Finite Group Databasebeta

← #35298914

Subgroups — #35298914

Order 21248.

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

normal subgroups only

This group has 412 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#24908292C211yesmaximal, characteristic
3#24906252C212yesmaximal, complemented
4#24905295C212yesmaximal, complemented
5C83×#1543C211yesmaximal, characteristic
6#19127779—22no—
7D4×C83⋊C8—22no—
8#19127754C2212yes—
9#19127725C2211yescharacteristic
10C2×#14581719—22no—
11#19127668—22no—
12C42×Dic83C2211yescharacteristic
13#19127454—22no—
14#19127452—22no—
15#532×C83C2211yescharacteristic
16#431×C83C2212yes—
17#333×C83C411yescharacteristic, complemented
18#333×C83C411yescharacteristic, complemented
19#331×C83C412yescomplemented
20C2×#13050574—22no—
21C2×#13050574—21no—
22C22×C83⋊C8—22no—
23C2×C4×Dic83—21no—
24C2×C4×Dic83—22no—
25C4⋊(C83⋊C8)—22no—
26C4⋊(C83⋊C8)D412yes—
27C4⋊(C83⋊C8)D412yes—
28#14581719—44no—
29C22⋊(C83⋊C8)—24no—
30C22⋊(C83⋊C8)—22no—
31C4×C83⋊C8—22no—
32C2×#41×C83D412yes—
33C4⋊Q8×C83C2×C411yescharacteristic
34#127×C83C2×C411yescharacteristic
35#124×C83Q812yes—
36#121×C83—22no—
37C4×D4×C83—22no—
38C2×C42×C83C2311yescharacteristic
39C4⋊C8×C83C2×C412yes—
40C4⋊C8×C83C2×C412yes—
41C22⋊C8×C83—22no—
42C4×C8×C83—22no—
43C22×Dic83—41no—
44#13050574—42no—
45#13050574—82no—
46C2×C83⋊C8—22no—
47C2×C83⋊C8—22no—
48C2×C83⋊C8—41no—
49C2×C83⋊C8—82no—
50C4×Dic83—22no—
51C4×Dic83—24no—
52C4×Dic83—42no—
53C4×Dic83—41no—
54#41×C83—42no—
55#41×C83—42no—
56C2×Q8×C83C4⋊C412yes—
57C2×D4×C83—22no—
58C2×D4×C83—22no—
59C2×D4×C83C4⋊C412yes—
60C22×C4×C83—22no—
61C22×C4×C83C2×D411yescharacteristic
62C22×C4×C83#4112yes—
63C2×C8×C83—22no—
64C2×C8×C83—22no—
65C4⋊C4×C83—22no—
66C22⋊C4×C83—24no—
67C42×C83C22×C411yescharacteristic
68C42×C83C2×D412yes—
69C42×C83C2×Q811yescharacteristic
70C23×C83—22no—
71C23×C83#12311yescharacteristic
72Q8×C83—42no—
73D4×C83—42no—
74D4×C83—42no—
75D4×C83—42no—
76D4×C83—82no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—42no—
83C2×C4×C83—42no—
84C2×C4×C83C2×C4⋊C411yescharacteristic
85C2×C4×C83C4×D412yes—
86C2×C4×C83#12011yescharacteristic
87C2×C4×C83C22⋊Q812yes—
88C2×Dic83—42no—
89C2×Dic83—44no—
90C8×C83—42no—
91C8×C83—42no—
92C83⋊C8—42no—
93C83⋊C8—42no—
94C83⋊C8—81no—
95C22×C83—21no—
96C22×C83—22no—
97C22×C83—42no—
98C22×C83—44no—
99C22×C83#38511yescharacteristic
100C4×C83—21no—
101C4×C83—22no—
102C4×C83—42no—
103C4×C83#44312yes—
104C4×C83#44412yes—
105C4×C83#45311yescharacteristic
106C4×C83#45511yescharacteristic
107Dic83—44no—
108#8935—831nomaximal, Sylow
109C2×C83—41no—
110C2×C83—82no—
111C2×C83#185611yescharacteristic
112C2×C83#186612yes—
113#2979—831no—
114#1962—831no—
115#1574—832no—
116#1543D8311yescharacteristic
117#1276—832no—
118C83#893511yescharacteristic, complemented, Sylow
