Finite Group Databasebeta

← #24917543

Subgroups — #24917543

Order 10624.

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

normal subgroups only

This group has 390 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#19128235C211yesmaximal, characteristic
2#19128031C211yesmaximal, characteristic, complemented
3#19128014C211yesmaximal, characteristic
4#19128008C212yesmaximal, complemented
5#19127923C212yesmaximal, complemented
6#19127857C211yesmaximal, characteristic, complemented
7#19127840C211yesmaximal, characteristic
8C4×#13050580C212yesmaximal, complemented
9C4×#13050577C212yesmaximal, complemented
10#19127754C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yes—
13C2×#13050577C2212yes—
14#14581786—21no—
15#14581784C2211yescharacteristic
16#14581780C2211yescharacteristic
17#14581775—21no—
18#14581771C2211yescharacteristic
19#14581766C2211yescharacteristic
20C4×Dic166C2211yescharacteristic
21C4×D332C2211yescharacteristic
22C4×C22⋊D83—21no—
23#14581749—21no—
24#14581725C2212yes—
25#14581723C2212yes—
26C4⋊(C83⋊C8)C2212yes—
27#14581716C2212yes—
28#14581716C2212yes—
29#14581715C2212yes—
30#14581712C2212yes—
31C22⋊(C83⋊C8)C2211yescharacteristic
32C22⋊(C83⋊C8)C2211yescharacteristic
33C4×C83⋊C8C2212yes—
34C22×Q8×C83C2211yescharacteristic
35C4×Q8×C83—21no—
36C4×Q8×C83—21no—
37C4×Q8×C83C2212yes—
38C4×Q8×C83C2212yes—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83C2211yescharacteristic
41C2×C42×C83—21no—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83—21no—
47#13050580—24no—
48#13050577—24no—
49C2×C83⋊C8C2312yes—
50C2×C83⋊C8C2312yes—
51C4⋊Dic83C2311yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53Dic83⋊C4—21no—
54Dic83⋊C4—21no—
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57C22⋊Dic83—21no—
58(C2×D83)⋊C4—21no—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—21no—
62C4×Dic83—21no—
63C2×Q8×C83—24no—
64C2×Q8×C83—21no—
65C2×Q8×C83C2312yes—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83D411yescharacteristic
69C22×C4×C83D411yescharacteristic
70C22×C4×C83C2311yescharacteristic
71C4⋊C4×C83—22no—
72C4⋊C4×C83—22no—
73C4⋊C4×C83—21no—
74C4⋊C4×C83C2312yes—
75C42×C83—22no—
76C42×C83—22no—
77C42×C83D411yescharacteristic
78C42×C83D411yescharacteristic
79C42×C83C2311yescharacteristic
80C42×C83C2311yescharacteristic
81C23×C83C2×D411yescharacteristic
82C22×D83—21no—
83Q8×C83—24no—
84Q8×C83—22no—
85Q8×C83#4114yes—
86C2×C4×C83—24no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—42no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—42no—
100C2×Dic83—21no—
101C2×Dic83—21no—
102C2×Dic83—21no—
103D332—22no—
104C4×D83—42no—
105Dic166—22no—
106C83⋊C8—22no—
107C83⋊C8—22no—
108C22×C83—21no—
109C22×C83—21no—
110C22×C83#13512yes—
111C22×C83C22×D411yescharacteristic
112C2×D83—41no—
113C2×D83—41no—
114C2×D83—41no—
115C4×C83—24no—
116C4×C83—21no—
117C4×C83—21no—
118C4×C83—21no—
119C4×C83—21no—
120C4×C83—42no—
121C4×C83#13712yes—
122C4×C83C2×#4112yes—
123Dic83—41no—
124Dic83—41no—
125Dic83—41no—
126C2×C83—22no—
127C2×C83#52511yescharacteristic
128C2×C83C2×#13511yescharacteristic
129C2×C83C2×#13711yescharacteristic
130D83—81no—
131#3108—831nomaximal, Sylow
