Finite Group Databasebeta

← #46822415

Subgroups — #46822415

Order 34432.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575481C211yesmaximal, characteristic, complemented
2C2×#22575475C211yesmaximal, characteristic, complemented
3#31771260C211yesmaximal, characteristic, complemented
4#31771251C211yesmaximal, characteristic, complemented
5#31770999C211yesmaximal, characteristic, complemented
6#31770863C211yesmaximal, characteristic, complemented
7#31770834C211yesmaximal, characteristic, complemented
8#31770821C211yesmaximal, characteristic, complemented
9#31770811C211yesmaximal, characteristic, complemented
10#31770769C211yesmaximal, characteristic, complemented
11#31770769C211yesmaximal, characteristic, complemented
12#31770758C211yesmaximal, characteristic, complemented
13#31770752C211yesmaximal, characteristic, complemented
14#31770718C211yesmaximal, characteristic, complemented
15#435×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C2×#16864402C2211yescharacteristic
18C2×#16864402C2211yescharacteristic, complemented
19C22×D1076C2211yescharacteristic
20#22575481—24no—
21#22575475—24no—
22C2×Dic1076C2211yescharacteristic, complemented
23C2×Dic1076C2211yescharacteristic, complemented
24C2×#16864391C2211yescharacteristic
25C2×#16864391C2211yescharacteristic
26C2×#16864391C2211yescharacteristic
27C2×#16864391C2211yescharacteristic
28C2×#16864391C2211yescharacteristic, complemented
29C2×D2152C2211yescharacteristic
30C2×D2152C2211yescharacteristic
31#22575453C2211yescharacteristic, complemented
32#22575441C2211yescharacteristic
33#22575438—21no—
34#22575435—21no—
35#22575435—21no—
36#22575432C2211yescharacteristic
37#22575432C2211yescharacteristic
38#22575432C2211yescharacteristic, complemented
39#22575427C2211yescharacteristic
40#22575426—21no—
41#22575424—21no—
42#22575421—21no—
43C4×D1076C2211yescharacteristic
44C4×D1076C2211yescharacteristic, complemented
45C4⋊C4×D269—21no—
46C22⋊C4×D269—21no—
47#22575385C2211yescharacteristic, complemented
48#22575378C2211yescharacteristic, complemented
49#22575378C2211yescharacteristic, complemented
50#22575378C2211yescharacteristic, complemented
51#22575375C2211yescharacteristic
52#22575375C2211yescharacteristic
53#22575375C2211yescharacteristic, complemented
54C2×M16×C269C2211yescharacteristic
55C22×C8×C269C2211yescharacteristic, complemented
56C4×D4×C269C2211yescharacteristic
57C4⋊C8×C269C2211yescharacteristic, complemented
58C22⋊C8×C269C2211yescharacteristic
59C22⋊C8×C269C2211yescharacteristic, complemented
60#97×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62#16864402—22no—
63#16864402—22no—
64#16864402—22no—
65#16864402—41no—
66#16864402—41no—
67#16864402—41no—
68C2×Dic538C2311yescharacteristic
69C2×Dic538C2311yescharacteristic
70C2×Dic538C2311yescharacteristic
71C2×D1076—22no—
72C2×D1076—21no—
73C2×D1076—41no—
74C2×D1076C2311yescharacteristic
75C2×D1076C2311yescharacteristic
76C2×D1076C2311yescharacteristic
77C2×C22⋊D269—21no—
78C2×C22⋊D269—21no—
79C2×C22⋊D269—21no—
80C2×C22⋊D269—21no—
81C2×C4×D269—21no—
82C2×C4×D269—21no—
83C2×C4×D269—21no—
84C2×C4×D269—21no—
85C2×C4×D269—21no—
86Dic1076—22no—
87Dic1076—41no—
88#16864391—22no—
89#16864391—22no—
90#16864391—22no—
91#16864391—22no—
92#16864391—41no—
93D2152—22no—
94D2152D414yescomplemented
95C4⋊Dic269—21no—
96C4⋊Dic269—21no—
97C4⋊Dic269C2311yescharacteristic
98C4⋊Dic269C2311yescharacteristic
99#16864381—21no—
100#16864381—21no—
101#16864381—21no—
102C22⋊Dic269—21no—
103C22⋊Dic269—21no—
104C22⋊Dic269—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379—21no—
108#16864379—21no—
109#16864379—21no—
110C2×D4×C269D411yescharacteristic
111C22×C4×C269C2311yescharacteristic
112C22×C4×C269C2311yescharacteristic
113M16×C269—22no—
