Finite Group Databasebeta

← #24918384

Subgroups — #24918384

Order 10624.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128249C211yesmaximal, characteristic
2C4⋊Q8×D83C211yesmaximal, characteristic, complemented
3#19128073C211yesmaximal, characteristic, complemented
4#19128052C211yesmaximal, characteristic, complemented
5#19128051C211yesmaximal, characteristic, complemented
6#19127904C214yesmaximal, complemented
7#19127838C214yesmaximal, complemented
8#19127752C211yesmaximal, characteristic, complemented
9#492×C83C211yesmaximal, characteristic, complemented
10C2×#13050589—21no—
11C2×Q8×D83C2212yes—
12C2×#13050578C2214yes—
13#14581792C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581791—21no—
16#14581791—21no—
17#14581790C2212yes—
18#14581779—21no—
19#14581763—21no—
20#14581763C2212yes—
21D4×Dic83—21no—
22D4×Dic83C2212yes—
23C4⋊C4×D83—21no—
24C4⋊C4×D83—21no—
25C42×D83C2211yescharacteristic
26Dic83⋊C8C2212yes—
27#14581717C2214yes—
28#14581716C2214yes—
29(C2×D83)⋊C8C2212yes—
30C4×C83⋊C8C2211yescharacteristic
31C2×SD16×C83C2214yes—
32C4⋊Q8×C83C2211yescharacteristic
33#127×C83C2211yescharacteristic
34C4×C8×C83C2211yescharacteristic
35#13050589—44no—
36C2×Dic166—21no—
37C2×Dic166—22no—
38C2×Dic166C2312yes—
39Q8×D83—44no—
40C2×C22⋊D83—22no—
41C22×Dic83—22no—
42C2×C4×D83—22no—
43C2×C4×D83D411yescharacteristic
44C2×C4×D83D411yescharacteristic
45C2×C4×D83C2311yescharacteristic
46#13050578—44no—
47C2×C83⋊C8C2312yes—
48C4⋊Dic83—21no—
49C4⋊Dic83—21no—
50C4⋊Dic83—21no—
51C4⋊Dic83C2312yes—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54C22⋊Dic83—22no—
55C22⋊Dic83—22no—
56C4×Dic83D411yescharacteristic
57C4×Dic83D411yescharacteristic
58C4×Dic83C2311yescharacteristic
59C2×Q8×C83C2312yes—
60C2×D4×C83—21no—
61C2×D4×C83—21no—
62C2×D4×C83C2312yes—
63SD16×C83—44no—
64C2×C8×C83C2312yes—
65C4⋊C4×C83—21no—
66C4⋊C4×C83—21no—
67C42×C83C2311yescharacteristic
68C23×C83—22no—
69C22×D83C2×D411yescharacteristic
70Q8×C83—24no—
71D4×C83—24no—
72D4×C83—42no—
73D4×C83—42no—
74C2×C4×C83—22no—
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2411yescharacteristic
78C22⋊D83—44no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—44no—
82C2×Dic83C2×D411yescharacteristic
83C2×Dic83C2×D411yescharacteristic
84C2×Dic83C2×D411yescharacteristic
85C4×D83—22no—
86C4×D83—22no—
87C4×D83—44no—
88C4×D83SD1614yescomplemented
89Dic166—24no—
90Dic166—44no—
91Dic166—42no—
92C8×C83—22no—
93C83⋊C8—22no—
94C22×C83—42no—
95C22×C83—42no—
96C22×C83—42no—
97C22×C83C22×D411yescharacteristic
98C2×D83—21no—
99C2×D83—21no—
100C2×D83C2×SD1612yes—
101C4×C83—42no—
102C4×C83C2×SD1612yes—
103C4×C83#13712yes—
104C4×C83H(5,𝔽2)12yes—
105Dic83—21no—
106Dic83—21no—
107Dic83—42no—
108Dic83—42no—
109Dic83C2×SD1612yes—
110C2×C83—82no—
111C2×C83#53511yescharacteristic
112C2×C83C22×SD1611yescharacteristic
