Finite Group Databasebeta

← #46820807

Subgroups — #46820807

Order 34432.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#31771115C211yesmaximal, characteristic
2C2×C4⋊C4×D269C211yesmaximal, characteristic, complemented
3#31770941C211yesmaximal, characteristic, complemented
4#31770940C211yesmaximal, characteristic, complemented
5#31770937C212yesmaximal, complemented
6#31770921C211yesmaximal, characteristic, complemented
7#31770917C212yesmaximal, complemented
8#31770644C212yesmaximal, complemented
9#31770603C212yesmaximal, complemented
10#31770567C211yesmaximal, characteristic, complemented
11#481×C269C211yesmaximal, characteristic, complemented
12C2×#16864401C2211yescharacteristic
13C22×C4×D269—21no—
14C22×C4×D269C2211yescharacteristic
15C2×#16864385C2212yes—
16#22575435—21no—
17#22575433C2211yescharacteristic
18#22575432C2211yescharacteristic
19#22575424C2211yescharacteristic
20#22575421—21no—
21D4×Dic269—21no—
22#22575413—21no—
23#22575411C2211yescharacteristic
24C2×C4⋊Dic269C2211yescharacteristic
25C2×#16864381—21no—
26C4⋊C4×D269—21no—
27C4⋊C4×D269—21no—
28C4⋊C4×D269C2212yes—
29C4⋊C4×D269C2212yes—
30#22575387C2212yescomplemented
31#22575386C2212yescomplemented
32#22575384C2212yescomplemented
33#22575378C2212yescomplemented
34#22575376C2212yescomplemented
35#22575374C2212yescomplemented
36C22⋊#14524130C2211yescharacteristic
37#22575365C2212yes—
38#121×C269C2211yescharacteristic
39C2×C4⋊C4×C269C2211yescharacteristic
40#106×C269C2212yescomplemented
41#102×C269C2212yescomplemented
42C22⋊C8×C269C2211yescharacteristic
43C23×D269D411yescharacteristic
44#16864401—24no—
45C2×Dic538C2311yescharacteristic
46C2×C22⋊D269—21no—
47C2×C22⋊D269—21no—
48C22×Dic269—21no—
49C22×Dic269—21no—
50C22×Dic269D411yescharacteristic
51C2×C4×D269—24no—
52C2×C4×D269—22no—
53C2×C4×D269—22no—
54C2×C4×D269—44no—
55C2×C4×D269D411yescharacteristic
56C2×C4×D269D411yescharacteristic
57C2×C4×D269C2311yescharacteristic
58C2×C4×D269C2311yescharacteristic
59#16864385—24no—
60C2×#14524130C2312yes—
61C4⋊Dic269—21no—
62C4⋊Dic269—21no—
63C4⋊Dic269C2312yes—
64C4⋊Dic269C2311yescharacteristic
65#16864381—22no—
66#16864381—22no—
67#16864381—21no—
68C22⋊Dic269—21no—
69C22⋊Dic269—21no—
70#16864379—21no—
71#16864379—21no—
72C4×Dic269—21no—
73C2×D4×C269—21no—
74C2×D4×C269C2311yescharacteristic
75C22×C4×C269—21no—
76C22×C4×C269C2311yescharacteristic
77C2×C8×C269C2312yes—
78C4⋊C4×C269—21no—
79C4⋊C4×C269C2312yes—
80C4⋊C4×C269C2311yescharacteristic
81C22⋊C4×C269—21no—
82C22×D269—22no—
83C22×D269—22no—
84C22×D269—22no—
85C22×D269C2×D411yescharacteristic
86C22×D269C2×D411yescharacteristic
87Dic538—22no—
88C22⋊D269—42no—
89C22⋊D269—42no—
90C2×Dic269—22no—
91C2×Dic269—21no—
92C2×Dic269—21no—
93C2×Dic269—22no—
94C2×Dic269—42no—
95C2×Dic269—42no—
96C2×Dic269C2×D411yescharacteristic
97C2×Dic269C2×D411yescharacteristic
98C4×D269—22no—
99C4×D269—22no—
100C4×D269—22no—
101C4×D269—44no—
102C4×D269—44no—
103C4×D269#4114yes—
104#14524130—22no—
105C23×C269—21no—
106C23×C269C2×D411yescharacteristic
107D4×C269—22no—
108D4×C269—42no—
109C2×C4×C269—22no—
110C2×C4×C269—21no—
111C2×C4×C269—22no—
112C2×C4×C269—42no—
113C2×C4×C269C2×D411yescharacteristic
114C2×C4×C269C2411yescharacteristic
115C8×C269—22no—
116C22×C269—21no—
117C22×C269—21no—
118C22×C269—41no—
119C22×C269—41no—
120C22×C269—41no—
121C22×C269#13712yes—
122C22×C269C22×D411yescharacteristic
123C2×D269—28no—
124C2×D269—21no—
125C2×D269—21no—
126C2×D269—21no—
127C2×D269—21no—
128C2×D269#12312yes—
129C2×D269AGL(1,ℤ/8)12yes—
130C4×C269—21no—
131C4×C269—42no—
132C4×C269—41no—
133C4×C269C2×#4112yes—
134Dic269—21no—
135Dic269—41no—
136Dic269—42no—
137Dic269—41no—
138Dic269#12312yes—
139C2×C269—22no—
140C2×C269—81no—
141C2×C269C2×#12311yescharacteristic
