Finite Group Databasebeta

← #44561899

Subgroups — #44561899

Order 32128.

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×#22548736C211yesmaximal, characteristic, complemented
2#30632356C211yesmaximal, characteristic, complemented
3#30632331C211yesmaximal, characteristic, complemented
4#30632075C211yesmaximal, characteristic, complemented
5#30631933C211yesmaximal, characteristic, complemented
6#30631920C211yesmaximal, characteristic, complemented
7#30631907C211yesmaximal, characteristic, complemented
8#30631880C211yesmaximal, characteristic, complemented
9#30631849C211yesmaximal, characteristic, complemented
10#30631831C211yesmaximal, characteristic, complemented
11#30631829C211yesmaximal, characteristic, complemented
12#30631821C211yesmaximal, characteristic, complemented
13#30631788C211yesmaximal, characteristic, complemented
14#30631756C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic, complemented
17C2×#16860809C2211yescharacteristic
18#22548736—24no—
19C2×#16860800C2211yescharacteristic, complemented
20C2×#16860799C2211yescharacteristic, complemented
21C2×#16860798C2211yescharacteristic, complemented
22C2×#16860795C2211yescharacteristic
23C2×#16860795C2211yescharacteristic
24C2×C8×D251C2211yescharacteristic, complemented
25#22548711—21no—
26#22548711C2211yescharacteristic, complemented
27#22548702C2211yescharacteristic
28#22548696—21no—
29#22548694C2211yescharacteristic
30#22548689—21no—
31#22548689C2211yescharacteristic
32#22548687—21no—
33#22548686—21no—
34#22548686C2211yescharacteristic
35#22548686C2211yescharacteristic, complemented
36#22548684—21no—
37Q8×Dic251C2211yescharacteristic, complemented
38D4×Dic251C2211yescharacteristic
39C4×C22⋊D251—21no—
40C4×C22⋊D251C2211yescharacteristic
41C2×C22⋊Dic251—21no—
42#22548649C2211yescharacteristic, complemented
43Dic251⋊C8C2211yescharacteristic, complemented
44#22548640C2211yescharacteristic, complemented
45#22548639C2211yescharacteristic, complemented
46#22548637C2211yescharacteristic, complemented
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548636C2211yescharacteristic, complemented
50C22⋊(C251⋊C8)C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52#22548625C2211yescharacteristic, complemented
53C2×#41×C251C2211yescharacteristic
54C22×D4×C251C2211yescharacteristic
55AGL(1,ℤ/8)×C251—24no—
56C2×SD16×C251C2211yescharacteristic
57C2×SD16×C251C2211yescharacteristic
58C2×D8×C251C2211yescharacteristic
59C2×D8×C251C2211yescharacteristic
60C2×M16×C251C2211yescharacteristic
61#16860811—22no—
62#16860811—41no—
63#16860809—22no—
64#16860809—41no—
65C2×Dic502C2311yescharacteristic
66C2×D1004—21no—
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74C2×C22⋊D251D411yescharacteristic
75C22×Dic251—21no—
76C22×Dic251—21no—
77C2×C4×D251—21no—
78C2×C4×D251C2311yescharacteristic
79#16860800—22no—
80#16860799—22no—
81#16860798—41no—
82#16860795—22no—
83#16860795—41no—
84C2×C251⋊C8C2311yescharacteristic
85C2×C251⋊C8C2311yescharacteristic
86C8×D251—22no—
87C4⋊Dic251—21no—
88C4⋊Dic251C2311yescharacteristic
89C4⋊Dic251C2311yescharacteristic
90Dic251⋊C4—21no—
91Dic251⋊C4D411yescharacteristic
92C22⋊Dic251—21no—
93C22⋊Dic251—21no—
94C22⋊Dic251—21no—
95C22⋊Dic251—21no—
96C22⋊Dic251—21no—
97C22⋊Dic251—21no—
98C22⋊Dic251D411yescharacteristic
99(C2×D251)⋊C4—21no—
100(C2×D251)⋊C4D411yescharacteristic
101C4×Dic251—21no—
102C4×Dic251C2311yescharacteristic
103C24×C251—21no—
104#41×C251—22no—
105#41×C251—41no—
106C2×Q8×C251C2311yescharacteristic
107C2×D4×C251—22no—
108C2×D4×C251—21no—
109C2×D4×C251—21no—
110C2×D4×C251—41no—
