Finite Group Databasebeta

← #35320997

Subgroups — #35320997

Order 21248.

490 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127783C211yesmaximal, characteristic
2C2×#19127526C211yesmaximal, characteristic, complemented
3C2×#19127503C211yesmaximal, characteristic
4C2×#19127501C211yesmaximal, characteristic
5#24908197C211yesmaximal, characteristic, complemented
6#24905540C211yesmaximal, characteristic, complemented
7C83×#1380C211yesmaximal, characteristic, complemented
8C22×D664C2211yescharacteristic
9C2×#14581754—21no—
10C22×C83⋊C16C2211yescharacteristic
11#19127783—22no—
12#19127783—22no—
13C2×#14581726C2211yescharacteristic
14C2×#14581712—21no—
15C2×C8×Dic83C2211yescharacteristic
16#19127600—21no—
17#19127526—22no—
18#19127503—22no—
19#19127501C414yescomplemented
20#19127463—21no—
21C22×C16×C83C2211yescharacteristic
22C2×#107×C83C2211yescharacteristic
23C2×#107×C83C2211yescharacteristic
24C22×D332—21no—
25C22×C4×D83—41no—
26C2×D664—21no—
27C2×D664—21no—
28C2×D664—21no—
29C2×D664C2×C412yes—
30C2×D664D412yes—
31C2×D664Q812yes—
32C22×C83⋊C8D411yescharacteristic
33#14581754—24no—
34#14581754—42no—
35C2×C4⋊Dic83—21no—
36C2×C4⋊Dic83—41no—
37C2×(C2×D83)⋊C4—41no—
38C2×C4×Dic83D411yescharacteristic
39C2×C83⋊C16—21no—
40C2×C83⋊C16—21no—
41C2×C83⋊C16C2×C412yescomplemented
42#14581726—21no—
43#14581726—21no—
44#14581712—24no—
45C8×Dic83—21no—
46C8×Dic83—21no—
47C83⋊#95—41no—
48C22×C8×C83C2311yescharacteristic
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C16×C83—21no—
52C2×C16×C83—21no—
53C2×C16×C83—21no—
54#107×C83—21no—
55#107×C83—21no—
56#107×C83C2×C412yescomplemented
57#107×C83D412yes—
58C23×D83—41no—
59C2×D332—22no—
60C2×D332—22no—
61C2×D332—22no—
62C2×D332—41no—
63C2×D332—41no—
64C2×D332—41no—
65C22×Dic83—41no—
66C22×Dic83D811yescharacteristic
67C22×Dic83D811yescharacteristic
68C2×C4×D83—44no—
69C2×C4×D83—82no—
70C2×C4×D83—82no—
71D664—22no—
72D664—22no—
73D664—22no—
74D664C4⋊C414yescomplemented
75C2×C83⋊C8—21no—
76C2×C83⋊C8—21no—
77C2×C83⋊C8—21no—
78C4⋊Dic83—41no—
79C4⋊Dic83—41no—
80C4⋊Dic83—82no—
81(C2×D83)⋊C4—44no—
82C4×Dic83—21no—
83C4×Dic83—21no—
84C83⋊C16—22no—
85C22×C4×C83—41no—
86C22×C4×C83—41no—
87C22×C4×C83C2×D411yescharacteristic
88C2×C8×C83C22×C411yescharacteristic
89C2×C8×C83C2×D411yescharacteristic
90C2×C8×C83C2×D411yescharacteristic
91C2×C8×C83C2×Q811yescharacteristic
92C2×C8×C83#4111yescharacteristic
93C2×C8×C83#4111yescharacteristic
94C4⋊C4×C83—22no—
95C4⋊C4×C83—22no—
96C4⋊C4×C83—41no—
97C4⋊C4×C83—41no—
98C16×C83—41no—
99C23×C83C2×D811yescharacteristic
100C22×D83—42no—
101C22×D83—42no—
102C22×D83—42no—
103C22×D83—81no—
104C22×D83—81no—
105C22×D83—81no—
106C22×D83—81no—
107C2×C4×C83—42no—
108C2×C4×C83—42no—
109C2×C4×C83—81no—
110C2×C4×C83—81no—
111C2×C4×C83—81no—
112C2×C4×C83—81no—
113C2×C4×C83C4×D411yescharacteristic
