Finite Group Databasebeta

← #44553018

Subgroups — #44553018

Order 32128.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548685C211yesmaximal, characteristic
2C2×#22548677C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D251C211yesmaximal, characteristic, complemented
4#30631589C211yesmaximal, characteristic, complemented
5#30631587C211yesmaximal, characteristic, complemented
6#30631580C211yesmaximal, characteristic, complemented
7#30631577C211yesmaximal, characteristic, complemented
8#30631576C211yesmaximal, characteristic, complemented
9#30631565C211yesmaximal, characteristic, complemented
10#30631565C211yesmaximal, characteristic, complemented
11#30631563C211yesmaximal, characteristic, complemented
12#30631561C211yesmaximal, characteristic, complemented
13#30631542C211yesmaximal, characteristic, complemented
14C4×(C2×D251)⋊C4C211yesmaximal, characteristic, complemented
15#388×C251C211yesmaximal, characteristic, complemented
16C24×D251C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C22×D1004C2211yescharacteristic
19C22×C22⋊D251C2211yescharacteristic
20C22×C4×D251C2211yescharacteristic
21C22×C4×D251C2211yescharacteristic
22C22×C4×D251C2211yescharacteristic, complemented
23#22548685C418yescomplemented
24#22548677—24no—
25C2×C4⋊Dic251C2211yescharacteristic, complemented
26C2×Dic251⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
30C2×(C2×D251)⋊C4C2211yescharacteristic
31C2×(C2×D251)⋊C4C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×(C2×D251)⋊C4C2211yescharacteristic
37C2×(C2×D251)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D251)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D251)⋊C4C2211yescharacteristic, complemented
40C22⋊C4×D251—24no—
41C2×C4×Dic251C2211yescharacteristic, complemented
42C2×C4×Dic251C2211yescharacteristic, complemented
43C251⋊#95C2211yescharacteristic, complemented
44C251⋊#95C2211yescharacteristic, complemented
45C251⋊#95C2211yescharacteristic, complemented
46C251⋊#95C2211yescharacteristic, complemented
47C2×C4⋊C4×C251C2211yescharacteristic, complemented
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic
51C2×C42×C251C2211yescharacteristic, complemented
52#95×C251C2211yescharacteristic, complemented
53#95×C251C2211yescharacteristic, complemented
54C23×D251—22no—
55C23×D251—22no—
56C23×D251—41no—
57C23×D251—41no—
58C23×D251—41no—
59C23×D251—41no—
60C23×D251C2×C414yescomplemented
61C23×D251C2311yescharacteristic
62C23×D251C2311yescharacteristic
63C23×D251C2311yescharacteristic
64C2×D1004—22no—
65C2×D1004—22no—
66C2×D1004—41no—
67C2×D1004—41no—
68C2×D1004C2×C414yes—
69C2×D1004C2×C414yescomplemented
70C2×C22⋊D251—41no—
71C2×C22⋊D251—41no—
72C2×C22⋊D251C2×C414yescomplemented
73C22×Dic251C2311yescharacteristic
74C22×Dic251C2311yescharacteristic
75C22×Dic251C2311yescharacteristic
76C22×Dic251C2311yescharacteristic
77C22×Dic251C2311yescharacteristic
78C2×C4×D251—22no—
79C2×C4×D251—22no—
80C2×C4×D251—41no—
81C2×C4×D251—41no—
82C2×C4×D251—41no—
83C2×C4×D251—41no—
84C2×C4×D251—41no—
85C2×C4×D251—41no—
86C2×C4×D251D414yescomplemented
87C4⋊Dic251—41no—
88Dic251⋊C4—41no—
89C22⋊Dic251—22no—
90C22⋊Dic251—41no—
91(C2×D251)⋊C4—22no—
92(C2×D251)⋊C4—22no—
93(C2×D251)⋊C4—22no—
