Finite Group Databasebeta

← #36335759

Subgroups — #36335759

Order 21376.

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

normal subgroups only

This group has 556 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×#19128859C212yesmaximal, complemented
2#24923941C211yesmaximal, characteristic, complemented
3D4×D668C211yesmaximal, characteristic, complemented
4#24923646C211yesmaximal, characteristic, complemented
5#24923523C212yesmaximal, complemented
6#24923469C211yesmaximal, characteristic, complemented
7#24923464C211yesmaximal, characteristic, complemented
8#24923417C212yesmaximal, complemented
9#24923404C212yesmaximal, complemented
10#24923348C211yesmaximal, characteristic, complemented
11C4×SD16×C167C211yesmaximal, characteristic, complemented
12C2×#14581951C2212yescomplemented
13C22×D668—21no—
14C2×D4×D167C2212yes—
15#19128859—28no—
16C2×#14581937C2212yescomplemented
17C2×D1336C2212yescomplemented
18C2×D1336C2212yescomplemented
19C2×#14581935C2212yes—
20C2×#14581933C2212yescomplemented
21#19128829—21no—
22#19128829C2211yescharacteristic
23#19128829C2211yescharacteristic, complemented
24#19128807—22no—
25#19128807C2212yes—
26#19128807C2212yescomplemented
27#19128806—21no—
28#19128802—22no—
29C4×D668—21no—
30C4×D668C2211yescharacteristic
31C4⋊(C167⋊C8)C2211yescharacteristic, complemented
32#19128752C2212yescomplemented
33#19128752C2212yescomplemented
34(C2×D167)⋊C8C2212yescomplemented
35C2×SD16×C167C2211yescharacteristic
36C4×Q8×C167C2211yescharacteristic, complemented
37C4×D4×C167C2211yescharacteristic
38#106×C167C2211yescharacteristic, complemented
39#103×C167C2211yescharacteristic, complemented
40#102×C167C2211yescharacteristic
41C4×C8×C167C2211yescharacteristic, complemented
42C23×D167—22no—
43#14581951—24no—
44#14581951—42no—
45C2×D668—22no—
46C2×D668—22no—
47C2×D668—22no—
48C2×D668—21no—
49C2×D668—21no—
50C2×D668—21no—
51C2×D668—21no—
52C2×D668—44no—
53C2×D668D411yescharacteristic
54C2×D668C2312yes—
55C2×D668C2312yes—
56C2×C22⋊D167—22no—
57D4×D167—24no—
58D4×D167—42no—
59C2×C4×D167—22no—
60C2×C4×D167C2312yes—
61#14581937—24no—
62D1336—24no—
63D1336—42no—
64#14581935—24no—
65#14581933—24no—
66C2×C167⋊C8C2312yes—
67C4⋊Dic167D411yescharacteristic
68(C2×D167)⋊C4—22no—
69(C2×D167)⋊C4—22no—
70(C2×D167)⋊C4D412yes—
71C2×Q8×C167C2311yescharacteristic
72C2×D4×C167C2311yescharacteristic
73C22×C4×C167—21no—
74SD16×C167D414yescomplemented
75C2×C8×C167C2311yescharacteristic
76C2×C8×C167C2311yescharacteristic
77C4⋊C4×C167—21no—
78C4⋊C4×C167C2311yescharacteristic
79C4⋊C4×C167C2311yescharacteristic
80C22⋊C4×C167—21no—
81C42×C167—21no—
82C42×C167C2311yescharacteristic
83C22×D167—22no—
84C22×D167—22no—
85C22×D167—44no—
86C22×D167—44no—
87C22×D167—82no—
88C22×D167C2×D412yes—
89D668—22no—
90D668—22no—
91D668—22no—
92D668—22no—
93D668—22no—
94D668—44no—
95D668—44no—
96D668—42no—
97D668—42no—
98D668—81no—
99D668—81no—
100D668—82no—
101C22⋊D167—44no—
102C2×Dic167C2×D412yes—
103C4×D167—22no—
104C4×D167—22no—
105C4×D167—44no—
106C167⋊C8—22no—
107C23×C167—21no—
108Q8×C167—21no—
109Q8×C167C2×D412yescomplemented
110D4×C167—21no—
111D4×C167C2×D412yes—
112C2×C4×C167—21no—
113C2×C4×C167—21no—
114C2×C4×C167—21no—
115C2×C4×C167—42no—
116C2×C4×C167C2×D411yescharacteristic
117C2×C4×C167C2×D411yescharacteristic
118C2×C4×C167C2411yescharacteristic
119C8×C167—21no—
120C8×C167C2×D412yescomplemented
121C22×C167—22no—
122C22×C167—41no—
123C22×C167—41no—
124C22×C167C22×D411yescharacteristic
125C2×D167—22no—
126C2×D167—22no—
127C2×D167—22no—
128C2×D167—42no—
129C2×D167—42no—
130C2×D167—42no—
131C2×D167—42no—
132C2×D167—42no—
133C2×D167—42no—
134C2×D167—88no—
135C4×C167—22no—
136C4×C167—21no—
137C4×C167—41no—
138C4×C167—41no—
139C4×C167AGL(1,ℤ/8)12yescomplemented
140C4×C167C22×D411yescharacteristic
141C4×C167H(5,𝔽2)11yescharacteristic
142Dic167—22no—
143C2×C167—42no—
144C2×C167D4211yescharacteristic
145C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C167#57611yescharacteristic
147D167—42no—
148D167—82no—
149D167—82no—
150C167#327411yescharacteristic, complemented, Sylow
151#3274—1671nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1672no—
153#550—1671no—
154D42—1671no—
155#493—1671no—
156#469—1672no—
157#460—1671no—
158#459—1671no—
159#449—1672no—
