Finite Group Databasebeta

← #46823451

Subgroups — #46823451

Order 34432.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771302C211yesmaximal, characteristic
2#31771297C211yesmaximal, characteristic, complemented
3#31771006C211yesmaximal, characteristic, complemented
4#31770943C211yesmaximal, characteristic, complemented
5#31770934C211yesmaximal, characteristic, complemented
6#31770927C211yesmaximal, characteristic, complemented
7#31770921C211yesmaximal, characteristic, complemented
8#31770914C211yesmaximal, characteristic, complemented
9#31770907C211yesmaximal, characteristic, complemented
10#31770848C211yesmaximal, characteristic, complemented
11SD16×Dic269C211yesmaximal, characteristic, complemented
12#31770681C211yesmaximal, characteristic, complemented
13#31770647C211yesmaximal, characteristic, complemented
14#31770646C211yesmaximal, characteristic, complemented
15#468×C269C211yesmaximal, characteristic, complemented
16C2×#16864385C2211yescharacteristic
17#22575448—21no—
18#22575446—21no—
19#22575444—21no—
20#22575444C2211yescharacteristic
21#22575444C2211yescharacteristic
22#22575437—21no—
23#22575434—21no—
24#22575429C2211yescharacteristic
25#22575424—21no—
26#22575424C2211yescharacteristic
27Q8×Dic269C2211yescharacteristic
28D4×Dic269C2211yescharacteristic
29C4×C22⋊D269C2211yescharacteristic
30#22575413—21no—
31#22575412C2211yescharacteristic
32#22575411C2211yescharacteristic
33#22575409—21no—
34C2×C4⋊Dic269C2211yescharacteristic
35C4⋊C4×D269C2211yescharacteristic
36#22575387C2211yescharacteristic
37#22575387C2211yescharacteristic
38#22575386C2211yescharacteristic
39#22575385C2211yescharacteristic
40#22575385C2211yescharacteristic
41#22575384C2211yescharacteristic
42#22575384C2211yescharacteristic
43#22575384C2211yescharacteristic
44#22575382C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575377C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×SD16×C269C2211yescharacteristic
53C2×M16×C269C2211yescharacteristic
54C22⋊Q8×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic
56#107×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59C2×C22⋊D269—21no—
60C2×C22⋊D269D411yescharacteristic
61C22×Dic269—21no—
62C22×Dic269—21no—
63C2×C4×D269—21no—
64C2×C4×D269C2311yescharacteristic
65#16864385—22no—
66C2×#14524130C2311yescharacteristic
67C2×#14524130C2311yescharacteristic
68C4⋊Dic269—21no—
69C4⋊Dic269—21no—
70C4⋊Dic269—21no—
71C4⋊Dic269—21no—
72C4⋊Dic269C2311yescharacteristic
73C4⋊Dic269C2311yescharacteristic
74C4⋊Dic269C2311yescharacteristic
75C4⋊Dic269C2311yescharacteristic
76#16864381—21no—
77#16864381—21no—
78#16864381—21no—
79#16864381—21no—
80#16864381D411yescharacteristic
81C22⋊Dic269—21no—
82C22⋊Dic269—21no—
83C22⋊Dic269—21no—
84C22⋊Dic269D411yescharacteristic
85#16864379—21no—
86#16864379—21no—
87#16864379D411yescharacteristic
88C4×Dic269—21no—
89C4×Dic269—21no—
90C4×Dic269—21no—
91C4×Dic269C2311yescharacteristic
92C2×Q8×C269C2311yescharacteristic
93C2×D4×C269—21no—
94C2×D4×C269C2311yescharacteristic
95C22×C4×C269C2311yescharacteristic
96SD16×C269—22no—
97M16×C269—22no—
98C2×C8×C269C2311yescharacteristic
99C2×C8×C269C2311yescharacteristic
100C4⋊C4×C269—21no—
101C4⋊C4×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22×D269C2×D411yescharacteristic
106C22⋊D269—41no—
107C22⋊D269—42no—
108C2×Dic269—21no—
109C2×Dic269—21no—
110C2×Dic269—21no—
111C2×Dic269—21no—
112C2×Dic269—42no—
113C2×Dic269—81no—
114C2×Dic269C2×D411yescharacteristic
115C2×Dic269C2×D411yescharacteristic
116C2×Dic269C2×D411yescharacteristic
117C4×D269—21no—
118C4×D269—21no—
119C4×D269—81no—
120#14524130—21no—
121#14524130—21no—
122C23×C269—21no—
123C23×C269C2×D411yescharacteristic
124Q8×C269—21no—
125Q8×C269—21no—
126D4×C269—21no—
127D4×C269—21no—
128D4×C269—81no—
129C2×C4×C269—21no—
130C2×C4×C269—21no—
131C2×C4×C269—21no—
132C2×C4×C269—21no—
133C2×C4×C269—21no—
134C2×C4×C269C2×D411yescharacteristic
135C2×C4×C269C2411yescharacteristic
136C8×C269—21no—
137C8×C269#4112yes—
138C22×C269—21no—
139C22×C269—21no—
140C22×C269—21no—
141C22×C269—41no—
142C22×C269—41no—
143C22×C269—41no—
144C22×C269C22×D411yescharacteristic
145C2×D269—21no—
146C2×D269—21no—
147C2×D269—21no—
148C4×C269—21no—
149C4×C269—41no—
150C4×C269—41no—
151C4×C269—41no—
152C4×C269C2×#4111yescharacteristic
