Finite Group Databasebeta

← #24919784

Subgroups — #24919784

Order 10624.

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

normal subgroups only

This group has 430 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#19128372C211yesmaximal, characteristic, complemented
2#19128049C211yesmaximal, characteristic, complemented
3#19128019C212yesmaximal, complemented
4#19127983C211yesmaximal, characteristic, complemented
5#19127952C212yesmaximal, complemented
6#19127896C211yesmaximal, characteristic, complemented
7#19127844C212yesmaximal, complemented
8C4×#13050578C211yesmaximal, characteristic, complemented
9#19127779C211yesmaximal, characteristic, complemented
10#19127700C212yesmaximal, complemented
11#547×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2212yes—
13C2×#13050578C2211yescharacteristic
14C2×#13050575C2212yes—
15C2×#13050574C2212yes—
16#14581789C2211yescharacteristic
17#14581788—21no—
18#14581766C2212yes—
19#14581765—22no—
20C4×Dic166C2211yescharacteristic
21#14581754—21no—
22#14581723C2212yes—
23#14581723C2211yescharacteristic
24C4⋊(C83⋊C8)C2211yescharacteristic
25#14581715C2212yes—
26#14581715C2211yescharacteristic
27#14581713C2211yescharacteristic
28#14581712C2212yes—
29#14581712C2212yes—
30#14581712C2212yes—
31C22⋊(C83⋊C8)C2212yes—
32C4×C83⋊C8C2211yescharacteristic
33C2×#41×C83—21no—
34C4⋊Q8×C83C2211yescharacteristic
35#123×C83—22no—
36C22⋊Q8×C83—21no—
37#121×C83C2212yes—
38C4×D4×C83C2211yescharacteristic
39C4×D4×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2212yes—
41#13050590—24no—
42#13050590—42no—
43C2×Dic166C2311yescharacteristic
44C2×D332—21no—
45C2×D332—21no—
46C2×D332C2312yes—
47C2×C22⋊D83—22no—
48C2×C22⋊D83—22no—
49C2×C4×D83—22no—
50#13050578—22no—
51#13050575—42no—
52#13050574—42no—
53C2×C83⋊C8C2311yescharacteristic
54C2×C83⋊C8C2312yes—
55C2×C83⋊C8C2311yescharacteristic
56C4⋊Dic83C2311yescharacteristic
57Dic83⋊C4—21no—
58(C2×D83)⋊C4—22no—
59C4×Dic83—21no—
60#41×C83—44no—
61C2×Q8×C83—21no—
62C2×D4×C83—22no—
63C2×D4×C83D411yescharacteristic
64C2×D4×C83C2311yescharacteristic
65C22×C4×C83—22no—
66C22×C4×C83—21no—
67C22×C4×C83C2312yes—
68C4⋊C4×C83—22no—
69C4⋊C4×C83—21no—
70C4⋊C4×C83D411yescharacteristic
71C4⋊C4×C83C2311yescharacteristic
72C4⋊C4×C83C2312yes—
73C22⋊C4×C83—21no—
74C22⋊C4×C83—22no—
75C22⋊C4×C83D412yes—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C23×C83C2×D412yes—
79C22×D83—22no—
80Q8×C83—42no—
81D4×C83—22no—
82D4×C83—21no—
83D4×C83—21no—
84D4×C83—44no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—21no—
88C2×C4×C83—22no—
89C2×C4×C83—44no—
90C2×C4×C83—42no—
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2×D412yes—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C22⋊D83—44no—
96C22⋊D83—82no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99D332—22no—
100D332—22no—
101D332—42no—
102D332—81no—
103C4×D83—44no—
104Dic166—21no—
105Dic166#4112yescomplemented
106C83⋊C8—21no—
107C83⋊C8—21no—
108C83⋊C8—22no—
109C22×C83—22no—
110C22×C83—22no—
111C22×C83—22no—
112C22×C83—41no—
113C22×C83—41no—
114C22×C83—41no—
115C22×C83C22×D411yescharacteristic
116C2×D83—42no—
117C2×D83—42no—
118C2×D83—42no—
119C4×C83—22no—
120C4×C83—21no—
121C4×C83—41no—
122C4×C83—42no—
123C4×C83AGL(1,ℤ/8)12yes—
124C4×C83C2×#4111yescharacteristic
125C4×C83H(5,𝔽2)11yescharacteristic
126Dic83—22no—
127Dic83—41no—
