Finite Group Databasebeta

← #46821364

Subgroups — #46821364

Order 34432.

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×#22575481C211yesmaximal, characteristic, complemented
2C2×#22575473C211yesmaximal, characteristic, complemented
3C22×#16864389C211yesmaximal, characteristic, complemented
4#31771179C211yesmaximal, characteristic, complemented
5#31771169C211yesmaximal, characteristic, complemented
6#31770998C212yesmaximal, complemented
7#31770991C212yesmaximal, complemented
8#31770990C212yesmaximal, complemented
9#31770976C212yesmaximal, complemented
10#31770563C211yesmaximal, characteristic, complemented
11C2×#137×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C22×D1076C2211yescharacteristic
14#22575481—24no—
15#22575473—24no—
16C2×#16864392C2211yescharacteristic, complemented
17C2×#16864391C2212yes—
18C2×#16864390C2211yescharacteristic, complemented
19C2×#16864389—24no—
20C2×#16864389C2212yes—
21C2×#16864389C2212yes—
22C2×#16864389C2211yescharacteristic, complemented
23C2×D2152C2212yes—
24C2×#16864387C2211yescharacteristic, complemented
25C22×#14524130C2211yescharacteristic
26#22575451C2212yescomplemented
27#22575449C2212yes—
28#22575442C2212yes—
29#22575442C2212yescomplemented
30#22575436—22no—
31#22575429—22no—
32#22575427—22no—
33#22575421—22no—
34#22575412C2211yescharacteristic
35#22575363C2212yescomplemented
36C8×Dic269C2212yescomplemented
37C2×#41×C269C2211yescharacteristic
38C22×Q8×C269C2211yescharacteristic
39#137×C269—24no—
40C2×Q16×C269C2212yescomplemented
41C2×SD16×C269C2212yescomplemented
42C2×M16×C269C2211yescharacteristic
43C23×D269—21no—
44#16864402—22no—
45#16864402—22no—
46C2×Dic538C2311yescharacteristic
47C2×D1076—22no—
48C2×D1076—22no—
49C2×D1076—21no—
50C2×D1076C2312yes—
51C2×D1076C2311yescharacteristic
52C2×C22⋊D269—24no—
53C2×C22⋊D269—21no—
54C2×C4×D269—21no—
55#16864392—22no—
56#16864391—24no—
57#16864390—22no—
58#16864389—24no—
59#16864389—24no—
60#16864389—22no—
61D2152—24no—
62#16864387—22no—
63C2×#14524130D414yescomplemented
64C2×#14524130C2311yescharacteristic
65C2×#14524130C2311yescharacteristic
66C4⋊Dic269D412yes—
67#16864381—24no—
68C22⋊Dic269D412yes—
69#16864379—24no—
70#16864379—24no—
71C4×Dic269C2312yes—
72#41×C269—22no—
73#41×C269—22no—
74C2×Q8×C269—21no—
75C2×Q8×C269—22no—
76C2×Q8×C269—22no—
77C2×Q8×C269C2311yescharacteristic
78C2×Q8×C269C2312yes—
79C2×D4×C269—21no—
80C2×D4×C269C2311yescharacteristic
81C22×C4×C269—21no—
82C22×C4×C269—21no—
83C22×C4×C269C2311yescharacteristic
84Q16×C269—24no—
85SD16×C269—24no—
86M16×C269D414yescomplemented
87C2×C8×C269C2312yes—
88C22×D269—21no—
89C22×D269—22no—
90C22×D269—42no—
91C22×D269—82no—
92Dic538—21no—
93Dic538—21no—
94D1076—21no—
95D1076—21no—
96D1076—22no—
97D1076—22no—
98D1076—42no—
99C22⋊D269—42no—
100C22⋊D269—84no—
101C2×Dic269—21no—
102C2×Dic269C2×D414yes—
103C4×D269—42no—
104#14524130C2×D412yes—
105#14524130C2×D412yes—
106C23×C269—21no—
107C23×C269C2×D411yescharacteristic
108Q8×C269—22no—
109Q8×C269—21no—
110Q8×C269—22no—
111Q8×C269—21no—
112Q8×C269—42no—
113D4×C269—21no—
114D4×C269—21no—
115D4×C269—42no—
116C2×C4×C269—21no—
117C2×C4×C269—22no—
118C2×C4×C269—42no—
119C2×C4×C269—42no—
120C2×C4×C269C2×D411yescharacteristic
121C2×C4×C269C2×D412yes—
122C2×C4×C269C2×D412yes—
123C2×C4×C269C2411yescharacteristic
124C8×C269C2×D414yes—
125C22×C269—22no—
126C22×C269—41no—
127C22×C269—42no—
128C22×C269#12712yes—
129C22×C269C22×D411yescharacteristic
130C2×D269—41no—
131C2×D269—42no—
132C2×D269—44no—
133C2×D269—42no—
134C2×D269—84no—
135C4×C269—41no—
136C4×C269—42no—
137C4×C269#12712yes—
138C4×C269C22×D411yescharacteristic
139C4×C269C22×D411yescharacteristic
140Dic269—24no—
141Dic269—41no—
142C2×C269—22no—
143C2×C269—81no—
144C2×C269C2×#12711yescharacteristic
145C2×C269#57712yes—
146D269—81no—
147D269—82no—
148C269#315411yescharacteristic, complemented, Sylow
