Finite Group Databasebeta

← #5037127

Subgroups — #5037127

Order 512.

3589 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2842C211yesmaximal, characteristic, complemented
2C2×#2656C211yesmaximal, characteristic, complemented
3#17091C212yesmaximal, complemented
4#17088C213yesmaximal, complemented
5#16555C216yesmaximal, complemented
6#15340C213yesmaximal, complemented
7#15315C216yesmaximal, complemented
8#15267C213yesmaximal, complemented
9#15257C213yesmaximal, complemented
10#14748C212yesmaximal, complemented
11#14618C211yesmaximal, characteristic, complemented
12C22×#121C2213yes—
13C22×#121C2213yescomplemented
14C22×#120C2211yescharacteristic
15C22×#120C2213yes—
16#2842—28no—
17#2656—28no—
18C2×#399C2213yes—
19C2×#399C2211yescharacteristic
20C2×#399C2213yescomplemented
21C2×#398C2213yescomplemented
22C2×#396C2212yes—
23C2×#396C2216yes—
24C2×#394C2216yes—
25C2×#394C2216yescomplemented
26C2×#392C2213yes—
27C2×#392C2216yes—
28C2×#392C2216yes—
29C2×#392C2211yescharacteristic, complemented
30C2×#390C2216yes—
31C2×#390C2213yescomplemented
32C2×#386C2216yes—
33C2×#385C2216yescomplemented
34C2×#379C2213yes—
35C22×#95C2211yescharacteristic, complemented
36#1443C2212yescomplemented
37#1441C2216yescomplemented
38#1440C2213yes—
39#1439C2216yes—
40#1439C2216yes—
41#1439C2216yescomplemented
42#1439C2216yescomplemented
43#1439C2216yescomplemented
44#1438C2216yes—
45#1438C2213yes—
46#1438C2213yes—
47#1438C2216yes—
48#1438C2213yescomplemented
49#1437C2213yes—
50#1436C2216yescomplemented
51#1436C2213yescomplemented
52C26C2311yescharacteristic
53C23×D4C2312yes—
54C23×D4C2313yes—
55C23×D4C2311yescharacteristic
56C23×D4C2316yes—
57C23×D4C2313yes—
58C23×D4C2313yes—
59C24×C4C2313yes—
60C2×#121—212no—
61C2×#121—212no—
62C2×#121—46no—
63C2×#121—46no—
64C2×#121—46no—
65C2×#121—46no—
66C2×#120—412no—
67C2×#120—46no—
68C2×#120—46no—
69C22×C4⋊C4C2313yes—
70C22×C4⋊C4C2316yes—
71C22×C4⋊C4C2316yes—
72C22×C4⋊C4C2313yes—
73C22×C4⋊C4C2313yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2316yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2316yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2316yes—
82C22×C22⋊C4C2313yes—
83C22×C22⋊C4C2313yes—
84C22×C22⋊C4C2316yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2311yescharacteristic
87C22×C22⋊C4C2313yes—
88C22×C42C2313yes—
89#399—212no—
90#399—212no—
91#399D418yescomplemented
92#398—46no—
93#396—224no—
94#396—28no—
95#394—412no—
96#394—412no—
97#392—212no—
98#392—24no—
99#392—412no—
100#392—412no—
101#390—412no—
102#390—46no—
103#386—412no—
104#385—412no—
105#379—212no—
106C2×#95—24no—
107C2×#95C2313yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2313yes—
112C2×#95C2316yes—
113C2×#95C2313yes—
114C2×#95C2316yes—
115C2×#95C2316yes—
116C2×#95C2311yescharacteristic
117C25—22no—
118C25—24no—
119C25—46no—
120C25—43no—
121C25—43no—
122C25C2413yes—
123C25C2411yescharacteristic
124C25C2412yes—
125C25C2411yescharacteristic
126C22×D4—22no—
127C22×D4—24no—
128C22×D4—26no—
129C22×D4—212no—
130C22×D4—26no—
131C22×D4—212no—
132C22×D4—26no—
133C22×D4—212no—
134C22×D4—26no—
135C22×D4—26no—
136C22×D4—26no—
137C22×D4—212no—
138C22×D4—46no—
139C22×D4—46no—
140C22×D4—46no—
141C22×D4—43no—
142C22×D4—46no—
143C22×D4—43no—
144C22×D4—43no—
145C22×D4—43no—
146C22×D4—43no—
147C22×D4—46no—
148C22×D4—46no—
149C22×D4—46no—
150C22×D4—46no—
