Finite Group Databasebeta

← #46820559

Subgroups — #46820559

Order 34432.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22575461C211yesmaximal, characteristic, complemented
2#31771183C211yesmaximal, characteristic, complemented
3#31771172C211yesmaximal, characteristic, complemented
4#31770855C212yesmaximal, complemented
5#31770848C212yesmaximal, complemented
6#31770838C212yesmaximal, complemented
7#31770831C212yesmaximal, complemented
8#31770647C211yesmaximal, characteristic, complemented
9#31770601C211yesmaximal, characteristic, complemented
10C2×#22575374C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13#22575461—24no—
14C2×#16864385C2212yescomplemented
15C22×#14524130C2211yescharacteristic
16C2×C8×D269C2212yescomplemented
17#22575449—21no—
18#22575440—21no—
19#22575432—21no—
20#22575429C2212yescomplemented
21#22575427—21no—
22#22575425—22no—
23#22575424C2212yes—
24#22575424C2212yes—
25#22575424C2212yescomplemented
26#22575422—22no—
27#22575412C2211yescharacteristic
28C2×C4⋊Dic269C2211yescharacteristic
29#22575387C2212yescomplemented
30#22575385C2212yescomplemented
31#22575377C2211yescharacteristic, complemented
32#22575377C2211yescharacteristic, complemented
33#22575374C2212yes—
34#22575374C2212yes—
35#22575374C2211yescharacteristic, complemented
36#22575374C2211yescharacteristic, complemented
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×Dic538D411yescharacteristic
46C2×D1076D411yescharacteristic
47C2×C22⋊D269—22no—
48C2×C22⋊D269—22no—
49C2×C22⋊D269—22no—
50C2×C22⋊D269D412yes—
51C22×Dic269—21no—
52C2×C4×D269C2312yes—
53#16864385—24no—
54C2×#14524130—22no—
55C2×#14524130C2311yescharacteristic
56C2×#14524130C2311yescharacteristic
57C8×D269—24no—
58C4⋊Dic269—21no—
59C4⋊Dic269C2312yes—
60C4⋊Dic269C2311yescharacteristic
61C4⋊Dic269C2311yescharacteristic
62#16864381—22no—
63C22⋊Dic269—22no—
64C22⋊Dic269—22no—
65C22⋊Dic269—22no—
66C22⋊Dic269—21no—
67#16864379—22no—
68C4×Dic269—21no—
69C24×C269—21no—
70#41×C269—22no—
71#41×C269—22no—
72C2×Q8×C269C2311yescharacteristic
73C2×D4×C269—21no—
74C2×D4×C269—22no—
75C2×D4×C269—21no—
76C2×D4×C269—22no—
77C2×D4×C269C2312yes—
78C2×D4×C269C2311yescharacteristic
79C22×C4×C269—21no—
80C22×C4×C269C2311yescharacteristic
81SD16×C269—24no—
82D8×C269—24no—
83M16×C269D414yescomplemented
84C2×C8×C269C2312yes—
85C22×D269C2×D412yes—
86Dic538—22no—
87D1076—22no—
88C22⋊D269—24no—
89C22⋊D269—82no—
90C22⋊D269—82no—
91C22⋊D269—82no—
92C2×Dic269—21no—
93C2×Dic269—21no—
94C2×Dic269—22no—
95C2×Dic269—81no—
96C2×Dic269C2×D412yes—
97C4×D269—22no—
98C4×D269—22no—
99#14524130—21no—
100#14524130—21no—
101C23×C269—21no—
102C23×C269—22no—
103C23×C269—42no—
104C23×C269—81no—
105C23×C269—81no—
106C23×C269C2×D411yescharacteristic
107Q8×C269—21no—
108Q8×C269—21no—
109D4×C269—21no—
110D4×C269—22no—
111D4×C269—22no—
112D4×C269—21no—
113D4×C269—42no—
114D4×C269—42no—
115C2×C4×C269—21no—
116C2×C4×C269—42no—
117C2×C4×C269C2×D411yescharacteristic
118C2×C4×C269C2×D412yes—
119C2×C4×C269#4112yes—
120C2×C4×C269C2411yescharacteristic
121C8×C269C2×D414yes—
122C22×C269—21no—
123C22×C269—21no—
124C22×C269—41no—
125C22×C269—42no—
126C22×C269—42no—
127C22×C269—41no—
128C22×C269—41no—
129C22×C269—42no—
130C22×C269—84no—
131C22×C269#12112yes—
132C22×C269C22×D411yescharacteristic
133C2×D269—22no—
134C2×D269—22no—
135C2×D269—22no—
136C4×C269—41no—
137C4×C269#12112yes—
138C4×C269C22×D411yescharacteristic
139C4×C269C2×#4111yescharacteristic
140Dic269—22no—
141Dic269—42no—
142Dic269—41no—
143Dic269—41no—
144C2×C269—22no—
145C2×C269—81no—
146C2×C269—82no—
147C2×C269C2×#12111yescharacteristic
148C2×C269#57611yescharacteristic
149C2×C269#57711yescharacteristic
150D269—42no—
151C269#306211yescharacteristic, complemented, Sylow
