Finite Group Databasebeta

← #35335458

Subgroups — #35335458

Order 21248.

1530 conjugacy classes of subgroups; only the 371 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 371 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
1C22×#14581786C211yesmaximal, characteristic
2C2×#19127569C211yesmaximal, characteristic, complemented
3#24905713C216yesmaximal, complemented
4#24905705C216yesmaximal
5C2×#384×C83C211yesmaximal, characteristic, complemented
6C2×#14581786—24no—
7C2×#14581786—24no—
8C2×#14581786C418yescomplemented
9C22×Dic83⋊C4C2213yes—
10C22×(C2×D83)⋊C4C2213yes—
11C22×(C2×D83)⋊C4C2214yes—
12#19127569—24no—
13C2×C83⋊#95C2216yes—
14C2×C83⋊#95C22112yes—
15C22×C4⋊C4×C83C2216yes—
16C22×C42×C83C2211yescharacteristic
17#384×C83—24no—
18C24×D83C2311yescharacteristic
19C23×Dic83C2314yes—
20C23×Dic83C2313yes—
21#14581786—232no—
22C2×Dic83⋊C4—26no—
23C2×Dic83⋊C4—26no—
24C2×Dic83⋊C4C2×C4112yes—
25C2×(C2×D83)⋊C4—26no—
26C2×(C2×D83)⋊C4—26no—
27C2×(C2×D83)⋊C4—412no—
28C2×(C2×D83)⋊C4C2×C4112yes—
29C83⋊#95—212no—
30C83⋊#95—412no—
31C23×C4×C83C2314yes—
32C23×C4×C83C2313yes—
33C2×C4⋊C4×C83—212no—
34C2×C4⋊C4×C83—412no—
35C2×C42×C83C2×C414yes—
36C2×C42×C83D414yes—
37C2×C42×C83D414yes—
38C23×D83—21no—
39C23×D83—21no—
40C23×D83—26no—
41C23×D83—26no—
42C23×D83C22×C412yes—
43C22×Dic83—23no—
44C22×Dic83—23no—
45C22×Dic83—26no—
46C22×Dic83—26no—
47C22×Dic83—24no—
48C22×Dic83—212no—
49C22×Dic83—212no—
50C22×Dic83C22×C416yes—
51Dic83⋊C4—224no—
52(C2×D83)⋊C4—224no—
53(C2×D83)⋊C4—416no—
54C24×C83C2411yescharacteristic
55C22×C4×C83—26no—
56C22×C4×C83—26no—
57C22×C4×C83—24no—
58C22×C4×C83—212no—
59C22×C4×C83—212no—
60C22×C4×C83C22×C416yes—
61C22×C4×C83C2×D416yes—
62C22×C4×C83C2×D416yes—
63C4⋊C4×C83—424no—
64C42×C83C22⋊C4116yescomplemented
65C23×C83C23×C411yescharacteristic
66C23×C83C22×D411yescharacteristic
67C23×C83C22×D411yescharacteristic
68C23×C83H(5,𝔽2)16yes—
69C23×C832−1+416yes—
70C22×D83—22no—
71C22×D83—212no—
72C22×D83—43no—
73C22×D83—43no—
74C22×D83—46no—
75C22×D83—44no—
76C22×D83—412no—
77C2×C4×C83—212no—
78C2×C4×C83—26no—
79C2×C4×C83—26no—
80C2×C4×C83—212no—
81C2×C4×C83—412no—
82C2×C4×C83—412no—
83C2×C4×C83—44no—
84C2×C4×C83C2×C22⋊C4112yes—
85C2×Dic83—26no—
86C2×Dic83—212no—
87C2×Dic83—43no—
88C2×Dic83—43no—
89C2×Dic83—46no—
90C2×Dic83—412no—
91C2×Dic83—44no—
92C2×Dic83—412no—
93C22×C83C22×C22⋊C411yescharacteristic
94C22×C83#52016yes—
95C22×C83#53513yes—
96C22×C83#53613yes—
97C22×C83#53716yes—
98C22×C83#56014yes—
99C22×C83#563112yes—
100C2×D83—46no—
101C2×D83—48no—
102C2×D83—81no—
103C2×D83—83no—
104C2×D83—84no—
105C4×C83—212no—
106C4×C83—84no—
107Dic83—46no—
108Dic83—84no—
109#14665—831nomaximal, Sylow
110C2×C83#231113yes—
111C2×C83#238314yes—
112C2×C83#281811yescharacteristic
113C2×C83#282113yes—
114C2×C83#282314yes—
115D83—82no—
116C22×#126—831no—