142C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C269C2×#13711yescharacteristic
144D269—22no—
145D269—22no—
146C269#309211yescharacteristic, complemented, Sylow
147#3092—2691nomaximal, Sylow
148#525—2691no—
149C22×C4⋊C4—2691no—
150#484—2691no—
151#484—2691no—
152#483—2692no—
153#481—2691no—
154#481D26911yescharacteristic, complemented
155#480—2692no—
156#428—2692no—
157#417—2692no—
158#407—2691no—
159C2×#41—2691no—
160C23×C4—2691no—
161C23×C4—5381no—
162C2×M16—2692no—
163#123—5381no—
164C22⋊Q8—2691no—
165C22⋊Q8—2691no—
166#121—2691no—
167#121C2×D26911yescharacteristic
168#120—5381no—
169C4×D4—5381no—
170C4×D4—5381no—
171#117—2691no—
172C2×C4⋊C4—2691no—
173C2×C4⋊C4—2692no—
174C2×C4⋊C4—2692no—
175C2×C4⋊C4—5381no—
176C2×C4⋊C4—5381no—
177C2×C4⋊C4—5381no—
178C2×C4⋊C4C2×D26911yescharacteristic
179#107—2692no—
180#107—2692no—
181#106—2692no—
182#106C2×D26912yescomplemented
183#103—2692no—
184#103—2692no—
185#102—2692no—
186#102C2×D26912yescomplemented
187C22⋊C8—2691no—
188C22⋊C8—2692no—
189C22⋊C8C2×D26911yescharacteristic
190C24—2691no—
191#41—5384no—
192C2×Q8—2691no—
193C2×D4—21no—
194C2×D4—5381no—
195C2×D4—5381no—
196C2×D4C22×D26911yescharacteristic
197C22×C4—21no—
198C22×C4—2691no—
199C22×C4—2691no—
200C22×C4—2691no—
201C22×C4—2691no—
202C22×C4—2691no—
203C22×C4—5384no—
204C22×C4—5382no—
205C22×C4—5382no—
206C22×C4—5381no—
207C22×C4—5381no—
208C22×C4—10764no—
209C22×C4C22×D26911yescharacteristic
210M16—5384no—
211C2×C8—2692no—
212C2×C8C22×D26912yes—
213C4⋊C4—21no—
214C4⋊C4—2692no—
215C4⋊C4—2691no—
216C4⋊C4—5382no—
217C4⋊C4—5382no—
218C4⋊C4—5381no—
219C4⋊C4—5381no—
220C4⋊C4—5381no—
221C4⋊C4C22×D26912yes—
222C4⋊C4C22×D26911yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—5381no—
225C22⋊C4—5381no—
226C22⋊C4—5381no—
227C22⋊C4—5381no—
228C42—5381no—
229C23—21no—
230C23—2691no—
231C23—2691no—
232C23—5382no—
233C23—5382no—
234C23—5382no—
235C23D4×D26911yescharacteristic
236Q8—5382no—
237D4—22no—
238D4—42no—
239D4—10762no—
240D4—10762no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—2691no—
246C2×C4—2694no—
247C2×C4—2691no—
248C2×C4—5382no—
249C2×C4—5382no—
250C2×C4—5382no—
251C2×C4—5382no—
252C2×C4—5381no—
253C2×C4—5381no—
254C2×C4—5382no—
255C2×C4—10764no—
256C2×C4—10764no—
257C2×C4—10762no—
258C2×C4—10762no—
259C2×C4D4×D26911yescharacteristic
260C2×C4C23×D26911yescharacteristic
261C8—22no—
262C8—5382no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—2692no—
269C22—2692no—
270C22—5388no—
271C22—5381no—
272C22—5381no—
273C22—5381no—
274C22—5381no—
275C22#2257548212yes—
276C22C2×D4×D26911yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—2692no—
281C4—5381no—
282C4—10761no—
283C4—10762no—
284C4—10761no—
285C4#41×D26912yes—
286C2—22no—
287C2—81no—
288C2—5382no—
289C2—5382no—
290C2#123×D26911yescharacteristic
291C2AGL(1,ℤ/8)×D26911yescharacteristic
292C2C2×#2257548211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#481×C269C2
socleC22×C269C22×D4
prime radical#31771115C2
hypercenter#481D269
hypocenterC269#3092
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D269C2semidirect
3. #31770941C2semidirect
4. #31770940C2semidirect
5. #31770937C2semidirect
6. #31770921C2semidirect
7. #31770917C2semidirect
8. #31770644C2semidirect
9. #31770603C2semidirect
10. #31770567C2semidirect
11. #481×C269C2semidirect
30. #22575387C22semidirect
31. #22575386C22semidirect
32. #22575384C22semidirect
33. #22575378C22semidirect
34. #22575376C22semidirect
35. #22575374C22semidirect
40. #106×C269C22semidirect
41. #102×C269C22semidirect
146. C269#3092semidirect
154. #481D269semidirect
182. #106C2×D269semidirect
186. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317711152
C2×C4⋊C4×D2692
#317709412
#317709402
#317709372
#317709212
#317709172
#317706442
#317706032
#317705672
#481×C2692
#3092269