Finite Group Databasebeta

← #24918697

Subgroups — #24918697

Order 10624.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128238C211yesmaximal, characteristic
3#19128108C212yesmaximal, complemented
4#19128062C212yesmaximal, complemented
5#19128035C212yesmaximal, complemented
6#19128021C212yesmaximal
7#19127986C211yesmaximal, characteristic, complemented
8#19127982C211yesmaximal, characteristic, complemented
9#19127978C211yesmaximal, characteristic, complemented
10#19127757C211yesmaximal, characteristic, complemented
11#475×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83—21no—
13#14581792C2211yescharacteristic
14#14581791—21no—
15#14581791C2212yes—
16#14581790—21no—
17#14581787—21no—
18#14581783C2211yescharacteristic
19#14581781C2211yescharacteristic
20#14581777—21no—
21#14581768C2211yescharacteristic
22#14581767C2211yescharacteristic
23C4×D332—21no—
24C4×D332C2211yescharacteristic
25#14581749—21no—
26C4⋊C4×D83—21no—
27C4⋊C4×D83C2212yes—
28C42×D83C2211yescharacteristic
29#14581726C2212yes—
30#14581725C2212yes—
31#14581723C2212yes—
32C4⋊(C83⋊C8)C2211yescharacteristic
33#14581716C2212yes—
34#14581714C2212yes—
35#14581712C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581702C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39C4×Q8×C83C2211yescharacteristic
40#106×C83C2212yes—
41C4⋊C8×C83C2211yescharacteristic
42#103×C83C2212yes—
43C2×Dic166—21no—
44C2×Dic166—22no—
45Q8×D83—44no—
46C2×D332—21no—
47C2×D332C2311yescharacteristic
48C2×C4×D83—22no—
49C2×C4×D83—21no—
50C2×C4×D83D411yescharacteristic
51C2×C4×D83D411yescharacteristic
52C2×C4×D83C2311yescharacteristic
53C2×C83⋊C8C2312yes—
54C4⋊Dic83—21no—
55C4⋊Dic83—21no—
56C4⋊Dic83C2311yescharacteristic
57C4⋊Dic83C2312yes—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60Dic83⋊C4—22no—
61Dic83⋊C4—22no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67C4×Dic83D411yescharacteristic
68C4×Dic83D411yescharacteristic
69C4×Dic83C2311yescharacteristic
70C2×Q8×C83—21no—
71C2×Q8×C83C2311yescharacteristic
72C2×C8×C83C2312yes—
73C4⋊C4×C83—21no—
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2311yescharacteristic
76C4⋊C4×C83C2312yes—
77C42×C83—21no—
78C42×C83C2311yescharacteristic
79C22×D83—21no—
80C22×D83C2×D411yescharacteristic
81Q8×C83—21no—
82Q8×C83—21no—
83Q8×C83—42no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2411yescharacteristic
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83C2×D411yescharacteristic
93C2×Dic83C2×D411yescharacteristic
94C2×Dic83C2×D411yescharacteristic
95D332—21no—
96D332—21no—
97D332—42no—
98C4×D83—22no—
99C4×D83—22no—
100C4×D83—21no—
101C4×D83—21no—
102C4×D83—44no—
103C4×D83—42no—
104Dic166—44no—
105Dic166—42no—
106C8×C83—22no—
107C83⋊C8—22no—
108C22×C83C22×D411yescharacteristic
109C2×D83—21no—
110C2×D83—21no—
111C2×D83—41no—
112C2×D83—41no—
113C2×D83—41no—
114C2×D83#13712yes—
115C4×C83—21no—
116C4×C83—41no—
117C4×C83—42no—
118C4×C83—41no—
119C4×C83AGL(1,ℤ/8)12yes—
120C4×C832−1+411yescharacteristic
121C4×C832−1+411yescharacteristic
122Dic83—21no—
123Dic83—21no—
124Dic83—21no—
125Dic83—42no—
126Dic83—41no—
127C2×C83#53611yescharacteristic
128C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C83C2×#13711yescharacteristic
