Finite Group Databasebeta

← #24907804

Subgroups — #24907804

Order 10624.

338 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22×#13050580C211yesmaximal, characteristic
2C2×#14581716C211yesmaximal, characteristic
3C2×#14581715C211yesmaximal, characteristic
4#19127607C211yesmaximal, characteristic
5#19127463C211yesmaximal, characteristic
6#19127456C211yesmaximal, characteristic
7C2×C4×Q8×C83C211yesmaximal, characteristic
8C22×Dic166C2211yescharacteristic
9C2×#13050580—22no—
10C2×#13050580—22no—
11C2×#13050580C414yescomplemented
12C22×C83⋊C8C2211yescharacteristic
13C22×C83⋊C8C2211yescharacteristic
14C2×C4⋊Dic83C2211yescharacteristic
15#14581716—22no—
16#14581715—22no—
17C83⋊#95—21no—
18C83⋊#95—21no—
19C22×Q8×C83C2211yescharacteristic
20C4×Q8×C83—22no—
21C4×Q8×C83—22no—
22C2×C4⋊C4×C83—21no—
23C2×C4⋊C4×C83C2211yescharacteristic
24C2×C42×C83—21no—
25C2×C42×C83C2211yescharacteristic
26C2×Dic166—21no—
27C2×Dic166—21no—
28C2×Dic166—21no—
29C2×Dic166—21no—
30C2×Dic166—21no—
31C2×Dic166C2×C412yes—
32C22×Dic83—21no—
33C22×Dic83—21no—
34#13050580—28no—
35C2×C83⋊C8—21no—
36C2×C83⋊C8—21no—
37C2×C83⋊C8—21no—
38C2×C83⋊C8—21no—
39C2×C83⋊C8—21no—
40C2×C83⋊C8C2×C412yes—
41C4⋊Dic83—21no—
42C4⋊Dic83—21no—
43C2×Q8×C83—21no—
44C2×Q8×C83—21no—
45C2×Q8×C83—21no—
46C2×Q8×C83C2×C412yes—
47C2×Q8×C83D412yes—
48C2×Q8×C83D412yes—
49C22×C4×C83—21no—
50C22×C4×C83—21no—
51C22×C4×C83D411yescharacteristic
52C22×C4×C83D411yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83—24no—
55C4⋊C4×C83—21no—
56C4⋊C4×C83D412yes—
57C42×C83—24no—
58C42×C83—21no—
59C42×C83—21no—
60C23×C83C2×D411yescharacteristic
61Q8×C83—22no—
62Q8×C83—22no—
63Q8×C83—22no—
64Q8×C83C22⋊C414yes—
65C2×C4×C83—22no—
66C2×C4×C83—21no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—21no—
74C2×C4×C83—41no—
75C2×C4×C83—41no—
76C2×C4×C83Q1612yescomplemented
77C2×C4×C83C22×C411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83#4111yescharacteristic
82C2×C4×C83#4111yescharacteristic
83C2×Dic83—22no—
84C2×Dic83—41no—
85C2×Dic83—41no—
86C2×Dic83—41no—
87C2×Dic83—41no—
88C2×Dic83—41no—
89Dic166—22no—
90Dic166—22no—
91Dic166—41no—
92Dic166—41no—
93C83⋊C8—22no—
94C83⋊C8—41no—
95C22×C83C4×D411yescharacteristic
96C22×C83#12011yescharacteristic
97C22×C83#12111yescharacteristic
98C22×C83C2×Q1611yescharacteristic
99C22×C83#13511yescharacteristic
100C22×C83#13711yescharacteristic
101C22×C83#13711yescharacteristic
102C4×C83—24no—
103C4×C83—22no—
104C4×C83—41no—
105C4×C83—42no—
106C4×C83C2×C22⋊C411yescharacteristic
107C4×C83C4×D411yescharacteristic
108C4×C83#12111yescharacteristic
109C4×C83#12311yescharacteristic
110Dic83—42no—
111Dic83—81no—
112C2×C83#38611yescharacteristic
113C2×C83C4×Q1611yescharacteristic
114C2×C83#44111yescharacteristic
115C2×C83#45111yescharacteristic
116C2×C83#45211yescharacteristic
117C2×C83#46211yescharacteristic
118C2×C83#46411yescharacteristic
119#1878—831nomaximal, Sylow
120C83#187811yescharacteristic, complemented, Sylow
