Finite Group Databasebeta

← #35324820

Subgroups — #35324820

Order 21248.

316 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24909158C211yesmaximal, characteristic, complemented
2#24909152C211yesmaximal, characteristic, complemented
3#24905636C212yesmaximal, complemented
4#24905630C212yesmaximal, complemented
5C83×#3221C211yesmaximal, characteristic
6#19127685—21no—
7C2×#14581711—21no—
8#19127500—21no—
9#19127500C2212yes—
10#19127500C2212yes—
11#19127498—21no—
12#540×C83C411yescharacteristic
13#536×C83C2211yescharacteristic
14#496×C83C411yescharacteristic
15#492×C83C2211yescharacteristic
16#491×C83C2211yescharacteristic
17#488×C83C411yescharacteristic
18#463×C83—21no—
19#461×C83—21no—
20#449×C83—21no—
21#448×C83—21no—
22#433×C83C411yescharacteristic
23C2×#13050574—41no—
24C2×#13050574—41no—
25#14581711—22no—
26#14581711—22no—
27#14581711—41no—
28#14581709—41no—
29C2×#41×C83D411yescharacteristic
30C22×Q8×C83D411yescharacteristic
31C2×SD16×C83—21no—
32C2×SD16×C83—21no—
33C2×SD16×C83—21no—
34C2×D8×C83—21no—
35C4⋊Q8×C83D411yescharacteristic
36C4⋊Q8×C83C2311yescharacteristic
37#127×C83C2×C411yescharacteristic
38#125×C83C2×C411yescharacteristic
39#124×C83C2×C411yescharacteristic
40#124×C83D411yescharacteristic
41#123×C83—41no—
42#123×C83—41no—
43C22⋊Q8×C83—41no—
44#121×C83—21no—
45#121×C83—41no—
46C4×D4×C83—21no—
47#117×C83C2×C411yescharacteristic
48C4⋊C8×C83—21no—
49#103×C83—21no—
50#103×C83—21no—
51#103×C83—21no—
52#102×C83—21no—
53C22⋊C8×C83—21no—
54#97×C83C2×C411yescharacteristic
55C4×C8×C83C2×C411yescharacteristic
56#13050574—42no—
57#13050574—42no—
58#13050574—81no—
59#13050574—81no—
60C2×C83⋊C8—81no—
61C2×C83⋊C8—81no—
62#41×C83—81no—
63C2×Q8×C83—21no—
64C2×Q8×C83—81no—
65C2×Q8×C83C22⋊C411yescharacteristic
66C2×Q8×C83C2×D411yescharacteristic
67C2×Q8×C83C2×D411yescharacteristic
68C2×D4×C83—21no—
69C2×D4×C83—21no—
70C2×D4×C83—41no—
71C2×D4×C83—41no—
72C2×D4×C83C22⋊C411yescharacteristic
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—41no—
76C22×C4×C83C22⋊C411yescharacteristic
77SD16×C83—81no—
78SD16×C83—81no—
79SD16×C83—81no—
80D8×C83—81no—
81C2×C8×C83—21no—
82C2×C8×C83—21no—
83C4⋊C4×C83—21no—
84C4⋊C4×C83—21no—
85C4⋊C4×C83—41no—
86C4⋊C4×C83—41no—
87C4⋊C4×C83—41no—
88C22⋊C4×C83—21no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—41no—
92C22⋊C4×C83—41no—
93C42×C83C22⋊C411yescharacteristic
94C42×C83C22×C411yescharacteristic
95C23×C83—21no—
96C23×C83—41no—
97C23×C83#9911yescharacteristic
98Q8×C83—41no—
99Q8×C83—41no—
100Q8×C83—41no—
101Q8×C83—81no—
102D4×C83—41no—
103D4×C83—41no—
104D4×C83—41no—
105D4×C83—81no—
106D4×C83—161no—
107D4×C83—161no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—161no—
120C2×C4×C83#9911yescharacteristic
121C2×C4×C83C2×C22⋊C411yescharacteristic
122C8×C83—41no—
123C8×C83—41no—
124C83⋊C8—82no—
125C83⋊C8—82no—
126C22×C83—21no—
127C22×C83—41no—
128C22×C83—41no—
129C22×C83—81no—
130C22×C83—81no—
131C22×C83—81no—
132C22×C83—81no—
133C22×C83C2×#9911yescharacteristic
134C4×C83—21no—
135C4×C83—21no—
136C4×C83—41no—
137C4×C83—41no—
138C4×C83—41no—
139C4×C83—41no—
140C4×C83—41no—
141C4×C83—81no—
142#13611—831nomaximal, Sylow
143C2×C83—21no—
144C2×C83—41no—
145C2×C83—81no—
