Finite Group Databasebeta

← #35297458

Subgroups — #35297458

Order 21248.

542 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
1#24907623C211yesmaximal, characteristic, complemented
2#24907463C212yesmaximal
3C2×#19127454C211yesmaximal, characteristic, complemented
4#24905347C212yesmaximal, complemented
5C2×C8×D4×C83C211yesmaximal, characteristic
6C2×#14581726—21no—
7C2×#14581726—21no—
8C2×C4⋊(C83⋊C8)C2212yes—
9C2×#14581713—22no—
10#19127571C2212yes—
11#19127463—22no—
12#19127454—22no—
13C23×C8×C83—21no—
14C2×C4×D4×C83C2211yescharacteristic
15C8×D4×C83—22no—
16C8×D4×C83C414yescomplemented
17C2×C4⋊C8×C83C2211yescharacteristic
18C2×C22⋊C8×C83—21no—
19C2×C4×C8×C83C2211yescharacteristic
20C22×C83⋊C8—22no—
21C2×C4⋊Dic83—22no—
22C2×C4⋊Dic83—22no—
23C2×C4⋊Dic83—22no—
24#14581726—42no—
25#14581726—42no—
26C4⋊(C83⋊C8)—22no—
27C4⋊(C83⋊C8)—22no—
28#14581713—44no—
29C22×D4×C83D411yescharacteristic
30C23×C4×C83—21no—
31C22×C8×C83—21no—
32C22×C8×C83—24no—
33C22×C8×C83—42no—
34C22×C8×C83—42no—
35C22×C8×C83D411yescharacteristic
36C22×C8×C83D411yescharacteristic
37C4×D4×C83—21no—
38C4×D4×C83—21no—
39C4×D4×C83C2×C412yes—
40C4×D4×C83Q812yes—
41C2×C4⋊C4×C83D411yescharacteristic
42C2×C22⋊C4×C83—21no—
43C2×C42×C83C2311yescharacteristic
44C4⋊C8×C83—21no—
45C4⋊C8×C83C2×C412yes—
46C22⋊C8×C83—24no—
47C4×C8×C83C2×C412yes—
48C4×C8×C83D412yes—
49C22×Dic83—42no—
50C2×C83⋊C8—42no—
51C2×C83⋊C8—42no—
52C2×C83⋊C8—42no—
53C4⋊Dic83—42no—
54C4⋊Dic83—42no—
55C4⋊Dic83—42no—
56C4⋊Dic83—42no—
57C4⋊Dic83—82no—
58C24×C83—21no—
59C2×D4×C83—21no—
60C2×D4×C83—21no—
61C2×D4×C83—21no—
62C2×D4×C83C4⋊C412yes—
63C2×D4×C83D812yes—
64C2×D4×C83Q1612yes—
65C22×C4×C83—21no—
66C22×C4×C83—22no—
67C22×C4×C83—22no—
68C22×C4×C83—41no—
69C22×C4×C83—41no—
70C22×C4×C83—41no—
71C22×C4×C83—41no—
72C22×C4×C83C2×D411yescharacteristic
73C22×C4×C83C2×D411yescharacteristic
74C22×C4×C83#4111yescharacteristic
75C2×C8×C83—21no—
76C2×C8×C83—21no—
77C2×C8×C83—22no—
78C2×C8×C83—41no—
79C2×C8×C83—41no—
80C2×C8×C83—48no—
81C2×C8×C83C22⋊C412yes—
82C2×C8×C83C22⋊C412yes—
83C2×C8×C83D812yes—
84C2×C8×C83SD1612yes—
85C4⋊C4×C83—21no—
86C4⋊C4×C83C4⋊C412yes—
87C22⋊C4×C83—22no—
88C22⋊C4×C83—22no—
89C42×C83C22×C411yescharacteristic
90C42×C83C2×D411yescharacteristic
91C42×C83C2×Q811yescharacteristic
92C42×C83#4111yescharacteristic
93C23×C83—22no—
94C23×C83—22no—
95C23×C83—22no—
96C23×C83—41no—
97C23×C83—41no—
98C23×C83—41no—
99C23×C83—41no—
100C23×C83#12111yescharacteristic
101D4×C83—22no—
102D4×C83—22no—
103D4×C83—22no—
104D4×C83#10714yescomplemented
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—22no—
108C2×C4×C83—41no—
109C2×C4×C83—41no—
110C2×C4×C83—42no—
111C2×C4×C83—42no—
112C2×C4×C83—42no—
113C2×C4×C83—42no—
114C2×C4×C83C2×C22⋊C411yescharacteristic
115C2×C4×C83C2×C4⋊C411yescharacteristic
116C2×C4×C83#11711yescharacteristic
117C2×C4×C83#12111yescharacteristic
118C2×C4×C83C22⋊Q811yescharacteristic
