Finite Group Databasebeta

← #1409129

Subgroups — #1409129

Order 512.

3513 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C213yesmaximal, complemented
2C2×#2425C213yesmaximal
3#17075C213yesmaximal, complemented
4#17059C213yesmaximal, complemented
5#16984C213yesmaximal, complemented
6#16807C213yesmaximal, complemented
7#15211C211yesmaximal, characteristic, complemented
8#15069C213yesmaximal, complemented
9#14936C213yesmaximal, complemented
10#14836C211yesmaximal, characteristic, complemented
11#14836C211yesmaximal, characteristic, complemented
12#14353C211yesmaximal, characteristic, complemented
13#14322C211yesmaximal, characteristic, complemented
14C2×#1444C211yesmaximal, characteristic, complemented
15D4×#95C211yesmaximal, characteristic, complemented
16C22×#123C2213yes—
17C22×#123C2213yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2213yes—
20C23×C4⋊C4C2213yes—
21#2486—224no—
22#2425—224no—
23C2×#399C2211yescharacteristic
24C2×#398C2216yes—
25C2×#398C2213yes—
26C2×#397C2213yes—
27C2×#397C2216yes—
28C2×#396C2213yes—
29C2×#394C2213yes—
30C2×#389C2213yes—
31C2×#389C2213yes—
32C2×#389C2213yes—
33C2×#388C2213yes—
34C2×#386C2213yes—
35C2×#386C2213yes—
36C2×#385C2213yes—
37C2×#381C2216yes—
38C2×#380C2213yes—
39C2×#380C2213yes—
40C2×#380C2213yes—
41C2×#380C2213yes—
42C22×#95C2211yescharacteristic
43C22×#95C2213yes—
44C22×#95C2211yescharacteristic
45#1445C2216yes—
46#1445C2213yes—
47#1445C2216yes—
48#1445C2213yes—
49#1445C2213yes—
50#1444C2211yescharacteristic
51#1444C2213yes—
52#1444C2211yescharacteristic
53#1444C2213yes—
54#1444C2213yes—
55#1444C2211yescharacteristic
56#1444C2213yes—
57#1444C2213yes—
58#1444C2213yes—
59#1444C2216yes—
60#1439C2216yes—
61#1439C2213yes—
62#1439C2213yes—
63#1439C2213yes—
64C4⋊#95C2213yes—
65C22⋊#95C2213yes—
66C22⋊#95C2213yes—
67C4×#95C2211yescharacteristic
68C23×D4C2311yescharacteristic
69C23×D4C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4C2313yes—
72C24×C4C2311yescharacteristic
73C24×C4C2313yes—
74C24×C4C2313yes—
75C2×#123—212no—
76C2×#123—224no—
77C2×#123—212no—
78C2×#123—412no—
79C2×#121—212no—
80C2×C4×D4—412no—
81C2×C4×D4D4124yes—
82C22×C4⋊C4—212no—
83C22×C4⋊C4—412no—
84C22×C4⋊C4C2313yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2313yes—
87C22×C4⋊C4C2313yes—
88C22×C4⋊C4C2313yes—
89C22×C4⋊C4C2313yes—
90C22×C4⋊C4C2316yes—
91C22×C4⋊C4C2313yes—
92C22×C4⋊C4C2313yes—
93C22×C4⋊C4C2313yes—
94C22×C4⋊C4C2313yes—
95C22×C4⋊C4C2313yes—
96C22×C4⋊C4C2313yes—
97C22×C22⋊C4C2313yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2313yes—
101C22×C22⋊C4C2313yes—
102C22×C22⋊C4C2313yes—
103C22×C22⋊C4C2313yes—
