Finite Group Databasebeta

← #24920373

Subgroups — #24920373

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#19128111C212yesmaximal, complemented
3#19128104C211yesmaximal, characteristic, complemented
4#19128005C212yesmaximal, complemented
5#19127993C211yesmaximal, characteristic, complemented
6#19127991C211yesmaximal, characteristic
7#19127981C212yesmaximal
8#19127832C211yesmaximal, characteristic, complemented
9#19127801C212yesmaximal
10C4×#13050578C211yesmaximal, characteristic
11#554×C83C211yesmaximal, characteristic
12C2×#13050578C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581792C2212yes—
15#14581791—21no—
16#14581791—22no—
17Q8×Dic83—21no—
18C4×Dic166C2211yescharacteristic
19C4×Dic166C2212yes—
20#14581725C2212yes—
21#14581725C2212yes—
22#14581725C2212yes—
23#14581723C2211yescharacteristic
24#14581723C2212yes—
25C4⋊(C83⋊C8)C2212yes—
26C4⋊(C83⋊C8)C2211yescharacteristic
27#14581715C2212yes—
28#14581715C2211yescharacteristic
29#14581713C2211yescharacteristic
30#14581713C2212yes—
31#14581703C2212yes—
32C4×C83⋊C8C2211yescharacteristic
33C4⋊Q8×C83C2211yescharacteristic
34#125×C83C2212yes—
35C22⋊Q8×C83—21no—
36C22⋊Q8×C83—22no—
37C4×Q8×C83C2211yescharacteristic
38C4×D4×C83C2212yes—
39C4×D4×C83C2211yescharacteristic
40C2×C4⋊C4×C83—21no—
41C2×Dic166—21no—
42C2×Dic166—22no—
43C2×Dic166C2311yescharacteristic
44#13050578Q814yescomplemented
45C2×C83⋊C8C2311yescharacteristic
46C2×C83⋊C8C2312yes—
47C2×C83⋊C8C2311yescharacteristic
48C4⋊Dic83—22no—
49C4⋊Dic83—21no—
50C4⋊Dic83C2312yes—
51C4⋊Dic83C2311yescharacteristic
52Dic83⋊C4—21no—
53Dic83⋊C4—22no—
54C4×Dic83—22no—
55C4×Dic83—21no—
56C2×Q8×C83—21no—
57C2×Q8×C83D411yescharacteristic
58C2×D4×C83C2311yescharacteristic
59C22×C4×C83—21no—
60C22×C4×C83—22no—
61C4⋊C4×C83—22no—
62C4⋊C4×C83—21no—
63C4⋊C4×C83—22no—
64C4⋊C4×C83D411yescharacteristic
65C4⋊C4×C83D412yes—
66C4⋊C4×C83C2311yescharacteristic
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—21no—
69C22⋊C4×C83—22no—
70C42×C83C2312yes—
71C42×C83C2311yescharacteristic
72C23×C83—21no—
73Q8×C83—22no—
74Q8×C83—81no—
75D4×C83—21no—
76D4×C83C2×Q812yes—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—42no—
80C2×C4×C83—82no—
81C2×C4×C83C2×D412yes—
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—21no—
87C2×Dic83—22no—
88C2×Dic83—21no—
89Dic166—21no—
90Dic166—82no—
91Dic166—81no—
92Dic166C2×Q812yes—
93C83⋊C8—21no—
94C83⋊C8—22no—
95C83⋊C8C2×Q812yes—
96C22×C83—22no—
97C22×C83—41no—
98C22×C83—41no—
99C22×C83C22×D411yescharacteristic
100C4×C83—22no—
101C4×C83—22no—
102C4×C83—21no—
103C4×C83—41no—
104C4×C83—42no—
105C4×C83#13712yes—
106C4×C83C22×Q811yescharacteristic
107C4×C832−1+411yescharacteristic
108Dic83—22no—
109Dic83—41no—
110Dic83—42no—
111C2×C83—42no—
112C2×C83D4×Q811yescharacteristic
113C2×C83C2×#13711yescharacteristic
114C2×C83#57611yescharacteristic
115#3385—831nomaximal, Sylow
116C83#338511yescharacteristic, complemented, Sylow
117Q82—831no—
118#554D8311yescharacteristic
119#501—832no—
120#500—831no—
121#479—832no—
122#477—831no—
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#125C2×D8312yes—
135C22⋊Q8—21no—
136C22⋊Q8—22no—
137C4×Q8—831no—
138C4×Q8—832no—
139C4×Q8—1661no—
140C4×Q8C2×D8311yescharacteristic
141C4×D4C2×D8311yescharacteristic
142C4×D4C2×D8312yes—
143C2×C4⋊C4—21no—
144#107—832no—
145#107—832no—
146#107—832no—
147#106—831no—
148#106—832no—
149C4⋊C8—832no—
150C4⋊C8—831no—
151#103—831no—
152#103—832no—
153#102—831no—
154#102—832no—
155#97—832no—
156C4×C8—831no—
157C2×Q8—21no—
158C2×Q8—831no—
159C2×Q8—1661no—
160C2×Q8—1662no—
161C2×Q8C22⋊D8311yescharacteristic
162C2×D4C22×D8311yescharacteristic
163C22×C4—21no—
164C22×C4—22no—
165SD16—834no—
166C2×C8—831no—
167C2×C8—831no—
168C2×C8—832no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—831no—
173C4⋊C4—832no—
174C4⋊C4—1661no—
175C4⋊C4—1662no—
176C4⋊C4—1662no—
177C4⋊C4—1661no—
178C4⋊C4C22⋊D8311yescharacteristic
179C4⋊C4C22⋊D8312yes—
180C4⋊C4C22×D8311yescharacteristic
181C4⋊C4C22×D8312yes—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C42—1662no—
185C42—1661no—
186C42C22×D8311yescharacteristic
187C42C22×D8312yes—
188C23—21no—
189Q8—22no—
190Q8—81no—
191Q8—832no—
192Q8—6642no—
193Q8—6641no—
194Q8—1661no—
195D4—21no—
196D4Q8×D8312yes—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—1662no—
202C2×C4—1661no—
203C2×C4—1661no—
204C2×C4C2×C22⋊D8312yes—
205C2×C4C2×C22⋊D8311yescharacteristic
206C2×C4C2×C22⋊D8312yes—
207C2×C4C2×C22⋊D8311yescharacteristic
208C2×C4C23×D8311yescharacteristic
209C8—832no—
210C8—1661no—
211C8—1662no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22C22×C22⋊D8311yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—42no—
221C4—3321no—
222C4—3322no—
223C4—1662no—
224C4#1458182212yes—
225C4C2×Q8×D8311yescharacteristic
226C4#1458184011yescharacteristic
227C2—42no—
228C2Q8×C22⋊D8311yescharacteristic
229C2C2×#1458182211yescharacteristic
230C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#554×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×C83C23
hypercenter#554D83
hypocenterC83#3385
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. Q8×Dic166C2semidirect
2. #19128111C2semidirect
3. #19128104C2semidirect
4. #19128005C2semidirect
5. #19127993C2semidirect
8. #19127832C2semidirect
44. #13050578Q8semidirect
116. C83#3385semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic1662
#191281112
#191281042
#191280052
#191279932
#191279912
#191279812
#191278322
#191278012
C4×#130505782
#554×C832
#338583