Finite Group Databasebeta

← #24920475

Subgroups — #24920475

Order 10624.

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

normal subgroups only

This group has 384 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
1#19128434C212yesmaximal, complemented
2#19128085C211yesmaximal, characteristic, complemented
3#19128084C211yesmaximal, characteristic, complemented
4#19128076C211yesmaximal, characteristic
5#19127916C212yesmaximal, complemented
6#19127915C212yesmaximal, complemented
7#19127900C212yesmaximal, complemented
8C8×Dic166C211yesmaximal, characteristic, complemented
9#19127795C211yesmaximal, characteristic, complemented
10#19127783C211yesmaximal, characteristic, complemented
11C4×D8×C83C211yesmaximal, characteristic, complemented
12C2×Dic332C2211yescharacteristic
13C2×#13050579C2212yes—
14C2×#13050578C2212yes—
15C2×D664C2211yescharacteristic
16C2×#13050575C2212yes—
17#14581788C2212yes—
18#14581780C2212yes—
19#14581779C2212yes—
20#14581778—22no—
21#14581775—22no—
22#14581774—22no—
23C4×Dic166C2211yescharacteristic
24#14581757C2212yes—
25#14581754C2212yes—
26C4×C22⋊D83—22no—
27C4⋊(C83⋊C8)C2211yescharacteristic
28Dic83⋊C8C2211yescharacteristic
29Dic83⋊C8C2211yescharacteristic
30#14581715C2212yes—
31#14581712C2212yes—
32C8×Dic83C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C2×D8×C83C2211yescharacteristic
35C4×D4×C83C2212yes—
36#107×C83C2211yescharacteristic
37#102×C83C2212yes—
38C4×C8×C83C2211yescharacteristic
39C2×Dic166D411yescharacteristic
40C2×Dic166C2312yes—
41C2×D332C2312yes—
42C2×C22⋊D83—22no—
43C2×C22⋊D83—22no—
44C2×C4×D83—22no—
45Dic332—22no—
46#13050579—42no—
47#13050578—42no—
48D664—22no—
49#13050575—42no—
50C2×C83⋊C8C2311yescharacteristic
51C2×C83⋊C8C2311yescharacteristic
52C4⋊Dic83D411yescharacteristic
53Dic83⋊C4—22no—
54Dic83⋊C4D411yescharacteristic
55Dic83⋊C4D411yescharacteristic
56C22⋊Dic83—22no—
57C22⋊Dic83—22no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—22no—
61C4×Dic83—22no—
62C4×Dic83C2311yescharacteristic
63C4×Dic83C2311yescharacteristic
64C2×D4×C83C2312yes—
65C22×C4×C83—22no—
66D8×C83—22no—
67C2×C8×C83C2311yescharacteristic
68C2×C8×C83C2311yescharacteristic
69C4⋊C4×C83C2312yes—
70C22⋊C4×C83—22no—
71C42×C83C2311yescharacteristic
72C23×C83—22no—
73C22×D83—22no—
74D4×C83—22no—
75D4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—82no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2411yescharacteristic
81C22⋊D83—44no—
82C22⋊D83—82no—
83C2×Dic83—22no—
84C2×Dic83C2×D411yescharacteristic
85C2×Dic83C2×D411yescharacteristic
86C2×Dic83C2×D411yescharacteristic
87C2×Dic83C2×D411yescharacteristic
88D332—22no—
89D332—22no—
90C4×D83—82no—
91Dic166—22no—
92Dic166—22no—
93Dic166—22no—
94C8×C83—21no—
95C8×C83#4112yes—
96C83⋊C8—21no—
97C83⋊C8—21no—
98C22×C83—42no—
99C22×C83—42no—
100C22×C83—42no—
101C22×C83C22×D411yescharacteristic
102C2×D83—42no—
103C2×D83—42no—
104C2×D83—42no—
105C4×C83—21no—
106C4×C83—21no—
107C4×C83—42no—
108C4×C83C2×#4111yescharacteristic
109C4×C83H(5,𝔽2)11yescharacteristic
110Dic83—21no—
111Dic83—21no—
112Dic83—21no—
113Dic83—42no—
114Dic83#13512yes—