160#447—1672no—
161C4×SD16D16711yescharacteristic, complemented
162#436—1671no—
163C2×#41—1672no—
164C22×D4—3341no—
165C22×D4—1672no—
166AGL(1,ℤ/8)—3348no—
167C2×SD16—1672no—
168C2×SD16C2×D16711yescharacteristic
169C2×D8—1672no—
170C2×D8—1672no—
171C2×D8—1672no—
172C2×M16—1672no—
173#127—3341no—
174#127—1671no—
175#127—1671no—
176#121—3341no—
177#121—3342no—
178#121—1672no—
179#121—1672no—
180#120—3342no—
181C4×Q8C2×D16711yescharacteristic, complemented
182C4×D4—3341no—
183C4×D4—1671no—
184C4×D4C2×D16711yescharacteristic
185#106C2×D16711yescharacteristic, complemented
186C4⋊C8—1671no—
187#103C2×D16711yescharacteristic, complemented
188#102—1672no—
189#102—1672no—
190#102C2×D16711yescharacteristic
191C22⋊C8—1672no—
192C4×C8C2×D16711yescharacteristic, complemented
193C24—3342no—
194#41—3344no—
195#41—6682no—
196C2×Q8C22×D16711yescharacteristic
197C2×D4—3342no—
198C2×D4—3342no—
199C2×D4—3342no—
200C2×D4—3344no—
201C2×D4—3342no—
202C2×D4—3341no—
203C2×D4—3341no—
204C2×D4—3341no—
205C2×D4—3341no—
206C2×D4—6684no—
207C2×D4—6682no—
208C2×D4—1671no—
209C2×D4—1672no—
210C2×D4—1672no—
211C2×D4C22×D16711yescharacteristic
212C22×C4—21no—
213C22×C4—3342no—
214C22×C4—1672no—
215SD16—3344no—
216SD16D66814yescomplemented
217D8—3344no—
218D8—3344no—
219D8—6682no—
220M16—3344no—
221C2×C8—1672no—
222C2×C8C22×D16711yescharacteristic
223C2×C8C22×D16711yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—1671no—
226C4⋊C4C22×D16711yescharacteristic
227C4⋊C4C22×D16711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—3342no—
230C22⋊C4—3342no—
231C22⋊C4—1672no—
232C42—21no—
233C42C22×D16711yescharacteristic
234C23—21no—
235C23—13362no—
236C23—3342no—
237C23—3342no—
238C23—6684no—
239C23—6684no—
240C23—1672no—
241Q8—21no—
242Q8C2×D66812yescomplemented
243D4—21no—
244D4—13361no—
245D4—13361no—
246D4—13362no—
247D4—3342no—
248D4—3342no—
249D4—3342no—
250D4—3342no—
251D4—3342no—
252D4—6684no—
253D4—6684no—
254D4—6684no—
255D4—6682no—
256D4—6682no—
257D4C2×D66812yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3342no—
263C2×C4—3342no—
264C2×C4—6684no—
265C2×C4—1672no—
266C2×C4D4×D16711yescharacteristic
267C2×C4D4×D16711yescharacteristic
268C2×C4C23×D16711yescharacteristic
269C8—21no—
270C8—3342no—
271C8C2×D66812yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—13368no—
276C22—3342no—
277C22—3342no—
278C22—3342no—
279C22—6682no—
280C22—6682no—
281C22—6682no—
282C22—6682no—
283C22—6682no—
284C22—6682no—
285C22C2×D4×D16711yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—3342no—
291C4#1912885912yescomplemented
292C4C22×D66811yescharacteristic
293C4#1912887811yescharacteristic
294C2—42no—
295C2—13362no—
296C2—13362no—
297C2—6682no—
298C2D4×D66811yescharacteristic
299C2C2×#1912885911yescharacteristic
300C2#2492407111yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC4×SD16×C167C2
socleC22×C167C22×D4
hypercenterC4×SD16D167
hypocenterC167#3274
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19128859C2semidirect
2. #24923941C2semidirect
3. D4×D668C2semidirect
4. #24923646C2semidirect
5. #24923523C2semidirect
6. #24923469C2semidirect
7. #24923464C2semidirect
8. #24923417C2semidirect
9. #24923404C2semidirect
10. #24923348C2semidirect
11. C4×SD16×C167C2semidirect
12. C2×#14581951C22semidirect
16. C2×#14581937C22semidirect
17. C2×D1336C22semidirect
18. C2×D1336C22semidirect
20. C2×#14581933C22semidirect
23. #19128829C22semidirect
26. #19128807C22semidirect
31. C4⋊(C167⋊C8)C22semidirect
32. #19128752C22semidirect
33. #19128752C22semidirect
34. (C2×D167)⋊C8C22semidirect
36. C4×Q8×C167C22semidirect
38. #106×C167C22semidirect
39. #103×C167C22semidirect
41. C4×C8×C167C22semidirect
74. SD16×C167D4semidirect
109. Q8×C167C2×D4semidirect
120. C8×C167C2×D4semidirect
139. C4×C167AGL(1,ℤ/8)semidirect
150. C167#3274semidirect
161. C4×SD16D167semidirect
181. C4×Q8C2×D167semidirect
185. #106C2×D167semidirect
187. #103C2×D167semidirect
192. C4×C8C2×D167semidirect
216. SD16D668semidirect
242. Q8C2×D668semidirect
271. C8C2×D668semidirect
291. C4#19128859semidirect

Maximal subgroups

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

subgroupindex
C2×#191288592
#249239412
D4×D6682
#249236462
#249235232
#249234692
#249234642
#249234172
#249234042
#249233482
C4×SD16×C1672
#3274167