Finite Group Databasebeta

← #44561911

Subgroups — #44561911

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×#22548739C211yesmaximal, characteristic, complemented
2C2×#22548734C211yesmaximal, characteristic, complemented
3#30632357C211yesmaximal, characteristic, complemented
4#30632332C211yesmaximal, characteristic, complemented
5#30632074C211yesmaximal, characteristic, complemented
6#30631932C211yesmaximal, characteristic, complemented
7#30631922C211yesmaximal, characteristic, complemented
8#30631910C211yesmaximal, characteristic, complemented
9#30631850C211yesmaximal, characteristic, complemented
10#30631833C211yesmaximal, characteristic, complemented
11#30631829C211yesmaximal, characteristic, complemented
12#30631819C211yesmaximal, characteristic, complemented
13#30631787C211yesmaximal, characteristic, complemented
14#30631757C211yesmaximal, characteristic, complemented
15#459×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×#16860809C2211yescharacteristic
18C2×#16860809C2211yescharacteristic, complemented
19C2×D4×D251C2211yescharacteristic
20#22548739—24no—
21#22548734—24no—
22C2×Dic1004C2211yescharacteristic, complemented
23C2×#16860799C2211yescharacteristic
24C2×#16860798C2211yescharacteristic
25C2×#16860798C2211yescharacteristic, complemented
26C2×#16860798C2211yescharacteristic, complemented
27C2×#16860795C2211yescharacteristic
28C2×#16860795C2211yescharacteristic
29C2×#16860793C2211yescharacteristic
30C2×C8×D251C2211yescharacteristic, complemented
31#22548711—21no—
32#22548703C2211yescharacteristic
33#22548699—21no—
34#22548695C2211yescharacteristic
35#22548690C2211yescharacteristic
36#22548689—21no—
37#22548687—21no—
38#22548687—21no—
39#22548686C2211yescharacteristic
40#22548686C2211yescharacteristic, complemented
41#22548685—21no—
42C4×Dic502C2211yescharacteristic, complemented
43C4×D1004C2211yescharacteristic
44D4×Dic251—21no—
45C2×(C2×D251)⋊C4—21no—
46#22548648C2211yescharacteristic, complemented
47#22548640C2211yescharacteristic, complemented
48#22548638C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic
50#22548636C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53#22548626C2211yescharacteristic, complemented
54C2×D8×C251C2211yescharacteristic
55C2×D8×C251C2211yescharacteristic
56#127×C251C2211yescharacteristic, complemented
57C4×D4×C251C2211yescharacteristic
58C4⋊C8×C251C2211yescharacteristic, complemented
59#102×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860809—22no—
63#16860809—22no—
64#16860809—22no—
65#16860809—41no—
66#16860809—41no—
67#16860809—41no—
68C2×Dic502C2311yescharacteristic
69C2×Dic502C2311yescharacteristic
70C2×D1004—21no—
71C2×D1004D411yescharacteristic
72C2×D1004C2311yescharacteristic
73C2×C22⋊D251—21no—
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76C2×C22⋊D251—21no—
77C2×C22⋊D251—21no—
78C2×C22⋊D251—21no—
79D4×D251—22no—
80D4×D251—41no—
81C22×Dic251—21no—
82C22×Dic251—21no—
83C22×Dic251—21no—
84C2×C4×D251C2311yescharacteristic
85C2×C4×D251C2311yescharacteristic
86Dic1004—22no—
87#16860799—22no—
88#16860798—22no—
89#16860798—41no—
90#16860798D414yescomplemented
91#16860795—22no—
92#16860795—41no—
93#16860793—22no—
94C2×C251⋊C8C2311yescharacteristic
95C2×C251⋊C8C2311yescharacteristic
96C8×D251—22no—
97C4⋊Dic251—21no—
98C4⋊Dic251D411yescharacteristic
99C4⋊Dic251C2311yescharacteristic
100Dic251⋊C4—21no—
101C22⋊Dic251—21no—
