Finite Group Databasebeta

← #35299348

Subgroups — #35299348

Order 21248.

458 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#19128071C211yesmaximal, characteristic
2#24908435C212yesmaximal
3C2×#19127443C211yesmaximal, characteristic
4#24905344C212yesmaximal
5C83×#2382C211yesmaximal, characteristic
6C2×#14581792C2211yescharacteristic
7#19128071—22no—
8#19128071C414yescomplemented
9C2×#14581715—22no—
10C2×#14581703C2212yes—
11C2×#14581703C2211yescharacteristic
12#19127463—22no—
13#19127463—22no—
14#19127443—22no—
15C2×#125×C83C2211yescharacteristic
16#389×C83—22no—
17#387×C83—22no—
18#387×C83—22no—
19#382×C83C2212yes—
20C22×Dic166—21no—
21C22×C83⋊C8—21no—
22C22×C83⋊C8—22no—
23#14581792—21no—
24#14581792—22no—
25#14581792C2×C412yes—
26C2×C4⋊Dic83—22no—
27#14581715—28no—
28#14581703—21no—
29#14581703—22no—
30#14581703—22no—
31#14581703C2×C412yes—
32#125×C83—22no—
33#125×C83C2×C412yes—
34#125×C83D412yes—
35C2×C4⋊C4×C83—22no—
36C2×C4⋊C4×C83—22no—
37C2×C4⋊C4×C83D412yes—
38C2×C4⋊C4×C83D412yes—
39C2×C4⋊C4×C83D412yes—
40C2×C42×C83—22no—
41C2×C42×C83C2311yescharacteristic
42#95×C83—22no—
43#95×C83—22no—
44#95×C83—22no—
45#95×C83—22no—
46C2×Dic166—22no—
47C2×Dic166—42no—
48C2×Dic166—42no—
49C2×Dic166—41no—
50C22×Dic83—41no—
51C2×C83⋊C8—24no—
52C2×C83⋊C8—22no—
53C2×C83⋊C8—41no—
54C2×C83⋊C8—41no—
55C2×C83⋊C8—42no—
56C2×C83⋊C8—42no—
57C4⋊Dic83—24no—
58C4⋊Dic83—42no—
59C22×C4×C83—22no—
60C22×C4×C83—22no—
61C22×C4×C83—22no—
62C22×C4×C83C2×D411yescharacteristic
63C22×C4×C83C2×D412yes—
64C4⋊C4×C83—24no—
65C4⋊C4×C83—24no—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—42no—
68C4⋊C4×C83—42no—
69C4⋊C4×C83—82no—
70C4⋊C4×C83C22⋊C414yes—
71C42×C83—82no—
72C42×C83C22×C411yescharacteristic
73C42×C83C2×D411yescharacteristic
74C42×C83#4112yes—
75C23×C83#12011yescharacteristic
76C2×C4×C83—21no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—24no—
81C2×C4×C83—42no—
82C2×C4×C83—42no—
83C2×C4×C83—42no—
84C2×C4×C83—42no—
85C2×C4×C83—42no—
86C2×C4×C83—42no—
87C2×C4×C83—42no—
88C2×C4×C83—42no—
89C2×C4×C83C2×C22⋊C411yescharacteristic
90C2×C4×C83C4×D412yes—
91C2×C4×C83#12112yes—
92C2×C4×C83#12311yescharacteristic
93C2×Dic83—42no—
94C2×Dic83—81no—
95C2×Dic83—81no—
96Dic166—44no—
97Dic166—82no—
98C83⋊C8—44no—
99C83⋊C8—42no—
100C22×C83#38611yescharacteristic
101C22×C83#45411yescharacteristic
102C22×C83#45411yescharacteristic
103C22×C83#45512yes—
104C22×C83#45612yes—
105C4×C83—22no—
106C4×C83—22no—
107C4×C83—21no—
108C4×C83—21no—
109C4×C83—44no—
110C4×C83—82no—
111C4×C83—82no—
112Dic83—82no—
113#9004—831nomaximal, Sylow
114C2×C83#166211yescharacteristic
115C2×C83#166311yescharacteristic
116C2×C83#166412yes—
117C2×C83#188212yes—
118C2×C83#188811yescharacteristic
119C2×#491—831no—
120#2382D8311yescharacteristic
121#1993—832no—
122C2×#330—831no—
123#1295—832no—
124C83#900411yescharacteristic, complemented, Sylow
125C2×C4⋊Q8—831no—
126C2×#125C2×D8311yescharacteristic
127#491—834no—
128#491—1662no—
129C2×#103—1662no—
130C2×#97—831no—
131C2×#97—832no—
132#389—22no—
133#387—22no—
134#387—22no—
135#382C2×D8312yes—
136#340—1662no—
137#340—1662no—
138#330—1662no—
139C22×Q8—1661no—
140C22×C8—1662no—
141C22×C8—1661no—
142C4⋊Q8—832no—
143C4⋊Q8—1661no—
144C4⋊Q8—1662no—
145#125—22no—
146#125C4×D8312yes—
147#125D33212yes—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—1662no—
151C2×C4⋊C4C22⋊D8312yes—
152C2×C4⋊C4C22⋊D8312yes—
153C2×C4⋊C4C22⋊D8312yes—
154C2×C42—22no—
155C2×C42C22×D8311yescharacteristic
156#103—1668no—
157#97—832no—
158#97—1662no—
159#97—1661no—
160#97—1662no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165C2×Q8—3322no—
166C2×Q8—3322no—
167C2×Q8—3321no—
168C2×Q8—1662no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—3321no—
173C22×C4C2×C22⋊D8312yes—
174C22×C4C2×C22⋊D8311yescharacteristic
175C2×C8—3321no—
176C2×C8—3321no—
177C2×C8—3322no—
178C2×C8—3322no—
179C2×C8—1664no—
180C2×C8—1662no—
181C4⋊C4—24no—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—82no—
187C4⋊C4—3322no—
188C4⋊C4—1664no—
189C4⋊C4(C2×D83)⋊C414yes—
190C42—82no—
191C42C2×C4×D8311yescharacteristic
192C42C2×D33211yescharacteristic
193C42#1305059012yes—
194C23#1458176111yescharacteristic
195Q8—3324no—
196Q8—6642no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—3322no—
211C2×C4—6641no—
212C2×C4—6641no—
213C2×C4C2×(C2×D83)⋊C411yescharacteristic
214C2×C4C4×C22⋊D8312yes—
215C2×C4#1458176612yes—
216C2×C4#1458177511yescharacteristic
217C8—3324no—
218C8—3322no—
219C22#1912758011yescharacteristic
220C22#1912787411yescharacteristic
221C22#1912787411yescharacteristic
222C22#1912787912yes—
223C22#1912788412yes—
224C4—22no—
225C4—22no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—82no—
230C4—82no—
231C4—6642no—
232C2#2490678611yescharacteristic
233C2#2490679111yescharacteristic
234C2#2490679712yes—
235C2#2490783212yes—
236C2#2490786611yescharacteristic
subgroupG/H
centerC23#14581761
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2382C2
socle, prime radicalC23×C83#120
cosocleC2×C42×C83C23
hypercenter#2382D83
hypocenterC83#9004
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2382

Complements

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

NH ≅ G/Ntype
8. #19128071C4semidirect
124. C83#9004semidirect

Maximal subgroups

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

subgroupindex
C2×#191280712
#249084352
C2×#191274432
#249053442
C83×#23822
#900483