94(C2×D251)⋊C4—22no—
95(C2×D251)⋊C4—22no—
96(C2×D251)⋊C4—41no—
97(C2×D251)⋊C4—41no—
98(C2×D251)⋊C4—41no—
99(C2×D251)⋊C4—41no—
100(C2×D251)⋊C4C2×C414yescomplemented
101(C2×D251)⋊C4C2×C414yescomplemented
102C4×Dic251—22no—
103C4×Dic251—41no—
104C24×C251C2311yescharacteristic
105C22×C4×C251C2311yescharacteristic
106C22×C4×C251C2311yescharacteristic
107C22×C4×C251C2311yescharacteristic
108C22×C4×C251C2311yescharacteristic
109C22×C4×C251C2311yescharacteristic
110C22×C4×C251C2311yescharacteristic
111C4⋊C4×C251—22no—
112C22⋊C4×C251—22no—
113C22⋊C4×C251—41no—
114C22⋊C4×C251C2×C414yescomplemented
115C42×C251—41no—
116C22×D251—21no—
117C22×D251—21no—
118C22×D251—21no—
119C22×D251—21no—
120C22×D251—21no—
121C22×D251—21no—
122C22×D251—21no—
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—21no—
127C22×D251—21no—
128C22×D251—21no—
129C22×D251—21no—
130C22×D251—21no—
131C22×D251—21no—
132C22×D251—28no—
133C22×D251—44no—
134C22×D251—44no—
135C22×D251—44no—
136C22×D251—44no—
137C22×D251—44no—
138C22×D251—44no—
139C22×D251C22×C412yes—
140C22×D251C22×C412yes—
141C22×D251C22×C412yes—
142C22×D251C2×D412yes—
143C22×D251#4112yes—
144D1004—28no—
145D1004—44no—
146C22⋊D251—44no—
147C2×Dic251—21no—
148C2×Dic251—21no—
149C2×Dic251—21no—
150C2×Dic251—21no—
151C2×Dic251—21no—
152C2×Dic251—21no—
153C2×Dic251—21no—
154C2×Dic251—21no—
155C2×Dic251—21no—
156C2×Dic251—21no—
157C2×Dic251—21no—
158C2×Dic251—21no—
159C2×Dic251—21no—
160C2×Dic251C22×C412yes—
161C2×Dic251C2×D412yes—
162C4×D251—44no—
163C4×D251—44no—
164C4×D251—44no—
165C23×C251—21no—
166C23×C251—21no—
167C23×C251—21no—
168C23×C251—21no—
169C23×C251—21no—
170C23×C251—21no—
171C23×C251C22×C412yes—
172C23×C251C2411yescharacteristic
173C2×C4×C251—21no—
174C2×C4×C251—21no—
175C2×C4×C251—21no—
176C2×C4×C251—21no—
177C2×C4×C251—21no—
178C2×C4×C251—21no—
179C2×C4×C251—21no—
180C2×C4×C251—21no—
181C2×C4×C251—21no—
182C2×C4×C251—21no—
183C2×C4×C251—21no—
184C2×C4×C251—21no—
185C2×C4×C251—21no—
186C2×C4×C251—21no—
187C2×C4×C251—21no—
188C2×C4×C251C22×C412yes—
189C2×C4×C251C22×C412yes—
190C2×C4×C251C2×D412yes—
191C22×C251—22no—
192C22×C251—22no—
193C22×C251—22no—
194C22×C251—41no—
195C22×C251—41no—
196C22×C251—41no—
197C22×C251—41no—
198C22×C251C23×C411yescharacteristic
199C22×C251C22×D411yescharacteristic
200C22×C251C2×#4111yescharacteristic
201C22×C251H(5,𝔽2)11yescharacteristic
202C22×C251H(5,𝔽2)11yescharacteristic
203C22×C251H(5,𝔽2)11yescharacteristic
204C22×C251H(5,𝔽2)11yescharacteristic
205C2×D251—22no—
206C2×D251—22no—
207C2×D251—22no—
208C2×D251—22no—
209C2×D251—22no—
210C2×D251—22no—
211C2×D251—22no—
212C2×D251—22no—
213C2×D251—22no—
214C2×D251—22no—
215C2×D251—22no—
216C2×D251—22no—
217C2×D251—416no—
218C2×D251—41no—
219C2×D251—41no—
220C2×D251—41no—
221C2×D251—41no—
222C2×D251—41no—
223C2×D251—41no—
224C2×D251—41no—
225C2×D251—41no—
226C2×D251C4×D414yes—
227C4×C251—22no—
228C4×C251—22no—
229C4×C251—22no—
230C4×C251—41no—
231C4×C251—41no—
232C4×C251—41no—
233Dic251—22no—
234Dic251—22no—
235Dic251—41no—
236Dic251—41no—
237Dic251—41no—
238C2×C251—42no—
239C2×C251C2×C4×D411yescharacteristic
240C2×C251#51811yescharacteristic
241C2×C251#51811yescharacteristic
242C2×C251#53411yescharacteristic
