Finite Group Databasebeta

← #1409131

Subgroups — #1409131

Order 512.

3655 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C213yesmaximal, complemented
2C2×#2488C213yesmaximal, complemented
3#17064C213yesmaximal, complemented
4#17064C213yesmaximal, complemented
5#16814C213yesmaximal, complemented
6#16692C213yesmaximal, complemented
7#15199C211yesmaximal, characteristic, complemented
8#15069C213yesmaximal, complemented
9#14926C213yesmaximal, complemented
10#14900C211yesmaximal, characteristic, complemented
11#14757C211yesmaximal, characteristic
12#14354C211yesmaximal, characteristic, complemented
13#14321C211yesmaximal, characteristic, complemented
14C2×#1444C211yesmaximal, characteristic, complemented
15#13809C211yesmaximal, characteristic, complemented
16C22×#123C2213yes—
17C22×#123C2216yes—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2213yes—
20C23×C4⋊C4C2213yes—
21#2501—224no—
22#2488—224no—
23C2×#400C2213yes—
24C2×#400C2213yes—
25C2×#399C2216yes—
26C2×#399C2216yes—
27C2×#397C2213yes—
28C2×#393C2213yes—
29C2×#392C2211yescharacteristic
30C2×#391C2213yes—
31C2×#390C2213yes—
32C2×#390C2213yes—
33C2×#388C2213yes—
34C2×#388C2213yes—
35C2×#386C2213yes—
36C2×#386C2213yes—
37C2×#384C2213yes—
38C2×#384C2213yes—
39C2×#382C2213yes—
40C2×#381C2213yes—
41C2×C4×C22⋊C4C2213yes—
42C2×#376C2211yescharacteristic
43C22×#95C2213yes—
44C22×#95C2211yescharacteristic
45#1444C2211yescharacteristic
46#1444C2213yes—
47#1444C2211yescharacteristic
48#1444C2213yes—
49#1444C2211yescharacteristic
50#1444C2213yes—
51#1444C2213yes—
52#1444C2213yes—
53#1444C2213yes—
54#1444C2216yes—
55#1439C2213yes—
56#1439C2216yes—
57#1439C2216yes—
58#1439C2216yes—
59#1439C2213yes—
60#1439C2213yes—
61#1438C2213yes—
62#1438C2213yes—
63#1438C2213yes—
64#1434C2213yes—
65#1434C2213yes—
66C22⋊#95C2213yes—
67C4×#95C2211yescharacteristic
68C23×Q8C2311yescharacteristic
69C23×D4C2313yes—
70C23×D4C2313yes—
71C24×C4C2313yes—
72C24×C4C2311yescharacteristic
73C24×C4C2313yes—
74C2×#123—212no—
75C2×#123—224no—
76C2×#123—224no—
77C2×#123—412no—
78C2×#123—412no—
79C2×C22⋊Q8—212no—
80C2×#121—212no—
81C2×#121—412no—
82C22×C4⋊C4—212no—
83C22×C4⋊C4—412no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2313yes—
86C22×C4⋊C4C2313yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2316yes—
90C22×C4⋊C4C2316yes—
91C22×C4⋊C4C2316yes—
92C22×C4⋊C4C2313yes—
93C22×C4⋊C4C2313yes—
94C22×C4⋊C4C2313yes—
95C22×C22⋊C4C2313yes—
96C22×C22⋊C4C2313yes—
97C22×C22⋊C4C2313yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2313yes—
100C22×C22⋊C4C2313yes—
101C22×C22⋊C4C2313yes—
102C22×C22⋊C4C2313yes—
103C22×C22⋊C4C2316yes—
104C22×C22⋊C4C2316yes—
