Finite Group Databasebeta

← #35304586

Subgroups — #35304586

Order 21248.

592 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24915815C211yesmaximal, characteristic
2#24907443C211yesmaximal, characteristic
3#24905342C214yesmaximal
4C83×#1434C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic, complemented
6C2×#14581771—21no—
7C2×C4⋊(C83⋊C8)—21no—
8C2×C22⋊(C83⋊C8)C2212yes—
9#19127607—21no—
10#19127599—21no—
11#19127599C412yescomplemented
12#19127571C411yescharacteristic, complemented
13#19127561—21no—
14#19127561C2212yes—
15#19127456—21no—
16C22×C42×C83C2211yescharacteristic
17C2×#95×C83—21no—
18C2×#95×C83—21no—
19C2×#95×C83C2212yes—
20C22×Dic166—21no—
21C22×C83⋊C8—22no—
22#14581792—42no—
23#14581771—44no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83C2×C412yes—
28C2×C4⋊Dic83C2×C412yes—
29C2×Dic83⋊C4—22no—
30C2×C22⋊Dic83—22no—
31C4⋊(C83⋊C8)—42no—
32C22⋊(C83⋊C8)—24no—
33C83⋊#95—22no—
34C83⋊#95—22no—
35C23×C4×C83—22no—
36C23×C4×C83D411yescharacteristic
37C23×C4×C83D411yescharacteristic
38C23×C4×C83C2311yescharacteristic
39C2×C42×C83—42no—
40C2×C42×C83C2×C411yescharacteristic
41C2×C42×C83C2×C411yescharacteristic
42C2×C42×C83D411yescharacteristic
43C2×C42×C83D411yescharacteristic
44#95×C83—22no—
45#95×C83—42no—
46#95×C83—42no—
47#95×C83Q814yescomplemented
48C2×Dic166—42no—
49C2×Dic166—42no—
50C2×Dic166—42no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53C2×C83⋊C8—42no—
54C2×C83⋊C8—42no—
55C2×C83⋊C8—42no—
56C4⋊Dic83—42no—
57C4⋊Dic83—42no—
58C4⋊Dic83—42no—
59C4⋊Dic83—42no—
60C4⋊Dic83—42no—
61Dic83⋊C4—44no—
62C22⋊Dic83—44no—
63C24×C83C2×D411yescharacteristic
64C22×C4×C83—22no—
65C22×C4×C83—22no—
66C22×C4×C83—24no—
67C22×C4×C83—41no—
68C22×C4×C83—41no—
69C22×C4×C83—41no—
70C22×C4×C83—41no—
71C22×C4×C83—41no—
72C22×C4×C83—42no—
73C22×C4×C83—42no—
74C22×C4×C83—42no—
75C22×C4×C83—42no—
76C22×C4×C83—42no—
77C22×C4×C83—42no—
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83SD1612yes—
83C22×C4×C83Q1612yes—
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83C2×Q812yes—
87C22×C4×C83#4112yes—
88C42×C83—22no—
89C42×C83—22no—
90C42×C83—41no—
91C42×C83—41no—
92C23×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—21no—
96C23×C83C2×C22⋊C411yescharacteristic
97C23×C83C22⋊Q812yes—
98C23×C83C2×SD1611yescharacteristic
99C23×C83C2×Q1611yescharacteristic
100C23×C83AGL(1,ℤ/8)12yes—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—42no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—44no—
119C2×C4×C83—44no—
120C2×C4×C83—44no—
121C2×C4×C83—44no—
122C2×C4×C83—82no—
123C2×C4×C83—82no—
124C2×C4×C83—82no—
125C2×C4×C83—82no—
126C2×C4×C83#9912yes—
127C2×C4×C83#10312yes—
128C2×C4×C83#10312yes—
129C2×C4×C83C4×Q812yes—
130C2×C4×C83#12411yescharacteristic
131C2×C4×C83C4⋊Q811yescharacteristic
132C2×Dic83—42no—
133C2×Dic83—42no—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83—42no—
137C2×Dic83—42no—
138Dic166—84no—
139C83⋊C8—82no—
140C22×C83—22no—
141C22×C83—22no—
142C22×C83—22no—
143C22×C83—22no—
144C22×C83—21no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—21no—
148C22×C83#39111yescharacteristic
149C22×C83C2×#9911yescharacteristic
150C22×C83#41011yescharacteristic
151C22×C83C2×#10311yescharacteristic
152C22×C83#41811yescharacteristic
153C22×C83#42111yescharacteristic
154C22×C83#42111yescharacteristic
155C22×C83#47612yes—
156C22×C83#47712yes—
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—21no—
160C4×C83—22no—
161C4×C83—22no—
162C4×C83—22no—
163C4×C83—41no—
164C4×C83—84no—
165Dic83—82no—
166Dic83—82no—
167#9911—831nomaximal, Sylow
168C2×C83—22no—
169C2×C83—22no—
170C2×C83#151011yescharacteristic
171C2×C83#151411yescharacteristic
172C2×C83#151811yescharacteristic
173C2×C83#152011yescharacteristic