115C2×C83—82no—
116C2×C83#55011yescharacteristic
117C2×C83C2×#13511yescharacteristic
118C2×C83#57611yescharacteristic
119D83—82no—
120#3396—831nomaximal, Sylow
121C83#339611yescharacteristic, complemented, Sylow
122#559—832no—
123#495—831no—
124#495—831no—
125#493—831no—
126#463—832no—
127#463—832no—
128#460—832no—
129C8×Q8—831no—
130C4×Q16—831no—
131C4×D8—831no—
132C4×D8D8311yescharacteristic, complemented
133C2×Q16—831no—
134C2×SD16—832no—
135C2×SD16—832no—
136C2×D8—832no—
137C2×D8—831no—
138C2×D8C2×D8311yescharacteristic
139#127—832no—
140#124—832no—
141#124—832no—
142#124—1662no—
143#123—1662no—
144#123—1662no—
145C4×Q8—831no—
146C4×Q8—832no—
147C4×D4—832no—
148C4×D4—1662no—
149C4×D4C2×D8312yes—
150#107C2×D8311yescharacteristic
151C4⋊C8—831no—
152C4⋊C8—831no—
153C4⋊C8—831no—
154#103—832no—
155#102—832no—
156#102C2×D8312yes—
157C4×C8—831no—
158C4×C8—831no—
159C4×C8C2×D8311yescharacteristic
160C2×Q8—831no—
161C2×Q8—832no—
162C2×D4—832no—
163C2×D4—1662no—
164C2×D4—1662no—
165C2×D4C22×D8312yes—
166C22×C4—22no—
167C22×C4—1662no—
168Q16—1662no—
169SD16—3322no—
170SD16—3322no—
171D8—22no—
172D8—3322no—
173D8—1662no—
174C2×C8—831no—
175C2×C8—831no—
176C2×C8C22×D8311yescharacteristic
177C2×C8C22×D8311yescharacteristic
178C4⋊C4—831no—
179C4⋊C4—831no—
180C4⋊C4—831no—
181C4⋊C4—1662no—
182C4⋊C4C22×D8312yes—
183C22⋊C4—22no—
184C22⋊C4—1662no—
185C22⋊C4—1662no—
186C22⋊C4—1662no—
187C22⋊C4—1662no—
188C22⋊C4—1662no—
189C42—831no—
190C42—831no—
191C42—1662no—
192C42C22×D8311yescharacteristic
193C23—22no—
194C23—1662no—
195Q8—1662no—
196Q8—1662no—
197Q8—1662no—
198D4—22no—
199D4—22no—
200D4—3324no—
201D4—6642no—
202D4—1662no—
203D4—1662no—
204C2×C4—22no—
205C2×C4—82no—
206C2×C4—831no—
207C2×C4—831no—
208C2×C4—831no—
209C2×C4—831no—
210C2×C4—6642no—
211C2×C4—1662no—
212C2×C4D4×D8311yescharacteristic
213C2×C4D4×D8311yescharacteristic
214C2×C4C23×D8311yescharacteristic
215C8—21no—
216C8—1661no—
217C8—1661no—
218C8#1305059012yes—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—3322no—
223C22—3322no—
224C22—3322no—
225C22C2×D4×D8311yescharacteristic
226C4—21no—
227C4—21no—
228C4—42no—
229C4—3322no—
230C4—832no—
231C4—1661no—
232C4—1661no—
233C4—1661no—
234C4C2×#1305059011yescharacteristic
235C4#1458183811yescharacteristic
236C2—82no—
237C2—6642no—
238C2#1912837211yescharacteristic
239C2#135×D8311yescharacteristic
240C2#1912850211yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×D8×C83C2
socleC22×C83C22×D4
prime radical#19128076C2
hypercenterC4×D8D83
hypocenterC83#3396
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #19128434C2semidirect
2. #19128085C2semidirect
3. #19128084C2semidirect
5. #19127916C2semidirect
6. #19127915C2semidirect
7. #19127900C2semidirect
8. C8×Dic166C2semidirect
9. #19127795C2semidirect
10. #19127783C2semidirect
11. C4×D8×C83C2semidirect
121. C83#3396semidirect
132. C4×D8D83semidirect

Maximal subgroups

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

subgroupindex
#191284342
#191280852
#191280842
#191280762
#191279162
#191279152
#191279002
C8×Dic1662
#191277952
#191277832
C4×D8×C832
#339683