243C2×C251#53511yescharacteristic
244C2×C251#53811yescharacteristic
245C2×C251#53811yescharacteristic
246D251—24no—
247D251—42no—
248D251—42no—
249C251#232111yescharacteristic, complemented, Sylow
250#2321—2511nomaximal, Sylow
251C2×#120—2511no—
252C2×C4×D4—2511no—
253C22×C22⋊C4—2511no—
254#390—2511no—
255#390—2511no—
256#388—2511no—
257#388—2511no—
258#388—2511no—
259#388D25111yescharacteristic, complemented
260#386—2511no—
261#386—2511no—
262#386—2511no—
263#385—2511no—
264#379—2511no—
265C4×C22⋊C4—2511no—
266C25—2511no—
267C22×D4—2511no—
268C22×D4—2511no—
269C22×D4—2511no—
270C23×C4—2511no—
271C23×C4—2511no—
272C23×C4—2511no—
273#120—2518no—
274C4×D4—5024no—
275C2×C4⋊C4—2511no—
276C2×C4⋊C4—2511no—
277C2×C4⋊C4C2×D25111yescharacteristic, complemented
278C2×C22⋊C4—5024no—
279C2×C22⋊C4—2511no—
280C2×C22⋊C4—2511no—
281C2×C22⋊C4—2511no—
282C2×C22⋊C4—2511no—
283C2×C22⋊C4—2511no—
284C2×C22⋊C4—2511no—
285C2×C22⋊C4—2511no—
286C2×C22⋊C4—2511no—
287C2×C22⋊C4—2511no—
288C2×C22⋊C4—2511no—
289C2×C22⋊C4—2511no—
290C2×C22⋊C4—2511no—
291C2×C22⋊C4—2511no—
292C2×C22⋊C4C2×D25111yescharacteristic
293C2×C22⋊C4C2×D25111yescharacteristic
294C2×C22⋊C4C2×D25111yescharacteristic
295C2×C42—2511no—
296C2×C42—2511no—
297C2×C42C2×D25111yescharacteristic, complemented
298#95—2511no—
299#95—2511no—
300#95—2511no—
301#95—2511no—
302#95C2×D25111yescharacteristic, complemented
303#95C2×D25111yescharacteristic, complemented
304C24—10041no—
305C24—10041no—
306C24—10041no—
307C24—10041no—
308C24—5022no—
309C24—5022no—
310C24—2514no—
311C24—2511no—
312C24—2511no—
313C24—2511no—
314C24C22×D25111yescharacteristic
315C2×D4—10041no—
316C2×D4—10041no—
317C2×D4—10041no—
318C2×D4—10041no—
319C2×D4—5022no—
320C2×D4—5022no—
321C2×D4—2514no—
322C2×D4—2514no—
323C2×D4—2514no—
324C22×C4—10041no—
325C22×C4—10041no—
326C22×C4—10041no—
327C22×C4—10041no—
328C22×C4—10041no—
329C22×C4—10041no—
330C22×C4—5022no—
331C22×C4—5022no—
332C22×C4—2514no—
333C22×C4—2511no—
334C22×C4—2511no—
335C22×C4—2511no—
336C22×C4—2511no—
337C22×C4—2511no—
338C22×C4C22×D25111yescharacteristic
339C22×C4C22×D25111yescharacteristic
340C22×C4C22×D25111yescharacteristic
341C22×C4C22×D25111yescharacteristic
342C22×C4C22×D25111yescharacteristic
343C22×C4C22×D25111yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—10041no—
346C4⋊C4—10041no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—10041no—
350C22⋊C4—10041no—
351C22⋊C4—10041no—
352C22⋊C4—10041no—
353C22⋊C4—10041no—
354C22⋊C4—5022no—
355C22⋊C4—5022no—
356C22⋊C4—5022no—
357C22⋊C4—5022no—
358C22⋊C4—5022no—
359C22⋊C4—5022no—
360C22⋊C4—2514no—
361C22⋊C4—2514no—
362C22⋊C4C4×D25114yescomplemented
363C42—41no—
364C42—10041no—
365C42—5022no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—10044no—
373C23—10044no—
374C23—10044no—
375C23—10044no—
376C23—10044no—
377C23—10044no—
378C23—5028no—
379C23—5021no—
380C23—5021no—
381C23—5021no—
382C23—5021no—
383C23—5021no—
384C23—5021no—
385C23—5021no—
386C23—5021no—
387C23—5021no—
388C23—5021no—
389C23—5021no—
390C23—5021no—
391C23—5021no—
392C23—5021no—
393C23—5021no—
394C23—5021no—
395C23—2512no—
396C23—2512no—
397C23—2512no—
398C23—2512no—
399C23—2512no—
400C23C2×C4×D25112yes—
