Finite Group Databasebeta

← #2745486

Subgroups — #2745486

Order 512.

3890 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2842C211yesmaximal, characteristic, complemented
2C22×#399C211yesmaximal, characteristic, complemented
3#17204C212yesmaximal, complemented
4#17136C216yesmaximal, complemented
5#16824C213yesmaximal, complemented
6#16298C213yesmaximal, complemented
7#14748C212yesmaximal, complemented
8#14734C213yesmaximal, complemented
9#14729C216yesmaximal, complemented
10#14636C211yesmaximal, characteristic, complemented
11#13867C213yesmaximal
12C24×D4C2211yescharacteristic
13C23×C4⋊C4C2213yescomplemented
14C23×C22⋊C4C2213yes—
15#2842—28no—
16C2×#399—28no—
17C2×#399C2213yes—
18C2×#399C2213yes—
19C2×#399C2211yescharacteristic
20C2×#399C2211yescharacteristic, complemented
21C2×#399C2213yescomplemented
22C2×#399C2213yescomplemented
23C2×#396C2212yescomplemented
24C2×#396C2216yescomplemented
25C2×#396C2216yescomplemented
26C2×#386C2216yes—
27C2×#386C2216yes—
28C2×#385C2216yescomplemented
29C2×#380C2213yescomplemented
30C2×#379C2213yes—
31C2×#379C2213yes—
32C2×#379C2213yes—
33C22×#95C2211yescharacteristic, complemented
34#1443C2212yescomplemented
35#1441C2216yescomplemented
36#1441C2216yescomplemented
37#1441C2216yescomplemented
38#1441C2216yescomplemented
39#1441C2216yescomplemented
40#1439C2213yes—
41#1439C2216yes—
42#1439C2216yescomplemented
43#1439C2216yescomplemented
44#1439C2216yescomplemented
45#1439C2213yescomplemented
46#1439C2216yescomplemented
47#1438C2216yes—
48#1438C2216yes—
49C22⋊#95C2213yes—
50C22⋊#95C2213yes—
51C22⋊#95C2213yescomplemented
52C26C2311yescharacteristic
53C26C2311yescharacteristic
54C23×D4—46no—
55C23×D4—46no—
56C23×D4Q818yes—
57C23×D4C2312yes—
58C23×D4C2312yes—
59C24×C4C2313yes—
60C24×C4C2313yes—
61C24×C4C2311yescharacteristic
62C22×C4⋊C4—212no—
63C22×C4⋊C4—46no—
64C22×C4⋊C4—46no—
65C22×C4⋊C4C2316yes—
66C22×C4⋊C4C2313yes—
67C22×C4⋊C4C2316yes—
68C22×C4⋊C4C2316yes—
69C22×C4⋊C4C2313yes—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2313yes—
72C22×C22⋊C4—212no—
73C22×C22⋊C4—46no—
74C22×C22⋊C4—46no—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2316yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2313yes—
82C22×C22⋊C4C2313yes—
83C22×C22⋊C4C2313yes—
84C22×C22⋊C4C2316yes—
85C22×C22⋊C4C2316yes—
86C22×C22⋊C4C2313yes—
87#399—212no—
88#399—212no—
89#399—212no—
90#399—24no—
91#399—46no—
92#399D418yescomplemented
93#396—224no—
94#396—28no—
95#396—412no—
96#386—412no—
97#386—412no—
98#385—412no—
99#380—46no—
100#379—212no—
101#379—46no—
102#379—46no—
103C2×#95—24no—
104C2×#95C2313yes—
105C2×#95C2313yes—
