Finite Group Databasebeta

← #24919765

Subgroups — #24919765

Order 10624.

420 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#19128355C211yesmaximal, characteristic, complemented
2#19128354C211yesmaximal, characteristic, complemented
3#19128053C211yesmaximal, characteristic, complemented
4#19128038C212yesmaximal, complemented
5#19127948C212yesmaximal, complemented
6#19127907C211yesmaximal, characteristic, complemented
7#19127847C212yesmaximal, complemented
8C4×#13050575C211yesmaximal, characteristic, complemented
9#19127778C211yesmaximal, characteristic, complemented
10#19127699C212yesmaximal, complemented
11#474×C83C211yesmaximal, characteristic, complemented
12C2×#13050591—21no—
13C2×#13050589C2212yes—
14C2×#13050579C2212yes—
15C2×#13050575C2211yescharacteristic
16C2×#13050573C2212yes—
17#14581792C2211yescharacteristic
18#14581777—22no—
19#14581776—22no—
20#14581772C2212yes—
21#14581768—21no—
22#14581767C2212yes—
23#14581766—21no—
24C4×D332C2211yescharacteristic
25C4×D332C2211yescharacteristic
26C2×C4⋊Dic83—21no—
27C4⋊C4×D83C2212yes—
28#14581725C2211yescharacteristic
29#14581717C2212yes—
30#14581715C2212yes—
31#14581715C2212yes—
32#14581713C2211yescharacteristic
33#14581712C2211yescharacteristic
34(C2×D83)⋊C8C2212yes—
35C4×C83⋊C8C2211yescharacteristic
36C4⋊Q8×C83C2211yescharacteristic
37C4×D4×C83C2211yescharacteristic
38#107×C83C2212yes—
39C4⋊C8×C83C2211yescharacteristic
40#102×C83C2212yes—
41#13050591—44no—
42#13050589—24no—
43#13050589—42no—
44C2×Dic166C2312yes—
45C2×D332—22no—
46C2×D332D411yescharacteristic
47C2×D332C2311yescharacteristic
48C2×C22⋊D83—22no—
49C22×Dic83—22no—
50C2×C4×D83—22no—
51C2×C4×D83—21no—
52C2×C4×D83C2312yes—
53#13050579—42no—
54#13050575—22no—
55#13050573—42no—
56C2×C83⋊C8C2311yescharacteristic
57C2×C83⋊C8C2311yescharacteristic
58C4⋊Dic83—21no—
59C4⋊Dic83—21no—
60C4⋊Dic83—22no—
61C4⋊Dic83D411yescharacteristic
62C4⋊Dic83C2311yescharacteristic
63Dic83⋊C4—22no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4D412yes—
68C2×Q8×C83—21no—
69C2×D4×C83C2311yescharacteristic
70C22×C4×C83—21no—
71C2×C8×C83C2312yes—
72C4⋊C4×C83—21no—
73C4⋊C4×C83C2312yes—
74C4⋊C4×C83C2311yescharacteristic
75C22⋊C4×C83—21no—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C22×D83—21no—
79C22×D83C2×D412yes—
80Q8×C83—42no—
81D4×C83—21no—
82D4×C83#4112yescomplemented
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—42no—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2411yescharacteristic
89C22⋊D83—44no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—44no—
93C2×Dic83C2×D412yes—
94D332—22no—
95D332—21no—
96D332—21no—
97D332—44no—
98C4×D83—22no—
99C4×D83—22no—
100C4×D83—44no—
101C4×D83—42no—
102Dic166—22no—
103Dic166—22no—
104C8×C83—22no—
105C83⋊C8—21no—
106C83⋊C8—21no—
107C22×C83—22no—
108C22×C83—41no—
109C22×C83—41no—
110C22×C83C22×D411yescharacteristic
111C2×D83—22no—
112C2×D83—22no—
113C2×D83—22no—
114C2×D83—41no—
115C2×D83—41no—
116C2×D83—41no—
117C4×C83—21no—
118C4×C83—42no—
119C4×C83—41no—
120C4×C83AGL(1,ℤ/8)12yes—
121C4×C83C2×#4111yescharacteristic
122C4×C83H(5,𝔽2)11yescharacteristic
123Dic83—22no—
124Dic83—42no—
125Dic83—41no—
126C2×C83—42no—
127C2×C83#54611yescharacteristic
128C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C83#57811yescharacteristic
