Finite Group Databasebeta

← #46822387

Subgroups — #46822387

Order 34432.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575481C211yesmaximal, characteristic, complemented
2C2×SD16×D269C211yesmaximal, characteristic, complemented
3#31771265C211yesmaximal, characteristic, complemented
4#31771239C211yesmaximal, characteristic, complemented
5#31770991C211yesmaximal, characteristic, complemented
6#31770842C211yesmaximal, characteristic, complemented
7#31770823C211yesmaximal, characteristic, complemented
8#31770804C211yesmaximal, characteristic, complemented
9#31770755C211yesmaximal, characteristic, complemented
10#31770748C211yesmaximal, characteristic, complemented
11#31770745C211yesmaximal, characteristic, complemented
12#31770740C211yesmaximal, characteristic, complemented
13#31770704C211yesmaximal, characteristic, complemented
14#31770674C211yesmaximal, characteristic, complemented
15#468×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic, complemented
17C2×Q8×D269C2211yescharacteristic, complemented
18C22×D1076C2211yescharacteristic
19C2×D4×D269—21no—
20C2×D4×D269C2211yescharacteristic
21#22575481—24no—
22C2×#16864391C2211yescharacteristic
23C2×#16864391C2211yescharacteristic, complemented
24C2×#16864390C2211yescharacteristic, complemented
25C2×#16864389C2211yescharacteristic
26C2×#16864389C2211yescharacteristic, complemented
27SD16×D269—24no—
28C2×D2152C2211yescharacteristic
29C2×D2152C2211yescharacteristic, complemented
30C2×#16864387C2211yescharacteristic, complemented
31C2×C8×D269C2211yescharacteristic, complemented
32#22575449C2211yescharacteristic
33#22575442C2211yescharacteristic, complemented
34#22575439—21no—
35#22575431—21no—
36#22575428C2211yescharacteristic
37#22575428C2211yescharacteristic, complemented
38#22575426—21no—
39#22575426—21no—
40#22575423—21no—
41#22575421—21no—
42#22575418C2211yescharacteristic, complemented
43#22575415—21no—
44#22575415C2211yescharacteristic
45C4×C22⋊D269C2211yescharacteristic
46#22575382C2211yescharacteristic, complemented
47#22575376C2211yescharacteristic, complemented
48#22575373C2211yescharacteristic
49#22575373C2211yescharacteristic, complemented
50#22575373C2211yescharacteristic, complemented
51C22⋊#14524130C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53#22575363C2211yescharacteristic, complemented
54C2×SD16×C269C2211yescharacteristic, complemented
55C2×M16×C269C2211yescharacteristic, complemented
56C22⋊Q8×C269C2211yescharacteristic, complemented
57#121×C269C2211yescharacteristic
58#107×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62C23×D269—21no—
63#16864402—22no—
64#16864402—41no—
65C2×Dic538—21no—
66C2×Dic538C2311yescharacteristic
67Q8×D269—22no—
68Q8×D269—41no—
69C2×D1076—22no—
70C2×D1076—21no—
71C2×D1076—21no—
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×C22⋊D269D411yescharacteristic
82D4×D269—22no—
83D4×D269—41no—
84D4×D269—44no—
85C2×C4×D269—21no—
86C2×C4×D269—21no—
87C2×C4×D269—21no—
88C2×C4×D269C2311yescharacteristic
89#16864391—22no—
90#16864391D414yescomplemented
91#16864390—22no—
92#16864389—22no—
93#16864389—41no—
94D2152—22no—
95D2152—22no—
96#16864387—41no—
97C2×#14524130C2311yescharacteristic
98C2×#14524130C2311yescharacteristic
99C8×D269—22no—
100#16864381—21no—
101#16864381D411yescharacteristic
102C22⋊Dic269D411yescharacteristic
103#16864379—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379—21no—
108#16864379D411yescharacteristic
109C4×Dic269—21no—
110C4×Dic269C2311yescharacteristic
111C2×Q8×C269C2311yescharacteristic
112C2×D4×C269—21no—
113C2×D4×C269C2311yescharacteristic
114C22×C4×C269C2311yescharacteristic
115SD16×C269—22no—
116M16×C269—22no—
117C2×C8×C269C2311yescharacteristic
118C2×C8×C269C2311yescharacteristic
119C4⋊C4×C269—21no—
120C4⋊C4×C269C2311yescharacteristic
121C4⋊C4×C269C2311yescharacteristic
122C22⋊C4×C269—21no—
123C22⋊C4×C269—21no—
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—21no—
