Finite Group Databasebeta

← #36333383

Subgroups — #36333383

Order 21376.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#14581950C211yesmaximal, characteristic, complemented
2C2×#19128865C211yesmaximal, characteristic, complemented
3C2×#19128860C211yesmaximal, characteristic, complemented
4#24923780C211yesmaximal, characteristic, complemented
5#24923433C214yesmaximal, complemented
6#24923416C214yesmaximal, complemented
7#24923279C211yesmaximal, characteristic, complemented
8#24923275C211yesmaximal, characteristic, complemented
9#407×C167C211yesmaximal, characteristic, complemented
10C2×#14581950—24no—
11C2×#14581950—24no—
12C2×#14581950—22no—
13C2×#14581950C2214yes—
14C2×#14581950C2211yescharacteristic
15C22×Dic334C2211yescharacteristic
16C22×D668C2211yescharacteristic
17C22×C22⋊D167—21no—
18C22×C4×D167—21no—
19#19128865—24no—
20#19128860—24no—
21C2×Dic668C2212yescomplemented
22C2×#14581939C2212yescomplemented
23C2×#14581939C2212yescomplemented
24C2×D1336C2212yescomplemented
25#19128815—21no—
26#19128811C2212yescomplemented
27#19128806C2212yescomplemented
28#19128801—21no—
29C4×C22⋊D167—22no—
30#19128791C2211yescharacteristic
31#19128757C2214yescomplemented
32#19128754C2214yescomplemented
33C23×C4×C167C2211yescharacteristic
34C2×M16×C167C2211yescharacteristic
35C2×M16×C167C2211yescharacteristic
36C22⋊C8×C167C2214yescomplemented
37C23×D167—21no—
38#14581950—216no—
39#14581950—28no—
40#14581950—28no—
41#14581950—22no—
42#14581950—22no—
43C2×Dic334—22no—
44C2×Dic334—21no—
45C2×Dic334D414yescomplemented
46C2×Dic334C2311yescharacteristic
47C2×Dic334C2312yes—
48C2×D668—22no—
49C2×D668—21no—
50C2×D668D414yescomplemented
51C2×D668C2311yescharacteristic
52C2×D668C2312yes—
53C2×C22⋊D167—28no—
54C2×C22⋊D167—24no—
55C2×C22⋊D167—21no—
56C2×C22⋊D167—22no—
57C22×Dic167—21no—
58C2×C4×D167—28no—
59C2×C4×D167—24no—
60C2×C4×D167—21no—
61Dic668—24no—
62#14581939—24no—
63#14581939—24no—
64D1336—24no—
65C4⋊Dic167C2312yes—
66Dic167⋊C4—22no—
67C22⋊Dic167—21no—
68C22⋊Dic167—22no—
69(C2×D167)⋊C4—22no—
70C4×Dic167—21no—
71C24×C167D411yescharacteristic
72C22×C4×C167—22no—
73C22×C4×C167—22no—
74C22×C4×C167D411yescharacteristic
75C22×C4×C167D412yes—
76C22×C4×C167C2312yes—
77C22×C4×C167C2311yescharacteristic
78M16×C167—22no—
79M16×C167—22no—
80C2×C8×C167C2312yes—
81C2×C8×C167C2312yes—
82C22×D167—24no—
83C22×D167—22no—
84C22×D167—21no—
85C22×D167—44no—
86Dic334—24no—
87Dic334—22no—
88Dic334—21no—
89Dic334—21no—
90Dic334C2×D414yes—
91D668—24no—
92D668—22no—
93D668—21no—
94D668—21no—
95D668C2×D414yes—
96C22⋊D167—216no—
97C22⋊D167—44no—
98C22⋊D167—42no—
99C2×Dic167—24no—
100C2×Dic167—22no—
101C2×Dic167—22no—
102C2×Dic167—21no—
103C4×D167—216no—
104C4×D167—42no—
105C23×C167—22no—
106C23×C167—42no—
107C23×C167C2×D412yes—
108C23×C167C2×D411yescharacteristic
109C2×C4×C167—22no—
110C2×C4×C167—24no—
111C2×C4×C167—22no—
112C2×C4×C167—22no—
113C2×C4×C167C2×D412yes—
114C2×C4×C167C2×D412yes—
115C2×C4×C167C2×D412yes—
116C2×C4×C167C2×D411yescharacteristic
117C2×C4×C167C2411yescharacteristic
118C8×C167—22no—
119C8×C167—22no—
120C22×C167—22no—
121C22×C167—22no—
122C22×C167—22no—
123C22×C167—22no—
124C22×C167—44no—
125C22×C167#12012yes—
126C22×C167C22×D411yescharacteristic
127C2×D167—24no—
128C2×D167—48no—
129C2×D167—44no—
130C2×D167—41no—
131C2×D167—42no—
132C4×C167—22no—
133C4×C167#12012yes—
134C4×C167C22×D411yescharacteristic
135C4×C167C22×D411yescharacteristic
136Dic167—24no—
137Dic167—41no—
138Dic167—42no—
139C2×C167—22no—
140C2×C167—42no—
141C2×C167C2×#12011yescharacteristic
142C2×C167#57512yes—
143D167—44no—
144D167—81no—
145C167#300511yescharacteristic, complemented, Sylow