111C2×D4×C251C2311yescharacteristic
112C2×D4×C251C2311yescharacteristic
113C2×D4×C251C2311yescharacteristic
114C22×C4×C251—21no—
115C22×C4×C251C2311yescharacteristic
116SD16×C251—22no—
117SD16×C251D414yescomplemented
118D8×C251—22no—
119D8×C251—22no—
120M16×C251—22no—
121C2×C8×C251C2311yescharacteristic
122C2×C8×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251C2×D411yescharacteristic
125Dic502—21no—
126Dic502—21no—
127D1004—21no—
128D1004—21no—
129D1004—42no—
130C22⋊D251—41no—
131C22⋊D251—42no—
132C22⋊D251—42no—
133C22⋊D251—81no—
134C22⋊D251—81no—
135C22⋊D251—81no—
136C22⋊D251—81no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—21no—
140C2×Dic251—42no—
141C2×Dic251—81no—
142C2×Dic251C2×D411yescharacteristic
143C2×Dic251C2×D411yescharacteristic
144C2×Dic251C2×D411yescharacteristic
145C4×D251—21no—
146C4×D251—21no—
147C4×D251—42no—
148C251⋊C8—21no—
149C251⋊C8—21no—
150C23×C251—21no—
151C23×C251—21no—
152C23×C251—21no—
153C23×C251—42no—
154C23×C251—42no—
155C23×C251—81no—
156C23×C251C2×D411yescharacteristic
157Q8×C251—21no—
158Q8×C251C2×D412yes—
159D4×C251—21no—
160D4×C251—21no—
161D4×C251—21no—
162D4×C251—21no—
163D4×C251—21no—
164D4×C251—42no—
165D4×C251—42no—
166D4×C251C2×D412yes—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—42no—
171C2×C4×C251C2×D411yescharacteristic
172C2×C4×C251C2411yescharacteristic
173C8×C251—21no—
174C8×C251C2×D412yes—
175C22×C251—21no—
176C22×C251—22no—
177C22×C251—21no—
178C22×C251—21no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251—84no—
188C22×C251C22×D411yescharacteristic
189C2×D251—21no—
190C2×D251—21no—
191C2×D251—21no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—41no—
195C4×C251—21no—
196C4×C251—22no—
197C4×C251C22×D411yescharacteristic
198C4×C251H(5,𝔽2)11yescharacteristic
199Dic251—21no—
200Dic251—21no—
201Dic251—21no—
202Dic251—41no—
203Dic251—41no—
204Dic251—41no—
205C2×C251—41no—
206C2×C251—42no—
207C2×C251—81no—
208C2×C251—81no—
209C2×C251D4211yescharacteristic
210C2×C251#57511yescharacteristic
211C2×C251#57611yescharacteristic
212D251—41no—
213D251—81no—
214C251#326811yescharacteristic, complemented, Sylow
215#3268—2511nomaximal, Sylow
216C2×AGL(1,ℤ/8)D25111yescharacteristic, complemented
217C2×#135—2511no—
218#550—2511no—
219#546—2511no—
220#496—2511no—
221#468—2511no—
222#466—2511no—
223#464—2511no—
224#459—2511no—
225#452—2511no—
226#449—2511no—
227#449—2511no—
228#447—2511no—
229#440—2511no—
230#435—2511no—
231C2×#41—2511no—
232C2×#41—2511no—
233C2×#41C2×D25111yescharacteristic
234C22×D4C2×D25111yescharacteristic
235AGL(1,ℤ/8)—24no—
236#135—5024no—
237C2×Q16—2511no—
238C2×SD16—2511no—
239C2×SD16—2511no—
240C2×SD16C2×D25111yescharacteristic
241C2×SD16C2×D25111yescharacteristic
242C2×D8—2511no—
243C2×D8—2511no—
244C2×D8C2×D25111yescharacteristic
245C2×D8C2×D25111yescharacteristic
246C2×M16C2×D25111yescharacteristic
247C22×C8—2511no—
248#127—5021no—
249#127—2511no—
250#124—2511no—
251#123—5021no—
252C22⋊Q8—2511no—
253#121—5021no—
254#121—5021no—
255#121—5021no—
256#121—2511no—
257#121—2511no—
258#121—2511no—
259#120—5021no—
260C4×Q8—2511no—
261C4×D4—5021no—
262C4×D4—2511no—
263C4×D4—2511no—
264C2×C22⋊C4—5021no—
265#107—2511no—
266C4⋊C8—2511no—
267#103—2511no—
268#103—2511no—
269#102—2511no—
270#102—2511no—
271#102—2511no—
272#102—2511no—
273C22⋊C8—2511no—
274C22⋊C8—2511no—
275#97—2511no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—10041no—
280#41—10041no—
281#41—5022no—
282#41—5022no—
283C2×Q8—2511no—
284C2×Q8C22×D25111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—5021no—
