Finite Group Databasebeta

← #44558361

Subgroups — #44558361

Order 32128.

712 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548685C211yesmaximal, characteristic, complemented
2C2×#22548683C211yesmaximal, characteristic, complemented
3#30631615C211yesmaximal, characteristic, complemented
4#30631614C211yesmaximal, characteristic, complemented
5#30631613C211yesmaximal, characteristic, complemented
6#30631605C211yesmaximal, characteristic, complemented
7#30631604C211yesmaximal, characteristic, complemented
8#30631603C211yesmaximal, characteristic, complemented
9#30631603C211yesmaximal, characteristic, complemented
10#30631602C211yesmaximal, characteristic, complemented
11#30631596C211yesmaximal, characteristic, complemented
12#30631596C211yesmaximal, characteristic, complemented
13#30631595C211yesmaximal, characteristic, complemented
14#30631542C211yesmaximal, characteristic, complemented
15#394×C251C211yesmaximal, characteristic, complemented
16C24×D251C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C22×D1004C2211yescharacteristic
19C22×C22⋊D251C2211yescharacteristic
20C22×C22⋊D251C2211yescharacteristic
21C22×C22⋊D251C2211yescharacteristic
22C22×C22⋊D251C2211yescharacteristic, complemented
23#22548685—24no—
24#22548683—24no—
25C2×C4⋊Dic251C2211yescharacteristic, complemented
26C2×Dic251⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic, complemented
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
38C2×(C2×D251)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D251)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D251)⋊C4C2211yescharacteristic, complemented
41C251⋊#95C2211yescharacteristic, complemented
42C251⋊#95C2211yescharacteristic, complemented
43C251⋊#95C2211yescharacteristic, complemented
44C251⋊#95C2211yescharacteristic, complemented
45C251⋊#95C2211yescharacteristic, complemented
46C22×D4×C251C2211yescharacteristic
47C2×C4⋊C4×C251C2211yescharacteristic, complemented
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic, complemented
51C2×C22⋊C4×C251C2211yescharacteristic, complemented
52#95×C251C2211yescharacteristic, complemented
53C23×D251—41no—
54C23×D251—41no—
55C23×D251—41no—
56C23×D251—41no—
57C23×D251—41no—
58C23×D251—41no—
59C23×D251D414yescomplemented
60C23×D251C2311yescharacteristic
61C23×D251C2311yescharacteristic
62C23×D251C2311yescharacteristic
63C2×D1004—22no—
64C2×D1004—22no—
65C2×D1004—41no—
66C2×D1004—41no—
67C2×D1004—41no—
68C2×D1004—41no—
69C2×C22⋊D251—22no—
70C2×C22⋊D251—22no—
71C2×C22⋊D251—22no—
72C2×C22⋊D251—41no—
73C2×C22⋊D251—41no—
74C2×C22⋊D251—41no—
75C2×C22⋊D251—41no—
76C2×C22⋊D251—41no—
77C2×C22⋊D251—41no—
78C2×C22⋊D251—41no—
79C2×C22⋊D251—41no—
80C2×C22⋊D251—41no—
81C22×Dic251C2311yescharacteristic
82C22×Dic251C2311yescharacteristic
83C22×Dic251C2311yescharacteristic
84C22×Dic251C2311yescharacteristic
85C22×Dic251C2311yescharacteristic
86C4⋊Dic251—41no—
87Dic251⋊C4—41no—
88C22⋊Dic251—22no—
89C22⋊Dic251—41no—
