Finite Group Databasebeta

← #24919196

Subgroups — #24919196

Order 10624.

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

normal subgroups only

This group has 494 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
1C2×#14581811C212yesmaximal, complemented
2#19128371C211yesmaximal, characteristic, complemented
3#19128355C211yesmaximal, characteristic, complemented
4#19128073C211yesmaximal, characteristic, complemented
5#19127929C212yesmaximal, complemented
6#19127912C211yesmaximal, characteristic, complemented
7#19127864C212yesmaximal, complemented
8#19127843C212yesmaximal, complemented
9C4×#13050578C211yesmaximal, characteristic, complemented
10#19127766C211yesmaximal, characteristic, complemented
11#459×C83C211yesmaximal, characteristic, complemented
12C2×#13050589C2212yes—
13C2×#13050589C2212yes—
14#14581811—28no—
15C2×Dic332C2212yes—
16C2×#13050578C2212yes—
17C2×#13050578C2212yes—
18C2×#13050578C2211yescharacteristic
19C2×C8×D83C2212yes—
20#14581792C2211yescharacteristic
21#14581788—21no—
22#14581776—22no—
23#14581772C2212yes—
24#14581766—21no—
25#14581764—22no—
26#14581763C2212yes—
27C4×Dic166C2211yescharacteristic
28C4×D332C2211yescharacteristic
29D4×Dic83—21no—
30C2×C4⋊Dic83—21no—
31#14581723C2211yescharacteristic
32#14581717C2212yes—
33#14581715C2211yescharacteristic
34#14581713C2211yescharacteristic
35(C2×D83)⋊C8C2212yes—
36C4×C83⋊C8C2211yescharacteristic
37C2×D8×C83C2212yes—
38#127×C83C2211yescharacteristic
39C4×D4×C83C2211yescharacteristic
40C4⋊C8×C83C2211yescharacteristic
41#102×C83C2212yes—
42#13050589—24no—
43#13050589—24no—
44#13050589—42no—
45#13050589—42no—
46C2×Dic166C2312yes—
47C2×Dic166C2311yescharacteristic
48C2×D332D411yescharacteristic
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C2×C22⋊D83—22no—
52C22×Dic83—22no—
53C22×Dic83—22no—
54C2×C4×D83C2312yes—
55Dic332—24no—
56#13050578—24no—
57#13050578—42no—
58#13050578D414yescomplemented
59C2×C83⋊C8C2311yescharacteristic
60C2×C83⋊C8C2311yescharacteristic
61C8×D83—24no—
62C4⋊Dic83—21no—
63C4⋊Dic83—21no—
64C4⋊Dic83—22no—
65C4⋊Dic83D411yescharacteristic
66C4⋊Dic83C2311yescharacteristic
67Dic83⋊C4—21no—
68C22⋊Dic83—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4D412yes—
71C4×Dic83—21no—
72C2×D4×C83—21no—
73C2×D4×C83—21no—
74C2×D4×C83C2312yes—
75C2×D4×C83C2311yescharacteristic
76C22×C4×C83—21no—
77D8×C83—24no—
78C2×C8×C83C2312yes—
79C4⋊C4×C83C2311yescharacteristic
80C22⋊C4×C83—21no—
81C42×C83C2311yescharacteristic
82C23×C83—21no—
83C23×C83—22no—
84C22×D83C2×D412yes—
85D4×C83—22no—
86D4×C83—21no—
87D4×C83—22no—
88D4×C83—42no—
89D4×C83—81no—
90D4×C83C2×D412yes—
91C2×C4×C83—21no—
92C2×C4×C83—42no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2411yescharacteristic
96C22⋊D83—44no—
97C22⋊D83—44no—
98C22⋊D83—82no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83—44no—
103C2×Dic83—44no—
104C2×Dic83C2×D412yes—
105D332—22no—
106C4×D83—22no—
107C4×D83—22no—
108Dic166—21no—
109Dic166—22no—
110Dic166—22no—
111Dic166C2×D412yes—
112C8×C83—22no—
113C83⋊C8—21no—
114C83⋊C8C2×D412yes—
115C22×C83—22no—
116C22×C83—41no—
117C22×C83—41no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83C22×D411yescharacteristic
122C2×D83—22no—
123C2×D83—22no—
124C2×D83—22no—
125C4×C83—21no—
126C4×C83—41no—
127C4×C83#13512yescomplemented
128C4×C83C22×D411yescharacteristic
129C4×C83H(5,𝔽2)11yescharacteristic
130Dic83—22no—
131Dic83—22no—
132Dic83—42no—
133Dic83—41no—
134C2×C83—42no—
135C2×C83—82no—
136C2×C83D4211yescharacteristic
