Finite Group Databasebeta

← #46822656

Subgroups — #46822656

Order 34432.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771420C211yesmaximal, characteristic, complemented
2#31771413C211yesmaximal, characteristic, complemented
3#31771412C211yesmaximal, characteristic, complemented
4#31770998C211yesmaximal, characteristic, complemented
5#31770878C211yesmaximal, characteristic, complemented
6#31770877C211yesmaximal, characteristic, complemented
7#31770870C211yesmaximal, characteristic, complemented
8#31770868C211yesmaximal, characteristic, complemented
9#31770788C211yesmaximal, characteristic, complemented
10#31770786C211yesmaximal, characteristic, complemented
11#31770781C211yesmaximal, characteristic, complemented
12#31770778C211yesmaximal, characteristic, complemented
13#31770727C211yesmaximal, characteristic, complemented
14#31770725C211yesmaximal, characteristic, complemented
15#578×C269C211yesmaximal, characteristic, complemented
16#22575500C2211yescharacteristic
17#22575499C2211yescharacteristic
18#22575485—21no—
19#22575484—21no—
20#22575484—21no—
21#22575484C2211yescharacteristic
22#22575484C2211yescharacteristic
23#22575483—21no—
24#22575483C2211yescharacteristic
25#22575483C2211yescharacteristic, complemented
26#22575478—21no—
27#22575478C2211yescharacteristic
28#22575478C2211yescharacteristic
29#22575477—21no—
30#22575474—21no—
31#22575473—21no—
32#22575473—21no—
33#22575473C2211yescharacteristic
34C2×#16864392—21no—
35C2×#16864392C2211yescharacteristic, complemented
36C2×#16864391C2211yescharacteristic
37C2×#16864389—21no—
38C2×#16864389C2211yescharacteristic
39SD16×D269—21no—
40#22575460C2211yescharacteristic
41#22575460C2211yescharacteristic, complemented
42#22575459C2211yescharacteristic
43#22575451C2211yescharacteristic, complemented
44#22575442C2211yescharacteristic
45#22575389C2211yescharacteristic, complemented
46#22575388C2211yescharacteristic, complemented
47#22575380C2211yescharacteristic
48#22575380C2211yescharacteristic, complemented
49#22575380C2211yescharacteristic, complemented
50#22575379C2211yescharacteristic
51#22575372C2211yescharacteristic, complemented
52#22575372C2211yescharacteristic, complemented
53#22575371C2211yescharacteristic, complemented
54#22575369C2211yescharacteristic
55#22575363C2211yescharacteristic, complemented
562−1+4×C269C2211yescharacteristic
572−1+4×C269C2211yescharacteristic, complemented
58#137×C269—21no—
59#137×C269—21no—
60#137×C269C2211yescharacteristic
61#137×C269C2211yescharacteristic, complemented
62#135×C269—21no—
63#135×C269C2211yescharacteristic
64C2×Q16×C269—21no—
65C2×Q16×C269C2211yescharacteristic, complemented
66#131×C269C2211yescharacteristic, complemented
67#16864403—21no—
68#16864402—21no—
69#16864402—21no—
70#16864402C2311yescharacteristic
71#16864401—21no—
72#16864401—41no—
73C2×Dic538—21no—
74C2×Dic538C2311yescharacteristic
75Q8×D269—21no—
76C2×D1076—21no—
77C2×D1076C2311yescharacteristic
78D4×D269—21no—
79D4×D269—41no—
80#16864392—21no—
81#16864392—21no—
82#16864392—21no—
83#16864392—21no—
84#16864392—21no—
85#16864392—21no—
86#16864392D411yescharacteristic
87#16864391—21no—
88#16864391—21no—
89#16864390—21no—
90#16864390—21no—
91#16864390D411yescharacteristic
92#16864389—21no—
93#16864389—21no—
94#16864389—21no—
95#16864389—21no—
96#16864389—21no—
97#16864389—21no—
98#16864389D411yescharacteristic
99#16864387—21no—
100#16864387—21no—
101#16864387D411yescharacteristic
102#16864386—21no—
103#16864386—21no—
104#16864386C2311yescharacteristic
105#16864386C2311yescharacteristic
106#16864386C2311yescharacteristic
107#16864385—21no—
108C2×#14524130—21no—
109C2×#14524130—21no—
110C2×#14524130C2311yescharacteristic
111C8×D269—21no—
112#16864381—41no—
113#16864379—41no—
114C4×Dic269C2311yescharacteristic
115#41×C269—21no—
116#41×C269—21no—
117#41×C269—21no—
118#41×C269—21no—
