Finite Group Databasebeta

← #24920185

Subgroups — #24920185

Order 10624.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128425C211yesmaximal, characteristic, complemented
2D4×Dic166C211yesmaximal, characteristic, complemented
3#19128062C212yesmaximal, complemented
4#19128051C211yesmaximal, characteristic, complemented
5#19127994C212yesmaximal, complemented
6#19127982C211yesmaximal, characteristic, complemented
7#19127918C212yesmaximal
8#19127828C211yesmaximal, characteristic, complemented
9#19127802C212yesmaximal, complemented
10C4×#13050575C211yesmaximal, characteristic
11#461×C83C211yesmaximal, characteristic, complemented
12C22×Dic166—21no—
13C2×#13050575C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581791C2212yes—
16#14581787—22no—
17#14581781C2212yes—
18#14581772—21no—
19#14581771—21no—
20#14581769—22no—
21Q8×Dic83C2212yes—
22C4×Dic166C2211yescharacteristic
23C4×D332C2211yescharacteristic
24D4×Dic83C2212yes—
25#14581750—21no—
26#14581726C2212yes—
27#14581725C2211yescharacteristic
28Dic83⋊C8C2212yes—
29#14581716C2212yes—
30#14581716C2212yes—
31#14581714C2212yes—
32#14581713C2211yescharacteristic
33#14581712C2211yescharacteristic
34#14581702C2212yes—
35C4×C83⋊C8C2211yescharacteristic
36C2×SD16×C83C2212yes—
37C4⋊Q8×C83C2211yescharacteristic
38C4×D4×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166—21no—
42C2×Dic166—22no—
43C2×Dic166—44no—
44C2×Dic166D411yescharacteristic
45C2×D332C2311yescharacteristic
46C22×Dic83—22no—
47C2×C4×D83—21no—
48#13050575—22no—
49C2×C83⋊C8C2311yescharacteristic
50C2×C83⋊C8C2311yescharacteristic
51C4⋊Dic83—21no—
52C4⋊Dic83—22no—
53C4⋊Dic83D411yescharacteristic
54C4⋊Dic83C2312yes—
55C4⋊Dic83C2311yescharacteristic
56Dic83⋊C4—22no—
57Dic83⋊C4D412yes—
58C22⋊Dic83—22no—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62C4×Dic83—22no—
63C4×Dic83C2312yes—
64C2×Q8×C83C2312yes—
65C2×D4×C83C2311yescharacteristic
66C22×C4×C83—21no—
67SD16×C83—24no—
68C2×C8×C83C2312yes—
69C4⋊C4×C83—21no—
70C4⋊C4×C83—21no—
71C4⋊C4×C83C2311yescharacteristic
72C22⋊C4×C83—21no—
73C42×C83C2311yescharacteristic
74C23×C83—21no—
75C22×D83—21no—
76Q8×C83—22no—
77Q8×C83—22no—
78D4×C83—21no—
79D4×C83#4112yes—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—42no—
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—21no—
87C2×Dic83—22no—
88C2×Dic83—44no—
89C2×Dic83C2×D412yes—
90C2×Dic83C2×D412yes—
91D332—21no—
92D332—21no—
93C4×D83—42no—
94Dic166—22no—
95Dic166—44no—
96Dic166—42no—
97C8×C83—22no—
98C83⋊C8—21no—
99C83⋊C8—21no—
100C22×C83—22no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83C22×D411yescharacteristic
104C2×D83—41no—
105C2×D83—41no—
106C2×D83—41no—
107C4×C83—21no—
108C4×C83—42no—
109C4×C83—41no—
110C4×C83AGL(1,ℤ/8)12yes—
111C4×C83C2×#4111yescharacteristic
112C4×C832−1+411yescharacteristic
113Dic83—22no—
114Dic83—22no—
115Dic83—42no—
116Dic83—41no—
117C2×C83—42no—
118C2×C83#54811yescharacteristic
119C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C83#57811yescharacteristic
121D83—81no—