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—41no—
96(C2×D251)⋊C4—41no—
97(C2×D251)⋊C4—41no—
98(C2×D251)⋊C4—41no—
99(C2×D251)⋊C4—41no—
100(C2×D251)⋊C4—41no—
101(C2×D251)⋊C4—41no—
102C24×C251C2311yescharacteristic
103C24×C251C2311yescharacteristic
104C2×D4×C251—22no—
105C2×D4×C251—41no—
106C2×D4×C251—41no—
107C22×C4×C251C2311yescharacteristic
108C22×C4×C251C2311yescharacteristic
109C22×C4×C251C2311yescharacteristic
110C22×C4×C251C2311yescharacteristic
111C22×C4×C251C2311yescharacteristic
112C4⋊C4×C251—41no—
113C22⋊C4×C251—22no—
114C22⋊C4×C251—41no—
115C22⋊C4×C251—41no—
116C22⋊C4×C251—41no—
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—21no—
133C22×D251—21no—
134C22×D251—21no—
135C22×D251—21no—
136C22×D251—44no—
137C22×D251—44no—
138C22×D251—44no—
139C22×D251—44no—
140C22×D251—44no—
141C22×D251—44no—
142C22×D251—44no—
143C22×D251C2×D412yes—
144C22×D251C2×D412yes—
145D1004—44no—
146D1004—44no—
147C22⋊D251—44no—
148C22⋊D251—44no—
149C22⋊D251—44no—
150C22⋊D251—44no—
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×Dic251—21no—
161C2×Dic251—21no—
162C2×Dic251—21no—
163C2×Dic251—21no—
164C2×Dic251—21no—
165C2×Dic251—21no—
166C23×C251—21no—
167C23×C251—21no—
168C23×C251—21no—
169C23×C251—21no—
170C23×C251—21no—
171C23×C251—21no—
172C23×C251—21no—
173C23×C251—21no—
174C23×C251—21no—
175C23×C251—21no—
176C23×C251—21no—
177C23×C251—21no—
178C23×C251—21no—
179C23×C251C2×D412yes—
180C23×C251C2411yescharacteristic
181D4×C251—44no—
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×C251—21no—
189C2×C4×C251—21no—
190C2×C4×C251—21no—
191C2×C4×C251—21no—
192C2×C4×C251—21no—
193C2×C4×C251—21no—
194C2×C4×C251—21no—
195C2×C4×C251—21no—
196C2×C4×C251—21no—
197C22×C251—22no—
198C22×C251—22no—
199C22×C251—22no—
200C22×C251—41no—
201C22×C251—41no—
202C22×C251—41no—
203C22×C251—41no—
204C22×C251—41no—
205C22×C251—41no—
206C22×C251—41no—
207C22×C251—41no—
208C22×C251—41no—
209C22×C251—41no—
210C22×C251—41no—
211C22×C251C22×D411yescharacteristic
212C22×C251H(5,𝔽2)11yescharacteristic
213C22×C251H(5,𝔽2)11yescharacteristic
214C22×C251H(5,𝔽2)11yescharacteristic
215C22×C251H(5,𝔽2)11yescharacteristic
216C22×C251H(5,𝔽2)11yescharacteristic
217C22×C251H(5,𝔽2)11yescharacteristic
218C2×D251—22no—
219C2×D251—22no—
220C2×D251—22no—
221C2×D251—22no—
222C2×D251—22no—
223C2×D251—22no—
224C2×D251—416no—
225C2×D251—41no—
226C2×D251—41no—
227C2×D251—41no—
228C2×D251—41no—
229C2×D251—41no—
230C2×D251—41no—
231C2×D251—41no—
232C2×D251—41no—
233C2×D251—41no—
234C2×D251—41no—
235C2×D251—41no—
236C2×D251—41no—
237C2×D251—41no—
238C2×D251—41no—
239C2×D251—41no—
240C4×C251—41no—
241C4×C251—41no—
242C4×C251—41no—
243C4×C251—41no—
244C4×C251—41no—
245Dic251—41no—
246Dic251—41no—
247Dic251—41no—
248Dic251—41no—
249Dic251—41no—
250C2×C251—42no—
251C2×C251—81no—
252C2×C251#53411yescharacteristic
253C2×C251#53411yescharacteristic
254C2×C251#53411yescharacteristic
255C2×C251#56011yescharacteristic
256C2×C251#56011yescharacteristic
257C2×C251#56011yescharacteristic
