Finite Group Databasebeta

← #5083051

Subgroups — #5083051

Order 512.

3645 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2675C211yesmaximal, characteristic, complemented
2C2×#2656C211yesmaximal, characteristic, complemented
3#17204C212yesmaximal, complemented
4#17088C213yesmaximal, complemented
5#16250C216yesmaximal, complemented
6#15950C213yesmaximal, complemented
7#15730C213yesmaximal, complemented
8#15310C213yesmaximal, complemented
9#15263C216yesmaximal, complemented
10#14748C212yesmaximal, complemented
11#14651C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#123C2213yes—
14C22×#121C2213yes—
15C22×#121C2213yes—
16C22×#121C2216yes—
17#2675—28no—
18#2656—28no—
19C2×#399C2211yescharacteristic
20C2×#399C2213yes—
21C2×#396C2212yes—
22C2×#394C2216yes—
23C2×#394C2213yes—
24C2×#392C2213yes—
25C2×#392C2213yes—
26C2×#392C2216yes—
27C2×#392C2216yes—
28C2×#392C2211yescharacteristic
29C2×#386C2213yes—
30C2×#386C2216yes—
31C2×#385C2213yes—
32C2×#385C2216yes—
33C22×#95C2211yescharacteristic
34#1443C2212yes—
35#1441C2216yes—
36#1441C2216yes—
37#1441C2216yes—
38#1440C2213yes—
39#1439C2216yes—
40#1439C2216yes—
41#1439C2216yes—
42#1439C2216yes—
43#1438C2216yes—
44#1438C2216yes—
45#1438C2213yes—
46#1438C2213yes—
47#1438C2213yes—
48#1438C2216yes—
49#1437C2213yes—
50#1436C2213yes—
51C22⋊#95C2216yes—
52C23×D4C2312yes—
53C23×D4C2313yes—
54C23×D4C2311yescharacteristic
55C23×D4C2313yes—
56C23×D4C2316yes—
57C23×D4C2311yescharacteristic
58C23×D4C2312yes—
59C23×D4C2313yes—
60C24×C4C2313yes—
61C24×C4C2313yes—
62C2×#127—46no—
63C2×#123—412no—
64C2×#123—46no—
65C2×#121—212no—
66C2×#121—224no—
67C2×#121—212no—
68C2×#121—412no—
69C2×#121—46no—
70C2×#121—412no—
71C2×#121—46no—
72C2×#121—46no—
73C2×#121—46no—
74C22×C4⋊C4C2313yes—
75C22×C4⋊C4C2316yes—
76C22×C4⋊C4C2316yes—
77C22×C4⋊C4C2313yes—
78C22×C22⋊C4C2313yes—
79C22×C22⋊C4C2316yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2316yes—
82C22×C22⋊C4C2316yes—
83C22×C22⋊C4C2316yes—
84C22×C22⋊C4C2313yes—
85C22×C22⋊C4C2316yes—
86C22×C22⋊C4C2316yes—
87C22×C22⋊C4C2313yes—
88C22×C22⋊C4C2313yes—
89C22×C22⋊C4C2313yes—
90C22×C42C2311yescharacteristic
91#399—212no—
92#399—24no—
93#396—28no—
94#394—412no—
95#394—46no—
96#392—212no—
97#392—212no—
98#392—412no—
99#392—412no—
100#392D418yescomplemented
101#386—412no—
102#386—46no—
103#385—212no—
104#385—412no—
105C2×#95—24no—
106C2×#95C2313yes—
107C2×#95C2316yes—
108C2×#95C2313yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2313yes—
112C2×#95C2316yes—
113C2×#95C2316yes—
114C2×#95C2316yes—
115C2×#95C2316yes—
116C2×#95C2311yescharacteristic
117C25C2412yes—
118C25C2411yescharacteristic
119C25C2413yes—
120C25C2411yescharacteristic
121C22×D4—26no—
122C22×D4—212no—
123C22×D4—26no—
124C22×D4—26no—
125C22×D4—212no—
126C22×D4—212no—
127C22×D4—26no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—26no—
131C22×D4—212no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—46no—
135C22×D4—46no—
136C22×D4—46no—
137C22×D4—43no—
138C22×D4—46no—
139C22×D4—43no—
140C22×D4—43no—
141C22×D4—43no—
142C22×D4—46no—
143C22×D4—46no—
144C22×D4—43no—
145C22×D4—46no—
