Finite Group Databasebeta

← #44561890

Subgroups — #44561890

Order 32128.

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×#22548739C211yesmaximal, characteristic, complemented
2#30632336C211yesmaximal, characteristic, complemented
3#30632325C211yesmaximal, characteristic, complemented
4#30632080C211yesmaximal, characteristic, complemented
5#30631942C211yesmaximal, characteristic, complemented
6#30631916C211yesmaximal, characteristic, complemented
7#30631904C211yesmaximal, characteristic, complemented
8#30631890C211yesmaximal, characteristic, complemented
9#30631849C211yesmaximal, characteristic, complemented
10#30631846C211yesmaximal, characteristic, complemented
11#30631836C211yesmaximal, characteristic, complemented
12#30631832C211yesmaximal, characteristic, complemented
13#30631799C211yesmaximal, characteristic, complemented
14#30631758C211yesmaximal, characteristic, complemented
15C2×#135×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18#22548739—24no—
19C2×#16860800C2211yescharacteristic, complemented
20C2×#16860799C2211yescharacteristic
21C2×#16860798C2211yescharacteristic
22C2×#16860797C2211yescharacteristic
23C2×#16860795C2211yescharacteristic
24C2×#16860793C2211yescharacteristic
25#22548713C2211yescharacteristic, complemented
26#22548704C2211yescharacteristic
27#22548698—21no—
28#22548698—21no—
29#22548696—21no—
30#22548694C2211yescharacteristic, complemented
31#22548691C2211yescharacteristic
32#22548691C2211yescharacteristic
33#22548689C2211yescharacteristic
34#22548687—21no—
35#22548687—21no—
36#22548683—21no—
37D4×Dic251C2211yescharacteristic
38D4×Dic251C2211yescharacteristic, complemented
39C4×C22⋊D251C2211yescharacteristic
40C2×Dic251⋊C4—21no—
41C2×(C2×D251)⋊C4—21no—
42#22548647C2211yescharacteristic, complemented
43Dic251⋊C8C2211yescharacteristic, complemented
44#22548640C2211yescharacteristic, complemented
45#22548639C2211yescharacteristic, complemented
46#22548639C2211yescharacteristic, complemented
47#22548637C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548636C2211yescharacteristic, complemented
50C22⋊(C251⋊C8)C2211yescharacteristic, complemented
51(C2×D251)⋊C8C2211yescharacteristic
52#22548625C2211yescharacteristic, complemented
53C2×#41×C251C2211yescharacteristic
54C2×#41×C251C2211yescharacteristic, complemented
55#135×C251—24no—
56C2×Q16×C251C2211yescharacteristic, complemented
57C2×SD16×C251C2211yescharacteristic
58C2×SD16×C251C2211yescharacteristic, complemented
59C2×D8×C251C2211yescharacteristic
60C22×C8×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860809—22no—
63#16860809—41no—
64C2×Dic502C2311yescharacteristic
65C2×D1004C2311yescharacteristic
66C2×C22⋊D251—21no—
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C2×C22⋊D251D411yescharacteristic
70D4×D251—22no—
71D4×D251—41no—
72C22×Dic251—21no—
73C22×Dic251—21no—
74C22×Dic251—21no—
75C2×C4×D251C2311yescharacteristic
76#16860800—41no—
77#16860799—22no—
78#16860798—22no—
79#16860797—41no—
80#16860795—22no—
81#16860793—22no—
82C2×C251⋊C8C2311yescharacteristic
83C2×C251⋊C8C2311yescharacteristic
84C4⋊Dic251C2311yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4—21no—
90Dic251⋊C4—21no—
91Dic251⋊C4D411yescharacteristic
92C22⋊Dic251—21no—
93C22⋊Dic251—21no—
94C22⋊Dic251—21no—
95C22⋊Dic251D411yescharacteristic
96(C2×D251)⋊C4—21no—
97(C2×D251)⋊C4—21no—
98(C2×D251)⋊C4—21no—
99(C2×D251)⋊C4—21no—
100(C2×D251)⋊C4—21no—
101(C2×D251)⋊C4D411yescharacteristic
102C4×Dic251C2311yescharacteristic
103#41×C251—22no—
104#41×C251—22no—
105#41×C251—41no—
106#41×C251—41no—
107C2×Q8×C251C2311yescharacteristic