119#41×C269—41no—
120#41×C269—41no—
121#41×C269C2311yescharacteristic
122#41×C269C2311yescharacteristic
123#41×C269C2311yescharacteristic
124C2×Q8×C269—21no—
125C2×Q8×C269—21no—
126C2×Q8×C269—21no—
127C2×Q8×C269—21no—
128C2×Q8×C269C2311yescharacteristic
129C2×Q8×C269C2311yescharacteristic
130Q16×C269—21no—
131Q16×C269—21no—
132Q16×C269—21no—
133Q16×C269—21no—
134Q16×C269D411yescharacteristic
135SD16×C269—21no—
136SD16×C269—21no—
137SD16×C269D411yescharacteristic
138SD16×C269D411yescharacteristic
139D8×C269D411yescharacteristic
140M16×C269—21no—
141M16×C269C2311yescharacteristic
142C2×C8×C269—21no—
143C2×C8×C269C2311yescharacteristic
144C22×D269—21no—
145C22×D269—41no—
146Dic538—21no—
147Dic538—21no—
148Dic538—41no—
149Dic538C2×D411yescharacteristic
150D1076—21no—
151D1076—21no—
152D1076—41no—
153D1076C2×D411yescharacteristic
154C22⋊D269—21no—
155C22⋊D269—41no—
156C22⋊D269—41no—
157C2×Dic269—21no—
158C2×Dic269—21no—
159C2×Dic269—41no—
160C4×D269—21no—
161C4×D269—41no—
162C4×D269—41no—
163#14524130—21no—
164#14524130—21no—
165#14524130—21no—
166#14524130C2×D411yescharacteristic
167#14524130C2×D411yescharacteristic
168Q8×C269—21no—
169Q8×C269—21no—
170Q8×C269—21no—
171Q8×C269—21no—
172Q8×C269—41no—
173Q8×C269—41no—
174Q8×C269C2×D411yescharacteristic
175Q8×C269C2×D411yescharacteristic
176Q8×C269C2×D411yescharacteristic
177D4×C269—21no—
178D4×C269—21no—
179D4×C269—21no—
180D4×C269—41no—
181D4×C269—41no—
182D4×C269C2×D411yescharacteristic
183D4×C269C2×D411yescharacteristic
184D4×C269C2×D411yescharacteristic
185C2×C4×C269—21no—
186C2×C4×C269—21no—
187C2×C4×C269—21no—
188C2×C4×C269—21no—
189C2×C4×C269—21no—
190C2×C4×C269—41no—
191C2×C4×C269—41no—
192C2×C4×C269—41no—
193C2×C4×C269—41no—
194C2×C4×C269C2411yescharacteristic
195C8×C269—21no—
196C8×C269C2×D411yescharacteristic
197C8×C269C2×D411yescharacteristic
198C22×C269—21no—
199C22×C269—21no—
200C22×C269—21no—
201C22×C269H(5,𝔽2)11yescharacteristic
202C2×D269—21no—
203C2×D269—41no—
204C2×D269—81no—
205C2×D269—161no—
206C4×C269—21no—
207C4×C269—21no—
208C4×C269—21no—
209C4×C269—41no—
210C4×C269—41no—
211C4×C269C22×D411yescharacteristic
212C4×C269C22×D411yescharacteristic
213Dic269—21no—
214Dic269—41no—
215Dic269—41no—
216C2×C269—21no—
217C2×C269—41no—
218C2×C269—41no—
219C2×C269—41no—
220C2×C269D4211yescharacteristic
221D269—41no—
222D269—81no—
223C269#328911yescharacteristic, complemented, Sylow
224#3289—2691nomaximal, Sylow
225#578—2691no—
226#578D26911yescharacteristic, complemented
227#577—2691no—
228#577—2691no—
229#497—2691no—
230#473—2691no—
231#473—2691no—
232#471—2691no—
233#471—2691no—
234#456—2691no—
235#456—2691no—
236#455—2691no—
237#454—2691no—
238#444—2691no—
239#444—2691no—
2402−1+4—2691no—
2412−1+4C2×D26911yescharacteristic
2422−1+4C2×D26911yescharacteristic, complemented
243H(5,𝔽2)—2691no—
244#137—21no—
245#137—21no—
246#137—2691no—
247#137—2691no—
248#137—2691no—
249#137—2691no—
250#137—5381no—
251#137—5381no—
252#137—5381no—
253#137—5381no—
254#137C2×D26911yescharacteristic
255#137C2×D26911yescharacteristic, complemented
256AGL(1,ℤ/8)—2691no—
257AGL(1,ℤ/8)—2691no—
258AGL(1,ℤ/8)—5381no—
259AGL(1,ℤ/8)—5381no—
260#135—21no—
261#135—2691no—
262#135—5381no—
263#135—5381no—
264#135—5381no—
265#135C2×D26911yescharacteristic
266C2×Q16—21no—
267C2×Q16—2691no—
268C2×Q16—5381no—
269C2×Q16C2×D26911yescharacteristic, complemented
270C2×SD16—2691no—
271C2×SD16—2691no—
272C2×SD16—5381no—
273C2×SD16—5381no—
274#131—2691no—
275#131—2691no—
276#131—2691no—
277#131C2×D26911yescharacteristic, complemented
278C4⋊Q8—2691no—
279#124—2691no—
280#108—2691no—
281#108—2691no—
282#104—2691no—
283#104—2691no—
284#104—2691no—
285#104—2691no—
286#101—2691no—
287#101—2691no—
288#101—2691no—
289#100—2691no—
290#97—2691no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—41no—
296#41—41no—
297#41—2691no—
298#41—5381no—
