Finite Group Databasebeta

← #35302004

Subgroups — #35302004

Order 21248.

504 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 111 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#24916530C211yesmaximal, characteristic
2#24906198C211yesmaximal, characteristic
3#24906148C212yesmaximal, complemented
4#24905353C212yesmaximal, complemented
5C83×#3152C211yesmaximal, characteristic
6#19127779—22no—
7D4×C83⋊C8—22no—
8#19127754C2212yes—
9#19127725C2212yes—
10#19127668—22no—
11#19127450—21no—
12#19127447—22no—
13#19127447—22no—
14#19127445—21no—
15C2×#137×C83—22no—
16#532×C83C2211yescharacteristic
17C2×C4⋊Q8×C83C2211yescharacteristic
18#497×C83C414yes—
19#494×C83—21no—
20#492×C83—21no—
21C4×M16×C83C2211yescharacteristic
22C2×#13050574—22no—
23C2×#13050574—22no—
24C22×C83⋊C8—22no—
25C4⋊(C83⋊C8)—22no—
26C4⋊(C83⋊C8)—22no—
27C4⋊(C83⋊C8)D412yes—
28C4⋊(C83⋊C8)D412yes—
29C22⋊(C83⋊C8)—24no—
30C22⋊(C83⋊C8)—22no—
31C4×C83⋊C8—22no—
32C2×#41×C83D412yes—
33C22×Q8×C83—21no—
34#137×C83—44no—
35#137×C83—44no—
36C2×Q16×C83—24no—
37C2×SD16×C83—24no—
38C2×M16×C83—21no—
39C4⋊Q8×C83—22no—
40C4⋊Q8×C83—21no—
41C4⋊Q8×C83C2×C412yes—
42C4⋊Q8×C83D411yescharacteristic
43#127×C83D411yescharacteristic
44#124×C83C2×C412yes—
45#121×C83—22no—
46C4×D4×C83—22no—
47C2×C4⋊C4×C83—22no—
48C2×C42×C83C2311yescharacteristic
49#97×C83C2×C412yes—
50C4×C8×C83—21no—
51#13050574—82no—
52#13050574—82no—
53C2×C83⋊C8—22no—
54C2×C83⋊C8—22no—
55C2×C83⋊C8—42no—
56C2×C83⋊C8—82no—
57#41×C83—42no—
58#41×C83—42no—
59C2×Q8×C83—22no—
60C2×Q8×C83—42no—
61C2×Q8×C83—42no—
62C2×Q8×C83—41no—
63C2×Q8×C83C22⋊C412yes—
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83C22⋊C412yes—
67C22×C4×C83—22no—
68C22×C4×C83—41no—
69C22×C4×C83C2×D411yescharacteristic
70C22×C4×C83#4112yes—
71Q16×C83—48no—
72SD16×C83—48no—
73M16×C83—22no—
74M16×C83—22no—
75C2×C8×C83—22no—
76C4⋊C4×C83—22no—
77C4⋊C4×C83—24no—
78C4⋊C4×C83—42no—
79C22⋊C4×C83—24no—
80C42×C83C22×C411yescharacteristic
81C42×C83C2×D412yes—
82C42×C83C2×D411yescharacteristic
83C23×C83—22no—
84C23×C83#12311yescharacteristic
85Q8×C83—42no—
86Q8×C83—44no—
87Q8×C83—81no—
88Q8×C83—81no—
89D4×C83—42no—
90D4×C83—42no—
91D4×C83—42no—
92D4×C83—82no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—42no—
99C2×C4×C83—42no—
100C2×C4×C83—42no—
101C2×C4×C83—82no—
102C2×C4×C83C2×C22⋊C411yescharacteristic
103C2×C4×C83C4×D412yes—
104C2×C4×C83#12011yescharacteristic
105C2×C4×C83#12112yes—
106C8×C83—24no—
107C83⋊C8—42no—
108C83⋊C8—42no—
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—82no—
119C4×C83#45311yescharacteristic
120C4×C83#45311yescharacteristic
121C4×C83#47112yes—
122C4×C83#47312yes—
123#9435—831nomaximal, Sylow
124C2×C83—41no—
125C2×C83—82no—
126C2×C83#188711yescharacteristic
127C2×C83#190812yes—
128#3152D8311yescharacteristic
