Finite Group Databasebeta

← #46822420

Subgroups — #46822420

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×#22575480C211yesmaximal, characteristic, complemented
2C2×#22575476C211yesmaximal, characteristic, complemented
3#31771277C211yesmaximal, characteristic, complemented
4#31771248C211yesmaximal, characteristic, complemented
5#31770995C211yesmaximal, characteristic, complemented
6#31770854C211yesmaximal, characteristic, complemented
7#31770840C211yesmaximal, characteristic, complemented
8#31770801C211yesmaximal, characteristic, complemented
9#31770766C211yesmaximal, characteristic, complemented
10#31770753C211yesmaximal, characteristic, complemented
11#31770749C211yesmaximal, characteristic, complemented
12#31770740C211yesmaximal, characteristic, complemented
13#31770703C211yesmaximal, characteristic, complemented
14#31770675C211yesmaximal, characteristic, complemented
15#464×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864403C2211yescharacteristic
18C2×#16864403C2211yescharacteristic, complemented
19C2×D4×D269C2211yescharacteristic
20#22575480—24no—
21#22575476—24no—
22C2×#16864389C2211yescharacteristic
23C2×#16864389C2211yescharacteristic, complemented
24C2×#16864389C2211yescharacteristic, complemented
25C2×D2152C2211yescharacteristic
26C2×D2152C2211yescharacteristic
27C2×#16864387C2211yescharacteristic
28C2×#16864387C2211yescharacteristic
29C2×#16864385C2211yescharacteristic
30C2×C8×D269C2211yescharacteristic, complemented
31#22575450—21no—
32#22575450C2211yescharacteristic, complemented
33#22575435—21no—
34#22575429C2211yescharacteristic
35#22575428—21no—
36#22575428—21no—
37#22575428C2211yescharacteristic
38#22575428C2211yescharacteristic, complemented
39#22575426—21no—
40#22575424C2211yescharacteristic
41#22575423—21no—
42C4×D1076—21no—
43C4×D1076C2211yescharacteristic
44C4×D1076C2211yescharacteristic
45C22⋊C4×D269—21no—
46#22575386C2211yescharacteristic, complemented
47#22575377C2211yescharacteristic, complemented
48#22575375C2211yescharacteristic, complemented
49#22575375C2211yescharacteristic, complemented
50#22575373C2211yescharacteristic
51#22575365C2211yescharacteristic
52#22575365C2211yescharacteristic, complemented
53#22575364C2211yescharacteristic, complemented
54C2×Q16×C269C2211yescharacteristic, complemented
55C2×SD16×C269C2211yescharacteristic
56#124×C269C2211yescharacteristic
57C4×Q8×C269C2211yescharacteristic, complemented
58C4⋊C8×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic
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×D1076—21no—
72C2×D1076—21no—
73C2×D1076—21no—
74C2×D1076—21no—
75C2×D1076D411yescharacteristic
76C2×D1076C2311yescharacteristic
77C2×D1076C2311yescharacteristic
78C2×D1076C2311yescharacteristic
79C2×C22⋊D269—21no—
80C2×C22⋊D269—21no—
81D4×D269—22no—
82D4×D269—41no—
83C2×C4×D269—21no—
84C2×C4×D269—21no—
85C2×C4×D269—21no—
86C2×C4×D269—21no—
87C2×C4×D269C2311yescharacteristic
88C2×C4×D269C2311yescharacteristic
89#16864389—22no—
90#16864389—41no—
91#16864389D414yescomplemented
92D2152—22no—
93D2152—22no—
94#16864387—22no—
95#16864387—41no—
96#16864385—22no—
97C2×#14524130C2311yescharacteristic
98C2×#14524130C2311yescharacteristic
99C8×D269—22no—
100C4⋊Dic269D411yescharacteristic
101C4⋊Dic269C2311yescharacteristic
102#16864381—21no—
103C22⋊Dic269—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379—21no—
108#16864379D411yescharacteristic
109#16864379D411yescharacteristic
110C2×Q8×C269C2311yescharacteristic