401C23C23×D25111yescharacteristic
402D4—10044no—
403D4—10044no—
404D4—5028no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—10044no—
421C2×C4—10044no—
422C2×C4—10044no—
423C2×C4—5021no—
424C2×C4—5021no—
425C2×C4—5021no—
426C2×C4—5021no—
427C2×C4—5021no—
428C2×C4—5021no—
429C2×C4—5021no—
430C2×C4—5021no—
431C2×C4—5021no—
432C2×C4—5021no—
433C2×C4—5021no—
434C2×C4—5021no—
435C2×C4—5021no—
436C2×C4—2512no—
437C2×C4—2512no—
438C2×C4C2×C4×D25112yes—
439C2×C4C2×C4×D25112yes—
440C2×C4D4×D25112yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—100416no—
449C22—10041no—
450C22—10041no—
451C22—10041no—
452C22—10041no—
453C22—10041no—
454C22—10041no—
455C22—10041no—
456C22—10041no—
457C22—5022no—
458C22—5022no—
459C22—5022no—
460C22—5022no—
461C22—5022no—
462C22—5022no—
463C22—5022no—
464C22—5022no—
465C22—5022no—
466C22—5022no—
467C22—5022no—
468C22—5022no—
469C22—2514no—
470C22C22×C4×D25111yescharacteristic
471C22C2×D4×D25111yescharacteristic
472C22#41×D25111yescharacteristic
473C22#2254876011yescharacteristic
474C22#2254876111yescharacteristic
475C22#2254876111yescharacteristic
476C22#2254876111yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—10041no—
484C4—10041no—
485C4—10041no—
486C4—5022no—
487C4—5022no—
488C2—42no—
489C2—10042no—
490C2—10042no—
491C2—5024no—
492C2C4×D4×D25111yescharacteristic
493C2#3063215911yescharacteristic
494C2#3063216111yescharacteristic
495C2#3063224511yescharacteristic
496C2#3063225511yescharacteristic
497C2#3063227311yescharacteristic
498C2#3063227611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#388×C251C2
socle, cosocleC23×C251C24
prime radicalC2×#22548685C2
hypercenter#388D251
hypocenterC251#2321
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#22548677C2semidirect
3. C2×C22⋊C4×D251C2semidirect
4. #30631589C2semidirect
5. #30631587C2semidirect
6. #30631580C2semidirect
7. #30631577C2semidirect
8. #30631576C2semidirect
9. #30631565C2semidirect
10. #30631565C2semidirect
11. #30631563C2semidirect
12. #30631561C2semidirect
13. #30631542C2semidirect
14. C4×(C2×D251)⋊C4C2semidirect
15. #388×C251C2semidirect
22. C22×C4×D251C22semidirect
23. #22548685C4semidirect
25. C2×C4⋊Dic251C22semidirect
26. C2×Dic251⋊C4C22semidirect
37. C2×(C2×D251)⋊C4C22semidirect
38. C2×(C2×D251)⋊C4C22semidirect
39. C2×(C2×D251)⋊C4C22semidirect
41. C2×C4×Dic251C22semidirect
42. C2×C4×Dic251C22semidirect
43. C251⋊#95C22semidirect
44. C251⋊#95C22semidirect
45. C251⋊#95C22semidirect
46. C251⋊#95C22semidirect
47. C2×C4⋊C4×C251C22semidirect
51. C2×C42×C251C22semidirect
52. #95×C251C22semidirect
53. #95×C251C22semidirect
60. C23×D251C2×C4semidirect
69. C2×D1004C2×C4semidirect
72. C2×C22⋊D251C2×C4semidirect
86. C2×C4×D251D4semidirect
100. (C2×D251)⋊C4C2×C4semidirect
101. (C2×D251)⋊C4C2×C4semidirect
114. C22⋊C4×C251C2×C4semidirect
249. C251#2321semidirect
259. #388D251semidirect
277. C2×C4⋊C4C2×D251semidirect
297. C2×C42C2×D251semidirect
302. #95C2×D251semidirect
303. #95C2×D251semidirect
362. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486852
C2×#225486772
C2×C22⋊C4×D2512
#306315892
#306315872
#306315802
#306315772
#306315762
#306315652
#306315632
#306315612
#306315422
C4×(C2×D251)⋊C42
#388×C2512
#2321251