Finite Group Databasebeta

← #24919179

Subgroups — #24919179

Order 10624.

484 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 131 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×#14581811C211yesmaximal, characteristic, complemented
2#19128350C212yesmaximal, complemented
3#19128079C211yesmaximal, characteristic, complemented
4#19127942C211yesmaximal, characteristic, complemented
5#19127939C211yesmaximal, characteristic, complemented
6#19127904C212yesmaximal, complemented
7#19127863C212yesmaximal, complemented
8#19127860C212yesmaximal, complemented
9D8×Dic83C211yesmaximal, characteristic, complemented
10C8×C22⋊D83C211yesmaximal, characteristic, complemented
11C2×#135×C83C211yesmaximal, characteristic, complemented
12C2×#13050589C2212yes—
13#14581811—24no—
14C2×Dic332C2211yescharacteristic
15C2×#13050580C2212yes—
16C2×#13050578C2212yes—
17C2×C8×D83C2211yescharacteristic
18#14581790C2212yes—
19#14581775—22no—
20#14581773—22no—
21#14581771C2212yes—
22#14581768C2212yes—
23#14581764—22no—
24D4×Dic83C2212yes—
25C4×C22⋊D83C2211yescharacteristic
26C2×Dic83⋊C4—22no—
27#14581726C2211yescharacteristic
28Dic83⋊C8C2211yescharacteristic
29#14581717C2212yes—
30#14581716C2212yes—
31#14581713C2212yes—
32C22⋊(C83⋊C8)C2211yescharacteristic
33(C2×D83)⋊C8C2211yescharacteristic
34C8×Dic83C2211yescharacteristic
35C2×#41×C83C2212yes—
36#135×C83—24no—
37C2×Q16×C83C2211yescharacteristic
38C2×SD16×C83C2212yes—
39C2×D8×C83C2211yescharacteristic
40C22×C8×C83C2211yescharacteristic
41#13050589—24no—
42#13050589—42no—
43C2×Dic166C2312yes—
44C2×C22⋊D83—22no—
45C2×C22⋊D83D411yescharacteristic
46C22×Dic83—22no—
47C22×Dic83—22no—
48C2×C4×D83C2311yescharacteristic
49Dic332—22no—
50#13050580—42no—
51#13050578—24no—
52C2×C83⋊C8C2311yescharacteristic
53C2×C83⋊C8C2311yescharacteristic
54C8×D83—22no—
55C4⋊Dic83C2312yes—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59Dic83⋊C4—22no—
60Dic83⋊C4D411yescharacteristic
61C22⋊Dic83—22no—
62C22⋊Dic83—22no—
63C22⋊Dic83D411yescharacteristic
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4D411yescharacteristic
66C4×Dic83C2311yescharacteristic
67#41×C83—24no—
68#41×C83—42no—
69C2×Q8×C83C2312yes—
70C2×D4×C83—22no—
71C2×D4×C83C2312yes—
72C22×C4×C83—22no—
73C22×C4×C83C2311yescharacteristic
74Q16×C83—22no—
75SD16×C83—24no—
76D8×C83D414yescomplemented
77C2×C8×C83—22no—
78C2×C8×C83C2311yescharacteristic
79C2×C8×C83C2311yescharacteristic
80C23×C83—22no—
81C23×C83C2×D411yescharacteristic
82C22×D83C2×D411yescharacteristic
83Q8×C83—22no—
84Q8×C83—22no—
85D4×C83—22no—
86D4×C83—44no—
87D4×C83C2×D414yes—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83—44no—
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2411yescharacteristic
94C22⋊D83—22no—
95C22⋊D83—44no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—44no—
99C2×Dic83—44no—
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102C2×Dic83C2×D411yescharacteristic
103C4×D83—21no—
104C4×D83—21no—
105Dic166—22no—
106Dic166—22no—
107C8×C83—21no—
108C8×C83C2×D412yes—
109C83⋊C8—21no—
110C83⋊C8—21no—
111C22×C83—24no—
112C22×C83—21no—
113C22×C83—21no—
114C22×C83—21no—
115C22×C83—42no—
116C22×C83—42no—
117C22×C83C22×D411yescharacteristic
118C2×D83—21no—
119C2×D83—21no—
120C2×D83—21no—
