Finite Group Databasebeta

← #46822579

Subgroups — #46822579

Order 34432.

546 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
1C22×#16864390C211yesmaximal, characteristic, complemented
2C2×SD16×D269C211yesmaximal, characteristic, complemented
3#31771268C211yesmaximal, characteristic, complemented
4D4×C22⋊D269C211yesmaximal, characteristic
5#31770976C211yesmaximal, characteristic, complemented
6#31770836C211yesmaximal, characteristic, complemented
7#31770833C211yesmaximal, characteristic, complemented
8#31770808C211yesmaximal, characteristic, complemented
9#31770804C211yesmaximal, characteristic, complemented
10#31770757C211yesmaximal, characteristic, complemented
11#31770746C211yesmaximal, characteristic, complemented
12#31770742C211yesmaximal, characteristic, complemented
13#31770696C211yesmaximal, characteristic, complemented
14#31770668C211yesmaximal, characteristic, complemented
15#450×C269C211yesmaximal, characteristic, complemented
16C22×Dic538C2211yescharacteristic, complemented
17C2×Q8×D269C2211yescharacteristic, complemented
18C22×C22⋊D269—21no—
19C2×D4×D269C2211yescharacteristic
20C2×#16864391C2211yescharacteristic, complemented
21C2×#16864391C2211yescharacteristic, complemented
22C2×#16864390—24no—
23C2×#16864390C2211yescharacteristic
24C2×#16864390C2211yescharacteristic
25C2×#16864390C2211yescharacteristic, complemented
26C2×#16864390C2211yescharacteristic, complemented
27C2×#16864389C2211yescharacteristic, complemented
28SD16×D269—24no—
29C22×#14524130C2211yescharacteristic, complemented
30C2×C8×D269C2211yescharacteristic, complemented
31#22575452—21no—
32#22575451C2211yescharacteristic, complemented
33#22575449C2211yescharacteristic
34#22575433C2211yescharacteristic, complemented
35#22575431—21no—
36#22575430—21no—
37#22575427C2211yescharacteristic
38#22575425—21no—
39#22575424C2211yescharacteristic
40#22575422—21no—
41#22575421—21no—
42#22575418C2211yescharacteristic, complemented
43D4×Dic269C2211yescharacteristic
44C4×C22⋊D269C2211yescharacteristic
45#22575413—21no—
46#22575384C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575378C2211yescharacteristic, complemented
49#22575378C2211yescharacteristic, complemented
50#22575377C2211yescharacteristic, complemented
51#22575373C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53C8×Dic269C2211yescharacteristic, 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—
62C2×Dic538—21no—
63C2×Dic538—22no—
64C2×Dic538—21no—
65C2×Dic538—41no—
66C2×Dic538C2311yescharacteristic
67C2×Dic538C2311yescharacteristic
68Q8×D269—22no—
69Q8×D269—41no—
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—44no—
77C2×C22⋊D269D411yescharacteristic
78D4×D269—22no—
79D4×D269—41no—
80C22×Dic269—21no—
81C22×Dic269—21no—
82C2×C4×D269—21no—
83C2×C4×D269C2311yescharacteristic
84#16864391—22no—
85#16864391—41no—
86#16864390—22no—
87#16864390—22no—
88#16864390—22no—
89#16864390D414yescomplemented
90#16864389—22no—
91C2×#14524130—22no—
92C2×#14524130C2311yescharacteristic
93C2×#14524130C2311yescharacteristic
94C8×D269—22no—
95C4⋊Dic269—21no—
96C4⋊Dic269C2311yescharacteristic
97#16864381—21no—
98#16864381—21no—
99#16864381—21no—
100#16864381D411yescharacteristic
101C22⋊Dic269—21no—
102C22⋊Dic269—21no—
103C22⋊Dic269D411yescharacteristic
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—42no—
126C22×D269—42no—
127C22×D269—81no—
128C22×D269C2×D411yescharacteristic
129Dic538—21no—
130Dic538—21no—
131Dic538—21no—
132Dic538—42no—
133Dic538—42no—
134Dic538C2×D412yescomplemented
135D1076—21no—
136D1076—21no—
137C22⋊D269—42no—
138C22⋊D269—42no—
139C22⋊D269—42no—
140C22⋊D269—42no—
141C22⋊D269—81no—
142C22⋊D269SD1614yescomplemented
143C2×Dic269—21no—
144C2×Dic269—21no—