149#3154—2691nomaximal, Sylow
150C2×#137D26911yescharacteristic, complemented
151C2×AGL(1,ℤ/8)—2691no—
152C2×#135—2691no—
153C22×SD16—2691no—
154#536—2691no—
155#535—2691no—
156#497—2692no—
157#496—2692no—
158#495—2692no—
159#492—2692no—
160#405—2691no—
161C2×#41—2691no—
162C2×#41C2×D26911yescharacteristic
163C22×Q8C2×D26911yescharacteristic
164C22×D4—2691no—
165#137—24no—
166AGL(1,ℤ/8)—5384no—
167#135—5384no—
168C2×Q16—2691no—
169C2×Q16C2×D26912yescomplemented
170C2×SD16—2692no—
171C2×SD16—2692no—
172C2×SD16—2691no—
173C2×SD16—2691no—
174C2×SD16—2692no—
175C2×SD16—5384no—
176C2×SD16C2×D26912yescomplemented
177C2×D8—2691no—
178C2×D8—2692no—
179C2×M16C2×D26911yescharacteristic
180C22×C8—2691no—
181C4⋊Q8—2692no—
182#127—2692no—
183#124—2692no—
184#124—2692no—
185#123—5382no—
186C22⋊Q8—5382no—
187#121—5382no—
188#120—5382no—
189#117—2691no—
190#97—2692no—
191C4×C8—2692no—
192C24—5381no—
193#41—22no—
194#41—22no—
195#41—5382no—
196#41—5382no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—2691no—
201C2×Q8C22×D26911yescharacteristic
202C2×Q8C22×D26912yes—
203C2×D4—21no—
204C2×D4—2691no—
205C2×D4—2692no—
206C2×D4—5384no—
207C2×D4—5381no—
208C2×D4—5382no—
209C2×D4—5381no—
210C2×D4—5382no—
211C2×D4C22×D26911yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—5381no—
215C22×C4C22×D26911yescharacteristic
216Q16—24no—
217Q16—5382no—
218SD16—24no—
219SD16—5382no—
220SD16—5384no—
221SD16—5384no—
222SD16—5382no—
223SD16—5384no—
224D8—5382no—
225D8—5384no—
226M16C22⋊D26914yescomplemented
227C2×C8—2694no—
228C2×C8—2691no—
229C2×C8—2691no—
230C2×C8C22×D26912yes—
231C4⋊C4—2692no—
232C4⋊C4—5384no—
233C22⋊C4—2692no—
234C22⋊C4—5384no—
235C22⋊C4—5384no—
236C42—2692no—
237C23—21no—
238C23—5381no—
239C23—5382no—
240C23—10762no—
241C23—21522no—
242C23D4×D26911yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—5381no—
249Q8—5381no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—5381no—
254D4—5382no—
255D4—5381no—
256D4—5382no—
257D4—10762no—
258D4—10762no—
259D4—21524no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—2694no—
265C2×C4—5381no—
266C2×C4—10762no—
267C2×C4D4×D26911yescharacteristic
268C2×C4D4×D26912yes—
269C2×C4C2×C22⋊D26912yes—
270C2×C4C23×D26911yescharacteristic
271C8—2692no—
272C8—2692no—
273C8C2×C22⋊D26914yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—10761no—
278C22—10762no—
279C22—10764no—
280C22—10762no—
281C22—21524no—
282C22#2257544912yes—
283C22C2×D4×D26911yescharacteristic
284C4—41no—
285C4—42no—
286C4—5384no—
287C4—10761no—
288C4#2257544912yes—
289C4C2×D4×D26911yescharacteristic
290C4C22×C22⋊D26911yescharacteristic
291C2—22no—
292C2—81no—
293C2—21522no—
294C2—21521no—
295C2C2×#2257544911yescharacteristic
296C2#3177141412yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#137×C269C2
socleC22×C269C22×D4
hypercenterC2×#137D269
hypocenterC269#3154
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22575481C2semidirect
2. C2×#22575473C2semidirect
3. C22×#16864389C2semidirect
4. #31771179C2semidirect
5. #31771169C2semidirect
6. #31770998C2semidirect
7. #31770991C2semidirect
8. #31770990C2semidirect
9. #31770976C2semidirect
10. #31770563C2semidirect
11. C2×#137×C269C2semidirect
16. C2×#16864392C22semidirect
18. C2×#16864390C22semidirect
22. C2×#16864389C22semidirect
24. C2×#16864387C22semidirect
26. #22575451C22semidirect
29. #22575442C22semidirect
35. #22575363C22semidirect
36. C8×Dic269C22semidirect
40. C2×Q16×C269C22semidirect
41. C2×SD16×C269C22semidirect
63. C2×#14524130D4semidirect
86. M16×C269D4semidirect
148. C269#3154semidirect
150. C2×#137D269semidirect
169. C2×Q16C2×D269semidirect
176. C2×SD16C2×D269semidirect
226. M16C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754812
C2×#225754732
C22×#168643892
#317711792
#317711692
#317709982
#317709912
#317709902
#317709762
#317705632
C2×#137×C2692
#3154269