Finite Group Databasebeta

← #35303885

Subgroups — #35303885

Order 21248.

406 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 127 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#24907635C211yesmaximal, characteristic
2#24906148C211yesmaximal, characteristic
3#24906144C211yesmaximal, characteristic
4#24906135C211yesmaximal, characteristic
5#24906135C211yesmaximal, characteristic
6#24905755C211yesmaximal, characteristic
7C2×#487×C83C211yesmaximal, characteristic
8#19127754C2211yescharacteristic
9#19127754C2211yescharacteristic
10C2×#14581728—22no—
11#19127725C2211yescharacteristic
12#19127725C2211yescharacteristic
13#19127450—22no—
14#19127447—22no—
15#19127447—22no—
16#19127447—22no—
17#19127423—22no—
18C2×C4⋊Q8×C83C2211yescharacteristic
19C2×C4⋊Q8×C83C2211yescharacteristic
20#487×C83—22no—
21#487×C83C414yes—
22C2×#103×C83—22no—
23C2×C4×C8×C83C2211yescharacteristic
24C2×#13050574—21no—
25C2×#13050574—21no—
26#14581728—44no—
27C4⋊(C83⋊C8)—21no—
28C4⋊(C83⋊C8)—21no—
29C4⋊(C83⋊C8)—21no—
30C4⋊(C83⋊C8)—21no—
31C4⋊(C83⋊C8)D412yes—
32C4⋊(C83⋊C8)D412yes—
33C22⋊(C83⋊C8)—21no—
34C22⋊(C83⋊C8)—21no—
35C4×C83⋊C8—21no—
36C4×C83⋊C8—21no—
37C22×Q8×C83—21no—
38C22×Q8×C83—21no—
39C22×C8×C83D412yes—
40C4⋊Q8×C83—21no—
41C4⋊Q8×C83—21no—
42C4⋊Q8×C83—21no—
43C4⋊Q8×C83—21no—
44C4⋊Q8×C83—21no—
45C4⋊Q8×C83—21no—
46C4⋊Q8×C83C2×C412yes—
47C4⋊Q8×C83C2×C412yes—
48C2×C4⋊C4×C83—21no—
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C4⋊C4×C83—21no—
52C2×C42×C83C2311yescharacteristic
53#103×C83—28no—
54C4×C8×C83C2×C412yes—
55C4×C8×C83D412yes—
56#13050574—42no—
57#13050574—42no—
58C2×C83⋊C8—22no—
59C2×C83⋊C8—22no—
60C2×C83⋊C8—41no—
61C2×C83⋊C8—41no—
62C2×Q8×C83—22no—
63C2×Q8×C83—22no—
64C2×Q8×C83—41no—
65C2×Q8×C83—41no—
66C2×Q8×C83—42no—
67C2×Q8×C83—42no—
68C2×Q8×C83—82no—
69C22×C4×C83—41no—
70C22×C4×C83—41no—
71C22×C4×C83C2×D411yescharacteristic
72C22×C4×C83#4111yescharacteristic
73C22×C4×C83#4111yescharacteristic
74C2×C8×C83—22no—
75C2×C8×C83—22no—
76C2×C8×C83C22⋊C414yes—
77C4⋊C4×C83—22no—
78C4⋊C4×C83—22no—
79C4⋊C4×C83—22no—
80C4⋊C4×C83—22no—
81C4⋊C4×C83—41no—
82C4⋊C4×C83—41no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—41no—
85C42×C83C22×C411yescharacteristic
86C42×C83C2×D411yescharacteristic
87C42×C83C2×D411yescharacteristic
88C42×C83C2×D411yescharacteristic
89C23×C83#12411yescharacteristic
90Q8×C83—44no—
91Q8×C83—82no—
92Q8×C83—82no—
93Q8×C83—81no—
94Q8×C83—81no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—42no—
98C2×C4×C83—42no—
99C2×C4×C83—82no—
100C2×C4×C83—82no—
101C2×C4×C83C2×C22⋊C411yescharacteristic
102C2×C4×C83C4×D411yescharacteristic
103C2×C4×C83C4×D411yescharacteristic
104C2×C4×C83#12111yescharacteristic
105C2×C4×C83#12111yescharacteristic
106C2×C4×C83#12711yescharacteristic
107C2×C4×C83AGL(1,ℤ/8)12yes—
108C2×C4×C83#13712yes—
109C8×C83—24no—
110C83⋊C8—42no—
111C83⋊C8—42no—
112C83⋊C8—81no—
113C83⋊C8—81no—
114C22×C83—21no—
115C22×C83—21no—
116C22×C83#39011yescharacteristic
117C22×C83#48912yes—
118C4×C83—21no—
119C4×C83—21no—
120C4×C83—82no—
121C4×C83—82no—
122C4×C83#41712yes—
123C4×C83#47112yes—
124C4×C83#47312yes—
