Finite Group Databasebeta

← #46821347

Subgroups — #46821347

Order 34432.

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×#22575486C211yesmaximal, characteristic, complemented
2C2×#22575483C211yesmaximal, characteristic, complemented
3C2×#22575479C211yesmaximal, characteristic, complemented
4#31771151C211yesmaximal, characteristic, complemented
5C2×#22575440C211yesmaximal, characteristic, complemented
6#31770996C212yesmaximal, complemented
7#31770993C212yesmaximal, complemented
8#31770988C212yesmaximal, complemented
9#31770986C212yesmaximal, complemented
10M16×Dic269C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C22×Dic538C2211yescharacteristic
14#22575486—24no—
15#22575483—24no—
16#22575479—24no—
17C2×Dic1076C2212yescomplemented
18C2×#16864392C2212yescomplemented
19C2×#16864391C2212yescomplemented
20C2×#16864390C2212yes—
21C2×#16864390C2212yescomplemented
22C2×#16864387C2212yes—
23C2×#16864386C2211yescharacteristic
24#22575451C2211yescharacteristic, complemented
25#22575449C2211yescharacteristic, complemented
26#22575442C2211yescharacteristic, complemented
27#22575440—22no—
28#22575440C2212yes—
29#22575440C2212yes—
30#22575440C2211yescharacteristic, complemented
31#22575425—22no—
32C4×C22⋊D269—22no—
33C2×C22⋊Dic269—22no—
34C2×C4×Dic269C2211yescharacteristic
35#22575363C2212yescomplemented
36C8×Dic269C2212yescomplemented
37C2×#41×C269C2211yescharacteristic
38C22×D4×C269C2211yescharacteristic
39AGL(1,ℤ/8)×C269—24no—
40C2×SD16×C269C2212yes—
41C2×D8×C269C2212yes—
42C2×M16×C269C2211yescharacteristic
43#16864402—22no—
44#16864402—22no—
45C2×Dic538—21no—
46C2×Dic538—22no—
47C2×Dic538—22no—
48C2×Dic538C2312yes—
49C2×Dic538C2311yescharacteristic
50C2×D1076C2311yescharacteristic
51C2×C22⋊D269—22no—
52C2×C22⋊D269—21no—
53C22×Dic269—21no—
54C22×Dic269D412yes—
55C2×C4×D269—21no—
56Dic1076—24no—
57#16864392—24no—
58#16864391—24no—
59#16864390—24no—
60#16864390—24no—
61#16864387—24no—
62#16864386D414yescomplemented
63C2×#14524130C2312yes—
64#16864381—22no—
65C22⋊Dic269—24no—
66C22⋊Dic269—24no—
67C22⋊Dic269—22no—
68#16864379—22no—
69C4×Dic269D412yes—
70C4×Dic269C2311yescharacteristic
71C4×Dic269C2311yescharacteristic
72C24×C269—21no—
73#41×C269—22no—
74#41×C269—22no—
75C2×Q8×C269C2311yescharacteristic
76C2×D4×C269—21no—
77C2×D4×C269—22no—
78C2×D4×C269—22no—
79C2×D4×C269—21no—
80C2×D4×C269C2312yes—
81C2×D4×C269C2311yescharacteristic
82C22×C4×C269—21no—
83C22×C4×C269C2311yescharacteristic
84SD16×C269—24no—
85D8×C269—24no—
86M16×C269D414yescomplemented
87C2×C8×C269C2312yes—
88C22×D269—21no—
89Dic538—21no—
90Dic538—22no—
91Dic538—22no—
92Dic538—21no—
93Dic538—42no—
94D1076—21no—
95D1076—21no—
96C22⋊D269—44no—
97C22⋊D269—42no—
98C2×Dic269—22no—
99C2×Dic269—21no—
100C2×Dic269—42no—
101C2×Dic269—42no—
102C2×Dic269C2×D412yes—
103C2×Dic269C2×D412yes—
104C4×D269—42no—
105#14524130C2×D414yes—
106C23×C269—21no—
107C23×C269—22no—
108C23×C269—44no—
109C23×C269—42no—
110C23×C269C2×D411yescharacteristic
111Q8×C269—21no—
112Q8×C269—21no—
113D4×C269—21no—
114D4×C269—22no—
115D4×C269—22no—
116D4×C269—21no—
117D4×C269—42no—
118D4×C269—42no—
119C2×C4×C269—21no—
120C2×C4×C269—42no—
121C2×C4×C269C2×D411yescharacteristic
122C2×C4×C269C2×D412yes—
123C2×C4×C269C2×D412yes—
124C2×C4×C269C2411yescharacteristic
125C8×C269C2×D414yes—
126C22×C269—22no—
127C22×C269—41no—
128C22×C269—42no—
129C22×C269—44no—
130C22×C269—42no—
131C22×C269—84no—
132C22×C269#12712yes—
133C22×C269C22×D411yescharacteristic
134C2×D269—41no—
135C2×D269—42no—
136C4×C269—41no—
137C4×C269#12712yes—
138C4×C269C22×D411yescharacteristic
139C4×C269C22×D411yescharacteristic
140Dic269—22no—
141Dic269—22no—
142Dic269—41no—
143Dic269—42no—
144C2×C269—22no—
145C2×C269—81no—
146C2×C269—82no—
147C2×C269C2×#12711yescharacteristic
148C2×C269#57512yes—
149D269—81no—
150C269#315311yescharacteristic, complemented, Sylow
