Finite Group Databasebeta

← #46820149

Subgroups — #46820149

Order 34432.

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

normal subgroups only

This group has 784 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×#22575498C211yesmaximal, characteristic, complemented
2C2×#22575479C211yesmaximal, characteristic, complemented
3#31771172C211yesmaximal, characteristic, complemented
4#31770747C214yesmaximal, complemented
5#31770737C214yesmaximal, complemented
6#31770610C211yesmaximal, characteristic, complemented
7#31770602C211yesmaximal, characteristic, complemented
8#31770571C211yesmaximal, characteristic, complemented
9C22×D8×C269C211yesmaximal, characteristic, complemented
10#22575498—24no—
11#22575498—24no—
12C2×#16864402C2211yescharacteristic
13C2×#16864402C2211yescharacteristic
14C2×#16864401—21no—
15C2×#16864401C2212yescomplemented
16C22×C22⋊D269—22no—
17C2×D4×D269—21no—
18C2×D4×D269C2212yescomplemented
19#22575479—24no—
20C2×#16864390C2212yescomplemented
21C2×#16864387C2212yescomplemented
22C2×#16864386C2211yescharacteristic, complemented
23#22575449—21no—
24#22575440—21no—
25#22575424C2214yescomplemented
26#22575422—22no—
27#22575412C2211yescharacteristic, complemented
28#22575378C2212yescomplemented
29#22575375C2212yescomplemented
30#22575374C2214yescomplemented
31#22575365C2214yescomplemented
32C22×D4×C269C2211yescharacteristic
33C22×D4×C269C2211yescharacteristic, complemented
34C2×D8×C269—24no—
35C2×D8×C269C2214yescomplemented
36C22×C8×C269C2211yescharacteristic, complemented
37C23×D269—22no—
38#16864402—22no—
39#16864402—22no—
40#16864402—22no—
41#16864402D414yescomplemented
42#16864401—28no—
43#16864401—24no—
44#16864401—24no—
45C2×Dic538D411yescharacteristic
46C2×Dic538C2311yescharacteristic
47C2×D1076D411yescharacteristic
48C2×D1076C2311yescharacteristic
49C2×C22⋊D269—216no—
50C2×C22⋊D269—22no—
51C2×C22⋊D269—24no—
52C2×C22⋊D269—22no—
53C2×C22⋊D269—21no—
54C2×C22⋊D269D412yes—
55D4×D269—28no—
56D4×D269—24no—
57D4×D269—24no—
58C22×Dic269—22no—
59C2×C4×D269—21no—
60C2×C4×D269C2312yes—
61#16864390—24no—
62#16864387—24no—
63#16864386—22no—
64C2×#14524130C2312yes—
65C4⋊Dic269C2312yes—
66C22⋊Dic269—24no—
67C22⋊Dic269—21no—
68C4×Dic269—21no—
69C24×C269—21no—
70C24×C269—21no—
71C2×D4×C269—21no—
72C2×D4×C269—22no—
73C2×D4×C269—22no—
74C2×D4×C269—22no—
75C2×D4×C269—21no—
76C2×D4×C269D414yescomplemented
77C2×D4×C269C2312yes—
78C2×D4×C269C2312yes—
79C22×C4×C269C2311yescharacteristic
80D8×C269—28no—
81C2×C8×C269—22no—
82C2×C8×C269C2312yes—
83C22×D269—21no—
84C22×D269—28no—
85C22×D269—84no—
86C22×D269C2×D412yes—
87Dic538—22no—
88Dic538—21no—
89Dic538C2×D412yes—
90D1076—22no—
91D1076—21no—
92D1076C2×D412yes—
93C22⋊D269—28no—
94C22⋊D269—28no—
95C22⋊D269—24no—
96C22⋊D269—24no—
97C22⋊D269—84no—
98C2×Dic269—28no—
99C2×Dic269—21no—
100C2×Dic269—22no—
101C2×Dic269C2×D412yes—
102C4×D269—24no—
103C4×D269—22no—
104C4×D269—22no—
105#14524130—22no—
106C23×C269—22no—
107C23×C269—24no—
108C23×C269—22no—
109C23×C269—42no—
110C23×C269—82no—
111C23×C269—82no—
112C23×C269C2×D411yescharacteristic
113D4×C269—24no—
114D4×C269—22no—
115D4×C269—22no—
116D4×C269—22no—
117D4×C269—42no—
118D4×C269C2×D414yes—
119C2×C4×C269C2×D411yescharacteristic
120C2×C4×C269C2×D412yes—
121C2×C4×C269C2×D412yes—
122C2×C4×C269C2411yescharacteristic
123C8×C269—22no—
124C22×C269—22no—
125C22×C269—24no—
126C22×C269—44no—
127C22×C269—42no—
128C22×C269—44no—
129C22×C269—84no—
130C22×C269—84no—
131C22×C269#12012yes—
132C22×C269C22×D411yescharacteristic
133C2×D269—22no—
134C2×D269—22no—
135C2×D269—24no—
136C2×D269—42no—
137C2×D269—88no—
138C4×C269#12012yes—
139C4×C269C22×D411yescharacteristic
140C4×C269C22×D411yescharacteristic
141Dic269—22no—
142Dic269—22no—
143Dic269—42no—
144C2×C269—22no—
145C2×C269—44no—
146C2×C269—82no—
147C2×C269C2×#12011yescharacteristic
148C2×C269#57612yes—
149D269—42no—
150D269—42no—
151C269#300811yescharacteristic, complemented, Sylow