117C2×#384D8311yescharacteristic, complemented
118C2×#383—831no—
119#1441—836no—
120#1439—836no—
121C83#1466511yescharacteristic, complemented, Sylow
122C2×#126—838no—
123C2×#126—1664no—
124C2×#126—1664no—
125C22×C4⋊C4—833no—
126C22×C4⋊C4C2×D8316yes—
127C22×C22⋊C4—834no—
128C22×C22⋊C4—833no—
129C22×C42C2×D8311yescharacteristic
130#384—24no—
131#383—1664no—
132C2×#95—8312no—
133C2×#95—836no—
134C25—831no—
135C23×C4—833no—
136C23×C4—834no—
137C23×C4C22×D8314yes—
138C23×C4C22×D8313yes—
139#126—16632no—
140C2×C4⋊C4—212no—
141C2×C4⋊C4—412no—
142C2×C4⋊C4—8312no—
143C2×C4⋊C4—1666no—
144C2×C4⋊C4—1666no—
145C2×C22⋊C4—33212no—
146C2×C22⋊C4—8312no—
147C2×C22⋊C4—1666no—
148C2×C22⋊C4—1666no—
149C2×C42C4×D8314yes—
150C2×C42D33214yes—
151C2×C42C22⋊D8314yes—
152#95—33212no—
153#95—16612no—
154C24—832no—
155C24—1661no—
156C24—1661no—
157C24—1666no—
158C24—1666no—
159C24C23×D8311yescharacteristic
160C22×C4—26no—
161C22×C4—26no—
162C22×C4—24no—
163C22×C4—212no—
164C22×C4—212no—
165C22×C4—836no—
166C22×C4—1664no—
167C22×C4—16612no—
168C22×C4—16612no—
169C22×C4—1663no—
170C22×C4—1663no—
171C22×C4—1666no—
172C22×C4—1666no—
173C22×C4C2×C4×D8316yes—
174C22×C4C2×C22⋊D8316yes—
175C22×C4C2×D33216yes—
176C4⋊C4—424no—
177C4⋊C4—16624no—
178C22⋊C4—33216no—
179C22⋊C4—16624no—
180C42(C2×D83)⋊C4116yescomplemented
181C23—3324no—
182C23—33212no—
183C23—3323no—
184C23—3323no—
185C23—3326no—
186C23—1662no—
187C23—16612no—
188C23C22×C4×D8311yescharacteristic
189C23C22×C22⋊D8311yescharacteristic
190C23C22×D33211yescharacteristic
191C23#1458183716yes—
192C23#1458184016yes—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—26no—
196C2×C4—212no—
197C2×C4—44no—
198C2×C4—412no—
199C2×C4—412no—
200C2×C4—3323no—
201C2×C4—3323no—
202C2×C4—3326no—
203C2×C4—3324no—
204C2×C4—33212no—
205C2×C4—33212no—
206C2×C4—1666no—
207C2×C4—16612no—
208C2×C4C2×(C2×D83)⋊C4112yes—
209C22—3328no—
210C22—3326no—
211C22—6641no—
212C22—6643no—
213C22—6644no—
214C22C22×(C2×D83)⋊C411yescharacteristic
215C22#1912818216yes—
216C22#1912826813yes—
217C22#1912827513yes—
218C22#1912828216yes—
219C22#1912843814yes—
220C22#19128450112yes—
221C4—212no—
222C4—84no—
223C4—3326no—
224C4—6644no—
225C2—6642no—
226C2#2491005613yes—
227C2#2491065614yes—
228C2#2491550411yescharacteristic
229C2#2491552213yes—
230C2#2491553714yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#384×C83C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenterC2×#384D83
hypocenterC83#14665
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. C2×#19127569C2semidirect
3. #24905713C2semidirect
5. C2×#384×C83C2semidirect
8. C2×#14581786C4semidirect
64. C42×C83C22⋊C4semidirect
117. C2×#384D83semidirect
121. C83#14665semidirect
180. C42(C2×D83)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145817862
C2×#191275692
#249057132
#249057052
C2×#384×C832
#1466583