Finite Group Databasebeta

← #24920375

Subgroups — #24920375

Order 10624.

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

normal subgroups only

This group has 340 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
1Q8×Dic166C211yesmaximal, characteristic, complemented
2#19128411C211yesmaximal, characteristic
3#19128110C212yesmaximal, complemented
4#19128106C211yesmaximal, characteristic, complemented
5#19128009C212yesmaximal, complemented
6#19127997C211yesmaximal, characteristic, complemented
7#19127992C211yesmaximal, characteristic
8#19127982C212yesmaximal
9#19127803C212yesmaximal
10C4×#13050579C211yesmaximal, characteristic
11#446×C83C211yesmaximal, characteristic, complemented
12C2×#13050579C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581792C2212yes—
16#14581791—22no—
17#14581791—21no—
18#14581784C2212yes—
19#14581770—22no—
20#14581768—21no—
21Q8×Dic83—21no—
22C4×Dic166C2211yescharacteristic
23C4×Dic166C2212yes—
24C4×D332C2212yes—
25C4×D332C2211yescharacteristic
26C4⋊C4×D83—21no—
27#14581726C2212yes—
28#14581726C2212yes—
29#14581726C2212yes—
30#14581724C2211yescharacteristic
31#14581724C2212yes—
32#14581717C2212yes—
33#14581717C2211yescharacteristic
34#14581714C2211yescharacteristic
35#14581714C2212yes—
36C4×Q8×C83C2211yescharacteristic
37C4⋊C8×C83C2212yes—
38C4⋊C8×C83C2211yescharacteristic
39#97×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41C2×Dic166—21no—
42C2×Dic166—21no—
43C2×Dic166—22no—
44C2×Dic166C2311yescharacteristic
45C2×D332C2311yescharacteristic
46C2×C4×D83—21no—
47C2×C4×D83—22no—
48#13050579Q814yescomplemented
49C4⋊Dic83—21no—
50C4⋊Dic83—22no—
51C4⋊Dic83—21no—
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2312yes—
54C4⋊Dic83C2312yes—
55C4⋊Dic83C2311yescharacteristic
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4—21no—
59Dic83⋊C4—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—21no—
62C4×Dic83—22no—
63C4×Dic83—21no—
64C2×Q8×C83D411yescharacteristic
65C2×C8×C83C2311yescharacteristic
66C2×C8×C83C2312yes—
67C2×C8×C83C2311yescharacteristic
68C4⋊C4×C83D412yes—
69C4⋊C4×C83D411yescharacteristic
70C42×C83C2312yes—
71C42×C83C2311yescharacteristic
72C22×D83—21no—
73Q8×C83—22no—
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D412yes—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2411yescharacteristic
79C2×Dic83—21no—
80C2×Dic83—22no—
81C2×Dic83—21no—
82C2×Dic83—21no—
83C2×Dic83—22no—
84D332—21no—
85D332C2×Q812yes—
86C4×D83—42no—
87C4×D83—82no—
88Dic166—21no—
89Dic166—81no—
90Dic166—82no—
91Dic166—81no—
92Dic166C2×Q812yes—
93C8×C83—21no—
94C8×C83—22no—
95C8×C83C2×Q812yes—
96C22×C83C22×D411yescharacteristic
97C2×D83—22no—
98C2×D83—41no—
99C2×D83—41no—
100C4×C83—22no—
101C4×C83—21no—
102C4×C83—22no—
103C4×C83#13712yes—
104C4×C83C22×Q811yescharacteristic
105C4×C832−1+411yescharacteristic
106Dic83—22no—
107Dic83—41no—
108Dic83—42no—
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#477—831no—
123#476—831no—
124#474—832no—
125#446D8311yescharacteristic, complemented
126#440—832no—
127C4×SD16—831no—
128C2×SD16—831no—
129C4⋊Q8—831no—
130C4⋊Q8—832no—
131C4⋊Q8—831no—
132C4⋊Q8—1661no—
133C4⋊Q8—1662no—
134#125—832no—
135C22⋊Q8—1661no—
136C22⋊Q8—1662no—
137C4×Q8—831no—
138C4×Q8—832no—
139C4×Q8—1661no—
140C4×Q8C2×D8311yescharacteristic
141C4×D4—831no—
142C4×D4—832no—
143C2×C4⋊C4—1661no—
144#107—832no—
145#107—832no—
146#107—832no—
147#106—831no—
148#106—832no—
149C4⋊C8C2×D8312yes—
150C4⋊C8C2×D8311yescharacteristic
151#103—831no—
152#103—832no—
153#102—831no—
154#102—832no—
155#97C2×D8312yes—
156C4×C8C2×D8311yescharacteristic
157C2×Q8—831no—
158C2×Q8—1661no—
159C2×Q8—1661no—
160C2×Q8—1662no—
161C2×Q8D33211yescharacteristic
162C2×D4—831no—
163C22×C4—1661no—
164C22×C4—1662no—
165SD16—834no—
166C2×C8C22×D8311yescharacteristic
167C2×C8C22×D8312yes—
168C2×C8C22×D8311yescharacteristic
169C4⋊C4—831no—
170C4⋊C4—832no—
171C4⋊C4—831no—
172C4⋊C4—832no—
173C4⋊C4—1662no—
174C4⋊C4—1661no—
175C4⋊C4—1661no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4—1661no—
179C4⋊C4—1662no—
180C4⋊C4D33211yescharacteristic
181C4⋊C4D33212yes—
182C22⋊C4—1661no—
183C22⋊C4—1662no—
184C42—1662no—
185C42—1661no—
186C42C22×D8311yescharacteristic
187C42C22×D8312yes—
188C23—1661no—
189Q8—22no—
190Q8—832no—
191Q8—6642no—
192Q8—6641no—
193Q8—6641no—
194Q8—1661no—
195D4—832no—
196D4—1661no—
197C2×C4—3322no—
198C2×C4—6642no—
199C2×C4—1662no—
200C2×C4—1661no—
201C2×C4—1661no—
202C2×C4—1662no—
203C2×C4—1661no—
204C2×C4C2×D33212yes—
205C2×C4C2×D33211yescharacteristic
206C2×C4C2×D33212yes—
207C2×C4C2×D33211yescharacteristic
208C2×C4C23×D8311yescharacteristic
209C8—21no—
210C8—22no—
211C8Q8×D8312yes—
212C22—3321no—
213C22—3321no—
214C22—1662no—
215C22C22×D33211yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—3321no—
220C4—3322no—
221C4—3321no—
222C4—3322no—
223C4—1662no—
224C4#1458182512yes—
225C4C2×Q8×D8311yescharacteristic
226C4#1458183911yescharacteristic
227C2—3322no—
228C2Q8×D33211yescharacteristic
229C2C2×#1458182511yescharacteristic
230C2#1912850611yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#446×C83C2
socleC22×C83C22×D4
prime radicalC2×D332C23
hypercenter#446D83
hypocenterC83#3385
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. Q8×Dic166C2semidirect
3. #19128110C2semidirect
4. #19128106C2semidirect
5. #19128009C2semidirect
6. #19127997C2semidirect
11. #446×C83C2semidirect
48. #13050579Q8semidirect
116. C83#3385semidirect
125. #446D83semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic1662
#191284112
#191281102
#191281062
#191280092
#191279972
#191279922
#191279822
#191278032
C4×#130505792
#446×C832
#338583