153C4×C2692−1+411yescharacteristic
154Dic269—21no—
155Dic269—21no—
156Dic269—21no—
157Dic269—41no—
158Dic269—41no—
159Dic269—41no—
160Dic269—41no—
161C2×C269—41no—
162C2×C269—81no—
163C2×C269#54711yescharacteristic
164C2×C269#57511yescharacteristic
165C2×C269#57711yescharacteristic
166D269—41no—
167C269#335111yescharacteristic, complemented, Sylow
168#3351—2691nomaximal, Sylow
169#553—2691no—
170#552—2691no—
171#499—2691no—
172#485—2691no—
173#483—2691no—
174#482—2691no—
175#481—2691no—
176#479—2691no—
177#478—2691no—
178#468—2691no—
179#468D26911yescharacteristic, complemented
180C4×SD16—2691no—
181#436—2691no—
182#428—2691no—
183#428—2691no—
184C2×SD16C2×D26911yescharacteristic
185C2×M16—2691no—
186C2×M16C2×D26911yescharacteristic
187#126—5381no—
188#126—5381no—
189#125—2691no—
190#125—2691no—
191#125—5381no—
192#123—5381no—
193#123—5381no—
194C22⋊Q8—2691no—
195C22⋊Q8C2×D26911yescharacteristic
196#121—2691no—
197#121—5381no—
198#121C2×D26911yescharacteristic
199C4×Q8—2691no—
200C4×D4—2691no—
201C4×D4—2691no—
202C4×D4—5381no—
203#117—2691no—
204#117—2691no—
205#117—5381no—
206C2×C4⋊C4—2691no—
207C2×C4⋊C4—2691no—
208#107—2691no—
209#107—2691no—
210#107—2691no—
211#107C2×D26911yescharacteristic
212#106—2691no—
213#106—2691no—
214#106—2691no—
215#106—2691no—
216#106—2691no—
217C4⋊C8—2691no—
218#103—2691no—
219#103—2691no—
220#103C2×D26911yescharacteristic
221#102—2691no—
222#102—2691no—
223#102C2×D26911yescharacteristic
224C22⋊C8—2691no—
225C22⋊C8—2691no—
226C4×C8—2691no—
227C2×Q8C22×D26911yescharacteristic
228C2×D4—21no—
229C2×D4—2691no—
230C2×D4—5381no—
231C2×D4C22×D26911yescharacteristic
232C22×C4—2691no—
233C22×C4—5381no—
234C22×C4—5381no—
235C22×C4—5381no—
236C22×C4C22×D26911yescharacteristic
237SD16—22no—
238M16—22no—
239M16—5382no—
240C2×C8—2691no—
241C2×C8—2691no—
242C2×C8C22×D26911yescharacteristic
243C2×C8C22×D26911yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—2691no—
246C4⋊C4—2691no—
247C4⋊C4—2691no—
248C4⋊C4—2691no—
249C4⋊C4—2691no—
250C4⋊C4—5381no—
251C4⋊C4—5381no—
252C4⋊C4—5381no—
253C4⋊C4—5381no—
254C4⋊C4—5381no—
255C4⋊C4—5381no—
256C4⋊C4—5381no—
257C4⋊C4—5381no—
258C4⋊C4C22×D26911yescharacteristic
259C4⋊C4C22×D26911yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—2691no—
263C22⋊C4—2691no—
264C22⋊C4—5381no—
265C22⋊C4—5381no—
266C22⋊C4—5381no—
267C22⋊C4—5381no—
268C22⋊C4—5381no—
269C42—2691no—
270C42—5381no—
271C42—5381no—
272C42—5381no—
273C23—21no—
274C23—2691no—
275C23D4×D26911yescharacteristic
276Q8—21no—
277Q8—21no—
278D4—21no—
279D4—21no—
280D4—81no—
281D4—10761no—
282D4—10762no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—2691no—
289C2×C4—2691no—
290C2×C4—2691no—
291C2×C4—5381no—
292C2×C4—5381no—
293C2×C4—5381no—
294C2×C4—5381no—
295C2×C4—5381no—
296C2×C4—5381no—
297C2×C4—10762no—
298C2×C4—21521no—
299C2×C4—21521no—
300C2×C4D4×D26911yescharacteristic
301C2×C4C23×D26911yescharacteristic
302C8—21no—
303C8—5381no—
304C8—5381no—
305C8#1686440112yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—5381no—
313C22—5381no—
314C22—5381no—
315C22C2×D4×D26911yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—5381no—
321C4—5381no—
322C4—5381no—
323C4—10761no—
324C4—10761no—
325C4—10761no—
326C4—10761no—
327C4C2×#1686440111yescharacteristic
328C4#2257550011yescharacteristic
329C2—41no—
330C2—81no—
331C2—10761no—
332C2#3177125511yescharacteristic
333C2#3177140411yescharacteristic
334C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#468×C269C2
socleC22×C269C22×D4
prime radical#31771302C2
hypercenter#468D269
hypocenterC269#3351
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #31771297C2semidirect
3. #31771006C2semidirect
4. #31770943C2semidirect
5. #31770934C2semidirect
6. #31770927C2semidirect
7. #31770921C2semidirect
8. #31770914C2semidirect
9. #31770907C2semidirect
10. #31770848C2semidirect
11. SD16×Dic269C2semidirect
12. #31770681C2semidirect
13. #31770647C2semidirect
14. #31770646C2semidirect
15. #468×C269C2semidirect
167. C269#3351semidirect
179. #468D269semidirect

Maximal subgroups

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

subgroupindex
#317713022
#317712972
#317710062
#317709432
#317709342
#317709272
#317709212
#317709142
#317709072
#317708482
SD16×Dic2692
#317706812
#317706472
#317706462
#468×C2692
#3351269