Finite Group Databasebeta

← #46822424

Subgroups — #46822424

Order 34432.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575477C211yesmaximal, characteristic, complemented
2C2×#22575474C211yesmaximal, characteristic, complemented
3#31771275C211yesmaximal, characteristic, complemented
4#31771248C211yesmaximal, characteristic, complemented
5#31770848C211yesmaximal, characteristic, complemented
6#31770838C211yesmaximal, characteristic, complemented
7#31770824C211yesmaximal, characteristic, complemented
8#31770797C211yesmaximal, characteristic, complemented
9#31770764C211yesmaximal, characteristic, complemented
10#31770749C211yesmaximal, characteristic, complemented
11#31770747C211yesmaximal, characteristic, complemented
12#31770737C211yesmaximal, characteristic, complemented
13#31770707C211yesmaximal, characteristic, complemented
14#31770678C211yesmaximal, characteristic, complemented
15#496×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864401C2211yescharacteristic
18C2×#16864401C2211yescharacteristic, complemented
19C2×D4×D269C2211yescharacteristic
20#22575477—24no—
21#22575474—24no—
22C2×#16864392C2211yescharacteristic, complemented
23C2×#16864391C2211yescharacteristic
24C2×#16864390C2211yescharacteristic
25C2×#16864387C2211yescharacteristic
26C2×#16864387C2211yescharacteristic
27C2×#16864385C2211yescharacteristic
28C2×C8×D269C2211yescharacteristic, complemented
29#22575449—21no—
30#22575435—21no—
31#22575429C2211yescharacteristic
32#22575426—21no—
33#22575425—21no—
34#22575425—21no—
35#22575424C2211yescharacteristic
36#22575424C2211yescharacteristic
37#22575424C2211yescharacteristic, complemented
38#22575423—21no—
39C4×Dic538C2211yescharacteristic, complemented
40C4×D1076C2211yescharacteristic
41C4×D1076C2211yescharacteristic
42D4×Dic269—21no—
43C22⋊C4×D269—21no—
44#22575387C2211yescharacteristic, complemented
45C4⋊#14524130C2211yescharacteristic, complemented
46#22575378C2211yescharacteristic, complemented
47#22575377C2211yescharacteristic, complemented
48#22575375C2211yescharacteristic
49#22575374C2211yescharacteristic
50#22575374C2211yescharacteristic, complemented
51#22575374C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic
53#22575365C2211yescharacteristic, complemented
54C2×SD16×C269C2211yescharacteristic, complemented
55C2×SD16×C269C2211yescharacteristic, complemented
56C2×D8×C269C2211yescharacteristic
57C2×D8×C269C2211yescharacteristic
58#127×C269C2211yescharacteristic, complemented
59#124×C269C2211yescharacteristic
60#97×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62#16864403—22no—
63#16864403—41no—
64#16864401—22no—
65#16864401—22no—
66#16864401—41no—
67#16864401—41no—
68C2×Dic538C2311yescharacteristic
69C2×D1076—21no—
70C2×D1076D411yescharacteristic
71C2×D1076C2311yescharacteristic
72C2×C22⋊D269—21no—
73C2×C22⋊D269—21no—
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76C2×C22⋊D269—21no—
77C2×C22⋊D269—21no—
78D4×D269—22no—
79D4×D269—41no—
80C22×Dic269—21no—
81C22×Dic269—21no—
82C2×C4×D269—21no—
83C2×C4×D269—21no—
84C2×C4×D269C2311yescharacteristic
85C2×C4×D269C2311yescharacteristic
86#16864392—22no—
87#16864391D414yescomplemented
88#16864390—22no—
89#16864387—22no—
90#16864387—22no—
91#16864385—22no—
92C2×#14524130C2311yescharacteristic
93C2×#14524130C2311yescharacteristic
94C8×D269—22no—
95C4⋊Dic269D411yescharacteristic
96C4⋊Dic269C2311yescharacteristic
97C4⋊Dic269C2311yescharacteristic
98#16864381—21no—
99#16864381—21no—
100C22⋊Dic269—21no—
101C22⋊Dic269—21no—
102C22⋊Dic269—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379D411yescharacteristic
106#16864379D411yescharacteristic
107C4×Dic269—21no—
108C2×Q8×C269C2311yescharacteristic
109C2×D4×C269—21no—
110C2×D4×C269—21no—