108C2×Q8×C251C2311yescharacteristic
109C2×D4×C251—21no—
110C2×D4×C251—21no—
111C2×D4×C251C2311yescharacteristic
112C2×D4×C251C2311yescharacteristic
113C22×C4×C251—21no—
114C22×C4×C251—21no—
115C22×C4×C251C2311yescharacteristic
116Q16×C251—22no—
117SD16×C251—22no—
118SD16×C251—22no—
119D8×C251D414yescomplemented
120C2×C8×C251—22no—
121C2×C8×C251C2311yescharacteristic
122C2×C8×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251—42no—
125C22×D251—42no—
126C22×D251—81no—
127C22×D251C2×D411yescharacteristic
128Dic502—21no—
129Dic502—21no—
130D1004—21no—
131D1004—21no—
132C22⋊D251—41no—
133C22⋊D251—42no—
134C22⋊D251—42no—
135C22⋊D251—81no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—42no—
140C2×Dic251—42no—
141C2×Dic251—81no—
142C2×Dic251C2×D411yescharacteristic
143C2×Dic251C2×D411yescharacteristic
144C2×Dic251C2×D411yescharacteristic
145C4×D251—21no—
146C4×D251—21no—
147C251⋊C8—21no—
148C251⋊C8—21no—
149C23×C251—21no—
150C23×C251—21no—
151C23×C251C2×D411yescharacteristic
152Q8×C251—21no—
153Q8×C251—21no—
154Q8×C251—21no—
155Q8×C251—21no—
156D4×C251—21no—
157D4×C251—21no—
158D4×C251—42no—
159D4×C251—42no—
160D4×C251C2×D412yes—
161D4×C251C2×D412yes—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—42no—
167C2×C4×C251—42no—
168C2×C4×C251C2×D411yescharacteristic
169C2×C4×C251C2411yescharacteristic
170C8×C251—21no—
171C8×C251C2×D412yes—
172C22×C251—21no—
173C22×C251—22no—
174C22×C251—22no—
175C22×C251—21no—
176C22×C251—21no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251C22×D411yescharacteristic
182C2×D251—21no—
183C2×D251—21no—
184C2×D251—21no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—84no—
189C4×C251—21no—
190C4×C251—41no—
191C4×C251—41no—
192C4×C251C22×D411yescharacteristic
193C4×C251H(5,𝔽2)11yescharacteristic
194Dic251—21no—
195Dic251—21no—
196Dic251—21no—
197Dic251—41no—
198Dic251—41no—
199Dic251—41no—
200C2×C251—42no—
201C2×C251—41no—
202C2×C251—42no—
203C2×C251D4211yescharacteristic
204C2×C251#57511yescharacteristic
205C2×C251#57711yescharacteristic
206D251—41no—
207D251—81no—
208C251#326711yescharacteristic, complemented, Sylow
209#3267—2511nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2511no—
211C2×#135D25111yescharacteristic, complemented
212#547—2511no—
213#546—2511no—
214#497—2511no—
215#470—2511no—
216#465—2511no—
217#463—2511no—
218#461—2511no—
219#452—2511no—
220#452—2511no—
221#450—2511no—
222#449—2511no—
223#442—2511no—
224#435—2511no—
225C2×#41—2511no—
226C2×#41C2×D25111yescharacteristic
227C2×#41C2×D25111yescharacteristic, complemented
228C22×D4—2511no—
229AGL(1,ℤ/8)—5024no—
230#135—24no—
231C2×Q16—2511no—
232C2×Q16C2×D25111yescharacteristic, complemented
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16—2511no—
236C2×SD16C2×D25111yescharacteristic
237C2×SD16C2×D25111yescharacteristic, complemented
238C2×D8—2511no—
239C2×D8C2×D25111yescharacteristic
240C2×M16—2511no—
241C22×C8C2×D25111yescharacteristic, complemented
242C4⋊Q8—2511no—
243#124—2511no—
244#123—5021no—
245#123—5021no—
246#123—5021no—
247C22⋊Q8—2511no—
248C22⋊Q8—2511no—
249C22⋊Q8—2511no—
250#121—5021no—
251#121—5021no—
252#121—2511no—
253#120—5021no—
254C4×D4—2511no—
255C4×D4—2511no—
256C4×D4—2511no—
257C2×C4⋊C4—5021no—
258C2×C22⋊C4—5021no—
259#106—2511no—
260C4⋊C8—2511no—
261#103—2511no—
262#103—2511no—
263#103—2511no—
264#102—2511no—
265#102—2511no—
266#102—2511no—
267C22⋊C8—2511no—
268C22⋊C8—2511no—
269#97—2511no—
270C24—5021no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—10041no—
276#41—5022no—
277C2×Q8—2511no—
278C2×Q8C22×D25111yescharacteristic
279C2×Q8C22×D25111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—10041no—