114C2×C8×C269—22no—
115C2×C8×C269C2311yescharacteristic
116C2×C8×C269C2311yescharacteristic
117C2×C8×C269C2311yescharacteristic
118C2×C8×C269C2311yescharacteristic
119C4⋊C4×C269D411yescharacteristic
120C22⋊C4×C269D411yescharacteristic
121C22⋊C4×C269D411yescharacteristic
122C42×C269C2311yescharacteristic
123C22×D269—21no—
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—42no—
127C22×D269—42no—
128C22×D269—81no—
129Dic538—21no—
130Dic538—21no—
131Dic538—21no—
132Dic538—21no—
133Dic538—21no—
134Dic538—21no—
135D1076—21no—
136D1076—21no—
137D1076—21no—
138D1076—21no—
139D1076—42no—
140D1076C2×D412yes—
141D1076C2×D412yes—
142C22⋊D269—42no—
143C22⋊D269—42no—
144C22⋊D269—42no—
145C22⋊D269—81no—
146C2×Dic269—21no—
147C2×Dic269—21no—
148C2×Dic269—21no—
149C2×Dic269—21no—
150C2×Dic269—21no—
151C4×D269—42no—
152C4×D269—42no—
153C4×D269—42no—
154C4×D269—42no—
155C4×D269—81no—
156C23×C269C2×D411yescharacteristic
157C23×C269C2×D411yescharacteristic
158D4×C269—41no—
159C2×C4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—21no—
163C2×C4×C269C2×D411yescharacteristic
164C2×C4×C269C2×D411yescharacteristic
165C2×C4×C269C2×D411yescharacteristic
166C2×C4×C269C2×D411yescharacteristic
167C2×C4×C269C2411yescharacteristic
168C8×C269—21no—
169C8×C269—21no—
170C8×C269—21no—
171C8×C269C2×D412yes—
172C22×C269—21no—
173C22×C269—21no—
174C22×C269—21no—
175C22×C269—21no—
176C22×C269—21no—
177C22×C269—21no—
178C22×C269C22×D411yescharacteristic
179C2×D269—22no—
180C2×D269—22no—
181C2×D269—41no—
182C2×D269—41no—
183C2×D269—41no—
184C2×D269—41no—
185C2×D269—41no—
186C2×D269—41no—
187C2×D269—41no—
188C2×D269—84no—
189C4×C269—21no—
190C4×C269—21no—
191C4×C269—21no—
192C4×C269—21no—
193C4×C269C22×D411yescharacteristic
194C4×C269H(5,𝔽2)11yescharacteristic
195Dic269—41no—
196Dic269—41no—
197Dic269—41no—
198Dic269—41no—
199Dic269—41no—
200C2×C269—41no—
201C2×C269—41no—
202C2×C269D4211yescharacteristic
203C2×C269#57511yescharacteristic
204C2×C269#57711yescharacteristic
205D269—42no—
206D269—42no—
207D269—81no—
208C269#326711yescharacteristic, complemented, Sylow
209#3267—2691nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2691no—
211C2×#135—2691no—
212#547—2691no—
213#546—2691no—
214#497—2691no—
215#470—2691no—
216#465—2691no—
217#463—2691no—
218#461—2691no—
219#452—2691no—
220#452—2691no—
221#450—2691no—
222#449—2691no—
223#442—2691no—
224#435D26911yescharacteristic, complemented
225C2×#41—2691no—
226C2×#41—2691no—
227C2×#41—2691no—
228C22×D4—2691no—
229AGL(1,ℤ/8)—5384no—
230#135—5384no—
231C2×Q16—2691no—
232C2×Q16—2691no—
233C2×SD16—2691no—
234C2×SD16—2691no—
235C2×SD16—2691no—
236C2×SD16—2691no—
237C2×SD16—2691no—
238C2×D8—2691no—
239C2×D8—2691no—
240C2×M16C2×D26911yescharacteristic
241C22×C8C2×D26911yescharacteristic, complemented
242C4⋊Q8—2691no—
243#124—2691no—
244#123—5381no—
245#123—5381no—
246#123—5381no—
247C22⋊Q8—2691no—
248C22⋊Q8—2691no—
249C22⋊Q8—2691no—
250#121—2691no—
251#121—5381no—
252#121—5381no—
253#120—5381no—
254C4×D4—2691no—
255C4×D4—2691no—
256C4×D4C2×D26911yescharacteristic
257C2×C4⋊C4—5381no—
258C2×C22⋊C4—5381no—
259#106—2691no—
260C4⋊C8C2×D26911yescharacteristic, complemented
261#103—2691no—
262#103—2691no—
263#103—2691no—
264#102—2691no—
265#102—2691no—
266#102—2691no—
267C22⋊C8C2×D26911yescharacteristic
268C22⋊C8C2×D26911yescharacteristic, complemented
269#97C2×D26911yescharacteristic, complemented
270C24—5381no—
271#41—5382no—
272#41—5382no—
273#41—5382no—
274#41—10761no—
275#41—10761no—
276#41—10761no—
277C2×Q8—2691no—
278C2×Q8—2691no—
279C2×Q8—2691no—
280C2×D4—2691no—
281C2×D4—2691no—
282C2×D4—2691no—
283C2×D4—5382no—
284C2×D4—5381no—
285C2×D4—5381no—
286C2×D4—5381no—
287C2×D4—5381no—
288C2×D4—5381no—
289C2×D4—10761no—
290C2×D4D107611yescharacteristic
