Finite Group Databasebeta

← #35302003

Subgroups — #35302003

Order 21248.

504 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 107 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#24917923C211yesmaximal, characteristic
2#24906198C211yesmaximal, characteristic
3#24906147C212yesmaximal, complemented
4#24905354C212yesmaximal, complemented
5C83×#2979C211yesmaximal, characteristic
6C2×#14581822—22no—
7C2×#14581792C2211yescharacteristic
8#19128096C414yes—
9#19128080—21no—
10#19128073—21no—
11#19127725C2212yes—
12C4×#13050574C2211yescharacteristic
13#19127450—21no—
14#19127448—22no—
15#19127448—22no—
16#19127445—21no—
17#532×C83C2211yescharacteristic
18#436×C83—22no—
19C8×D4×C83—22no—
20#431×C83C2212yes—
21#408×C83—22no—
22C22×Dic166—21no—
23#14581822—44no—
24#14581822—44no—
25C2×#13050580—24no—
26C2×#13050578—24no—
27C2×#13050574—21no—
28C2×#13050574—22no—
29#14581792—22no—
30#14581792—21no—
31#14581792C2×C412yes—
32C2×C4⋊Dic83—22no—
33C4⋊(C83⋊C8)—22no—
34C4⋊(C83⋊C8)—22no—
35#14581703C2×C412yes—
36C4×C83⋊C8—21no—
37C2×#41×C83D412yes—
38C2×M16×C83—22no—
39C22×C8×C83—22no—
40C4⋊Q8×C83D411yescharacteristic
41#127×C83D411yescharacteristic
42#124×C83C2×C412yes—
43#121×C83—22no—
44C4×D4×C83—22no—
45C2×C42×C83C2311yescharacteristic
46C4⋊C8×C83D412yes—
47C4⋊C8×C83D412yes—
48C22⋊C8×C83—24no—
49C22⋊C8×C83—22no—
50C4×C8×C83—22no—
51C2×Dic166—22no—
52C2×Dic166—42no—
53C2×Dic166—42no—
54C2×Dic166—41no—
55C22×Dic83—41no—
56#13050580—48no—
57#13050578—48no—
58#13050574—22no—
59#13050574—22no—
60#13050574—82no—
61C2×C83⋊C8—22no—
62C2×C83⋊C8—42no—
63C4⋊Dic83—24no—
64C4⋊Dic83—42no—
65#41×C83—42no—
66#41×C83—42no—
67C2×Q8×C83C22⋊C412yes—
68C2×D4×C83—22no—
69C2×D4×C83—22no—
70C2×D4×C83C22⋊C412yes—
71C22×C4×C83—22no—
72C22×C4×C83C2×D411yescharacteristic
73C22×C4×C83#4112yes—
74M16×C83—82no—
75C2×C8×C83—22no—
76C2×C8×C83—22no—
77C2×C8×C83—82no—
78C4⋊C4×C83—22no—
79C22⋊C4×C83—24no—
80C42×C83C22×C411yescharacteristic
81C42×C83C2×D412yes—
82C42×C83C2×D411yescharacteristic
83C23×C83—22no—
84C23×C83#12311yescharacteristic
85Q8×C83—42no—
86D4×C83—42no—
87D4×C83—42no—
88D4×C83—42no—
89D4×C83—82no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—42no—
96C2×C4×C83—42no—
97C2×C4×C83C2×C22⋊C411yescharacteristic
98C2×C4×C83C4×D412yes—
99C2×C4×C83#12011yescharacteristic
100C2×C4×C83#12112yes—
101C2×Dic83—42no—
102C2×Dic83—82no—
103Dic166—44no—
104Dic166—81no—
105Dic166—81no—
106C8×C83—42no—
107C8×C83—42no—
108C83⋊C8—24no—
109C83⋊C8—82no—
110C22×C83—21no—
111C22×C83—22no—
112C22×C83—42no—
113C22×C83—44no—
114C22×C83#38611yescharacteristic
115C4×C83—21no—
116C4×C83—22no—
117C4×C83—42no—
118C4×C83#45311yescharacteristic
119C4×C83#45311yescharacteristic
120C4×C83#47112yes—
121C4×C83#47312yes—
122Dic83—82no—
123#9435—831nomaximal, Sylow
124C2×C83—41no—
125C2×C83—82no—
126C2×C83#188711yescharacteristic
