Finite Group Databasebeta

← #24920320

Subgroups — #24920320

Order 10624.

370 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 133 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#19128431C211yesmaximal, characteristic, complemented
2#19128366C211yesmaximal, characteristic
3#19128098C212yesmaximal, complemented
4#19127990C212yesmaximal
5#19127988C211yesmaximal, characteristic
6#19127985C212yesmaximal, complemented
7#19127983C211yesmaximal, characteristic, complemented
8C8×Dic166C211yesmaximal, characteristic, complemented
9C4×#13050579C211yesmaximal, characteristic
10#19127788C212yesmaximal
11#498×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83—21no—
13C2×#13050579C2211yescharacteristic
14#14581791—21no—
15#14581791C2212yes—
16#14581791C2212yes—
17#14581790—22no—
18#14581770—22no—
19#14581768—21no—
20C4×Dic166C2211yescharacteristic
21C4×Dic166C2211yescharacteristic
22#14581757—21no—
23#14581757C2212yes—
24C4×D332C2211yescharacteristic
25#14581754C2212yes—
26#14581724C2211yescharacteristic
27#14581723C2212yes—
28#14581723C2212yes—
29C4⋊(C83⋊C8)C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581717C2211yescharacteristic
32#14581715C2212yes—
33#14581714C2211yescharacteristic
34#14581712C2212yes—
35C8×Dic83C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4⋊Q8×C83C2211yescharacteristic
38#106×C83C2212yes—
39#106×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41C2×Dic166—22no—
42C2×Dic166—22no—
43C2×Dic166D411yescharacteristic
44C2×Dic166C2311yescharacteristic
45Q8×D83—44no—
46C2×D332C2311yescharacteristic
47C2×C4×D83—21no—
48C2×C4×D83—22no—
49#13050579Q814yescomplemented
50C2×C83⋊C8C2312yes—
51C4⋊Dic83D411yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—21no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—22no—
61C4×Dic83—21no—
62C4×Dic83—22no—
63C4×Dic83C2312yes—
64C2×Q8×C83—21no—
65C2×Q8×C83—21no—
66C2×C8×C83C2311yescharacteristic
67C2×C8×C83C2311yescharacteristic
68C4⋊C4×C83—21no—
69C4⋊C4×C83—21no—
70C4⋊C4×C83C2312yes—
71C4⋊C4×C83C2312yes—
72C42×C83C2311yescharacteristic
73C22×D83—21no—
74Q8×C83—42no—
75Q8×C83—81no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2411yescharacteristic
81C2×Dic83—21no—
82C2×Dic83—21no—
83C2×Dic83—21no—
84C2×Dic83C2×D412yes—
85C2×Dic83C2×D412yes—
86D332—21no—
87D332C2×Q812yes—
88C4×D83—42no—
89C4×D83—44no—
90Dic166—21no—
91Dic166—44no—
92Dic166—82no—
93Dic166SD1614yescomplemented
94Dic166C2×Q812yescomplemented
95C8×C83—21no—
96C8×C83C2×Q812yescomplemented
97C83⋊C8—22no—
98C22×C83C22×D411yescharacteristic
99C2×D83—22no—
100C2×D83—41no—
101C2×D83—41no—
102C4×C83—21no—
103C4×C83—42no—
104C4×C83—42no—
105C4×C83C2×SD1612yes—
106C4×C83C22×Q811yescharacteristic
107C4×C832−1+411yescharacteristic
108Dic83—22no—
109Dic83—22no—
110Dic83—41no—
111Dic83—41no—
112Dic83C2×SD1614yes—
113C2×C83D4×Q811yescharacteristic
114C2×C83C22×SD1611yescharacteristic
115C2×C83#57711yescharacteristic
116D83—42no—
117Q8×SD16—831nomaximal, Sylow
