Finite Group Databasebeta

← #46822378

Subgroups — #46822378

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×#22575479C211yesmaximal, characteristic, complemented
2C2×SD16×D269C211yesmaximal, characteristic, complemented
3#31771265C211yesmaximal, characteristic, complemented
4D4×C22⋊D269C211yesmaximal, characteristic, complemented
5#31770991C211yesmaximal, characteristic, complemented
6#31770853C211yesmaximal, characteristic, complemented
7#31770840C211yesmaximal, characteristic, complemented
8#31770822C211yesmaximal, characteristic, complemented
9#31770804C211yesmaximal, characteristic, complemented
10#31770752C211yesmaximal, characteristic, complemented
11#31770742C211yesmaximal, characteristic, complemented
12#31770740C211yesmaximal, characteristic, complemented
13#31770702C211yesmaximal, characteristic, complemented
14#31770679C211yesmaximal, characteristic, complemented
15#450×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic, complemented
17C2×Q8×D269C2211yescharacteristic, complemented
18C22×C22⋊D269—21no—
19C2×D4×D269C2211yescharacteristic
20#22575479—24no—
21C2×#16864391C2211yescharacteristic, complemented
22C2×#16864390C2211yescharacteristic
23C2×#16864390C2211yescharacteristic, complemented
24C2×#16864389C2211yescharacteristic, complemented
25SD16×D269—24no—
26C2×D2152C2211yescharacteristic, complemented
27C2×#16864387C2211yescharacteristic
28C2×#16864387C2211yescharacteristic, complemented
29C2×#16864386C2211yescharacteristic, complemented
30C2×C8×D269C2211yescharacteristic, complemented
31#22575449C2211yescharacteristic
32#22575442C2211yescharacteristic, complemented
33#22575439—21no—
34#22575431—21no—
35#22575428C2211yescharacteristic, complemented
36#22575427C2211yescharacteristic
37#22575426—21no—
38#22575425—21no—
39#22575424C2211yescharacteristic
40#22575422—21no—
41#22575421—21no—
42#22575418C2211yescharacteristic, complemented
43D4×Dic269C2211yescharacteristic
44#22575415—21no—
45C4×C22⋊D269C2211yescharacteristic
46#22575386C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575378C2211yescharacteristic, complemented
49#22575377C2211yescharacteristic, complemented
50#22575375C2211yescharacteristic, complemented
51#22575373C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53#22575363C2211yescharacteristic, complemented
54C22×D4×C269C2211yescharacteristic
55C2×SD16×C269C2211yescharacteristic
56C2×SD16×C269C2211yescharacteristic, complemented
57C22⋊Q8×C269C2211yescharacteristic, complemented
58#102×C269C2211yescharacteristic
59#102×C269C2211yescharacteristic, complemented
60C22⋊C8×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62#16864402—22no—
63#16864402—41no—
64C2×Dic538—21no—
65C2×Dic538C2311yescharacteristic
66Q8×D269—22no—
67Q8×D269—41no—
68C2×D1076—21no—
69C2×D1076C2311yescharacteristic
70C2×D1076C2311yescharacteristic
71C2×C22⋊D269—21no—
72C2×C22⋊D269—21no—
73C2×C22⋊D269—21no—
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76C2×C22⋊D269—21no—
77C2×C22⋊D269—44no—
78C2×C22⋊D269D411yescharacteristic
79D4×D269—22no—
80D4×D269—41no—
81C22×Dic269—21no—
82C2×C4×D269—21no—
83C2×C4×D269—21no—
84C2×C4×D269C2311yescharacteristic
85#16864391—22no—
86#16864390—22no—
87#16864390D414yescomplemented
88#16864389—22no—
89D2152—41no—
90#16864387—22no—
91#16864387—22no—
92#16864386—22no—
93C2×#14524130C2311yescharacteristic
94C2×#14524130C2311yescharacteristic
95C8×D269—22no—
96C4⋊Dic269C2311yescharacteristic
97#16864381—21no—
98#16864381D411yescharacteristic
99C22⋊Dic269—21no—
100C22⋊Dic269—21no—
101C22⋊Dic269D411yescharacteristic
102#16864379—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379D411yescharacteristic
107C4×Dic269—21no—
