteristic
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—3321no—
301C2×C8—3321no—
302C2×C8—3321no—
303C2×C8—3321no—
304C2×C8—3321no—
305C2×C8—3321no—
306C2×C8—1661no—
307C2×C8—1661no—
308C2×C8—1661no—
309C2×C8—1661no—
310C2×C8—1661no—
311C2×C8—1661no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4#1305057712yes—
320C42—41no—
321C42—41no—
322C42D4×D8311yescharacteristic
323C42C2×D33211yescharacteristic
324C42#1305058911yescharacteristic
325C42#1305059011yescharacteristic
326C23—3322no—
327C23—6641no—
328C23—6641no—
329C23—6641no—
330C23—6641no—
331C23—6641no—
332C23—6641no—
333C23#1458176411yescharacteristic
334Q8—82no—
335Q8—82no—
336D4—3322no—
337D4—3322no—
338D4—3322no—
339D4—3322no—
340D4—6642no—
341D4—6642no—
342D4—6641no—
343D4—6641no—
344D4—6641no—
345D4—6641no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4#1458176211yescharacteristic
358C2×C4#1458176311yescharacteristic
359C2×C4#1458176611yescharacteristic
360C2×C4#1458177411yescharacteristic
361C2×C4#1458177611yescharacteristic
362C2×C4#1458177811yescharacteristic
363C2×C4C2×#1305057711yescharacteristic
364C2×C4#1458181211yescharacteristic
365C2×C4#1458181311yescharacteristic
366C2×C4#1458181511yescharacteristic
367C2×C4#1458181711yescharacteristic
368C2×C4#1458181911yescharacteristic
369C2×C4#1458182311yescharacteristic
370C2×C4#1458182411yescharacteristic
371C8—81no—
372C8—3321no—
373C8—3321no—
374C8—6641no—
375C8—6641no—
376C22—13281no—
377C22—13281no—
378C22—13281no—
379C22—13281no—
380C22—6642no—
381C22—6642no—
382C22—6642no—
383C22#1912761711yescharacteristic
384C22#1912791811yescharacteristic
385C22#1912792511yescharacteristic
386C22#1912793211yescharacteristic
387C22#1912794911yescharacteristic
388C22#1912796911yescharacteristic
389C22#1912796911yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—82no—
394C4—81no—
395C4#1912784911yescharacteristic
396C4#1912786611yescharacteristic
397C4#1912790111yescharacteristic
398C4#1912790311yescharacteristic
399C4#1912791311yescharacteristic
400C4#1912792311yescharacteristic
401C4#1912805711yescharacteristic
402C4#1912806211yescharacteristic
403C2—13282no—
404C2#2490687411yescharacteristic
405C2#2490688111yescharacteristic
406C2#2490699811yescharacteristic
407C2#2490700511yescharacteristic
408C2#2490870111yescharacteristic
409C2#2490871811yescharacteristic
410C2#2490877811yescharacteristic
subgroupG/H
centerC23#14581764
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1987C2
socleC23×C83#121
prime radicalC2×#14581789C22
hypercenter#1987D83
hypocenterC83#10539
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. #24908182C2semidirect
4. C2×#19127450C2semidirect
6. #24905297C2semidirect
7. C83×#1987C2semidirect
207. #1987D83semidirect
213. C83#10539semidirect

Maximal subgroups

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

subgroupindex
C2×#191280492
#249083902
#249081822
C2×#191274502
C2×#191274442
#249052972
C83×#19872
#1053983