146C22×D4—46no—
147C22×D4—43no—
148C22×D4—46no—
149C22×D4—46no—
150C22×D4—46no—
151C22×D4—46no—
152C22×D4—46no—
153C22×D4—43no—
154C22×D4C2×D4112yes—
155C23×C4—26no—
156C23×C4—26no—
157C23×C4—26no—
158C23×C4—26no—
159C23×C4—212no—
160C23×C4—46no—
161C23×C4—43no—
162C23×C4—46no—
163C23×C4—46no—
164C23×C4—43no—
165C23×C4C2416yes—
166C23×C4C2413yes—
167C23×C4C2413yes—
168C23×C4C2411yescharacteristic
169C23×C4C2413yes—
170C23×C4C2416yes—
171C23×C4C2412yes—
172#127—416no—
173#123—448no—
174#121—496no—
175#121—448no—
176#121—448no—
177C2×C4⋊C4—212no—
178C2×C4⋊C4—26no—
179C2×C4⋊C4—26no—
180C2×C4⋊C4—212no—
181C2×C4⋊C4—26no—
182C2×C4⋊C4—212no—
183C2×C4⋊C4—212no—
184C2×C4⋊C4—46no—
185C2×C4⋊C4—43no—
186C2×C4⋊C4—43no—
187C2×C4⋊C4—43no—
188C2×C4⋊C4—46no—
189C2×C22⋊C4—26no—
190C2×C22⋊C4—212no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—212no—
193C2×C22⋊C4—212no—
194C2×C22⋊C4—26no—
195C2×C22⋊C4—212no—
196C2×C22⋊C4—212no—
197C2×C22⋊C4—212no—
198C2×C22⋊C4—212no—
199C2×C22⋊C4—212no—
200C2×C22⋊C4—212no—
201C2×C22⋊C4—43no—
202C2×C22⋊C4—43no—
203C2×C22⋊C4—46no—
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—43no—
211C2×C22⋊C4—46no—
212C2×C22⋊C4—46no—
213C2×C22⋊C4—43no—
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⋊C4—46no—
221C2×C22⋊C4C2×D4112yes—
222C2×C42—43no—
223#95—26no—
224#95—26no—
225#95—212no—
226#95—46no—
227#95—46no—
228#95—46no—
229#95—46no—
230#95—46no—
231#95—46no—
232#95—43no—
233#95C2×D414yes—
234C24—21no—
235C24—26no—
236C24—23no—
237C24—22no—
238C24—26no—
239C24—26no—
240C24—23no—
241C24—23no—
242C24—26no—
243C24—26no—
244C24—26no—
245C24—26no—
246C24—22no—
247C24—21no—
248C24—26no—
249C24—26no—
250C24—26no—
251C24—23no—
252C24—26no—
253C24—23no—
254C24—26no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24C22×D416yes—
259C24H(5,𝔽2)16yes—
260C24C2511yescharacteristic
261C2×D4—412no—
262C2×D4—424no—
263C2×D4—424no—
264C2×D4—412no—
265C2×D4—412no—
266C2×D4—424no—
267C2×D4—412no—
268C2×D4—412no—
269C2×D4—424no—
270C2×D4—424no—
271C2×D4—44no—
272C2×D4—48no—
273C2×D4—44no—
274C2×D4—48no—
275C2×D4—412no—
276C2×D4—86no—
277C2×D4—812no—
278C2×D4—812no—
279C2×D4—812no—
280C2×D4—86no—
281C2×D4—812no—
282C2×D4—812no—
283C2×D4—812no—
284C2×D4—86no—
285C2×D4—812no—
286C2×D4—86no—
287C2×D4—86no—
288C2×D4—812no—
289C2×D4—812no—
290C2×D4—812no—
291C2×D4—812no—
292C2×D4—86no—
293C2×D4—812no—
294C2×D4—86no—
295C22×C4—23no—
296C22×C4—26no—
297C22×C4—26no—
298C22×C4—26no—
299C22×C4—26no—
300C22×C4—26no—
301C22×C4—26no—
302C22×C4—26no—
303C22×C4—26no—
304C22×C4—26no—
305C22×C4—23no—
306C22×C4—26no—
307C22×C4—23no—
308C22×C4—26no—
309C22×C4—23no—
310C22×C4—23no—
311C22×C4—23no—
312C22×C4—26no—
313C22×C4—224no—
314C22×C4—23no—
315C22×C4—26no—
316C22×C4—26no—
317C22×C4—23no—
318C22×C4—26no—
319C22×C4—26no—
320C22×C4—26no—
321C22×C4—26no—
322C22×C4—26no—
323C22×C4—26no—
324C22×C4—23no—
325C22×C4—26no—
326C22×C4—412no—
327C22×C4—424no—
328C22×C4—812no—
329C22×C4—86no—
330C22×C4—812no—
331C22×C4—86no—
332C22×C4—86no—
333C22×C4—812no—
334C22×C4—86no—
335C22×C4C22×D412yes—
336C22×C4C22×D416yes—
337C22×C4H(5,𝔽2)112yes—