152#3008—2691nomaximal, Sylow
153C2×H(5,𝔽2)—2691no—
154C2×AGL(1,ℤ/8)—2691no—
155C22×D8D26911yescharacteristic, complemented
156#535—2691no—
157#449—2694no—
158#447—2694no—
159#418—2691no—
160#416—2691no—
161#408—2691no—
162H(5,𝔽2)—5384no—
163H(5,𝔽2)—5384no—
164C2×#41—2691no—
165C2×#41—2692no—
166C2×#41—2691no—
167C2×#41—5381no—
168C22×D4—2692no—
169C22×D4—5381no—
170C22×D4—5382no—
171C22×D4C2×D26911yescharacteristic
172C22×D4C2×D26911yescharacteristic, complemented
173AGL(1,ℤ/8)—5384no—
174C2×SD16—2692no—
175C2×D8—24no—
176C2×D8—2692no—
177C2×D8C2×D26914yescomplemented
178C2×M16—2691no—
179C22×C8C2×D26911yescharacteristic, complemented
180#127—5381no—
181#124—5381no—
182#121—2694no—
183#120—5382no—
184#117—2691no—
185#103—2692no—
186#102—2694no—
187#102—2692no—
188C22⋊C8—2694no—
189C24—21no—
190C24—21no—
191C24—5382no—
192#41—2694no—
193#41—5384no—
194#41—5388no—
195#41—5384no—
196#41—5382no—
197#41—5382no—
198#41—5382no—
199C2×Q8—2691no—
200C2×Q8—2691no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—2692no—
207C2×D4—2691no—
208C2×D4—2691no—
209C2×D4—5382no—
210C2×D4—5384no—
211C2×D4—5388no—
212C2×D4—5382no—
213C2×D4—53816no—
214C2×D4—5381no—
215C2×D4—5384no—
216C2×D4—5384no—
217C2×D4C22⋊D26914yescomplemented
218C2×D4C22×D26912yes—
219C2×D4C22×D26912yes—
220C22×C4—2692no—
221C22×C4—5382no—
222C22×C4—5381no—
223C22×C4C22×D26911yescharacteristic
224SD16—5384no—
225D8—28no—
226D8—5384no—
227M16—5382no—
228C2×C8—22no—
229C2×C8—2692no—
230C2×C8C22×D26912yes—
231C4⋊C4—2692no—
232C22⋊C4—5384no—
233C22⋊C4—5381no—
234C42—5381no—
235C23—22no—
236C23—24no—
237C23—22no—
238C23—42no—
239C23—82no—
240C23—82no—
241C23—2692no—
242C23—5388no—
243C23—5381no—
244C23—21524no—
245C23D4×D26911yescharacteristic
246Q8—2692no—
247Q8—5382no—
248Q8—5381no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—2692no—
255D4—5388no—
256D4—5388no—
257D4—5384no—
258D4—5384no—
259D4—5382no—
260D4—5381no—
261D4—21524no—
262D4C2×C22⋊D26914yes—
263C2×C4—2692no—
264C2×C4—5388no—
265C2×C4—5381no—
266C2×C4—5384no—
267C2×C4—5382no—
268C2×C4—5382no—
269C2×C4—5382no—
270C2×C4D4×D26911yescharacteristic
271C2×C4D4×D26912yes—
272C2×C4C2×C22⋊D26912yes—
273C2×C4C23×D26911yescharacteristic
274C8—22no—
275C8—5382no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—44no—
280C22—42no—
281C22—84no—
282C22—84no—
283C22—5384no—
284C22—5382no—
285C22—5382no—
286C22—10762no—
287C22—21528no—
288C22#2257542112yes—
289C22C2×D4×D26911yescharacteristic
290C4—5382no—
291C4—5382no—
292C4—10762no—
293C4#2257542112yes—
294C4C2×D4×D26911yescharacteristic
295C4C22×C22⋊D26911yescharacteristic
296C2—22no—
297C2—44no—
298C2—82no—
299C2—10762no—
300C2—10762no—
301C2C2×#2257542111yescharacteristic
302C2#3177140612yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC22×D8×C269C2
socleC22×C269C22×D4
hypercenterC22×D8D269
hypocenterC269#3008
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#22575498C2semidirect
2. C2×#22575479C2semidirect
3. #31771172C2semidirect
4. #31770747C2semidirect
5. #31770737C2semidirect
6. #31770610C2semidirect
7. #31770602C2semidirect
8. #31770571C2semidirect
9. C22×D8×C269C2semidirect
15. C2×#16864401C22semidirect
18. C2×D4×D269C22semidirect
20. C2×#16864390C22semidirect
21. C2×#16864387C22semidirect
22. C2×#16864386C22semidirect
25. #22575424C22semidirect
27. #22575412C22semidirect
28. #22575378C22semidirect
29. #22575375C22semidirect
30. #22575374C22semidirect
31. #22575365C22semidirect
33. C22×D4×C269C22semidirect
35. C2×D8×C269C22semidirect
36. C22×C8×C269C22semidirect
41. #16864402D4semidirect
76. C2×D4×C269D4semidirect
151. C269#3008semidirect
155. C22×D8D269semidirect
172. C22×D4C2×D269semidirect
177. C2×D8C2×D269semidirect
179. C22×C8C2×D269semidirect
217. C2×D4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754982
C2×#225754792
#317711722
#317707472
#317707372
#317706102
#317706022
#317705712
C22×D8×C2692
#3008269