Finite Group Databasebeta

← #44562068

Subgroups — #44562068

Order 32128.

504 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×#22548735C211yesmaximal, characteristic, complemented
2C22×#16860798C211yesmaximal, characteristic, complemented
3#30632346C211yesmaximal, characteristic, complemented
4#30632069C211yesmaximal, characteristic, complemented
5#30631918C211yesmaximal, characteristic, complemented
6#30631914C211yesmaximal, characteristic, complemented
7#30631906C211yesmaximal, characteristic, complemented
8#30631902C211yesmaximal, characteristic, complemented
9#30631849C211yesmaximal, characteristic, complemented
10#30631842C211yesmaximal, characteristic, complemented
11#30631830C211yesmaximal, characteristic, complemented
12#30631821C211yesmaximal, characteristic, complemented
13C4×#16860798C211yesmaximal, characteristic, complemented
14D4×C251⋊C8C211yesmaximal, characteristic, complemented
15#546×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic, complemented
17C22×Dic502C2211yescharacteristic
18#22548735—24no—
19C2×#16860800C2211yescharacteristic, complemented
20C2×#16860800C2211yescharacteristic, complemented
21C2×#16860798—24no—
22C2×#16860798C2211yescharacteristic
23C2×#16860798C2211yescharacteristic
24C2×#16860798C2211yescharacteristic
25C2×#16860798C2211yescharacteristic
26C2×#16860797C2211yescharacteristic, complemented
27C2×#16860795C2211yescharacteristic
28C2×#16860795C2211yescharacteristic
29C22×C251⋊C8C2211yescharacteristic
30C22×C251⋊C8C2211yescharacteristic, complemented
31#22548703C2211yescharacteristic, complemented
32#22548701—21no—
33#22548694C2211yescharacteristic, complemented
34#22548693—21no—
35#22548689C2211yescharacteristic
36#22548687—21no—
37C4×Dic502C2211yescharacteristic, complemented
38(C22⋊D251)⋊C4—21no—
39#22548646C2211yescharacteristic, complemented
40C4⋊(C251⋊C8)C2211yescharacteristic, complemented
41#22548639C2211yescharacteristic, complemented
42#22548638C2211yescharacteristic, complemented
43#22548638C2211yescharacteristic, complemented
44#22548636C2211yescharacteristic
45#22548636C2211yescharacteristic
46#22548635C2211yescharacteristic, complemented
47C22⋊(C251⋊C8)C2211yescharacteristic
48C22⋊(C251⋊C8)C2211yescharacteristic, complemented
49C4×C251⋊C8C2211yescharacteristic, complemented
50C2×#41×C251C2211yescharacteristic
51C22×D4×C251C2211yescharacteristic
52#124×C251C2211yescharacteristic
53#123×C251—21no—
54C22⋊Q8×C251C2211yescharacteristic
55#121×C251—21no—
56#121×C251C2211yescharacteristic
57#120×C251—21no—
58C4×D4×C251C2211yescharacteristic
59C4×D4×C251C2211yescharacteristic
60C2×C22⋊C4×C251—21no—
61#16860810—22no—
62#16860810—41no—
63C2×Dic502—22no—
64C2×Dic502—21no—
65C2×Dic502—41no—
66C2×Dic502C2311yescharacteristic
67C2×Dic502C2311yescharacteristic
68C2×D1004C2311yescharacteristic
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C22×Dic251—21no—
72C2×C4×D251—21no—
73#16860800—22no—
74#16860800—41no—
75#16860798—22no—
76#16860798—22no—
77#16860798—22no—
78#16860798D414yescomplemented
79#16860797—22no—
80#16860795—22no—
81#16860795—41no—
82C2×C251⋊C8—22no—
83C2×C251⋊C8—22no—
84C2×C251⋊C8C2311yescharacteristic
85C2×C251⋊C8C2311yescharacteristic
86C2×C251⋊C8C2311yescharacteristic
87C2×C251⋊C8C2311yescharacteristic
88C4⋊Dic251C2311yescharacteristic
89Dic251⋊C4—21no—
90Dic251⋊C4—21no—
91C22⋊Dic251—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95C4×Dic251—21no—
96C24×C251—21no—
97#41×C251—22no—
98#41×C251—41no—
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251—22no—
101C2×D4×C251—21no—
102C2×D4×C251—21no—