121C4×C83—21no—
122C4×C83—42no—
123C4×C83C22×D411yescharacteristic
124C4×C83H(5,𝔽2)11yescharacteristic
125Dic83—21no—
126Dic83—21no—
127Dic83—42no—
128Dic83—42no—
129Dic83#13512yescomplemented
130C2×C83—41no—
131C2×C83—44no—
132C2×C83D4211yescharacteristic
133C2×C83C2×#13511yescharacteristic
134C2×C83#57811yescharacteristic
135D83—41no—
136#3283—831nomaximal, Sylow
137C83#328311yescharacteristic, complemented, Sylow
138C2×#135—831no—
139C2×#135D8311yescharacteristic, complemented
140#547—832no—
141#494—831no—
142#467—831no—
143#467—831no—
144#461—832no—
145#452—832no—
146#452—832no—
147C4×D8—831no—
148C8×D4—831no—
149C2×#41—832no—
150C2×#41C2×D8312yes—
151#135—24no—
152#135—1664no—
153C2×Q16—832no—
154C2×Q16—831no—
155C2×Q16C2×D8311yescharacteristic
156C2×SD16—832no—
157C2×SD16C2×D8312yes—
158C2×D8C2×D8311yescharacteristic
159C22×C8—831no—
160C22×C8C2×D8311yescharacteristic
161C4⋊Q8—832no—
162#123—1662no—
163#123—1662no—
164C22⋊Q8—832no—
165C22⋊Q8—832no—
166#121—1662no—
167C4×D4—831no—
168C4×D4—832no—
169C2×C4⋊C4—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—24no—
179#41—42no—
180#41—3322no—
181#41—1664no—
182C2×Q8—832no—
183C2×Q8C22×D8312yes—
184C2×D4—22no—
185C2×D4—831no—
186C2×D4—1662no—
187C2×D4C22×D8312yes—
188C22×C4—22no—
189C22×C4—831no—
190C22×C4—1662no—
191C22×C4—1662no—
192C22×C4C22×D8311yescharacteristic
193Q16—22no—
194Q16—3322no—
195Q16—1662no—
196SD16—24no—
197SD16—1664no—
198D8C22⋊D8314yescomplemented
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—
209C4⋊C4—1662no—
210C4⋊C4—1662no—
211C22⋊C4—831no—
212C22⋊C4—831no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C22⋊C4—1662no—
216C42—831no—
217C23—22no—
218C23—831no—
219C23D4×D8311yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—1662no—
223Q8—1662no—
224D4—22no—
225D4—44no—
226D4—3324no—
227D4—1662no—
228D4C2×C22⋊D8314yes—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—44no—
233C2×C4—3324no—
234C2×C4—3324no—
235C2×C4—831no—
236C2×C4—831no—
237C2×C4—831no—
238C2×C4—1662no—
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—24no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—42no—
254C22—1661no—
255C22—1661no—
256C22—1661no—
257C22C2×D4×D8311yescharacteristic
258C4—21no—
259C4—42no—
260C4—3322no—
261C4—3322no—
262C4—832no—
263C4—1661no—
264C4—1661no—
265C4C22×C22⋊D8311yescharacteristic
266C4#1458183711yescharacteristic
267C2—41no—
268C2—44no—
269C2—3321no—
270C2D4×C22⋊D8311yescharacteristic
271C2#135×D8311yescharacteristic
272C2#1912851511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#135×C83C2
socleC22×C83C22×D4
hypercenterC2×#135D83
hypocenterC83#3283
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#14581811C2semidirect
2. #19128350C2semidirect
3. #19128079C2semidirect
4. #19127942C2semidirect
5. #19127939C2semidirect
6. #19127904C2semidirect
7. #19127863C2semidirect
8. #19127860C2semidirect
9. D8×Dic83C2semidirect
10. C8×C22⋊D83C2semidirect
11. C2×#135×C83C2semidirect
76. D8×C83D4semidirect
129. Dic83#135semidirect
137. C83#3283semidirect
139. C2×#135D83semidirect
198. D8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818112
#191283502
#191280792
#191279422
#191279392
#191279042
#191278632
#191278602
D8×Dic832
C8×C22⋊D832
C2×#135×C832
#328383