Finite Group Databasebeta

← #46822409

Subgroups — #46822409

Order 34432.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575480C211yesmaximal, characteristic, complemented
2C2×#22575476C211yesmaximal, characteristic, complemented
3#31771258C211yesmaximal, characteristic, complemented
4#31771248C211yesmaximal, characteristic, complemented
5#31770860C211yesmaximal, characteristic, complemented
6#31770831C211yesmaximal, characteristic, complemented
7#31770817C211yesmaximal, characteristic, complemented
8#31770807C211yesmaximal, characteristic, complemented
9#31770766C211yesmaximal, characteristic, complemented
10#31770766C211yesmaximal, characteristic, complemented
11#31770754C211yesmaximal, characteristic, complemented
12#31770749C211yesmaximal, characteristic, complemented
13#31770718C211yesmaximal, characteristic, complemented
14#31770678C211yesmaximal, characteristic, complemented
15#497×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864403C2211yescharacteristic
18C2×#16864403C2211yescharacteristic, complemented
19C2×D4×D269C2211yescharacteristic
20#22575480—24no—
21#22575476—24no—
22C2×#16864389C2211yescharacteristic
23C2×#16864389C2211yescharacteristic
24C2×#16864389C2211yescharacteristic, complemented
25C2×D2152C2211yescharacteristic
26C2×#16864387C2211yescharacteristic
27C2×#16864385C2211yescharacteristic
28C2×C8×D269C2211yescharacteristic, complemented
29#22575438—21no—
30#22575438—21no—
31#22575435—21no—
32#22575429C2211yescharacteristic
33#22575429C2211yescharacteristic
34#22575429C2211yescharacteristic, complemented
35#22575428—21no—
36#22575426—21no—
37#22575424C2211yescharacteristic
38#22575423—21no—
39C4×D1076C2211yescharacteristic
40C4×D1076C2211yescharacteristic
41C4×D1076C2211yescharacteristic, complemented
42C4⋊C4×D269—21no—
43C22⋊C4×D269—21no—
44#22575385C2211yescharacteristic, complemented
45C4⋊#14524130C2211yescharacteristic, complemented
46#22575377C2211yescharacteristic, complemented
47#22575377C2211yescharacteristic, complemented
48#22575377C2211yescharacteristic, complemented
49#22575375C2211yescharacteristic
50#22575375C2211yescharacteristic, complemented
51#22575374C2211yescharacteristic
52#22575365C2211yescharacteristic
53#22575365C2211yescharacteristic, complemented
54C2×Q16×C269C2211yescharacteristic, complemented
55C2×Q16×C269C2211yescharacteristic, complemented
56C2×SD16×C269C2211yescharacteristic
57C2×SD16×C269C2211yescharacteristic
58C4⋊Q8×C269C2211yescharacteristic, complemented
59#124×C269C2211yescharacteristic
60#97×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62#16864403—22no—
63#16864403—22no—
64#16864403—22no—
65#16864403—41no—
66#16864403—41no—
67#16864403—41no—
68C2×D1076—21no—
69C2×D1076—21no—
70C2×D1076—21no—
71C2×D1076D411yescharacteristic
72C2×D1076C2311yescharacteristic
73C2×D1076C2311yescharacteristic
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76D4×D269—22no—
77D4×D269—41no—
78C2×C4×D269—21no—
79C2×C4×D269—21no—
80C2×C4×D269—21no—
81C2×C4×D269—21no—
82C2×C4×D269—21no—
83C2×C4×D269C2311yescharacteristic
84C2×C4×D269C2311yescharacteristic
85#16864389—22no—
86#16864389—22no—
87#16864389—22no—
88D2152D414yescomplemented
89#16864387—22no—
90#16864385—22no—
91C2×#14524130C2311yescharacteristic
92C2×#14524130C2311yescharacteristic
93C8×D269—22no—
94C4⋊Dic269D411yescharacteristic
95C4⋊Dic269C2311yescharacteristic
96C4⋊Dic269C2311yescharacteristic
97#16864381—21no—
98#16864381—21no—
99#16864381—21no—
100C22⋊Dic269—21no—
101#16864379—21no—
102#16864379—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379D411yescharacteristic
107#16864379D411yescharacteristic
108C2×Q8×C269C2311yescharacteristic