111C2×Q8×C269C2311yescharacteristic
112C2×D4×C269C2311yescharacteristic
113Q16×C269—22no—
114SD16×C269—22no—
115C2×C8×C269C2311yescharacteristic
116C2×C8×C269C2311yescharacteristic
117C4⋊C4×C269—21no—
118C4⋊C4×C269C2311yescharacteristic
119C22⋊C4×C269—21no—
120C22⋊C4×C269—21no—
121C42×C269—21no—
122C42×C269C2311yescharacteristic
123C22×D269—21no—
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—42no—
127C22×D269—42no—
128C22×D269—81no—
129C22×D269C2×D411yescharacteristic
130C22×D269C2×D411yescharacteristic
131D1076—21no—
132D1076—21no—
133D1076—21no—
134D1076—21no—
135D1076—21no—
136D1076—41no—
137D1076—42no—
138D1076—42no—
139D1076—42no—
140D1076—42no—
141D1076—81no—
142D1076—81no—
143D1076—81no—
144D1076C2×D412yes—
145C22⋊D269—42no—
146C22⋊D269—81no—
147C2×Dic269—21no—
148C2×Dic269C2×D411yescharacteristic
149C2×Dic269C2×D411yescharacteristic
150C4×D269—21no—
151C4×D269—21no—
152C4×D269—21no—
153C4×D269—21no—
154C4×D269—42no—
155C4×D269—42no—
156C4×D269—42no—
157C4×D269—81no—
158#14524130—21no—
159#14524130C2×D412yes—
160C23×C269—21no—
161Q8×C269—21no—
162Q8×C269—21no—
163Q8×C269—21no—
164Q8×C269C2×D412yes—
165D4×C269—21no—
166D4×C269—21no—
167C2×C4×C269—21no—
168C2×C4×C269—21no—
169C2×C4×C269—21no—
170C2×C4×C269C2×D411yescharacteristic
171C2×C4×C269C2×D411yescharacteristic
172C2×C4×C269C2411yescharacteristic
173C8×C269—21no—
174C8×C269—21no—
175C22×C269—41no—
176C22×C269—41no—
177C22×C269—41no—
178C22×C269C22×D411yescharacteristic
179C2×D269—21no—
180C2×D269—21no—
181C2×D269—22no—
182C2×D269—21no—
183C2×D269—21no—
184C2×D269—21no—
185C2×D269—21no—
186C2×D269—41no—
187C2×D269—41no—
188C2×D269—41no—
189C2×D269—41no—
190C2×D269—41no—
191C2×D269—41no—
192C2×D269—41no—
193C2×D269—41no—
194C2×D269—84no—
195C4×C269—22no—
196C4×C269—21no—
197C4×C269—21no—
198C4×C269—41no—
199C4×C269—41no—
200C4×C269C22×D411yescharacteristic
201C4×C269H(5,𝔽2)11yescharacteristic
202Dic269—21no—
203Dic269—21no—
204Dic269—41no—
205C2×C269—81no—
206C2×C269D4211yescharacteristic
207C2×C269#57511yescharacteristic
208C2×C269#57611yescharacteristic
209D269—41no—
210D269—41no—
211D269—42no—
212D269—81no—
213D269—81no—
214C269#326811yescharacteristic, complemented, Sylow
215#3268—2691nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2691no—
217C2×#135—2691no—
218#550—2691no—
219#546—2691no—
220#496—2691no—
221#468—2691no—
222#466—2691no—
223#464D26911yescharacteristic, complemented
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×Q16C2×D26911yescharacteristic, complemented
238C2×SD16—2691no—
239C2×SD16—2691no—
240C2×SD16—2691no—
241C2×SD16C2×D26911yescharacteristic
242C2×D8—2691no—
243C2×D8—2691no—
244C2×D8—2691no—
245C2×D8—2691no—
246C2×M16—2691no—
247C22×C8—2691no—
248#127—2691no—
249#127—5381no—
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×Q8C2×D26911yescharacteristic, complemented
261C4×D4—2691no—
262C4×D4—2691no—
263C4×D4—5381no—
264C2×C22⋊C4—5381no—
265#107—2691no—
266C4⋊C8C2×D26911yescharacteristic, complemented
267#103—2691no—
268#103C2×D26911yescharacteristic, complemented
269#102—2691no—
270#102—2691no—
271#102—2691no—
272#102C2×D26911yescharacteristic
273C22⋊C8—2691no—
274C22⋊C8—2691no—
275#97—2691no—
276C24—5381no—
277#41—5382no—
278#41—5382no—
279#41—5382no—
280#41—10761no—
281#41—10761no—
282#41—10761no—
283C2×Q8C22×D26911yescharacteristic
284C2×Q8C22×D26911yescharacteristic
285C2×D4—2691no—
286C2×D4—2691no—
287C2×D4—2691no—
288C2×D4—2691no—
289C2×D4—5381no—
290C2×D4—5381no—
291C2×D4—5381no—
