Finite Group Databasebeta

← #35307090

Subgroups — #35307090

Order 21248.

614 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 113 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#24918584C212yesmaximal
2#24908291C211yesmaximal, characteristic
3#24906135C212yesmaximal, complemented
4#24905296C211yesmaximal, characteristic, complemented
5C83×#2868C211yesmaximal, characteristic, complemented
6#19128251C2211yescharacteristic
7#19128104—22no—
8#19128049—22no—
9#19128014—24no—
10#19127978—24no—
11#19127754C2212yes—
12#19127725C2211yescharacteristic
13C2×#14581720—22no—
14#19127447—24no—
15C2×C4⋊Q8×C83—22no—
16C2×C4⋊Q8×C83C2212yes—
17#384×C83—24no—
18C43×C83C2211yescharacteristic
19C2×#13050590—21no—
20C2×#13050574—21no—
21#14581792D411yescharacteristic
22#14581789D411yescharacteristic
23#14581780—21no—
24#14581766—22no—
25C4×D332—22no—
26#14581725—28no—
27C4⋊(C83⋊C8)—22no—
28C4⋊(C83⋊C8)D414yes—
29#14581720—44no—
30#14581712—28no—
31C22⋊(C83⋊C8)—22no—
32C4×C83⋊C8—22no—
33C22×Q8×C83—22no—
34C4⋊Q8×C83—22no—
35C4⋊Q8×C83—22no—
36C4⋊Q8×C83—22no—
37C4⋊Q8×C83—22no—
38C4⋊Q8×C83—44no—
39C4⋊Q8×C83—44no—
40C2×C4⋊C4×C83—28no—
41C2×C4⋊C4×C83—22no—
42C2×C4⋊C4×C83D414yes—
43C2×C42×C83—22no—
44C2×C42×C83D412yes—
45C2×C42×C83C2311yescharacteristic
46#13050590—42no—
47#13050590—42no—
48C2×Dic166—21no—
49C2×D332—22no—
50C2×D332—21no—
51C2×C22⋊D83—41no—
52C2×C4×D83—41no—
53#13050574—42no—
54C2×C83⋊C8—24no—
55C2×C83⋊C8—41no—
56C4⋊Dic83—22no—
57(C2×D83)⋊C4—42no—
58C2×Q8×C83—42no—
59C2×Q8×C83—42no—
60C2×Q8×C83—44no—
61C2×Q8×C83—44no—
62C22×C4×C83—22no—
63C22×C4×C83—24no—
64C22×C4×C83C2×D412yes—
65C22×C4×C83C2×D411yescharacteristic
66C4⋊C4×C83—24no—
67C4⋊C4×C83—24no—
68C4⋊C4×C83—416no—
69C4⋊C4×C83—42no—
70C4⋊C4×C83—42no—
71C42×C83—28no—
72C42×C83—22no—
73C42×C83D814yes—
74C42×C83C2×D411yescharacteristic
75C42×C83C2×D412yes—
76C42×C83#4111yescharacteristic
77C23×C83#12011yescharacteristic
78C22×D83—41no—
79Q8×C83—84no—
80Q8×C83—82no—
81Q8×C83—82no—
82C2×C4×C83—28no—
83C2×C4×C83—24no—
84C2×C4×C83—24no—
85C2×C4×C83—44no—
86C2×C4×C83—48no—
87C2×C4×C83#12012yes—
88C2×C4×C83#12111yescharacteristic
89C2×C4×C83#12112yes—
90C2×C4×C83#12711yescharacteristic
91C2×C4×C83C2×D812yes—
92C2×C4×C83AGL(1,ℤ/8)12yes—
93C22⋊D83—82no—
94C2×Dic83—41no—
95D332—44no—
96D332—42no—
97C4×D83—81no—
98C4×D83—81no—
99Dic166—42no—
100C83⋊C8—44no—
101C83⋊C8—81no—
102C22×C83—22no—
103C22×C83#39211yescharacteristic
104C22×C83#44712yes—
105C2×D83—82no—
106C2×D83—81no—
107C4×C83—28no—
108C4×C83—22no—
109C4×C83—84no—
110C4×C83#45614yes—
111C4×C83#45912yes—
112C4×C83#49311yescharacteristic
113C4×C83#49611yescharacteristic
114Dic83—81no—
115#10289—831nomaximal, Sylow
116C2×C83—22no—
117C2×C83#200512yes—
118C2×C83#201111yescharacteristic
119D83—161no—