104C22×C22⋊C4C2313yes—
105C22×C22⋊C4C2313yes—
106C22×C22⋊C4C2313yes—
107C22×C22⋊C4C2313yes—
108C22×C42C2313yes—
109#399—24no—
110#398—212no—
111#398—224no—
112#397—212no—
113#397—224no—
114#396—212no—
115#394—212no—
116#389—212no—
117#389—212no—
118#389—212no—
119#388—212no—
120#386—212no—
121#386—212no—
122#385—212no—
123#381—224no—
124#380—212no—
125#380—212no—
126#380—212no—
127#380—212no—
128C2×#95—212no—
129C2×#95—24no—
130C2×#95—24no—
131C2×#95C2313yes—
132C2×#95C2313yes—
133C2×#95C2316yes—
134C2×#95C2313yes—
135C2×#95C2316yes—
136C2×#95C2316yes—
137C2×#95C2313yes—
138C2×#95C2316yes—
139C2×#95C2316yes—
140C2×#95C2313yes—
141C2×#95C2313yes—
142C2×#95C2313yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2313yes—
146C2×#95C2313yes—
147C2×#95C2316yes—
148C2×#95C2311yescharacteristic
149C25C2411yescharacteristic
150C25C2411yescharacteristic
151C25C2411yescharacteristic
152C22×D4—26no—
153C22×D4—22no—
154C22×D4—26no—
155C22×D4—26no—
156C22×D4—22no—
157C22×D4—26no—
158C22×D4—22no—
159C22×D4—43no—
160C22×D4C2×D4112yes—
161C23×C4—212no—
162C23×C4—26no—
163C23×C4—26no—
164C23×C4—26no—
165C23×C4—26no—
166C23×C4—212no—
167C23×C4—26no—
168C23×C4—26no—
169C23×C4—212no—
170C23×C4—26no—
171C23×C4—26no—
172C23×C4—22no—
173C23×C4—46no—
174C23×C4—46no—
175C23×C4—46no—
176C23×C4—43no—
177C23×C4C2×D4112yes—
178C23×C4C2×D4112yes—
179C23×C4C2413yes—
180C23×C4C2413yes—
181C23×C4C2413yes—
182C23×C4C2413yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2413yes—
186C23×C4C2413yes—
187C23×C4C2413yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2413yes—
191#123—448no—
192#123—448no—
193#121—416no—
194C4×D4—448no—
195C2×C4⋊C4—26no—
196C2×C4⋊C4—26no—
197C2×C4⋊C4—212no—
198C2×C4⋊C4—26no—
199C2×C4⋊C4—26no—
200C2×C4⋊C4—26no—
201C2×C4⋊C4—26no—
202C2×C4⋊C4—26no—
203C2×C4⋊C4—212no—
204C2×C4⋊C4—212no—
205C2×C4⋊C4—26no—
206C2×C4⋊C4—26no—
207C2×C4⋊C4—26no—
208C2×C4⋊C4—26no—
209C2×C4⋊C4—26no—
210C2×C4⋊C4—26no—
211C2×C4⋊C4—26no—
212C2×C4⋊C4—46no—
213C2×C4⋊C4—46no—
214C2×C4⋊C4—46no—
215C2×C4⋊C4—46no—
216C2×C4⋊C4—43no—
217C2×C4⋊C4—46no—
218C2×C4⋊C4—46no—
219C2×C4⋊C4—46no—
220C2×C4⋊C4—46no—
221C2×C4⋊C4—448no—
222C2×C4⋊C4—46no—
223C2×C4⋊C4C2×D4112yes—
224C2×C22⋊C4—212no—
225C2×C22⋊C4—26no—
226C2×C22⋊C4—212no—
227C2×C22⋊C4—26no—
228C2×C22⋊C4—212no—
229C2×C22⋊C4—212no—
230C2×C22⋊C4—26no—
231C2×C22⋊C4—26no—
232C2×C22⋊C4—26no—
233C2×C22⋊C4—26no—
234C2×C22⋊C4—26no—
235C2×C22⋊C4—26no—
236C2×C22⋊C4—26no—
237C2×C22⋊C4—26no—
238C2×C22⋊C4—26no—
239C2×C22⋊C4—46no—
240C2×C22⋊C4—46no—
241C2×C22⋊C4—46no—
242C2×C22⋊C4—46no—
243C2×C22⋊C4C2×D4112yes—
244C2×C22⋊C4C2×D4112yes—
245C2×C42—46no—
246C2×C42C2×D4112yes—