106C2×#95C2316yes—
107C2×#95C2313yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2316yes—
112C2×#95C2316yes—
113C2×#95C2316yes—
114C2×#95C2316yes—
115C2×#95C2311yescharacteristic
116C25—22no—
117C25—22no—
118C25—26no—
119C25—26no—
120C25—22no—
121C25—22no—
122C25—43no—
123C25—43no—
124C25—43no—
125C25—43no—
126C25—43no—
127C25—43no—
128C25C2×Q814yes—
129C25C2×Q814yes—
130C25C2412yes—
131C25C2411yescharacteristic
132C25C2411yescharacteristic
133C25C2411yescharacteristic
134C22×D4—212no—
135C22×D4—212no—
136C22×D4—46no—
137C22×D4—46no—
138C22×D4—448no—
139C22×D4—448no—
140C22×D4—416no—
141C22×D4C2×Q818yes—
142C22×D4C2×Q818yes—
143C23×C4—26no—
144C23×C4—26no—
145C23×C4—26no—
146C23×C4—212no—
147C23×C4—26no—
148C23×C4—212no—
149C23×C4—26no—
150C23×C4—46no—
151C23×C4—43no—
152C23×C4—43no—
153C23×C4—43no—
154C23×C4—46no—
155C23×C4C2×Q814yes—
156C23×C4C2416yes—
157C23×C4C2416yes—
158C23×C4C2413yes—
159C23×C4C2413yes—
160C23×C4C2413yes—
161C23×C4C2412yes—
162C23×C4C2413yes—
163C2×C4⋊C4—212no—
164C2×C4⋊C4—26no—
165C2×C4⋊C4—212no—
166C2×C4⋊C4—212no—
167C2×C4⋊C4—26no—
168C2×C4⋊C4—212no—
169C2×C4⋊C4—43no—
170C2×C4⋊C4—43no—
171C2×C4⋊C4—46no—
172C2×C4⋊C4—43no—
173C2×C4⋊C4—43no—
174C2×C4⋊C4—448no—
175C2×C4⋊C4—46no—
176C2×C4⋊C4—46no—
177C2×C4⋊C4—43no—
178C2×C4⋊C4—43no—
179C2×C4⋊C4—46no—
180C2×C4⋊C4—46no—
181C2×C4⋊C4—46no—
182C2×C4⋊C4—43no—
183C2×C4⋊C4C2×D4112yescomplemented
184C2×C22⋊C4—26no—
185C2×C22⋊C4—212no—
186C2×C22⋊C4—212no—
187C2×C22⋊C4—212no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—26no—
190C2×C22⋊C4—26no—
191C2×C22⋊C4—26no—
192C2×C22⋊C4—26no—
193C2×C22⋊C4—26no—
194C2×C22⋊C4—212no—
195C2×C22⋊C4—212no—
196C2×C22⋊C4—212no—
197C2×C22⋊C4—43no—
198C2×C22⋊C4—43no—
199C2×C22⋊C4—43no—
200C2×C22⋊C4—43no—
201C2×C22⋊C4—43no—
202C2×C22⋊C4—43no—
203C2×C22⋊C4—448no—
204C2×C22⋊C4—46no—
205C2×C22⋊C4—46no—
206C2×C22⋊C4—43no—
207C2×C22⋊C4—46no—
208C2×C22⋊C4—43no—
209C2×C22⋊C4—43no—
210C2×C22⋊C4—43no—
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—
221#95—26no—
222#95—26no—
223#95—46no—
224#95—46no—
225#95—46no—
226#95—46no—
227#95—46no—
228#95—46no—
229#95—46no—
230#95—46no—
231#95—43no—
232#95C2×D414yescomplemented
233C24—21no—
234C24—23no—
235C24—21no—
236C24—23no—
237C24—23no—
238C24—23no—
239C24—23no—
240C24—23no—
241C24—28no—
242C24—21no—
243C24—22no—
244C24—23no—
245C24—26no—
246C24—26no—
247C24—26no—
248C24—23no—
249C24—23no—
250C24—21no—
251C24—22no—
252C24—26no—
253C24—23no—
254C24—224no—
255C24—28no—
256C24—412no—
257C24—412no—
258C24—412no—
259C24—412no—
260C24—412no—
261C24—412no—
262C24—412no—
263C24—44no—
264C24—412no—
265C24—412no—