109C2×Q8×C269C2311yescharacteristic
110C2×Q8×C269C2311yescharacteristic
111C2×D4×C269C2311yescharacteristic
112Q16×C269—22no—
113Q16×C269—41no—
114SD16×C269—22no—
115SD16×C269—41no—
116C2×C8×C269C2311yescharacteristic
117C2×C8×C269C2311yescharacteristic
118C4⋊C4×C269—21no—
119C4⋊C4×C269—21no—
120C22⋊C4×C269—21no—
121C22⋊C4×C269—21no—
122C42×C269C2311yescharacteristic
123C22×D269—21no—
124C22×D269—21no—
125C22×D269—42no—
126C22×D269—42no—
127C22×D269—81no—
128C22×D269C2×D411yescharacteristic
129C22×D269C2×D411yescharacteristic
130D1076—21no—
131D1076—21no—
132D1076—41no—
133D1076—42no—
134D1076—42no—
135D1076—42no—
136D1076C2×D412yes—
137D1076C2×D412yes—
138C22⋊D269—42no—
139C22⋊D269—81no—
140C2×Dic269—21no—
141C2×Dic269—21no—
142C2×Dic269C2×D411yescharacteristic
143C2×Dic269C2×D411yescharacteristic
144C4×D269—21no—
145C4×D269—21no—
146C4×D269—21no—
147C4×D269—21no—
148C4×D269—42no—
149C4×D269—42no—
150C4×D269—42no—
151C4×D269—42no—
152C4×D269—81no—
153#14524130—21no—
154#14524130—21no—
155C23×C269—21no—
156Q8×C269—21no—
157Q8×C269—21no—
158Q8×C269—21no—
159Q8×C269—21no—
160Q8×C269—21no—
161Q8×C269—21no—
162D4×C269—21no—
163D4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—21no—
166C2×C4×C269—21no—
167C2×C4×C269C2×D411yescharacteristic
168C2×C4×C269C2×D411yescharacteristic
169C2×C4×C269C2411yescharacteristic
170C8×C269—21no—
171C8×C269C2×D412yes—
172C22×C269—41no—
173C22×C269—41no—
174C22×C269—41no—
175C22×C269C22×D411yescharacteristic
176C2×D269—21no—
177C2×D269—21no—
178C2×D269—22no—
179C2×D269—22no—
180C2×D269—21no—
181C2×D269—21no—
182C2×D269—21no—
183C2×D269—21no—
184C2×D269—41no—
185C2×D269—41no—
186C2×D269—41no—
187C2×D269—41no—
188C2×D269—84no—
189C4×C269—21no—
190C4×C269—21no—
191C4×C269—41no—
192C4×C269—41no—
193C4×C269—41no—
194C4×C269C22×D411yescharacteristic
195C4×C269H(5,𝔽2)11yescharacteristic
196Dic269—21no—
197Dic269—21no—
198Dic269—41no—
199Dic269—41no—
200C2×C269—81no—
201C2×C269D4211yescharacteristic
202C2×C269#57511yescharacteristic
203C2×C269#57711yescharacteristic
204D269—41no—
205D269—41no—
206D269—42no—
207D269—42no—
208C269#326711yescharacteristic, complemented, Sylow
209#3267—2691nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2691no—
211C2×#135—2691no—
212#547—2691no—
213#546—2691no—
214#497D26911yescharacteristic, complemented
215#470—2691no—
216#465—2691no—
217#463—2691no—
218#461—2691no—
219#452—2691no—
220#452—2691no—
221#450—2691no—
222#449—2691no—
223#442—2691no—
224#435—2691no—
225C2×#41—2691no—
226C2×#41—2691no—
227C2×#41—2691no—
228C22×D4—2691no—
229AGL(1,ℤ/8)—5384no—
230#135—5384no—
231C2×Q16C2×D26911yescharacteristic, complemented
232C2×Q16C2×D26911yescharacteristic, complemented
233C2×SD16—2691no—
234C2×SD16—2691no—
235C2×SD16—2691no—
236C2×SD16C2×D26911yescharacteristic
237C2×SD16C2×D26911yescharacteristic
238C2×D8—2691no—
239C2×D8—2691no—
240C2×M16—2691no—
241C22×C8—2691no—
242C4⋊Q8C2×D26911yescharacteristic, complemented
243#124C2×D26911yescharacteristic
244#123—5381no—
245#123—5381no—
246#123—5381no—
247C22⋊Q8—2691no—
248C22⋊Q8—2691no—
249C22⋊Q8—2691no—
250#121—2691no—
251#121—5381no—
252#121—5381no—
253#120—5381no—
254C4×D4—2691no—
255C4×D4—2691no—
256C4×D4—2691no—
257C2×C4⋊C4—5381no—
258C2×C22⋊C4—5381no—
259#106—2691no—
260C4⋊C8—2691no—
261#103—2691no—
262#103—2691no—
263#103—2691no—
264#102—2691no—
265#102—2691no—
266#102—2691no—
267C22⋊C8—2691no—
268C22⋊C8—2691no—
269#97C2×D26911yescharacteristic, complemented
270C24—5381no—
271#41—5382no—
272#41—5382no—
273#41—5382no—
274#41—10761no—
275#41—10761no—
276#41—10761no—
277C2×Q8C22×D26911yescharacteristic
278C2×Q8C22×D26911yescharacteristic
279C2×Q8C22×D26911yescharacteristic
280C2×D4—2691no—