114C2×C4×C83#12011yescharacteristic
115C2×C4×C83C22⋊Q811yescharacteristic
116C2×C4×C83#13511yescharacteristic
117C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C83#13711yescharacteristic
119C2×Dic83—21no—
120C2×Dic83—21no—
121C2×Dic83—21no—
122C2×Dic83—21no—
123C2×Dic83—21no—
124C2×Dic83—81no—
125C2×Dic83—81no—
126C2×Dic83—81no—
127C2×Dic83D1612yes—
128D332—24no—
129D332—42no—
130D332—42no—
131D332—42no—
132C4×D83—88no—
133C8×C83C2×C4⋊C411yescharacteristic
134C8×C83C4×D411yescharacteristic
135C8×C83C22⋊Q811yescharacteristic
136C8×C83#12311yescharacteristic
137C83⋊C8—41no—
138C22×C83C4×D811yescharacteristic
139C22×C83#44711yescharacteristic
140C22×C83#47411yescharacteristic
141C22×C83C2×D1611yescharacteristic
142C22×C83#50811yescharacteristic
143C22×C83#50911yescharacteristic
144C22×C83#51011yescharacteristic
145C2×D83—44no—
146C2×D83—82no—
147C2×D83—82no—
148C2×D83—82no—
149C2×D83—82no—
150C2×D83—82no—
151C2×D83—82no—
152C4×C83—82no—
153C4×C83—82no—
154C4×C83#38511yescharacteristic
155C4×C83#44011yescharacteristic
156C4×C83#45211yescharacteristic
157C4×C83#47811yescharacteristic
158Dic83—22no—
159Dic83—41no—
160Dic83—161no—
161#12935—831nomaximal, Sylow
162C2×C83#186211yescharacteristic
163C2×C83C4×D1611yescharacteristic
164C2×C83#217511yescharacteristic
165C2×C83#218411yescharacteristic
166C2×C83#218911yescharacteristic
167C2×C83#222411yescharacteristic
168C2×C83#222811yescharacteristic
169D83—84no—
170C2×C4×D8—831no—
171C2×#366—831no—
172C2×#357—831no—
173C2×#357—831no—
174#1942—831no—
175#1380D8311yescharacteristic, complemented
176#1374—831no—
177C83#1293511yescharacteristic, complemented, Sylow
178C22×D8—831no—
179C2×C4×D4—1661no—
180C22×C16—831no—
181C22×C16C2×D8311yescharacteristic
182C4×D8—1662no—
183C4×D8—1662no—
184C2×#107—831no—
185C2×#107C2×D8311yescharacteristic
186C2×#107C2×D8311yescharacteristic
187C2×#102—1661no—
188C2×C4×C8—831no—
189#389—1661no—
190#366—1662no—
191#357—834no—
192#357—1662no—
193#340—1661no—
194C22×D4—1661no—
195C23×C4—3321no—
196C2×D8—832no—
197C2×D8—832no—
198C2×D8—832no—
199C2×D8—1661no—
200C2×D8—1661no—
201C2×D8—1661no—
202C22×C8—831no—
203C22×C8C22×D8311yescharacteristic
204C4×D4—3322no—
205C4×D4—1664no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—3321no—
209C2×C4⋊C4—1661no—
210C2×C22⋊C4—3321no—
211C2×C42—831no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—832no—
216C2×C16—1661no—
217C2×C16—1661no—
218#107—21no—
219#107—21no—
220#107—1661no—
221#107—1661no—
222#107C4×D8312yescomplemented
223#107D33212yes—
224#102—1664no—
225C4×C8—1661no—
226C4×C8—1661no—
227#95—3321no—
228C24—3321no—
229C2×D4—3321no—
230C2×D4—3321no—
231C2×D4—3321no—
232C2×D4—1662no—
233C2×D4—1662no—
234C2×D4—1662no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—3324no—
238C22×C4—3321no—
239C22×C4—831no—
