Finite Group Databasebeta

← #24918386

Subgroups — #24918386

Order 10624.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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#19128246C211yesmaximal, characteristic
2C2×#14581790C211yesmaximal, characteristic, complemented
3#19128072C212yesmaximal, complemented
4#19128050C212yesmaximal, complemented
5#19127905C212yesmaximal, complemented
6#19127904C212yesmaximal, complemented
7#19127842C211yesmaximal, characteristic, complemented
8#19127840C211yesmaximal, characteristic, complemented
9#19127839C211yesmaximal, characteristic, complemented
10#19127751C211yesmaximal, characteristic, complemented
11#448×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—21no—
13C22×Dic166C2211yescharacteristic
14C2×#13050579C2212yes—
15C2×#13050578C2212yes—
16C2×#13050577C2212yes—
17#14581791C2211yescharacteristic
18#14581790—21no—
19#14581790—21no—
20#14581790C2212yes—
21#14581790C2212yes—
22#14581788C2211yescharacteristic
23#14581778—21no—
24#14581766C2211yescharacteristic
25#14581764—21no—
26D4×Dic83C2211yescharacteristic
27#14581754C2211yescharacteristic
28(C22⋊D83)⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×C4×Dic83C2211yescharacteristic
32Dic83⋊C8C2212yes—
33#14581717C2212yes—
34#14581716C2212yes—
35#14581715C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yes—
38C22×Q8×C83C2211yescharacteristic
39C2×SD16×C83C2212yes—
40#121×C83C2211yescharacteristic
41#103×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43#13050589—44no—
44C2×Dic166—21no—
45C2×Dic166—21no—
46C2×Dic166—42no—
47C2×Dic166C2312yes—
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—21no—
50C2×C22⋊D83—21no—
51C2×C22⋊D83—21no—
52C22×Dic83—21no—
53C22×Dic83—21no—
54C22×Dic83D411yescharacteristic
55C22×Dic83D411yescharacteristic
56C2×C4×D83—21no—
57#13050579—42no—
58#13050578—42no—
59#13050577—42no—
60C2×C83⋊C8C2312yes—
61C4⋊Dic83C2311yescharacteristic
62Dic83⋊C4—21no—
63Dic83⋊C4—22no—
64Dic83⋊C4—22no—
65Dic83⋊C4—22no—
66Dic83⋊C4—22no—
67C22⋊Dic83—21no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—21no—
70C4×Dic83D411yescharacteristic
71C4×Dic83D411yescharacteristic
72C4×Dic83C2311yescharacteristic
73C4×Dic83C2311yescharacteristic
74C2×Q8×C83—21no—
75C2×Q8×C83—42no—
76C2×Q8×C83C2312yes—
77C2×D4×C83—21no—
78C2×D4×C83C2311yescharacteristic
79C22×C4×C83—21no—
80C22×C4×C83C2311yescharacteristic
81SD16×C83—42no—
82C2×C8×C83C2312yes—
83C4⋊C4×C83C2311yescharacteristic
84C22⋊C4×C83—21no—
85C23×C83—21no—
86C23×C83C2×D411yescharacteristic
87C22×D83—21no—
88Q8×C83—22no—
89Q8×C83—22no—
90Q8×C83—42no—
91D4×C83—22no—
92D4×C83—42no—
93C2×C4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—42no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—42no—
100C22⋊D83—42no—
101C22⋊D83—42no—
102C2×Dic83—22no—
103C2×Dic83—21no—
104C2×Dic83—22no—
105C2×Dic83—42no—
106C2×Dic83—42no—
107C2×Dic83SD1614yescomplemented
108C2×Dic83C2×D411yescharacteristic
109C2×Dic83C2×D411yescharacteristic
110C2×Dic83C2×D411yescharacteristic
111C2×Dic83C2×D411yescharacteristic
112D332—22no—
113C4×D83—42no—
114Dic166—22no—
115Dic166—22no—
116Dic166—42no—
117Dic166—42no—
118C8×C83—22no—
119C83⋊C8—22no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83—41no—
125C22×C83C2×SD1612yes—
126C22×C83C22×D411yescharacteristic
127C2×D83—41no—
128C2×D83—41no—
129C2×D83—41no—
130C4×C83—21no—
131C4×C83—42no—
132C4×C83—41no—
133C4×C83H(5,𝔽2)12yes—
134Dic83—21no—
135Dic83—42no—
136Dic83—41no—
137Dic83C2×SD1612yes—
138Dic83C2×SD1612yes—
139Dic83#13712yes—
