Finite Group Databasebeta

← #36364037

Subgroups — #36364037

Order 22144.

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

normal subgroups only

This group has 564 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×#19149712C211yesmaximal, characteristic, complemented
2C2×#19149699C211yesmaximal, characteristic, complemented
3C22×#14583814C211yesmaximal, characteristic, complemented
4#26012062C211yesmaximal, characteristic, complemented
5#26012058C211yesmaximal, characteristic, complemented
6#26011882C212yesmaximal, complemented
7#26011879C212yesmaximal, complemented
8#26011874C212yesmaximal, complemented
9#26011862C212yesmaximal, complemented
10#26011449C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C22×Dic346C2211yescharacteristic
14#19149712—24no—
15#19149699—24no—
16C2×Dic692C2212yescomplemented
17C2×#14583816C2211yescharacteristic, complemented
18C2×#14583815C2212yescomplemented
19C2×#14583814—24no—
20C2×#14583814C2212yes—
21C2×#14583814C2212yes—
22C2×#14583814C2211yescharacteristic, complemented
23C2×#14583813C2211yescharacteristic, complemented
24C2×#14583811C2211yescharacteristic, complemented
25C22×#13063436C2211yescharacteristic
26#19149677C2212yescomplemented
27#19149675C2212yes—
28#19149666C2212yes—
29#19149666C2212yescomplemented
30#19149660—22no—
31#19149658—22no—
32#19149650—22no—
33#19149648—22no—
34#19149638C2211yescharacteristic
35#19149589C2212yescomplemented
36C8×Dic173C2212yescomplemented
37C2×#41×C173C2211yescharacteristic
38C22×D4×C173C2211yescharacteristic
39AGL(1,ℤ/8)×C173—24no—
40C2×SD16×C173C2212yes—
41C2×D8×C173C2212yes—
42C2×M16×C173C2211yescharacteristic
43#14583826—22no—
44#14583826—22no—
45C2×Dic346—21no—
46C2×Dic346—22no—
47C2×Dic346—22no—
48C2×Dic346C2311yescharacteristic
49C2×Dic346C2312yes—
50C2×D692C2311yescharacteristic
51C2×C22⋊D173—24no—
52C2×C22⋊D173—21no—
53C22×Dic173—21no—
54C2×C4×D173—21no—
55Dic692—24no—
56#14583816—22no—
57#14583815—24no—
58#14583814—24no—
59#14583814—24no—
60#14583814—22no—
61#14583813—22no—
62#14583811—22no—
63C2×#13063436D414yescomplemented
64C2×#13063436C2311yescharacteristic
65C2×#13063436C2311yescharacteristic
66C4⋊Dic173D412yes—
67#14583805—24no—
68C22⋊Dic173—24no—
69C22⋊Dic173—24no—
70C22⋊Dic173D412yes—
71C4×Dic173C2312yes—
72C24×C173—21no—
73#41×C173—22no—
74#41×C173—22no—
75C2×Q8×C173C2311yescharacteristic
76C2×D4×C173—21no—
77C2×D4×C173—22no—
78C2×D4×C173—22no—
79C2×D4×C173—21no—
80C2×D4×C173C2311yescharacteristic
81C2×D4×C173C2312yes—
82C22×C4×C173—21no—
83C22×C4×C173C2311yescharacteristic
84SD16×C173—24no—
85D8×C173—24no—
86M16×C173D414yescomplemented
87C2×C8×C173C2312yes—
88C22×D173—21no—
89Dic346—22no—
90Dic346—21no—
91Dic346—22no—
92Dic346—21no—
93Dic346—42no—
94D692—21no—
95D692—21no—
96C22⋊D173—42no—
97C22⋊D173—84no—
98C2×Dic173—21no—
99C2×Dic173—22no—
100C2×Dic173—42no—
101C2×Dic173C2×D414yes—
102C4×D173—42no—
103#13063436C2×D412yes—
104#13063436C2×D412yes—
105C23×C173—22no—
106C23×C173—21no—
107C23×C173—42no—
108C23×C173—82no—
109C23×C173C2×D411yescharacteristic
110Q8×C173—21no—
111Q8×C173—21no—
112D4×C173—21no—
113D4×C173—22no—
114D4×C173—21no—
115D4×C173—22no—
116D4×C173—42no—
117D4×C173—42no—
118C2×C4×C173—21no—
119C2×C4×C173—42no—
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173C2×D412yes—
122C2×C4×C173C2×D412yes—
123C2×C4×C173C2411yescharacteristic
124C8×C173C2×D414yes—
125C22×C173—22no—
126C22×C173—41no—
127C22×C173—42no—
128C22×C173—42no—
129C22×C173—44no—
130C22×C173—84no—
131C22×C173#12712yes—
132C22×C173C22×D411yescharacteristic
133C2×D173—41no—
134C2×D173—42no—
135C4×C173—41no—
136C4×C173#12712yes—
137C4×C173C22×D411yescharacteristic
138C4×C173C22×D411yescharacteristic
139Dic173—24no—
140Dic173—41no—
141Dic173—42no—
142C2×C173—22no—
143C2×C173—81no—
144C2×C173—82no—
145C2×C173C2×#12711yescharacteristic
146C2×C173#57712yes—
