Finite Group Databasebeta

← #46820161

Subgroups — #46820161

Order 34432.

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

normal subgroups only

This group has 752 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×#22575499C211yesmaximal, characteristic
2C22×#16864389C211yesmaximal, characteristic, complemented
3#31771179C211yesmaximal, characteristic, complemented
4#31770766C214yesmaximal, complemented
5#31770754C214yesmaximal, complemented
6#31770611C211yesmaximal, characteristic, complemented
7#31770601C211yesmaximal, characteristic, complemented
8#31770573C211yesmaximal, characteristic, complemented
9C2×#137×C269C211yesmaximal, characteristic, complemented
10#22575499—24no—
11#22575499—24no—
12C2×#16864403C2212yes—
13C2×#16864402—21no—
14C2×#16864402C2211yescharacteristic
15C22×D1076—21no—
16C22×D1076C2211yescharacteristic
17C2×D4×D269—22no—
18C2×D4×D269C2212yes—
19C2×#16864389—24no—
20C2×#16864389C2214yescomplemented
21C22×#14524130C2211yescharacteristic, complemented
22#22575451—21no—
23#22575442—21no—
24#22575436—22no—
25#22575429C2214yescomplemented
26#22575412C2211yescharacteristic, complemented
27#22575377C2212yescomplemented
28#22575375C2214yescomplemented
29#22575374C2212yescomplemented
30#22575365C2214yescomplemented
31C2×#41×C269C2211yescharacteristic
32C22×Q8×C269C2211yescharacteristic, complemented
33#137×C269—24no—
34C2×Q16×C269C2212yescomplemented
35C2×SD16×C269C2212yescomplemented
36C2×M16×C269C2211yescharacteristic, complemented
37C23×D269—22no—
38C23×D269—21no—
39#16864403—24no—
40#16864403—24no—
41#16864402—28no—
42#16864402—22no—
43#16864402—22no—
44C2×Dic538D411yescharacteristic
45C2×D1076—28no—
46C2×D1076—22no—
47C2×D1076—21no—
48C2×D1076—22no—
49C2×D1076D411yescharacteristic
50C2×D1076D414yes—
51C2×D1076C2312yes—
52C2×C22⋊D269—22no—
53C2×C22⋊D269D412yes—
54D4×D269—216no—
55D4×D269—24no—
56D4×D269—24no—
57C2×C4×D269—22no—
58C2×C4×D269C2312yes—
59#16864389—28no—
60C2×#14524130—22no—
61C2×#14524130C2312yes—
62C4⋊Dic269C2312yes—
63#16864381—24no—
64C22⋊Dic269—21no—
65#16864379—24no—
66C4×Dic269—21no—
67#41×C269—22no—
68#41×C269D414yes—
69C2×Q8×C269—21no—
70C2×Q8×C269—22no—
71C2×Q8×C269—22no—
72C2×Q8×C269C2312yes—
73C2×Q8×C269C2311yescharacteristic
74C2×D4×C269—21no—
75C2×D4×C269C2311yescharacteristic
76C22×C4×C269—21no—
77C22×C4×C269—21no—
78C22×C4×C269C2311yescharacteristic
79Q16×C269—24no—
80SD16×C269—24no—
81M16×C269—22no—
82C2×C8×C269C2312yes—
83C22×D269—22no—
84C22×D269—28no—
85C22×D269—24no—
86C22×D269—84no—
87C22×D269—82no—
88C22×D269C2×D412yes—
89Dic538—22no—
90D1076—28no—
91D1076—22no—
92D1076—24no—
93D1076—22no—
94D1076C2×D414yes—
95C22⋊D269—28no—
96C22⋊D269—24no—
97C2×Dic269—22no—
98C2×Dic269C2×D412yes—
99C4×D269—28no—
100C4×D269—22no—
101C4×D269—22no—
102#14524130—22no—
103C23×C269—21no—
104C23×C269C2×D411yescharacteristic
105Q8×C269—21no—
106Q8×C269—22no—
107Q8×C269—22no—
108Q8×C269—42no—
109Q8×C269C2×D412yes—
110D4×C269—24no—
111D4×C269—21no—
112D4×C269C2×D412yes—
113C2×C4×C269—21no—
114C2×C4×C269—24no—
115C2×C4×C269—22no—
116C2×C4×C269—42no—
117C2×C4×C269C2×D411yescharacteristic
118C2×C4×C269C2×D412yes—
119C2×C4×C269C2×D412yes—
120C2×C4×C269C2411yescharacteristic
121C8×C269—22no—
122C22×C269—22no—
123C22×C269—22no—
124C22×C269—42no—
125C22×C269#12012yes—
126C22×C269C22×D411yescharacteristic
127C2×D269—22no—
128C2×D269—24no—
129C2×D269—24no—
130C2×D269—44no—
131C2×D269—84no—
132C2×D269—88no—
133C4×C269—22no—
134C4×C269—42no—
135C4×C269#12012yes—
136C4×C269C22×D411yescharacteristic
137C4×C269C22×D411yescharacteristic
138Dic269—22no—
139Dic269—42no—
140C2×C269—22no—
141C2×C269—42no—
142C2×C269C2×#12011yescharacteristic
143C2×C269#57712yes—
144D269—42no—
145D269—44no—
146C269#300911yescharacteristic, complemented, Sylow