127C22×D269—42no—
128C22×D269—42no—
129C22×D269—42no—
130C22×D269—42no—
131C22×D269—81no—
132C22×D269—81no—
133C22×D269C2×D411yescharacteristic
134Dic538—21no—
135Dic538—42no—
136Dic538C2×D412yescomplemented
137D1076—21no—
138D1076—21no—
139D1076—21no—
140D1076—21no—
141D1076—21no—
142D1076—42no—
143D1076—42no—
144D1076—42no—
145D1076C2×D412yes—
146C22⋊D269—22no—
147C22⋊D269—42no—
148C22⋊D269—42no—
149C22⋊D269—42no—
150C22⋊D269—81no—
151C2×Dic269—21no—
152C2×Dic269C2×D411yescharacteristic
153C2×Dic269C2×D411yescharacteristic
154C2×Dic269C2×D411yescharacteristic
155C4×D269—21no—
156C4×D269—21no—
157C4×D269—42no—
158C4×D269—42no—
159C4×D269—42no—
160#14524130—21no—
161#14524130—21no—
162C23×C269—21no—
163C23×C269C2×D411yescharacteristic
164Q8×C269—21no—
165Q8×C269—21no—
166D4×C269—21no—
167D4×C269—21no—
168D4×C269—42no—
169C2×C4×C269—21no—
170C2×C4×C269—21no—
171C2×C4×C269—21no—
172C2×C4×C269—21no—
173C2×C4×C269—21no—
174C2×C4×C269C2×D411yescharacteristic
175C2×C4×C269C2411yescharacteristic
176C8×C269—21no—
177C8×C269C2×D412yescomplemented
178C22×C269—21no—
179C22×C269—21no—
180C22×C269—21no—
181C22×C269—41no—
182C22×C269—41no—
183C22×C269—41no—
184C22×C269C22×D411yescharacteristic
185C2×D269—21no—
186C2×D269—21no—
187C2×D269—22no—
188C2×D269—41no—
189C2×D269—41no—
190C2×D269—41no—
191C2×D269—41no—
192C2×D269—41no—
193C2×D269—41no—
194C2×D269—41no—
195C2×D269—41no—
196C2×D269—84no—
197C2×D269—84no—
198C2×D269AGL(1,ℤ/8)12yescomplemented
199C4×C269—21no—
200C4×C269—41no—
201C4×C269—41no—
202C4×C269—41no—
203C4×C269C22×D411yescharacteristic
204C4×C269H(5,𝔽2)11yescharacteristic
205Dic269—21no—
206Dic269—22no—
207Dic269—21no—
208Dic269—21no—
209C2×C269—41no—
210C2×C269—81no—
211C2×C269D4211yescharacteristic
212C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C269#57711yescharacteristic
214D269—22no—
215D269—42no—
216D269—81no—
217D269—81no—
218C269#326511yescharacteristic, complemented, Sylow
219#3265—2691nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2691no—
221C22×SD16—2691no—
222#548—2691no—
223D42—2691no—
224#496—2691no—
225#468D26911yescharacteristic, complemented
226#466—2691no—
227#463—2691no—
228#460—2691no—
229#450—2691no—
230#449—2691no—
231#448—2691no—
232#447—2691no—
233#440—2691no—
234#435—2691no—
235C2×#41—2691no—
236C22×Q8—2691no—
237C22×D4—2691no—
238C22×D4—2691no—
239C22×D4—5381no—
240AGL(1,ℤ/8)—5384no—
241C2×SD16—2691no—
242C2×SD16—2691no—
243C2×SD16—2691no—
244C2×SD16—2691no—
245C2×SD16—2691no—
246C2×SD16—5384no—
247C2×SD16C2×D26911yescharacteristic, complemented
248C2×D8—2691no—
249C2×D8—2691no—
250C2×D8—2691no—
251C2×M16C2×D26911yescharacteristic, complemented
252C22×C8—2691no—
253#127—2691no—
254#124—2691no—
255#124—5381no—
256C22⋊Q8—5381no—
257C22⋊Q8C2×D26911yescharacteristic, complemented
258#121—2691no—
259#121—2691no—
260#121—5381no—
261#121—5381no—
262#121C2×D26911yescharacteristic
263#120—5381no—
264#120—5381no—
265C4×Q8—2691no—
266C4×D4—2691no—
267C4×D4—2691no—
268C4×D4—5381no—
269#107C2×D26911yescharacteristic, complemented
270C4⋊C8—2691no—
271#103—2691no—
272#103C2×D26911yescharacteristic, complemented
273#102—2691no—
274#102—2691no—
275#102—2691no—
276#102C2×D26911yescharacteristic, complemented
277C22⋊C8—2691no—
278C22⋊C8—2691no—
279#97—2691no—
280C24—5381no—
281C24—5381no—
282#41—5382no—
283#41—10761no—
284C2×Q8—2691no—
285C2×Q8—5381no—
286C2×Q8—5382no—
287C2×Q8—10761no—
288C2×Q8C22×D26911yescharacteristic
289C2×D4—21no—
290C2×D4—2691no—
291C2×D4—2691no—
292C2×D4—2691no—
293C2×D4—2691no—
294C2×D4—5382no—
295C2×D4—5381no—
296C2×D4—5381no—
297C2×D4—5381no—
298C2×D4—5381no—
299C2×D4—5381no—
300C2×D4—5382no—
301C2×D4—5381no—
302C2×D4—5381no—
303C2×D4—10761no—
304C2×D4—10761no—
305C2×D4—10764no—
306C2×D4C22×D26911yescharacteristic
307C22×C4—2691no—
308C22×C4—5381no—
309C22×C4—5381no—
310C22×C4—5381no—
311C22×C4C22×D26911yescharacteristic