151C22×D4—46no—
152C22×D4—46no—
153C22×D4—46no—
154C23×C4—26no—
155C23×C4—26no—
156C23×C4—26no—
157C23×C4—46no—
158C23×C4—46no—
159C23×C4C2416yes—
160C23×C4C2411yescharacteristic
161C23×C4C2412yes—
162C23×C4C2413yes—
163C23×C4C2413yes—
164C23×C4C2416yes—
165C23×C4C2413yes—
166#121—448no—
167#121—448no—
168#120—448no—
169#120—416no—
170C2×C4⋊C4—26no—
171C2×C4⋊C4—26no—
172C2×C4⋊C4—212no—
173C2×C4⋊C4—26no—
174C2×C4⋊C4—212no—
175C2×C4⋊C4—43no—
176C2×C4⋊C4—43no—
177C2×C4⋊C4—43no—
178C2×C4⋊C4—46no—
179C2×C4⋊C4—46no—
180C2×C4⋊C4—43no—
181C2×C4⋊C4—46no—
182C2×C4⋊C4—46no—
183C2×C4⋊C4—43no—
184C2×C4⋊C4C2×D4112yes—
185C2×C22⋊C4—212no—
186C2×C22⋊C4—212no—
187C2×C22⋊C4—26no—
188C2×C22⋊C4—212no—
189C2×C22⋊C4—212no—
190C2×C22⋊C4—212no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—212no—
193C2×C22⋊C4—26no—
194C2×C22⋊C4—212no—
195C2×C22⋊C4—212no—
196C2×C22⋊C4—212no—
197C2×C22⋊C4—212no—
198C2×C22⋊C4—46no—
199C2×C22⋊C4—43no—
200C2×C22⋊C4—43no—
201C2×C22⋊C4—46no—
202C2×C22⋊C4—43no—
203C2×C22⋊C4—43no—
204C2×C22⋊C4—46no—
205C2×C22⋊C4—43no—
206C2×C22⋊C4—43no—
207C2×C22⋊C4—46no—
208C2×C22⋊C4—46no—
209C2×C22⋊C4—46no—
210C2×C22⋊C4—46no—
211C2×C22⋊C4—46no—
212C2×C22⋊C4—46no—
213C2×C22⋊C4—46no—
214C2×C22⋊C4—46no—
215C2×C22⋊C4—46no—
216C2×C22⋊C4—46no—
217C2×C22⋊C4—46no—
218C2×C22⋊C4—46no—
219C2×C22⋊C4—46no—
220C2×C22⋊C4C2×D4112yes—
221C2×C42—46no—
222C2×C42—43no—
223#95—26no—
224#95—26no—
225#95—212no—
226#95—212no—
227#95—212no—
228#95—46no—
229#95—46no—
230#95—43no—
231#95—46no—
232#95C2×D414yes—
233C24—21no—
234C24—26no—
235C24—23no—
236C24—22no—
237C24—28no—
238C24—22no—
239C24—26no—
240C24—26no—
241C24—26no—
242C24—23no—
243C24—23no—
244C24—26no—
245C24—26no—
246C24—22no—
247C24—22no—
248C24—23no—
249C24—26no—
250C24—26no—
251C24—26no—
252C24—23no—
253C24—26no—
254C24—23no—
255C24—26no—
256C24—26no—
257C24—44no—
258C24—412no—
259C24—86no—
260C24—86no—
261C24—812no—
262C24—812no—
263C24—812no—
264C24—86no—
265C24—86no—
266C24C22×D412yes—
267C24H(5,𝔽2)16yes—
268C242−1+414yes—
269C24C2511yescharacteristic
270C2×D4—412no—
271C2×D4—424no—
272C2×D4—412no—
273C2×D4—424no—
274C2×D4—424no—
275C2×D4—412no—
276C2×D4—412no—
277C2×D4—412no—
278C2×D4—424no—
279C2×D4—44no—
280C2×D4—48no—
281C2×D4—412no—
282C2×D4—812no—
283C2×D4—812no—
284C2×D4—86no—
285C2×D4—812no—
286C2×D4—812no—
287C2×D4—812no—
288C2×D4—86no—
289C2×D4—812no—
290C2×D4—812no—
291C2×D4—86no—
292C2×D4—86no—
293C2×D4—812no—
294C2×D4—812no—
295C2×D4—812no—
296C2×D4—812no—
297C2×D4—86no—
298C22×C4—23no—
299C22×C4—26no—
300C22×C4—26no—
301C22×C4—26no—
302C22×C4—26no—
303C22×C4—26no—
304C22×C4—26no—
305C22×C4—26no—
306C22×C4—26no—
307C22×C4—23no—
308C22×C4—26no—
309C22×C4—26no—
310C22×C4—23no—
311C22×C4—26no—
312C22×C4—23no—
313C22×C4—23no—
314C22×C4—23no—
315C22×C4—26no—
316C22×C4—224no—
317C22×C4—26no—
318C22×C4—21no—
319C22×C4—26no—
320C22×C4—26no—
321C22×C4—23no—
322C22×C4—26no—
323C22×C4—26no—
324C22×C4—26no—
325C22×C4—26no—
326C22×C4—26no—
327C22×C4—26no—
328C22×C4—23no—
329C22×C4—22no—
330C22×C4—812no—
331C22×C4—86no—
332C22×C4—86no—
333C22×C4—812no—
334C22×C4C22×D416yes—
335C22×C4C22×D416yes—
336C22×C4H(5,𝔽2)112yes—
337C22×C4H(5,𝔽2)16yes—
338C4⋊C4—412no—
339C4⋊C4—412no—
340C4⋊C4—86no—
341C4⋊C4—812no—
342C4⋊C4—812no—
343C22⋊C4—412no—
344C22⋊C4—412no—
345C22⋊C4—424no—
346C22⋊C4—424no—
347C22⋊C4—424no—
348C22⋊C4—412no—