102C22⋊Dic251—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4D411yescharacteristic
108(C2×D251)⋊C4D411yescharacteristic
109C4×Dic251—21no—
110C2×D4×C251—21no—
111C2×D4×C251—21no—
112C2×D4×C251C2311yescharacteristic
113C2×D4×C251C2311yescharacteristic
114C2×D4×C251C2311yescharacteristic
115C22×C4×C251—21no—
116D8×C251—22no—
117D8×C251—22no—
118C2×C8×C251C2311yescharacteristic
119C2×C8×C251C2311yescharacteristic
120C4⋊C4×C251C2311yescharacteristic
121C22⋊C4×C251—21no—
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251—42no—
125C22×D251—42no—
126C22×D251—81no—
127C22×D251C2×D411yescharacteristic
128C22×D251C2×D411yescharacteristic
129Dic502—21no—
130Dic502—21no—
131Dic502—21no—
132Dic502C2×D412yes—
133D1004—21no—
134D1004—21no—
135D1004—41no—
136D1004—81no—
137C22⋊D251—42no—
138C22⋊D251—42no—
139C22⋊D251—42no—
140C22⋊D251—42no—
141C22⋊D251—81no—
142C22⋊D251—81no—
143C2×Dic251—21no—
144C2×Dic251—21no—
145C2×Dic251—21no—
146C2×Dic251—42no—
147C2×Dic251—42no—
148C2×Dic251—42no—
149C2×Dic251C2×D411yescharacteristic
150C2×Dic251C2×D411yescharacteristic
151C4×D251—21no—
152C4×D251—21no—
153C4×D251—21no—
154C4×D251—21no—
155C251⋊C8—21no—
156C251⋊C8C2×D412yes—
157C23×C251—21no—
158C23×C251—21no—
159C23×C251—21no—
160D4×C251—21no—
161D4×C251—21no—
162D4×C251—21no—
163D4×C251—21no—
164D4×C251—21no—
165D4×C251—42no—
166D4×C251—81no—
167D4×C251C2×D412yes—
168C2×C4×C251—21no—
169C2×C4×C251—81no—
170C2×C4×C251C2×D411yescharacteristic
171C2×C4×C251C2×D411yescharacteristic
172C2×C4×C251C2411yescharacteristic
173C8×C251—21no—
174C8×C251—21no—
175C22×C251—22no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251C22×D411yescharacteristic
185C2×D251—21no—
186C2×D251—21no—
187C2×D251—21no—
188C2×D251—21no—
189C2×D251—21no—
190C2×D251—21no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—84no—
195C4×C251—21no—
196C4×C251—21no—
197C4×C251—41no—
198C4×C251C22×D411yescharacteristic
199C4×C251H(5,𝔽2)11yescharacteristic
200Dic251—21no—
201Dic251—21no—
202Dic251—22no—
203Dic251—41no—
204Dic251—41no—
205C2×C251—42no—
206C2×C251—81no—
207C2×C251—81no—
208C2×C251D4211yescharacteristic
209C2×C251#57511yescharacteristic
210C2×C251#57611yescharacteristic
211D251—41no—
212D251—41no—
213D251—81no—
214C251#326811yescharacteristic, complemented, Sylow
215#3268—2511nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2511no—
217C2×#135—2511no—
218#550—2511no—
219#546—2511no—
220#496—2511no—
221#468—2511no—
222#466—2511no—
223#464—2511no—
224#459D25111yescharacteristic, complemented
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×#41—2511no—
234C22×D4—2511no—
235AGL(1,ℤ/8)—5024no—
236#135—5024no—
237C2×Q16—2511no—
238C2×SD16—2511no—
239C2×SD16—2511no—
240C2×SD16—2511no—
241C2×SD16—2511no—
242C2×D8—2511no—
243C2×D8—2511no—
244C2×D8C2×D25111yescharacteristic
245C2×D8C2×D25111yescharacteristic
246C2×M16—2511no—
247C22×C8—2511no—
248#127—5021no—
249#127C2×D25111yescharacteristic, complemented
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×D4C2×D25111yescharacteristic
264C2×C22⋊C4—5021no—
265#107—2511no—
266C4⋊C8C2×D25111yescharacteristic, complemented
267#103—2511no—
268#103—2511no—
269#102—2511no—
270#102—2511no—
271#102C2×D25111yescharacteristic, complemented
272#102C2×D25111yescharacteristic, complemented
273C22⋊C8—2511no—
274C22⋊C8—2511no—
275#97—2511no—
276C24—5021no—
277#41—10041no—
278#41—10041no—
279#41—10041no—
280#41—5022no—
281#41—5022no—