247#95—26no—
248#95—26no—
249#95—26no—
250#95—26no—
251#95—212no—
252#95—212no—
253#95—26no—
254#95—212no—
255#95—212no—
256#95—212no—
257#95—26no—
258#95—26no—
259#95—26no—
260#95—26no—
261#95—26no—
262#95—26no—
263#95—46no—
264#95#4114yes—
265C24—21no—
266C24—23no—
267C24—21no—
268C24—23no—
269C24—21no—
270C24—23no—
271C24—23no—
272C24—21no—
273C24—21no—
274C24—23no—
275C24—23no—
276C24—21no—
277C24—23no—
278C24—23no—
279C24—21no—
280C24—21no—
281C24—23no—
282C24C22×D416yes—
283C24C22×D416yes—
284C24H(5,𝔽2)16yes—
285C242−1+412yes—
286C24C2511yescharacteristic
287C2×D4—224no—
288C2×D4—224no—
289C2×D4—412no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—412no—
293C2×D4—412no—
294C2×D4—86no—
295C2×D4#127116yes—
296C22×C4—23no—
297C22×C4—26no—
298C22×C4—21no—
299C22×C4—26no—
300C22×C4—224no—
301C22×C4—23no—
302C22×C4—23no—
303C22×C4—23no—
304C22×C4—26no—
305C22×C4—26no—
306C22×C4—26no—
307C22×C4—26no—
308C22×C4—23no—
309C22×C4—23no—
310C22×C4—23no—
311C22×C4—23no—
312C22×C4—26no—
313C22×C4—26no—
314C22×C4—23no—
315C22×C4—224no—
316C22×C4—224no—
317C22×C4—23no—
318C22×C4—26no—
319C22×C4—26no—
320C22×C4—23no—
321C22×C4—23no—
322C22×C4—23no—
323C22×C4—26no—
324C22×C4—26no—
325C22×C4—23no—
326C22×C4—23no—
327C22×C4—23no—
328C22×C4—26no—
329C22×C4—26no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—26no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—26no—
341C22×C4—23no—
342C22×C4—23no—
343C22×C4—412no—
344C22×C4—412no—
345C22×C4—412no—
346C22×C4—424no—
347C22×C4—424no—
348C22×C4—412no—
349C22×C4—44no—
350C22×C4—412no—
351C22×C4—424no—
352C22×C4—412no—
353C22×C4—424no—
354C22×C4—412no—
355C22×C4—812no—
356C22×C4—812no—
357C22×C4—86no—
358C22×C4C22×D416yes—
359C22×C4C22×D416yes—
360C22×C4C22×D416yes—
361C22×C4C22×D416yes—
362C22×C4C22×D416yes—
363C22×C4C2×#4116yes—
364C22×C42−1+412yes—
365C22×C42−1+416yes—
366C4⋊C4—412no—
367C4⋊C4—412no—
368C4⋊C4—412no—
369C4⋊C4—412no—
370C4⋊C4—412no—
371C4⋊C4—424no—
372C4⋊C4—412no—
373C4⋊C4—412no—
374C4⋊C4—44no—
375C4⋊C4—412no—
376C4⋊C4—412no—
377C4⋊C4—412no—
378C4⋊C4—86no—
379C22⋊C4—224no—
380C22⋊C4—412no—
381C22⋊C4—412no—
382C22⋊C4—412no—
383C22⋊C4—44no—
384C22⋊C4—412no—
385C22⋊C4—412no—
386C22⋊C4—412no—
387C22⋊C4—44no—
388C22⋊C4—412no—
389C22⋊C4—86no—
390C42—412no—
391C23—26no—
392C23—26no—
393C23—26no—
394C23—26no—
395C23—26no—
396C23—22no—
397C23—212no—
398C23—26no—
399C23—26no—
400C23—26no—
401C23—26no—
402C23—26no—
403C23—26no—
404C23—26no—
405C23—26no—
406C23—43no—
407C23—41no—
408C23—46no—
409C23—43no—
410C23—43no—
411C23—41no—
412C23—43no—
413C23—43no—
414C23—43no—
415C23—43no—
416C23—43no—
