Finite Group Databasebeta

← #35301800

Subgroups — #35301800

Order 21248.

514 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 123 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#24916504C211yesmaximal, characteristic
2#24907455C211yesmaximal, characteristic
3#24906144C212yesmaximal, complemented
4#24905366C212yesmaximal, complemented
5C83×#2563C211yesmaximal, characteristic
6#19127779—24no—
7#19127754C2212yes—
8#19127725C2212yes—
9C2×#14581709—21no—
10#19127665—22no—
11#19127470—21no—
12#19127450—24no—
13C2×C4⋊Q8×C83C2211yescharacteristic
14C2×C4×D4×C83C2211yescharacteristic
15#393×C83C414yes—
16#390×C83—21no—
17#382×C83—21no—
18#380×C83—22no—
19C4×C4⋊C4×C83C2211yescharacteristic
20C2×#13050574—24no—
21C2×#13050574—22no—
22C4⋊(C83⋊C8)—22no—
23C4⋊(C83⋊C8)—22no—
24C4⋊(C83⋊C8)D414yes—
25#14581709—42no—
26C22⋊(C83⋊C8)—24no—
27C22⋊(C83⋊C8)—22no—
28C4×C83⋊C8—22no—
29C22×Q8×C83C2×C412yes—
30C22×D4×C83D411yescharacteristic
31C23×C4×C83D412yes—
32C4⋊Q8×C83—22no—
33C4⋊Q8×C83—22no—
34C4×D4×C83—22no—
35C4×D4×C83—22no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83D411yescharacteristic
40C2×C22⋊C4×C83—24no—
41C2×C22⋊C4×C83C2×C412yes—
42C2×C42×C83—21no—
43C2×C42×C83C2311yescharacteristic
44#95×C83—24no—
45#95×C83C2×C412yes—
46#13050574—42no—
47#13050574—42no—
48#13050574—84no—
49C2×C83⋊C8—24no—
50C2×C83⋊C8—42no—
51C24×C83C22⋊C412yes—
52C2×Q8×C83—24no—
53C2×Q8×C83—82no—
54C2×D4×C83—22no—
55C2×D4×C83—21no—
56C2×D4×C83—21no—
57C2×D4×C83—22no—
58C22×C4×C83—22no—
59C22×C4×C83—22no—
60C22×C4×C83—22no—
61C22×C4×C83—22no—
62C22×C4×C83—44no—
63C22×C4×C83C22⋊C412yes—
64C22×C4×C83C22×C411yescharacteristic
65C22×C4×C83C2×D411yescharacteristic
66C22×C4×C83C2×D411yescharacteristic
67C4⋊C4×C83—22no—
68C4⋊C4×C83—42no—
69C4⋊C4×C83—41no—
70C4⋊C4×C83—42no—
71C4⋊C4×C83—41no—
72C22⋊C4×C83—24no—
73C22⋊C4×C83—84no—
74C42×C83—41no—
75C42×C83—41no—
76C42×C83C2×D412yes—
77C42×C83#4112yes—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—42no—
82C23×C83—42no—
83C23×C83C2×C22⋊C411yescharacteristic
84Q8×C83—84no—
85D4×C83—42no—
86D4×C83—42no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—41no—
90C2×C4×C83—44no—
91C2×C4×C83—42no—
92C2×C4×C83—42no—
93C2×C4×C83—44no—
94C2×C4×C83—42no—
95C2×C4×C83—42no—
96C2×C4×C83—42no—
97C2×C4×C83—82no—
98C2×C4×C83C4×D412yes—
99C2×C4×C83#12011yescharacteristic
100C2×C4×C83#12112yes—
101C2×C4×C83#12311yescharacteristic
102C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C83#13711yescharacteristic
104C2×C4×C83#13712yes—
105C83⋊C8—44no—
106C83⋊C8—82no—
107C22×C83—21no—
108C22×C83—21no—
109C22×C83—44no—
110C22×C83—42no—
111C22×C83—42no—
112C22×C83—42no—
113C22×C83—82no—
114C22×C83#38611yescharacteristic
115C22×C83#41711yescharacteristic
116C22×C83#41911yescharacteristic
117C4×C83—21no—
118C4×C83—22no—
119C4×C83—42no—
120C4×C83—82no—
121C4×C83—82no—
122C4×C83#45512yes—
123C4×C83#46812yes—
