Finite Group Databasebeta

← #46820404

Subgroups — #46820404

Order 34432.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575485C211yesmaximal, characteristic, complemented
2C2×SD16×D269C211yesmaximal, characteristic, complemented
3#31771173C211yesmaximal, characteristic
4#31771081C211yesmaximal, characteristic, complemented
5#31770830C211yesmaximal, characteristic, complemented
6#31770829C211yesmaximal, characteristic, complemented
7#31770824C211yesmaximal, characteristic, complemented
8#31770806C211yesmaximal, characteristic, complemented
9#31770805C211yesmaximal, characteristic, complemented
10#31770804C211yesmaximal, characteristic, complemented
11#31770803C211yesmaximal, characteristic, complemented
12#31770630C211yesmaximal, characteristic, complemented
13#31770608C211yesmaximal, characteristic, complemented
14#31770600C211yesmaximal, characteristic, complemented
15#464×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic
18C2×D4×D269C2211yescharacteristic
19#22575485—24no—
20C2×#16864392C2211yescharacteristic, complemented
21C2×#16864391C2211yescharacteristic
22C2×#16864391C2211yescharacteristic, complemented
23C2×#16864390C2211yescharacteristic
24C2×#16864389C2211yescharacteristic
25C2×#16864389C2211yescharacteristic, complemented
26SD16×D269—24no—
27C2×#16864385C2211yescharacteristic
28C2×C8×D269C2211yescharacteristic
29#22575450C2211yescharacteristic
30#22575449C2211yescharacteristic
31#22575442C2211yescharacteristic
32#22575426—21no—
33#22575426—21no—
34#22575423—21no—
35#22575423—21no—
36Q8×Dic269C2211yescharacteristic, complemented
37C4×Dic538—21no—
38C4×Dic538C2211yescharacteristic, complemented
39#22575418—21no—
40#22575418C2211yescharacteristic, complemented
41#22575411—21no—
42#22575411C2211yescharacteristic
43#22575410—21no—
44#22575410C2211yescharacteristic
45C4⋊#14524130C2211yescharacteristic, complemented
46#22575382C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575377C2211yescharacteristic, complemented
49#22575375C2211yescharacteristic
50#22575375C2211yescharacteristic, complemented
51#22575374C2211yescharacteristic
52#22575373C2211yescharacteristic
53#22575373C2211yescharacteristic, complemented
54C2×Q16×C269C2211yescharacteristic, complemented
55C2×SD16×C269C2211yescharacteristic
56#124×C269C2211yescharacteristic
57C4×Q8×C269C2211yescharacteristic, complemented
58C4⋊C8×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic
61C23×D269—21no—
62#16864403—22no—
63#16864403—22no—
64C2×Dic538—21no—
65C2×Dic538C2311yescharacteristic
66Q8×D269—22no—
67Q8×D269D414yescomplemented
68C2×D1076—21no—
69C2×D1076—21no—
70C2×D1076—21no—
71C2×D1076C2311yescharacteristic
72C2×D1076C2311yescharacteristic
73C2×C22⋊D269—21no—
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76D4×D269—22no—
77D4×D269—22no—
78C2×C4×D269—21no—
79C2×C4×D269—21no—
80C2×C4×D269C2311yescharacteristic
81#16864392—22no—
82#16864391—22no—
83#16864391—22no—
84#16864390—22no—
85#16864389—22no—
86#16864389—22no—
87#16864385—22no—
88C2×#14524130C2311yescharacteristic
89C2×#14524130C2311yescharacteristic
90C8×D269—22no—
91C4⋊Dic269—21no—
92C4⋊Dic269C2311yescharacteristic
93#16864381—21no—
94#16864381—21no—
95#16864381D411yescharacteristic
96#16864381D411yescharacteristic
97#16864379—21no—
98#16864379—21no—
99#16864379—21no—
100#16864379D411yescharacteristic
101#16864379D411yescharacteristic
102C4×Dic269—21no—
103C4×Dic269—21no—
104C4×Dic269—21no—
105C4×Dic269—21no—
106C4×Dic269C2311yescharacteristic
107C2×Q8×C269C2311yescharacteristic