146C2×C83—161no—
147C2×C83#212711yescharacteristic
148#3221D8311yescharacteristic
149#2135—831no—
150#2133—831no—
151#1413—832no—
152#1410—832no—
153C83#1361111yescharacteristic, complemented, Sylow
154#540Dic8311yescharacteristic
155#536C2×D8311yescharacteristic
156#496Dic8311yescharacteristic
157#492C2×D8311yescharacteristic
158#491C2×D8311yescharacteristic
159#488Dic8311yescharacteristic
160#463—21no—
161#461—21no—
162#449—21no—
163#448—21no—
164#433Dic8311yescharacteristic
165#413—1661no—
166C2×#101—1661no—
167#356—832no—
168#356—832no—
169#356—1661no—
170#355—1661no—
171C2×#41C22⋊D8311yescharacteristic
172C22×Q8C22⋊D8311yescharacteristic
173C2×SD16—21no—
174C2×SD16—21no—
175C2×SD16—21no—
176C2×D8—21no—
177C2×M16—3321no—
178C2×M16—3321no—
179C4⋊Q8C22⋊D8311yescharacteristic
180C4⋊Q8C22×D8311yescharacteristic
181#127C2×Dic8311yescharacteristic
182#125C2×Dic8311yescharacteristic
183#124C2×Dic8311yescharacteristic
184#124C22⋊D8311yescharacteristic
185#123—41no—
186#123—41no—
187C22⋊Q8—41no—
188#121—21no—
189#121—41no—
190C4×D4—21no—
191#117C2×Dic8311yescharacteristic
192C4⋊C8—21no—
193#103—21no—
194#103—21no—
195#103—21no—
196#102—21no—
197#101—3321no—
198#101—1662no—
199#101—1662no—
200#100—3321no—
201C22⋊C8—21no—
202#97C2×Dic8311yescharacteristic
203C4×C8C2×Dic8311yescharacteristic
204#41—81no—
205C2×Q8—21no—
206C2×Q8—81no—
207C2×Q8C22⋊Dic8311yescharacteristic
208C2×Q8C2×C22⋊D8311yescharacteristic
209C2×Q8C2×C22⋊D8311yescharacteristic
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4C22⋊Dic8311yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4C22⋊Dic8311yescharacteristic
219SD16—81no—
220SD16—81no—
221SD16—81no—
222D8—81no—
223M16—3322no—
224M16—3322no—
225M16—6641no—
226M16—6641no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—6641no—
230C2×C8—6641no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—21no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C42C22⋊Dic8311yescharacteristic
242C42C22×Dic8311yescharacteristic
243C23—21no—
244C23—41no—
245C23#1458170711yescharacteristic
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—81no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—81no—
254D4—161no—
255D4—161no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—161no—
268C2×C4#1458170711yescharacteristic
269C2×C4C2×C22⋊Dic8311yescharacteristic
270C8—41no—
271C8—41no—
272C8—6642no—
273C8—6642no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22C2×#1458170711yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—161no—
294C2#2490913211yescharacteristic
subgroupG/H
centerC2#24909132
commutatorC42×C83C22×C4
FrattiniC4⋊Q8C22×D83
Fitting, prime radicalC83×#3221C2
socleC2×C83#2127
cosocleC4⋊Q8×C83C23
hypercenter#3221D83
hypocenterC83#13611
LCSG ▷ C42×C83 ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #3221

Complements

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

NH ≅ G/Ntype
1. #24909158C2semidirect
2. #24909152C2semidirect
3. #24905636C2semidirect
4. #24905630C2semidirect
153. C83#13611semidirect

Maximal subgroups

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

subgroupindex
#249091582
#249091522
#249056362
#249056302
C83×#32212
#1361183