120#3226—832no—
121#2868D8311yescharacteristic, complemented
122#1962—831no—
123#1546—832no—
124#1276—831no—
125C83#1028911yescharacteristic, complemented, Sylow
126#532—831no—
127C2×C4⋊Q8—22no—
128C2×C4⋊Q8C2×D8312yes—
129#500—1662no—
130#486—1662no—
131#480—1664no—
132#474—1664no—
133#431—832no—
134#423—831no—
135C2×#104—1662no—
136#384—24no—
137C43C2×D8311yescharacteristic
138#332—1664no—
139C2×#41—1661no—
140C22×Q8—22no—
141C2×M16—1661no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C4⋊Q8—831no—
149#127—831no—
150#124—1661no—
151#121—1662no—
152C4×D4—1662no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊D8314yes—
156C2×C42—22no—
157C2×C42C22⋊D8312yes—
158C2×C42C22×D8311yescharacteristic
159#107—1668no—
160C4⋊C8—834no—
161C4⋊C8—1662no—
162#104—3324no—
163#102—1668no—
164C22⋊C8—1662no—
165C4×C8—1662no—
166#41—3322no—
167#41—3322no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—1661no—
173C2×D4—3321no—
174C2×D4—1662no—
175C2×D4—1661no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—3321no—
179C22×C4C2×C22⋊D8312yes—
180C22×C4C2×C22⋊D8311yescharacteristic
181M16—3322no—
182C2×C8—3321no—
183C2×C8—1664no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—416no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—1662no—
190C22⋊C4—3322no—
191C42—28no—
192C42—22no—
193C42#1305057514yes—
194C42D4×D8311yescharacteristic
195C42D4×D8312yes—
196C42#1305058911yescharacteristic
197C23—3321no—
198C23#1458176111yescharacteristic
199Q8—82no—
200Q8—82no—
201Q8—84no—
202Q8—3322no—
203D4—3324no—
204D4—3322no—
205D4—6642no—
206C2×C4—28no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—48no—
210C2×C4—44no—
211C2×C4—3321no—
212C2×C4—6641no—
213C2×C4—6641no—
214C2×C4#1458176012yes—
215C2×C4#1458176311yescharacteristic
216C2×C4#1458176412yes—
217C2×C4#1458178811yescharacteristic
218C2×C4C2×#1305057512yes—
219C2×C4#1458181812yes—
220C8—3324no—
221C8—6641no—
222C22—22no—
223C22—6642no—
224C22—6641no—
225C22#1912760911yescharacteristic
226C22#1912783512yes—
227C4—28no—
228C4—22no—
229C4—84no—
230C4—6641no—
231C4#1912788314yes—
232C4#1912789312yes—
233C4#1912807711yescharacteristic
234C4#1912808811yescharacteristic
235C2—22no—
236C2—13281no—
237C2#2490849912yes—
238C2#2490853411yescharacteristic
subgroupG/H
centerC22#19127609
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2868C2
socleC22×C83#392
prime radical#19128251C22
hypercenter#2868D83
hypocenterC83#10289
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
3. #24906135C2semidirect
4. #24905296C2semidirect
5. C83×#2868C2semidirect
121. #2868D83semidirect
125. C83#10289semidirect

Maximal subgroups

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

subgroupindex
#249185842
#249082912
#249061352
#249052962
C83×#28682
#1028983