Finite Group Databasebeta

← #35313699

Subgroups — #35313699

Order 21248.

392 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 87 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#24910181C211yesmaximal, characteristic
2#24905785C212yesmaximal
3#24905406C212yesmaximal
4#24905397C211yesmaximal, characteristic
5C83×#1319C211yesmaximal, characteristic
6C2×#14581790C411yescharacteristic
7C2×Q8×Dic83C411yescharacteristic
8C2×#14581757C2211yescharacteristic
9#19127751—22no—
10#19127607C411yescharacteristic
11#19127588—22no—
12#19127583—22no—
13#19127566C2211yescharacteristic
14#19127547C411yescharacteristic
15#19127427C2212yes—
16#19127426—22no—
17#391×C83C2211yescharacteristic
18#324×C83C2212yes—
19C22×Dic166C2×C411yescharacteristic
20#14581790—22no—
21#14581790C814yescomplemented
22Q8×Dic83—42no—
23#14581757—42no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83C2×C411yescharacteristic
26C2×Dic83⋊C4—22no—
27C2×Dic83⋊C4D412yes—
28C2×C4×Dic83—21no—
29C2×C4×Dic83C2×C411yescharacteristic
30C2×C4×Dic83C2×C411yescharacteristic
31C2×C4×Dic83D412yes—
32Dic83⋊C8—42no—
33C22⋊(C83⋊C8)—22no—
34C8×Dic83—42no—
35C83⋊#95—21no—
36C83⋊#95—22no—
37C83⋊#95—21no—
38C83⋊#95—22no—
39C22×Q8×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83C2311yescharacteristic
41C2×C42×C83C2×C411yescharacteristic
42C22⋊C8×C83—22no—
43#95×C83—22no—
44C2×Dic166—41no—
45C2×Dic166—41no—
46C2×Dic166C2×C812yes—
47C2×Dic166M1612yes—
48C22×Dic83—21no—
49C22×Dic83—21no—
50C22×Dic83C22⋊C412yes—
51C22×Dic83C22⋊C411yescharacteristic
52C22×Dic83C22⋊C411yescharacteristic
53C2×C83⋊C8—42no—
54C4⋊Dic83—42no—
55C4⋊Dic83—41no—
56Dic83⋊C4—28no—
57Dic83⋊C4—42no—
58C4×Dic83—21no—
59C4×Dic83—22no—
60C4×Dic83—22no—
61C4×Dic83—22no—
62C4×Dic83—42no—
63C4×Dic83C2×C812yes—
64C2×Q8×C83—21no—
65C2×Q8×C83—41no—
66C2×Q8×C83—41no—
67C2×Q8×C83C2×C812yes—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83C22×C411yescharacteristic
71C22×C4×C83C2×D412yes—
72C2×C8×C83—42no—
73C4⋊C4×C83—41no—
74C42×C83—41no—
75C23×C83C2×C22⋊C411yescharacteristic
76Q8×C83—42no—
77Q8×C83—42no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—41no—
83C2×C4×C83—41no—
84C2×C4×C83—42no—
85C2×C4×C83—41no—
86C2×C4×C83—41no—
87C2×C4×C83—41no—
88C2×C4×C83C22×C811yescharacteristic
89C2×C4×C83C2×M1611yescharacteristic
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—21no—
94C2×Dic83—41no—
95C2×Dic83—41no—
96C2×Dic83—41no—
97C2×Dic83—41no—
98C2×Dic83—42no—
99C2×Dic83—41no—
100C2×Dic83C22⋊C814yes—
101C2×Dic83#9912yes—
102Dic166—42no—
103Dic166—42no—
104C8×C83—82no—
105C83⋊C8—82no—
106C22×C83—21no—
107C22×C83—21no—
108C22×C83C2×C22⋊C811yescharacteristic
109C22×C83C2×#9911yescharacteristic
110C22×C83#41311yescharacteristic
111C4×C83—21no—
112C4×C83—21no—
113C4×C83—42no—
114C4×C83—42no—
115C4×C83—81no—
116Dic83—22no—
117Dic83—24no—
118Dic83—42no—
119Dic83—41no—
120Dic83—81no—
121#11482—831nomaximal, Sylow
122C2×C83—21no—
123C2×C83—21no—
124C2×C83#145911yescharacteristic
125C2×C83#212911yescharacteristic
126C2×C83#213311yescharacteristic
127#2327—831no—
128#1461—832no—
129#1324—832no—
130#1319—831no—
131#1319D8311yescharacteristic
132C83#1148211yescharacteristic, complemented, Sylow
133C2×C4⋊Q8—831no—
134C2×C4×Q8—831no—
135C2×C4×Q8—831no—
136#431—1662no—
137#391—831no—
138#391C2×D8311yescharacteristic
139#387—1662no—
140#387—1662no—
141#382—831no—
142#376—831no—
143#324—832no—
144#324—1662no—
145#324C2×D8312yes—
146C22×Q8—831no—
147C22×Q8C4×D8311yescharacteristic
148C4⋊Q8—834no—
149C4⋊Q8—1662no—
150C4×Q8—3322no—
151C4×Q8—3322no—
152C2×C4⋊C4—832no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—1661no—
155C2×C4⋊C4—1662no—
156C2×C4⋊C4C22×D8311yescharacteristic
157C2×C42—831no—
158C2×C42—831no—
159C2×C42—832no—
160C2×C42—1661no—
161C2×C42C4×D8311yescharacteristic
162C4⋊C8—3322no—
163C22⋊C8—22no—
164C22⋊C8—1662no—
165C4×C8—3322no—
166#95—22no—
167#95—1661no—
168#95—1661no—
169#95—1662no—
170#95—1662no—
171C2×Q8—21no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—3321no—
175C2×Q8—3321no—
176C2×Q8—832no—
177C2×Q8—832no—
178C2×Q8C8×D8312yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—831no—
182C22×C4—831no—
183C22×C4—832no—
184C22×C4—1661no—
185C22×C4—1661no—
186C22×C4C2×C4×D8311yescharacteristic
187C22×C4D4×D8312yes—
188C2×C8—42no—
189C2×C8—3322no—
190C4⋊C4—41no—
191C4⋊C4—3321no—
192C4⋊C4—3322no—
193C4⋊C4—3322no—
194C4⋊C4—1668no—
195C42—41no—
196C42—3322no—
197C42—832no—
198C42—1662no—
199C42—1662no—
200C42—1662no—
201C42—1661no—
202C23C22⋊C4×D8311yescharacteristic
203Q8—42no—
204Q8—42no—
205Q8—3322no—
206Q8—3322no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—3321no—
218C2×C4—3321no—
219C2×C4—3321no—
220C2×C4—3322no—
221C2×C4—3321no—
222C2×C4—3321no—
223C2×C4—834no—
224C2×C4—832no—
225C2×C4—1662no—
226C2×C4—1661no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4C2×C8×D8311yescharacteristic
230C2×C4M16×D8311yescharacteristic
231C8—82no—
232C8—6642no—
233C22—21no—
234C22—21no—
235C22C22⋊C8×D8311yescharacteristic
236C22#99×D8311yescharacteristic
237C22#1912768211yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—3322no—
244C4—3321no—
245C4—6641no—
246C4—1664no—
247C4—1662no—
248C2—21no—
249C2—21no—
250C2#2490577811yescharacteristic
251C2#2490913611yescharacteristic
252C2#2490915111yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1319C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1319D83
hypocenterC83#11482
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
21. #14581790C8semidirect
132. C83#11482semidirect

Maximal subgroups

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

subgroupindex
#249101812
#249057852
#249054062
#249053972
C83×#13192
#1148283