417C23—43no—
418C23—43no—
419C23—41no—
420C23—46no—
421C23—43no—
422C23—43no—
423C23C2×#12714yes—
424C23C2×#12714yes—
425C23#534112yes—
426C23#537112yes—
427C23C23×D413yes—
428C23C22×#4111yescharacteristic
429C23C2×H(5,𝔽2)13yes—
430C23C2×2−1+411yescharacteristic
431C23C2×2−1+413yes—
432C23C2×2−1+411yescharacteristic
433C23#58513yes—
434D4—232no—
435D4—88no—
436D4—88no—
437C2×C4—26no—
438C2×C4—26no—
439C2×C4—26no—
440C2×C4—26no—
441C2×C4—212no—
442C2×C4—212no—
443C2×C4—26no—
444C2×C4—26no—
445C2×C4—26no—
446C2×C4—26no—
447C2×C4—26no—
448C2×C4—26no—
449C2×C4—26no—
450C2×C4—26no—
451C2×C4—26no—
452C2×C4—26no—
453C2×C4—448no—
454C2×C4—41no—
455C2×C4—43no—
456C2×C4—46no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—46no—
461C2×C4—46no—
462C2×C4—43no—
463C2×C4—43no—
464C2×C4—41no—
465C2×C4—43no—
466C2×C4—43no—
467C2×C4—41no—
468C2×C4—43no—
469C2×C4—43no—
470C2×C4—46no—
471C2×C4—46no—
472C2×C4—43no—
473C2×C4—43no—
474C2×C4—43no—
475C2×C4—46no—
476C2×C4—46no—
477C2×C4—43no—
478C2×C4—43no—
479C2×C4—46no—
480C2×C4—43no—
481C2×C4—46no—
482C2×C4—43no—
483C2×C4—43no—
484C2×C4—46no—
485C2×C4—43no—
486C2×C4—43no—
487C2×C4—46no—
488C2×C4—824no—
489C2×C4—88no—
490C2×C4—824no—
491C2×C4C2×#12714yes—
492C2×C4#547112yes—
493C22—212no—
494C22—24no—
495C22—212no—
496C22—212no—
497C22—212no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—46no—
503C22—46no—
504C22—46no—
505C22—46no—
506C22—81no—
507C22—83no—
508C22—81no—
509C22—83no—
510C22—83no—
511C22—83no—
512C22—81no—
513C22#265618yes—
514C22C22×#12711yescharacteristic
515C22C2×#53413yes—
516C22C2×#53713yes—
517C22C2×#54713yes—
518C22#348513yes—
519C22#348913yes—
520C22#349913yes—
521C22#351313yes—
522C22#352311yescharacteristic
523C22#353516yes—
524C22#354413yes—
525C22#356313yes—
526C4—24no—
527C4—212no—
528C4—46no—
529C4—46no—
530C4—46no—
531C4—42no—
532C4—83no—
533C4—83no—
534C4—81no—
535C4—83no—
536C4—83no—
537C4—81no—
538C2—28no—
539C2—44no—
540C2—161no—
541C2C2×#265611yescharacteristic
542C2#3606011yescharacteristic
543C2#3617911yescharacteristic
544C2#3803813yes—
545C2#3804313yes—
546C2#4137313yes—
547C2#4229513yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2486C2semidirect
3. #17075C2semidirect
4. #17059C2semidirect
5. #16984C2semidirect
6. #16807C2semidirect
7. #15211C2semidirect
8. #15069C2semidirect
9. #14936C2semidirect
10. #14836C2semidirect
11. #14836C2semidirect
12. #14353C2semidirect
13. #14322C2semidirect
14. C2×#1444C2semidirect
15. D4×#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#24252
#170752
#170592
#169842
#168072
#152112
#150692
#149362
#148362
#143532
#143222
C2×#14442
D4×#952