281C2×D4—2691no—
282C2×D4—2691no—
283C2×D4—5381no—
284C2×D4—5381no—
285C2×D4—5381no—
286C2×D4—5381no—
287C2×D4—5382no—
288C2×D4—5381no—
289C2×D4—10761no—
290C2×D4C22×D26911yescharacteristic
291C22×C4—2691no—
292C22×C4—2691no—
293C22×C4—5381no—
294C22×C4—5381no—
295C22×C4—5381no—
296C22×C4—5381no—
297C22×C4—5381no—
298Q16—22no—
299Q16—41no—
300SD16—22no—
301SD16—41no—
302SD16—5382no—
303SD16—5382no—
304SD16—5382no—
305D8—2694no—
306D8—5382no—
307M16—5382no—
308C2×C8—2691no—
309C2×C8—2691no—
310C2×C8—5382no—
311C2×C8C22×D26911yescharacteristic
312C2×C8C22×D26911yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—2691no—
316C4⋊C4—2691no—
317C4⋊C4—2691no—
318C4⋊C4—5381no—
319C4⋊C4—5381no—
320C4⋊C4—5381no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—2691no—
324C22⋊C4—2691no—
325C22⋊C4—5381no—
326C22⋊C4—5381no—
327C22⋊C4—5381no—
328C22⋊C4—5381no—
329C22⋊C4—5381no—
330C22⋊C4—5381no—
331C42C22×D26911yescharacteristic
332C23—21no—
333C23—2691no—
334C23—2691no—
335C23—5381no—
336C23—5381no—
337C23—10762no—
338C23—10762no—
339C23—21521no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—2692no—
349D4—2692no—
350D4—5381no—
351D4—5381no—
352D4—10761no—
353D4—10762no—
354D4—10762no—
355D4—10762no—
356D4—10762no—
357D4—21521no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—2691no—
362C2×C4—2691no—
363C2×C4—5381no—
364C2×C4—5381no—
365C2×C4—5381no—
366C2×C4—5381no—
367C2×C4—5381no—
368C2×C4—5381no—
369C2×C4—10762no—
370C2×C4—10762no—
371C2×C4—10762no—
372C2×C4—10762no—
373C2×C4—21521no—
374C2×C4D4×D26911yescharacteristic
375C2×C4D4×D26911yescharacteristic
376C2×C4C23×D26911yescharacteristic
377C8—21no—
378C8—5381no—
379C8—5381no—
380C8D4×D26912yes—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—5381no—
385C22—5381no—
386C22—5381no—
387C22—5381no—
388C22—5381no—
389C22—5381no—
390C22—5382no—
391C22—5382no—
392C22—10761no—
393C22—10761no—
394C22—10761no—
395C22—10761no—
396C22—21524no—
397C22C2×D4×D26911yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—5381no—
404C4—5381no—
405C4—10761no—
406C4—10761no—
407C4C2×D4×D26911yescharacteristic
408C4#2257549811yescharacteristic
409C2—81no—
410C2—10761no—
411C2—10761no—
412C2—10762no—
413C2—10762no—
414C2#3177124011yescharacteristic
415C2#3177140411yescharacteristic
416C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#497×C269C2
socleC22×C269C22×D4
hypercenter#497D269
hypocenterC269#3267
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#22575480C2semidirect
2. C2×#22575476C2semidirect
3. #31771258C2semidirect
4. #31771248C2semidirect
5. #31770860C2semidirect
6. #31770831C2semidirect
7. #31770817C2semidirect
8. #31770807C2semidirect
9. #31770766C2semidirect
10. #31770766C2semidirect
11. #31770754C2semidirect
12. #31770749C2semidirect
13. #31770718C2semidirect
14. #31770678C2semidirect
15. #497×C269C2semidirect
18. C2×#16864403C22semidirect
24. C2×#16864389C22semidirect
28. C2×C8×D269C22semidirect
34. #22575429C22semidirect
41. C4×D1076C22semidirect
44. #22575385C22semidirect
45. C4⋊#14524130C22semidirect
46. #22575377C22semidirect
47. #22575377C22semidirect
48. #22575377C22semidirect
50. #22575375C22semidirect
53. #22575365C22semidirect
54. C2×Q16×C269C22semidirect
55. C2×Q16×C269C22semidirect
58. C4⋊Q8×C269C22semidirect
60. #97×C269C22semidirect
88. D2152D4semidirect
208. C269#3267semidirect
214. #497D269semidirect
231. C2×Q16C2×D269semidirect
232. C2×Q16C2×D269semidirect
242. C4⋊Q8C2×D269semidirect
269. #97C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754802
C2×#225754762
#317712582
#317712482
#317708602
#317708312
#317708172
#317708072
#317707662
#317707542
#317707492
#317707182
#317706782
#497×C2692
#3267269