129#2979—831no—
130#1560—831no—
131#1549—832no—
132#1299—832no—
133C83#943511yescharacteristic, complemented, Sylow
134C2×#137—22no—
135#532C2×D8311yescharacteristic
136C2×C4⋊Q8C2×D8311yescharacteristic
137#497Dic8314yes—
138#494—21no—
139#492—21no—
140#436—1662no—
141C8×D4—1662no—
142#431—832no—
143#423—832no—
144#408—1662no—
145C4×M16C2×D8311yescharacteristic
146#333—1661no—
147#332—1662no—
148#332—1662no—
149#331—1661no—
150C2×#41C22⋊D8312yes—
151C22×Q8—21no—
152#137—44no—
153#137—44no—
154C2×Q16—24no—
155C2×SD16—24no—
156C2×M16—21no—
157C2×M16—1662no—
158C2×M16—1662no—
159C22×C8—1662no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8C2×Dic8312yes—
163C4⋊Q8C22⋊D8311yescharacteristic
164#127C22⋊D8311yescharacteristic
165#124C2×Dic8312yes—
166#121—22no—
167C4×D4—22no—
168C2×C4⋊C4—22no—
169C2×C42C22×D8311yescharacteristic
170C4⋊C8—832no—
171C4⋊C8—832no—
172C4⋊C8—1662no—
173C4⋊C8—1662no—
174C22⋊C8—1664no—
175C22⋊C8—1662no—
176#97C2×Dic8312yes—
177C4×C8—21no—
178C4×C8—1662no—
179#41—42no—
180#41—42no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8C22⋊Dic8312yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4C22⋊Dic8312yes—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4C2×C22⋊D8311yescharacteristic
192C22×C4#1305058912yes—
193Q16—48no—
194SD16—48no—
195M16—22no—
196M16—22no—
197M16—6642no—
198M16—6642no—
199C2×C8—22no—
200C2×C8—3322no—
201C2×C8—6642no—
202C2×C8—1662no—
203C2×C8—1662no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4—42no—
207C22⋊C4—24no—
208C42C22×Dic8311yescharacteristic
209C42D4×D8312yes—
210C42C2×C22⋊D8311yescharacteristic
211C23—22no—
212C23#1458177311yescharacteristic
213Q8—42no—
214Q8—44no—
215Q8—81no—
216Q8—81no—
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—42no—
229C2×C4—82no—
230C2×C4C2×C22⋊Dic8311yescharacteristic
231C2×C4D4×Dic8312yes—
232C2×C4#1458176011yescharacteristic
233C2×C4#1458176412yes—
234C8—24no—
235C8—3322no—
236C8—3322no—
237C8—6642no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—44no—
242C22#1912758111yescharacteristic
243C4—21no—
244C4—22no—
245C4—42no—
246C4—82no—
247C4#1912786811yescharacteristic
248C4#1912786811yescharacteristic
249C4#1912796612yes—
250C4#1912797412yes—
251C2—41no—
252C2—82no—
253C2#2490786211yescharacteristic
254C2#2490798012yes—
subgroupG/H
centerC22#19127581
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#3152C2
socleC22×C83#386
prime radical#532×C83C22
cosocleC2×C42×C83C23
hypercenter#3152D83
hypocenterC83#9435
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
3. #24906148C2semidirect
4. #24905353C2semidirect
133. C83#9435semidirect

Maximal subgroups

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

subgroupindex
#249165302
#249061982
#249061482
#249053532
C83×#31522
#943583