258C2×C251H(3,𝔽4)11yescharacteristic
259D251—42no—
260D251—42no—
261D251—81no—
262C251#281211yescharacteristic, complemented, Sylow
263#2812—2511nomaximal, Sylow
264C2×#120—2511no—
265C2×#120—2511no—
266#397—2511no—
267#397—2511no—
268#397—2511no—
269#394—2511no—
270#394—2511no—
271#394—2511no—
272#394—2511no—
273#394—2511no—
274#394D25111yescharacteristic, complemented
275#392—2511no—
276#392—2511no—
277#392—2511no—
278#379—2511no—
279C25—2511no—
280C22×D4—2511no—
281C22×D4—2511no—
282C22×D4—2511no—
283C22×D4—2511no—
284C22×D4—2511no—
285C22×D4—2511no—
286C22×D4C2×D25111yescharacteristic
287#120—5024no—
288#120—5024no—
289C2×C4⋊C4—2511no—
290C2×C4⋊C4—2511no—
291C2×C4⋊C4C2×D25111yescharacteristic, complemented
292C2×C22⋊C4—2511no—
293C2×C22⋊C4—2511no—
294C2×C22⋊C4—2511no—
295C2×C22⋊C4—2511no—
296C2×C22⋊C4—2511no—
297C2×C22⋊C4—2511no—
298C2×C22⋊C4—2511no—
299C2×C22⋊C4—2511no—
300C2×C22⋊C4—2511no—
301C2×C22⋊C4—2511no—
302C2×C22⋊C4—2511no—
303C2×C22⋊C4—2511no—
304C2×C22⋊C4—2511no—
305C2×C22⋊C4—2511no—
306C2×C22⋊C4C2×D25111yescharacteristic
307C2×C22⋊C4C2×D25111yescharacteristic
308C2×C22⋊C4C2×D25111yescharacteristic, complemented
309C2×C22⋊C4C2×D25111yescharacteristic, complemented
310#95—2511no—
311#95—2511no—
312#95—2511no—
313#95—2511no—
314#95—2511no—
315#95C2×D25111yescharacteristic, complemented
316C24—10041no—
317C24—10041no—
318C24—10041no—
319C24—10041no—
320C24—10041no—
321C24—10041no—
322C24—2514no—
323C24—2511no—
324C24—2511no—
325C24—2511no—
326C24C22×D25111yescharacteristic
327C24C22×D25111yescharacteristic
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—10041no—
332C2×D4—10041no—
333C2×D4—10041no—
334C2×D4—10041no—
335C2×D4—10041no—
336C2×D4—10041no—
337C2×D4—10041no—
338C2×D4—10041no—
339C2×D4—10041no—
340C2×D4—10041no—
341C2×D4—10041no—
342C2×D4—10041no—
343C2×D4—10041no—
344C2×D4—5022no—
345C2×D4—5022no—
346C2×D4—5022no—
347C2×D4—5022no—
348C2×D4—5022no—
349C22×C4—2511no—
350C22×C4—2511no—
351C22×C4—2511no—
352C22×C4—2511no—
353C22×C4—2511no—
354C22×C4C22×D25111yescharacteristic
355C22×C4C22×D25111yescharacteristic
356C22×C4C22×D25111yescharacteristic
357C22×C4C22×D25111yescharacteristic
358C22×C4C22×D25111yescharacteristic
359C4⋊C4—41no—
360C4⋊C4—10041no—
361C4⋊C4—10041no—
362C22⋊C4—22no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—10041no—
367C22⋊C4—10041no—
368C22⋊C4—10041no—
369C22⋊C4—10041no—
370C22⋊C4—10041no—
371C22⋊C4—10041no—
372C22⋊C4—10041no—
373C22⋊C4—10041no—
374C22⋊C4—10041no—
375C22⋊C4—5022no—
376C22⋊C4—5022no—
377C22⋊C4—5022no—
378C22⋊C4—5022no—
379C22⋊C4—5022no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—10044no—
394C23—10044no—
395C23—10044no—
396C23—10044no—
397C23—10044no—
398C23—10044no—
399C23—10044no—
400C23—5021no—
401C23—5021no—
402C23—5021no—
403C23—5021no—
404C23—5021no—
405C23—5021no—
406C23—5021no—
407C23—5021no—
408C23—5021no—
409C23—5021no—
410C23—5021no—
411C23—5021no—
412C23—5021no—
413C23—5021no—
414C23—5021no—
415C23—5021no—
