Finite Group Databasebeta

← #24918118

Subgroups — #24918118

Order 10624.

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

normal subgroups only

This group has 338 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#19128309C212yesmaximal
2#19128102C211yesmaximal, characteristic, complemented
3#19128079C211yesmaximal, characteristic, complemented
4#19128068C212yesmaximal, complemented
5#19127958C212yesmaximal
6#19127943C211yesmaximal, characteristic
7#19127939C211yesmaximal, characteristic
8#19127806C212yesmaximal, complemented
9C4×Dic332C211yesmaximal, characteristic, complemented
10#19127655C211yesmaximal, characteristic
11C4×Q16×C83C211yesmaximal, characteristic, complemented
12C2×Dic332C2211yescharacteristic
13C2×#13050580C2212yes—
14C2×#13050573C2211yescharacteristic
15C2×C8×D83C2211yescharacteristic
16#14581790C2212yes—
17#14581787—22no—
18#14581786—22no—
19#14581784—22no—
20#14581783C2212yes—
21#14581782—22no—
22#14581772C2212yes—
23#14581768C2212yes—
24C4×Dic166C2212yes—
25#14581749C2211yescharacteristic
26#14581726C2211yescharacteristic
27#14581723C2212yes—
28#14581717C2212yes—
29#14581716C2212yes—
30#14581715C2212yes—
31#14581703C2211yescharacteristic
32#14581702C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C2×Q16×C83C2211yescharacteristic
35C4×Q8×C83C2212yes—
36#107×C83C2211yescharacteristic
37#103×C83C2212yes—
38C4×C8×C83C2211yescharacteristic
39C2×Dic166C2312yes—
40C2×C4×D83C2311yescharacteristic
41Dic332—22no—
42#13050580—24no—
43#13050573—22no—
44C2×C83⋊C8C2311yescharacteristic
45C2×C83⋊C8C2311yescharacteristic
46C8×D83—22no—
47C4⋊Dic83—22no—
48C4⋊Dic83—22no—
49C4⋊Dic83C2312yes—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55Dic83⋊C4D411yescharacteristic
56Dic83⋊C4D411yescharacteristic
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4D411yescharacteristic
60(C2×D83)⋊C4D411yescharacteristic
61C4×Dic83—22no—
62C4×Dic83C2311yescharacteristic
63C2×Q8×C83C2312yes—
64Q16×C83—22no—
65C2×C8×C83C2311yescharacteristic
66C2×C8×C83C2311yescharacteristic
67C4⋊C4×C83—22no—
68C4⋊C4×C83C2312yes—
69C42×C83—22no—
70C42×C83C2311yescharacteristic
71C22×D83C2×D411yescharacteristic
72Q8×C83—22no—
73Q8×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2411yescharacteristic
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83C2×D411yescharacteristic
82C2×Dic83C2×D411yescharacteristic
83C2×Dic83C2×D411yescharacteristic
84C4×D83—21no—
85C4×D83—21no—
86Dic166—22no—
87Dic166—22no—
88C8×C83—21no—
89C8×C83#4112yes—
90C83⋊C8—21no—
91C83⋊C8#4112yes—
92C22×C83C22×D411yescharacteristic
93C2×D83—21no—
94C2×D83—21no—
95C2×D83—21no—
96C4×C83—21no—
97C4×C83—21no—
98C4×C83—42no—
99C4×C83—42no—
100C4×C83C2×#4111yescharacteristic
101C4×C83C2×#4111yescharacteristic
102Dic83—21no—
103Dic83—21no—
104Dic83—21no—
105Dic83—42no—
106Dic83—42no—
107C2×C83#53211yescharacteristic
108C2×C83#57811yescharacteristic
109C2×C83#57811yescharacteristic
110D83—41no—
111#3177—831nomaximal, Sylow
112C83#317711yescharacteristic, complemented, Sylow
113#541—832no—
114#499—831no—
115#494—831no—
116#491—832no—
117#470—832no—
118#467—831no—
119#467—831no—
120#441—832no—
121C4×Q16—831no—
122C4×Q16D8311yescharacteristic, complemented
123#405—831no—
124C2×Q16—832no—
125C2×Q16—831no—
126C2×Q16C2×D8311yescharacteristic
127C2×M16—831no—
128C22×C8—831no—
129C4⋊Q8—832no—
130#126—1662no—
131#126—1662no—
132#125—832no—
133#125—1662no—
134#125—1662no—
135C22⋊Q8—832no—
136C22⋊Q8—832no—
137C4×Q8—832no—
138C4×Q8C2×D8312yes—
139#117—831no—
140#107—831no—
141#107C2×D8311yescharacteristic
142#106—832no—
143#103—832no—
144#103—832no—
145#103—832no—
146#103C2×D8312yes—
147#97—831no—
148#97—831no—
149C4×C8—831no—
150C4×C8C2×D8311yescharacteristic
151C2×Q8—832no—
152C2×Q8C22×D8312yes—
153C22×C4—831no—
154Q16—22no—
155Q16—1664no—
156Q16—1662no—
157M16—1662no—
158C2×C8—831no—
159C2×C8—831no—
160C2×C8—1662no—
161C2×C8C22×D8311yescharacteristic
162C2×C8C22×D8311yescharacteristic
163C4⋊C4—22no—
164C4⋊C4—831no—
165C4⋊C4—831no—
166C4⋊C4—832no—
167C4⋊C4—1662no—
168C4⋊C4—1662no—
169C4⋊C4—1662no—
170C4⋊C4—1662no—
171C4⋊C4—1662no—
172C4⋊C4—1662no—
173C4⋊C4—1662no—
174C4⋊C4C22×D8312yes—
175C22⋊C4—831no—
176C22⋊C4—831no—
177C22⋊C4—1662no—
178C22⋊C4—1662no—
179C42—22no—
180C42—831no—
181C42—1662no—
182C42C22×D8311yescharacteristic
183C23—831no—
184Q8—22no—
185Q8—22no—
186Q8—1662no—
187Q8—1662no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—831no—
191C2×C4—831no—
192C2×C4—831no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1661no—
196C2×C4—1661no—
197C2×C4D4×D8311yescharacteristic
198C2×C4D4×D8311yescharacteristic
199C2×C4C23×D8311yescharacteristic
200C8—21no—
201C8—832no—
202C8—1661no—
203C8#1305059012yes—
204C22—1661no—
205C22—1661no—
206C22—1661no—
207C22C2×D4×D8311yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—3322no—
213C4—3322no—
214C4—1661no—
215C4—1661no—
216C4—1661no—
217C4#41×D8311yescharacteristic
218C4C2×#1305059011yescharacteristic
219C2—3321no—
220C2#1912824811yescharacteristic
221C2#1912851411yescharacteristic
222C2#1912851511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×Q16×C83C2
socleC22×C83C22×D4
prime radicalC22×D83C2×D4
hypercenterC4×Q16D83
hypocenterC83#3177
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. #19128102C2semidirect
3. #19128079C2semidirect
4. #19128068C2semidirect
8. #19127806C2semidirect
9. C4×Dic332C2semidirect
11. C4×Q16×C83C2semidirect
112. C83#3177semidirect
122. C4×Q16D83semidirect

Maximal subgroups

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

subgroupindex
#191283092
#191281022
#191280792
#191280682
#191279582
#191279432
#191279392
#191278062
C4×Dic3322
#191276552
C4×Q16×C832
#317783