111C2×D4×C269C2311yescharacteristic
112C2×D4×C269C2311yescharacteristic
113C2×D4×C269C2311yescharacteristic
114SD16×C269—22no—
115SD16×C269—41no—
116D8×C269—22no—
117D8×C269—41no—
118C2×C8×C269C2311yescharacteristic
119C2×C8×C269C2311yescharacteristic
120C22⋊C4×C269—21no—
121C22⋊C4×C269—21no—
122C42×C269C2311yescharacteristic
123C22×D269—21no—
124C22×D269—42no—
125C22×D269—42no—
126C22×D269—81no—
127C22×D269C2×D411yescharacteristic
128C22×D269C2×D411yescharacteristic
129Dic538—21no—
130Dic538C2×D412yes—
131D1076—21no—
132D1076—41no—
133D1076—42no—
134D1076C2×D412yes—
135C22⋊D269—42no—
136C22⋊D269—42no—
137C22⋊D269—42no—
138C22⋊D269—81no—
139C22⋊D269—81no—
140C22⋊D269—81no—
141C2×Dic269—21no—
142C2×Dic269—21no—
143C2×Dic269—21no—
144C2×Dic269—42no—
145C2×Dic269—42no—
146C2×Dic269C2×D411yescharacteristic
147C2×Dic269C2×D411yescharacteristic
148C4×D269—21no—
149C4×D269—21no—
150C4×D269—21no—
151C4×D269—21no—
152C4×D269—42no—
153C4×D269—81no—
154#14524130—21no—
155#14524130—21no—
156C23×C269—21no—
157C23×C269—21no—
158C23×C269—21no—
159Q8×C269—21no—
160Q8×C269—21no—
161D4×C269—21no—
162D4×C269—21no—
163D4×C269—21no—
164D4×C269—21no—
165D4×C269—21no—
166D4×C269—21no—
167D4×C269—42no—
168D4×C269—81no—
169C2×C4×C269—21no—
170C2×C4×C269C2×D411yescharacteristic
171C2×C4×C269C2×D411yescharacteristic
172C2×C4×C269C2411yescharacteristic
173C8×C269—21no—
174C8×C269C2×D412yes—
175C22×C269—41no—
176C22×C269—41no—
177C22×C269—41no—
178C22×C269—41no—
179C22×C269—41no—
180C22×C269—41no—
181C22×C269—41no—
182C22×C269—41no—
183C22×C269—41no—
184C22×C269C22×D411yescharacteristic
185C2×D269—21no—
186C2×D269—21no—
187C2×D269—22no—
188C2×D269—21no—
189C2×D269—21no—
190C2×D269—21no—
191C2×D269—21no—
192C2×D269—41no—
193C2×D269—41no—
194C2×D269—84no—
195C4×C269—21no—
196C4×C269—21no—
197C4×C269—41no—
198C4×C269C22×D411yescharacteristic
199C4×C269H(5,𝔽2)11yescharacteristic
200Dic269—21no—
201Dic269—21no—
202Dic269—22no—
203Dic269—41no—
204Dic269—41no—
205C2×C269—81no—
206C2×C269—81no—
207C2×C269—81no—
208C2×C269D4211yescharacteristic
209C2×C269#57511yescharacteristic
210C2×C269#57611yescharacteristic
211D269—41no—
212D269—41no—
213D269—42no—
214C269#326811yescharacteristic, complemented, Sylow
215#3268—2691nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2691no—
217C2×#135—2691no—
218#550—2691no—
219#546—2691no—
220#496D26911yescharacteristic, complemented
221#468—2691no—
222#466—2691no—
223#464—2691no—
224#459—2691no—
225#452—2691no—
226#449—2691no—
227#449—2691no—
228#447—2691no—
229#440—2691no—
230#435—2691no—
231C2×#41—2691no—
232C2×#41—2691no—
233C2×#41—2691no—
234C22×D4—2691no—
235AGL(1,ℤ/8)—5384no—
236#135—5384no—
237C2×Q16—2691no—
238C2×SD16—2691no—
239C2×SD16—2691no—
240C2×SD16C2×D26911yescharacteristic, complemented
241C2×SD16C2×D26911yescharacteristic, complemented
242C2×D8—2691no—
243C2×D8—2691no—
244C2×D8C2×D26911yescharacteristic
245C2×D8C2×D26911yescharacteristic
246C2×M16—2691no—
247C22×C8—2691no—
248#127—5381no—
249#127C2×D26911yescharacteristic, complemented
250#124C2×D26911yescharacteristic
251#123—5381no—
252C22⋊Q8—2691no—
253#121—2691no—
254#121—2691no—
255#121—2691no—
256#121—5381no—
257#121—5381no—
258#121—5381no—
259#120—5381no—
260C4×Q8—2691no—
261C4×D4—2691no—
262C4×D4—2691no—
263C4×D4—5381no—
264C2×C22⋊C4—5381no—
265#107—2691no—
266C4⋊C8—2691no—
267#103—2691no—
268#103—2691no—
269#102—2691no—
270#102—2691no—
271#102—2691no—
272#102—2691no—
273C22⋊C8—2691no—
274C22⋊C8—2691no—
275#97C2×D26911yescharacteristic, complemented
276C24—5381no—
277#41—5382no—
278#41—5382no—
279#41—5382no—
280#41—10761no—
281#41—10761no—
282#41—10761no—
283C2×Q8—2691no—
284C2×Q8C22×D26911yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2691no—