103C2×D4×C251—21no—
104C2×D4×C251—41no—
105C2×D4×C251D411yescharacteristic
106C2×D4×C251C2311yescharacteristic
107C2×D4×C251C2311yescharacteristic
108C22×C4×C251—21no—
109C22×C4×C251—21no—
110C22×C4×C251C2311yescharacteristic
111C22×C4×C251C2311yescharacteristic
112C4⋊C4×C251—21no—
113C4⋊C4×C251D411yescharacteristic
114C4⋊C4×C251C2311yescharacteristic
115C22⋊C4×C251—21no—
116C22⋊C4×C251—21no—
117C22⋊C4×C251—21no—
118C22⋊C4×C251—21no—
119C22⋊C4×C251—21no—
120C22⋊C4×C251D411yescharacteristic
121C22⋊C4×C251D411yescharacteristic
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124Dic502—21no—
125Dic502—21no—
126Dic502—21no—
127Dic502—42no—
128Dic502C2×D412yes—
129D1004—21no—
130D1004—21no—
131C22⋊D251—42no—
132C22⋊D251—42no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—42no—
137C4×D251—42no—
138C251⋊C8—21no—
139C251⋊C8—21no—
140C251⋊C8—21no—
141C251⋊C8C2×D412yes—
142C23×C251—21no—
143C23×C251—21no—
144C23×C251—42no—
145C23×C251—42no—
146C23×C251—81no—
147C23×C251C2×D411yescharacteristic
148C23×C251C2×D411yescharacteristic
149Q8×C251—21no—
150Q8×C251—21no—
151D4×C251—22no—
152D4×C251—21no—
153D4×C251—21no—
154D4×C251—21no—
155D4×C251—42no—
156D4×C251—42no—
157D4×C251—81no—
158D4×C251C2×D412yes—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—42no—
166C2×C4×C251—81no—
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2×D411yescharacteristic
169C2×C4×C251C2×D411yescharacteristic
170C2×C4×C251C2×D411yescharacteristic
171C2×C4×C251C2411yescharacteristic
172C22×C251—21no—
173C22×C251—21no—
174C22×C251—21no—
175C22×C251—22no—
176C22×C251—21no—
177C22×C251—21no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—84no—
184C22×C251#13512yescomplemented
185C22×C251C22×D411yescharacteristic
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—41no—
189C4×C251—21no—
190C4×C251—21no—
191C4×C251—21no—
192C4×C251—21no—
193C4×C251—41no—
194C4×C251—41no—
195C4×C251C22×D411yescharacteristic
196C4×C251H(5,𝔽2)11yescharacteristic
197Dic251—22no—
198Dic251—41no—
199Dic251—41no—
200C2×C251—22no—
201C2×C251—41no—
202C2×C251—42no—
203C2×C251—81no—
204C2×C251D4211yescharacteristic
205C2×C251C2×#13511yescharacteristic
206C2×C251#57711yescharacteristic
207D251—81no—
208C251#328211yescharacteristic, complemented, Sylow
209#3282—2511nomaximal, Sylow
210C2×#135—2511no—
211C22×SD16—2511no—
212#548—2511no—
213#546D25111yescharacteristic, complemented
214#495—2511no—
215#465—2511no—
216#465—2511no—
217#464—2511no—
218#463—2511no—
219#452—2511no—
220#451—2511no—
221#449—2511no—
222#447—2511no—
223C4×SD16—2511no—
224C8×D4—2511no—
225C2×#41—2511no—
226C2×#41C2×D25111yescharacteristic
227C22×Q8—2511no—
228C22×D4C2×D25111yescharacteristic
229#135—5024no—
230C2×Q16—2511no—
231C2×Q16—2511no—
232C2×SD16—5024no—
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16—2511no—
236C2×SD16—2511no—
237C2×SD16—2511no—
238C2×D8—2511no—
239C2×D8—2511no—
240C22×C8—2511no—
241C22×C8—2511no—
242#124—5021no—
243#124—2511no—
244#124C2×D25111yescharacteristic
245#123—21no—
246C22⋊Q8—5021no—
247C22⋊Q8—2511no—
248C22⋊Q8C2×D25111yescharacteristic
249#121—21no—
250#121—5021no—
251#121—2511no—
252#121C2×D25111yescharacteristic
253#120—21no—
254C4×Q8—2511no—
255C4×D4—5021no—
256C4×D4C2×D25111yescharacteristic
257C4×D4C2×D25111yescharacteristic
258C2×C22⋊C4—21no—
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—
269C4×C8—2511no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—10041no—
274#41—5022no—
275C2×Q8—10041no—
276C2×Q8—5022no—
277C2×Q8—5021no—
278C2×Q8—2511no—
279C2×Q8—2511no—
280C2×Q8C22×D25111yescharacteristic