119C2×C4×C83C22⋊Q811yescharacteristic
120C2×C4×C83C2×D811yescharacteristic
121C2×C4×C83C2×D811yescharacteristic
122C2×C4×C83C2×SD1611yescharacteristic
123C2×C4×C83C2×Q1611yescharacteristic
124C2×C4×C83#13511yescharacteristic
125C2×C4×C83#13511yescharacteristic
126C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C83#13711yescharacteristic
128C2×Dic83—82no—
129C2×Dic83—82no—
130C2×Dic83—82no—
131C8×C83—22no—
132C8×C83—42no—
133C8×C83#10214yescomplemented
134C83⋊C8—82no—
135C22×C83—24no—
136C22×C83—42no—
137C22×C83—42no—
138C22×C83—42no—
139C22×C83—42no—
140C22×C83—42no—
141C22×C83—42no—
142C22×C83#38011yescharacteristic
143C22×C83#45911yescharacteristic
144C22×C83#46111yescharacteristic
145C22×C83#46611yescharacteristic
146C22×C83#46711yescharacteristic
147C22×C83#47211yescharacteristic
148C22×C83#47311yescharacteristic
149C4×C83—21no—
150C4×C83—21no—
151C4×C83—42no—
152C4×C83C2×#10211yescharacteristic
153C4×C83#41711yescharacteristic
154C4×C83C2×#10711yescharacteristic
155C4×C83#42711yescharacteristic
156C4×C83#46611yescharacteristic
157C4×C83#46711yescharacteristic
158C4×C83#47411yescharacteristic
159C4×C83#47611yescharacteristic
160Dic83—162no—
161#8644—831nomaximal, Sylow
162C2×C83—44no—
163C2×C83#166711yescharacteristic
164C2×C83#166911yescharacteristic
165C2×C83#167311yescharacteristic
166C2×C83#167511yescharacteristic
167C2×C83#180211yescharacteristic
168C2×C83#180811yescharacteristic
169C2×C83#181611yescharacteristic
170C2×C8×D4D8311yescharacteristic
171#1845—831no—
172#1802—832no—
173C2×#335—831no—
174#1296—832no—
175C83#864411yescharacteristic, complemented, Sylow
176C23×C8—21no—
177C2×C4×D4C2×D8311yescharacteristic
178C8×D4—22no—
179C8×D4Dic8314yescomplemented
180C2×#107—1661no—
181C2×#107—1661no—
182C2×C4⋊C8—832no—
183C2×C4⋊C8C2×D8311yescharacteristic
184C2×#102—1662no—
185C2×C22⋊C8—21no—
186C2×C4×C8C2×D8311yescharacteristic
187#384—832no—
188#340—1662no—
189#335—1662no—
190C22×D4D33211yescharacteristic
191C23×C4—21no—
192C22×C8—21no—
193C22×C8—24no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—1662no—
197C22×C8C22⋊D8311yescharacteristic
198C22×C8C22⋊D8311yescharacteristic
199C4×D4—21no—
200C4×D4—21no—
201C4×D4Dic16612yes—
202C4×D4C2×Dic8312yes—
203C2×C4⋊C4—1662no—
204C2×C4⋊C4—1662no—
205C2×C4⋊C4—1662no—
206C2×C4⋊C4D33211yescharacteristic
207C2×C22⋊C4—21no—
208C2×C42C22×D8311yescharacteristic
209#107—3322no—
210#107—3322no—
211C4⋊C8—21no—
212C4⋊C8—1662no—
213C4⋊C8—1662no—
214C4⋊C8C2×Dic8312yes—
215#102—3324no—
216C22⋊C8—24no—
217C4×C8C2×Dic8312yes—
218C4×C8C22⋊D8312yes—
219C24—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4C4⋊Dic8312yes—
224C2×D4D66412yes—
225C2×D4Dic33212yes—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—3322no—
234C22×C4C2×C22⋊D8311yescharacteristic
235C22×C4C2×D33211yescharacteristic
236C22×C4#1305059011yescharacteristic
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—22no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—48no—