130D83—22no—
131D83—81no—
132#3239—831nomaximal, Sylow
133C83#323911yescharacteristic, complemented, Sylow
134C2×C4⋊Q8—831no—
135#529—831no—
136#501—832no—
137#489—832no—
138#484—832no—
139#481—832no—
140#475—831no—
141#475D8311yescharacteristic, complemented
142#474—831no—
143#474—831no—
144#432—831no—
145C22×Q8—1661no—
146C4⋊Q8—832no—
147C4⋊Q8—831no—
148C4⋊Q8—1661no—
149C4⋊Q8—1661no—
150C4⋊Q8C2×D8311yescharacteristic
151#126—1661no—
152#125—831no—
153#124—831no—
154#123—1661no—
155C22⋊Q8—831no—
156#121—831no—
157C4×Q8C2×D8311yescharacteristic
158C4×D4—831no—
159C4×D4—1661no—
160#117—1661no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—1661no—
163C2×C42—831no—
164#107—832no—
165#107—832no—
166#106—832no—
167#106C2×D8312yes—
168C4⋊C8—831no—
169C4⋊C8C2×D8311yescharacteristic
170#103—832no—
171#103C2×D8312yes—
172#102—832no—
173#102—832no—
174C22⋊C8—832no—
175#97—832no—
176C2×Q8—21no—
177C2×Q8—3324no—
178C2×Q8—1662no—
179C2×Q8—1661no—
180C2×Q8C22×D8311yescharacteristic
181C2×D4—831no—
182C2×D4—1661no—
183C22×C4—831no—
184C22×C4—831no—
185C22×C4—831no—
186C22×C4—1662no—
187C22×C4—1661no—
188C2×C8—832no—
189C2×C8C22×D8312yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—832no—
193C4⋊C4—831no—
194C4⋊C4—1662no—
195C4⋊C4—1662no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1661no—
199C4⋊C4—1661no—
200C4⋊C4C22×D8312yes—
201C4⋊C4C22×D8311yescharacteristic
202C22⋊C4—1661no—
203C22⋊C4—1661no—
204C22⋊C4—1661no—
205C22⋊C4—1661no—
206C22⋊C4—1661no—
207C42—21no—
208C42—831no—
209C42—831no—
210C42—831no—
211C42C22×D8311yescharacteristic
212C23—831no—
213C23—1661no—
214Q8—21no—
215Q8—21no—
216Q8—42no—
217Q8—3324no—
218Q8—3322no—
219D4—3322no—
220D4—1661no—
221D4—1661no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—3324no—
226C2×C4—3322no—
227C2×C4—831no—
228C2×C4—831no—
229C2×C4—831no—
230C2×C4—1662no—
231C2×C4—1661no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1661no—
235C2×C4—1661no—
236C2×C4D4×D8311yescharacteristic
237C2×C4D4×D8311yescharacteristic
238C2×C4C23×D8311yescharacteristic
239C8—22no—
240C8—1662no—
241C22—3321no—
242C22—3321no—
243C22—3321no—
244C22—832no—
245C22—1661no—
246C22—1661no—
247C22C2×D4×D8311yescharacteristic
248C4—21no—
249C4—42no—
250C4—41no—
251C4—41no—
252C4—3322no—
253C4—3321no—
254C4—1661no—
255C4—1661no—
256C4—1661no—
257C4#1458181912yes—
258C4#1458183911yescharacteristic
259C4#1458184011yescharacteristic
260C2—6641no—
261C2—1662no—
262C2#1912827411yescharacteristic
263C2C2×#1458181911yescharacteristic
264C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#475×C83C2
socleC22×C83C22×D4
prime radical#14581767C22
hypercenter#475D83
hypocenterC83#3239
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D83C2semidirect
3. #19128108C2semidirect
4. #19128062C2semidirect
5. #19128035C2semidirect
7. #19127986C2semidirect
8. #19127982C2semidirect
9. #19127978C2semidirect
10. #19127757C2semidirect
11. #475×C83C2semidirect
133. C83#3239semidirect
141. #475D83semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D832
#191282382
#191281082
#191280622
#191280352
#191280212
#191279862
#191279822
#191279782
#191277572
#475×C832
#323983