125C4×C83#49611yescharacteristic
126C4×C83#49711yescharacteristic
127#9775—831nomaximal, Sylow
128C2×C83—22no—
129C2×C83#195911yescharacteristic
130C2×C83#197711yescharacteristic
131C2×C83#197911yescharacteristic
132C2×#487D8311yescharacteristic
133#1848—831no—
134#1549—831no—
135#1548—831no—
136#1546—831no—
137#1546—831no—
138#1452—831no—
139C83#977511yescharacteristic, complemented, Sylow
140C2×C4⋊Q8C2×D8311yescharacteristic
141C2×C4⋊Q8C2×D8311yescharacteristic
142#487—22no—
143#487Dic8314yes—
144#431—831no—
145#431—831no—
146C2×#108—1662no—
147#423—831no—
148#423—831no—
149C2×#103—22no—
150C2×C4×C8C2×D8311yescharacteristic
151#333—1662no—
152#332—1662no—
153#332—1662no—
154#332—1662no—
155#322—1662no—
156C22×Q8—21no—
157C22×Q8—21no—
158C2×M16—1661no—
159C2×M16—1661no—
160C22×C8C22⋊D8312yes—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C2×Dic8312yes—
168C4⋊Q8C2×Dic8312yes—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C42C22×D8311yescharacteristic
174#108—3324no—
175C4⋊C8—832no—
176C4⋊C8—832no—
177C4⋊C8—1661no—
178C4⋊C8—1661no—
179C4⋊C8—1661no—
180C4⋊C8—1661no—
181#103—28no—
182C22⋊C8—1661no—
183C22⋊C8—1661no—
184C4×C8—1661no—
185C4×C8—1661no—
186C4×C8C2×Dic8312yes—
187C4×C8C22⋊D8312yes—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—82no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4C2×C22⋊D8311yescharacteristic
198C22×C4#1305058911yescharacteristic
199C22×C4#1305058911yescharacteristic
200M16—3322no—
201M16—3322no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—3321no—
205C2×C8—3321no—
206C2×C8—1662no—
207C2×C8—1662no—
208C2×C8C22⋊Dic8314yes—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C42C22×Dic8311yescharacteristic
218C42D4×D8311yescharacteristic
219C42D4×D8311yescharacteristic
220C42C2×C22⋊D8311yescharacteristic
221C23#1458177911yescharacteristic
222Q8—44no—
223Q8—82no—
224Q8—81no—
225Q8—81no—
226Q8—82no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4C2×C22⋊Dic8311yescharacteristic
234C2×C4D4×Dic8311yescharacteristic
235C2×C4D4×Dic8311yescharacteristic
236C2×C4#1458176311yescharacteristic
237C2×C4#1458176311yescharacteristic
238C2×C4#1458178811yescharacteristic
239C2×C4#1458181712yes—
240C2×C4#1458182212yes—
241C8—24no—
242C8—3322no—
243C8—3322no—
244C8—6641no—
245C8—6641no—
246C22—21no—
247C22—21no—
248C22#1912760411yescharacteristic
249C22#1912806012yes—
250C4—21no—
251C4—21no—
252C4—82no—
253C4—82no—
254C4#1912770212yes—
255C4#1912796112yes—
256C4#1912797312yes—
257C4#1912809111yescharacteristic
258C4#1912809611yescharacteristic
259C2—22no—
260C2#2490828011yescharacteristic
261C2#2490836811yescharacteristic
262C2#2490837911yescharacteristic
subgroupG/H
centerC22#19127604
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC2×#487×C83C2
socleC22×C83#390
prime radicalC23×C83#124
cosocleC2×C42×C83C23
hypercenterC2×#487D83
hypocenterC83#9775
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
139. C83#9775semidirect

Maximal subgroups

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

subgroupindex
#249076352
#249061482
#249061442
#249061352
#249057552
C2×#487×C832
#977583