Finite Group Databasebeta

← #24919833

Subgroups — #24919833

Order 10624.

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

normal subgroups only

This group has 342 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#19128392C212yesmaximal
2#19128103C211yesmaximal, characteristic, complemented
3#19128040C212yesmaximal, complemented
4#19128030C212yesmaximal, complemented
5#19128008C212yesmaximal, complemented
6#19127939C211yesmaximal, characteristic, complemented
7#19127939C211yesmaximal, characteristic, complemented
8C8×D332C211yesmaximal, characteristic
9#19127738C211yesmaximal, characteristic, complemented
10#107×D83C211yesmaximal, characteristic, complemented
11C4×Q16×C83C211yesmaximal, characteristic, complemented
12C2×C8×D83C2211yescharacteristic
13C2×C8×D83C2211yescharacteristic
14#14581786—22no—
15#14581784C2212yes—
16#14581782—22no—
17#14581777—22no—
18#14581768C2212yes—
19#14581768C2212yes—
20C4×D332C2211yescharacteristic
21#14581750C2212yes—
22#14581749—22no—
23C4⋊C4×D83C2212yes—
24#14581726C2211yescharacteristic
25#14581726C2211yescharacteristic
26#14581725C2212yes—
27#14581724C2212yes—
28#14581723C2212yes—
29C4⋊(C83⋊C8)C2211yescharacteristic
30#14581716C2212yes—
31#14581716C2212yes—
32(C2×D83)⋊C8C2211yescharacteristic
33(C2×D83)⋊C8C2211yescharacteristic
34C2×Q16×C83C2211yescharacteristic
35C4×Q8×C83C2212yes—
36#107×C83C2211yescharacteristic
37#103×C83C2212yes—
38C4×C8×C83C2211yescharacteristic
39C2×D332D411yescharacteristic
40C2×C4×D83—22no—
41C2×C4×D83C2311yescharacteristic
42C2×C4×D83C2311yescharacteristic
43C2×C83⋊C8C2311yescharacteristic
44C2×C83⋊C8C2311yescharacteristic
45C8×D83—22no—
46C8×D83—22no—
47C4⋊Dic83D411yescharacteristic
48C4⋊Dic83C2312yes—
49C4⋊Dic83C2312yes—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4D411yescharacteristic
59(C2×D83)⋊C4D411yescharacteristic
60C4×Dic83—22no—
61C2×Q8×C83C2312yes—
62Q16×C83—22no—
63C2×C8×C83C2311yescharacteristic
64C2×C8×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C4⋊C4×C83C2312yes—
67C42×C83—22no—
68C42×C83C2311yescharacteristic
69C22×D83C2×D411yescharacteristic
70C22×D83C2×D411yescharacteristic
71Q8×C83—22no—
72Q8×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2411yescharacteristic
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83C2×D411yescharacteristic
81C2×Dic83C2×D411yescharacteristic
82D332—22no—
83C4×D83—21no—
84C4×D83—21no—
85C4×D83—21no—
86C4×D83—21no—
87C4×D83—44no—
88C8×C83—21no—
89C8×C83#4112yes—
90C83⋊C8—21no—
91C83⋊C8—21no—
92C22×C83C22×D411yescharacteristic
93C2×D83—21no—
94C2×D83—21no—
95C2×D83—21no—
96C2×D83—21no—
97C2×D83—21no—
98C2×D83#13512yes—
99C4×C83—21no—
100C4×C83—21no—
101C4×C83—42no—
102C4×C83—42no—
103C4×C83C2×#4111yescharacteristic
104C4×C832−1+411yescharacteristic
105Dic83—21no—
106Dic83—21no—
107Dic83—42no—
108Dic83—42no—
109C2×C83#54711yescharacteristic
110C2×C83C2×#13511yescharacteristic
111C2×C83#57811yescharacteristic
112D83—22no—
113D83—41no—
114#3336—831nomaximal, Sylow
115C83#333611yescharacteristic, complemented, Sylow
116#552—832no—
117#499—831no—
118#485—832no—
119#483—832no—
120#479—832no—
121#467—831no—
122#467—831no—
123C4×Q16D8311yescharacteristic, complemented
124C8×D4—831no—
125#427—831no—
126C2×#107—831no—
127C2×Q16C2×D8311yescharacteristic
128C22×C8—831no—
129C22×C8—831no—
130#126—1662no—
131#125—832no—
132#125—1662no—
133#123—1662no—
134C22⋊Q8—832no—
135C22⋊Q8—832no—
136C4×Q8C2×D8312yes—
137C4×D4—831no—
138#117—832no—
139#117—1662no—
140C2×C4⋊C4—832no—
141#107—831no—
142#107—831no—
143#107—832no—
144#107C2×D8311yescharacteristic
145#106—832no—
146#106—832no—
147C4⋊C8—831no—
148#103—832no—
149#103—832no—
150#103C2×D8312yes—
151C22⋊C8—831no—
152C22⋊C8—831no—
153C4×C8C2×D8311yescharacteristic
154C2×Q8C22×D8312yes—
155C2×D4—831no—
156C22×C4—831no—
157C22×C4—831no—
158C22×C4—1662no—
159Q16—22no—
160C2×C8—831no—
161C2×C8—831no—
162C2×C8—1662no—
163C2×C8—1662no—
164C2×C8C22×D8311yescharacteristic
165C2×C8C22×D8311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—831no—
168C4⋊C4—832no—
169C4⋊C4—832no—
170C4⋊C4—1662no—
171C4⋊C4—1662no—
172C4⋊C4—1662no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4C22×D8312yes—
176C22⋊C4—831no—
177C22⋊C4—831no—
178C22⋊C4—1662no—
179C22⋊C4—1662no—
180C22⋊C4—1662no—
181C42—22no—
182C42—1662no—
183C42C22×D8311yescharacteristic
184C23—831no—
185C23—831no—
186Q8—22no—
187Q8—22no—
188D4—1662no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—3324no—
192C2×C4—831no—
193C2×C4—831no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4—1661no—
197C2×C4—1661no—
198C2×C4—1661no—
199C2×C4—1661no—
200C2×C4D4×D8311yescharacteristic
201C2×C4D4×D8311yescharacteristic
202C2×C4C23×D8311yescharacteristic
203C8—21no—
204C8—1661no—
205C8—1661no—
206C8#1305059012yes—
207C22—832no—
208C22—1661no—
209C22—1661no—
210C22—1661no—
211C22—1661no—
212C22—1661no—
213C22C2×D4×D8311yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—42no—
218C4—3322no—
219C4—3322no—
220C4—1661no—
221C4—1661no—
222C4C2×#1305059011yescharacteristic
223C4#1458183911yescharacteristic
224C2—3321no—
225C2—1662no—
226C2#1912835611yescharacteristic
227C2#135×D8311yescharacteristic
228C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×Q16×C83C2
socleC22×C83C22×D4
prime radicalC2×D332D4
hypercenterC4×Q16D83
hypocenterC83#3336
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #19128103C2semidirect
3. #19128040C2semidirect
4. #19128030C2semidirect
5. #19128008C2semidirect
6. #19127939C2semidirect
7. #19127939C2semidirect
9. #19127738C2semidirect
10. #107×D83C2semidirect
11. C4×Q16×C83C2semidirect
115. C83#3336semidirect
123. C4×Q16D83semidirect

Maximal subgroups

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

subgroupindex
#191283922
#191281032
#191280402
#191280302
#191280082
#191279392
C8×D3322
#191277382
#107×D832
C4×Q16×C832
#333683