Finite Group Databasebeta

← #44553209

Subgroups — #44553209

Order 32128.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548672C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D251C211yesmaximal, characteristic
3C2×C22⋊C4×D251C211yesmaximal, characteristic
4#30631581C211yesmaximal, characteristic, complemented
5#30631580C211yesmaximal, characteristic, complemented
6#30631575C211yesmaximal, characteristic, complemented
7#30631561C211yesmaximal, characteristic, complemented
8#30631558C211yesmaximal, characteristic, complemented
9#30631549C211yesmaximal, characteristic, complemented
10#30631544C211yesmaximal, characteristic, complemented
11#30631542C211yesmaximal, characteristic
12C4×(C2×D251)⋊C4C211yesmaximal, characteristic, complemented
13#30631534C211yesmaximal, characteristic, complemented
14#30631534C211yesmaximal, characteristic, complemented
15#388×C251C211yesmaximal, characteristic, complemented
16C24×D251C2211yescharacteristic
17C22×C4×D251C2211yescharacteristic
18C22×C4×D251C2211yescharacteristic
19C22×C4×D251C2211yescharacteristic
20C22×C4×D251C2211yescharacteristic
21#22548672—24no—
22C2×C4⋊Dic251C2211yescharacteristic, complemented
23C2×Dic251⋊C4C2211yescharacteristic, complemented
24C2×Dic251⋊C4C2211yescharacteristic, complemented
25C2×C22⋊Dic251C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×(C2×D251)⋊C4C2211yescharacteristic
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
36C22⋊C4×D251—24no—
37C22⋊C4×D251C418yescomplemented
38C2×C4×Dic251C2211yescharacteristic, complemented
39C2×C4×Dic251C2211yescharacteristic, complemented
40C2×C4×Dic251C2211yescharacteristic, complemented
41C251⋊#95C2211yescharacteristic, complemented
42C251⋊#95C2211yescharacteristic, 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—22no—
57C23×D251—22no—
58C23×D251—41no—
59C23×D251—41no—
60C23×D251C2×C414yes—
61C23×D251C2311yescharacteristic
62C23×D251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C2×C4×D251—22no—
70C2×C4×D251—22no—
71C2×C4×D251—22no—
72C2×C4×D251—22no—
73C2×C4×D251—22no—
74C2×C4×D251—22no—
75C2×C4×D251—41no—
76C2×C4×D251—41no—
77C2×C4×D251—41no—
78C2×C4×D251—41no—
79C2×C4×D251C2×C414yescomplemented
80C2×C4×D251C2×C414yescomplemented
81C4⋊Dic251—41no—
82Dic251⋊C4—22no—
83Dic251⋊C4—22no—
84C22⋊Dic251—22no—
85C22⋊Dic251—41no—
86C22⋊Dic251C2×C414yescomplemented
87(C2×D251)⋊C4—22no—
88(C2×D251)⋊C4—22no—
89(C2×D251)⋊C4—22no—
90(C2×D251)⋊C4—22no—
91(C2×D251)⋊C4—22no—
92(C2×D251)⋊C4C2×C414yes—
93(C2×D251)⋊C4C2×C414yes—
94(C2×D251)⋊C4D414yes—
95C4×Dic251—22no—
96C4×Dic251—41no—
97C4×Dic251—41no—
98C24×C251C2311yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102C22×C4×C251C2311yescharacteristic
103C22×C4×C251C2311yescharacteristic
104C22×C4×C251C2311yescharacteristic
105C4⋊C4×C251—41no—
106C22⋊C4×C251—22no—
107C22⋊C4×C251—41no—
108C22⋊C4×C251C2×C414yescomplemented
109C42×C251—22no—
110C22×D251—21no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—21no—
116C22×D251—21no—
117C22×D251—21no—
118C22×D251—28no—
119C22×D251—28no—
120C22×D251—44no—
121C22×D251—44no—
122C22×D251—44no—
123C22×D251—44no—
124C22×D251—44no—
125C22×D251C22×C412yes—
126C22×D251C22×C412yes—
127C22×D251C2×D412yes—
128C22×D251#4112yes—
129C22×D251#4112yes—
130C22×D251#4112yes—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—21no—
140C2×Dic251—21no—
141C2×Dic251—21no—
142C2×Dic251—21no—
143C2×Dic251—21no—
144C2×Dic251—21no—
145C2×Dic251—21no—
146C2×Dic251C22×C412yes—
147C2×Dic251C22×C412yes—
148C2×Dic251C2×D412yes—
149C4×D251—28no—
150C4×D251—44no—
151C4×D251—44no—
