Finite Group Databasebeta

← #35308630

Subgroups — #35308630

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×#19128048C211yesmaximal, characteristic
2#24908410C211yesmaximal, characteristic, complemented
3#24908389C211yesmaximal, characteristic
4#24908181C211yesmaximal, characteristic, complemented
5C2×#19127449C211yesmaximal, characteristic, complemented
6C2×#19127444C211yesmaximal, characteristic
7C83×#1277C211yesmaximal, characteristic, complemented
8C2×#14581792C2211yescharacteristic
9C2×#14581789C2211yescharacteristic
10#19128048—22no—
11#19128048—22no—
12C2×C4⋊(C83⋊C8)C2211yescharacteristic
13C2×Dic83⋊C8—21no—
14C2×#14581714—21no—
15C2×#14581714—21no—
16C2×#14581714—21no—
17C2×#14581714—21no—
18C4×C4⋊Dic83C2211yescharacteristic
19#19127464—21no—
20#19127464—21no—
21#19127455—21no—
22#19127449—22no—
23#19127444—22no—
24C2×C4⋊C8×C83C2211yescharacteristic
25C2×C4⋊C8×C83C2211yescharacteristic
26C2×C4×C8×C83C2211yescharacteristic
27C22×Dic166—21no—
28C22×D332—21no—
29C22×D332—21no—
30C22×D332—21no—
31C22×C83⋊C8—21no—
32#14581792—21no—
33#14581792—21no—
34#14581792—21no—
35#14581792D412yes—
36#14581789—21no—
37#14581789—21no—
38#14581789—21no—
39#14581789D412yes—
40C2×C4⋊Dic83—21no—
41C2×C4⋊Dic83—21no—
42C2×C4⋊Dic83D411yescharacteristic
43C2×C4⋊Dic83D411yescharacteristic
44C2×C4×Dic83—21no—
45C4⋊(C83⋊C8)—21no—
46C4⋊(C83⋊C8)—21no—
47Dic83⋊C8—42no—
48#14581714—24no—
49#14581714—24no—
50#14581714—42no—
51#14581714—42no—
52C83⋊#95—21no—
53C22×C8×C83—21no—
54C22×C8×C83—21no—
55C22×C8×C83D411yescharacteristic
56C22×C8×C83D411yescharacteristic
57C2×C42×C83C2311yescharacteristic
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4×C8×C83—21no—
63C4×C8×C83—21no—
64C23×D83—41no—
65C2×Dic166—22no—
66C2×Dic166—41no—
67C2×Dic166—81no—
68C2×Dic166—81no—
69C2×D332—22no—
70C2×D332—22no—
71C2×D332—22no—
72C2×D332—41no—
73C2×D332—41no—
74C2×D332—41no—
75C2×D332—41no—
76C2×D332—41no—
77C2×D332—41no—
78C2×D332—41no—
79C2×D332—41no—
80C2×D332—41no—
81C2×D332—41no—
82C2×D332—41no—
83C2×D332—81no—
84C2×D332—81no—
85C22×Dic83—21no—
86C22×Dic83—21no—
87C22×Dic83—41no—
88C2×C83⋊C8—41no—
89C2×C83⋊C8—41no—
90C2×C83⋊C8—41no—
91C4⋊Dic83—21no—
92C4⋊Dic83—22no—
93C4⋊Dic83—22no—
94C4⋊Dic83—21no—
95C4⋊Dic83—21no—
96C4⋊Dic83—41no—
97C4⋊Dic83—41no—
98C4⋊Dic83SD1612yes—
99C4×Dic83—41no—
100C4×Dic83—41no—
101C22×C4×C83C2×D411yescharacteristic
102C22×C4×C83C2×D411yescharacteristic
103C22×C4×C83#4111yescharacteristic
104C2×C8×C83—21no—
105C2×C8×C83—21no—
106C2×C8×C83—21no—
107C2×C8×C83—21no—
108C2×C8×C83—21no—
109C2×C8×C83—21no—
110C2×C8×C83—41no—
111C2×C8×C83—41no—
112C2×C8×C83—41no—
113C2×C8×C83—41no—
114C2×C8×C83—41no—
115C2×C8×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—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83#12011yescharacteristic
131C2×C4×C83#12111yescharacteristic
