Finite Group Databasebeta

← #35324208

Subgroups — #35324208

Order 21248.

362 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127503C211yesmaximal, characteristic
2C2×#19127501C211yesmaximal, characteristic
3#24908327C211yesmaximal, characteristic
4#24907626C211yesmaximal, characteristic, complemented
5#24905554C211yesmaximal, characteristic, complemented
6#24905540C211yesmaximal, characteristic, complemented
7C83×#1380C211yesmaximal, characteristic, complemented
8C22×D664C2211yescharacteristic
9C22×C83⋊C16C2211yescharacteristic
10C2×#14581726C2211yescharacteristic
11C2×#14581723—21no—
12C2×#14581712—21no—
13C2×C8×Dic83C2211yescharacteristic
14#19127602—21no—
15#19127566—21no—
16#19127503—22no—
17#19127501—22no—
18C22×C16×C83C2211yescharacteristic
19C2×#107×C83C2211yescharacteristic
20C2×#107×C83C2211yescharacteristic
21C22×D332—21no—
22C2×D664—21no—
23C2×D664—21no—
24C2×D664—21no—
25C2×D664—21no—
26C2×D664—21no—
27C2×D664D412yes—
28C22×C83⋊C8D411yescharacteristic
29C2×C4⋊Dic83—21no—
30C2×(C2×D83)⋊C4—41no—
31C2×(C2×D83)⋊C4—41no—
32C2×C4×Dic83D411yescharacteristic
33C2×C83⋊C16—21no—
34C2×C83⋊C16—21no—
35C2×C83⋊C16—21no—
36#14581726—21no—
37#14581726—21no—
38#14581723—42no—
39#14581712—42no—
40C8×Dic83—21no—
41C8×Dic83—21no—
42C83⋊#95—41no—
43C83⋊#95—41no—
44C22×C8×C83C2311yescharacteristic
45C2×C4⋊C4×C83—21no—
46C2×C4⋊C4×C83—21no—
47C2×C16×C83—21no—
48C2×C16×C83—21no—
49C2×C16×C83—21no—
50#107×C83—21no—
51#107×C83—21no—
52#107×C83—21no—
53#107×C83—21no—
54C23×D83—41no—
55C2×D332—22no—
56C2×D332—41no—
57C2×D332—41no—
58C2×D332—41no—
59C2×D332—41no—
60C2×D332—41no—
61C22×Dic83—41no—
62C22×Dic83D811yescharacteristic
63C22×Dic83D811yescharacteristic
64D664—22no—
65D664—22no—
66D664—41no—
67D664—41no—
68C2×C83⋊C8—21no—
69C2×C83⋊C8—21no—
70C2×C83⋊C8SD1612yes—
71C4⋊Dic83—41no—
72C4⋊Dic83—41no—
73(C2×D83)⋊C4—82no—
74(C2×D83)⋊C4—82no—
75C4×Dic83—21no—
76C4×Dic83—21no—
77C83⋊C16—41no—
78C22×C4×C83—41no—
79C22×C4×C83—41no—
80C22×C4×C83C2×D411yescharacteristic
81C2×C8×C83C2×D411yescharacteristic
82C2×C8×C83#4111yescharacteristic
83C2×C8×C83#4111yescharacteristic
84C2×C8×C83#4111yescharacteristic
85C2×C8×C83#4111yescharacteristic
86C2×C8×C83#4111yescharacteristic
87C4⋊C4×C83—41no—
88C4⋊C4×C83—41no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—41no—
91C16×C83—41no—
92C23×C83C2×D811yescharacteristic
93C22×D83—42no—
94C22×D83—81no—
95C22×D83—81no—
96C22×D83—81no—
97C22×D83—81no—
98C22×D83—81no—
99C22×D83—81no—
100C2×C4×C83—81no—
101C2×C4×C83—81no—
102C2×C4×C83—81no—
103C2×C4×C83—81no—
104C2×C4×C83—81no—
105C2×C4×C83—81no—
106C2×C4×C83#12111yescharacteristic
107C2×C4×C83#12311yescharacteristic
108C2×C4×C83#12411yescharacteristic
109C2×C4×C83C2×SD1611yescharacteristic
110C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C83#13711yescharacteristic
112C2×Dic83—21no—
113C2×Dic83—21no—
114C2×Dic83—21no—
115C2×Dic83—21no—
116C2×Dic83—21no—
117C2×Dic83—21no—
118C2×Dic83—81no—
119C2×Dic83—81no—
120C2×Dic83—81no—
121D332—42no—
122D332—42no—
123D332—81no—
124D332—81no—
125C8×C83#12311yescharacteristic
126C8×C83#12311yescharacteristic
127C8×C83#12611yescharacteristic
128C8×C83#12611yescharacteristic
129C83⋊C8—22no—
130C22×C83#45911yescharacteristic
131C22×C83#48011yescharacteristic
132C22×C83#48611yescharacteristic
133C22×C83#50811yescharacteristic
134C22×C83#50811yescharacteristic
135C22×C83#50911yescharacteristic
136C22×C83#50911yescharacteristic
137C2×D83—82no—
138C2×D83—82no—
139C2×D83—82no—
140C2×D83—161no—
141C2×D83—161no—
142C2×D83—161no—
143C2×D83—161no—
144C4×C83—161no—
145C4×C83—161no—
146C4×C83#39711yescharacteristic
147C4×C83#46111yescharacteristic
148C4×C83#48111yescharacteristic
149C4×C83#48911yescharacteristic
150Dic83—41no—
151Dic83—41no—
152Dic83—161no—
153#13482—831nomaximal, Sylow
154C2×C83#207111yescharacteristic
155C2×C83#221011yescharacteristic
156C2×C83#221011yescharacteristic
157C2×C83#223411yescharacteristic
158C2×C83#223411yescharacteristic
159C2×C83#224211yescharacteristic
