Finite Group Databasebeta

← #24920322

Subgroups — #24920322

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
1Q8×Dic166C211yesmaximal, characteristic, complemented
2#19128366C211yesmaximal, characteristic
3#19128098C212yesmaximal, complemented
4#19128097C211yesmaximal, characteristic, complemented
5#19127992C211yesmaximal, characteristic
6#19127990C212yesmaximal
7#19127984C212yesmaximal, complemented
8#19127825C211yesmaximal, characteristic, complemented
9C4×#13050577C211yesmaximal, characteristic
10#19127786C212yesmaximal
11#475×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83—21no—
13C2×#13050577C2211yescharacteristic
14#14581792—21no—
15#14581792C2211yescharacteristic
16#14581791—22no—
17#14581791C2212yes—
18#14581791C2212yes—
19#14581770—22no—
20#14581768—21no—
21Q8×Dic83C2212yes—
22C4×Dic166—21no—
23C4×Dic166C2211yescharacteristic
24C4×D332C2211yescharacteristic
25#14581754C2212yes—
26#14581724C2212yes—
27#14581723C2212yes—
28#14581723C2212yes—
29#14581723C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581716C2211yescharacteristic
32#14581714C2212yes—
33#14581712C2211yescharacteristic
34C4×C83⋊C8C2211yescharacteristic
35C8×Dic83C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4×Q8×C83C2211yescharacteristic
38#106×C83C2212yes—
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166—22no—
42C2×Dic166—21no—
43C2×Dic166—22no—
44C2×Dic166D411yescharacteristic
45Q8×D83—44no—
46C2×D332C2311yescharacteristic
47C2×C4×D83—21no—
48C2×C4×D83—22no—
49#13050577Q814yescomplemented
50C2×C83⋊C8C2311yescharacteristic
51C2×C83⋊C8C2311yescharacteristic
52C4⋊Dic83—21no—
53C4⋊Dic83—22no—
54C4⋊Dic83D411yescharacteristic
55C4⋊Dic83C2311yescharacteristic
56C4⋊Dic83C2312yes—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59Dic83⋊C4D412yes—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—22no—
62C4×Dic83—22no—
63C4×Dic83C2312yes—
64C2×Q8×C83—21no—
65C2×Q8×C83C2311yescharacteristic
66C2×C8×C83C2312yes—
67C4⋊C4×C83—21no—
68C4⋊C4×C83—21no—
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2312yes—
71C42×C83—21no—
72C42×C83C2311yescharacteristic
73C22×D83—21no—
74Q8×C83—21no—
75Q8×C83—42no—
76Q8×C83C2×Q812yescomplemented
77C2×C4×C83—21no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2411yescharacteristic
83C2×Dic83—22no—
84C2×Dic83—21no—
85C2×Dic83C2×D412yes—
86C2×Dic83C2×D412yes—
87D332—21no—
88D332C2×Q812yes—
89C4×D83—42no—
90C4×D83—44no—
91Dic166—44no—
92Dic166—82no—
93Dic166—81no—
94Dic166SD1614yescomplemented
95C8×C83—22no—
96C83⋊C8—21no—
97C83⋊C8C2×Q812yescomplemented
98C22×C83C22×D411yescharacteristic
99C2×D83—22no—
100C2×D83—41no—
101C2×D83—41no—
102C4×C83—22no—
103C4×C83—21no—
104C4×C83—42no—
105C4×C83—41no—
106C4×C83C2×SD1612yes—
107C4×C83C22×Q811yescharacteristic
108C4×C832−1+411yescharacteristic
109Dic83—22no—
110Dic83—41no—
111Dic83—42no—
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—831no—
122#498—832no—
123#476—832no—
124#476—831no—
125#475—832no—
126#475D8311yescharacteristic, complemented
127C8×Q8—831no—
128C4×SD16—831no—
129C4×SD16—832no—
130C22×Q8—1661no—
131C2×SD16—831no—
132C4⋊Q8—832no—
133C4⋊Q8—832no—
134C4⋊Q8—831no—
135C4⋊Q8—1661no—
136C4⋊Q8—1662no—
137C4⋊Q8C2×D8311yescharacteristic
138C22⋊Q8—1662no—
139C22⋊Q8—1661no—
140C4×Q8—831no—
141C4×Q8—832no—
142C4×Q8—1661no—
143C4×Q8C2×D8311yescharacteristic
144C4×D4—832no—
145C4×D4—831no—
146#106—832no—
147#106—832no—
148#106—831no—
149#106—832no—
150#106C2×D8312yes—
151C4⋊C8—832no—
152C4⋊C8C2×D8311yescharacteristic
153#103—831no—
154#103C2×D8312yes—
155#102—832no—
156#102—831no—
157C4×C8—832no—
158C4×C8—831no—
159C2×Q8—21no—
160C2×Q8—3324no—
161C2×Q8—831no—
162C2×Q8—1662no—
163C2×Q8—1662no—
164C2×Q8—1661no—
165C2×Q8C22×D8311yescharacteristic
166C2×D4—831no—
167C22×C4—1662no—
168C22×C4—1661no—
169SD16—834no—
170C2×C8—831no—
171C2×C8—831no—
172C2×C8C22×D8312yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—832no—
176C4⋊C4—831no—
177C4⋊C4—832no—
178C4⋊C4—831no—
179C4⋊C4—1662no—
180C4⋊C4—1662no—
181C4⋊C4—1662no—
182C4⋊C4—1661no—
183C4⋊C4C22×D8311yescharacteristic
184C4⋊C4C22×D8312yes—
185C22⋊C4—1662no—
186C22⋊C4—1661no—
187C42—21no—
188C42—832no—
189C42—1662no—
190C42C22×D8311yescharacteristic
191C23—1661no—
192Q8—21no—
193Q8—42no—
194Q8—3324no—
195Q8—834no—
196Q8—6641no—
197Q8—6642no—
198Q8Q8×D8312yescomplemented
199D4—832no—
200D4—1661no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—3324no—
205C2×C4—3322no—
206C2×C4—832no—
207C2×C4—832no—
208C2×C4—1662no—
209C2×C4—1661no—
210C2×C4D4×D8311yescharacteristic
211C2×C4D4×D8311yescharacteristic
212C2×C4C23×D8311yescharacteristic
213C8—22no—
214C8—832no—
215C8—1661no—
216C22—3321no—
217C22—3321no—
218C22—1662no—
219C22C2×D4×D8311yescharacteristic
220C4—22no—
221C4—21no—
222C4—42no—
223C4—41no—
224C4—3321no—
225C4—3322no—
226C4—834no—
227C4—1662no—
228C4SD16×D8312yes—
229C4C2×Q8×D8311yescharacteristic
230C4#1458183911yescharacteristic
231C2—3322no—
232C2#1912836411yescharacteristic
233C2C2×SD16×D8311yescharacteristic
234C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#475×C83C2
socleC22×C83C22×D4
prime radicalC2×D332C23
hypercenter#475D83
hypocenterC83Q8×SD16
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. Q8×Dic166C2semidirect
3. #19128098C2semidirect
4. #19128097C2semidirect
7. #19127984C2semidirect
8. #19127825C2semidirect
11. #475×C83C2semidirect
49. #13050577Q8semidirect
76. Q8×C83C2×Q8semidirect
94. Dic166SD16semidirect
97. C83⋊C8C2×Q8semidirect
118. C83Q8×SD16semidirect
126. #475D83semidirect
198. Q8Q8×D83semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic1662
#191283662
#191280982
#191280972
#191279922
#191279902
#191279842
#191278252
C4×#130505772
#191277862
#475×C832
Q8×SD1683