266C24—44no—
267C24—86no—
268C24—812no—
269C24—86no—
270C24—86no—
271C24—86no—
272C24—86no—
273C24—86no—
274C24—86no—
275C24—812no—
276C24—86no—
277C24C22×D412yes—
278C24C22×Q814yes—
279C24C22×Q812yes—
280C24C22×Q812yes—
281C24C22×Q812yes—
282C24H(5,𝔽2)16yes—
283C24H(5,𝔽2)16yes—
284C242−1+412yes—
285C24C2511yescharacteristic
286C2×D4—248no—
287C2×D4—424no—
288C2×D4—4128no—
289C2×D4—48no—
290C2×D4—424no—
291C2×D4—424no—
292C2×D4—48no—
293C22×C4—224no—
294C22×C4—23no—
295C22×C4—26no—
296C22×C4—26no—
297C22×C4—26no—
298C22×C4—26no—
299C22×C4—23no—
300C22×C4—26no—
301C22×C4—23no—
302C22×C4—23no—
303C22×C4—23no—
304C22×C4—23no—
305C22×C4—26no—
306C22×C4—26no—
307C22×C4—26no—
308C22×C4—26no—
309C22×C4—23no—
310C22×C4—26no—
311C22×C4—26no—
312C22×C4—26no—
313C22×C4—23no—
314C22×C4—26no—
315C22×C4—26no—
316C22×C4—26no—
317C22×C4—26no—
318C22×C4—26no—
319C22×C4—26no—
320C22×C4—26no—
321C22×C4—26no—
322C22×C4—23no—
323C22×C4—26no—
324C22×C4—26no—
325C22×C4—26no—
326C22×C4—26no—
327C22×C4—412no—
328C22×C4—424no—
329C22×C4—424no—
330C22×C4—412no—
331C22×C4—424no—
332C22×C4—412no—
333C22×C4—44no—
334C22×C4—412no—
335C22×C4—412no—
336C22×C4—412no—
337C22×C4—86no—
338C22×C4—812no—
339C22×C4C22×D416yes—
340C22×C4C22×D416yes—
341C22×C4C22×Q814yes—
342C22×C4H(5,𝔽2)16yes—
343C4⋊C4—412no—
344C4⋊C4—424no—
345C4⋊C4—412no—
346C4⋊C4—86no—
347C4⋊C4—812no—
348C4⋊C4—812no—
349C4⋊C4—86no—
350C22⋊C4—412no—
351C22⋊C4—412no—
352C22⋊C4—412no—
353C22⋊C4—412no—
354C22⋊C4—424no—
355C22⋊C4—812no—
356C22⋊C4—812no—
357C22⋊C4—812no—
358C22⋊C4—86no—
359C22⋊C4—86no—
360C22⋊C4—812no—
361C22⋊C4—812no—
362C23—22no—
363C23—26no—
364C23—26no—
365C23—26no—
366C23—26no—
367C23—212no—
368C23—26no—
369C23—26no—
370C23—26no—
371C23—26no—
372C23—26no—
373C23—26no—
374C23—26no—
375C23—26no—
376C23—26no—
377C23—212no—
378C23—26no—
379C23—22no—
380C23—24no—
381C23—26no—
382C23—212no—
383C23—448no—
384C23—41no—
385C23—43no—
386C23—43no—
387C23—46no—
388C23—43no—
389C23—416no—
390C23—416no—
391C23—43no—
392C23—46no—
393C23—46no—
394C23—43no—
395C23—41no—
396C23—42no—
397C23—43no—
398C23—46no—
399C23—46no—
400C23—43no—
401C23—46no—
402C23—46no—
403C23—46no—
404C23—43no—
405C23—46no—
406C23—43no—
407C23—46no—
408C23—46no—
409C23—43no—
410C23—824no—
411C23—824no—
412C23—88no—
413C23—824no—
414C23—824no—
415C23—88no—
416C23—824no—
417C23—88no—
418C23—824no—
419C23—824no—
420C23—88no—
421C23#543112yes—
422C23D4×Q814yes—
423C23#560112yes—
424C23#56014yes—
425C23C23×D411yescharacteristic
426C23C23×Q811yescharacteristic
427C23C2×H(5,𝔽2)13yes—
428C23C2×H(5,𝔽2)13yes—
429C23C2×H(5,𝔽2)13yes—
430C23C2×2−1+411yescharacteristic
