Finite Group Databasebeta

← #24908656

Subgroups — #24908656

Order 10624.

316 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×#14581723C211yesmaximal, characteristic, complemented
2C2×#14581716C211yesmaximal, characteristic, complemented
3C2×#14581716C211yesmaximal, characteristic, complemented
4C2×C22⋊(C83⋊C8)C211yesmaximal, characteristic
5#19127561C211yesmaximal, characteristic
6#19127463C211yesmaximal, characteristic, complemented
7C2×C22⋊Q8×C83C211yesmaximal, characteristic
8C22×C83⋊C8C2211yescharacteristic
9C22×C83⋊C8C2211yescharacteristic
10C2×C4⋊Dic83C2211yescharacteristic
11C2×C4⋊Dic83C2211yescharacteristic
12C2×C22⋊Dic83—21no—
13#14581723—22no—
14#14581716—22no—
15#14581716—22no—
16C22⋊(C83⋊C8)—22no—
17C83⋊#95—21no—
18C22×Q8×C83C2211yescharacteristic
19C23×C4×C83C2211yescharacteristic
20C22⋊Q8×C83—22no—
21C22⋊Q8×C83—22no—
22C2×C4⋊C4×C83—21no—
23C2×C4⋊C4×C83C2211yescharacteristic
24C2×C22⋊C4×C83—21no—
25C22×Dic83—21no—
26C22×Dic83—21no—
27C2×C83⋊C8—21no—
28C2×C83⋊C8—21no—
29C2×C83⋊C8—21no—
30C2×C83⋊C8—21no—
31C2×C83⋊C8—21no—
32C2×C83⋊C8D412yes—
33C4⋊Dic83—21no—
34C4⋊Dic83—21no—
35C4⋊Dic83—21no—
36C4⋊Dic83—21no—
37C22⋊Dic83—42no—
38C24×C83D411yescharacteristic
39C2×Q8×C83—21no—
40C2×Q8×C83—21no—
41C2×Q8×C83—21no—
42C2×Q8×C83—21no—
43C2×Q8×C83—21no—
44C2×Q8×C83D412yes—
45C22×C4×C83—21no—
46C22×C4×C83—21no—
47C22×C4×C83—21no—
48C22×C4×C83—21no—
49C22×C4×C83—21no—
50C22×C4×C83—21no—
51C22×C4×C83—21no—
52C22×C4×C83—21no—
53C22×C4×C83D411yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—24no—
56C4⋊C4×C83—21no—
57C4⋊C4×C83D412yes—
58C22⋊C4×C83—24no—
59C23×C83—21no—
60C23×C83—21no—
61C23×C83—21no—
62C23×C83—21no—
63C23×C83—21no—
64C23×C83—21no—
65C23×C83SD1612yes—
66C23×C83C2×D411yescharacteristic
67Q8×C83—22no—
68Q8×C83—22no—
69Q8×C83—41no—
70Q8×C83—41no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—21no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—41no—
77C2×C4×C83—41no—
78C2×C4×C83—41no—
79C2×C4×C83—41no—
80C2×C4×C83—41no—
81C2×C4×C83—41no—
82C2×C4×C83—41no—
83C2×C4×C83—41no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83#4111yescharacteristic
88C2×C4×C83#4111yescharacteristic
89C2×C4×C83#4111yescharacteristic
90C2×Dic83—41no—
91C2×Dic83—41no—
92C2×Dic83—41no—
93C2×Dic83—41no—
94C2×Dic83—41no—
95C2×Dic83—41no—
96C83⋊C8—22no—
97C83⋊C8—41no—
98C22×C83—22no—
99C22×C83—22no—
100C22×C83—22no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83—41no—
104C22×C83—41no—
105C22×C83#12011yescharacteristic
106C22×C83#12111yescharacteristic
107C22×C83#12311yescharacteristic
108C22×C83C2×SD1611yescharacteristic
109C22×C83#13511yescharacteristic
110C22×C83#13711yescharacteristic
111C22×C83#13711yescharacteristic
112C4×C83—41no—
113C4×C83—42no—
114C4×C83—42no—
115C4×C83#12111yescharacteristic
116C4×C83#12111yescharacteristic
117C4×C83#12311yescharacteristic
118C4×C83#12411yescharacteristic
119Dic83—81no—
120Dic83—81no—
121C2×C83—42no—
122C2×C83#39411yescharacteristic
123C2×C83#44811yescharacteristic
124C2×C83#45211yescharacteristic