105C22×C22⋊C4C2316yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C42C2313yes—
108C22×C42C2313yes—
109#400—212no—
110#400—212no—
111#399—224no—
112#399—224no—
113#397—212no—
114#393—212no—
115#392—24no—
116#391—212no—
117#390—212no—
118#390—212no—
119#388—212no—
120#388—212no—
121#386—212no—
122#386—212no—
123#384—212no—
124#384—212no—
125#382—212no—
126#381—212no—
127C4×C22⋊C4D4124yescomplemented
128#376—24no—
129C2×#95—212no—
130C2×#95—24no—
131C2×#95C2313yes—
132C2×#95C2313yes—
133C2×#95C2313yes—
134C2×#95C2313yes—
135C2×#95C2313yes—
136C2×#95C2313yes—
137C2×#95C2313yes—
138C2×#95C2313yes—
139C2×#95C2313yes—
140C2×#95C2313yes—
141C2×#95C2313yes—
142C2×#95C2313yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2313yes—
146C2×#95C2316yes—
147C2×#95C2313yes—
148C2×#95C2313yes—
149C2×#95C2311yescharacteristic
150C25C2413yes—
151C25C2411yescharacteristic
152C22×Q8—26no—
153C22×Q8—22no—
154C22×Q8—43no—
155C22×D4—212no—
156C22×D4—26no—
157C22×D4—26no—
158C22×D4—212no—
159C22×D4—26no—
160C22×D4—26no—
161C22×D4—26no—
162C22×D4—26no—
163C22×D4—46no—
164C22×D4—46no—
165C23×C4—212no—
166C23×C4—26no—
167C23×C4—26no—
168C23×C4—212no—
169C23×C4—26no—
170C23×C4—26no—
171C23×C4—212no—
172C23×C4—26no—
173C23×C4—26no—
174C23×C4—22no—
175C23×C4—46no—
176C23×C4—43no—
177C23×C4C2×D4112yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2413yes—
181C23×C4C2413yes—
182C23×C4C2411yescharacteristic
183C23×C4C2413yes—
184C23×C4C2413yes—
185C23×C4C2413yes—
186C23×C4C2413yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2413yes—
191#123—496no—
192#123—448no—
193C22⋊Q8—416no—
194#121—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—212no—
206C2×C4⋊C4—212no—
207C2×C4⋊C4—26no—
208C2×C4⋊C4—46no—
209C2×C4⋊C4—46no—
210C2×C4⋊C4—46no—
211C2×C4⋊C4—46no—
212C2×C4⋊C4—46no—
213C2×C4⋊C4—46no—
214C2×C4⋊C4—46no—
215C2×C4⋊C4—46no—
216C2×C4⋊C4—46no—
217C2×C4⋊C4—448no—
218C2×C22⋊C4—26no—
219C2×C22⋊C4—26no—
220C2×C22⋊C4—26no—
221C2×C22⋊C4—212no—
222C2×C22⋊C4—26no—
223C2×C22⋊C4—26no—
224C2×C22⋊C4—212no—
225C2×C22⋊C4—212no—
226C2×C22⋊C4—212no—
227C2×C22⋊C4—212no—
228C2×C22⋊C4—26no—
229C2×C22⋊C4—26no—
230C2×C22⋊C4—26no—
231C2×C22⋊C4—26no—
232C2×C22⋊C4—26no—
233C2×C22⋊C4—26no—
234C2×C22⋊C4—26no—
235C2×C22⋊C4—43no—
236C2×C22⋊C4—46no—
237C2×C22⋊C4—46no—
238C2×C22⋊C4—46no—
239C2×C22⋊C4—46no—
240C2×C22⋊C4—46no—
241C2×C22⋊C4—46no—
242C2×C22⋊C4—46no—
243C2×C22⋊C4—46no—
244C2×C22⋊C4C2×D4112yes—
245C2×C22⋊C4C2×D4112yes—
246C2×C42—26no—
247C2×C42—26no—
248C2×C42—46no—
249C2×C42C2×D4112yes—
250C2×C42C2×D4112yes—
251#95—26no—
252#95—26no—
253#95—26no—
254#95—26no—
255#95—26no—
256#95—26no—
257#95—26no—
258#95—26no—
259#95—26no—