243C2×C8—3322no—
244C2×C8—3322no—
245C2×C8—3322no—
246C2×C8C22⋊Dic8312yes—
247C2×C8C22⋊Dic8312yes—
248C2×C8#1305057512yes—
249C2×C8#1305057812yes—
250C4⋊C4—21no—
251C4⋊C4—3322no—
252C4⋊C4—3322no—
253C4⋊C4—3322no—
254C4⋊C4—3322no—
255C4⋊C4—6642no—
256C4⋊C4C4⋊Dic8312yes—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C42C22×Dic8311yescharacteristic
260C42C2×C22⋊D8311yescharacteristic
261C42C2×Dic16611yescharacteristic
262C42#1305059011yescharacteristic
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23#1458176611yescharacteristic
271D4—22no—
272D4—22no—
273D4—22no—
274D4#1458172614yescomplemented
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—6642no—
285C2×C4—6642no—
286C2×C4—6642no—
287C2×C4C2×C22⋊Dic8311yescharacteristic
288C2×C4C2×C4⋊Dic8311yescharacteristic
289C2×C4#1458175111yescharacteristic
290C2×C4#1458176611yescharacteristic
291C2×C4#1458177111yescharacteristic
292C2×C4#1458177111yescharacteristic
293C2×C4C2×#1305057511yescharacteristic
294C2×C4C2×D66411yescharacteristic
295C2×C4C2×#1305057811yescharacteristic
296C2×C4C2×Dic33211yescharacteristic
297C2×C4#1458181411yescharacteristic
298C2×C4#1458181411yescharacteristic
299C2×C4#1458181711yescharacteristic
300C2×C4#1458182211yescharacteristic
301C8—22no—
302C8—42no—
303C8—6642no—
304C8#1458171314yescomplemented
305C22—24no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22#1912756111yescharacteristic
313C22#1912789611yescharacteristic
314C22#1912790611yescharacteristic
315C22#1912793711yescharacteristic
316C22#1912794211yescharacteristic
317C22#1912797111yescharacteristic
318C22#1912797611yescharacteristic
319C4—21no—
320C4—21no—
321C4—42no—
322C4—13282no—
323C4C2×#1458171311yescharacteristic
324C4#1912770211yescharacteristic
325C4C2×#1458172611yescharacteristic
326C4#1912773911yescharacteristic
327C4#1912793711yescharacteristic
328C4#1912794211yescharacteristic
329C4#1912798111yescharacteristic
330C4#1912799111yescharacteristic
331C2—44no—
332C2#2490681811yescharacteristic
333C2#2490682811yescharacteristic
334C2#2490685211yescharacteristic
335C2#2490686211yescharacteristic
336C2#2490746311yescharacteristic
337C2#2490749211yescharacteristic
338C2#2490752711yescharacteristic
subgroupG/H
centerC23#14581766
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC2×C8×D4×C83C2
socleC23×C83#121
prime radicalC22×D4×C83D4
cosocleC2×C42×C83C23
hypercenterC2×C8×D4D83
hypocenterC83#8644
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. #24907623C2semidirect
3. C2×#19127454C2semidirect
4. #24905347C2semidirect
16. C8×D4×C83C4semidirect
104. D4×C83#107semidirect
133. C8×C83#102semidirect
175. C83#8644semidirect
179. C8×D4Dic83semidirect
274. D4#14581726semidirect
304. C8#14581713semidirect

Maximal subgroups

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

subgroupindex
#249076232
#249074632
C2×#191274542
#249053472
C2×C8×D4×C832
#864483