145C2×Dic269—21no—
146C2×Dic269—42no—
147C2×Dic269—42no—
148C2×Dic269C2×D411yescharacteristic
149C2×Dic269C2×D411yescharacteristic
150C2×Dic269C2×D411yescharacteristic
151C4×D269—21no—
152C4×D269—21no—
153C4×D269—42no—
154#14524130—21no—
155#14524130C2×D412yescomplemented
156C23×C269—21no—
157C23×C269—21no—
158C23×C269—42no—
159C23×C269—42no—
160C23×C269—81no—
161C23×C269C2×D411yescharacteristic
162Q8×C269—21no—
163Q8×C269—21no—
164D4×C269—21no—
165D4×C269—21no—
166D4×C269—21no—
167D4×C269—42no—
168D4×C269C2×D412yes—
169C2×C4×C269—21no—
170C2×C4×C269—21no—
171C2×C4×C269—21no—
172C2×C4×C269—21no—
173C2×C4×C269C2×D411yescharacteristic
174C2×C4×C269C2411yescharacteristic
175C8×C269—21no—
176C8×C269—21no—
177C22×C269—21no—
178C22×C269—21no—
179C22×C269—22no—
180C22×C269—41no—
181C22×C269—41no—
182C22×C269—41no—
183C22×C269—41no—
184C22×C269—41no—
185C22×C269—84no—
186C22×C269C2×SD1612yescomplemented
187C22×C269C22×D411yescharacteristic
188C2×D269—21no—
189C2×D269—21no—
190C2×D269—41no—
191C2×D269—41no—
192C2×D269—41no—
193C2×D269—84no—
194C2×D269C2×SD1612yescomplemented
195C4×C269—21no—
196C4×C269—41no—
197C4×C269—41no—
198C4×C269C22×D411yescharacteristic
199C4×C269H(5,𝔽2)11yescharacteristic
200Dic269—21no—
201Dic269—22no—
202Dic269—21no—
203Dic269—41no—
204Dic269—41no—
205Dic269C2×SD1612yescomplemented
206C2×C269—22no—
207C2×C269—42no—
208C2×C269—81no—
209C2×C269D4211yescharacteristic
210C2×C269C22×SD1611yescharacteristic
211C2×C269#57711yescharacteristic
212D269—22no—
213D269—81no—
214C269D4×SD1611yescharacteristic, complemented, Sylow
215D4×SD16—2691nomaximal, Sylow
216C22×SD16—2691no—
217C22×SD16—2691no—
218D4×Q8—2691no—
219D42—2691no—
220#492—2691no—
221#465—2691no—
222#465—2691no—
223#461—2691no—
224#460—2691no—
225#450—2691no—
226#450D26911yescharacteristic, complemented
227#448—2691no—
228#448—2691no—
229C4×SD16—2691no—
230C8×D4—2691no—
231C22×Q8—2691no—
232C22×Q8—2691no—
233C22×D4—2691no—
234C22×D4—5381no—
235C22×D4C2×D26911yescharacteristic
236C2×SD16—2691no—
237C2×SD16—2691no—
238C2×SD16—2691no—
239C2×SD16—2691no—
240C2×SD16—2691no—
241C2×SD16—2691no—
242C2×SD16—2691no—
243C2×SD16—5384no—
244C2×SD16—5384no—
245C2×SD16C2×D26911yescharacteristic
246C2×SD16C2×D26911yescharacteristic, complemented
247C22×C8—2691no—
248C22×C8—2691no—
249C4⋊Q8—2691no—
250C4⋊Q8—5381no—
251#127—2691no—
252C22⋊Q8—2691no—
253C22⋊Q8—5381no—
254C22⋊Q8—5381no—
255C22⋊Q8C2×D26911yescharacteristic, complemented
256#121—2691no—
257#121—2691no—
258#121—5381no—
259#120—5381no—
260#120—5381no—
261C4×Q8—2691no—
262C4×D4—2691no—
263C4×D4—2691no—
264C4×D4—5381no—
265#106—2691no—
266C4⋊C8—2691no—
267#103—2691no—
268#103—2691no—
269#103—2691no—
270#102—2691no—
271#102C2×D26911yescharacteristic
272#102C2×D26911yescharacteristic, complemented
273C22⋊C8—2691no—
274C22⋊C8C2×D26911yescharacteristic, complemented
275C4×C8—2691no—
276C24—21no—
277C24—5381no—
278C2×Q8—2691no—
279C2×Q8—2691no—
280C2×Q8—5381no—
281C2×Q8—5382no—
282C2×Q8—5381no—
283C2×Q8—5382no—
284C2×Q8—10761no—
285C2×Q8—10761no—
286C2×Q8C22×D26911yescharacteristic
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2691no—
291C2×D4—2691no—
292C2×D4—5381no—
293C2×D4—5381no—
294C2×D4—5381no—
295C2×D4—5382no—
296C2×D4—5381no—
297C2×D4—5381no—
298C2×D4—10761no—
299C2×D4—10764no—
300C2×D4C22×D26911yescharacteristic
301C2×D4C22×D26911yescharacteristic
302C22×C4—2691no—
303C22×C4—5381no—
304C22×C4—5381no—
305C22×C4—5381no—
306C22×C4C22×D26911yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—2694no—
310SD16—5382no—
311SD16—5382no—
312SD16—5382no—
313SD16—5382no—
314SD16—5382no—
315SD16—10761no—
316C2×C8—2691no—
317C2×C8—2691no—
318C2×C8—5382no—
319C2×C8—5382no—