260#95—26no—
261#95—26no—
262#95—26no—
263#95—26no—
264#95—26no—
265#95—26no—
266#95—46no—
267#95C2×D4112yes—
268#95C2×D4112yes—
269#95#4114yes—
270C24—21no—
271C24—23no—
272C24—21no—
273C24—23no—
274C24—23no—
275C24—26no—
276C24—23no—
277C24—23no—
278C24—26no—
279C24—23no—
280C24—23no—
281C24—23no—
282C24—26no—
283C24—26no—
284C24—23no—
285C24C22×D416yes—
286C24H(5,𝔽2)12yes—
287C24H(5,𝔽2)16yes—
288C24C2511yescharacteristic
289C2×Q8—412no—
290C2×Q8—44no—
291C2×Q8—86no—
292C2×D4—412no—
293C2×D4—412no—
294C2×D4—412no—
295C2×D4—424no—
296C2×D4—424no—
297C2×D4—412no—
298C2×D4—412no—
299C2×D4—412no—
300C2×D4—812no—
301C2×D4—812no—
302C22×C4—21no—
303C22×C4—26no—
304C22×C4—23no—
305C22×C4—21no—
306C22×C4—23no—
307C22×C4—26no—
308C22×C4—224no—
309C22×C4—23no—
310C22×C4—23no—
311C22×C4—23no—
312C22×C4—23no—
313C22×C4—23no—
314C22×C4—23no—
315C22×C4—23no—
316C22×C4—26no—
317C22×C4—26no—
318C22×C4—23no—
319C22×C4—23no—
320C22×C4—23no—
321C22×C4—23no—
322C22×C4—26no—
323C22×C4—26no—
324C22×C4—23no—
325C22×C4—224no—
326C22×C4—23no—
327C22×C4—23no—
328C22×C4—23no—
329C22×C4—23no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—26no—
333C22×C4—26no—
334C22×C4—26no—
335C22×C4—21no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—23no—
339C22×C4—26no—
340C22×C4—26no—
341C22×C4—26no—
342C22×C4—23no—
343C22×C4—23no—
344C22×C4—23no—
345C22×C4—23no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—23no—
349C22×C4—23no—
350C22×C4—412no—
351C22×C4—424no—
352C22×C4—424no—
353C22×C4—412no—
354C22×C4—44no—
355C22×C4—412no—
356C22×C4—424no—
357C22×C4—424no—
358C22×C4—412no—
359C22×C4—412no—
360C22×C4—86no—
361C22×C4C22×D416yes—
362C22×C4C22×D416yes—
363C22×C4C22×D416yes—
364C22×C4C22×D416yes—
365C22×C4C22×D416yes—
366C22×C4C22×D416yes—
367C22×C4C2×#4116yes—
368C22×C42−1+412yes—
369C22×C42−1+416yes—
370C4⋊C4—412no—
371C4⋊C4—412no—
372C4⋊C4—424no—
373C4⋊C4—412no—
374C4⋊C4—424no—
375C4⋊C4—424no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—412no—
380C4⋊C4—412no—
381C22⋊C4—224no—
382C22⋊C4—412no—
383C22⋊C4—424no—
384C22⋊C4—424no—
385C22⋊C4—424no—
386C22⋊C4—44no—
387C22⋊C4—412no—
388C22⋊C4—412no—
389C22⋊C4—412no—
390C22⋊C4—86no—
391C22⋊C4—86no—
392C22⋊C4#127116yes—
393C42—224no—
394C42—412no—
395C23—26no—
396C23—26no—
397C23—26no—
398C23—26no—
399C23—26no—
400C23—22no—
401C23—212no—
402C23—26no—
403C23—26no—
404C23—43no—
405C23—46no—
406C23—46no—
407C23—43no—
408C23—43no—
409C23—43no—
410C23—46no—
411C23—43no—
412C23—46no—
413C23—43no—
414C23—46no—
415C23—43no—
416C23—46no—
417C23—43no—
418C23—46no—
419C23—43no—
420C23—46no—
421C23—46no—
422C23—46no—
423C23—46no—
424C23—43no—
425C23—43no—