299#41—5381no—
300#41—5381no—
301#41—5381no—
302#41—10761no—
303#41C22×D26911yescharacteristic
304#41C22×D26911yescharacteristic
305#41C22×D26911yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—2691no—
311C2×Q8—5381no—
312C2×Q8—5381no—
313C2×Q8C22×D26911yescharacteristic
314C2×Q8C22×D26911yescharacteristic
315C2×D4—2691no—
316C2×D4—5381no—
317C2×D4—5381no—
318C2×D4—10761no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—2691no—
324Q16—5381no—
325Q16—5381no—
326Q16—5381no—
327Q16—5381no—
328Q16—5381no—
329Q16—5381no—
330Q16C22⋊D26911yescharacteristic
331SD16—21no—
332SD16—21no—
333SD16—2691no—
334SD16—2691no—
335SD16—5381no—
336SD16—5381no—
337SD16—5381no—
338SD16—5381no—
339SD16—5381no—
340SD16—5381no—
341SD16—5381no—
342SD16—5381no—
343SD16—5381no—
344SD16—5381no—
345SD16C22⋊D26911yescharacteristic
346SD16C22⋊D26911yescharacteristic
347D8—2691no—
348D8—5381no—
349D8—5381no—
350D8C22⋊D26911yescharacteristic
351M16—21no—
352M16—2691no—
353M16—2691no—
354M16—2691no—
355M16—5381no—
356M16—5381no—
357M16—5381no—
358M16C22×D26911yescharacteristic
359C2×C8—21no—
360C2×C8—2691no—
361C2×C8—5381no—
362C2×C8—5381no—
363C2×C8—5381no—
364C2×C8C22×D26911yescharacteristic
365C4⋊C4—10761no—
366C22⋊C4—10761no—
367C42—2691no—
368C23—5381no—
369C23—10761no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—41no—
376Q8—2691no—
377Q8—5381no—
378Q8—5381no—
379Q8—10761no—
380Q8D4×D26911yescharacteristic
381Q8C2×C22⋊D26911yescharacteristic
382Q8C2×C22⋊D26911yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—41no—
387D4—41no—
388D4—2691no—
389D4—5381no—
390D4—5381no—
391D4—5381no—
392D4—10761no—
393D4—10761no—
394D4—10761no—
395D4D4×D26911yescharacteristic
396D4C2×C22⋊D26911yescharacteristic
397D4C2×C22⋊D26911yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—5381no—
408C2×C4—5381no—
409C2×C4—5381no—
410C2×C4—10761no—
411C2×C4—10761no—
412C2×C4—10761no—
413C2×C4C23×D26911yescharacteristic
414C8—21no—
415C8—2691no—
416C8—2691no—
417C8—5381no—
418C8—5381no—
419C8—5381no—
420C8C2×C22⋊D26911yescharacteristic
421C8C2×C22⋊D26911yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—5381no—
426C22—10761no—
427C22—21521no—
428C22—43041no—
429C22#2257549811yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—5381no—
436C4—10761no—
437C4—10761no—
438C4C2×D4×D26911yescharacteristic
439C4C22×C22⋊D26911yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—10761no—
445C2—21521no—
446C2D4×C22⋊D26911yescharacteristic
subgroupG/H
centerC2D4×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#578×C269C2
socleC2×C269D42
hypercenter#578D269
hypocenterC269#3289
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #31771420C2semidirect
2. #31771413C2semidirect
3. #31771412C2semidirect
4. #31770998C2semidirect
5. #31770878C2semidirect
6. #31770877C2semidirect
7. #31770870C2semidirect
8. #31770868C2semidirect
9. #31770788C2semidirect
10. #31770786C2semidirect
11. #31770781C2semidirect
12. #31770778C2semidirect
13. #31770727C2semidirect
14. #31770725C2semidirect
15. #578×C269C2semidirect
25. #22575483C22semidirect
35. C2×#16864392C22semidirect
41. #22575460C22semidirect
43. #22575451C22semidirect
45. #22575389C22semidirect
46. #22575388C22semidirect
48. #22575380C22semidirect
49. #22575380C22semidirect
51. #22575372C22semidirect
52. #22575372C22semidirect
53. #22575371C22semidirect
55. #22575363C22semidirect
57. 2−1+4×C269C22semidirect
61. #137×C269C22semidirect
65. C2×Q16×C269C22semidirect
66. #131×C269C22semidirect
223. C269#3289semidirect
226. #578D269semidirect
242. 2−1+4C2×D269semidirect
255. #137C2×D269semidirect
269. C2×Q16C2×D269semidirect
277. #131C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714202
#317714132
#317714122
#317709982
#317708782
#317708772
#317708702
#317708682
#317707882
#317707862
#317707812
#317707782
#317707272
#317707252
#578×C2692
#3289269