Finite Group Databasebeta

← #24918694

Subgroups — #24918694

Order 10624.

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

normal subgroups only

This group has 362 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
1C4⋊Q8×D83C211yesmaximal, characteristic, complemented
2#19128236C211yesmaximal, characteristic
3#19128111C211yesmaximal, characteristic, complemented
4#19128063C211yesmaximal, characteristic, complemented
5#19128061C211yesmaximal, characteristic, complemented
6#19128038C212yesmaximal, complemented
7#19128021C212yesmaximal
8#19127985C212yesmaximal, complemented
9#19127980C212yesmaximal, complemented
10#19127756C211yesmaximal, characteristic, complemented
11#501×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83—21no—
13#14581792C2211yescharacteristic
14#14581791—21no—
15#14581791C2212yes—
16#14581790—21no—
17#14581787—21no—
18#14581780C2211yescharacteristic
19#14581777—21no—
20#14581772C2211yescharacteristic
21#14581767C2211yescharacteristic
22#14581757C2211yescharacteristic
23#14581754—21no—
24#14581754C2211yescharacteristic
25#14581750—21no—
26C4⋊C4×D83—21no—
27C4⋊C4×D83C2212yes—
28C42×D83C2211yescharacteristic
29#14581725C2212yes—
30#14581723C2212yes—
31Dic83⋊C8C2212yes—
32#14581717C2212yes—
33#14581715C2212yes—
34#14581714C2212yes—
35#14581712C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581703C2211yescharacteristic
38C4⋊Q8×C83C2211yescharacteristic
39#125×C83C2211yescharacteristic
40#107×C83C2212yes—
41#106×C83C2212yes—
42#97×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166—22no—
45C2×Dic166C2311yescharacteristic
46Q8×D83—44no—
47C2×D332—21no—
48C2×D332C2311yescharacteristic
49C2×C4×D83—22no—
50C2×C4×D83—21no—
51C2×C4×D83D411yescharacteristic
52C2×C4×D83D411yescharacteristic
53C2×C4×D83C2311yescharacteristic
54C2×C83⋊C8C2312yes—
55C4⋊Dic83—21no—
56C4⋊Dic83—21no—
57C4⋊Dic83C2312yes—
58Dic83⋊C4—21no—
59Dic83⋊C4—22no—
60Dic83⋊C4—22no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66C4×Dic83—21no—
67C4×Dic83D411yescharacteristic
68C4×Dic83D411yescharacteristic
69C4×Dic83C2311yescharacteristic
70C2×Q8×C83—21no—
71C2×C8×C83C2312yes—
72C4⋊C4×C83—21no—
73C4⋊C4×C83—21no—
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2311yescharacteristic
76C4⋊C4×C83C2311yescharacteristic
77C4⋊C4×C83C2312yes—
78C42×C83C2311yescharacteristic
79C22×D83—21no—
80C22×D83C2×D411yescharacteristic
81Q8×C83—42no—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2411yescharacteristic
88C2×Dic83—21no—
89C2×Dic83—22no—
90C2×Dic83C2×D411yescharacteristic
91C2×Dic83C2×D411yescharacteristic
92C2×Dic83C2×D411yescharacteristic
93D332—22no—
94D332—42no—
95C4×D83—22no—
96C4×D83—22no—
97C4×D83—22no—
98C4×D83—44no—
99C4×D83—42no—
100Dic166—22no—
101Dic166—42no—
102Dic166—44no—
103C8×C83—22no—
104C83⋊C8—22no—
105C22×C83C22×D411yescharacteristic
106C2×D83—21no—
107C2×D83—21no—
108C2×D83—41no—
109C2×D83—41no—
110C2×D83—41no—
111C2×D83#13712yes—
112C4×C83—21no—
113C4×C83—21no—
114C4×C83—42no—
115C4×C83—41no—
116C4×C83—41no—
117C4×C832−1+412yes—
118Dic83—21no—
119Dic83—21no—
120Dic83—41no—
121Dic83—42no—
122Dic83AGL(1,ℤ/8)12yes—
