Finite Group Databasebeta

← #24919204

Subgroups — #24919204

Order 10624.

482 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581815C211yesmaximal, characteristic, complemented
2#19128362C212yesmaximal, complemented
3#19128086C211yesmaximal, characteristic, complemented
4#19127939C211yesmaximal, characteristic, complemented
5#19127937C211yesmaximal, characteristic, complemented
6#19127925C212yesmaximal, complemented
7#19127862C212yesmaximal, complemented
8#19127840C212yesmaximal, complemented
9Q16×Dic83C211yesmaximal, characteristic, complemented
10C8×C22⋊D83C211yesmaximal, characteristic, complemented
11C22×Q16×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2212yes—
13#14581815—24no—
14C2×#13050577C2212yes—
15C2×#13050577C2212yes—
16C2×D664C2211yescharacteristic
17C2×C8×D83C2211yescharacteristic
18#14581781—22no—
19#14581781C2212yes—
20#14581768—22no—
21#14581768C2212yes—
22#14581766—22no—
23#14581766C2212yes—
24Q8×Dic83C2212yes—
25C4×C22⋊D83—22no—
26C4×C22⋊D83C2211yescharacteristic
27#14581726C2211yescharacteristic
28Dic83⋊C8C2211yescharacteristic
29#14581716C2212yes—
30#14581716C2212yes—
31#14581714C2212yes—
32C22⋊(C83⋊C8)C2211yescharacteristic
33(C2×D83)⋊C8C2211yescharacteristic
34C8×Dic83C2211yescharacteristic
35C22×Q8×C83C2212yes—
36C2×Q16×C83—24no—
37C2×Q16×C83C2211yescharacteristic
38C2×Q16×C83C2212yes—
39C2×Q16×C83C2211yescharacteristic
40C22×C8×C83C2211yescharacteristic
41#13050591—24no—
42#13050591—42no—
43C2×D332—22no—
44C2×D332C2312yes—
45C2×C22⋊D83—22no—
46C2×C22⋊D83D411yescharacteristic
47C2×C4×D83—22no—
48C2×C4×D83C2311yescharacteristic
49#13050577—24no—
50#13050577—42no—
51D664—22no—
52C2×C83⋊C8C2311yescharacteristic
53C2×C83⋊C8C2311yescharacteristic
54C8×D83—22no—
55C4⋊Dic83—22no—
56C4⋊Dic83C2312yes—
57Dic83⋊C4—22no—
58Dic83⋊C4D411yescharacteristic
59C22⋊Dic83D411yescharacteristic
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4D411yescharacteristic
65C4×Dic83—22no—
66C4×Dic83C2311yescharacteristic
67C2×Q8×C83—22no—
68C2×Q8×C83—24no—
69C2×Q8×C83—42no—
70C2×Q8×C83C2312yes—
71C2×Q8×C83C2312yes—
72C22×C4×C83—22no—
73C22×C4×C83C2311yescharacteristic
74Q16×C83—24no—
75Q16×C83—22no—
76Q16×C83D414yescomplemented
77C2×C8×C83—22no—
78C2×C8×C83C2311yescharacteristic
79C2×C8×C83C2311yescharacteristic
80C23×C83C2×D411yescharacteristic
81C22×D83—22no—
82C22×D83C2×D411yescharacteristic
83Q8×C83—22no—
84Q8×C83—22no—
85Q8×C83—22no—
86Q8×C83—44no—
87Q8×C83C2×D414yes—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83—44no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C22⋊D83—22no—
96C22⋊D83—44no—
97C2×Dic83—22no—
98C2×Dic83C2×D411yescharacteristic
99C2×Dic83C2×D411yescharacteristic
100C2×Dic83C2×D411yescharacteristic
101D332—22no—
102D332—22no—
103D332—44no—
104C4×D83—21no—
105C4×D83—21no—
106C4×D83—44no—
107C8×C83—21no—
108C8×C83C2×D412yes—
109C83⋊C8—21no—
110C83⋊C8—21no—
111C22×C83—21no—
112C22×C83—21no—
113C22×C83#13512yescomplemented
114C22×C83C22×D411yescharacteristic
115C2×D83—21no—
116C2×D83—21no—
117C2×D83—21no—
118C2×D83—42no—
119C2×D83—42no—
120C2×D83—42no—
121C4×C83—24no—
122C4×C83—21no—
123C4×C83—42no—
124C4×C83C22×D411yescharacteristic
125C4×C83H(5,𝔽2)11yescharacteristic
126Dic83—21no—
127Dic83—21no—
128Dic83—21no—
129Dic83—42no—
130C2×C83—22no—
131C2×C83D4211yescharacteristic
132C2×C83C2×#13511yescharacteristic
133C2×C83#57811yescharacteristic