152#3062—2691nomaximal, Sylow
153C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
154C2×#131—2691no—
155#537—2691no—
156#535—2691no—
157#469—2692no—
158#468—2692no—
159#466—2692no—
160#465—2692no—
161#428—2691no—
162#416—2691no—
163C2×#102—2691no—
164C2×#41—2691no—
165C2×#41C2×D26911yescharacteristic
166C22×D4C2×D26911yescharacteristic
167AGL(1,ℤ/8)—24no—
168C2×SD16C2×D26912yes—
169C2×D8C2×D26912yes—
170#131—5384no—
171C2×M16—2692no—
172C2×M16C2×D26911yescharacteristic
173C22×C8—2691no—
174C22×C8—2692no—
175#127—5381no—
176#124—5381no—
177C22⋊Q8—2692no—
178C22⋊Q8—5381no—
179#121—2692no—
180#121—2692no—
181#121—2692no—
182#121—5381no—
183#121—5382no—
184#120—5382no—
185#117—2691no—
186C2×C4⋊C4—2691no—
187#107—2692no—
188#106—2692no—
189#103—2691no—
190#103—2691no—
191#102—2692no—
192#102—2692no—
193#102—2691no—
194#102—2691no—
195C24—21no—
196#41—22no—
197#41—22no—
198#41—5382no—
199#41—5382no—
200C2×Q8—2691no—
201C2×Q8C22×D26911yescharacteristic
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—2692no—
207C2×D4—2691no—
208C2×D4—5382no—
209C2×D4—5382no—
210C2×D4—5382no—
211C2×D4C22×D26911yescharacteristic
212C2×D4C22×D26912yes—
213C22×C4—21no—
214C22×C4—2692no—
215C22×C4—5381no—
216C22×C4C22×D26911yescharacteristic
217SD16—24no—
218D8—24no—
219M16—5384no—
220M16C22⋊D26914yescomplemented
221C2×C8—2691no—
222C2×C8—2691no—
223C2×C8—5382no—
224C2×C8—5384no—
225C2×C8C22×D26912yes—
226C4⋊C4—2692no—
227C4⋊C4—2691no—
228C4⋊C4—2691no—
229C4⋊C4—5382no—
230C4⋊C4—5381no—
231C22⋊C4—5382no—
232C22⋊C4—5382no—
233C22⋊C4—5382no—
234C22⋊C4—5382no—
235C22⋊C4—5381no—
236C42—5381no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—81no—
241C23—81no—
242C23—2692no—
243C23D4×D26911yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—5382no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—21no—
251D4—42no—
252D4—42no—
253D4—5384no—
254D4—5382no—
255D4—21522no—
256D4—21522no—
257D4—21522no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—2692no—
261C2×C4—5382no—
262C2×C4—5382no—
263C2×C4—5381no—
264C2×C4—5381no—
265C2×C4—5382no—
266C2×C4—21521no—
267C2×C4D4×D26911yescharacteristic
268C2×C4C2×C22⋊D26912yes—
269C2×C4#1686440112yes—
270C2×C4C23×D26911yescharacteristic
271C8—5381no—
272C8—5381no—
273C8C2×C22⋊D26914yes—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—84no—
283C22—5382no—
284C22—5382no—
285C22—5382no—
286C22#2257542412yes—
287C22C2×D4×D26911yescharacteristic
288C4—41no—
289C4—5382no—
290C4—10762no—
291C4—10761no—
292C4—10761no—
293C4#2257542412yes—
294C4C22×C22⋊D26911yescharacteristic
295C4C2×#1686440111yescharacteristic
296C2—22no—
297C2—82no—
298C2—81no—
299C2—10762no—
300C2C2×#2257542411yescharacteristic
301C2#3177140911yescharacteristic
302C2#3177141511yescharacteristic
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#3062
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×#22575461C2semidirect
2. #31771183C2semidirect
3. #31771172C2semidirect
4. #31770855C2semidirect
5. #31770848C2semidirect
6. #31770838C2semidirect
7. #31770831C2semidirect
8. #31770647C2semidirect
9. #31770601C2semidirect
10. C2×#22575374C2semidirect
11. C2×AGL(1,ℤ/8)×C269C2semidirect
14. C2×#16864385C22semidirect
16. C2×C8×D269C22semidirect
20. #22575429C22semidirect
25. #22575424C22semidirect
29. #22575387C22semidirect
30. #22575385C22semidirect
31. #22575377C22semidirect
32. #22575377C22semidirect
35. #22575374C22semidirect
36. #22575374C22semidirect
83. M16×C269D4semidirect
151. C269#3062semidirect
153. C2×AGL(1,ℤ/8)D269semidirect
220. M16C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754612
#317711832
#317711722
#317708552
#317708482
#317708382
#317708312
#317706472
#317706012
C2×#225753742
C2×AGL(1,ℤ/8)×C2692
#3062269