108C4×Dic269C2311yescharacteristic
109C24×C269—21no—
110C2×Q8×C269C2311yescharacteristic
111C2×D4×C269—22no—
112C2×D4×C269—21no—
113C2×D4×C269—41no—
114C2×D4×C269C2311yescharacteristic
115C2×D4×C269C2311yescharacteristic
116C22×C4×C269C2311yescharacteristic
117SD16×C269—22no—
118SD16×C269—41no—
119C2×C8×C269C2311yescharacteristic
120C2×C8×C269C2311yescharacteristic
121C4⋊C4×C269—21no—
122C4⋊C4×C269C2311yescharacteristic
123C22⋊C4×C269—21no—
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—42no—
127C22×D269—42no—
128C22×D269—81no—
129C22×D269C2×D411yescharacteristic
130Dic538—21no—
131Dic538—42no—
132Dic538C2×D412yescomplemented
133D1076—21no—
134D1076—21no—
135D1076—21no—
136D1076—21no—
137D1076—42no—
138C22⋊D269—22no—
139C22⋊D269—42no—
140C22⋊D269—42no—
141C22⋊D269—42no—
142C22⋊D269—42no—
143C22⋊D269—42no—
144C22⋊D269—81no—
145C2×Dic269—21no—
146C2×Dic269—21no—
147C2×Dic269—42no—
148C2×Dic269C2×D411yescharacteristic
149C2×Dic269C2×D411yescharacteristic
150C2×Dic269C2×D411yescharacteristic
151C4×D269—21no—
152C4×D269—21no—
153C4×D269—42no—
154C4×D269—42no—
155#14524130—21no—
156#14524130C2×D412yescomplemented
157C23×C269—21no—
158C23×C269—21no—
159C23×C269—42no—
160C23×C269—42no—
161C23×C269—81no—
162C23×C269C2×D411yescharacteristic
163Q8×C269—21no—
164Q8×C269—21no—
165D4×C269—21no—
166D4×C269—21no—
167D4×C269—21no—
168D4×C269—42no—
169D4×C269C2×D412yes—
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×C269—21no—
178C22×C269—21no—
179C22×C269—22no—
180C22×C269—21no—
181C22×C269—21no—
182C22×C269—41no—
183C22×C269—41no—
184C22×C269—41no—
185C22×C269—41no—
186C22×C269—41no—
187C22×C269—84no—
188C22×C269C22×D411yescharacteristic
189C2×D269—21no—
190C2×D269—21no—
191C2×D269—41no—
192C2×D269—41no—
193C2×D269—41no—
194C2×D269—41no—
195C2×D269—41no—
196C2×D269—41no—
197C2×D269—84no—
198C2×D269AGL(1,ℤ/8)12yescomplemented
199C4×C269—21no—
200C4×C269—41no—
201C4×C269—41no—
202C4×C269C22×D411yescharacteristic
203C4×C269H(5,𝔽2)11yescharacteristic
204Dic269—21no—
205Dic269—22no—
206Dic269—21no—
207Dic269—21no—
208Dic269—41no—
209C2×C269—42no—
210C2×C269—41no—
211C2×C269—81no—
212C2×C269D4211yescharacteristic
213C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C269#57711yescharacteristic
215D269—22no—
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#468—2691no—
226#466—2691no—
227#463—2691no—
228#460—2691no—
229#450D26911yescharacteristic, complemented
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—5381no—
239C22×D4C2×D26911yescharacteristic
240AGL(1,ℤ/8)—5384no—
241C2×SD16—2691no—
242C2×SD16—2691no—
243C2×SD16—2691no—
244C2×SD16—2691no—
245C2×SD16—5384no—
246C2×SD16C2×D26911yescharacteristic
247C2×SD16C2×D26911yescharacteristic, complemented
248C2×D8—2691no—
249C2×D8—2691no—
250C2×D8—2691no—
251C2×M16—2691no—
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—2691no—
261#121—5381no—
262#121—5381no—
263#120—5381no—
264#120—5381no—
265C4×Q8—2691no—
266C4×D4—2691no—
267C4×D4—2691no—
268C4×D4—5381no—
269#107—2691no—
270C4⋊C8—2691no—
271#103—2691no—
272#103—2691no—
273#102—2691no—
274#102—2691no—
275#102C2×D26911yescharacteristic
276#102C2×D26911yescharacteristic, complemented
277C22⋊C8—2691no—
278C22⋊C8C2×D26911yescharacteristic, complemented
279#97—2691no—
280C24—21no—
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—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—2691no—
293C2×D4—2691no—
294C2×D4—2691no—
295C2×D4—5381no—
296C2×D4—5381no—
297C2×D4—5381no—
298C2×D4—5381no—
299C2×D4—5382no—
300C2×D4—5381no—
301C2×D4—5381no—
302C2×D4—5381no—
303C2×D4—10761no—
304C2×D4—10764no—
305C2×D4C22×D26911yescharacteristic
306C2×D4C22×D26911yescharacteristic
307C22×C4—2691no—