338C22×C4H(5,𝔽2)16yes—
339C22×C42−1+414yes—
340C4⋊C4—424no—
341C4⋊C4—412no—
342C4⋊C4—424no—
343C4⋊C4—412no—
344C22⋊C4—224no—
345C22⋊C4—412no—
346C22⋊C4—424no—
347C22⋊C4—424no—
348C22⋊C4—424no—
349C22⋊C4—412no—
350C22⋊C4—424no—
351C22⋊C4—412no—
352C22⋊C4—812no—
353C22⋊C4—812no—
354C22⋊C4—86no—
355C22⋊C4—812no—
356C42—44no—
357C23—26no—
358C23—212no—
359C23—212no—
360C23—26no—
361C23—26no—
362C23—26no—
363C23—212no—
364C23—212no—
365C23—26no—
366C23—43no—
367C23—41no—
368C23—43no—
369C23—43no—
370C23—46no—
371C23—43no—
372C23—43no—
373C23—46no—
374C23—46no—
375C23—46no—
376C23—46no—
377C23—46no—
378C23—46no—
379C23—46no—
380C23—43no—
381C23—43no—
382C23—46no—
383C23—46no—
384C23—46no—
385C23—46no—
386C23—46no—
387C23—46no—
388C23—43no—
389C23—43no—
390C23—46no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—43no—
396C23—46no—
397C23—46no—
398C23—46no—
399C23—46no—
400C23—46no—
401C23—46no—
402C23—46no—
403C23—46no—
404C23—46no—
405C23—43no—
406C23—43no—
407C23—46no—
408C23—46no—
409C23—46no—
410C23#56014yes—
411C23C23×D411yescharacteristic
412C23C2×H(5,𝔽2)13yes—
413C23C2×H(5,𝔽2)16yes—
414C23C2×H(5,𝔽2)13yes—
415C23C2×2−1+412yes—
416D4—88no—
417D4—816no—
418D4—824no—
419D4—824no—
420D4—848no—
421D4—88no—
422D4—816no—
423D4—824no—
424C2×C4—212no—
425C2×C4—26no—
426C2×C4—26no—
427C2×C4—212no—
428C2×C4—26no—
429C2×C4—212no—
430C2×C4—26no—
431C2×C4—212no—
432C2×C4—212no—
433C2×C4—26no—
434C2×C4—46no—
435C2×C4—46no—
436C2×C4—46no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—43no—
440C2×C4—46no—
441C2×C4—46no—
442C2×C4—46no—
443C2×C4—46no—
444C2×C4—43no—
445C2×C4—46no—
446C2×C4—43no—
447C2×C4—46no—
448C2×C4—43no—
449C2×C4—43no—
450C2×C4—46no—
451C2×C4—46no—
452C2×C4—46no—
453C2×C4—46no—
454C2×C4—46no—
455C2×C4—46no—
456C2×C4—43no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—824no—
460C2×C4—824no—
461C22—24no—
462C22—28no—
463C22—46no—
464C22—412no—
465C22—46no—
466C22—46no—
467C22—412no—
468C22—46no—
469C22—412no—
470C22—46no—
471C22—86no—
472C22—86no—
473C22—86no—
474C22—86no—
475C22—86no—
476C22—83no—
477C22—86no—
478C22—86no—
479C22—86no—
480C22—86no—
481C22—83no—
482C22—81no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22#348413yes—
488C22#348413yes—
489C22#349911yescharacteristic
490C22C2×#56011yescharacteristic
491C22#353213yes—
492C22#353616yes—
493C22#353716yes—
494C22#354313yes—
495C22#354313yes—
496C22#354816yes—
497C4—412no—
498C4—46no—
499C4—46no—
500C4—42no—
501C4—44no—
502C4—86no—
503C4—83no—
504C2—44no—
505C2—86no—
506C2—162no—
507C2—161no—
508C2#4632411yescharacteristic
509C2#4642313yes—
510C2#4782713yes—
511C2#5247212yes—
512C2#5801016yes—
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×#2675C2semidirect
2. C2×#2656C2semidirect
3. #17204C2semidirect
4. #17088C2semidirect
5. #16250C2semidirect
6. #15950C2semidirect
7. #15730C2semidirect
8. #15310C2semidirect
9. #15263C2semidirect
10. #14748C2semidirect
11. #14651C2semidirect
100. #392D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26752
C2×#26562
#172042
#170882
#162502
#159502
#157302
#153102
#152632
#147482
#146512