113C2×C83C2×#13711yescharacteristic
114D83—22no—
115#3206—831nomaximal, Sylow
116C83#320611yescharacteristic, complemented, Sylow
117#532—831no—
118C2×C4⋊Q8—831no—
119#492—831no—
120#492D8311yescharacteristic, complemented
121#487—831no—
122#487—831no—
123#461—834no—
124#448—834no—
125#431—831no—
126C2×#41—1661no—
127C22×Q8—832no—
128C2×SD16—834no—
129C2×SD16C2×D8314yes—
130C4⋊Q8—832no—
131C4⋊Q8—831no—
132C4⋊Q8—831no—
133C4⋊Q8—1661no—
134C4⋊Q8—1661no—
135C4⋊Q8C2×D8311yescharacteristic
136#127C2×D8311yescharacteristic
137#124—1661no—
138#121—832no—
139#121—1661no—
140C4×D4—832no—
141C4×D4—1661no—
142C2×C4⋊C4—1661no—
143C2×C4⋊C4—1661no—
144C2×C42—831no—
145C4⋊C8—832no—
146#103—834no—
147#103—834no—
148C22⋊C8—832no—
149C4×C8—831no—
150C4×C8C2×D8311yescharacteristic
151#41—3324no—
152C2×Q8—3324no—
153C2×Q8—832no—
154C2×Q8—1662no—
155C2×Q8—1661no—
156C2×Q8C22×D8312yes—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—1662no—
160C2×D4C22×D8312yes—
161C22×C4—831no—
162C22×C4—831no—
163C22×C4—831no—
164C22×C4—1662no—
165C22×C4—1662no—
166SD16—44no—
167SD16—3324no—
168C2×C8—832no—
169C2×C8C22×D8312yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—832no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4—1661no—
176C4⋊C4—1661no—
177C4⋊C4—1661no—
178C22⋊C4—1662no—
179C22⋊C4—1662no—
180C42—831no—
181C42—831no—
182C42—831no—
183C42C22×D8311yescharacteristic
184C23—22no—
185C23—831no—
186Q8—24no—
187Q8—3324no—
188Q8—3322no—
189Q8—1664no—
190D4—24no—
191D4—42no—
192D4—42no—
193D4—3324no—
194C2×C4—22no—
195C2×C4—3324no—
196C2×C4—3324no—
197C2×C4—831no—
198C2×C4—834no—
199C2×C4—831no—
200C2×C4—831no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4D4×D8311yescharacteristic
206C2×C4D4×D8311yescharacteristic
207C2×C4C23×D8311yescharacteristic
208C8—22no—
209C8—1662no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—832no—
214C22—1661no—
215C22—1661no—
216C22C2×D4×D8311yescharacteristic
217C4—42no—
218C4—3322no—
219C4—3322no—
220C4—832no—
221C4—1661no—
222C4—1661no—
223C4SD16×D8312yes—
224C4#1458182112yes—
225C4#1458183712yes—
226C2—82no—
227C2—1662no—
228C2#1912826411yescharacteristic
229C2C2×SD16×D8311yescharacteristic
230C2C2×#1458182111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#492×C83C2
socleC22×C83C22×D4
prime radical#19128249C2
hypercenter#492D83
hypocenterC83#3206
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D83C2semidirect
3. #19128073C2semidirect
4. #19128052C2semidirect
5. #19128051C2semidirect
6. #19127904C2semidirect
7. #19127838C2semidirect
8. #19127752C2semidirect
9. #492×C83C2semidirect
88. C4×D83SD16semidirect
116. C83#3206semidirect
120. #492D83semidirect

Maximal subgroups

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

subgroupindex
#191282492
C4⋊Q8×D832
#191280732
#191280522
#191280512
#191279042
#191278382
#191277522
#492×C832
#320683