Finite Group Databasebeta

← #24920372

Subgroups — #24920372

Order 10624.

340 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic166C211yesmaximal, characteristic, complemented
2#19128412C211yesmaximal, characteristic
3#19128109C212yesmaximal, complemented
4#19128104C211yesmaximal, characteristic, complemented
5#19128006C212yesmaximal, complemented
6#19127992C211yesmaximal, characteristic
7#19127980C212yesmaximal
8#19127828C211yesmaximal, characteristic, complemented
9#19127804C212yesmaximal
10C4×#13050577C211yesmaximal, characteristic
11#477×C83C211yesmaximal, characteristic, complemented
12C2×#13050577C2211yescharacteristic
13#14581792—21no—
14#14581792C2211yescharacteristic
15#14581791—22no—
16#14581791C2212yes—
17#14581783C2212yes—
18#14581772—21no—
19#14581768—22no—
20Q8×Dic83C2212yes—
21C4×Dic166—21no—
22C4×Dic166C2211yescharacteristic
23C4×D332C2211yescharacteristic
24#14581754C2212yes—
25C4⋊C4×D83—21no—
26#14581725C2212yes—
27#14581725C2212yes—
28#14581724C2212yes—
29#14581723C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581716C2211yescharacteristic
32#14581714C2212yes—
33#14581712C2211yescharacteristic
34#14581702C2212yes—
35C4×C83⋊C8C2211yescharacteristic
36C4⋊Q8×C83C2211yescharacteristic
37C4×Q8×C83C2211yescharacteristic
38#107×C83C2212yes—
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166—21no—
42C2×Dic166—22no—
43C2×Dic166D411yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C4×D83—21no—
46C2×C4×D83—22no—
47#13050577Q814yescomplemented
48C2×C83⋊C8C2311yescharacteristic
49C2×C83⋊C8C2311yescharacteristic
50C4⋊Dic83—21no—
51C4⋊Dic83—22no—
52C4⋊Dic83—22no—
53C4⋊Dic83D411yescharacteristic
54C4⋊Dic83C2311yescharacteristic
55C4⋊Dic83C2312yes—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—22no—
61C4×Dic83—22no—
62C4×Dic83C2312yes—
63C2×Q8×C83—21no—
64C2×Q8×C83C2311yescharacteristic
65C2×C8×C83C2312yes—
66C4⋊C4×C83—21no—
67C4⋊C4×C83—21no—
68C4⋊C4×C83C2312yes—
69C4⋊C4×C83C2311yescharacteristic
70C42×C83—21no—
71C42×C83C2311yescharacteristic
72C22×D83—21no—
73Q8×C83—21no—
74Q8×C83—81no—
75Q8×C83C2×Q812yes—
76C2×C4×C83—21no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2411yescharacteristic
82C2×Dic83—22no—
83C2×Dic83—21no—
84C2×Dic83C2×D412yes—
85C2×Dic83C2×D412yes—
86D332—21no—
87D332C2×Q812yes—
88C4×D83—42no—
89C4×D83—82no—
90Dic166—22no—
91Dic166—82no—
92Dic166—81no—
93C8×C83—22no—
94C83⋊C8—21no—
95C83⋊C8C2×Q812yes—
96C22×C83C22×D411yescharacteristic
97C2×D83—22no—
98C2×D83—41no—
99C2×D83—41no—
100C4×C83—22no—
101C4×C83—21no—
102C4×C83—41no—
103C4×C83—42no—
104C4×C83#13712yes—
105C4×C83C22×Q811yescharacteristic
106C4×C832−1+411yescharacteristic
107Dic83—22no—
108Dic83—22no—
109Dic83—41no—
110Dic83—42no—
111C2×C83D4×Q811yescharacteristic
112C2×C83C2×#13711yescharacteristic
113C2×C83#57611yescharacteristic
114D83—42no—
115#3385—831nomaximal, Sylow
116C83#338511yescharacteristic, complemented, Sylow
117Q82—831no—
118#554—831no—
119#501—832no—
120#500—831no—
121#479—832no—
122#477D8311yescharacteristic, complemented
123#476—831no—
124#474—832no—
125#446—831no—
126#440—832no—
127C4×SD16—831no—
128C2×SD16—831no—
129C4⋊Q8—832no—
130C4⋊Q8—831no—
131C4⋊Q8—1661no—
132C4⋊Q8—1662no—
133C4⋊Q8C2×D8311yescharacteristic
134#125—832no—
135C22⋊Q8—1661no—
136C22⋊Q8—1662no—
137C4×Q8—831no—
138C4×Q8—832no—
139C4×Q8—1661no—
140C4×Q8C2×D8311yescharacteristic
141C4×D4—832no—
142C4×D4—831no—
143C2×C4⋊C4—1661no—
144#107—832no—
145#107—832no—
146#107C2×D8312yes—
147#106—832no—
148#106—831no—
149C4⋊C8—832no—
150C4⋊C8C2×D8311yescharacteristic
151#103—831no—
152#103C2×D8312yes—
153#102—832no—
154#102—831no—
155#97—832no—
156C4×C8—831no—
157C2×Q8—21no—
158C2×Q8—831no—
159C2×Q8—1662no—
160C2×Q8—1661no—
161C2×Q8C22×D8311yescharacteristic
162C2×D4—831no—
163C22×C4—1662no—
164C22×C4—1661no—
165SD16—834no—
166C2×C8—831no—
167C2×C8—831no—
168C2×C8C22×D8312yes—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—831no—
172C4⋊C4—832no—
173C4⋊C4—832no—
174C4⋊C4—831no—
175C4⋊C4—1662no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4—1662no—
179C4⋊C4—1661no—
180C4⋊C4C22×D8311yescharacteristic
181C4⋊C4C22×D8312yes—
182C22⋊C4—1662no—
183C22⋊C4—1661no—
184C42—21no—
185C42—832no—
186C42—1662no—
187C42C22×D8311yescharacteristic
188C23—1661no—
189Q8—21no—
190Q8—81no—
191Q8—6641no—
192Q8—6642no—
193Q8—1662no—
194Q8Q8×D8312yes—
195D4—832no—
196D4—1661no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—3322no—
201C2×C4—832no—
202C2×C4—832no—
203C2×C4—6642no—
204C2×C4—1662no—
205C2×C4—1661no—
206C2×C4D4×D8311yescharacteristic
207C2×C4D4×D8311yescharacteristic
208C2×C4C23×D8311yescharacteristic
209C8—22no—
210C8—832no—
211C8—1661no—
212C22—3321no—
213C22—3321no—
214C22—1662no—
215C22C2×D4×D8311yescharacteristic
216C4—22no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—3321no—
221C4—3322no—
222C4—1662no—
223C4—1662no—
224C4#1458182412yes—
225C4C2×Q8×D8311yescharacteristic
226C4#1458183911yescharacteristic
227C2—3322no—
228C2#1912836411yescharacteristic
229C2C2×#1458182411yescharacteristic
230C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#477×C83C2
socleC22×C83C22×D4
prime radicalC2×D332C23
hypercenter#477D83
hypocenterC83#3385
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. Q8×Dic166C2semidirect
3. #19128109C2semidirect
4. #19128104C2semidirect
5. #19128006C2semidirect
8. #19127828C2semidirect
11. #477×C83C2semidirect
47. #13050577Q8semidirect
116. C83#3385semidirect
122. #477D83semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic1662
#191284122
#191281092
#191281042
#191280062
#191279922
#191279802
#191278282
#191278042
C4×#130505772
#477×C832
#338583