292C2×D4—5381no—
293C2×D4—5381no—
294C2×D4—5381no—
295C2×D4—5382no—
296C2×D4—5381no—
297C2×D4—5381no—
298C2×D4—5381no—
299C2×D4—10761no—
300C2×D4C22×D26911yescharacteristic
301C22×C4—2691no—
302C22×C4—2691no—
303C22×C4—5381no—
304C22×C4—5381no—
305C22×C4—5381no—
306C22×C4—5381no—
307Q16—22no—
308SD16—22no—
309SD16—2694no—
310SD16—5382no—
311SD16—10761no—
312D8—5382no—
313D8—5382no—
314D8—5382no—
315D8—10761no—
316M16—5382no—
317C2×C8—2691no—
318C2×C8—2691no—
319C2×C8—5382no—
320C2×C8C22×D26911yescharacteristic
321C2×C8C22×D26911yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—2691no—
324C4⋊C4—2691no—
325C4⋊C4—5381no—
326C4⋊C4C22×D26911yescharacteristic
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—21no—
337C42C22×D26911yescharacteristic
338C23—21no—
339C23—2691no—
340C23—2691no—
341C23—5381no—
342C23—5381no—
343C23—5381no—
344C23—10762no—
345C23—10762no—
346C23—21521no—
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8D4×D26912yes—
351D4—21no—
352D4—21no—
353D4—2692no—
354D4—5381no—
355D4—5381no—
356D4—5381no—
357D4—5381no—
358D4—5381no—
359D4—10761no—
360D4—10762no—
361D4—10762no—
362D4—10762no—
363D4—10762no—
364D4—10762no—
365D4—21521no—
366D4—21521no—
367D4—21521no—
368D4—21521no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—2691no—
373C2×C4—2691no—
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—21no—
388C8—2692no—
389C8—5381no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—5381no—
394C22—5381no—
395C22—5381no—
396C22—5381no—
397C22—5381no—
398C22—5381no—
399C22—5382no—
400C22—10761no—
401C22—10761no—
402C22—10761no—
403C22—10761no—
404C22—10761no—
405C22—10761no—
406C22—10761no—
407C22—10761no—
408C22—21524no—
409C22C2×D4×D26911yescharacteristic
410C4—21no—
411C4—21no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—5381no—
416C4—5381no—
417C4—10761no—
418C4C2×D4×D26911yescharacteristic
419C4#2257549911yescharacteristic
420C2—81no—
421C2—10761no—
422C2—10761no—
423C2—10762no—
424C2—21521no—
425C2—21521no—
426C2#3177123911yescharacteristic
427C2#3177140411yescharacteristic
428C2#3177140711yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#464×C269C2
socleC22×C269C22×D4
hypercenter#464D269
hypocenterC269#3268
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#22575480C2semidirect
2. C2×#22575476C2semidirect
3. #31771277C2semidirect
4. #31771248C2semidirect
5. #31770995C2semidirect
6. #31770854C2semidirect
7. #31770840C2semidirect
8. #31770801C2semidirect
9. #31770766C2semidirect
10. #31770753C2semidirect
11. #31770749C2semidirect
12. #31770740C2semidirect
13. #31770703C2semidirect
14. #31770675C2semidirect
15. #464×C269C2semidirect
18. C2×#16864403C22semidirect
23. C2×#16864389C22semidirect
24. C2×#16864389C22semidirect
30. C2×C8×D269C22semidirect
32. #22575450C22semidirect
38. #22575428C22semidirect
46. #22575386C22semidirect
47. #22575377C22semidirect
48. #22575375C22semidirect
49. #22575375C22semidirect
52. #22575365C22semidirect
53. #22575364C22semidirect
54. C2×Q16×C269C22semidirect
57. C4×Q8×C269C22semidirect
58. C4⋊C8×C269C22semidirect
59. #103×C269C22semidirect
91. #16864389D4semidirect
214. C269#3268semidirect
223. #464D269semidirect
237. C2×Q16C2×D269semidirect
260. C4×Q8C2×D269semidirect
266. C4⋊C8C2×D269semidirect
268. #103C2×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
#317712772
#317712482
#317709952
#317708542
#317708402
#317708012
#317707662
#317707532
#317707492
#317707402
#317707032
#317706752
#464×C2692
#3268269