312SD16—22no—
313SD16—2694no—
314SD16—5382no—
315SD16—5382no—
316SD16—5382no—
317SD16—10761no—
318D8—5382no—
319D8—5382no—
320D8—10761no—
321M16—22no—
322C2×C8—2691no—
323C2×C8—2691no—
324C2×C8—5382no—
325C2×C8C22×D26911yescharacteristic
326C2×C8C22×D26911yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2691no—
329C4⋊C4—5381no—
330C4⋊C4C22×D26911yescharacteristic
331C4⋊C4C22×D26911yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—2691no—
335C22⋊C4—2691no—
336C22⋊C4—5381no—
337C22⋊C4—5381no—
338C22⋊C4—5381no—
339C22⋊C4—5381no—
340C22⋊C4—5381no—
341C42—2691no—
342C42—5381no—
343C23—21no—
344C23—2691no—
345C23—5381no—
346C23—5381no—
347C23—5381no—
348C23—10762no—
349C23—10762no—
350C23—10762no—
351C23—10762no—
352C23—21521no—
353C23—21521no—
354C23D4×D26911yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2692no—
358Q8—5381no—
359Q8—10762no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—2692no—
364D4—5382no—
365D4—5381no—
366D4—5381no—
367D4—5381no—
368D4—5381no—
369D4—5381no—
370D4—10762no—
371D4—10762no—
372D4—10762no—
373D4—10762no—
374D4—10762no—
375D4—10762no—
376D4—21521no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—2691no—
383C2×C4—2691no—
384C2×C4—2691no—
385C2×C4—5381no—
386C2×C4—5381no—
387C2×C4—5381no—
388C2×C4—10762no—
389C2×C4—10762no—
390C2×C4—10762no—
391C2×C4D4×D26911yescharacteristic
392C2×C4C23×D26911yescharacteristic
393C8—21no—
394C8—5381no—
395C8—5381no—
396C8D4×D26912yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—2692no—
404C22—5381no—
405C22—5381no—
406C22—5382no—
407C22—10761no—
408C22—10761no—
409C22—10761no—
410C22—10761no—
411C22—10761no—
412C22—10761no—
413C22—10761no—
414C22—10761no—
415C22—21524no—
416C22—21524no—
417C22C2×D4×D26911yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—5381no—
423C4—5381no—
424C4—5382no—
425C4—5381no—
426C4C2×D4×D26911yescharacteristic
427C4#2257549911yescharacteristic
428C2—41no—
429C2—81no—
430C2—5382no—
431C2—10762no—
432C2—21521no—
433C2—21521no—
434C2#3177123911yescharacteristic
435C2AGL(1,ℤ/8)×D26911yescharacteristic
436C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#468×C269C2
socleC22×C269C22×D4
hypercenter#468D269
hypocenterC269#3265
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
1. C2×#22575481C2semidirect
2. C2×SD16×D269C2semidirect
3. #31771265C2semidirect
4. #31771239C2semidirect
5. #31770991C2semidirect
6. #31770842C2semidirect
7. #31770823C2semidirect
8. #31770804C2semidirect
9. #31770755C2semidirect
10. #31770748C2semidirect
11. #31770745C2semidirect
12. #31770740C2semidirect
13. #31770704C2semidirect
14. #31770674C2semidirect
15. #468×C269C2semidirect
16. C2×#16864402C22semidirect
17. C2×Q8×D269C22semidirect
23. C2×#16864391C22semidirect
24. C2×#16864390C22semidirect
26. C2×#16864389C22semidirect
29. C2×D2152C22semidirect
30. C2×#16864387C22semidirect
31. C2×C8×D269C22semidirect
33. #22575442C22semidirect
37. #22575428C22semidirect
42. #22575418C22semidirect
46. #22575382C22semidirect
47. #22575376C22semidirect
49. #22575373C22semidirect
50. #22575373C22semidirect
51. C22⋊#14524130C22semidirect
52. #22575365C22semidirect
53. #22575363C22semidirect
54. C2×SD16×C269C22semidirect
55. C2×M16×C269C22semidirect
56. C22⋊Q8×C269C22semidirect
58. #107×C269C22semidirect
59. #103×C269C22semidirect
60. #102×C269C22semidirect
90. #16864391D4semidirect
136. Dic538C2×D4semidirect
177. C8×C269C2×D4semidirect
198. C2×D269AGL(1,ℤ/8)semidirect
218. C269#3265semidirect
225. #468D269semidirect
247. C2×SD16C2×D269semidirect
251. C2×M16C2×D269semidirect
257. C22⋊Q8C2×D269semidirect
269. #107C2×D269semidirect
272. #103C2×D269semidirect
276. #102C2×D269semidirect
396. C8D4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754812
C2×SD16×D2692
#317712652
#317712392
#317709912
#317708422
#317708232
#317708042
#317707552
#317707482
#317707452
#317707402
#317707042
#317706742
#468×C2692
#3265269