130D83—42no—
131D83—81no—
132#3330—831nomaximal, Sylow
133C83#333011yescharacteristic, complemented, Sylow
134#547—831no—
135#547—831no—
136#487—831no—
137#484—832no—
138#474D8311yescharacteristic, complemented
139#468—832no—
140#461—831no—
141#449—832no—
142C4×D8—831no—
143#436—831no—
144#417—832no—
145C2×#41—832no—
146C2×#41—1661no—
147C2×SD16—832no—
148C2×D8—831no—
149C2×M16—832no—
150C4⋊Q8—831no—
151C4⋊Q8C2×D8311yescharacteristic
152#123—1662no—
153#123—1662no—
154C22⋊Q8—832no—
155C22⋊Q8—1661no—
156#121—832no—
157#121—1661no—
158C4×D4—831no—
159C4×D4—831no—
160C4×D4C2×D8311yescharacteristic
161C2×C4⋊C4—832no—
162C2×C4⋊C4—1661no—
163#107—831no—
164#107C2×D8312yes—
165C4⋊C8C2×D8311yescharacteristic
166#103—832no—
167#103—832no—
168#103—832no—
169#102—831no—
170#102—831no—
171#102C2×D8312yes—
172C22⋊C8—832no—
173C4×C8—831no—
174#41—3324no—
175#41—3322no—
176#41—1664no—
177C2×Q8—21no—
178C2×Q8—832no—
179C2×D4—831no—
180C2×D4—831no—
181C2×D4—1662no—
182C2×D4—1662no—
183C2×D4C22×D8311yescharacteristic
184C22×C4—21no—
185C22×C4—832no—
186C22×C4—1662no—
187C22×C4—1662no—
188C22×C4—1661no—
189SD16—3322no—
190D8—1662no—
191M16—3322no—
192C2×C8—831no—
193C2×C8—831no—
194C2×C8C22×D8312yes—
195C4⋊C4—21no—
196C4⋊C4—831no—
197C4⋊C4—831no—
198C4⋊C4—1662no—
199C4⋊C4—1662no—
200C4⋊C4—1661no—
201C4⋊C4—1661no—
202C4⋊C4C22×D8311yescharacteristic
203C4⋊C4C22×D8312yes—
204C22⋊C4—21no—
205C22⋊C4—832no—
206C22⋊C4—1662no—
207C22⋊C4—1662no—
208C22⋊C4—1661no—
209C42C22×D8311yescharacteristic
210C23—21no—
211C23—832no—
212C23—1661no—
213Q8—42no—
214Q8—1662no—
215Q8—1662no—
216D4—21no—
217D4—3324no—
218D4—3324no—
219D4—1662no—
220D4—1661no—
221D4—1661no—
222D4#1305059112yescomplemented
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—3324no—
227C2×C4—3324no—
228C2×C4—3322no—
229C2×C4—832no—
230C2×C4—1661no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4D4×D8311yescharacteristic
235C2×C4D4×D8311yescharacteristic
236C2×C4C23×D8311yescharacteristic
237C8—22no—
238C8—1661no—
239C8—1661no—
240C22—22no—
241C22—41no—
242C22—41no—
243C22—3321no—
244C22—3321no—
245C22—3321no—
246C22—1662no—
247C22—1662no—
248C22—1662no—
249C22C2×D4×D8311yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—3322no—
254C4—3321no—
255C4—1662no—
256C4#1458181612yes—
257C4C2×#1305059111yescharacteristic
258C4#1458183811yescharacteristic
259C2—42no—
260C2—3322no—
261C2—6641no—
262C2#1912834211yescharacteristic
263C2C2×#1458181611yescharacteristic
264C2#1912851711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#474×C83C2
socleC22×C83C22×D4
hypercenter#474D83
hypocenterC83#3330
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #19128355C2semidirect
2. #19128354C2semidirect
3. #19128053C2semidirect
4. #19128038C2semidirect
5. #19127948C2semidirect
6. #19127907C2semidirect
7. #19127847C2semidirect
8. C4×#13050575C2semidirect
9. #19127778C2semidirect
10. #19127699C2semidirect
11. #474×C83C2semidirect
82. D4×C83#41semidirect
133. C83#3330semidirect
138. #474D83semidirect
222. D4#13050591semidirect

Maximal subgroups

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

subgroupindex
#191283552
#191283542
#191280532
#191280382
#191279482
#191279072
#191278472
C4×#130505752
#191277782
#191276992
#474×C832
#333083