134D83—41no—
135D83—82no—
136#3285—831nomaximal, Sylow
137C83#328511yescharacteristic, complemented, Sylow
138C2×#135—831no—
139C22×Q16D8311yescharacteristic, complemented
140#548—832no—
141#495—831no—
142#467—831no—
143#466—831no—
144#464—832no—
145#452—832no—
146#448—832no—
147C4×Q16—831no—
148C8×D4—831no—
149C2×#41—832no—
150C22×Q8C2×D8312yes—
151#135—1664no—
152C2×Q16—24no—
153C2×Q16C2×D8312yes—
154C2×Q16C2×D8311yescharacteristic
155C2×Q16C2×D8311yescharacteristic
156C2×SD16—832no—
157C2×SD16—832no—
158C2×D8—831no—
159C22×C8—831no—
160C22×C8C2×D8311yescharacteristic
161#124—832no—
162#124—1662no—
163C22⋊Q8—832no—
164C22⋊Q8—1662no—
165#121—832no—
166#121—1662no—
167C4×Q8—832no—
168C4×D4—831no—
169C4×D4—1662no—
170#107—831no—
171C4⋊C8—831no—
172#103—832no—
173#103—832no—
174#102—832no—
175C22⋊C8—831no—
176C22⋊C8—831no—
177C4×C8—831no—
178#41—3322no—
179#41—1664no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8C22×D8312yes—
184C2×Q8C22×D8312yes—
185C2×D4—831no—
186C2×D4—832no—
187C2×D4—1662no—
188C2×D4—1662no—
189C22×C4—22no—
190C22×C4—831no—
191C22×C4—1662no—
192C22×C4C22×D8311yescharacteristic
193Q16—24no—
194Q16—22no—
195Q16C22⋊D8314yescomplemented
196SD16—3322no—
197SD16—1664no—
198D8—1662no—
199C2×C8—22no—
200C2×C8—831no—
201C2×C8—831no—
202C2×C8—1662no—
203C2×C8C22×D8311yescharacteristic
204C2×C8C22×D8311yescharacteristic
205C4⋊C4—831no—
206C4⋊C4—832no—
207C4⋊C4—1662no—
208C4⋊C4—1662no—
209C22⋊C4—831no—
210C22⋊C4—831no—
211C22⋊C4—1662no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C42—831no—
216C42—1662no—
217C23—831no—
218C23—1662no—
219C23D4×D8311yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—44no—
224Q8C2×C22⋊D8314yes—
225D4—3324no—
226D4—3324no—
227D4—1662no—
228D4—1662no—
229D4—1662no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—3324no—
236C2×C4—831no—
237C2×C4—831no—
238C2×C4—831no—
239C2×C4—1662no—
240C2×C4—1661no—
241C2×C4—1661no—
242C2×C4D4×D8311yescharacteristic
243C2×C4C23×D8311yescharacteristic
244C8—21no—
245C8—1661no—
246C8—1661no—
247C8C2×C22⋊D8312yes—
248C22—21no—
249C22—21no—
250C22—3322no—
251C22—3322no—
252C22—3322no—
253C22—1661no—
254C22—1661no—
255C22—1661no—
256C22#1458181512yescomplemented
257C22C2×D4×D8311yescharacteristic
258C4—21no—
259C4—24no—
260C4—42no—
261C4—3322no—
262C4—1661no—
263C4—1661no—
264C4—1661no—
265C4C22×C22⋊D8311yescharacteristic
266C4#1458183711yescharacteristic
267C2—22no—
268C2—3321no—
269C2—6642no—
270C2D4×C22⋊D8311yescharacteristic
271C2C2×#1458181511yescharacteristic
272C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC22×Q16×C83C2
socleC22×C83C22×D4
hypercenterC22×Q16D83
hypocenterC83#3285
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#14581815C2semidirect
2. #19128362C2semidirect
3. #19128086C2semidirect
4. #19127939C2semidirect
5. #19127937C2semidirect
6. #19127925C2semidirect
7. #19127862C2semidirect
8. #19127840C2semidirect
9. Q16×Dic83C2semidirect
10. C8×C22⋊D83C2semidirect
11. C22×Q16×C83C2semidirect
76. Q16×C83D4semidirect
113. C22×C83#135semidirect
137. C83#3285semidirect
139. C22×Q16D83semidirect
195. Q16C22⋊D83semidirect
256. C22#14581815semidirect

Maximal subgroups

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

subgroupindex
C2×#145818152
#191283622
#191280862
#191279392
#191279372
#191279252
#191278622
#191278402
Q16×Dic832
C8×C22⋊D832
C22×Q16×C832
#328583