Finite Group Databasebeta

← #36334600

Subgroups — #36334600

Order 21376.

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

normal subgroups only

This group has 572 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
1C2×#19128865C211yesmaximal, characteristic, complemented
2C2×#19128862C211yesmaximal, characteristic, complemented
3C2×#19128858C211yesmaximal, characteristic, complemented
4#24923815C211yesmaximal, characteristic, complemented
5C2×#19128819C211yesmaximal, characteristic, complemented
6#24923660C212yesmaximal, complemented
7#24923657C212yesmaximal, complemented
8#24923652C212yesmaximal, complemented
9#24923650C212yesmaximal, complemented
10M16×Dic167C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C167C211yesmaximal, characteristic, complemented
12C2×#14581950C2211yescharacteristic
13C22×Dic334C2211yescharacteristic
14#19128865—24no—
15#19128862—24no—
16#19128858—24no—
17C2×Dic668C2212yescomplemented
18C2×#14581940C2212yescomplemented
19C2×#14581939C2212yescomplemented
20C2×#14581938C2212yes—
21C2×#14581938C2212yescomplemented
22C2×#14581935C2212yes—
23C2×#14581934C2211yescharacteristic
24#19128830C2211yescharacteristic, complemented
25#19128828C2211yescharacteristic, complemented
26#19128821C2211yescharacteristic, complemented
27#19128819—22no—
28#19128819C2212yes—
29#19128819C2212yes—
30#19128819C2211yescharacteristic, complemented
31#19128804—22no—
32C4×C22⋊D167—22no—
33C2×C22⋊Dic167—22no—
34C2×C4×Dic167C2211yescharacteristic
35#19128742C2212yescomplemented
36C8×Dic167C2212yescomplemented
37C2×#41×C167C2211yescharacteristic
38C22×D4×C167C2211yescharacteristic
39AGL(1,ℤ/8)×C167—24no—
40C2×SD16×C167C2212yes—
41C2×D8×C167C2212yes—
42C2×M16×C167C2211yescharacteristic
43#14581950—22no—
44#14581950—22no—
45C2×Dic334—21no—
46C2×Dic334—22no—
47C2×Dic334—22no—
48C2×Dic334C2311yescharacteristic
49C2×Dic334C2312yes—
50C2×D668C2311yescharacteristic
51C2×C22⋊D167—22no—
52C2×C22⋊D167—21no—
53C22×Dic167—21no—
54C22×Dic167D412yes—
55C2×C4×D167—21no—
56Dic668—24no—
57#14581940—24no—
58#14581939—24no—
59#14581938—24no—
60#14581938—24no—
61#14581935—24no—
62#14581934D414yescomplemented
63C2×C167⋊C8C2312yes—
64Dic167⋊C4—22no—
65C22⋊Dic167—24no—
66C22⋊Dic167—24no—
67C22⋊Dic167—22no—
68(C2×D167)⋊C4—22no—
69C4×Dic167D412yes—
70C4×Dic167C2311yescharacteristic
71C4×Dic167C2311yescharacteristic
72C24×C167—21no—
73#41×C167—22no—
74#41×C167—22no—
75C2×Q8×C167C2311yescharacteristic
76C2×D4×C167—21no—
77C2×D4×C167—22no—
78C2×D4×C167—22no—
79C2×D4×C167—21no—
80C2×D4×C167C2312yes—
81C2×D4×C167C2311yescharacteristic
82C22×C4×C167—21no—
83C22×C4×C167C2311yescharacteristic
84SD16×C167—24no—
85D8×C167—24no—
86M16×C167D414yescomplemented
87C2×C8×C167C2312yes—
88C22×D167—21no—
89Dic334—21no—
90Dic334—22no—
91Dic334—22no—
92Dic334—21no—
93Dic334—42no—
94D668—21no—
95D668—21no—
96C22⋊D167—44no—
97C22⋊D167—42no—
98C2×Dic167—22no—
99C2×Dic167—21no—
100C2×Dic167—42no—
101C2×Dic167—42no—
102C2×Dic167C2×D412yes—
103C2×Dic167C2×D412yes—
104C4×D167—42no—
105C167⋊C8C2×D414yes—
106C23×C167—21no—
107C23×C167—22no—
108C23×C167—44no—
109C23×C167—42no—
110C23×C167C2×D411yescharacteristic
111Q8×C167—21no—
112Q8×C167—21no—
113D4×C167—21no—
114D4×C167—22no—
115D4×C167—22no—
116D4×C167—21no—
117D4×C167—42no—
118D4×C167—42no—
119C2×C4×C167—21no—
120C2×C4×C167—42no—
121C2×C4×C167C2×D411yescharacteristic
122C2×C4×C167C2×D412yes—
123C2×C4×C167C2×D412yes—
124C2×C4×C167C2411yescharacteristic
125C8×C167C2×D414yes—
126C22×C167—22no—
127C22×C167—41no—
128C22×C167—42no—
129C22×C167—44no—
130C22×C167—42no—
131C22×C167—84no—
132C22×C167#12712yes—
133C22×C167C22×D411yescharacteristic
134C2×D167—41no—
135C2×D167—42no—
136C4×C167—41no—
137C4×C167#12712yes—
138C4×C167C22×D411yescharacteristic
139C4×C167C22×D411yescharacteristic
140Dic167—22no—
141Dic167—22no—
142Dic167—41no—
143Dic167—42no—
144C2×C167—22no—
145C2×C167—81no—
146C2×C167—82no—