152C4×D251—44no—
153C23×C251—21no—
154C23×C251—21no—
155C23×C251—21no—
156C23×C251—21no—
157C23×C251—21no—
158C23×C251—21no—
159C23×C251C22×C412yes—
160C23×C251C2411yescharacteristic
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—21no—
173C2×C4×C251—21no—
174C2×C4×C251—21no—
175C2×C4×C251—21no—
176C2×C4×C251C22×C412yes—
177C2×C4×C251C22×C412yes—
178C2×C4×C251C2×D412yes—
179C22×C251—22no—
180C22×C251—22no—
181C22×C251—22no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251C23×C411yescharacteristic
187C22×C251C22×D411yescharacteristic
188C22×C251C2×#4111yescharacteristic
189C22×C251C2×#4111yescharacteristic
190C22×C251C2×#4111yescharacteristic
191C22×C251H(5,𝔽2)11yescharacteristic
192C22×C251H(5,𝔽2)11yescharacteristic
193C2×D251—22no—
194C2×D251—22no—
195C2×D251—22no—
196C2×D251—22no—
197C2×D251—22no—
198C2×D251—22no—
199C2×D251—22no—
200C2×D251—22no—
201C2×D251—22no—
202C2×D251—22no—
203C2×D251—22no—
204C2×D251—22no—
205C2×D251—416no—
206C2×D251#11714yescomplemented
207C2×D251C4×D414yes—
208C4×C251—22no—
209C4×C251—22no—
210C4×C251—22no—
211C4×C251—41no—
212C4×C251—41no—
213C4×C251—41no—
214Dic251—22no—
215Dic251—22no—
216Dic251—22no—
217Dic251—41no—
218Dic251—41no—
219Dic251—41no—
220C2×C251—42no—
221C2×C251C2×#11711yescharacteristic
222C2×C251C2×C4×D411yescharacteristic
223C2×C251#51811yescharacteristic
224C2×C251#54611yescharacteristic
225C2×C251#54611yescharacteristic
226C2×C251#55111yescharacteristic
227C2×C251#55111yescharacteristic
228D251—24no—
229D251—24no—
230C251#234211yescharacteristic, complemented, Sylow
231#2342—2511nomaximal, Sylow
232C2×#117—2511no—
233C22×C22⋊C4—2511no—
234C22×C22⋊C4—2511no—
235#388—2511no—
236#388—2511no—
237#388—2511no—
238#388D25111yescharacteristic, complemented
239#385—2511no—
240#385—2511no—
241#381—2511no—
242#380—2511no—
243#379—2511no—
244C4×C22⋊C4—2511no—
245C4×C22⋊C4—2511no—
246C4×C22⋊C4—2511no—
247C25—2511no—
248C23×C4—2511no—
249C23×C4—2511no—
250C23×C4—2511no—
251C23×C4—2511no—
252#117—5024no—
253C2×C4⋊C4—2511no—
254C2×C4⋊C4—2511no—
255C2×C4⋊C4—2511no—
256C2×C4⋊C4C2×D25111yescharacteristic, complemented
257C2×C22⋊C4—5024no—
258C2×C22⋊C4—2511no—
259C2×C22⋊C4—2511no—
260C2×C22⋊C4—2511no—
261C2×C22⋊C4—2511no—
262C2×C22⋊C4—2511no—
263C2×C22⋊C4—2511no—
264C2×C22⋊C4—2511no—
265C2×C22⋊C4—2511no—
266C2×C22⋊C4—2511no—
267C2×C22⋊C4—2511no—
268C2×C22⋊C4—2511no—
269C2×C22⋊C4—2518no—
270C2×C22⋊C4C2×D25111yescharacteristic
271C2×C22⋊C4C2×D25111yescharacteristic
272C2×C22⋊C4C2×D25111yescharacteristic
273C2×C42—2511no—
274C2×C42—2511no—
275C2×C42—2511no—
276C2×C42C2×D25111yescharacteristic, complemented
277#95—2511no—
278#95—2511no—
279#95—2511no—
280#95—2511no—
281#95—2511no—
282#95—2511no—
283#95C2×D25111yescharacteristic, complemented
284#95C2×D25111yescharacteristic, complemented
285C24—10041no—
286C24—10041no—
287C24—5022no—
288C24—5022no—
289C24—5022no—
290C24—5022no—
291C24—2514no—
292C24—2511no—
293C24—2511no—
294C24C22×D25111yescharacteristic
295C22×C4—10041no—
296C22×C4—10041no—
297C22×C4—10041no—
298C22×C4—10041no—
299C22×C4—5022no—
300C22×C4—5022no—
301C22×C4—5022no—
302C22×C4—5022no—
303C22×C4—5022no—
304C22×C4—5022no—
305C22×C4—2514no—
306C22×C4—2511no—
307C22×C4—2511no—
308C22×C4—2511no—
309C22×C4—2511no—
310C22×C4—2511no—
311C22×C4—2511no—
312C22×C4—2514no—
313C22×C4C22×D25111yescharacteristic
314C22×C4C22×D25111yescharacteristic
315C22×C4C22×D25111yescharacteristic
316C22×C4C22×D25111yescharacteristic
317C22×C4C22×D25111yescharacteristic
318C22×C4C22×D25111yescharacteristic
319C4⋊C4—41no—
