Finite Group Databasebeta

← #46821361

Subgroups — #46821361

Order 34432.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22575481C211yesmaximal, characteristic, complemented
3C2×#22575475C211yesmaximal, characteristic, complemented
4C22×#16864391C211yesmaximal, characteristic, complemented
5#31771181C211yesmaximal, characteristic, complemented
6#31771174C211yesmaximal, characteristic, complemented
7#31770999C212yesmaximal, complemented
8#31770994C212yesmaximal, complemented
9#31770989C212yesmaximal, complemented
10#31770978C212yesmaximal, complemented
11#405×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C2×#16864402C2211yescharacteristic
14C22×Dic538C2211yescharacteristic
15C22×D1076C2211yescharacteristic
16#22575486—24no—
17#22575481—24no—
18#22575475—24no—
19C2×Dic1076C2212yescomplemented
20C2×Dic1076C2211yescharacteristic, complemented
21C2×#16864391—24no—
22C2×#16864391C2212yes—
23C2×#16864391C2212yes—
24C2×#16864391C2212yes—
25C2×#16864391C2212yescomplemented
26C2×#16864391C2211yescharacteristic, complemented
27C2×#16864391C2211yescharacteristic, complemented
28C2×D2152C2212yes—
29C2×D2152C2211yescharacteristic, complemented
30#22575453C2212yescomplemented
31#22575450C2212yes—
32#22575441C2212yes—
33#22575441C2212yescomplemented
34#22575437—22no—
35#22575433—22no—
36#22575428—22no—
37#22575423—22no—
38C2×M16×C269C2211yescharacteristic
39C22×C8×C269C2211yescharacteristic
40#117×C269C2211yescharacteristic
41#97×C269C2212yescomplemented
42C4×C8×C269C2212yescomplemented
43C23×D269—21no—
44#16864402—22no—
45#16864402—22no—
46#16864402—22no—
47#16864402—22no—
48C2×Dic538—21no—
49C2×Dic538—22no—
50C2×Dic538—22no—
51C2×Dic538C2311yescharacteristic
52C2×Dic538C2312yes—
53C2×Dic538C2311yescharacteristic
54C2×D1076—22no—
55C2×D1076—22no—
56C2×D1076—21no—
57C2×D1076—24no—
58C2×D1076C2312yes—
59C2×D1076C2311yescharacteristic
60C2×D1076C2311yescharacteristic
61C2×C22⋊D269—21no—
62C2×C22⋊D269—21no—
63C22×Dic269—21no—
64C2×C4×D269—21no—
65C2×C4×D269—21no—
66Dic1076—22no—
67Dic1076—24no—
68#16864391—24no—
69#16864391—24no—
70#16864391—24no—
71#16864391—22no—
72#16864391—22no—
73#16864391—24no—
74D2152—24no—
75D2152—22no—
76C4⋊Dic269—24no—
77#16864379—24no—
78#16864379—24no—
79C22×C4×C269C2311yescharacteristic
80M16×C269D414yescomplemented
81C2×C8×C269D414yescomplemented
82C2×C8×C269C2312yes—
83C2×C8×C269C2311yescharacteristic
84C2×C8×C269C2311yescharacteristic
85C4⋊C4×C269D412yes—
86C22⋊C4×C269D412yes—
87C42×C269C2312yes—
88C22×D269—21no—
89C22×D269—22no—
90C22×D269—21no—
91C22×D269—42no—
92C22×D269—82no—
93Dic538—22no—
94Dic538—21no—
95Dic538—22no—
96Dic538—21no—
97Dic538—21no—
98Dic538—21no—
99Dic538—42no—
100D1076—21no—
101D1076—22no—
102D1076—21no—
103D1076—21no—
104D1076—22no—
105D1076—21no—
106D1076—42no—
107D1076—84no—
108C22⋊D269—42no—
109C22⋊D269—42no—
110C2×Dic269—21no—
111C2×Dic269—22no—
112C2×Dic269—21no—
113C2×Dic269—42no—
114C4×D269—42no—
115C4×D269—42no—
116C23×C269C2×D411yescharacteristic
117C2×C4×C269C2×D414yes—
118C2×C4×C269C2×D411yescharacteristic
119C2×C4×C269C2×D412yes—
120C2×C4×C269C2×D412yes—
121C2×C4×C269C2411yescharacteristic
122C8×C269C2×D412yes—
123C8×C269C2×D412yes—
124C8×C269C2×D414yes—
125C22×C269—22no—
126C22×C269#12712yes—
127C22×C269C22×D411yescharacteristic
128C2×D269—41no—
129C2×D269—41no—
130C2×D269—42no—
131C2×D269—42no—
132C2×D269—44no—
133C2×D269—42no—
134C2×D269—84no—
135C4×C269—24no—
136C4×C269#12712yes—
137C4×C269C22×D411yescharacteristic
138C4×C269C22×D411yescharacteristic
139Dic269—41no—
140Dic269—41no—
141Dic269—42no—
142C2×C269—22no—
143C2×C269C2×#12711yescharacteristic
144C2×C269#57712yes—
145D269—81no—
146D269—81no—
147D269—82no—
148C269#315411yescharacteristic, complemented, Sylow
149#3154—2691nomaximal, Sylow