132C83#310811yescharacteristic, complemented, Sylow
133#529—831no—
134C2×C4×Q8D8311yescharacteristic, complemented
135#483—831no—
136#480—831no—
137#479—832no—
138#464—832no—
139#451—831no—
140#448—831no—
141C4×Q16—832no—
142C4×SD16—832no—
143#431—831no—
144C22×Q8C2×D8311yescharacteristic
145C2×Q16—832no—
146C2×SD16—832no—
147#126—1661no—
148#125—831no—
149#124—831no—
150#123—1661no—
151C22⋊Q8—831no—
152#121—831no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—831no—
156C4×Q8C2×D8312yes—
157C4×Q8C2×D8312yes—
158C4×D4—831no—
159C4×D4—1661no—
160#117—1661no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×D8311yescharacteristic
163C2×C42—21no—
164C2×C42C2×D8311yescharacteristic
165#107—832no—
166#106—832no—
167C4⋊C8—832no—
168#103—832no—
169#103—832no—
170#103—832no—
171#102—832no—
172C22⋊C8—831no—
173C22⋊C8—831no—
174C4×C8—832no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—831no—
178C2×Q8C22×D8312yes—
179C2×D4—831no—
180C2×D4—1661no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—1661no—
184C22×C4C22⋊D8311yescharacteristic
185C22×C4C22⋊D8311yescharacteristic
186C22×C4C22×D8311yescharacteristic
187Q16—1664no—
188SD16—1664no—
189C2×C8—832no—
190C2×C8—832no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—831no—
195C4⋊C4—831no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1661no—
199C4⋊C4—1661no—
200C4⋊C4C22×D8312yes—
201C22⋊C4—1661no—
202C22⋊C4—1661no—
203C22⋊C4—1661no—
204C22⋊C4—1661no—
205C22⋊C4—1661no—
206C42—22no—
207C42—22no—
208C42—1661no—
209C42C22⋊D8311yescharacteristic
210C42C22⋊D8311yescharacteristic
211C42C22×D8311yescharacteristic
212C42C22×D8311yescharacteristic
213C23—1661no—
214C23C2×C22⋊D8311yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—1662no—
218Q8#1305059014yes—
219D4—3322no—
220D4—1662no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—3322no—
229C2×C4—1661no—
230C2×C4—1661no—
231C2×C4—1661no—
232C2×C4C2×C22⋊D8311yescharacteristic
233C2×C4C2×C22⋊D8311yescharacteristic
234C2×C4C2×C22⋊D8311yescharacteristic
235C2×C4C2×C22⋊D8311yescharacteristic
236C2×C4C2×C22⋊D8311yescharacteristic
237C2×C4C23×D8311yescharacteristic
238C8—1662no—
239C8—1662no—
240C22—21no—
241C22—21no—
242C22—3321no—
243C22—3321no—
244C22—3321no—
245C22#1458181212yes—
246C22C22×C22⋊D8311yescharacteristic
247C4—24no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—42no—
253C4—3321no—
254C4—3321no—
255C4—3321no—
256C4#1458182312yes—
257C4C2×#1305059012yes—
258C2—22no—
259C2—6641no—
260C2#1912821211yescharacteristic
261C2C2×#1458181211yescharacteristic
262C2C2×#1458182311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C4×Q8×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenterC2×C4×Q8D83
hypocenterC83#3108
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #19128031C2semidirect
4. #19128008C2semidirect
5. #19127923C2semidirect
6. #19127857C2semidirect
8. C4×#13050580C2semidirect
9. C4×#13050577C2semidirect
10. #19127754C2semidirect
11. C2×C4×Q8×C83C2semidirect
132. C83#3108semidirect
134. C2×C4×Q8D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191280312
#191280142
#191280082
#191279232
#191278572
#191278402
C4×#130505802
C4×#130505772
#191277542
C2×C4×Q8×C832
#310883