137C2×C83C2×#13511yescharacteristic
138C2×C83#57711yescharacteristic
139D83—42no—
140#3284—831nomaximal, Sylow
141C83#328411yescharacteristic, complemented, Sylow
142C2×#135—832no—
143#550—831no—
144#547—831no—
145#492—831no—
146#465—832no—
147#462—831no—
148#459D8311yescharacteristic, complemented
149#452—832no—
150#449—832no—
151C4×SD16—831no—
152C8×D4—831no—
153C2×#41—832no—
154C2×#41—832no—
155#135—1668no—
156C2×Q16—832no—
157C2×SD16—832no—
158C2×SD16—831no—
159C2×SD16—832no—
160C2×D8C2×D8312yes—
161C22×C8—832no—
162C4⋊Q8—831no—
163#127—1661no—
164#127C2×D8311yescharacteristic
165#123—1662no—
166C22⋊Q8—832no—
167#121—832no—
168#121—1662no—
169#121—1661no—
170C4×Q8—831no—
171C4×D4—831no—
172C4×D4—1661no—
173C4×D4C2×D8311yescharacteristic
174C2×C4⋊C4—1661no—
175#106—831no—
176C4⋊C8C2×D8311yescharacteristic
177#103—832no—
178#103—831no—
179#102—831no—
180#102C2×D8312yes—
181C22⋊C8—832no—
182C4×C8—831no—
183#41—3322no—
184#41—3322no—
185#41—1664no—
186#41—1664no—
187C2×Q8—831no—
188C2×Q8—832no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—831no—
192C2×D4—1662no—
193C2×D4—1662no—
194C2×D4—1662no—
195C2×D4C22×D8312yes—
196C2×D4C22×D8311yescharacteristic
197C22×C4—21no—
198C22×C4—832no—
199C22×C4—1662no—
200C22×C4—1662no—
201Q16—1664no—
202SD16—3322no—
203SD16—834no—
204SD16—1664no—
205D8—24no—
206C2×C8—831no—
207C2×C8—831no—
208C2×C8—1664no—
209C2×C8C22×D8312yes—
210C4⋊C4—831no—
211C4⋊C4—831no—
212C4⋊C4—1662no—
213C4⋊C4—1661no—
214C4⋊C4—1661no—
215C4⋊C4—1661no—
216C4⋊C4C22×D8311yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—832no—
219C22⋊C4—1662no—
220C22⋊C4—1662no—
221C42—1661no—
222C42C22×D8311yescharacteristic
223C23—21no—
224C23—22no—
225C23—832no—
226Q8—832no—
227Q8—1662no—
228Q8—1662no—
229Q8—1661no—
230D4—22no—
231D4—21no—
232D4—22no—
233D4—42no—
234D4—81no—
235D4—3324no—
236D4—3324no—
237D4—6642no—
238D4—1662no—
239D4D4×D8312yes—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—3324no—
243C2×C4—3324no—
244C2×C4—832no—
245C2×C4—1661no—
246C2×C4—1662no—
247C2×C4—1661no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4D4×D8311yescharacteristic
251C2×C4D4×D8311yescharacteristic
252C2×C4C23×D8311yescharacteristic
253C8—22no—
254C8—832no—
255C8—1661no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—1662no—
263C22—1662no—
264C22—1662no—
265C22C2×D4×D8311yescharacteristic
266C4—21no—
267C4—41no—
268C4—3322no—
269C4—3321no—
270C4—1662no—
271C4—1662no—
272C4#1458181112yescomplemented
273C4C2×D4×D8311yescharacteristic
274C4#1458183811yescharacteristic
275C2—42no—
276C2—82no—
277C2—3322no—
278C2#1912833511yescharacteristic
279C2C2×#1458181111yescharacteristic
280C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#459×C83C2
socleC22×C83C22×D4
hypercenter#459D83
hypocenterC83#3284
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#14581811C2semidirect
2. #19128371C2semidirect
3. #19128355C2semidirect
4. #19128073C2semidirect
5. #19127929C2semidirect
6. #19127912C2semidirect
7. #19127864C2semidirect
8. #19127843C2semidirect
9. C4×#13050578C2semidirect
10. #19127766C2semidirect
11. #459×C83C2semidirect
58. #13050578D4semidirect
127. C4×C83#135semidirect
141. C83#3284semidirect
148. #459D83semidirect
272. C4#14581811semidirect

Maximal subgroups

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

subgroupindex
C2×#145818112
#191283712
#191283552
#191280732
#191279292
#191279122
#191278642
#191278432
C4×#130505782
#191277662
#459×C832
#328483