118C83Q8×SD1611yescharacteristic, complemented, Sylow
119Q82—831no—
120D4×Q8—831no—
121#498—832no—
122#498D8311yescharacteristic, complemented
123#476—831no—
124#476—832no—
125#475—831no—
126#475—832no—
127C8×Q8—831no—
128C4×SD16—832no—
129C4×SD16—831no—
130C22×Q8—1661no—
131C2×SD16—831no—
132C4⋊Q8—832no—
133C4⋊Q8—832no—
134C4⋊Q8—1661no—
135C4⋊Q8—1662no—
136C4⋊Q8C2×D8311yescharacteristic
137C4⋊Q8C2×D8311yescharacteristic
138C22⋊Q8—1662no—
139C22⋊Q8—1661no—
140C4×Q8—831no—
141C4×Q8—832no—
142C4×Q8—831no—
143C4×Q8—1661no—
144C4×D4—832no—
145C4×D4—831no—
146#106—831no—
147#106—832no—
148#106—832no—
149#106C2×D8312yes—
150#106C2×D8312yes—
151C4⋊C8—832no—
152C4⋊C8—831no—
153#103—831no—
154#103—832no—
155#102—831no—
156#102—832no—
157C4×C8—832no—
158C4×C8C2×D8311yescharacteristic
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—3324no—
162C2×Q8—831no—
163C2×Q8—831no—
164C2×Q8—1662no—
165C2×Q8—1662no—
166C2×D4—831no—
167C22×C4—1662no—
168C22×C4—1661no—
169SD16—834no—
170C2×C8—832no—
171C2×C8C22×D8311yescharacteristic
172C2×C8C22×D8311yescharacteristic
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—832no—
176C4⋊C4—831no—
177C4⋊C4—831no—
178C4⋊C4—831no—
179C4⋊C4—1662no—
180C4⋊C4—1662no—
181C4⋊C4—1662no—
182C4⋊C4—1661no—
183C4⋊C4C22×D8312yes—
184C4⋊C4C22×D8312yes—
185C22⋊C4—1662no—
186C22⋊C4—1661no—
187C42—832no—
188C42—1662no—
189C42—1661no—
190C42C22×D8311yescharacteristic
191C23—1661no—
192Q8—42no—
193Q8—81no—
194Q8—3324no—
195Q8—834no—
196Q8—832no—
197Q8—6642no—
198Q8—1661no—
199D4—832no—
200D4—1661no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—3324no—
204C2×C4—3322no—
205C2×C4—832no—
206C2×C4—832no—
207C2×C4—1661no—
208C2×C4—1661no—
209C2×C4—1661no—
210C2×C4D4×D8311yescharacteristic
211C2×C4D4×D8311yescharacteristic
212C2×C4C23×D8311yescharacteristic
213C8—21no—
214C8—1662no—
215C8Q8×D8312yescomplemented
216C22—3321no—
217C22—3321no—
218C22—1662no—
219C22C2×D4×D8311yescharacteristic
220C4—21no—
221C4—42no—
222C4—42no—
223C4—3321no—
224C4—3321no—
225C4—834no—
226C4—1662no—
227C4—1662no—
228C4SD16×D8312yes—
229C4C2×Q8×D8311yescharacteristic
230C4#1458184011yescharacteristic
231C2—3322no—
232C2#1912836611yescharacteristic
233C2C2×SD16×D8311yescharacteristic
234C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#498×C83C2
socleC22×C83C22×D4
prime radicalC2×D332C23
hypercenter#498D83
hypocenterC83Q8×SD16
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #19128431C2semidirect
3. #19128098C2semidirect
6. #19127985C2semidirect
7. #19127983C2semidirect
8. C8×Dic166C2semidirect
11. #498×C83C2semidirect
49. #13050579Q8semidirect
93. Dic166SD16semidirect
94. Dic166C2×Q8semidirect
96. C8×C83C2×Q8semidirect
118. C83Q8×SD16semidirect
122. #498D83semidirect
215. C8Q8×D83semidirect

Maximal subgroups

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

subgroupindex
#191284312
#191283662
#191280982
#191279902
#191279882
#191279852
#191279832
C8×Dic1662
C4×#130505792
#191277882
#498×C832
Q8×SD1683