320C4⋊C4—10041no—
321C4⋊C4—5022no—
322C4⋊C4—5022no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—10041no—
326C22⋊C4—5022no—
327C22⋊C4—5022no—
328C22⋊C4—5022no—
329C22⋊C4—5022no—
330C22⋊C4—5022no—
331C22⋊C4—5022no—
332C22⋊C4—2514no—
333C22⋊C4—2514no—
334C22⋊C4—2514no—
335C22⋊C4—2514no—
336C22⋊C4C4×D25114yescomplemented
337C42—22no—
338C42—10041no—
339C42—10041no—
340C42—5022no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—10044no—
348C23—10044no—
349C23—10044no—
350C23—10044no—
351C23—10044no—
352C23—5028no—
353C23—5028no—
354C23—5021no—
355C23—5021no—
356C23—5021no—
357C23—5021no—
358C23—5021no—
359C23—5021no—
360C23—5021no—
361C23—5021no—
362C23—2512no—
363C23—2512no—
364C23—2512no—
365C23—2512no—
366C23—2512no—
367C23—2512no—
368C23C2×C4×D25112yes—
369C23C23×D25111yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—10044no—
386C2×C4—10044no—
387C2×C4—10044no—
388C2×C4—5028no—
389C2×C4—5021no—
390C2×C4—5021no—
391C2×C4—5021no—
392C2×C4—5021no—
393C2×C4—5021no—
394C2×C4—5021no—
395C2×C4—5021no—
396C2×C4—5021no—
397C2×C4—5021no—
398C2×C4—5021no—
399C2×C4—5021no—
400C2×C4—5021no—
401C2×C4—5021no—
402C2×C4—5021no—
403C2×C4—5021no—
404C2×C4—2512no—
405C2×C4—2512no—
406C2×C4—2512no—
407C2×C4C2×C4×D25112yes—
408C2×C4C2×C4×D25112yes—
409C2×C4D4×D25112yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—100416no—
418C22—5022no—
419C22—5022no—
420C22—5022no—
421C22—5022no—
422C22—5022no—
423C22—5022no—
424C22—5022no—
425C22—5022no—
426C22—5022no—
427C22—5022no—
428C22—5022no—
429C22—5022no—
430C22—2514no—
431C22—2514no—
432C22C22×C4×D25111yescharacteristic
433C22C2×D4×D25111yescharacteristic
434C22#41×D25111yescharacteristic
435C22#41×D25111yescharacteristic
436C22#41×D25111yescharacteristic
437C22#2254876011yescharacteristic
438C22#2254876111yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—10041no—
446C4—10041no—
447C4—10041no—
448C4—5022no—
449C4—5022no—
450C4—5022no—
451C2—42no—
452C2—5024no—
453C2—5024no—
454C2#117×D25111yescharacteristic
455C2C4×D4×D25111yescharacteristic
456C2#3063215911yescharacteristic
457C2#3063232911yescharacteristic
458C2#3063233411yescharacteristic
459C2#3063236211yescharacteristic
460C2#3063236711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#388×C251C2
socle, cosocleC23×C251C24
prime radicalC24×D251C22
hypercenter#388D251
hypocenterC251#2342
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#22548672C2semidirect
4. #30631581C2semidirect
5. #30631580C2semidirect
6. #30631575C2semidirect
7. #30631561C2semidirect
8. #30631558C2semidirect
9. #30631549C2semidirect
10. #30631544C2semidirect
12. C4×(C2×D251)⋊C4C2semidirect
13. #30631534C2semidirect
14. #30631534C2semidirect
15. #388×C251C2semidirect
22. C2×C4⋊Dic251C22semidirect
23. C2×Dic251⋊C4C22semidirect
24. C2×Dic251⋊C4C22semidirect
37. C22⋊C4×D251C4semidirect
38. C2×C4×Dic251C22semidirect
39. C2×C4×Dic251C22semidirect
40. C2×C4×Dic251C22semidirect
41. C251⋊#95C22semidirect
42. C251⋊#95C22semidirect
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
79. C2×C4×D251C2×C4semidirect
80. C2×C4×D251C2×C4semidirect
86. C22⋊Dic251C2×C4semidirect
108. C22⋊C4×C251C2×C4semidirect
206. C2×D251#117semidirect
230. C251#2342semidirect
238. #388D251semidirect
256. C2×C4⋊C4C2×D251semidirect
276. C2×C42C2×D251semidirect
283. #95C2×D251semidirect
284. #95C2×D251semidirect
336. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486722
C2×C22⋊C4×D2512
#306315812
#306315802
#306315752
#306315612
#306315582
#306315492
#306315442
#306315422
C4×(C2×D251)⋊C42
#306315342
#388×C2512
#2342251