146#3005—1671nomaximal, Sylow
147C22×#41—1671no—
148C2×#137—1671no—
149C2×AGL(1,ℤ/8)—1671no—
150#525—1671no—
151#452—1674no—
152#449—1674no—
153#419—1671no—
154#418—1671no—
155#407D16711yescharacteristic, complemented
156C2×#41—3344no—
157C2×#41—3344no—
158C2×#41—3342no—
159C2×#41—1671no—
160C2×#41—1674no—
161C22×Q8—1671no—
162C22×D4—3341no—
163C22×D4—1671no—
164C23×C4—3341no—
165C23×C4C2×D16711yescharacteristic
166#137—3344no—
167AGL(1,ℤ/8)—3344no—
168C2×Q16—1672no—
169C2×SD16—1672no—
170C2×SD16—1672no—
171C2×D8—1672no—
172C2×M16C2×D16711yescharacteristic
173C2×M16C2×D16711yescharacteristic
174#123—3341no—
175C22⋊Q8—1672no—
176#121—1672no—
177#120—3341no—
178C4×D4—3342no—
179#117—1671no—
180#103—1674no—
181#102—1674no—
182C22⋊C8C2×D16714yescomplemented
183C24—3341no—
184C24D66811yescharacteristic
185#41—33416no—
186#41—3348no—
187#41—3348no—
188#41—3342no—
189#41—3342no—
190C2×Q8—3342no—
191C2×Q8—3341no—
192C2×Q8—1674no—
193C2×Q8—1671no—
194C2×Q8—1672no—
195C2×D4—3348no—
196C2×D4—3344no—
197C2×D4—3342no—
198C2×D4—3341no—
199C2×D4—3342no—
200C2×D4—3341no—
201C2×D4—1674no—
202C2×D4—1671no—
203C2×D4—1672no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—3348no—
207C22×C4—3341no—
208C22×C4—3344no—
209C22×C4—3341no—
210C22×C4D66812yes—
211C22×C4D66811yescharacteristic
212C22×C4C22×D16712yes—
213C22×C4C22×D16711yescharacteristic
214Q16—3344no—
215SD16—3344no—
216SD16—3344no—
217D8—3344no—
218M16—22no—
219M16—22no—
220C2×C8C22×D16712yes—
221C2×C8C22×D16712yes—
222C4⋊C4—3342no—
223C4⋊C4—1672no—
224C22⋊C4—3341no—
225C22⋊C4—3342no—
226C22⋊C4—3342no—
227C42—3341no—
228C23—22no—
229C23—42no—
230C23—3342no—
231C23—3344no—
232C23—3341no—
233C23—6684no—
234C23C2×D66812yes—
235C23C2×D66811yescharacteristic
236Q8—3342no—
237Q8—3344no—
238Q8—3341no—
239Q8—3341no—
240Q8—1674no—
241D4—33416no—
242D4—3342no—
243D4—3341no—
244D4—3341no—
245D4—3344no—
246D4—6684no—
247D4—6682no—
248D4—1674no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—33416no—
254C2×C4—3342no—
255C2×C4—3344no—
256C2×C4—3342no—
257C2×C4—3341no—
258C2×C4—6682no—
259C2×C4D4×D16712yes—
260C2×C4D4×D16712yes—
261C2×C4C2×D66812yes—
262C2×C4C2×D66811yescharacteristic
263C2×C4C23×D16711yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—3344no—
272C22—6688no—
273C22—6681no—
274C22—6682no—
275C22—6684no—
276C22#1912880212yes—
277C22C22×D66811yescharacteristic
278C4—22no—
279C4—3344no—
280C4—6681no—
281C4—6682no—
282C4#1912880212yes—
283C4C2×D4×D16711yescharacteristic
284C4C2×D4×D16711yescharacteristic
285C2—22no—
286C2—42no—
287C2—13361no—
288C2—6684no—
289C2C2×#1912880211yescharacteristic
290C2#2492406912yes—
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#407×C167C2
socleC22×C167C22×D4
hypercenter#407D167
hypocenterC167#3005
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#14581950C2semidirect
2. C2×#19128865C2semidirect
3. C2×#19128860C2semidirect
4. #24923780C2semidirect
5. #24923433C2semidirect
6. #24923416C2semidirect
7. #24923279C2semidirect
8. #24923275C2semidirect
9. #407×C167C2semidirect
21. C2×Dic668C22semidirect
22. C2×#14581939C22semidirect
23. C2×#14581939C22semidirect
24. C2×D1336C22semidirect
26. #19128811C22semidirect
27. #19128806C22semidirect
31. #19128757C22semidirect
32. #19128754C22semidirect
36. C22⋊C8×C167C22semidirect
45. C2×Dic334D4semidirect
50. C2×D668D4semidirect
145. C167#3005semidirect
155. #407D167semidirect
182. C22⋊C8C2×D167semidirect

Maximal subgroups

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

subgroupindex
C22×#145819502
C2×#191288652
C2×#191288602
#249237802
#249234332
#249234162
#249232792
#249232752
#407×C1672
#3005167