119#532C2×D8311yescharacteristic
120#436—1662no—
121C8×D4—1662no—
122#431—832no—
123#431C2×D8312yes—
124#423—831no—
125C2×#104—1662no—
126#408—1662no—
127C43—831no—
128#335—1662no—
129#334—1662no—
130#333Dic8311yescharacteristic, complemented
131#333Dic8311yescharacteristic, complemented
132#331Dic8312yescomplemented
133C2×#41D33212yes—
134C2×M16—1661no—
135C2×M16—1662no—
136C22×C8—1662no—
137C4⋊Q8C2×Dic8311yescharacteristic
138#127C2×Dic8311yescharacteristic
139#124Dic16612yes—
140#121—22no—
141C4×D4—22no—
142C2×C42—1661no—
143C2×C42—1662no—
144C2×C42C22×D8311yescharacteristic
145C4⋊C8—832no—
146C4⋊C8—832no—
147C4⋊C8—1662no—
148C4⋊C8C2×Dic8312yes—
149C4⋊C8C2×Dic8312yes—
150#104—3324no—
151C22⋊C8—22no—
152C22⋊C8—1664no—
153C22⋊C8—1662no—
154C4×C8—22no—
155C4×C8—1662no—
156#41—42no—
157#41—42no—
158C2×Q8C4⋊Dic8312yes—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4C4⋊Dic8312yes—
162C22×C4—22no—
163C22×C4—3321no—
164C22×C4C2×D33211yescharacteristic
165C22×C4#1305058912yes—
166M16—3322no—
167M16—6642no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—3321no—
171C2×C8—6642no—
172C2×C8—1662no—
173C2×C8—1662no—
174C4⋊C4—22no—
175C22⋊C4—24no—
176C42—3322no—
177C42—3321no—
178C42—1662no—
179C42—1664no—
180C42C22×Dic8311yescharacteristic
181C42D4×D8312yes—
182C42C2×Dic16611yescharacteristic
183C23—22no—
184C23#1458177611yescharacteristic
185Q8—42no—
186D4—42no—
187D4—42no—
188D4—42no—
189D4—82no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—3322no—
198C2×C4—3324no—
199C2×C4C2×C4⋊Dic8311yescharacteristic
200C2×C4D4×Dic8312yes—
201C2×C4#1458176211yescharacteristic
202C2×C4#1458176912yes—
203C8—42no—
204C8—42no—
205C8—3322no—
206C8—3322no—
207C8—6641no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—44no—
212C22#1912757611yescharacteristic
213C4—21no—
214C4—22no—
215C4—42no—
216C4—3324no—
217C4#1912781812yes—
218C4#1912782312yes—
219C4#1912787111yescharacteristic
220C4#1912788111yescharacteristic
221C2—41no—
222C2—82no—
223C2#2490767511yescharacteristic
224C2#2490774012yes—
subgroupG/H
centerC2×C4#14581762
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1543C2
socleC22×C83#385
prime radical#532×C83C22
cosocleC2×C42×C83C23
hypercenter#1543D83
hypocenterC83#8935
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1543

Complements

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

NH ≅ G/Ntype
3. #24906252C2semidirect
4. #24905295C2semidirect
17. #333×C83C4semidirect
18. #333×C83C4semidirect
19. #331×C83C4semidirect
118. C83#8935semidirect
130. #333Dic83semidirect
131. #333Dic83semidirect
132. #331Dic83semidirect

Maximal subgroups

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

subgroupindex
#249165302
#249082922
#249062522
#249052952
C83×#15432
#893583