125C2×C83#46511yescharacteristic
126C2×C83#47011yescharacteristic
127C2×C83#48311yescharacteristic
128C2×C83#48511yescharacteristic
129#2035—831nomaximal, Sylow
130C83#203511yescharacteristic, complemented, Sylow
131C2×C22⋊Q8D8311yescharacteristic
132C2×#106—831no—
133C2×#103—831no—
134C2×#103—831no—
135C2×C22⋊C8—831no—
136#380—831no—
137#340—831no—
138C22×Q8C2×D8311yescharacteristic
139C23×C4C2×D8311yescharacteristic
140C22×C8—831no—
141C22×C8—831no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—831no—
146C2×C4⋊C4—831no—
147C2×C4⋊C4C2×D8311yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—1661no—
150#106—1662no—
151#103—1662no—
152#103—1662no—
153C22⋊C8—1662no—
154#95—1661no—
155C24C22⋊D8311yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8C22⋊D8312yes—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—1661no—
171C22×C4—1661no—
172C22×C4C22⋊D8311yescharacteristic
173C22×C4C22×D8311yescharacteristic
174C2×C8—832no—
175C2×C8—1661no—
176C2×C8—1661no—
177C2×C8—1661no—
178C2×C8—1661no—
179C2×C8—1661no—
180C4⋊C4—24no—
181C4⋊C4—21no—
182C4⋊C4—1661no—
183C4⋊C4—1661no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4C22⋊D8312yes—
187C22⋊C4—24no—
188C22⋊C4—3322no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#1305057712yes—
196C23C2×C22⋊D8311yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—3321no—
215C2×C4—3321no—
216C2×C4—3321no—
217C2×C4—3321no—
218C2×C4—3321no—
219C2×C4—3321no—
220C2×C4D4×D8311yescharacteristic
221C2×C4C2×C22⋊D8311yescharacteristic
222C2×C4C2×C22⋊D8311yescharacteristic
223C2×C4#1305058911yescharacteristic
224C2×C4#1305058911yescharacteristic
225C2×C4#1305058911yescharacteristic
226C8—3321no—
227C8—1662no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22#1458176111yescharacteristic
236C22#1458176411yescharacteristic
237C22#1458177311yescharacteristic
238C22C2×#1305057711yescharacteristic
239C22#1458181211yescharacteristic
240C22#1458182211yescharacteristic
241C22#1458182311yescharacteristic
242C4—42no—
243C4—41no—
244C4—42no—
245C4—6641no—
246C4—6641no—
247C4#1458176311yescharacteristic
248C4#1458176411yescharacteristic
249C4#1458177311yescharacteristic
250C4#1458177911yescharacteristic
251C2—42no—
252C2#1912761511yescharacteristic
253C2#1912784011yescharacteristic
254C2#1912786311yescharacteristic
255C2#1912793211yescharacteristic
256C2#1912795711yescharacteristic
257C2#1912802911yescharacteristic
258C2#1912803911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D83
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
FittingC2×C22⋊Q8×C83C2
socleC23×C83C2×D4
prime radicalC24×C83D4
hypercenterC2×C22⋊Q8D83
hypocenterC83#2035
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14581723C2semidirect
2. C2×#14581716C2semidirect
3. C2×#14581716C2semidirect
6. #19127463C2semidirect
130. C83#2035semidirect

Maximal subgroups

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

subgroupindex
C2×#145817232
C2×#145817162
C2×C22⋊(C83⋊C8)2
#191275612
#191274632
C2×C22⋊Q8×C832
#203583