288C2×D4—2691no—
289C2×D4—5381no—
290C2×D4—5381no—
291C2×D4—5382no—
292C2×D4—5381no—
293C2×D4—5381no—
294C2×D4—5381no—
295C2×D4—5381no—
296C2×D4—5381no—
297C2×D4—10761no—
298C2×D4C22×D26911yescharacteristic
299C2×D4C22×D26911yescharacteristic
300C2×D4C22×D26911yescharacteristic
301C22×C4—2691no—
302C22×C4—2691no—
303C22×C4—5381no—
304C22×C4—5381no—
305C22×C4—5381no—
306C22×C4—5381no—
307Q16—5382no—
308SD16—22no—
309SD16—41no—
310SD16—2694no—
311SD16—5382no—
312D8—22no—
313D8—41no—
314D8—5382no—
315D8—5382no—
316M16—5382no—
317C2×C8—2691no—
318C2×C8—2691no—
319C2×C8—5382no—
320C2×C8C22×D26911yescharacteristic
321C2×C8C22×D26911yescharacteristic
322C4⋊C4—2691no—
323C4⋊C4—2691no—
324C4⋊C4—2691no—
325C4⋊C4—5381no—
326C4⋊C4—5381no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—2691no—
330C22⋊C4—2691no—
331C22⋊C4—5381no—
332C22⋊C4—5381no—
333C22⋊C4—5381no—
334C22⋊C4—5381no—
335C22⋊C4—5381no—
336C42—5381no—
337C42C22×D26911yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2691no—
342C23—2691no—
343C23—5381no—
344C23—10762no—
345C23—10762no—
346C23—21521no—
347Q8—21no—
348Q8—21no—
349Q8—2692no—
350Q8—5381no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—2692no—
360D4—5381no—
361D4—10761no—
362D4—10762no—
363D4—10762no—
364D4—10762no—
365D4—10762no—
366D4—21521no—
367D4—21521no—
368D4—21521no—
369C2×C4—21no—
370C2×C4—2691no—
371C2×C4—2691no—
372C2×C4—5381no—
373C2×C4—5381no—
374C2×C4—5381no—
375C2×C4—5381no—
376C2×C4—5381no—
377C2×C4—5381no—
378C2×C4—5381no—
379C2×C4—10762no—
380C2×C4—10762no—
381C2×C4—10762no—
382C2×C4—21521no—
383C2×C4D4×D26911yescharacteristic
384C2×C4D4×D26911yescharacteristic
385C2×C4C23×D26911yescharacteristic
386C8—21no—
387C8—5381no—
388C8—5381no—
389C8D4×D26912yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—5381no—
400C22—5381no—
401C22—5381no—
402C22—5381no—
403C22—5381no—
404C22—5381no—
405C22—5382no—
406C22—10761no—
407C22—10761no—
408C22—21524no—
409C22C2×D4×D26911yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5381no—
414C4—5381no—
415C4—5382no—
416C4—10761no—
417C4—10761no—
418C4C2×D4×D26911yescharacteristic
419C4#2257549811yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—10761no—
424C2—10761no—
425C2—10762no—
426C2#3177124011yescharacteristic
427C2#3177140111yescharacteristic
428C2#3177140911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#496×C269C2
socleC22×C269C22×D4
hypercenter#496D269
hypocenterC269#3268
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22575477C2semidirect
2. C2×#22575474C2semidirect
3. #31771275C2semidirect
4. #31771248C2semidirect
5. #31770848C2semidirect
6. #31770838C2semidirect
7. #31770824C2semidirect
8. #31770797C2semidirect
9. #31770764C2semidirect
10. #31770749C2semidirect
11. #31770747C2semidirect
12. #31770737C2semidirect
13. #31770707C2semidirect
14. #31770678C2semidirect
15. #496×C269C2semidirect
18. C2×#16864401C22semidirect
22. C2×#16864392C22semidirect
28. C2×C8×D269C22semidirect
37. #22575424C22semidirect
39. C4×Dic538C22semidirect
44. #22575387C22semidirect
45. C4⋊#14524130C22semidirect
46. #22575378C22semidirect
47. #22575377C22semidirect
50. #22575374C22semidirect
51. #22575374C22semidirect
53. #22575365C22semidirect
54. C2×SD16×C269C22semidirect
55. C2×SD16×C269C22semidirect
58. #127×C269C22semidirect
60. #97×C269C22semidirect
87. #16864391D4semidirect
214. C269#3268semidirect
220. #496D269semidirect
240. C2×SD16C2×D269semidirect
241. C2×SD16C2×D269semidirect
249. #127C2×D269semidirect
275. #97C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754772
C2×#225754742
#317712752
#317712482
#317708482
#317708382
#317708242
#317707972
#317707642
#317707492
#317707472
#317707372
#317707072
#317706782
#496×C2692
#3268269