160C2×C83#224411yescharacteristic
161D83—162no—
162C2×#357—831no—
163C2×#357—831no—
164#1970—831no—
165#1846—831no—
166#1380—831no—
167#1380D8311yescharacteristic, complemented
168#1374—831no—
169C83#1348211yescharacteristic, complemented, Sylow
170C22×D8—831no—
171C22×C16—831no—
172C22×C16C2×D8311yescharacteristic
173C2×#107—831no—
174C2×#107C2×D8311yescharacteristic
175C2×#107C2×D8311yescharacteristic
176C2×#106—1661no—
177C2×#102—1661no—
178C2×C4×C8—831no—
179#390—1661no—
180#382—1661no—
181#357—1662no—
182#357—1662no—
183C22×D4—1661no—
184C2×D8—832no—
185C2×D8—1661no—
186C2×D8—1661no—
187C2×D8—1661no—
188C2×D8—1661no—
189C2×D8—1661no—
190C22×C8—831no—
191C22×C8C22×D8311yescharacteristic
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—1661no—
195C2×C22⋊C4—3321no—
196C2×C22⋊C4—3321no—
197C2×C42—831no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—1661no—
202C2×C16—1661no—
203C2×C16—1661no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—1661no—
209#107—1661no—
210#106—3322no—
211#102—3322no—
212C4×C8—1661no—
213C4×C8—1661no—
214#95—3321no—
215#95—3321no—
216C24—3321no—
217C2×D4—3321no—
218C2×D4—3321no—
219C2×D4—3321no—
220C2×D4—3321no—
221C2×D4—3321no—
222C2×D4—1662no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—3321no—
226C22×C4—831no—
227C22×C4—831no—
228C22×C4D4×D8311yescharacteristic
229D8—3321no—
230D8—3321no—
231D8—1662no—
232D8—1662no—
233C2×C8—832no—
234C2×C8—1661no—
235C2×C8—1661no—
236C2×C8D4×D8311yescharacteristic
237C2×C8#1305058911yescharacteristic
238C2×C8#1305059011yescharacteristic
239C2×C8#1305059011yescharacteristic
240C2×C8#1305059011yescharacteristic
241C2×C8#1305059111yescharacteristic
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—3321no—
247C4⋊C4—3321no—
248C22⋊C4—6642no—
249C22⋊C4—6642no—
250C42—1661no—
251C42—1661no—
252C16—41no—
253C16—3321no—
254C23—3322no—
255C23—6641no—
256C23—6641no—
257C23—6641no—
258C23—6641no—
259C23—6641no—
260C23—6641no—
261C23D8×D8311yescharacteristic
262D4—3322no—
263D4—3322no—
264D4—6641no—
265D4—6641no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—6641no—
273C2×C4—6641no—
274C2×C4—6641no—
275C2×C4—1661no—
276C2×C4—1661no—
277C2×C4—1661no—
278C2×C4—1661no—
279C2×C4—1661no—
280C2×C4—1661no—
281C2×C4#1458176511yescharacteristic
282C2×C4#1458177711yescharacteristic
283C2×C4#1458177811yescharacteristic
284C2×C4SD16×D8311yescharacteristic
285C2×C4#1458181911yescharacteristic
286C2×C4#1458182411yescharacteristic
287C8—1662no—
288C8#1458177411yescharacteristic
289C8#1458177711yescharacteristic
290C8#1458178611yescharacteristic
291C8#1458178711yescharacteristic
292C22—13281no—
293C22—13281no—
294C22—13281no—
295C22—13281no—
296C22—6642no—
297C22—6642no—
298C22—6642no—
299C22#1912789511yescharacteristic
300C22#1912801611yescharacteristic
301C22#1912804611yescharacteristic
302C22#1912813211yescharacteristic
303C22#1912813411yescharacteristic
304C22#1912813511yescharacteristic
305C22#1912813811yescharacteristic
306C4—161no—
307C4—161no—
308C4—13281no—
309C4—3321no—
310C4—3321no—
311C4#1912763011yescharacteristic
312C4#1912790511yescharacteristic
313C4#1912802111yescharacteristic
314C4#1912806211yescharacteristic
315C2—13282no—
316C2#2490887011yescharacteristic
317C2#2490951711yescharacteristic
318C2#2490952111yescharacteristic
319C2#2490965611yescharacteristic
320C2#2490965711yescharacteristic
321C2#2490969611yescharacteristic
322C2#2490970311yescharacteristic
subgroupG/H
centerC23D8×D83
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC83×#1380C2
socleC23×C83C2×D8
prime radicalC22×D664C22
hypercenter#1380D83
hypocenterC83#13482
LCSG ▷ C22×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
4. #24907626C2semidirect
5. #24905554C2semidirect
6. #24905540C2semidirect
7. C83×#1380C2semidirect
167. #1380D83semidirect
169. C83#13482semidirect

Maximal subgroups

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

subgroupindex
C2×#191275032
C2×#191275012
#249083272
#249076262
#249055542
#249055402
C83×#13802
#1348283