124C4×C83#47012yes—
125#9399—831nomaximal, Sylow
126C2×C83—21no—
127C2×C83—21no—
128C2×C83—82no—
129C2×C83#188511yescharacteristic
130C2×C83#189411yescharacteristic
131C2×C83#191411yescharacteristic
132#2975—831no—
133#2563D8311yescharacteristic
134#1800—831no—
135#1548—832no—
136#1303—832no—
137C83#939911yescharacteristic, complemented, Sylow
138C2×C4⋊Q8C2×D8311yescharacteristic
139C2×C4×D4C2×D8311yescharacteristic
140#436—1664no—
141#431—832no—
142#423—832no—
143C2×#100—1661no—
144#407—1662no—
145#393Dic8314yes—
146#390—21no—
147#382—21no—
148#380—22no—
149C4×C4⋊C4C2×D8311yescharacteristic
150#343—1661no—
151#333—1664no—
152C22×Q8C2×Dic8312yes—
153C22×D4C22⋊D8311yescharacteristic
154C23×C4C22⋊D8312yes—
155C2×M16—1662no—
156C2×M16—1664no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4×D4—22no—
160C4×D4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C22⋊D8311yescharacteristic
165C2×C22⋊C4—24no—
166C2×C22⋊C4C2×Dic8312yes—
167C2×C42—21no—
168C2×C42C22×D8311yescharacteristic
169C4⋊C8—834no—
170C4⋊C8—1662no—
171C4⋊C8—1662no—
172#100—3322no—
173C22⋊C8—1664no—
174C22⋊C8—1662no—
175C4×C8—1662no—
176#95—24no—
177#95C2×Dic8312yes—
178C24C22⋊Dic8312yes—
179C2×Q8—24no—
180C2×Q8—82no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—44no—
190C22×C4C22⋊Dic8312yes—
191C22×C4C22×Dic8311yescharacteristic
192C22×C4C2×C22⋊D8311yescharacteristic
193C22×C4C2×C22⋊D8311yescharacteristic
194M16—3322no—
195M16—3322no—
196M16—6644no—
197C2×C8—3322no—
198C2×C8—1664no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—41no—
204C22⋊C4—24no—
205C22⋊C4—84no—
206C42—41no—
207C42—41no—
208C42D4×D8312yes—
209C42#1305058912yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊Dic8311yescharacteristic
216Q8—84no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4D4×Dic8312yes—
231C2×C4#1458176011yescharacteristic
232C2×C4#1458176412yes—
233C2×C4#1458177311yescharacteristic
234C2×C4#1458181711yescharacteristic
235C2×C4#1458182211yescharacteristic
236C2×C4#1458182312yes—
237C8—3324no—
238C8—6642no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#1912758111yescharacteristic
247C22#1912770211yescharacteristic
248C22#1912771011yescharacteristic
249C4—21no—
250C4—22no—
251C4—42no—
252C4—82no—
253C4—82no—
254C4#1912787812yes—
255C4#1912794912yes—
256C4#1912795712yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2490785211yescharacteristic
261C2#2490789711yescharacteristic
262C2#2490801111yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2563C2
socleC22×C83#386
prime radicalC22×D4×C83D4
cosocleC2×C42×C83C23
hypercenter#2563D83
hypocenterC83#9399
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #24906144C2semidirect
4. #24905366C2semidirect
137. C83#9399semidirect

Maximal subgroups

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

subgroupindex
#249165042
#249074552
#249061442
#249053662
C83×#25632
#939983