308C22×C4—5381no—
309C22×C4—5381no—
310C22×C4—5381no—
311C22×C4C22×D26911yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—2694no—
315SD16—5382no—
316SD16—5382no—
317SD16—5382no—
318D8—5382no—
319D8—5382no—
320D8—10761no—
321M16—5382no—
322C2×C8—2691no—
323C2×C8—2691no—
324C2×C8—5382no—
325C2×C8C22×D26911yescharacteristic
326C2×C8C22×D26911yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2691no—
329C4⋊C4—2691no—
330C4⋊C4—5381no—
331C4⋊C4C22×D26911yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—2691no—
334C22⋊C4—2691no—
335C22⋊C4—5381no—
336C22⋊C4—5381no—
337C22⋊C4—5381no—
338C22⋊C4—5381no—
339C22⋊C4—5381no—
340C22⋊C4—5381no—
341C42—2691no—
342C42—5381no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—2691no—
349C23—5381no—
350C23—5381no—
351C23—10762no—
352C23—10762no—
353C23—21521no—
354C23D4×D26911yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2692no—
358Q8—5381no—
359Q8—10762no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—5382no—
365D4—5381no—
366D4—5381no—
367D4—5381no—
368D4—5381no—
369D4—10762no—
370D4—10762no—
371D4—10762no—
372D4—10762no—
373D4—10762no—
374D4—10762no—
375D4—21521no—
376D4D4×D26912yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—2691no—
382C2×C4—2691no—
383C2×C4—2691no—
384C2×C4—5381no—
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—21no—
395C8—2692no—
396C8—5381no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—2692no—
408C22—5381no—
409C22—5381no—
410C22—10761no—
411C22—10761no—
412C22—10761no—
413C22—10761no—
414C22—10761no—
415C22—10761no—
416C22—21524no—
417C22C2×D4×D26911yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—5381no—
422C4—5381no—
423C4—5382no—
424C4—5381no—
425C4—10761no—
426C4C2×D4×D26911yescharacteristic
427C4#2257549811yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—5382no—
432C2—21521no—
433C2—21521no—
434C2#3177123711yescharacteristic
435C2AGL(1,ℤ/8)×D26911yescharacteristic
436C2#3177141111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#450×C269C2
socleC22×C269C22×D4
hypercenter#450D269
hypocenterC269#3265
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#22575479C2semidirect
2. C2×SD16×D269C2semidirect
3. #31771265C2semidirect
4. D4×C22⋊D269C2semidirect
5. #31770991C2semidirect
6. #31770853C2semidirect
7. #31770840C2semidirect
8. #31770822C2semidirect
9. #31770804C2semidirect
10. #31770752C2semidirect
11. #31770742C2semidirect
12. #31770740C2semidirect
13. #31770702C2semidirect
14. #31770679C2semidirect
15. #450×C269C2semidirect
16. C2×#16864402C22semidirect
17. C2×Q8×D269C22semidirect
21. C2×#16864391C22semidirect
23. C2×#16864390C22semidirect
24. C2×#16864389C22semidirect
26. C2×D2152C22semidirect
28. C2×#16864387C22semidirect
29. C2×#16864386C22semidirect
30. C2×C8×D269C22semidirect
32. #22575442C22semidirect
35. #22575428C22semidirect
42. #22575418C22semidirect
46. #22575386C22semidirect
47. #22575382C22semidirect
48. #22575378C22semidirect
49. #22575377C22semidirect
50. #22575375C22semidirect
51. #22575373C22semidirect
52. #22575365C22semidirect
53. #22575363C22semidirect
56. C2×SD16×C269C22semidirect
57. C22⋊Q8×C269C22semidirect
59. #102×C269C22semidirect
60. C22⋊C8×C269C22semidirect
87. #16864390D4semidirect
132. Dic538C2×D4semidirect
156. #14524130C2×D4semidirect
198. C2×D269AGL(1,ℤ/8)semidirect
218. C269#3265semidirect
229. #450D269semidirect
247. C2×SD16C2×D269semidirect
257. C22⋊Q8C2×D269semidirect
276. #102C2×D269semidirect
278. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754792
C2×SD16×D2692
#317712652
D4×C22⋊D2692
#317709912
#317708532
#317708402
#317708222
#317708042
#317707522
#317707422
#317707402
#317707022
#317706792
#450×C2692
#3265269