240C22×C4—831no—
241C22×C4—6642no—
242C22×C4—6642no—
243C22×C4D4×D8311yescharacteristic
244D8—834no—
245D8—1662no—
246D8—1662no—
247D8—1662no—
248C2×C8—1661no—
249C2×C8—1661no—
250C2×C8—1661no—
251C2×C8C2×C4×D8311yescharacteristic
252C2×C8D4×D8311yescharacteristic
253C2×C8C2×D33211yescharacteristic
254C2×C8Q8×D8311yescharacteristic
255C2×C8#1305058911yescharacteristic
256C2×C8#1305059011yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—3321no—
262C4⋊C4—3321no—
263C4⋊C4—6642no—
264C22⋊C4—3324no—
265C42—1661no—
266C42—1661no—
267C16—41no—
268C16—1662no—
269C23—3322no—
270C23—3322no—
271C23—3322no—
272C23—6641no—
273C23—6641no—
274C23—6641no—
275C23—6641no—
276C23D8×D8311yescharacteristic
277D4—3322no—
278D4—3322no—
279D4—3322no—
280D4—1664no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—832no—
288C2×C4—6648no—
289C2×C4—6641no—
290C2×C4—6641no—
291C2×C4—6641no—
292C2×C4—1661no—
293C2×C4—1661no—
294C2×C4—1661no—
295C2×C4—1661no—
296C2×C4—1661no—
297C2×C4(C22⋊D83)⋊C411yescharacteristic
298C2×C4#1458176211yescharacteristic
299C2×C4#1458177011yescharacteristic
300C2×C4#1458181311yescharacteristic
301C2×C4#1458181911yescharacteristic
302C2×C4#1458182411yescharacteristic
303C8—3321no—
304C8C4⋊C4×D8311yescharacteristic
305C8C4×D33211yescharacteristic
306C8#1458177211yescharacteristic
307C8#1458177411yescharacteristic
308C22—3324no—
309C22—6642no—
310C22—6642no—
311C22—6642no—
312C22—6642no—
313C22—6642no—
314C22—6642no—
315C22#1912778111yescharacteristic
316C22#1912783611yescharacteristic
317C22#1912798011yescharacteristic
318C22D16×D8311yescharacteristic
319C22#1912813211yescharacteristic
320C22#1912813811yescharacteristic
321C22#1912814311yescharacteristic
322C4—82no—
323C4—82no—
324C4—13281no—
325C4—3321no—
326C4—1662no—
327C4#1912757511yescharacteristic
328C4#1912780411yescharacteristic
329C4#1912786611yescharacteristic
330C4#1912800011yescharacteristic
331C2—6644no—
332C2#2490771111yescharacteristic
333C2#2490930711yescharacteristic
334C2#2490933011yescharacteristic
335C2#2490937611yescharacteristic
336C2#2490940611yescharacteristic
337C2#2490959911yescharacteristic
338C2#2490961911yescharacteristic
subgroupG/H
centerC23D8×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1380C2
socleC23×C83C2×D8
prime radicalC22×D664C22
cosocleC22×C8×C83C23
hypercenter#1380D83
hypocenterC83#12935
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#19127526C2semidirect
5. #24908197C2semidirect
6. #24905540C2semidirect
7. C83×#1380C2semidirect
19. #19127501C4semidirect
41. C2×C83⋊C16C2×C4semidirect
56. #107×C83C2×C4semidirect
74. D664C4⋊C4semidirect
175. #1380D83semidirect
177. C83#12935semidirect
222. #107C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191277832
C2×#191275262
C2×#191275032
C2×#191275012
#249081972
#249055402
C83×#13802
#1293583