291C22×C4—5381no—
292C22×C4—5381no—
293C22×C4—5381no—
294C22×C4—5381no—
295C22×C4—5381no—
296C22×C4C22×D26911yescharacteristic
297C22×C4C22×D26911yescharacteristic
298Q16—5382no—
299Q16—10761no—
300SD16—5382no—
301SD16—5382no—
302SD16—5382no—
303SD16—5382no—
304SD16—10761no—
305D8—2694no—
306D8—5382no—
307M16—22no—
308C2×C8—22no—
309C2×C8C22×D26911yescharacteristic
310C2×C8C22×D26911yescharacteristic
311C2×C8C22×D26911yescharacteristic
312C2×C8C22×D26911yescharacteristic
313C4⋊C4—2691no—
314C4⋊C4—2691no—
315C4⋊C4—5381no—
316C4⋊C4—5381no—
317C4⋊C4—5381no—
318C4⋊C4—5381no—
319C4⋊C4—5381no—
320C4⋊C4D107611yescharacteristic
321C22⋊C4—5381no—
322C22⋊C4—5381no—
323C22⋊C4—5381no—
324C22⋊C4—5381no—
325C22⋊C4—5381no—
326C22⋊C4—5381no—
327C22⋊C4—5381no—
328C22⋊C4—5381no—
329C22⋊C4D107611yescharacteristic
330C22⋊C4D107611yescharacteristic
331C42C22×D26911yescharacteristic
332C23—5381no—
333C23—5381no—
334C23—5381no—
335C23—10762no—
336C23—10762no—
337C23—21521no—
338C23C2×D107611yescharacteristic
339C23C2×D107611yescharacteristic
340Q8—5381no—
341Q8—5381no—
342Q8—5381no—
343Q8—5381no—
344Q8—5381no—
345Q8—5381no—
346D4—41no—
347D4—2692no—
348D4—2692no—
349D4—5381no—
350D4—5381no—
351D4—5381no—
352D4—5381no—
353D4—10762no—
354D4—10762no—
355D4—10762no—
356D4—10762no—
357D4—21521no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—5381no—
363C2×C4—5381no—
364C2×C4—5381no—
365C2×C4—5381no—
366C2×C4—5381no—
367C2×C4—10762no—
368C2×C4—10762no—
369C2×C4—10762no—
370C2×C4—10762no—
371C2×C4—21521no—
372C2×C4C2×D107611yescharacteristic
373C2×C4C2×D107611yescharacteristic
374C2×C4C2×D107611yescharacteristic
375C2×C4C2×D107611yescharacteristic
376C2×C4C23×D26911yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D26912yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—5382no—
388C22—5382no—
389C22—10761no—
390C22—10761no—
391C22—10761no—
392C22—10761no—
393C22—10761no—
394C22—10761no—
395C22—10761no—
396C22—21524no—
397C22C22×D107611yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—10761no—
403C4—10761no—
404C4—10761no—
405C4—10761no—
406C4—10761no—
407C4C2×D4×D26911yescharacteristic
408C4#2257549911yescharacteristic
409C2—41no—
410C2—41no—
411C2—10762no—
412C2—10762no—
413C2—21521no—
414C2D4×D107611yescharacteristic
415C2#3177140511yescharacteristic
416C2#3177141711yescharacteristic
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#435×C269C2
socleC22×C269C22×D4
hypercenter#435D269
hypocenterC269#3267
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22575481C2semidirect
2. C2×#22575475C2semidirect
3. #31771260C2semidirect
4. #31771251C2semidirect
5. #31770999C2semidirect
6. #31770863C2semidirect
7. #31770834C2semidirect
8. #31770821C2semidirect
9. #31770811C2semidirect
10. #31770769C2semidirect
11. #31770769C2semidirect
12. #31770758C2semidirect
13. #31770752C2semidirect
14. #31770718C2semidirect
15. #435×C269C2semidirect
18. C2×#16864402C22semidirect
22. C2×Dic1076C22semidirect
23. C2×Dic1076C22semidirect
28. C2×#16864391C22semidirect
31. #22575453C22semidirect
38. #22575432C22semidirect
44. C4×D1076C22semidirect
47. #22575385C22semidirect
48. #22575378C22semidirect
49. #22575378C22semidirect
50. #22575378C22semidirect
53. #22575375C22semidirect
55. C22×C8×C269C22semidirect
57. C4⋊C8×C269C22semidirect
59. C22⋊C8×C269C22semidirect
60. #97×C269C22semidirect
94. D2152D4semidirect
208. C269#3267semidirect
224. #435D269semidirect
241. C22×C8C2×D269semidirect
260. C4⋊C8C2×D269semidirect
268. C22⋊C8C2×D269semidirect
269. #97C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754812
C2×#225754752
#317712602
#317712512
#317709992
#317708632
#317708342
#317708212
#317708112
#317707692
#317707582
#317707522
#317707182
#435×C2692
#3267269