283C2×D4—5021no—
284C2×D4—5021no—
285C2×D4—5022no—
286C2×D4—5021no—
287C2×D4—2511no—
288C2×D4—2511no—
289C2×D4C22×D25111yescharacteristic
290C2×D4C22×D25111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—5021no—
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4—2511no—
297C22×C4C22×D25111yescharacteristic
298Q16—22no—
299Q16—10041no—
300SD16—22no—
301SD16—22no—
302SD16—10041no—
303SD16—5022no—
304SD16—5022no—
305D8—5022no—
306D8C22⋊D25114yescomplemented
307M16—5022no—
308C2×C8—22no—
309C2×C8—2511no—
310C2×C8—2511no—
311C2×C8C22×D25111yescharacteristic
312C2×C8C22×D25111yescharacteristic
313C4⋊C4—5021no—
314C4⋊C4—5021no—
315C4⋊C4—5021no—
316C4⋊C4—5021no—
317C4⋊C4—5021no—
318C4⋊C4—2511no—
319C4⋊C4—2511no—
320C4⋊C4—2511no—
321C22⋊C4—5021no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4—5021no—
329C22⋊C4—2511no—
330C22⋊C4—2511no—
331C42—2511no—
332C23—21no—
333C23—21no—
334C23—20081no—
335C23—10042no—
336C23—10042no—
337C23—5021no—
338C23—2511no—
339C23D4×D25111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5021no—
345Q8—5021no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—42no—
350D4—20081no—
351D4—10042no—
352D4—10042no—
353D4—10041no—
354D4—5021no—
355D4—5021no—
356D4C2×C22⋊D25112yes—
357D4C2×C22⋊D25112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—20081no—
365C2×C4—10042no—
366C2×C4—10042no—
367C2×C4—5021no—
368C2×C4—5021no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4—2511no—
373C2×C4—2511no—
374C2×C4—2511no—
375C2×C4D4×D25111yescharacteristic
376C2×C4C23×D25111yescharacteristic
377C8—21no—
378C8—5021no—
379C8—5021no—
380C8C2×C22⋊D25112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—20084no—
391C22—10041no—
392C22—10041no—
393C22—10041no—
394C22—5021no—
395C22—5021no—
396C22—5021no—
397C22C2×D4×D25111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—10041no—
402C4—10041no—
403C4—10041no—
404C4—5021no—
405C4—5021no—
406C4—5021no—
407C4C22×C22⋊D25111yescharacteristic
408C4#2254876011yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—20081no—
413C2—10041no—
414C2D4×C22⋊D25111yescharacteristic
415C2#3063248411yescharacteristic
416C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#135×C251C2
socleC22×C251C22×D4
hypercenterC2×#135D251
hypocenterC251#3267
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. #30632336C2semidirect
3. #30632325C2semidirect
4. #30632080C2semidirect
5. #30631942C2semidirect
6. #30631916C2semidirect
7. #30631904C2semidirect
8. #30631890C2semidirect
9. #30631849C2semidirect
10. #30631846C2semidirect
11. #30631836C2semidirect
12. #30631832C2semidirect
13. #30631799C2semidirect
14. #30631758C2semidirect
15. C2×#135×C251C2semidirect
19. C2×#16860800C22semidirect
25. #22548713C22semidirect
30. #22548694C22semidirect
38. D4×Dic251C22semidirect
42. #22548647C22semidirect
43. Dic251⋊C8C22semidirect
44. #22548640C22semidirect
45. #22548639C22semidirect
46. #22548639C22semidirect
49. #22548636C22semidirect
50. C22⋊(C251⋊C8)C22semidirect
52. #22548625C22semidirect
54. C2×#41×C251C22semidirect
56. C2×Q16×C251C22semidirect
58. C2×SD16×C251C22semidirect
60. C22×C8×C251C22semidirect
119. D8×C251D4semidirect
208. C251#3267semidirect
211. C2×#135D251semidirect
227. C2×#41C2×D251semidirect
232. C2×Q16C2×D251semidirect
237. C2×SD16C2×D251semidirect
241. C22×C8C2×D251semidirect
306. D8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487392
#306323362
#306323252
#306320802
#306319422
#306319162
#306319042
#306318902
#306318492
#306318462
#306318362
#306318322
#306317992
#306317582
C2×#135×C2512
#3267251