132C2×C4×C83#12111yescharacteristic
133C2×C4×C83#12311yescharacteristic
134C2×C4×C83#12311yescharacteristic
135C2×C4×C83#12411yescharacteristic
136C2×C4×C83C2×SD1611yescharacteristic
137C2×C4×C83#13511yescharacteristic
138C2×C4×C83#13511yescharacteristic
139C2×C4×C83#13511yescharacteristic
140C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C83#13711yescharacteristic
143C2×C4×C83#13711yescharacteristic
144C2×Dic83—22no—
145C2×Dic83—41no—
146C2×Dic83—41no—
147C2×Dic83—42no—
148C2×Dic83—41no—
149C2×Dic83—41no—
150C2×Dic83—41no—
151C2×Dic83—81no—
152C2×Dic83—81no—
153D332—42no—
154D332—42no—
155D332—42no—
156D332—42no—
157D332—82no—
158D332—82no—
159D332—81no—
160D332—81no—
161D332—81no—
162D332—81no—
163Dic166—82no—
164Dic166—82no—
165C8×C83—41no—
166C8×C83—41no—
167C8×C83—81no—
168C8×C83—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#44911yescharacteristic
187C4×C83#45211yescharacteristic
188C4×C83#46011yescharacteristic
189C4×C83#46111yescharacteristic
190C4×C83#46311yescharacteristic
191C4×C83#46411yescharacteristic
192C4×C83#48811yescharacteristic
193C4×C83#48911yescharacteristic
194Dic83—42no—
195Dic83—81no—
196Dic83—82no—
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#1987—831no—
208#1982—831no—
209#1939—831no—
210C2×#333—831no—
211C2×#331—831no—
212#1277D8311yescharacteristic, complemented
213C83#1053911yescharacteristic, complemented, Sylow
214C2×C4⋊Q8—831no—
215C2×#127—831no—
216#486—1662no—
217#486—1662no—
218C2×C4⋊C8—831no—
219C2×C4⋊C8—1661no—
220C2×C4⋊C8C2×D8311yescharacteristic
221C2×C4⋊C8C2×D8311yescharacteristic
222C2×#102—1661no—
223C2×#102—1661no—
224C2×#102—1661no—
225C2×#102—1661no—
226C2×C4×C8C2×D8311yescharacteristic
227C4×C4⋊C4—831no—
228#340—1661no—
229#340—1661no—
230#336—1661no—
231#333—1662no—
232#331—1662no—
233C22×Q8—1661no—
234C22×D4—1661no—
235C22×D4—1661no—
236C22×D4—1661no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—1661no—
240C22×C8D33211yescharacteristic
241C22×C8D33211yescharacteristic
242C4⋊Q8—832no—
243C4⋊Q8—1661no—
244C4⋊Q8—1661no—
245C4⋊Q8—1661no—
246#127—832no—
247#127—1661no—
248#127—1661no—
249#127—1661no—
250C2×C4⋊C4—831no—
251C2×C4⋊C4—831no—
252C2×C4⋊C4—1661no—
253C2×C4⋊C4—1661no—
254C2×C42—1661no—
255C2×C42C22×D8311yescharacteristic
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—3322no—
261C4⋊C8—1661no—
262C4⋊C8—1661no—
263#102—3322no—
264#102—3322no—
265#102—1664no—
266#102—1664no—
267C4×C8—21no—
268C4×C8—21no—
269#95—1661no—
270C24—3321no—
271C2×Q8—3321no—
272C2×Q8—6641no—
273C2×Q8—6641no—
274C2×Q8—1662no—
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—3321no—
292C22×C4—1661no—
293C22×C4—1661no—
294C22×C4D4×D8311yescharacteristic
295C22×C4C2×D33211yescharacteristic
296C22×C4#1305059111yescharacteristic
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—3321no—