320C2×C8C22×D26911yescharacteristic
321C2×C8C22×D26911yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—2691no—
324C4⋊C4—2691no—
325C4⋊C4—5381no—
326C4⋊C4—5381no—
327C4⋊C4—5381no—
328C4⋊C4—5381no—
329C4⋊C4C22×D26911yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—2691no—
332C22⋊C4—2691no—
333C22⋊C4—5381no—
334C22⋊C4—5381no—
335C22⋊C4—5381no—
336C22⋊C4—5381no—
337C42—2691no—
338C42—5381no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—2691no—
345C23—5381no—
346C23—10762no—
347C23—10762no—
348C23—21521no—
349C23D4×D26911yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—2692no—
353Q8—5381no—
354Q8—5381no—
355Q8—5381no—
356Q8—10762no—
357Q8—10762no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—2694no—
363D4—5381no—
364D4—5381no—
365D4—10762no—
366D4—10762no—
367D4—10762no—
368D4—10762no—
369D4—21521no—
370D4D4×D26912yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—2691no—
376C2×C4—2691no—
377C2×C4—2691no—
378C2×C4—5381no—
379C2×C4—5381no—
380C2×C4—5381no—
381C2×C4—5381no—
382C2×C4—5381no—
383C2×C4—10762no—
384C2×C4—10762no—
385C2×C4—10762no—
386C2×C4D4×D26911yescharacteristic
387C2×C4C23×D26911yescharacteristic
388C8—21no—
389C8—21no—
390C8—2692no—
391C8—5381no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—84no—
401C22—2692no—
402C22—5381no—
403C22—5381no—
404C22—10761no—
405C22—10761no—
406C22—10761no—
407C22—21524no—
408C22SD16×D26912yescomplemented
409C22C2×D4×D26911yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—2692no—
414C4—5381no—
415C4—5381no—
416C4—5382no—
417C4—10761no—
418C4—10761no—
419C4C2×D4×D26911yescharacteristic
420C4#2257549811yescharacteristic
421C2—22no—
422C2—42no—
423C2—81no—
424C2—5382no—
425C2—21521no—
426C2#3177123711yescharacteristic
427C2C2×SD16×D26911yescharacteristic
428C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#450×C269C2
socleC22×C269C22×D4
prime radicalD4×C22⋊D269C2
hypercenter#450D269
hypocenterC269D4×SD16
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. C22×#16864390C2semidirect
2. C2×SD16×D269C2semidirect
3. #31771268C2semidirect
5. #31770976C2semidirect
6. #31770836C2semidirect
7. #31770833C2semidirect
8. #31770808C2semidirect
9. #31770804C2semidirect
10. #31770757C2semidirect
11. #31770746C2semidirect
12. #31770742C2semidirect
13. #31770696C2semidirect
14. #31770668C2semidirect
15. #450×C269C2semidirect
16. C22×Dic538C22semidirect
17. C2×Q8×D269C22semidirect
20. C2×#16864391C22semidirect
21. C2×#16864391C22semidirect
25. C2×#16864390C22semidirect
26. C2×#16864390C22semidirect
27. C2×#16864389C22semidirect
29. C22×#14524130C22semidirect
30. C2×C8×D269C22semidirect
32. #22575451C22semidirect
34. #22575433C22semidirect
42. #22575418C22semidirect
46. #22575384C22semidirect
47. #22575382C22semidirect
48. #22575378C22semidirect
49. #22575378C22semidirect
50. #22575377C22semidirect
51. #22575373C22semidirect
52. #22575365C22semidirect
53. C8×Dic269C22semidirect
56. C2×SD16×C269C22semidirect
57. C22⋊Q8×C269C22semidirect
59. #102×C269C22semidirect
60. C22⋊C8×C269C22semidirect
89. #16864390D4semidirect
134. Dic538C2×D4semidirect
142. C22⋊D269SD16semidirect
155. #14524130C2×D4semidirect
186. C22×C269C2×SD16semidirect
194. C2×D269C2×SD16semidirect
205. Dic269C2×SD16semidirect
214. C269D4×SD16semidirect
226. #450D269semidirect
246. C2×SD16C2×D269semidirect
255. C22⋊Q8C2×D269semidirect
272. #102C2×D269semidirect
274. C22⋊C8C2×D269semidirect
408. C22SD16×D269semidirect

Maximal subgroups

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

subgroupindex
C22×#168643902
C2×SD16×D2692
#317712682
D4×C22⋊D2692
#317709762
#317708362
#317708332
#317708082
#317708042
#317707572
#317707462
#317707422
#317706962
#317706682
#450×C2692
D4×SD16269