121C22×Q16—831no—
122C2×C4×Q8D8311yescharacteristic
123C2×#103—831no—
124C2×#103—831no—
125#391—831no—
126#340—831no—
127#336—831no—
128C22×Q8—831no—
129C22×Q8C2×D8311yescharacteristic
130C2×Q16—834no—
131C2×Q16—1662no—
132C2×Q16—1662no—
133C22×C8—831no—
134C22×C8—831no—
135C4×Q8—22no—
136C4×Q8—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—831no—
139C2×C4⋊C4C2×D8311yescharacteristic
140C2×C42—21no—
141C2×C42C2×D8311yescharacteristic
142#103—1662no—
143#103—1662no—
144#95—1661no—
145#95—1661no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—832no—
150C2×Q8—1661no—
151C2×Q8—1661no—
152C2×Q8—1661no—
153C2×Q8—1661no—
154C2×Q8—1661no—
155C2×Q8C4×D8312yes—
156C2×Q8D33212yes—
157C2×Q8C22⋊D8312yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1661no—
161C22×C4—1661no—
162C22×C4C22⋊D8311yescharacteristic
163C22×C4C22⋊D8311yescharacteristic
164C22×C4C22×D8311yescharacteristic
165Q16—1668no—
166C2×C8—832no—
167C2×C8—1661no—
168C2×C8—1661no—
169C2×C8—1661no—
170C2×C8—1661no—
171C2×C8—1661no—
172C4⋊C4—24no—
173C4⋊C4—21no—
174C4⋊C4—1661no—
175C4⋊C4—1661no—
176C4⋊C4C22⋊D8312yes—
177C42—24no—
178C42—21no—
179C42—21no—
180C23C2×C22⋊D8311yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—22no—
184Q8—3321no—
185Q8—3321no—
186Q8—1662no—
187Q8—1662no—
188Q8(C2×D83)⋊C414yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—3321no—
201C2×C4—3321no—
202C2×C4—3321no—
203C2×C4—3321no—
204C2×C4—3321no—
205C2×C4—1662no—
206C2×C4#1305058012yescomplemented
207C2×C4C2×C4×D8311yescharacteristic
208C2×C4C2×C22⋊D8311yescharacteristic
209C2×C4C2×C22⋊D8311yescharacteristic
210C2×C4C2×D33211yescharacteristic
211C2×C4#1305059011yescharacteristic
212C2×C4#1305059011yescharacteristic
213C8—3321no—
214C8—1662no—
215C22C4×C22⋊D8311yescharacteristic
216C22#1458176111yescharacteristic
217C22#1458176611yescharacteristic
218C22C2×#1305058011yescharacteristic
219C22#1458181211yescharacteristic
220C22#1458182211yescharacteristic
221C22#1458182311yescharacteristic
222C4—24no—
223C4—22no—
224C4—41no—
225C4—42no—
226C4—3322no—
227C4—6641no—
228C4C2×(C2×D83)⋊C411yescharacteristic
229C4C4×C22⋊D8311yescharacteristic
230C4#1458176611yescharacteristic
231C4#1458177511yescharacteristic
232C2#1912758011yescharacteristic
233C2C4×#1305058011yescharacteristic
234C2#1912780611yescharacteristic
235C2#1912785711yescharacteristic
236C2#1912786311yescharacteristic
237C2#1912791111yescharacteristic
238C2#1912792311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC2×C4×Q8×C83C2
socle, prime radicalC23×C83C2×D4
cosocleC22×C4×C83C23
hypercenterC2×C4×Q8D83
hypocenterC83#1878
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
11. C2×#13050580C4semidirect
76. C2×C4×C83Q16semidirect
120. C83#1878semidirect
206. C2×C4#13050580semidirect

Maximal subgroups

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

subgroupindex
C22×#130505802
C2×#145817162
C2×#145817152
#191276072
#191274632
#191274562
C2×C4×Q8×C832
#187883