147#3009—2691nomaximal, Sylow
148C2×H(5,𝔽2)—2691no—
149C2×#137D26911yescharacteristic, complemented
150C22×SD16—2691no—
151#536—2691no—
152#452—2694no—
153#450—2694no—
154#418—2691no—
155#416—2691no—
156#408—2691no—
157H(5,𝔽2)—5384no—
158H(5,𝔽2)—5384no—
159C2×#41—2691no—
160C2×#41—2692no—
161C2×#41—5381no—
162C2×#41C2×D26911yescharacteristic
163C22×Q8C2×D26911yescharacteristic, complemented
164C22×D4—2692no—
165C22×D4—2691no—
166C22×D4—5381no—
167C22×D4—5382no—
168#137—24no—
169C2×Q16C2×D26912yescomplemented
170C2×SD16—2694no—
171C2×SD16—5384no—
172C2×SD16C2×D26912yescomplemented
173C2×M16C2×D26911yescharacteristic, complemented
174C22×C8—2691no—
175C4⋊Q8—5381no—
176#124—5381no—
177#123—5382no—
178C22⋊Q8—2694no—
179#117—2691no—
180#103—2692no—
181#102—2692no—
182#102—2694no—
183C22⋊C8—2694no—
184C24—5382no—
185C24—5381no—
186#41—22no—
187#41—5388no—
188#41—5384no—
189#41—5384no—
190#41—5382no—
191#41—5382no—
192#41C22⋊D26914yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—2691no—
197C2×Q8C22×D26912yes—
198C2×Q8C22×D26911yescharacteristic
199C2×D4—21no—
200C2×D4—2692no—
201C2×D4—2694no—
202C2×D4—2691no—
203C2×D4—2692no—
204C2×D4—5382no—
205C2×D4—5388no—
206C2×D4—5382no—
207C2×D4—5384no—
208C2×D4—53816no—
209C2×D4—5384no—
210C2×D4—5381no—
211C2×D4—5382no—
212C2×D4C22×D26911yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2692no—
216C22×C4—5382no—
217C22×C4C22×D26911yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—5388no—
221M16—22no—
222C2×C8—2692no—
223C2×C8—5382no—
224C2×C8C22×D26912yes—
225C4⋊C4—2692no—
226C4⋊C4—5384no—
227C22⋊C4—5384no—
228C22⋊C4—5381no—
229C42—5381no—
230C23—21no—
231C23—2692no—
232C23—5388no—
233C23—5382no—
234C23—5384no—
235C23—21522no—
236C23—21524no—
237C23D4×D26911yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—5382no—
243Q8C2×C22⋊D26912yes—
244D4—24no—
245D4—21no—
246D4—2694no—
247D4—5388no—
248D4—5388no—
249D4—5384no—
250D4—5384no—
251D4—5382no—
252D4—5382no—
253D4C2×C22⋊D26912yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—2692no—
259C2×C4—5388no—
260C2×C4—5382no—
261C2×C4—5382no—
262C2×C4—5382no—
263C2×C4D4×D26911yescharacteristic
264C2×C4D4×D26912yes—
265C2×C4C2×C22⋊D26912yes—
266C2×C4C23×D26911yescharacteristic
267C8—22no—
268C8—5382no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—5384no—
273C22—5384no—
274C22—5382no—
275C22—10764no—
276C22—21524no—
277C22—21528no—
278C22#2257542112yes—
279C22C2×D4×D26911yescharacteristic
280C4—22no—
281C4—42no—
282C4—5382no—
283C4—10762no—
284C4#2257542112yes—
285C4C2×D4×D26911yescharacteristic
286C4C22×C22⋊D26911yescharacteristic
287C2—22no—
288C2—42no—
289C2—10762no—
290C2—10764no—
291C2C2×#2257542111yescharacteristic
292C2#3177141412yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#137×C269C2
socleC22×C269C22×D4
prime radicalC2×#22575499C2
hypercenterC2×#137D269
hypocenterC269#3009
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
2. C22×#16864389C2semidirect
3. #31771179C2semidirect
4. #31770766C2semidirect
5. #31770754C2semidirect
6. #31770611C2semidirect
7. #31770601C2semidirect
8. #31770573C2semidirect
9. C2×#137×C269C2semidirect
20. C2×#16864389C22semidirect
21. C22×#14524130C22semidirect
25. #22575429C22semidirect
26. #22575412C22semidirect
27. #22575377C22semidirect
28. #22575375C22semidirect
29. #22575374C22semidirect
30. #22575365C22semidirect
32. C22×Q8×C269C22semidirect
34. C2×Q16×C269C22semidirect
35. C2×SD16×C269C22semidirect
36. C2×M16×C269C22semidirect
146. C269#3009semidirect
149. C2×#137D269semidirect
163. C22×Q8C2×D269semidirect
169. C2×Q16C2×D269semidirect
172. C2×SD16C2×D269semidirect
173. C2×M16C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
C22×#168643892
#317711792
#317707662
#317707542
#317706112
#317706012
#317705732
C2×#137×C2692
#3009269