127C2×C83#190812yes—
128#3152—831no—
129#2979D8311yescharacteristic
130#1560—831no—
131#1549—832no—
132#1299—832no—
133C83#943511yescharacteristic, complemented, Sylow
134C2×#137—1662no—
135#532C2×D8311yescharacteristic
136C2×C4⋊Q8—831no—
137#497—834no—
138#494—1661no—
139#492—1661no—
140#436—22no—
141C8×D4—22no—
142#431C2×D8312yes—
143#423—832no—
144#408—22no—
145C4×M16—831no—
146#333—1661no—
147#332—1662no—
148#332—1662no—
149#331—1661no—
150C2×#41C22⋊D8312yes—
151C22×Q8—1661no—
152#137—3324no—
153#137—3324no—
154C2×Q16—1664no—
155C2×SD16—1664no—
156C2×M16—22no—
157C2×M16—1662no—
158C2×M16—1661no—
159C22×C8—22no—
160C4⋊Q8—832no—
161C4⋊Q8—1662no—
162C4⋊Q8—1661no—
163C4⋊Q8D33211yescharacteristic
164#127D33211yescharacteristic
165#124C4×D8312yes—
166#121—22no—
167C4×D4—22no—
168C2×C4⋊C4—1662no—
169C2×C42C22×D8311yescharacteristic
170C4⋊C8—1662no—
171C4⋊C8—1662no—
172C4⋊C8C22⋊D8312yes—
173C4⋊C8C22⋊D8312yes—
174C22⋊C8—24no—
175C22⋊C8—22no—
176#97—832no—
177C4×C8—22no—
178C4×C8—1661no—
179#41—42no—
180#41—42no—
181C2×Q8—3322no—
182C2×Q8—3322no—
183C2×Q8—3321no—
184C2×Q8—1662no—
185C2×Q8(C2×D83)⋊C412yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4(C2×D83)⋊C412yes—
189C22×C4—22no—
190C22×C4—3321no—
191C22×C4C2×C22⋊D8311yescharacteristic
192C22×C4#1305059012yes—
193Q16—3328no—
194SD16—3328no—
195M16—82no—
196M16—6642no—
197M16—1662no—
198M16—1662no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—82no—
202C2×C8—3322no—
203C2×C8—1662no—
204C4⋊C4—22no—
205C4⋊C4—3322no—
206C4⋊C4—1664no—
207C22⋊C4—24no—
208C42C2×C4×D8311yescharacteristic
209C42C2×C22⋊D8312yes—
210C42C2×D33211yescharacteristic
211C23—22no—
212C23#1458177511yescharacteristic
213Q8—42no—
214Q8—3324no—
215Q8—6641no—
216Q8—6641no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—82no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—3322no—
229C2×C4—6642no—
230C2×C4C2×(C2×D83)⋊C411yescharacteristic
231C2×C4C4×C22⋊D8312yes—
232C2×C4#1458176111yescharacteristic
233C2×C4#1458176612yes—
234C8—42no—
235C8—42no—
236C8—6642no—
237C8—1664no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—44no—
242C22#1912758011yescharacteristic
243C4—21no—
244C4—22no—
245C4—42no—
246C4—6642no—
247C4#1912786911yescharacteristic
248C4#1912786911yescharacteristic
249C4#1912796412yes—
250C4#1912797612yes—
251C2—41no—
252C2—82no—
253C2#2490786111yescharacteristic
254C2#2490797812yes—
subgroupG/H
centerC22#19127580
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2979C2
socleC22×C83#386
prime radical#532×C83C22
cosocleC2×C42×C83C23
hypercenter#2979D83
hypocenterC83#9435
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
3. #24906147C2semidirect
4. #24905354C2semidirect
133. C83#9435semidirect

Maximal subgroups

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

subgroupindex
#249179232
#249061982
#249061472
#249053542
C83×#29792
#943583