151#3153—2691nomaximal, Sylow
152C2×#137—2691no—
153C2×#137—2691no—
154C2×AGL(1,ℤ/8)—2691no—
155C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
156#532—2691no—
157C2×#124—2691no—
158#497—2692no—
159#496—2692no—
160#495—2692no—
161#495—2692no—
162C4×M16—2691no—
163C2×#41—2691no—
164C2×#41C2×D26911yescharacteristic
165C22×Q8—2691no—
166C22×D4C2×D26911yescharacteristic
167#137—5384no—
168#137—5384no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—5384no—
171C2×Q16—2692no—
172C2×Q16—2692no—
173C2×SD16—2692no—
174C2×SD16—2692no—
175C2×SD16—2692no—
176C2×SD16C2×D26912yes—
177C2×D8—2692no—
178C2×D8C2×D26912yes—
179C2×M16—2691no—
180C2×M16C2×D26911yescharacteristic
181C4⋊Q8—2691no—
182#127—2691no—
183#124—2691no—
184#124—2692no—
185#124—2692no—
186#124—2691no—
187#124—5382no—
188#121—5382no—
189C4×D4—5382no—
190C2×C22⋊C4—5382no—
191C2×C42—2691no—
192#97—2692no—
193C4×C8—2692no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—5382no—
198#41—5382no—
199C2×Q8—2692no—
200C2×Q8—2691no—
201C2×Q8—5381no—
202C2×Q8—5382no—
203C2×Q8—5382no—
204C2×Q8C22×D26911yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2691no—
210C2×D4—5382no—
211C2×D4—5381no—
212C2×D4C22×D26912yes—
213C2×D4C22×D26911yescharacteristic
214C22×C4—21no—
215C22×C4—2692no—
216C22×C4—5381no—
217C22×C4—5381no—
218C22×C4C22×D26911yescharacteristic
219Q16—5384no—
220Q16—5384no—
221SD16—24no—
222SD16—5384no—
223SD16—5384no—
224SD16—5384no—
225D8—24no—
226D8—5384no—
227M16—2694no—
228M16C22⋊D26914yescomplemented
229C2×C8—2692no—
230C2×C8C22×D26912yes—
231C4⋊C4—5382no—
232C22⋊C4—5382no—
233C22⋊C4—5384no—
234C22⋊C4—5384no—
235C22⋊C4—5382no—
236C42—2692no—
237C42—2691no—
238C42—2691no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—5381no—
244C23D4×D26911yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—5381no—
248Q8—5382no—
249Q8—5382no—
250Q8—5381no—
251Q8—10762no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—5381no—
259D4—5381no—
260D4—10764no—
261D4—10762no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—2692no—
265C2×C4—2692no—
266C2×C4—5382no—
267C2×C4—5381no—
268C2×C4—10762no—
269C2×C4—10762no—
270C2×C4—10762no—
271C2×C4D4×D26911yescharacteristic
272C2×C4D4×D26912yes—
273C2×C4C2×C22⋊D26912yes—
274C2×C4C23×D26911yescharacteristic
275C8—2694no—
276C8C2×C22⋊D26914yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—10761no—
284C22—10762no—
285C22#2257544912yes—
286C22C2×D4×D26911yescharacteristic
287C4—41no—
288C4—5382no—
289C4—5382no—
290C4—10761no—
291C4—10762no—
292C4#2257544912yes—
293C4C2×D4×D26911yescharacteristic
294C4C22×C22⋊D26911yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—21521no—
299C2C2×#2257544911yescharacteristic
300C2#3177140112yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×AGL(1,ℤ/8)×C269C2
socleC22×C269C22×D4
hypercenterC2×AGL(1,ℤ/8)D269
hypocenterC269#3153
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
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×#22575486C2semidirect
2. C2×#22575483C2semidirect
3. C2×#22575479C2semidirect
4. #31771151C2semidirect
5. C2×#22575440C2semidirect
6. #31770996C2semidirect
7. #31770993C2semidirect
8. #31770988C2semidirect
9. #31770986C2semidirect
10. M16×Dic269C2semidirect
11. C2×AGL(1,ℤ/8)×C269C2semidirect
17. C2×Dic1076C22semidirect
18. C2×#16864392C22semidirect
19. C2×#16864391C22semidirect
21. C2×#16864390C22semidirect
24. #22575451C22semidirect
25. #22575449C22semidirect
26. #22575442C22semidirect
30. #22575440C22semidirect
35. #22575363C22semidirect
36. C8×Dic269C22semidirect
62. #16864386D4semidirect
86. M16×C269D4semidirect
150. C269#3153semidirect
155. C2×AGL(1,ℤ/8)D269semidirect
228. M16C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754862
C2×#225754832
C2×#225754792
#317711512
C2×#225754402
#317709962
#317709932
#317709882
#317709862
M16×Dic2692
C2×AGL(1,ℤ/8)×C2692
#3153269