174C2×C83#198711yescharacteristic
175C2×C83#199511yescharacteristic
176C2×C83#199611yescharacteristic
177#2857—831no—
178#1797—831no—
179#1434D8311yescharacteristic
180#1294—834no—
181C83#991111yescharacteristic, complemented, Sylow
182C2×C4⋊Q8—831no—
183C2×C22⋊Q8—1661no—
184C22×C42C2×D8311yescharacteristic
185C2×C4⋊C8—1661no—
186C2×C22⋊C8—832no—
187#391—1661no—
188#389—832no—
189#389—1661no—
190#384—831no—
191#380—832no—
192#380—1661no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95C2×D8312yes—
196#336—1661no—
197C22×Q8—1661no—
198C23×C4—22no—
199C23×C4C22⋊D8311yescharacteristic
200C23×C4C22⋊D8311yescharacteristic
201C23×C4C22×D8311yescharacteristic
202C22×C8—1662no—
203C4⋊Q8—3322no—
204C22⋊Q8—3324no—
205C2×C4⋊C4—832no—
206C2×C4⋊C4—832no—
207C2×C4⋊C4—1661no—
208C2×C4⋊C4—1662no—
209C2×C4⋊C4—1661no—
210C2×C4⋊C4—1661no—
211C2×C22⋊C4—1662no—
212C2×C42—42no—
213C2×C42C4×D8311yescharacteristic
214C2×C42C4×D8311yescharacteristic
215C2×C42D33211yescharacteristic
216C2×C42D33211yescharacteristic
217C4⋊C8—3322no—
218C22⋊C8—1664no—
219#95—22no—
220#95—42no—
221#95—42no—
222#95—1662no—
223#95—1662no—
224#95Dic16614yescomplemented
225C24C2×C22⋊D8311yescharacteristic
226C2×Q8—3322no—
227C2×Q8—3322no—
228C2×Q8—3322no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—1662no—
244C22×C4—1662no—
245C22×C4(C2×D83)⋊C411yescharacteristic
246C22×C4(C2×D83)⋊C411yescharacteristic
247C22×C4(C2×D83)⋊C411yescharacteristic
248C22×C4(C2×D83)⋊C411yescharacteristic
249C22×C4#1305057812yes—
250C22×C4#1305058012yes—
251C22×C4C2×C4×D8311yescharacteristic
252C22×C4C2×D33211yescharacteristic
253C22×C4C2×Dic16612yes—
254C22×C4#1305059012yes—
255C2×C8—3322no—
256C2×C8—3322no—
257C2×C8—3322no—
258C4⋊C4—3322no—
259C4⋊C4—3322no—
260C4⋊C4—3324no—
261C4⋊C4—3322no—
262C4⋊C4—3322no—
263C4⋊C4—3322no—
264C22⋊C4—3324no—
265C42—22no—
266C42—22no—
267C42—41no—
268C42—41no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23C2×(C2×D83)⋊C411yescharacteristic
274C23#1458177112yes—
275C23C2×#1305057811yescharacteristic
276C23C2×#1305058011yescharacteristic
277C23#1458181712yes—
278Q8—6644no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—3322no—
305C2×C4—3322no—
306C2×C4—3322no—
307C2×C4—3322no—
308C2×C4—3322no—
309C2×C4—3322no—
310C2×C4#1458170612yes—
311C2×C4#1458171512yes—
312C2×C4#1458171512yes—
313C2×C4C4×Dic16612yes—
314C2×C4#1458178011yescharacteristic
315C2×C4#1458179211yescharacteristic
316C8—6642no—
317C22—22no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22#1912760711yescharacteristic
326C22C2×#1458170611yescharacteristic
327C22#1912767411yescharacteristic
328C22C2×#1458171511yescharacteristic
329C22#1912770511yescharacteristic
330C22#1912771911yescharacteristic
331C22#1912771911yescharacteristic
332C22#1912799112yes—
333C22#1912799612yes—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—21no—
338C4—21no—
339C4—22no—
340C4—41no—
341C4—84no—
342C4—6642no—
343C4—6642no—
344C2—22no—
345C2—22no—
346C2#2490599111yescharacteristic
347C2#2490600911yescharacteristic
348C2#2490602611yescharacteristic
349C2#2490603411yescharacteristic
350C2#2490841111yescharacteristic
351C2#2490844411yescharacteristic
352C2#2490844811yescharacteristic
subgroupG/H
centerC23C2×(C2×D83)⋊C4
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1434C2
socleC23×C83C2×C22⋊C4
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenter#1434D83
hypocenterC83#9911
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
5. C2×#14581792C4semidirect
11. #19127599C4semidirect
12. #19127571C4semidirect
47. #95×C83Q8semidirect
181. C83#9911semidirect
224. #95Dic166semidirect

Maximal subgroups

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

subgroupindex
#249158152
#249074432
#249053422
C83×#14342
#991183