Finite Group Databasebeta

← #24917907

Subgroups — #24917907

Order 10624.

564 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14581815C212yesmaximal, complemented
2C2×#14581813C212yesmaximal, complemented
3#19128252C211yesmaximal, characteristic, complemented
4#19128250C211yesmaximal, characteristic, complemented
5#19128094C212yesmaximal, complemented
6#19128093C211yesmaximal, characteristic, complemented
7#19128091C211yesmaximal, characteristic, complemented
8#19128090C211yesmaximal, characteristic, complemented
9#19128087C212yesmaximal, complemented
10#97×D83C211yesmaximal, characteristic, complemented
11#497×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2212yes—
13C2×#13050591C2211yescharacteristic
14C2×#13050589C2211yescharacteristic
15#14581815—28no—
16#14581813—28no—
17C2×#13050580C2212yes—
18C2×#13050579C2212yes—
19C2×#13050577C2212yes—
20C2×#13050577C2212yes—
21C2×D664C2212yes—
22C2×#13050575C2212yes—
23C2×C8×D83C2212yes—
24#14581790C2211yescharacteristic
25#14581789C2211yescharacteristic
26#14581788C2211yescharacteristic
27#14581781C2212yes—
28#14581780C2211yescharacteristic
29#14581767—22no—
30#14581765—22no—
31#14581754—22no—
32(C22⋊D83)⋊C4—22no—
33C42×D83C2211yescharacteristic
34#14581703C2211yescharacteristic
35#14581702C2212yes—
36C2×Q16×C83C2212yes—
37C2×SD16×C83C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39#124×C83C2211yescharacteristic
40#97×C83C2211yescharacteristic
41#13050591—24no—
42#13050591—24no—
43#13050591—24no—
44#13050589—24no—
45C2×Dic166C2311yescharacteristic
46C2×D332—21no—
47C2×D332—22no—
48C2×D332—22no—
49C2×D332C2311yescharacteristic
50C2×D332C2312yes—
51C2×C22⋊D83—21no—
52C2×C22⋊D83—22no—
53C22×Dic83—21no—
54C2×C4×D83—21no—
55C2×C4×D83—22no—
56C2×C4×D83D412yes—
57C2×C4×D83C2311yescharacteristic
58#13050580—24no—
59#13050579—24no—
60#13050577—24no—
61#13050577—24no—
62D664—24no—
63#13050575—24no—
64C2×C83⋊C8C2312yes—
65C8×D83D418yescomplemented
66Dic83⋊C4—22no—
67(C2×D83)⋊C4—22no—
68(C2×D83)⋊C4—24no—
69C4×Dic83D412yes—
70C4×Dic83C2311yescharacteristic
71C2×Q8×C83C2312yes—
72C2×Q8×C83C2311yescharacteristic
73C2×D4×C83C2311yescharacteristic
74Q16×C83—24no—
75SD16×C83—24no—
76C2×C8×C83C2312yes—
77C4⋊C4×C83—22no—
78C22⋊C4×C83—22no—
79C42×C83C2311yescharacteristic
80C23×C83—21no—
81C22×D83—21no—
82C22×D83—22no—
83C22×D83C2×D411yescharacteristic
84Q8×C83—22no—
85Q8×C83—22no—
86Q8×C83—22no—
87D4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83C2×D412yes—
91C2×C4×C83C2411yescharacteristic
92C22⋊D83—44no—
93C22⋊D83—42no—
94C2×Dic83—21no—
95C2×Dic83—42no—
96C2×Dic83C2×D412yes—
97C2×Dic83C2×D411yescharacteristic
98D332—22no—
99D332—22no—
100D332—22no—
101D332—44no—
102D332—42no—
103D332—44no—
104C4×D83—42no—
105C4×D83—42no—
106C4×D83—44no—
107C4×D83C2×D414yes—
108Dic166—22no—
109C8×C83C2×D414yes—
110C83⋊C8C2×D414yes—
111C22×C83—41no—
112C22×C83—42no—
113C22×C83C22×D411yescharacteristic
114C2×D83—22no—
115C2×D83—42no—
116C2×D83—41no—
117C2×D83—44no—
118C2×D83—42no—
119C2×D83#12712yes—
120C4×C83—22no—
121C4×C83—41no—
122C4×C83—42no—
123C4×C83C22×D412yes—
124Dic83—22no—
125Dic83—41no—
126Dic83#12712yes—
127C2×C83—81no—