310C2×C8—3321no—
311C2×C8—3321no—
312C4⋊C4—3321no—
313C4⋊C4—3321no—
314C4⋊C4—832no—
315C4⋊C4—1662no—
316C4⋊C4—1662no—
317C4⋊C4—1661no—
318C4⋊C4—1661no—
319C4⋊C4—1661no—
320C42—3321no—
321C42—3321no—
322C42D4×D8311yescharacteristic
323C42D4×D8311yescharacteristic
324C42#1305059011yescharacteristic
325C42#1305059011yescharacteristic
326C23—3322no—
327C23—6641no—
328C23—6641no—
329C23—6641no—
330C23—6641no—
331C23—6641no—
332C23—6641no—
333C23#1458176711yescharacteristic
334Q8—6642no—
335Q8—6642no—
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—3321no—
349C2×C4—3321no—
350C2×C4—3322no—
351C2×C4—3321no—
352C2×C4—3321no—
353C2×C4—3321no—
354C2×C4—6641no—
355C2×C4—6641no—
356C2×C4—1662no—
357C2×C4#1458176211yescharacteristic
358C2×C4#1458176511yescharacteristic
359C2×C4#1458176511yescharacteristic
360C2×C4#1458177711yescharacteristic
361C2×C4#1458177711yescharacteristic
362C2×C4#1458178011yescharacteristic
363C2×C4SD16×D8311yescharacteristic
364C2×C4#1458181311yescharacteristic
365C2×C4#1458181411yescharacteristic
366C2×C4#1458181411yescharacteristic
367C2×C4#1458181611yescharacteristic
368C2×C4#1458182011yescharacteristic
369C2×C4#1458182411yescharacteristic
370C2×C4#1458182511yescharacteristic
371C8—41no—
372C8—41no—
373C8—81no—
374C8—81no—
375C8—6641no—
376C22—13281no—
377C22—13281no—
378C22—13281no—
379C22—13281no—
380C22—6642no—
381C22—6642no—
382C22—6642no—
383C22#1912761911yescharacteristic
384C22#1912791711yescharacteristic
385C22#1912792411yescharacteristic
386C22#1912793111yescharacteristic
387C22#1912794811yescharacteristic
388C22#1912797211yescharacteristic
389C22#1912797211yescharacteristic
390C4—21no—
391C4—21no—
392C4—3322no—
393C4—6642no—
394C4—6641no—
395C4#1912784811yescharacteristic
396C4#1912786511yescharacteristic
397C4#1912790011yescharacteristic
398C4#1912790511yescharacteristic
399C4#1912791511yescharacteristic
400C4#1912792611yescharacteristic
401C4#1912805611yescharacteristic
402C4#1912806111yescharacteristic
403C2—13282no—
404C2#2490687311yescharacteristic
405C2#2490688011yescharacteristic
406C2#2490699711yescharacteristic
407C2#2490700411yescharacteristic
408C2#2490870511yescharacteristic
409C2#2490872011yescharacteristic
410C2#2490878211yescharacteristic
subgroupG/H
centerC23#14581767
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1277C2
socleC23×C83#121
prime radicalC2×#14581789C22
hypercenter#1277D83
hypocenterC83#10539
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. #24908410C2semidirect
4. #24908181C2semidirect
5. C2×#19127449C2semidirect
7. C83×#1277C2semidirect
212. #1277D83semidirect
213. C83#10539semidirect

Maximal subgroups

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

subgroupindex
C2×#191280482
#249084102
#249083892
#249081812
C2×#191274492
C2×#191274442
C83×#12772
#1053983