108C2×Q8×C269C2311yescharacteristic
109C2×D4×C269C2311yescharacteristic
110Q16×C269—22no—
111SD16×C269—22no—
112C2×C8×C269C2311yescharacteristic
113C2×C8×C269C2311yescharacteristic
114C4⋊C4×C269—21no—
115C4⋊C4×C269C2311yescharacteristic
116C22⋊C4×C269—21no—
117C22⋊C4×C269—21no—
118C42×C269—21no—
119C42×C269C2311yescharacteristic
120C22×D269—21no—
121C22×D269—21no—
122C22×D269—42no—
123C22×D269—81no—
124C22×D269—81no—
125C22×D269C2×D411yescharacteristic
126Dic538—21no—
127Dic538—24no—
128Dic538C2×D412yes—
129D1076—21no—
130D1076—21no—
131D1076—21no—
132D1076—21no—
133D1076—42no—
134D1076—81no—
135D1076—81no—
136C22⋊D269—42no—
137C22⋊D269—81no—
138C22⋊D269—81no—
139C2×Dic269—21no—
140C2×Dic269—21no—
141C2×Dic269C2×D411yescharacteristic
142C2×Dic269C2×D411yescharacteristic
143C2×Dic269C2×D411yescharacteristic
144C4×D269—24no—
145C4×D269—42no—
146C4×D269C2×D412yes—
147C4×D269#4112yes—
148#14524130—21no—
149#14524130—21no—
150C23×C269—21no—
151Q8×C269—21no—
152Q8×C269—21no—
153Q8×C269—21no—
154Q8×C269C2×D412yes—
155D4×C269—21no—
156D4×C269—21no—
157C2×C4×C269—21no—
158C2×C4×C269—21no—
159C2×C4×C269—21no—
160C2×C4×C269C2×D411yescharacteristic
161C2×C4×C269C2×D411yescharacteristic
162C2×C4×C269C2411yescharacteristic
163C8×C269—21no—
164C8×C269—21no—
165C22×C269—41no—
166C22×C269—41no—
167C22×C269—41no—
168C22×C269C22×D411yescharacteristic
169C2×D269—21no—
170C2×D269—21no—
171C2×D269—41no—
172C2×D269—41no—
173C2×D269—41no—
174C2×D269—41no—
175C2×D269—41no—
176C2×D269—41no—
177C2×D269—84no—
178C2×D269#12112yes—
179C4×C269—22no—
180C4×C269—21no—
181C4×C269—21no—
182C4×C269—41no—
183C4×C269—41no—
184C4×C269C22×D411yescharacteristic
185C4×C269C2×#4111yescharacteristic
186Dic269—22no—
187Dic269—21no—
188Dic269—21no—
189Dic269—41no—
190Dic269#12112yes—
191C2×C269—81no—
192C2×C269C2×#12111yescharacteristic
193C2×C269#57711yescharacteristic
194C2×C269#57711yescharacteristic
195D269—22no—
196D269—81no—
197D269—81no—
198C269#304311yescharacteristic, complemented, Sylow
199#3043—2691nomaximal, Sylow
200C2×#137—2691no—
201C22×SD16—2691no—
202#535—2691no—
203#519—2691no—
204#464—2691no—
205#464—2691no—
206#464—2691no—
207#464D26911yescharacteristic, complemented
208#460—2691no—
209#460—2691no—
210#460—2691no—
211#460—2691no—
212#424—2691no—
213#418—2691no—
214#416—2691no—
215C2×#41—2691no—
216C22×Q8—2691no—
217C22×D4—2691no—
218#137—5384no—
219C2×Q16—2691no—
220C2×Q16C2×D26911yescharacteristic, complemented
221C2×SD16—2691no—
222C2×SD16—2691no—
223C2×SD16—2691no—
224C2×SD16—2691no—
225C2×SD16—2691no—
226C2×SD16—5384no—
227C2×SD16C2×D26911yescharacteristic
228C2×M16—2691no—
229C22×C8—2691no—
230#127—2691no—
231#127—2691no—
232#124—2691no—
233#124C2×D26911yescharacteristic
234#121—5381no—
235#121—5381no—
236#120—5381no—
237#120—5381no—
238C4×Q8—2691no—
239C4×Q8—2691no—
240C4×Q8—2691no—
241C4×Q8—5381no—
242C4×Q8—5381no—
243C4×Q8C2×D26911yescharacteristic, complemented
244#117—2691no—
245#117—2691no—
246#117—5381no—
247#117—5381no—
248C4⋊C8—2691no—
249C4⋊C8—2691no—
250C4⋊C8—2691no—
251C4⋊C8C2×D26911yescharacteristic, complemented
252#103—2691no—
253#103C2×D26911yescharacteristic, complemented
254#102—2691no—
255#102—2691no—
256#102—2691no—
257#102—2691no—
258#102—2691no—
259#102C2×D26911yescharacteristic
260C24—5381no—
261#41—5382no—
262#41—5382no—
263C2×Q8—2694no—
264C2×Q8—2691no—
265C2×Q8—5381no—
266C2×Q8—5382no—
267C2×Q8C22×D26911yescharacteristic
268C2×Q8C22×D26911yescharacteristic
269C2×D4—2691no—
270C2×D4—2691no—
271C2×D4—5381no—
272C2×D4—5382no—
273C2×D4—5382no—