128C2×C83—42no—
129C2×C83—81no—
130C2×C83#54611yescharacteristic
131C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C83#57711yescharacteristic
133D83—82no—
134#3332—831nomaximal, Sylow
135C83#333211yescharacteristic, complemented, Sylow
136#550—831no—
137#547D8311yescharacteristic, complemented
138#486—831no—
139#481—832no—
140#475—831no—
141#469—832no—
142#459—831no—
143#449—832no—
144C4×SD16—831no—
145#436—831no—
146#417—832no—
147C2×#41—21no—
148C2×#41—832no—
149C2×SD16—831no—
150C2×D8—832no—
151C2×M16—832no—
152C4⋊Q8C2×D8311yescharacteristic
153#127—831no—
154#127—1661no—
155#123—22no—
156C22⋊Q8—21no—
157#121—832no—
158#121—1662no—
159#121C2×D8312yes—
160C4×Q8—831no—
161C4×D4—1661no—
162C4×D4C2×D8311yescharacteristic
163C4×D4C2×D8311yescharacteristic
164C2×C4⋊C4C2×D8312yes—
165#106—831no—
166#106—832no—
167C4⋊C8—831no—
168#103—831no—
169#103—832no—
170#102—831no—
171#102—832no—
172#102—832no—
173#102—832no—
174C22⋊C8—832no—
175C4×C8—831no—
176#41—44no—
177#41—3322no—
178#41—1664no—
179C2×Q8—21no—
180C2×Q8—831no—
181C2×D4—22no—
182C2×D4—832no—
183C2×D4—1662no—
184C2×D4—1662no—
185C2×D4—1661no—
186C2×D4—1661no—
187C2×D4C22⋊D8311yescharacteristic
188C2×D4C22×D8311yescharacteristic
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—1662no—
192C22×C4C22×D8312yes—
193SD16—1662no—
194D8—3322no—
195M16—3322no—
196C2×C8—831no—
197C2×C8—831no—
198C2×C8—832no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—831no—
202C4⋊C4—1661no—
203C4⋊C4C22⋊D8311yescharacteristic
204C4⋊C4C22×D8311yescharacteristic
205C4⋊C4C22×D8312yes—
206C22⋊C4—21no—
207C22⋊C4—22no—
208C22⋊C4—1662no—
209C22⋊C4C22⋊D8312yes—
210C42—1661no—
211C42C22×D8311yescharacteristic
212C23—21no—
213C23—1662no—
214C23C2×C22⋊D8312yes—
215Q8—42no—
216Q8—832no—
217Q8—1661no—
218D4—22no—
219D4—21no—
220D4—21no—
221D4—44no—
222D4—3324no—
223D4—3322no—
224D4—6641no—
225D4—6642no—
226D4—1662no—
227D4—1662no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—3324no—
235C2×C4—1661no—
236C2×C4—1661no—
237C2×C4C2×C22⋊D8311yescharacteristic
238C2×C4C2×C22⋊D8312yes—
239C2×C4C2×C22⋊D8311yescharacteristic
240C2×C4C23×D8311yescharacteristic
241C8—1661no—
242C8—1661no—
243C8—1662no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—3322no—
251C22—3322no—
252C22—3322no—
253C22C22×C22⋊D8311yescharacteristic
254C4—22no—
255C4—21no—
256C4—41no—
257C4—42no—
258C4—3321no—
259C4—1662no—
260C4#1458181812yes—
261C4#41×D8311yescharacteristic
262C4#1458183811yescharacteristic
263C2—42no—
264C2—81no—
265C2—6642no—
266C2#1912833911yescharacteristic
267C2C2×#1458181811yescharacteristic
268C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#547×C83C2
socleC22×C83C22×D4
hypercenter#547D83
hypocenterC83#3332
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #19128372C2semidirect
2. #19128049C2semidirect
3. #19128019C2semidirect
4. #19127983C2semidirect
5. #19127952C2semidirect
6. #19127896C2semidirect
7. #19127844C2semidirect
8. C4×#13050578C2semidirect
9. #19127779C2semidirect
10. #19127700C2semidirect
11. #547×C83C2semidirect
105. Dic166#41semidirect
135. C83#3332semidirect
137. #547D83semidirect

Maximal subgroups

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

subgroupindex
#191283722
#191280492
#191280192
#191279832
#191279522
#191278962
#191278442
C4×#130505782
#191277792
#191277002
#547×C832
#333283