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—5021no—
287C2×D4—5021no—
288C2×D4—2511no—
289C2×D4C22⋊D25111yescharacteristic
290C2×D4C22×D25111yescharacteristic
291C2×D4C22×D25111yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4C22×D25111yescharacteristic
297C22×C4C22×D25111yescharacteristic
298Q16—10041no—
299Q16—5022no—
300SD16—5022no—
301SD16—5022no—
302SD16—5022no—
303SD16—5022no—
304SD16—2514no—
305D8—10041no—
306D8—5022no—
307C2×C8—5022no—
308C2×C8—5022no—
309C2×C8—2511no—
310C2×C8—2511no—
311C2×C8—2511no—
312C2×C8—2511no—
313C4⋊C4—21no—
314C4⋊C4—5021no—
315C4⋊C4—5021no—
316C4⋊C4—2511no—
317C4⋊C4C22⋊D25111yescharacteristic
318C4⋊C4C22×D25111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4C22⋊D25111yescharacteristic
329C22⋊C4C22⋊D25111yescharacteristic
330C42—5021no—
331C42C22×D25111yescharacteristic
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—5021no—
338C23C2×C22⋊D25111yescharacteristic
339C23C2×C22⋊D25111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—10042no—
343Q8—5021no—
344Q8—5021no—
345Q8—5021no—
346Q8—2512no—
347D4—22no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—42no—
353D4—81no—
354D4—10042no—
355D4—10042no—
356D4—5021no—
357D4—5021no—
358D4D4×D25112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—10042no—
368C2×C4—10042no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4C2×C22⋊D25111yescharacteristic
373C2×C4C2×C22⋊D25111yescharacteristic
374C2×C4C2×C22⋊D25111yescharacteristic
375C2×C4C2×C22⋊D25111yescharacteristic
376C2×C4C23×D25111yescharacteristic
377C8—5021no—
378C8—5021no—
379C8—5021no—
380C8—2512no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—22no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—84no—
393C22—10041no—
394C22—10041no—
395C22—10041no—
396C22#2254873512yescomplemented
397C22C22×C22⋊D25111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—10041no—
405C4—10041no—
406C4—5022no—
407C4C2×D4×D25111yescharacteristic
408C4#2254876011yescharacteristic
409C2—22no—
410C2—41no—
411C2—42no—
412C2—81no—
413C2—20081no—
414C2D4×C22⋊D25111yescharacteristic
415C2C2×#2254873511yescharacteristic
416C2#3063249811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#546×C251C2
socleC22×C251C22×D4
hypercenter#546D251
hypocenterC251#3282
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#22548735C2semidirect
2. C22×#16860798C2semidirect
3. #30632346C2semidirect
4. #30632069C2semidirect
5. #30631918C2semidirect
6. #30631914C2semidirect
7. #30631906C2semidirect
8. #30631902C2semidirect
9. #30631849C2semidirect
10. #30631842C2semidirect
11. #30631830C2semidirect
12. #30631821C2semidirect
13. C4×#16860798C2semidirect
14. D4×C251⋊C8C2semidirect
15. #546×C251C2semidirect
16. C2×#16860810C22semidirect
19. C2×#16860800C22semidirect
20. C2×#16860800C22semidirect
26. C2×#16860797C22semidirect
30. C22×C251⋊C8C22semidirect
31. #22548703C22semidirect
33. #22548694C22semidirect
37. C4×Dic502C22semidirect
39. #22548646C22semidirect
40. C4⋊(C251⋊C8)C22semidirect
41. #22548639C22semidirect
42. #22548638C22semidirect
43. #22548638C22semidirect
46. #22548635C22semidirect
48. C22⋊(C251⋊C8)C22semidirect
49. C4×C251⋊C8C22semidirect
78. #16860798D4semidirect
184. C22×C251#135semidirect
208. C251#3282semidirect
213. #546D251semidirect
396. C22#22548735semidirect

Maximal subgroups

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

subgroupindex
C2×#225487352
C22×#168607982
#306323462
#306320692
#306319182
#306319142
#306319062
#306319022
#306318492
#306318422
#306318302
#306318212
C4×#168607982
D4×C251⋊C82
#546×C2512
#3282251