122#3367—831nomaximal, Sylow
123C83#336711yescharacteristic, complemented, Sylow
124#556—831no—
125D4×Q8—831no—
126#489—832no—
127#487—831no—
128#477—832no—
129#474—831no—
130#463—832no—
131#461D8311yescharacteristic, complemented
132#446—831no—
133#440—832no—
134C4×D8—831no—
135C22×Q8—1661no—
136C2×SD16C2×D8312yes—
137C2×D8—831no—
138C4⋊Q8—832no—
139C4⋊Q8—831no—
140C4⋊Q8C2×D8311yescharacteristic
141#126—1662no—
142#124—832no—
143C22⋊Q8—1662no—
144C22⋊Q8—1661no—
145C22⋊Q8—1661no—
146C4×Q8—831no—
147C4×Q8—832no—
148C4×D4—832no—
149C4×D4—831no—
150C4×D4C2×D8311yescharacteristic
151#117—1661no—
152#107—832no—
153#107—831no—
154C4⋊C8—832no—
155C4⋊C8C2×D8311yescharacteristic
156#103—832no—
157#103—832no—
158#103C2×D8312yes—
159#102—831no—
160#102—832no—
161#102—831no—
162#97—832no—
163C4×C8—831no—
164C2×Q8—3324no—
165C2×Q8—831no—
166C2×Q8—1662no—
167C2×Q8—1661no—
168C2×Q8C22×D8312yes—
169C2×D4—831no—
170C2×D4C22×D8311yescharacteristic
171C22×C4—21no—
172C22×C4—1662no—
173C22×C4—1661no—
174SD16—24no—
175D8—1662no—
176C2×C8—831no—
177C2×C8—831no—
178C2×C8C22×D8312yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—831no—
182C4⋊C4—832no—
183C4⋊C4—832no—
184C4⋊C4—831no—
185C4⋊C4—1662no—
186C4⋊C4—1662no—
187C4⋊C4—1661no—
188C4⋊C4C22×D8311yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—1662no—
191C22⋊C4—1662no—
192C22⋊C4—1662no—
193C22⋊C4—1661no—
194C42—832no—
195C42—1662no—
196C42C22×D8311yescharacteristic
197C23—21no—
198C23—1661no—
199Q8—22no—
200Q8—22no—
201Q8—3324no—
202Q8—3322no—
203Q8—1662no—
204D4—21no—
205D4—1661no—
206D4—1661no—
207D4#1305058912yes—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—3324no—
212C2×C4—3322no—
213C2×C4—832no—
214C2×C4—832no—
215C2×C4—1661no—
216C2×C4—1662no—
217C2×C4D4×D8311yescharacteristic
218C2×C4D4×D8311yescharacteristic
219C2×C4C23×D8311yescharacteristic
220C8—22no—
221C8—1661no—
222C8—1661no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—3321no—
227C22—3321no—
228C22—3321no—
229C22C2×D4×D8311yescharacteristic
230C4—21no—
231C4—42no—
232C4—41no—
233C4—3322no—
234C4—3321no—
235C4—1662no—
236C4—1662no—
237C4#1458181912yes—
238C4C2×#1305058911yescharacteristic
239C4#1458183911yescharacteristic
240C2—42no—
241C2—6641no—
242C2#1912835711yescharacteristic
243C2C2×#1458181911yescharacteristic
244C2#1912851711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#461×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050575C22
hypercenter#461D83
hypocenterC83#3367
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #19128425C2semidirect
2. D4×Dic166C2semidirect
3. #19128062C2semidirect
4. #19128051C2semidirect
5. #19127994C2semidirect
6. #19127982C2semidirect
8. #19127828C2semidirect
9. #19127802C2semidirect
11. #461×C83C2semidirect
123. C83#3367semidirect
131. #461D83semidirect

Maximal subgroups

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

subgroupindex
#191284252
D4×Dic1662
#191280622
#191280512
#191279942
#191279822
#191279182
#191278282
#191278022
C4×#130505752
#461×C832
#336783