150C2×#137—2691no—
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#405D26911yescharacteristic, complemented
161C2×#41—2691no—
162C2×#41—2691no—
163C22×Q8—2691no—
164C22×D4—2691no—
165#137—5384no—
166AGL(1,ℤ/8)—5384no—
167#135—5384no—
168C2×Q16—2691no—
169C2×Q16—2692no—
170C2×SD16—2692no—
171C2×SD16—2692no—
172C2×SD16—2691no—
173C2×SD16—2691no—
174C2×SD16—2692no—
175C2×SD16—2692no—
176C2×SD16—5384no—
177C2×D8—2691no—
178C2×D8—2692no—
179C2×M16C2×D26911yescharacteristic
180C22×C8C2×D26911yescharacteristic
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#117C2×D26911yescharacteristic
190#97C2×D26912yescomplemented
191C4×C8C2×D26912yescomplemented
192C24—5381no—
193#41—5382no—
194#41—5382no—
195#41—5382no—
196#41—5382no—
197C2×Q8—2691no—
198C2×Q8—2692no—
199C2×Q8—2691no—
200C2×Q8—5381no—
201C2×Q8—5382no—
202C2×Q8—5382no—
203C2×D4—2691no—
204C2×D4—2692no—
205C2×D4—2691no—
206C2×D4—5381no—
207C2×D4—5381no—
208C2×D4—5382no—
209C2×D4—5382no—
210C2×D4—5381no—
211C2×D4—5384no—
212C22×C4—5381no—
213C22×C4—5381no—
214C22×C4—5381no—
215C22×C4C22×D26911yescharacteristic
216Q16—5382no—
217Q16—5384no—
218SD16—5384no—
219SD16—5384no—
220SD16—5382no—
221SD16—5382no—
222SD16—5384no—
223SD16—5384no—
224D8—5382no—
225D8—5384no—
226M16D107614yescomplemented
227C2×C8D107614yescomplemented
228C2×C8C22×D26911yescharacteristic
229C2×C8C22×D26912yes—
230C2×C8C22×D26911yescharacteristic
231C4⋊C4—5384no—
232C4⋊C4D107612yes—
233C22⋊C4—5384no—
234C22⋊C4—5384no—
235C22⋊C4D107612yes—
236C42C22×D26912yes—
237C23—5381no—
238C23—5382no—
239C23—5381no—
240C23—10762no—
241C23—21522no—
242C23C2×D107611yescharacteristic
243Q8—5382no—
244Q8—5381no—
245Q8—5382no—
246Q8—5381no—
247Q8—5381no—
248Q8—5381no—
249Q8—10762no—
250D4—5381no—
251D4—5382no—
252D4—5381no—
253D4—5381no—
254D4—5382no—
255D4—5381no—
256D4—10762no—
257D4—10762no—
258D4—10762no—
259D4—21524no—
260C2×C4—5381no—
261C2×C4—5382no—
262C2×C4—5381no—
263C2×C4—10762no—
264C2×C4—10762no—
265C2×C4—10762no—
266C2×C4C2×D107614yes—
267C2×C4C2×D107611yescharacteristic
268C2×C4C2×D107612yes—
269C2×C4C2×D107612yes—
270C2×C4C23×D26911yescharacteristic
271C8C2×D107612yes—
272C8C2×D107612yes—
273C8C2×D107614yes—
274C22—22no—
275C22—10761no—
276C22—10761no—
277C22—10762no—
278C22—10762no—
279C22—10764no—
280C22—10762no—
281C22—21524no—
282C22#2257545012yes—
283C22C22×D107611yescharacteristic
284C4—24no—
285C4—10761no—
286C4—10761no—
287C4—10762no—
288C4#2257545012yes—
289C4C22×D107611yescharacteristic
290C4C22×D107611yescharacteristic
291C2—22no—
292C2—21522no—
293C2—21521no—
294C2—21521no—
295C2C2×#2257545011yescharacteristic
296C2#3177141712yes—
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#405×C269C2
socleC22×C269C22×D4
hypercenter#405D269
hypocenterC269#3154
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#22575486C2semidirect
2. C2×#22575481C2semidirect
3. C2×#22575475C2semidirect
4. C22×#16864391C2semidirect
5. #31771181C2semidirect
6. #31771174C2semidirect
7. #31770999C2semidirect
8. #31770994C2semidirect
9. #31770989C2semidirect
10. #31770978C2semidirect
11. #405×C269C2semidirect
19. C2×Dic1076C22semidirect
20. C2×Dic1076C22semidirect
25. C2×#16864391C22semidirect
26. C2×#16864391C22semidirect
27. C2×#16864391C22semidirect
29. C2×D2152C22semidirect
30. #22575453C22semidirect
33. #22575441C22semidirect
41. #97×C269C22semidirect
42. C4×C8×C269C22semidirect
80. M16×C269D4semidirect
81. C2×C8×C269D4semidirect
148. C269#3154semidirect
160. #405D269semidirect
190. #97C2×D269semidirect
191. C4×C8C2×D269semidirect
226. M16D1076semidirect
227. C2×C8D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225754862
C2×#225754812
C2×#225754752
C22×#168643912
#317711812
#317711742
#317709992
#317709942
#317709892
#317709782
#405×C2692
#3154269