282#41—5022no—
283C2×Q8—2511no—
284C2×Q8—2511no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—10041no—
288C2×D4—5021no—
289C2×D4—5021no—
290C2×D4—5021no—
291C2×D4—5021no—
292C2×D4—5021no—
293C2×D4—5021no—
294C2×D4—5022no—
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×C4—2511no—
307Q16—5022no—
308SD16—10041no—
309SD16—5022no—
310SD16—5022no—
311SD16—2514no—
312D8—22no—
313D8—22no—
314D8—10041no—
315D8—5022no—
316M16—5022no—
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⋊C4C22×D25111yescharacteristic
327C22⋊C4—21no—
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—
337C42C22×D25111yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—20081no—
342C23—10042no—
343C23—10042no—
344C23—5021no—
345C23—2511no—
346C23—2511no—
347Q8—5021no—
348Q8—5021no—
349Q8—5021no—
350Q8—2512no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—20081no—
359D4—20081no—
360D4—20081no—
361D4—10041no—
362D4—10042no—
363D4—10042no—
364D4—10042no—
365D4—10042no—
366D4—5021no—
367D4—5021no—
368D4D4×D25112yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—10042no—
372C2×C4—10042no—
373C2×C4—10042no—
374C2×C4—5021no—
375C2×C4—5021no—
376C2×C4—5021no—
377C2×C4—5021no—
378C2×C4—5021no—
379C2×C4—5021no—
380C2×C4—5021no—
381C2×C4—2511no—
382C2×C4—2511no—
383C2×C4D4×D25111yescharacteristic
384C2×C4D4×D25111yescharacteristic
385C2×C4C23×D25111yescharacteristic
386C8—21no—
387C8—21no—
388C8—5021no—
389C8—2512no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—20084no—
400C22—10041no—
401C22—10041no—
402C22—10041no—
403C22—5021no—
404C22—5021no—
405C22—5021no—
406C22—5021no—
407C22—5021no—
408C22—5021no—
409C22C2×D4×D25111yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—10041no—
414C4—10041no—
415C4—5021no—
416C4—5021no—
417C4—5022no—
418C4C2×D4×D25111yescharacteristic
419C4#2254876111yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—20081no—
424C2—10041no—
425C2—10041no—
426C2#3063232111yescharacteristic
427C2#3063248311yescharacteristic
428C2#3063248811yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#459×C251C2
socleC22×C251C22×D4
hypercenter#459D251
hypocenterC251#3268
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. C2×#22548734C2semidirect
3. #30632357C2semidirect
4. #30632332C2semidirect
5. #30632074C2semidirect
6. #30631932C2semidirect
7. #30631922C2semidirect
8. #30631910C2semidirect
9. #30631850C2semidirect
10. #30631833C2semidirect
11. #30631829C2semidirect
12. #30631819C2semidirect
13. #30631787C2semidirect
14. #30631757C2semidirect
15. #459×C251C2semidirect
18. C2×#16860809C22semidirect
22. C2×Dic1004C22semidirect
25. C2×#16860798C22semidirect
26. C2×#16860798C22semidirect
30. C2×C8×D251C22semidirect
40. #22548686C22semidirect
42. C4×Dic502C22semidirect
46. #22548648C22semidirect
47. #22548640C22semidirect
48. #22548638C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. #22548626C22semidirect
56. #127×C251C22semidirect
58. C4⋊C8×C251C22semidirect
59. #102×C251C22semidirect
60. #102×C251C22semidirect
90. #16860798D4semidirect
214. C251#3268semidirect
224. #459D251semidirect
249. #127C2×D251semidirect
266. C4⋊C8C2×D251semidirect
271. #102C2×D251semidirect
272. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487392
C2×#225487342
#306323572
#306323322
#306320742
#306319322
#306319222
#306319102
#306318502
#306318332
#306318292
#306318192
#306317872
#306317572
#459×C2512
#3268251