123C2×C83#53611yescharacteristic
124C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C83C2×#13711yescharacteristic
126D83—22no—
127D83—81no—
128#3239—831nomaximal, Sylow
129C83#323911yescharacteristic, complemented, Sylow
130C2×C4⋊Q8—831no—
131#529—831no—
132#501—831no—
133#501D8311yescharacteristic, complemented
134#489—831no—
135#489—831no—
136#484—832no—
137#481—832no—
138#475—832no—
139#474—832no—
140#432—831no—
141C22×Q8—1661no—
142C4⋊Q8—832no—
143C4⋊Q8—831no—
144C4⋊Q8—1661no—
145C4⋊Q8—1661no—
146C4⋊Q8C2×D8311yescharacteristic
147#126—1661no—
148#125C2×D8311yescharacteristic
149#124—831no—
150#123—1661no—
151C22⋊Q8—831no—
152#121—831no—
153C4×Q8—831no—
154C4×D4—831no—
155C4×D4—1661no—
156#117—1661no—
157C2×C4⋊C4—832no—
158C2×C4⋊C4—1661no—
159C2×C42—831no—
160#107—832no—
161#107C2×D8312yes—
162#106—832no—
163#106C2×D8312yes—
164C4⋊C8—832no—
165#103—832no—
166#103—832no—
167#102—832no—
168#102—832no—
169C22⋊C8—832no—
170#97—831no—
171#97C2×D8311yescharacteristic
172C2×Q8—21no—
173C2×Q8—3324no—
174C2×Q8—831no—
175C2×Q8—1662no—
176C2×Q8—1661no—
177C2×D4—831no—
178C2×D4—1661no—
179C22×C4—831no—
180C22×C4—831no—
181C22×C4—831no—
182C22×C4—1661no—
183C22×C4—1662no—
184C2×C8—832no—
185C2×C8C22×D8312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—832no—
190C4⋊C4—1662no—
191C4⋊C4—1662no—
192C4⋊C4—1661no—
193C4⋊C4—1661no—
194C4⋊C4—1661no—
195C4⋊C4C22×D8312yes—
196C4⋊C4C22×D8311yescharacteristic
197C4⋊C4C22×D8311yescharacteristic
198C22⋊C4—1661no—
199C22⋊C4—1661no—
200C22⋊C4—1661no—
201C22⋊C4—1661no—
202C22⋊C4—1661no—
203C42—831no—
204C42—831no—
205C42—831no—
206C42—1661no—
207C42C22×D8311yescharacteristic
208C23—831no—
209C23—1661no—
210Q8—42no—
211Q8—3322no—
212Q8—3324no—
213Q8—1662no—
214D4—3322no—
215D4—1662no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—3324no—
220C2×C4—3322no—
221C2×C4—831no—
222C2×C4—831no—
223C2×C4—831no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1661no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4D4×D8311yescharacteristic
230C2×C4D4×D8311yescharacteristic
231C2×C4C23×D8311yescharacteristic
232C8—22no—
233C8—1662no—
234C22—3321no—
235C22—3321no—
236C22—3321no—
237C22—832no—
238C22—1661no—
239C22—1661no—
240C22C2×D4×D8311yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—3322no—
247C4—3321no—
248C4—832no—
249C4—1661no—
250C4—1661no—
251C4#1458184012yes—
252C2—6641no—
253C2—1662no—
254C2#1912827611yescharacteristic
255C2AGL(1,ℤ/8)×D8311yescharacteristic
256C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#501×C83C2
socleC22×C83C22×D4
prime radical#14581767C22
hypercenter#501D83
hypocenterC83#3239
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D83C2semidirect
3. #19128111C2semidirect
4. #19128063C2semidirect
5. #19128061C2semidirect
6. #19128038C2semidirect
8. #19127985C2semidirect
9. #19127980C2semidirect
10. #19127756C2semidirect
11. #501×C83C2semidirect
129. C83#3239semidirect
133. #501D83semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D832
#191282362
#191281112
#191280632
#191280612
#191280382
#191280212
#191279852
#191279802
#191277562
#501×C832
#323983