290C2×D4—5021no—
291C2×D4—5021no—
292C2×D4—5021no—
293C2×D4—5021no—
294C2×D4—5021no—
295C2×D4—5021no—
296C2×D4—2511no—
297C2×D4—2511no—
298C2×D4C22×D25111yescharacteristic
299C2×D4C22×D25111yescharacteristic
300C2×D4C22×D25111yescharacteristic
301C22×C4—21no—
302C22×C4—5021no—
303C22×C4—5021no—
304C22×C4—5021no—
305C22×C4—2511no—
306C22×C4C22×D25111yescharacteristic
307Q16—5022no—
308SD16—22no—
309SD16—10041no—
310SD16—5022no—
311SD16C22⋊D25114yescomplemented
312D8—22no—
313D8—22no—
314D8—10041no—
315D8—5022no—
316M16—22no—
317C2×C8—5022no—
318C2×C8—2511no—
319C2×C8—2511no—
320C2×C8C22×D25111yescharacteristic
321C2×C8C22×D25111yescharacteristic
322C4⋊C4—5021no—
323C4⋊C4—5021no—
324C4⋊C4—2511no—
325C4⋊C4—2511no—
326C4⋊C4—2511no—
327C22⋊C4—5021no—
328C22⋊C4—5021no—
329C22⋊C4—5021no—
330C22⋊C4—5021no—
331C22⋊C4—5021no—
332C22⋊C4—5021no—
333C22⋊C4—5021no—
334C22⋊C4—2511no—
335C22⋊C4—2511no—
336C42—5021no—
337C42—2511no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—5021no—
345C23—2511no—
346C23D4×D25111yescharacteristic
347Q8—21no—
348Q8—5021no—
349Q8—5021no—
350Q8C2×C22⋊D25112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—20081no—
359D4—20081no—
360D4—20081no—
361D4—20081no—
362D4—10042no—
363D4—10042no—
364D4—10041no—
365D4—10042no—
366D4—5021no—
367D4—5021no—
368D4C2×C22⋊D25112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—20081no—
374C2×C4—10042no—
375C2×C4—10042no—
376C2×C4—5021no—
377C2×C4—5021no—
378C2×C4—5021no—
379C2×C4—5021no—
380C2×C4—5021no—
381C2×C4—2511no—
382C2×C4—2511no—
383C2×C4—2511no—
384C2×C4D4×D25111yescharacteristic
385C2×C4C23×D25111yescharacteristic
386C8—21no—
387C8—5021no—
388C8—5021no—
389C8C2×C22⋊D25112yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—10041no—
404C22—10041no—
405C22—10041no—
406C22—5021no—
407C22—5021no—
408C22—5021no—
409C22C2×D4×D25111yescharacteristic
410C4—21no—
411C4—22no—
412C4—10041no—
413C4—10041no—
414C4—10041no—
415C4—5021no—
416C4—5021no—
417C4—5021no—
418C4C22×C22⋊D25111yescharacteristic
419C4#2254876011yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2—20081no—
425C2—10041no—
426C2D4×C22⋊D25111yescharacteristic
427C2#3063248311yescharacteristic
428C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×AGL(1,ℤ/8)×C251C2
socleC22×C251C22×D4
hypercenterC2×AGL(1,ℤ/8)D251
hypocenterC251#3268
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22548736C2semidirect
2. #30632356C2semidirect
3. #30632331C2semidirect
4. #30632075C2semidirect
5. #30631933C2semidirect
6. #30631920C2semidirect
7. #30631907C2semidirect
8. #30631880C2semidirect
9. #30631849C2semidirect
10. #30631831C2semidirect
11. #30631829C2semidirect
12. #30631821C2semidirect
13. #30631788C2semidirect
14. #30631756C2semidirect
15. C2×AGL(1,ℤ/8)×C251C2semidirect
16. C2×#16860811C22semidirect
19. C2×#16860800C22semidirect
20. C2×#16860799C22semidirect
21. C2×#16860798C22semidirect
24. C2×C8×D251C22semidirect
26. #22548711C22semidirect
35. #22548686C22semidirect
37. Q8×Dic251C22semidirect
42. #22548649C22semidirect
43. Dic251⋊C8C22semidirect
44. #22548640C22semidirect
45. #22548639C22semidirect
46. #22548637C22semidirect
49. #22548636C22semidirect
51. (C2×D251)⋊C8C22semidirect
52. #22548625C22semidirect
117. SD16×C251D4semidirect
214. C251#3268semidirect
216. C2×AGL(1,ℤ/8)D251semidirect
311. SD16C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487362
#306323562
#306323312
#306320752
#306319332
#306319202
#306319072
#306318802
#306318492
#306318312
#306318292
#306318212
#306317882
#306317562
C2×AGL(1,ℤ/8)×C2512
#3268251