426C23—46no—
427C23C2×#12714yes—
428C23#535112yes—
429C23#535112yes—
430C23C23×D413yes—
431C23C22×#4111yescharacteristic
432C23C2×H(5,𝔽2)13yes—
433C23C2×H(5,𝔽2)11yescharacteristic
434C23C2×2−1+413yes—
435C23C2×2−1+411yescharacteristic
436C23#58513yes—
437Q8—88no—
438D4—824no—
439D4—824no—
440C2×C4—26no—
441C2×C4—26no—
442C2×C4—26no—
443C2×C4—26no—
444C2×C4—26no—
445C2×C4—26no—
446C2×C4—212no—
447C2×C4—212no—
448C2×C4—26no—
449C2×C4—26no—
450C2×C4—26no—
451C2×C4—26no—
452C2×C4—26no—
453C2×C4—26no—
454C2×C4—26no—
455C2×C4—26no—
456C2×C4—26no—
457C2×C4—26no—
458C2×C4—448no—
459C2×C4—41no—
460C2×C4—43no—
461C2×C4—43no—
462C2×C4—46no—
463C2×C4—43no—
464C2×C4—46no—
465C2×C4—46no—
466C2×C4—43no—
467C2×C4—43no—
468C2×C4—46no—
469C2×C4—41no—
470C2×C4—43no—
471C2×C4—43no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—43no—
475C2×C4—46no—
476C2×C4—46no—
477C2×C4—46no—
478C2×C4—46no—
479C2×C4—46no—
480C2×C4—46no—
481C2×C4—43no—
482C2×C4—43no—
483C2×C4—43no—
484C2×C4—43no—
485C2×C4—43no—
486C2×C4—43no—
487C2×C4—43no—
488C2×C4—43no—
489C2×C4—43no—
490C2×C4—43no—
491C2×C4—43no—
492C2×C4—43no—
493C2×C4—88no—
494C2×C4—824no—
495C2×C4C2×#12714yes—
496C2×C4C2×#12714yes—
497C2×C4#547112yes—
498C22—212no—
499C22—24no—
500C22—212no—
501C22—212no—
502C22—212no—
503C22—24no—
504C22—86no—
505C22—83no—
506C22—83no—
507C22—83no—
508C22—86no—
509C22—83no—
510C22—86no—
511C22—83no—
512C22—86no—
513C22—83no—
514C22—83no—
515C22#265718yes—
516C22C22×#12711yescharacteristic
517C22C2×#53513yes—
518C22C2×#53513yes—
519C22C2×#54713yes—
520C22#348513yes—
521C22#348913yes—
522C22#349913yes—
523C22#351313yes—
524C22#352211yescharacteristic
525C22#353516yes—
526C22#354413yes—
527C22#355913yes—
528C4—24no—
529C4—212no—
530C4—24no—
531C4—46no—
532C4—46no—
533C4—46no—
534C4—42no—
535C4—83no—
536C4—83no—
537C4—83no—
538C4—81no—
539C4—81no—
540C4—81no—
541C2—28no—
542C2—163no—
543C2C2×#265711yescharacteristic
544C2#3606011yescharacteristic
545C2#3617911yescharacteristic
546C2#3803913yes—
547C2#3804113yes—
548C2#4137513yes—
549C2#4229713yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14757C2
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×#2501C2semidirect
2. C2×#2488C2semidirect
3. #17064C2semidirect
4. #17064C2semidirect
5. #16814C2semidirect
6. #16692C2semidirect
7. #15199C2semidirect
8. #15069C2semidirect
9. #14926C2semidirect
10. #14900C2semidirect
12. #14354C2semidirect
13. #14321C2semidirect
14. C2×#1444C2semidirect
15. #13809C2semidirect
127. C4×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#24882
#170642
#168142
#166922
#151992
#150692
#149262
#149002
#147572
#143542
#143212
C2×#14442
#138092