140C2×C83—22no—
141C2×C83—81no—
142C2×C83#53511yescharacteristic
143C2×C83C22×SD1611yescharacteristic
144C2×C83C2×#13711yescharacteristic
145D83—81no—
146#3206—831nomaximal, Sylow
147C83#320611yescharacteristic, complemented, Sylow
148#532—831no—
149C2×C4⋊Q8—831no—
150#492—832no—
151#487—832no—
152#461—832no—
153#461—832no—
154#448—831no—
155#448—831no—
156#448—831no—
157#448D8311yescharacteristic, complemented
158#431—831no—
159C2×#41—1661no—
160C22×Q8—831no—
161C22×Q8C2×D8311yescharacteristic
162C2×SD16—832no—
163C2×SD16—832no—
164C2×SD16—832no—
165C2×SD16C2×D8312yes—
166C4⋊Q8—831no—
167C4⋊Q8—832no—
168C4⋊Q8—832no—
169C4⋊Q8—1661no—
170C4⋊Q8—1661no—
171#127—831no—
172#124—1661no—
173#121—831no—
174#121—1661no—
175#121C2×D8311yescharacteristic
176C4×D4—831no—
177C4×D4—831no—
178C4×D4—1661no—
179C2×C4⋊C4—1661no—
180C2×C4⋊C4—1661no—
181C2×C42—831no—
182C4⋊C8—832no—
183#103—832no—
184#103—832no—
185#103—832no—
186#103C2×D8312yes—
187C22⋊C8—831no—
188C22⋊C8C2×D8311yescharacteristic
189C4×C8—832no—
190#41—3324no—
191C2×Q8—21no—
192C2×Q8—42no—
193C2×Q8—3322no—
194C2×Q8—832no—
195C2×Q8—1661no—
196C2×Q8—1661no—
197C2×Q8C22×D8312yes—
198C2×D4—21no—
199C2×D4—831no—
200C2×D4—1661no—
201C2×D4—1661no—
202C2×D4—1661no—
203C2×D4C22×D8311yescharacteristic
204C22×C4—21no—
205C22×C4—831no—
206C22×C4—831no—
207C22×C4—1661no—
208C22×C4—1661no—
209C22×C4—1661no—
210C22×C4C22×D8311yescharacteristic
211SD16—42no—
212SD16—3322no—
213SD16—3322no—
214SD16—3322no—
215C2×C8—832no—
216C2×C8C22×D8312yes—
217C4⋊C4—831no—
218C4⋊C4—1662no—
219C4⋊C4—1662no—
220C4⋊C4—1662no—
221C4⋊C4—1662no—
222C4⋊C4—1661no—
223C4⋊C4C22×D8311yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—1661no—
226C22⋊C4—1661no—
227C22⋊C4—1661no—
228C42—831no—
229C42—831no—
230C42—831no—
231C42—831no—
232C23—21no—
233C23—1661no—
234C23D4×D8311yescharacteristic
235Q8—22no—
236Q8—22no—
237Q8—42no—
238Q8—3322no—
239Q8—3322no—
240Q8—1662no—
241Q8—1662no—
242D4—22no—
243D4—42no—
244D4—3322no—
245D4—3322no—
246D4—3322no—
247D4—1662no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—3322no—
253C2×C4—3322no—
254C2×C4—3322no—
255C2×C4—834no—
256C2×C4—831no—
257C2×C4—831no—
258C2×C4—831no—
259C2×C4—831no—
260C2×C4—1661no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4D4×D8311yescharacteristic
264C2×C4C23×D8311yescharacteristic
265C8—22no—
266C8—1662no—
267C22—21no—
268C22—21no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—3321no—
273C22—3321no—
274C22—3321no—
275C22SD16×D8312yes—
276C22C2×D4×D8311yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—3321no—
281C4—3322no—
282C4—832no—
283C4—832no—
284C4—832no—
285C4—1661no—
286C4#1458183712yes—
287C2—22no—
288C2—81no—
289C2—6641no—
290C2#1912826611yescharacteristic
291C2C2×SD16×D8311yescharacteristic
292C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#448×C83C2
socleC22×C83C22×D4
prime radical#19128246C2
hypercenter#448D83
hypocenterC83#3206
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C2×#14581790C2semidirect
3. #19128072C2semidirect
4. #19128050C2semidirect
5. #19127905C2semidirect
6. #19127904C2semidirect
7. #19127842C2semidirect
8. #19127840C2semidirect
9. #19127839C2semidirect
10. #19127751C2semidirect
11. #448×C83C2semidirect
107. C2×Dic83SD16semidirect
147. C83#3206semidirect
157. #448D83semidirect

Maximal subgroups

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

subgroupindex
#191282462
C2×#145817902
#191280722
#191280502
#191279052
#191279042
#191278422
#191278402
#191278392
#191277512
#448×C832
#320683