128C2×C83C2×#12711yescharacteristic
129C2×C83#57512yes—
130D83—22no—
131D83—82no—
132D83—81no—
133#3150—831nomaximal, Sylow
134C83#315011yescharacteristic, complemented, Sylow
135C2×#135—832no—
136C2×#135—832no—
137#532—831no—
138#532—831no—
139#497—832no—
140#497—831no—
141#497D8311yescharacteristic, complemented
142#496—832no—
143#496—831no—
144#496—831no—
145C2×#97—831no—
146C2×#41—832no—
147C2×#41—831no—
148C2×#41—831no—
149#135—1668no—
150#135—1668no—
151C2×Q16—832no—
152C2×Q16C2×D8312yes—
153C2×SD16—832no—
154C2×SD16—832no—
155C2×SD16—832no—
156C2×SD16C2×D8312yes—
157C2×D8—832no—
158C2×D8—832no—
159C22×C8—832no—
160C4⋊Q8—831no—
161C4⋊Q8C2×D8311yescharacteristic
162#127—831no—
163#127—831no—
164#124—832no—
165#124—831no—
166#124C2×D8311yescharacteristic
167#121—1662no—
168#121—1662no—
169C4×D4—1662no—
170C4×D4—1662no—
171C2×C42—831no—
172#97—832no—
173#97—831no—
174#97C2×D8311yescharacteristic
175#41—1664no—
176#41—1664no—
177#41—1664no—
178#41—1664no—
179C2×Q8—831no—
180C2×Q8C22×D8312yes—
181C2×Q8C22×D8311yescharacteristic
182C2×D4—832no—
183C2×D4—831no—
184C2×D4—1661no—
185C2×D4—1662no—
186C2×D4—1662no—
187C2×D4—1661no—
188C2×D4—1662no—
189C2×D4C22×D8311yescharacteristic
190C22×C4—832no—
191C22×C4—831no—
192C22×C4—1661no—
193C22×C4—1662no—
194C22×C4—1661no—
195Q16—24no—
196Q16—1664no—
197SD16—24no—
198SD16—1664no—
199SD16—1664no—
200SD16—1664no—
201D8—1664no—
202D8—1664no—
203C2×C8—832no—
204C2×C8—838no—
205C2×C8C22×D8312yes—
206C4⋊C4—22no—
207C4⋊C4—1662no—
208C22⋊C4—22no—
209C22⋊C4—1664no—
210C22⋊C4—1662no—
211C42—832no—
212C42—831no—
213C42C22×D8311yescharacteristic
214C23—21no—
215C23—831no—
216C23—1662no—
217C23—1661no—
218Q8—22no—
219Q8—22no—
220Q8—22no—
221Q8—1662no—
222D4—22no—
223D4—3324no—
224D4—3322no—
225D4—3324no—
226D4—3322no—
227D4—3324no—
228D4—1662no—
229D4—1662no—
230D4—1662no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—3322no—
234C2×C4—3322no—
235C2×C4—3324no—
236C2×C4—3322no—
237C2×C4—831no—
238C2×C4—834no—
239C2×C4—832no—
240C2×C4—1661no—
241C2×C4D4×D8312yes—
242C2×C4C23×D8311yescharacteristic
243C8—834no—
244C8D4×D8314yes—
245C22—41no—
246C22—42no—
247C22—3322no—
248C22—3321no—
249C22—3324no—
250C22—3322no—
251C22—832no—
252C22—1662no—
253C22C2×D4×D8311yescharacteristic
254C4—22no—
255C4—41no—
256C4—42no—
257C4—3321no—
258C4—832no—
259C4—1662no—
260C4C2×D4×D8312yes—
261C2—81no—
262C2—6642no—
263C2—6641no—
264C2—1662no—
265C2#127×D8311yescharacteristic
266C2#1912850012yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#497×C83C2
socleC22×C83C22×D4
hypercenter#497D83
hypocenterC83#3150
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#14581815C2semidirect
2. C2×#14581813C2semidirect
3. #19128252C2semidirect
4. #19128250C2semidirect
5. #19128094C2semidirect
6. #19128093C2semidirect
7. #19128091C2semidirect
8. #19128090C2semidirect
9. #19128087C2semidirect
10. #97×D83C2semidirect
11. #497×C83C2semidirect
65. C8×D83D4semidirect
134. C83#3150semidirect
141. #497D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818152
C2×#145818132
#191282522
#191282502
#191280942
#191280932
#191280912
#191280902
#191280872
#97×D832
#497×C832
#315083