147C2×C167C2×#12711yescharacteristic
148C2×C167#57512yes—
149D167—81no—
150C167#315311yescharacteristic, complemented, Sylow
151#3153—1671nomaximal, Sylow
152C2×#137—1671no—
153C2×#137—1671no—
154C2×AGL(1,ℤ/8)—1671no—
155C2×AGL(1,ℤ/8)D16711yescharacteristic, complemented
156#532—1671no—
157C2×#124—1671no—
158#497—1672no—
159#496—1672no—
160#495—1672no—
161#495—1672no—
162C4×M16—1671no—
163C2×#41—1671no—
164C2×#41C2×D16711yescharacteristic
165C22×Q8—1671no—
166C22×D4C2×D16711yescharacteristic
167#137—3344no—
168#137—3344no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—3344no—
171C2×Q16—1672no—
172C2×Q16—1672no—
173C2×SD16—1672no—
174C2×SD16—1672no—
175C2×SD16—1672no—
176C2×SD16C2×D16712yes—
177C2×D8—1672no—
178C2×D8C2×D16712yes—
179C2×M16—1671no—
180C2×M16C2×D16711yescharacteristic
181C4⋊Q8—1671no—
182#127—1671no—
183#124—3342no—
184#124—1671no—
185#124—1672no—
186#124—1672no—
187#124—1671no—
188#121—3342no—
189C4×D4—3342no—
190C2×C22⋊C4—3342no—
191C2×C42—1671no—
192#97—1672no—
193C4×C8—1672no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—3342no—
198#41—3342no—
199C2×Q8—3341no—
200C2×Q8—3342no—
201C2×Q8—3342no—
202C2×Q8—1672no—
203C2×Q8—1671no—
204C2×Q8C22×D16711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3341no—
210C2×D4—3342no—
211C2×D4—1671no—
212C2×D4C22×D16712yes—
213C2×D4C22×D16711yescharacteristic
214C22×C4—21no—
215C22×C4—3341no—
216C22×C4—3341no—
217C22×C4—1672no—
218C22×C4C22×D16711yescharacteristic
219Q16—3344no—
220Q16—3344no—
221SD16—24no—
222SD16—3344no—
223SD16—3344no—
224SD16—3344no—
225D8—24no—
226D8—3344no—
227M16—1674no—
228M16C22⋊D16714yescomplemented
229C2×C8—1672no—
230C2×C8C22×D16712yes—
231C4⋊C4—3342no—
232C22⋊C4—3342no—
233C22⋊C4—3344no—
234C22⋊C4—3344no—
235C22⋊C4—3342no—
236C42—1672no—
237C42—1671no—
238C42—1671no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—3341no—
244C23D4×D16711yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—3341no—
248Q8—3342no—
249Q8—3342no—
250Q8—3341no—
251Q8—6682no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3341no—
259D4—3341no—
260D4—6684no—
261D4—6682no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—3342no—
265C2×C4—3341no—
266C2×C4—6682no—
267C2×C4—6682no—
268C2×C4—6682no—
269C2×C4—1672no—
270C2×C4—1672no—
271C2×C4D4×D16712yes—
272C2×C4D4×D16711yescharacteristic
273C2×C4C2×C22⋊D16712yes—
274C2×C4C23×D16711yescharacteristic
275C8—1674no—
276C8C2×C22⋊D16714yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—6681no—
284C22—6682no—
285C22#1912882812yes—
286C22C2×D4×D16711yescharacteristic
287C4—41no—
288C4—3342no—
289C4—3342no—
290C4—6681no—
291C4—6682no—
292C4#1912882812yes—
293C4C2×D4×D16711yescharacteristic
294C4C22×C22⋊D16711yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—13361no—
299C2C2×#1912882811yescharacteristic
300C2#2492406512yes—
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC2×AGL(1,ℤ/8)×C167C2
socleC22×C167C22×D4
hypercenterC2×AGL(1,ℤ/8)D167
hypocenterC167#3153
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19128865C2semidirect
2. C2×#19128862C2semidirect
3. C2×#19128858C2semidirect
4. #24923815C2semidirect
5. C2×#19128819C2semidirect
6. #24923660C2semidirect
7. #24923657C2semidirect
8. #24923652C2semidirect
9. #24923650C2semidirect
10. M16×Dic167C2semidirect
11. C2×AGL(1,ℤ/8)×C167C2semidirect
17. C2×Dic668C22semidirect
18. C2×#14581940C22semidirect
19. C2×#14581939C22semidirect
21. C2×#14581938C22semidirect
24. #19128830C22semidirect
25. #19128828C22semidirect
26. #19128821C22semidirect
30. #19128819C22semidirect
35. #19128742C22semidirect
36. C8×Dic167C22semidirect
62. #14581934D4semidirect
86. M16×C167D4semidirect
150. C167#3153semidirect
155. C2×AGL(1,ℤ/8)D167semidirect
228. M16C22⋊D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288652
C2×#191288622
C2×#191288582
#249238152
C2×#191288192
#249236602
#249236572
#249236522
#249236502
M16×Dic1672
C2×AGL(1,ℤ/8)×C1672
#3153167