349C22⋊C4—48no—
350C22⋊C4—44no—
351C22⋊C4—812no—
352C22⋊C4—812no—
353C22⋊C4—812no—
354C22⋊C4—812no—
355C22⋊C4—86no—
356C22⋊C4—812no—
357C42—86no—
358C23—26no—
359C23—212no—
360C23—212no—
361C23—26no—
362C23—26no—
363C23—22no—
364C23—24no—
365C23—26no—
366C23—212no—
367C23—212no—
368C23—41no—
369C23—43no—
370C23—43no—
371C23—46no—
372C23—43no—
373C23—46no—
374C23—46no—
375C23—41no—
376C23—42no—
377C23—46no—
378C23—43no—
379C23—46no—
380C23—46no—
381C23—46no—
382C23—46no—
383C23—46no—
384C23—46no—
385C23—46no—
386C23—46no—
387C23—46no—
388C23—43no—
389C23—43no—
390C23—46no—
391C23—43no—
392C23—43no—
393C23—43no—
394C23—46no—
395C23—43no—
396C23—46no—
397C23—43no—
398C23—46no—
399C23—46no—
400C23—46no—
401C23—46no—
402C23—46no—
403C23—46no—
404C23—46no—
405C23—46no—
406C23—46no—
407C23—46no—
408C23—46no—
409C23—43no—
410C23—43no—
411C23—824no—
412C23—848no—
413C23—824no—
414C23—88no—
415C23—816no—
416C23#56014yes—
417C23C23×D411yescharacteristic
418C23C2×H(5,𝔽2)13yes—
419C23C2×H(5,𝔽2)13yes—
420C23C2×H(5,𝔽2)16yes—
421C23C2×2−1+412yes—
422D4—824no—
423D4—848no—
424D4—824no—
425D4—824no—
426D4—88no—
427D4—816no—
428C2×C4—26no—
429C2×C4—26no—
430C2×C4—212no—
431C2×C4—26no—
432C2×C4—212no—
433C2×C4—212no—
434C2×C4—26no—
435C2×C4—212no—
436C2×C4—212no—
437C2×C4—26no—
438C2×C4—46no—
439C2×C4—43no—
440C2×C4—43no—
441C2×C4—46no—
442C2×C4—46no—
443C2×C4—46no—
444C2×C4—46no—
445C2×C4—46no—
446C2×C4—43no—
447C2×C4—46no—
448C2×C4—43no—
449C2×C4—46no—
450C2×C4—43no—
451C2×C4—46no—
452C2×C4—46no—
453C2×C4—46no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—46no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—46no—
461C2×C4—46no—
462C2×C4—43no—
463C2×C4—824no—
464C22—24no—
465C22—28no—
466C22—46no—
467C22—412no—
468C22—46no—
469C22—46no—
470C22—412no—
471C22—412no—
472C22—46no—
473C22—42no—
474C22—44no—
475C22—832no—
476C22—86no—
477C22—86no—
478C22—86no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—86no—
483C22—83no—
484C22—86no—
485C22—86no—
486C22—86no—
487C22—83no—
488C22—86no—
489C22—83no—
490C22—86no—
491C22#348413yes—
492C22#348611yescharacteristic
493C22#349813yes—
494C22C2×#56011yescharacteristic
495C22#353213yes—
496C22#353616yes—
497C22#353616yes—
498C22#353716yes—
499C22#354313yes—
500C22#354313yes—
501C4—412no—
502C4—46no—
503C4—46no—
504C4—46no—
505C4—86no—
506C4—81no—
507C4—82no—
508C2—44no—
509C2—84no—
510C2—82no—
511C2—163no—
512C2#4632411yescharacteristic
513C2#4635613yes—
514C2#5064613yes—
515C2#5247212yes—
516C2#5639716yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2842C2semidirect
2. C2×#2656C2semidirect
3. #17091C2semidirect
4. #17088C2semidirect
5. #16555C2semidirect
6. #15340C2semidirect
7. #15315C2semidirect
8. #15267C2semidirect
9. #15257C2semidirect
10. #14748C2semidirect
11. #14618C2semidirect
13. C22×#121C22semidirect
20. C2×#399C22semidirect
21. C2×#398C22semidirect
25. C2×#394C22semidirect
29. C2×#392C22semidirect
31. C2×#390C22semidirect
33. C2×#385C22semidirect
35. C22×#95C22semidirect
36. #1443C22semidirect
37. #1441C22semidirect
41. #1439C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
48. #1438C22semidirect
50. #1436C22semidirect
51. #1436C22semidirect
91. #399D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28422
C2×#26562
#170912
#170882
#165552
#153402
#153152
#152672
#152572
#147482
#146182