274C2×D4—5381no—
275C2×D4—5381no—
276C2×D4—5381no—
277C2×D4—5381no—
278C2×D4—5381no—
279C2×D4C22×D26911yescharacteristic
280C22×C4—2691no—
281C22×C4—5381no—
282C22×C4—5381no—
283Q16—22no—
284Q16—5382no—
285SD16—22no—
286SD16—5382no—
287SD16—5382no—
288SD16—5382no—
289SD16—5382no—
290SD16—5382no—
291M16—5382no—
292C2×C8—2691no—
293C2×C8—2691no—
294C2×C8—5382no—
295C2×C8C22×D26911yescharacteristic
296C2×C8C22×D26911yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—2691no—
299C4⋊C4—2691no—
300C4⋊C4—2691no—
301C4⋊C4—5381no—
302C4⋊C4—5381no—
303C4⋊C4—5381no—
304C4⋊C4C22×D26911yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—2691no—
308C22⋊C4—2691no—
309C22⋊C4—5381no—
310C22⋊C4—5381no—
311C22⋊C4—5381no—
312C42—21no—
313C42—2691no—
314C42—5381no—
315C42—5381no—
316C42—5381no—
317C42—5381no—
318C42C22×D26911yescharacteristic
319C23—21no—
320C23—2691no—
321C23—5381no—
322C23—5381no—
323C23—10762no—
324C23—21521no—
325C23—21521no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—2692no—
330Q8—5384no—
331Q8—5381no—
332Q8D4×D26912yes—
333D4—21no—
334D4—21no—
335D4—5381no—
336D4—5381no—
337D4—5381no—
338D4—5381no—
339D4—10762no—
340D4—10762no—
341D4—21521no—
342D4—21521no—
343D4—21521no—
344D4—21521no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—2691no—
349C2×C4—2692no—
350C2×C4—2692no—
351C2×C4—2691no—
352C2×C4—2691no—
353C2×C4—5384no—
354C2×C4—5381no—
355C2×C4—5381no—
356C2×C4—10762no—
357C2×C4D4×D26911yescharacteristic
358C2×C4D4×D26911yescharacteristic
359C2×C4C23×D26911yescharacteristic
360C8—21no—
361C8—21no—
362C8—5381no—
363C8—5381no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—2692no—
368C22—5381no—
369C22—5381no—
370C22—10761no—
371C22—10761no—
372C22—10761no—
373C22—10761no—
374C22—10761no—
375C22—10761no—
376C22—21524no—
377C22C2×D4×D26911yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—2692no—
384C4—5381no—
385C4—5381no—
386C4—5382no—
387C4—10761no—
388C4C2×D4×D26911yescharacteristic
389C4#41×D26911yescharacteristic
390C2—81no—
391C2—5382no—
392C2—21521no—
393C2—21521no—
394C2#121×D26911yescharacteristic
395C2#3177141211yescharacteristic
396C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#464×C269C2
socleC22×C269C22×D4
prime radical#31771173C2
hypercenter#464D269
hypocenterC269#3043
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#22575485C2semidirect
2. C2×SD16×D269C2semidirect
4. #31771081C2semidirect
5. #31770830C2semidirect
6. #31770829C2semidirect
7. #31770824C2semidirect
8. #31770806C2semidirect
9. #31770805C2semidirect
10. #31770804C2semidirect
11. #31770803C2semidirect
12. #31770630C2semidirect
13. #31770608C2semidirect
14. #31770600C2semidirect
15. #464×C269C2semidirect
20. C2×#16864392C22semidirect
22. C2×#16864391C22semidirect
25. C2×#16864389C22semidirect
36. Q8×Dic269C22semidirect
38. C4×Dic538C22semidirect
40. #22575418C22semidirect
45. C4⋊#14524130C22semidirect
46. #22575382C22semidirect
47. #22575382C22semidirect
48. #22575377C22semidirect
50. #22575375C22semidirect
53. #22575373C22semidirect
54. C2×Q16×C269C22semidirect
57. C4×Q8×C269C22semidirect
58. C4⋊C8×C269C22semidirect
59. #103×C269C22semidirect
67. Q8×D269D4semidirect
198. C269#3043semidirect
207. #464D269semidirect
220. C2×Q16C2×D269semidirect
243. C4×Q8C2×D269semidirect
251. C4⋊C8C2×D269semidirect
253. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754852
C2×SD16×D2692
#317711732
#317710812
#317708302
#317708292
#317708242
#317708062
#317708052
#317708042
#317708032
#317706302
#317706082
#317706002
#464×C2692
#3043269