147D173—81no—
148C173#315411yescharacteristic, complemented, Sylow
149#3154—1731nomaximal, Sylow
150C2×#137—1731no—
151C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
152C2×#135—1731no—
153C22×SD16—1731no—
154#536—1731no—
155#535—1731no—
156#497—1732no—
157#496—1732no—
158#495—1732no—
159#492—1732no—
160#405—1731no—
161C2×#41—1731no—
162C2×#41C2×D17311yescharacteristic
163C22×Q8—1731no—
164C22×D4C2×D17311yescharacteristic
165#137—3464no—
166AGL(1,ℤ/8)—24no—
167#135—3464no—
168C2×Q16—1731no—
169C2×Q16—1732no—
170C2×SD16—3464no—
171C2×SD16—1731no—
172C2×SD16—1732no—
173C2×SD16—1732no—
174C2×SD16—1731no—
175C2×SD16—1732no—
176C2×SD16C2×D17312yes—
177C2×D8—1731no—
178C2×D8C2×D17312yes—
179C2×M16C2×D17311yescharacteristic
180C22×C8—1731no—
181C4⋊Q8—1732no—
182#127—1732no—
183#124—1732no—
184#124—1732no—
185#123—3462no—
186C22⋊Q8—3462no—
187#121—3462no—
188#120—3462no—
189#117—1731no—
190#97—1732no—
191C4×C8—1732no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—3462no—
196#41—3462no—
197C2×Q8—3461no—
198C2×Q8—3462no—
199C2×Q8—3462no—
200C2×Q8—1731no—
201C2×Q8—1732no—
202C2×Q8C22×D17311yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—3461no—
208C2×D4—3464no—
209C2×D4—1731no—
210C2×D4C22×D17311yescharacteristic
211C2×D4C22×D17312yes—
212C22×C4—21no—
213C22×C4—3461no—
214C22×C4—3461no—
215C22×C4C22×D17311yescharacteristic
216Q16—3462no—
217Q16—3464no—
218SD16—24no—
219SD16—3464no—
220SD16—3464no—
221SD16—3462no—
222SD16—3462no—
223SD16—3464no—
224D8—24no—
225D8—3462no—
226M16C22⋊D17314yescomplemented
227C2×C8—1734no—
228C2×C8—1731no—
229C2×C8—1731no—
230C2×C8C22×D17312yes—
231C4⋊C4—3464no—
232C4⋊C4—1732no—
233C22⋊C4—3464no—
234C22⋊C4—3464no—
235C22⋊C4—1732no—
236C42—1732no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—3461no—
242C23D4×D17311yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—3462no—
246Q8—3461no—
247Q8—3462no—
248Q8—3461no—
249Q8—6922no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—3461no—
257D4—3461no—
258D4—13844no—
259D4—6922no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3461no—
263C2×C4—3462no—
264C2×C4—1734no—
265C2×C4—6922no—
266C2×C4—6922no—
267C2×C4D4×D17311yescharacteristic
268C2×C4D4×D17312yes—
269C2×C4C2×C22⋊D17312yes—
270C2×C4C23×D17311yescharacteristic
271C8—1732no—
272C8—1732no—
273C8C2×C22⋊D17314yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—6921no—
281C22—6922no—
282C22#1914967512yes—
283C22C2×D4×D17311yescharacteristic
284C4—41no—
285C4—3464no—
286C4—6921no—
287C4—6922no—
288C4#1914967512yes—
289C4C2×D4×D17311yescharacteristic
290C4C22×C22⋊D17311yescharacteristic
291C2—22no—
292C2—81no—
293C2—82no—
294C2—13841no—
295C2C2×#1914967511yescharacteristic
296C2#2601230112yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173C22×D4
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173#3154
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
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×#19149712C2semidirect
2. C2×#19149699C2semidirect
3. C22×#14583814C2semidirect
4. #26012062C2semidirect
5. #26012058C2semidirect
6. #26011882C2semidirect
7. #26011879C2semidirect
8. #26011874C2semidirect
9. #26011862C2semidirect
10. #26011449C2semidirect
11. C2×AGL(1,ℤ/8)×C173C2semidirect
16. C2×Dic692C22semidirect
17. C2×#14583816C22semidirect
18. C2×#14583815C22semidirect
22. C2×#14583814C22semidirect
23. C2×#14583813C22semidirect
24. C2×#14583811C22semidirect
26. #19149677C22semidirect
29. #19149666C22semidirect
35. #19149589C22semidirect
36. C8×Dic173C22semidirect
63. C2×#13063436D4semidirect
86. M16×C173D4semidirect
148. C173#3154semidirect
151. C2×AGL(1,ℤ/8)D173semidirect
226. M16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C2×#191496992
C22×#145838142
#260120622
#260120582
#260118822
#260118792
#260118742
#260118622
#260114492
C2×AGL(1,ℤ/8)×C1732
#3154173