416C23—5021no—
417C23—5021no—
418C23—5021no—
419C23—2512no—
420C23—2512no—
421C23D4×D25112yes—
422C23C23×D25111yescharacteristic
423D4—44no—
424D4—10044no—
425D4—10044no—
426D4—10044no—
427D4—10044no—
428D4—10044no—
429D4—10044no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—5021no—
446C2×C4—5021no—
447C2×C4—5021no—
448C2×C4—5021no—
449C2×C4—5021no—
450C2×C4—5021no—
451C2×C4—5021no—
452C2×C4—5021no—
453C2×C4—5021no—
454C2×C4—5021no—
455C2×C4—5021no—
456C2×C4—5021no—
457C2×C4—5021no—
458C2×C4—5021no—
459C2×C4—5021no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—100416no—
475C22—10041no—
476C22—10041no—
477C22—10041no—
478C22—10041no—
479C22—10041no—
480C22—10041no—
481C22—10041no—
482C22—10041no—
483C22—10041no—
484C22—10041no—
485C22—10041no—
486C22—10041no—
487C22—10041no—
488C22—10041no—
489C22—10041no—
490C22—5022no—
491C22—5022no—
492C22—5022no—
493C22—5022no—
494C22—5022no—
495C22—5022no—
496C22C2×D4×D25111yescharacteristic
497C22#2254876011yescharacteristic
498C22#2254876011yescharacteristic
499C22#2254876011yescharacteristic
500C22#2254876111yescharacteristic
501C22#2254876111yescharacteristic
502C22#2254876111yescharacteristic
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—10041no—
509C4—10041no—
510C4—10041no—
511C4—10041no—
512C4—10041no—
513C2—42no—
514C2—81no—
515C2—20081no—
516C2—10042no—
517C2—10042no—
518C2#3063224511yescharacteristic
519C2#3063224711yescharacteristic
520C2#3063224811yescharacteristic
521C2#3063242511yescharacteristic
522C2#3063242511yescharacteristic
523C2#3063242711yescharacteristic
524C2#3063242911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
hypercenter#394D251
hypocenterC251#2812
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#22548685C2semidirect
2. C2×#22548683C2semidirect
3. #30631615C2semidirect
4. #30631614C2semidirect
5. #30631613C2semidirect
6. #30631605C2semidirect
7. #30631604C2semidirect
8. #30631603C2semidirect
9. #30631603C2semidirect
10. #30631602C2semidirect
11. #30631596C2semidirect
12. #30631596C2semidirect
13. #30631595C2semidirect
14. #30631542C2semidirect
15. #394×C251C2semidirect
22. C22×C22⋊D251C22semidirect
25. C2×C4⋊Dic251C22semidirect
26. C2×Dic251⋊C4C22semidirect
29. C2×C22⋊Dic251C22semidirect
38. C2×(C2×D251)⋊C4C22semidirect
39. C2×(C2×D251)⋊C4C22semidirect
40. C2×(C2×D251)⋊C4C22semidirect
41. C251⋊#95C22semidirect
42. C251⋊#95C22semidirect
43. C251⋊#95C22semidirect
44. C251⋊#95C22semidirect
45. C251⋊#95C22semidirect
47. C2×C4⋊C4×C251C22semidirect
50. C2×C22⋊C4×C251C22semidirect
51. C2×C22⋊C4×C251C22semidirect
52. #95×C251C22semidirect
59. C23×D251D4semidirect
262. C251#2812semidirect
274. #394D251semidirect
291. C2×C4⋊C4C2×D251semidirect
308. C2×C22⋊C4C2×D251semidirect
309. C2×C22⋊C4C2×D251semidirect
315. #95C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486852
C2×#225486832
#306316152
#306316142
#306316132
#306316052
#306316042
#306316032
#306316022
#306315962
#306315952
#306315422
#394×C2512
#2812251