431C23#58513yes—
432D4—432no—
433D4—432no—
434C2×C4—212no—
435C2×C4—26no—
436C2×C4—26no—
437C2×C4—26no—
438C2×C4—212no—
439C2×C4—212no—
440C2×C4—212no—
441C2×C4—448no—
442C2×C4—416no—
443C2×C4—46no—
444C2×C4—43no—
445C2×C4—43no—
446C2×C4—43no—
447C2×C4—42no—
448C2×C4—46no—
449C2×C4—43no—
450C2×C4—43no—
451C2×C4—46no—
452C2×C4—43no—
453C2×C4—46no—
454C2×C4—43no—
455C2×C4—46no—
456C2×C4—46no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—46no—
461C2×C4—43no—
462C2×C4—46no—
463C2×C4—46no—
464C2×C4—46no—
465C2×C4—46no—
466C2×C4—46no—
467C2×C4—46no—
468C2×C4—46no—
469C2×C4—46no—
470C2×C4—46no—
471C2×C4—46no—
472C2×C4—43no—
473C2×C4—824no—
474C22—24no—
475C22—212no—
476C22—212no—
477C22—212no—
478C22—212no—
479C22—24no—
480C22—212no—
481C22—42no—
482C22—42no—
483C22—46no—
484C22—46no—
485C22—46no—
486C22—42no—
487C22—46no—
488C22—412no—
489C22—42no—
490C22—44no—
491C22—412no—
492C22—46no—
493C22—832no—
494C22—832no—
495C22—81no—
496C22—82no—
497C22—83no—
498C22—86no—
499C22—83no—
500C22—86no—
501C22—81no—
502C22—82no—
503C22#284218yescomplemented
504C22C2×#54313yes—
505C22C2×D4×Q811yescharacteristic
506C22#348413yes—
507C22#348813yes—
508C22#350113yes—
509C22C2×#56011yescharacteristic
510C22C2×#56013yes—
511C22#353313yes—
512C22#353416yes—
513C22#353513yes—
514C22#353613yes—
515C22#354313yes—
516C4—44no—
517C4—46no—
518C4—46no—
519C4—46no—
520C4—86no—
521C4—86no—
522C4—83no—
523C2—28no—
524C2—44no—
525C2—82no—
526C2—84no—
527C2C2×#284211yescharacteristic
528C2#3768413yes—
529C2#4061213yes—
530C2#4545513yes—
531C2#4632411yescharacteristic
532C2#5254813yes—
533C2#5274211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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. C22×#399C2semidirect
3. #17204C2semidirect
4. #17136C2semidirect
5. #16824C2semidirect
6. #16298C2semidirect
7. #14748C2semidirect
8. #14734C2semidirect
9. #14729C2semidirect
10. #14636C2semidirect
13. C23×C4⋊C4C22semidirect
20. C2×#399C22semidirect
21. C2×#399C22semidirect
22. C2×#399C22semidirect
23. C2×#396C22semidirect
24. C2×#396C22semidirect
25. C2×#396C22semidirect
28. C2×#385C22semidirect
29. C2×#380C22semidirect
33. C22×#95C22semidirect
34. #1443C22semidirect
35. #1441C22semidirect
36. #1441C22semidirect
37. #1441C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
51. C22⋊#95C22semidirect
92. #399D4semidirect
183. C2×C4⋊C4C2×D4semidirect
232. #95C2×D4semidirect
503. C22#2842semidirect

Maximal subgroups

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

subgroupindex
C2×#28422
C22×#3992
#172042
#171362
#168242
#162982
#147482
#147342
#147292
#146362
#138672